Toward Massive Detection of Planets around M Dwarfs Using the Radial Velocity Technique

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Title:
Toward Massive Detection of Planets around M Dwarfs Using the Radial Velocity Technique
Physical Description:
1 online resource (185 p.)
Language:
english
Creator:
Wang, Ji
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Astronomy
Committee Chair:
Ge, Jian
Committee Members:
Wan, Xiaoke
Ford, Eric B
Telesco, Charles M
Muller, Guido

Subjects

Subjects / Keywords:
astronomy -- exoplanet -- planet
Astronomy -- Dissertations, Academic -- UF
Genre:
Astronomy thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
M dwarfs are the least massive but the most common type of stars in the solar neighborhood. Discoveries of M dwarf planets would lead to a complete understanding of planet formation and evolution around stars of different types. Radial velocity (RV) technique is one of the leading technologies that detect exoplanets and the RV technique favors detection of planet in the habitable zone of an M dwarf. The dispersed fixed delay interferometer (DFDI) method is one branch of the RV technique that uses an interferometer to boost Doppler sensitivity of a spectrograph with a given resolution. I systematically studied the comparison between the DFDI method and the traditional high-resolution spectrograph method (the DE method). My work provides a guidance for future exoplanet survey: 1, a survey of a large sample of stars should adopt the DFDI method, which enables both adequate RV precision and high survey efficiency; 2, high precision low-mass exoplanet search should adopt the DE method with a high resolution spectrograph. I concluded that NIR observation of mid-late type M dwarfs is the most realistic and likely approach in the search for a habitable Earth-like planet. Current M dwarf planet survey in the NIR faces two severe challenges, telluric contamination and lack of a precise wavelength calibration source. I developed the  binary mask cross correlation technique and the telluric standard star method in the NIR to eliminate telluric contamination. I also developed an precise and stable wavelength calibration source, i.e., the Sine source, which provides a promising candidate for NIR wavelength calibrator. All these software and hardware developments will pave the way for the next wave of massive detection of M dwarf planets using instruments such as the FIRST. The MARVELS project is multi-object planet survey with the DFDI method. Owing to my work in MARVELS interferometer group delay calibration, over 250 binary and a dozen of brown dwarfs have been discovered and they provide a valuable insight of formation and evolution of low-mass stellar companion and brown dwarf. I have envisioned an M-dwarf planet survey and provided a concept study of such survey.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Ji Wang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Ge, Jian.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2012
System ID:
UFE0044382:00001


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TOWARDMASSIVEDETECTIONOFPLANETSAROUNDMDWARFSUSINGTHERADIALVELOCITYTECHNIQUEByJIWANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012JiWang 2

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Idedicateittomyparentsandmywife 3

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ACKNOWLEDGMENTS Iwouldliketothankmyparents,whohavegivenalltheyhavetoraiseandeducateme.Theytaughtmehowtobecomeagoodman,amanwithskills,faithandintegrity.Iwanttothankmyadvisor,Dr.JianGe,forhisvaluableadvicesandcontinuoussupportthroughoutmygraduatestudyattheUniversityofFlorida.Ilearnedfromhimthe3-Rstandard,i.e.,responsive,reliableandresponsible.The3-Rstandardhasbecomemyguidanceandwillkeepbenetingmeinmyfuturecareer.HetaughtmehowtobecomealeaderintheeldofAstronomywithvisionanddetermination.IwanttothankDr.EricFordforhisadvicesonacademiclife.Duringourcollaborationontheeccentricitypaper,Ihavelearnedtheentireprocessofpublishingapaper.Itishispatienceandmeticulousnessthatsetmeanexampleofhowtobecomearesponsibleadvisorandanexceptionalresearcher.IwanttothankDr.XiaokeWanforhishelpinmylaboratorywork.Heisapreciseandcarefulexperimentalist.Hetaughtmebasicsonopticsandhands-onexperienceinthelab.IalsothankhimforhisadvicesonhowtosurviveasaChineseintheUS.Ithankmywife,HeHuang,forherunconditionalsupportforme.IfIhavetogivearea-sonforhersupport,thatislove,themostamazingthingintheworld.ItisherencouragementthatmotivatesmetostriveformoreandmakesmetobelievethatIcandobetter.Itishersacricingsupportathomethatlessensmyburdenasafamilymemberandfreesupmoreofmytimethatisdevotedtoresearch.Sheisthepersonthatmademestarttorealizethepowerofloveandthefaiththattogetherwecanovercomeeveryadversityandachieveonegoalandthenanotherinlife.TherearenumerousteachersthatIwanttoexpressmygratitudesto.Mr.ZhenlongOu,themathteacherinmyjuniorschool,hetaughtmetobemoreaggressiveandhardworkingwhenInaivelythoughtthatIcansuccessjustwithmysmartness.Mayherestinpeace!Mrs.XiaomeiTangismyChineseteacherinjuniorschool.IwanttothankherforhervoluntarytutoringandfreedinnerswhenIwaspreparingtheentranceexamforhighschool.Sheismorelikeamothertomethanateacherwhoiswillingtotakecareofeverythingforme.Mr. 4

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ZhifangLiismyhighschoolmathteacher.IwanttothankherformotivatingmetobethebeststudentbylettingmeknowthatIamnotherfavoritestudent.Shealsotaughtmethatstudyshouldnotbeeverythingforaperson,thereareotherthingsthatmakemylifecolorfulanddiversiedsuchasmusic,sportsandsoon.ItbecamemorevaluableanadviceafterIwasadmittedintheUniversityofScienceandTechnologyofChina(USTC),whereIfoundmyselflivingahappierlifethanmostofmyclassmates.IwouldliketothankmyadvisorsatUSTC,Dr.TingguiWang,Dr.FuzhenChenandDr.XuKong.Theyareexcellentprofessorsinastronomyresearchandwouldbegoodexamplesformyupcomingpathinastronomy.TheentireETgroupledbyDr.JianGehashelpedmealotinnishingmydissertation.IwouldliketothankformermemberDr.SuvrathMahadevan,whoisnowanassistantprofessoratPennStateUniversity.HeisalwaystherewheneverIneedhelponMdwarfplanetsciencenomatterhowbusyheis.IthankDr.JullianvanEykenforhisvaluablediscussiononDFDItheory.IwanttospeciallythankDr.JustinCrepp,whoisgoingtostartanewchapterofhislifeattheUniversityofNotreDameasanassistantprofessor.IthankhimforthevaluableexperienceIgainedfromourlaboratoryworkforhigh-contrastimagingofexoplanet.IthankhimforthegoodtimewehadatUF,PasadenaandYellowstone.IalsowanttothankhimforhisgenerousofferwhenIwaslookingforajob.IthankDr.ScottFleming,Dr.NathanDeLeeandDr.BrianLeeforhisadvicesonmyprojectsandobservationproposals.Iwishallthegroupmembersthebestintheirfutureresearchesandlives.Duringmysixyears'studyattheUniversityofFlorida,Iamsothankfultohavemanyfriendsaccompanied.IthankPengchengGuo,akaPC,forallhishelpwhenIrstcameandsettleddownhereandthegoodtimeplayingpool,badmintonandbasketballandswimming.Hetaughtmetobeagentlemanandtobelessemotionalnomatterwhathappens.Dr.BoZhaocameheretwoyearsaheadofmeandtreatedmelikehislittlebrother.IthankDanLiforbeingthededicatedshingpartner.Dr.PengJiangisanelderalumnifromUSTCwhokeepsmemotivatedandIamalwaysencouragedbyhimtothinkdeeperintoaproblem.I 5

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thankLiangChang,aka,Liangge,forallthetimewespenttogetheronlytryingtokilltime.IthankDr.JiweiXieandhiswifefortheirfunnystories.IthankBoMaforalwaysbeingtheguywhoismadefunof.Wewouldbelesshappierwithoutyourexistence.PC,me,Peng,JiweiandBomadetheoldAstronomyFive,thebasketballteamthatdefeatedotherteamseffortlessly.Withpartoftheoldteamleavingandnewbloodcomingin,thenewAstronomyFiveisformedwithmembersbeingPC,me,Bo,RuiandShuo.ThenewAstronomyFiveisyoungerandmoresophisticatedingameexecutionanditisonitswaytobethebestbasketballteamintheChinesecommunitywithinGainesvillearea.BesidesmyChinesefriends,Iwouldliketothankmyinternationalfriends(USfriendsincluded).IthankSoungchulandMarkforalltheluncheswehadtogether.Wehadsuchagoodtimeindiscussingrecentprojects,keepingeachothermotivatedandexchangingideasontopicsthatinterestedallofus.IthankCraigforkeepingmetbypushingmetonishonemoreuphillstairsandthenanotherinstadiumrunning.IthankCatherineforhelpingmeoutoftroubleinregistrationanddepartmentalaffairs.IthankDr.CullenBlakeforvaluablediscussionsandreferenceletters.IthankDr.RobertWittenmyerforhismentalsupportfromAustralia,theotherendoftheEarth.IthankHaliJakemanforproofreadingallmypapersandforherhelpduringStephanie'squalifyingexam.Iwouldlikethankmyfriendsgrowingupwith,ZhiZeng,LiZhouandLiqinHuang,fortheirgenerosityintreatingmealseverytimeIgobacktoChina.IcannotthankenoughbecauseIamsuchablessedperson. 6

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 4 LISTOFTABLES ....................................... 11 LISTOFFIGURES ...................................... 12 ABSTRACT .......................................... 15 CHAPTER 1BACKGROUNDANDMOTIVATION .......................... 17 1.1TheChronicleofExoplanetsSearch ....................... 17 1.2DetectionMethods ................................ 17 1.2.1TheRadialVelocityTechnique ...................... 17 1.2.2TheTransitMethod ............................ 18 1.2.3DirectImaging ............................... 18 1.2.4TheTimingTechnique .......................... 18 1.2.5Microlensing ................................ 19 1.2.6Astrometry ................................. 19 1.3ConventionalSpectrographandtheDispersedFixedDelayInterferometer .. 19 1.4MajorQuestionsToBeAnswered ........................ 20 1.5PlanetsAroundMDwarfs ............................ 24 1.5.1EssentialFactsAboutM-dwarf ...................... 24 1.5.2M-dwarfPlanets .............................. 25 1.5.3SurveysandResults ........................... 25 2FUNDAMENTALPERFORMANCEOFTHEDFDIMETHOD ............ 28 2.1Introduction .................................... 28 2.2MethodologyofCalculatingPhoton-limitedRVMeasurementUncertainty ... 33 2.2.1Photon-limitedRVUncertaintyofDE .................. 34 2.2.2Photon-limitedRVUncertaintyofDFDI ................. 35 2.3ComparisonbetweenDEandOptimizedDFDI ................. 36 2.3.1OptimizedDFDI .............................. 36 2.3.2InuenceofSpectralResolutionR .................... 39 2.3.3InuenceofDetectorPixelNumbers ................... 44 2.3.4InuenceofMulti-ObjectObservations ................. 47 2.3.5InuenceofProjectedRotationalVelocityVsini ............ 50 2.4SummaryandDiscussion ............................ 50 2.4.1QFactorsforDFDI,DEandFTS .................... 50 2.4.2ApplicationofDFDI ............................ 54 7

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3COMPREHENSIVESIMULATIONSFORHABITABLEPLANETSEARCHINTHENIR ........................................... 56 3.1Introduction .................................... 56 3.2SimulationMethodology ............................. 58 3.2.1HighResolutionSyntheticSpectra .................... 58 3.2.2RVCalibrationSources .......................... 60 3.2.3StellarNoise ................................ 62 3.2.4TelluricLinesContamination ....................... 64 3.3Results ...................................... 66 3.3.1RVCalibrationUncertainty ........................ 66 3.3.2OptimalSpectralBandForRVMeasurements ............. 67 3.3.2.1StellarSpectralQuality ..................... 68 3.3.2.2StellarSpectralQuality+StellarRotation ........... 69 3.3.2.3StellarSpectralQuality+RVCalibrationSource ........ 72 3.3.2.4StellarSpectralQuality+RVCalibrationSource+Atmosphere 74 3.3.2.5ComparisonstoPreviousWork ................. 76 3.3.3CurrentPrecisionvs.SignalofanEarth-likePlanetinHabitableZone 79 3.3.3.1StellarSpectralQuality ..................... 80 3.3.3.2StellarSpectralQuality+RVCalibrationSource+Atmosphere 82 3.3.3.3StellarSpectralQuality+RVCalibrationSource+StellarNoise ............................... 84 3.4SummaryandDiscussion ............................ 87 4PLANETSEARCHAROUNDMDWARFS ...................... 92 4.1Introduction .................................... 92 4.1.1CurrentStatus ............................... 92 4.1.2Challenges ................................ 92 4.1.2.1Atmophsere ........................... 92 4.1.2.2WavelengthCalibrationSources ................ 93 4.2TacklingAdversitiesinNIRRVMeasurement .................. 93 4.2.1SoftwareAdvancement .......................... 93 4.2.1.1PreciseTelluricLinesRemoval ................. 93 4.2.1.2BinaryMaskCrossCorrelation ................. 95 4.2.2HardwareAdvancement ......................... 96 4.3M-dwarfPlanetSearchandCharacterization-Results ............. 99 4.3.1TelluricLineRVStability ......................... 99 4.3.2RVMeasurementsofaReferenceStar-GJ411 ............. 100 4.4M-dwarfPlanetSearchandCharacterization-FutureWorks .......... 101 4.4.1SearchingForPlanetsAroundMDwarfswithEXPERT ......... 101 4.4.2Multi-BandStudyofRadialVelocityInducedbyStellarActivitywithEXPERT .................................. 104 4.4.3Mid-LateTypeMDwarfPlanetSurveyUsingFIRST .......... 107 4.4.3.1ScienceJustication ....................... 108 4.4.3.2TargetSelection ......................... 109 8

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4.4.3.3PlanetYieldPrediction ..................... 110 5ACCURATEGROUPDELAYMEASUREMENTFORRVINSTRUMENTSUSINGTHEDFDIMETHOD .................................. 115 5.1Introduction .................................... 115 5.2GDMeasurementUsingWhiteLightCombs .................. 119 5.2.1Method .................................. 119 5.2.2DataReduction .............................. 120 5.2.3GDMeasurementResults ........................ 121 5.2.4GDMeasurementErrorAnalysis ..................... 124 5.3GDCalibration:ObservinganRVReferenceStar ............... 127 5.3.1Method .................................. 127 5.3.2GDCalibrationPrecision ......................... 128 5.4ImplementationofMeasuredGDinAstronomicalObservations ........ 128 5.5SummariesandDiscussions ........................... 130 5.5.1Summaries ................................ 130 5.5.2Discussions ................................ 130 5.5.2.1WhiteLightComb(WLC)Method ................ 130 5.5.2.2ReferenceStar(RS)Method .................. 132 5.5.2.3AFutureM-DwarfSurveyWiththeDFDIMethod ....... 132 6ECCENTRICITYDISTRIBUTIONFORSHORT-PERIODEXOPLANETS ...... 134 6.1Introduction .................................... 134 6.2Method ...................................... 134 6.2.1BayesianOrbitalAnalysisofIndividualPlanet .............. 135 6.2.2)]TJ /F1 11.357 Tf 9.93 0 Td[(AnalysisofIndividualSystems ..................... 137 6.3ResultsforIndividualPlanets .......................... 140 6.3.1Comparison:StandardMCMCandReferences ............. 141 6.3.2Comparison:StandardMCMCand)]TJ /F1 11.357 Tf 9.93 0 Td[(Analysis ............. 143 6.3.3Discussionof)]TJ /F1 11.357 Tf 9.93 0 Td[(Analysis ......................... 144 6.4TidalInteractionBetweenStarandPlanet .................... 147 6.5EccentricityDistribution .............................. 154 6.6Discussion .................................... 162 6.7Conclusion .................................... 164 7SUMMARY,CONCLUSIONANDCONTRIBUTION .................. 169 7.1Chapter2 ..................................... 169 7.2Chapter3 ..................................... 170 7.3Chapter4 ..................................... 170 7.4Chapter5 ..................................... 171 7.5Chapter6 ..................................... 171 REFERENCES ........................................ 172 9

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BIOGRAPHICALSKETCH .................................. 181 10

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LISTOFTABLES Table page 2-1OPDchoiceasafunctionofRandVsiniatdifferentTe .............. 38 2-2PowerLawIndexasafunctionofSpectralResolutionR(Te=2400K) ..... 41 2-3SpectralResolutionandwavelengthcoverageonagivendetector ......... 46 2-4Q00comparisonofDFDIandDEasafunctionofVsini ............... 50 3-1Denitionofobservationalbandpasses ........................ 61 3-2RVuncertaintiescausedbycalibrationsourcesatdifferentspectralresolutions .. 68 3-3Photon-limitedRVuncertaintiesbasedonstellarspectralqualityatdifferentspec-tralresolutionsfordifferentspectraltypes ....................... 70 3-4SpectralType,correspondingTe,andtypicalstellarrotationVsini ......... 72 3-5ComparisonofQfactorsfromourresultsto Bouchyetal. ( 2001 ) .......... 77 3-6ComparisonofpredictedRVprecisionbetweenourresultsto Reinersetal. ( 2010 ) 78 3-7ComparisonofpredictedRVprecisionbetweenourresultsto Rodleretal. ( 2011 ) 79 3-8RequiredS/NfordetectionofanEarth-likePlanetintheHZasafunctionofspec-traltype ......................................... 81 3-9Twoexamplesoftelluriccontamination ........................ 83 3-10Predictionvs.HARPSobservation ........................... 84 5-1GDmeasurementresultsasafunctionofspectrumnumber(GD(#)=C0+C1#+C2#2)andstandarddeviation(GD)atdifferentfrequencies() .......... 127 5-2MARVELSpredictedRVuncertainty(atanaverageS/Nof100)vs.Te ...... 128 5-3ComparisonbetweentwomethodsofGDmeasurementandcalibration ...... 130 6-1ComparisonofEccentricitiesCalculatedFromDifferentMethods .......... 148 6-2Two-sampleK-Stestresult ............................... 155 6-3Bayesiananalysisresults ................................ 160 6-4CatalogofShort-PeriodSingle-PlanetSystems .................... 165 11

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LISTOFFIGURES Figure page 1-1IllustrationofconventionalspectrographandtheDFDImethod ........... 21 1-2IllustrationofDopplersensitivityforaconventionalspectrograph .......... 22 1-3IllustrationofDopplersensitivityfortheDFDImethod ................ 23 2-1DFDIlayoutdiagram .................................. 30 2-2DFDIillustration ..................................... 30 2-3ExamplesofM-dwarfspectra ............................. 31 2-4PowerspectrumofaM-dwarfspectrum ........................ 32 2-5OptimalOPDvs.VsiniandR ............................. 37 2-6Qfactorgainvs.spectralresolution .......................... 40 2-7Qfactorvs.spectralresolution ............................. 42 2-8ImprovementofQDFDIoverQDEvs.spectralresolution ................ 43 2-9ComparisonofQ0IRETandQ0DEatdifferentspectralresolutions ............ 46 2-10ComparisonofQ00DFDIandQ00DFDI,R=100,000atdifferentR ................ 48 2-11ComparisonofQ00DEandQ00DE,R=100,000atdifferentspectralresolutions ........ 49 2-12QDFDIvs.Vsini ..................................... 51 3-1ComparisonsbetweensyntheticandobservedM-dwarfspectra ........... 59 3-2RVcalibrationuncertaintiesvs.spectralresolutions ................. 67 3-3RVprecisionbasedonspectralqualityfactor ..................... 69 3-4RVprecisionbasedonspectralqualityfactorandtypicalstellarrotation ....... 71 3-5RVprecisionbasedonspectralqualityfactorandRVcalibrationuncertainties ... 73 3-6RVprecisionconsideringspectralqualityfactor,RVcalibrationuncertaintiesandtelluriccontamination .................................. 75 3-7ThepercentagecontributionofRVuncertaintyinducedbytelluriccontamination .. 77 3-8RVprecisionsconsideringspectralqualityfactorataS/Nof425 .......... 81 3-9RVprecision(R=120,000)consideringspectralqualityfactor(S/N=425),RVcali-brationuncertaintiesandtelluriccontamination .................... 83 12

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3-10RVprecisionconsideringspectralqualityfactor(S/N=425),RVcalibrationuncer-taintiesandstellarnoise ................................ 85 3-11RMSerrorofKeplerianorbitttingforplanetsdetectedbyHARPSsince2004 ... 87 4-1Comparisonbetweentwospectrabeforeandafterremovingtelluriclines ...... 94 4-2Comparisonbetweenanobservedstellarspectra(GJ411,telluriclinesremoved)andasyntheticspectrum ................................ 95 4-3Telluriclineremovalresidualis2.7% ......................... 96 4-4Anexampleofbinarymasktemplate ......................... 97 4-5Applicationofthesinesourceasanabsorptioncell ................. 99 4-6Applicationofthesinesourceasanemissionlampforsimultaneouswavelengthcalibration ........................................ 100 4-7Sinsourcedemonstrationexperiment ......................... 101 4-8TelluriclinesRVstability ................................ 102 4-9RVmeasurementsforGJ411 ............................. 103 4-10RVmeasuredinIbandusingDEMofEXPERTforKEP11859158 ......... 106 4-11RVmeasuredinVbandusingDEMofEXPERTforKEP11859158 ......... 107 4-12VandJbanddistributionforFIRSTsurveytargets .................. 110 4-13TedistributionforFIRSTsurveytargets ....................... 111 4-14PredictedRVmeasurementprecisionfortheFIRSTsurvey ............. 112 4-15ThepredictedsurveycompletenesscontoursbasedonobservationstrategyandRVprecisionforthepessimisticcase ......................... 113 4-16ThepredictedsurveycompletenesscontoursbasedonobservationstrategyandRVprecisionforthebaselinecase ........................... 114 5-1IllustrationoftheDFDImethod ............................. 117 5-2SimulatedWLCsofaninterferometer ......................... 120 5-3PhaseofsimulatedWLCs ............................... 121 5-4Thenormalizeduxandvisibility()asafunctionoffrequency ........... 122 5-5TopandsideviewofanindividualberbeamfeedingoftheMARVELSinterfer-ometer .......................................... 123 13

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5-6Whitelightcombsphaseasafunctionoffrequency ................. 124 5-7Measuredgroupdelayasafunctionbernumber .................. 125 5-8GDasafunctionoffrequencyatdifferentbernumbers ............... 126 5-9RVsofHIP14810(barycentricvelocitynotcorrected)overaperiodof70days ... 129 6-1ExamplesofhowcredibleintervalsofstandardMCMCanalysisarecalculatedus-ingposteriordistributionofe .............................. 137 6-2ContoursofposteriordistributioninhandkspaceforHD68988 .......... 139 6-3)]TJ /F1 11.357 Tf 9.93 0 Td[(asafunctionofeccentricityeforHD68988 ..................... 141 6-4ComparisonamongstandardMCMCanalysis,)]TJ /F1 11.357 Tf 9.93 0 Td[(andlysisandpreviousreferences 142 6-5Cumulativedistributionsfunctions(CDFs)ofeccentricitiesfromdifferentmethods 146 6-6Distributionofshort-periodsingle-planetsystemsin(e,age=circ)space ....... 150 6-7Cumulativedistributionfunctionofeccentricity .................... 154 6-8MarginalizedprobabilitydensityfunctionsofparametersforanalyticaleccentricitydistributionwithamixtureofexponentialandRayleighpdfs ............. 157 6-9Marginalizedprobabilitydensityfunctionsofparametersforanalyticaleccentricitydistributionwithamixtureofexponentialanduniformpdf .............. 158 6-10Cumulativedistributionsfunctions(cdf)ofeccentricitiesfromdifferentmethods ... 159 6-11Distributionofshort-periodsingle-planetsystemsinperiod-eccentricityspace ... 162 14

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTOWARDMASSIVEDETECTIONOFPLANETSAROUNDMDWARFSUSINGTHERADIALVELOCITYTECHNIQUEByJiWangAugust2012Chair:JianGeMajor:AstronomyMdwarfsaretheleastmassivebutthemostcommontypeofstarsinthesolarneigh-borhood.DiscoveriesofMdwarfplanetswouldleadtoacompleteunderstandingofplanetformationandevolutionaroundstarsofdifferenttypes.Radialvelocity(RV)techniqueisoneoftheleadingtechnologiesthatdetectexoplanetsandtheRVtechniquefavorsdetectionofplanetinthehabitablezoneofanMdwarf.Thedispersedxeddelayinterferometer(DFDI)methodisonebranchoftheRVtechniquethatusesaninterferometertoboostDopplersensitivityofaspectrographwithagivenresolution.IsystematicallystudiedthecomparisonbetweentheDFDImethodandthetraditionalhigh-resolutionspectrographmethod(theDEmethod).Myworkprovidesaguidanceforfutureexoplanetsurvey:1,asurveyofalargesampleofstarsshouldadopttheDFDImethod,whichenablesbothadequateRVprecisionandhighsurveyefciency;2,highprecisionlow-massexoplanetsearchshouldadopttheDEmethodwithahighresolutionspectrograph.IconcludedthatNIRobservationofmid-latetypeMdwarfsisthemostrealisticandlikelyapproachinthesearchforahabitableEarth-likeplanet.CurrentMdwarfplanetsurveyintheNIRfacestwoseverechallenges,telluriccon-taminationandlackofaprecisewavelengthcalibrationsource.IdevelopedthebinarymaskcrosscorrelationtechniqueandthetelluricstandardstarmethodintheNIRtoeliminatetelluriccontamination.Ialsodevelopedanpreciseandstablewavelengthcalibrationsource,i.e.,theSinesource,whichprovidesapromisingcandidateforNIRwavelengthcalibrator.Allthesesoftwareandhardwaredevelopmentswillpavethewayforthenextwaveofmassive 15

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detectionofMdwarfplanetsusinginstrumentssuchastheFIRST.TheMARVELSprojectismulti-objectplanetsurveywiththeDFDImethod.OwingtomyworkinMARVELSinter-ferometergroupdelaycalibration,over250binaryandadozenofbrowndwarfshavebeendiscoveredandtheyprovideavaluableinsightofformationandevolutionoflow-massstellarcompanionandbrowndwarf.IhaveenvisionedanM-dwarfplanetsurveyandprovidedaconceptstudyofsuchsurvey. 16

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CHAPTER1BACKGROUNDANDMOTIVATION 1.1TheChronicleofExoplanetsSearchThesearchforplanetsaroundotherstars(i.e.,exoplanets)wasrstproposedby Struve ( 1952 ). vandeKamp ( 1963 )claimedadetectionanastrometricwobbleofBarnard'sstarwhichislaterdisputedby Gatewood&Eichhorn ( 1973 ). Grifn ( 1973 )demonstrated30ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1stellarradialvelocitymeasurementprecisionusingtelluriclinesforwavelengthreference.Itwasnotuntil1992thattherstexoplanetaroundapulsarwasdetectedbyusingthepulsartimingtechnique( Wolszczan&Frail 1992 ).Therstexoplanetaroundamainsequencestarwas51Pegasidiscoveredby Mayor&Queloz ( 1995 ).Morediscoveriesofexoplanetsquicklyfollowed( Marcy&Butler 1996 ),andtheeldofexoplanetstartedtogaintremendousattentionsincethen.AsofFeb2012,therearemorethan700exoplanetsdetectedbyavarietyofmethods1. 1.2DetectionMethods 1.2.1TheRadialVelocityTechniqueInastar-planetsystem,astarandaplanetorbitaroundtheircommoncenterofmass.Thewobbleofthestarcanbedetectedbymeasuringthestellarvelocityalonglineofsight,i.e.,theradialvelocity(RV).RVisusuallymeasuredbymonitoringtinyshiftsofstellarabsorptionlineswithahigh-resolutionspectrograph.Most(morethan70%)ofcurrentlyknownexoplanetsarediscoveredbytheRVtechnique2.RVmeasurementsprecisionof1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1hasbeenroutinelyobtained( Bouchyetal. 2009 ; Howardetal. 2010b )withinstrumentssuchasHARPS( Mayoretal. 2003 )andHIRES( Vogtetal. 1994 ).Incomparison,Jupitercauses12.5ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RVwobblewhiletheEarthinduces10cms)]TJ /F5 7.97 Tf 5.07 0 Td[(1annualRVvariationinoursolarsystem.Withaincreasingtimebaseline,exoplanetswith 1 http://exoplanet.eu/ 2 http://exoplanets.org/ 17

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semi-majoraxisof6AU(orbitalperiod5000day)havebeendetected( Fischeretal. 2008 ; Jonesetal. 2010 ). 1.2.2TheTransitMethodStarbrightnessdropswhenanobjectblocksaportionofstarlightduringatransitevent.Aperiodicalstardimmingeventmaybeindicativeofanorbitingexoplanet.Thetransitmethodispowerfultoolinstudyingpropertiesofanexoplanetsuchasradius,absorptionandtransmissionspectrumofatmosphereandsoon.Ground-basedtransitingobservationhasyieldedmanydiscoveriesincludingasuper-Eartharoundalowmassstar( Charbonneauetal. 2009 ).Asuper-Earthreferstoaplanetwithmassroughly2-10timesoftheEarth.Space-basedtransitingplanetssurveysuchasCoRot( Baglin 2003 )andKepler( Boruckietal. 2011a c )overcometheatmosphericturbulence-inducednoiseinprecisionphotometricmeasurements.Manyinterestingexoplanetsystemsarediscoveredincludingsuper-Earths( Batalhaetal. 2011 ; Legeretal. 2009 ),habitableplanet( Boruckietal. 2012 )andEarth-likeplanetsinamultiplesystem( Gautieretal. 2011 ). 1.2.3DirectImagingExoplanetscanbedirectlyimagedbybothground-basedandspace-basedtelescopes.Becauseofthehighbrightnesscontrastbetweenastarandanexoplanet,itisextremelydifculttodirectlyimageanclose-inexoplanet.Therefore,mostofcurrentlyknowndirectly-imagedexoplanetsaremorethan10AUawayhoststars.Thedirectly-imagedexoplanetscomplementexoplanetsdiscoveredbytheRVtechniqueandwiththetransitmethodinthediscoveryparameterspace. 1.2.4TheTimingTechniqueSomeastronomicalphenomenaexhibitstringentrepeatabilityintimedomain.Theexistenceofanexoplanetmaybebetrayedbyasmallbutdetectableperturbationoftherepeatability.Forexample, Wolszczan&Frail ( 1992 )discovered3planetsaroundapulsarPSR-1257-12bymeasuringthetimingvariationofitsrotationperiod, Qianetal. ( 2012 )madeatentativediscoveryofaJupiter-likeplanetaroundabinarystarwiththeeclipsing 18

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binarytimingtechnique.Similarideawasproposedinthemeasurementoftransittimingvariation( Ford&Holman 2007 ; Holman&Murray 2005 ),andthistechniquehasalsoresultedinseveraldiscoveies,e.g., Ballardetal. ( 2011 ). 1.2.5MicrolensingMicrolensingispredictedbyEinsteinwithhisgeneraltheoryofrelativity.WhenaeldstarisalignedwiththelineofsightbetweentheEarthandadistantstar,thebrightnessofthedistantstarexperiencesasuddenboostbecauseoftheeldstar(lensingstar).Moreinterestingly,ifthelensingstarhasanorbitingplanet,anerspikeduetotheplanetwillbeobservedtogetherwithmajorspikeduetothelensingstarinacontinuousphotometrymeasurement.Themicrolensingmethodiscapableofdetectinganexoplanetatagreatdistancethatisoutofreachforotherdetectionmethods,whicharetypicallylimitedwithinseveralhundredparsecawayfromtheEarth.Onelatestdetectionbythismethodisreportedby Batistaetal. ( 2011 ) 1.2.6AstrometryStarpositionisperiodicallyperturbedbyasurroundingplanet.Apreciseastrometrymeasurementofanearbystarwouldbeagoodcandidatemethodfordetectinganexoplanet.LikethedirectimagingmethoditcomplementstheRVtechniqueandthetransitmethodinthediscoveryparameterspaceforitssensitivitytoplanetsfarawayfromhoststars.UnliketheRVandthetransitmethods,astrometrycanindependentlymeasureplanetmasswithouttheaidfromothermethods.However,noexoplanethasbeendetectedbythismethodyet. 1.3ConventionalSpectrographandtheDispersedFixedDelayInterferometerTwomajormethodsexistfortheRVtechniqueforexoplanetdetection,conventionalspectrograph Marcy&Butler ( 1996 ); Mayor&Queloz ( 1995 )andtheDispersedFixedDelayInterferometer(DFDI)method( Erskine 2003 ; Erskine&Ge 2000 ; Ge 2002 ; Geetal. 2002 ).TheRVisobtainedbydirectlymeasuringthemovementofstellarspectrallinesalongthedispersiondirectionforaconventionalspectrograph(seebottomofFig. 1-1 ).Incomparison,theDFDImeasuresthemovementofverticaluxdistributioninorderto 19

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measureRV(seetoprightofFig. 1-1 ).Theverticaluxdistributioniscreatedbystellarabsorptionlinesandwhitelightcombsgeneratedbyaxeddelayinterferometer.Stellarabsorptionlinesareusuallyverynarrow(withalinewidthof0.1),ahighresolutionspectrographwithR50,000isusuallyrequiredtoresolvethoselinesinordertopreciselydeterminedthelinemovement.Suchahighresolutionspectrographrequiresalonglightpathandthereforelargeandexpensive.Itisdifculttorealizesuchdesignundertightnancialbudgetandspaceconstraintforanewequipment.Incontrast,thexeddelayinterferometerinaDFDIinstrumentprovidesanextraspectralresolvingpowerandenablesanDFDIinstrumentwithlowspectralresolutiontohaveaequivalentDopplersensitivitywithaconventionalspectrographatahighresolution.Therefore,aDFDIinstrumentcanbemadeatalowercostandmorecompact.What'smore,themulti-objectcapabilityofaDFDIinstrumentmakesitattractiveforfuturelargeareaDopplerplanetsurveys( Ge 2002 ; Wangetal. 2011 ). Erskine ( 2003 )usedasimpleexampletoillustratetheadvantageoftheDFDImethodoveraconventionalspectrograph.Iwillbrieyintroducetheillustrationanditsmaincon-clusion.Readercanrefertohisoriginalpaperformoredetails.Foraconventionalspectro-graph,asillustratedinFig. 1-2 ,anintrinsicabsorptionlineisblurredbythelimitedspectralresolution.ThereactionfunctiontoasmallfrequencyshiftduetoDopplershiftistheslopeoftheblurredprole.Accordingto Erskine ( 2003 ),theS/NforaxedDopplershiftofDisp 2nDHi=p Ai(Ao=Ai)3=2.Incontrast,fortheDFDImethodwithaxeddelayinterferometeraddedintotheopticalpathpriortothedispersingelement(asillustratedinFig. 1-3 ),thedominantmeasurablebecomeslinedepthchangeinsteadoflinecentroidmovement.TheS/NforaxedDopplershiftfortheDFDImethodis,accordingtoEquation10in Erskine ( 2003 ),p nDHi=2p Ai(Ao=Ai)1=2.Therefore,theratioofS/NinducedbyaxedDopplershift,(S=N)DE=(S=N)DFDIis2p 2=(Ao=Ai).Forexample,Aois1at5000atR=5,000,(S=N)DFDIisafactorof3.5higherthan(S=N)DEforaxedDopplershift. 20

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Figure1-1. Topleft:asamplespectrumoftheDFDImethod.Thespectrumiscomposedofastellarabsorptionlineandwhitelightcombsgeneratedbyaxeddelayinterferometer.Topright:Fluxdistributionofverticaldirection.Bottom:uxdistributionofhorizontaldirection,i.e.,dispersiondirection. 1.4MajorQuestionsToBeAnsweredFacingthenumberandthediversityofcurrentknownexoplanets,wecannothelpwonderingmanyquestions.Amongthem,twosequentialquestionsmaybethemostfrequent.Howcommonareexoplanetsandhowcommonislifeonotherexoplanets?Withcontinuouslyadvancingtechnologies,webegintohaveareasonablehandlingoftherstquestion.Forthesecondone,wemaynotbeabletoansweruntilapopulationofhabitableexoplanetshasbeendiscovered.However,westartseeingthetipoficebergafteracoupleofpotentialhabitableworldsaredetected( Boruckietal. 2012 ; Charbonneauetal. 2009 ). 21

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Figure1-2. IllustrationofDopplersensitivityforaconventionalspectrograph(Fig.6in Erskine ( 2003 )).a)Intrinsicabsorptionline.b)Anabsorptionlineafterblurringduetolimitedspectralresolution.c)ReactionfunctiontoasmallfrequencyshiftduetoDopplereffect,i.e.,theslopeoflineproleinb). Planetoccurrencerateisacomplicatedissueinvolvingmanydependencessuchasstellarmetallicity,planetandstellarmass.Thereisawell-establishedplanet-metallicitycorre-lationindicatingthatoccurrenceraterisesfrom3%for[Fe/H]0to25%for[Fe/H]+0.4( Fis-cher&Valenti 2005 ).Asmeasurementprecisionkeepsincreasing,apopulationofexoplan-ets,suchassuper-Earthsandsub-Neptunes(2M20M),startstobeprobed.Thispopulationtogetherwithotherplanetswithlowermassarewhatiscalledlow-massex-oplanets.Thestudyofplanetoccurrencehasbeenfocusedonlow-massexoplanetsaroundsolar-typestars.ScientistsusingHARPS( Mayoretal. 2003 )estimateda30-50%planetoccurrenceforsuper-Earths( Lovisetal. 2009 ; Mayoretal. 2009a ; Udry 2010 ).Another 22

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Figure1-3. IllustrationofDopplersensitivityfortheDFDImethod(Fig.6in Erskine ( 2003 )).a)Intrinsicabsorptionline.b)Interferometercombs.c)Productofintrinsicabsorptionlineandinterferometercombs.d)Afterblurring,combsaresmoothedandbecomehalfcontinuum,Biteareabecomesauxdip.e)ReactionfunctiontoasmallfrequencyshiftdueDopplereffect.Unlikeaconventionalspectrograph,uxdipasshownind)isamoredominantmeasurablethanuxchangealongdispersiondirection. 23

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RVsurveyyieldedaslightlyloweroccurrencerate20%,whichisreportedby Howardetal. ( 2010b ).Theoccurrencerateisevenlower,i.e.,130.8%( Howardetal. 2011 )or19%( Youdin 2011 ),accordingtorecently-releasedKeplerdata( Boruckietal. 2011c ).Discrepanciesamongdifferentsurveysspursspeculationsandeffortstoexplain( Wolfgang&Laughlin 2011 ),however,acompleteunderstandingwouldnotbeobtaineduntilwefullyunderstandthebiasandcompletenessofeachsurveyandstriveforbettermeasurementprecisioninthefuture. Johnsonetal. ( 2010a )studiedthecorrelationbetweenplanetoccurrenceandstellarmassandfoundapositivecorrelationcharacterizedasarisefrom3%at0.5Mto14%at2.0M.Occurrencerateoflow-massexoplanetsaroundstarsotherthansolar-typestarsisrarelymentioneduntilveryrecently. Bonlsetal. ( 2011b )foundthatsuper-EarthsareabundantaroundMdwarfs(35%)andtheoccurrencerateforhabitableplanetsis41+54)]TJ /F5 7.97 Tf 5.07 0 Td[(13%foraM-dwarfsampleintheirsurvey.Incomparison, Howardetal. ( 2010b )foundthisnumbertobe23+16)]TJ /F5 7.97 Tf 5.07 0 Td[(10%forsolar-typestars.Thelargeerrorbarsfromtheirreportsindicatetheamountofeffortrequiredforconstrainingthelow-massplanetoccurrencerateasafunctionofstellarmass.Otherquestionsareasinterestingas,ifnotmoreinterestingthan,thosementionedabove.Forexample,whatisthemassdistributionofexoplanet?Thisquestionhelpsustodistinguishbetweenplanetsandotherobjectssuchasbrowndwarfsandstarsandtoexplaintheobservedbrowndwarfdesert( Marcyetal. 2005 ).Whatistheeccentricitydistributionofexoplanetsandwhatcanbeinferredfromit?Whatisthestatisticsofmultipleplanetarysystems?Allthesestatisticalinformationshelptoconstrainandrenetheoreticalmodels( Ida&Lin 2005 ; Mordasinietal. 2009 )thateventuallyprovidecompleteandaccuratepictureofplanetformationandevolution. 1.5PlanetsAroundMDwarfsWithhundredsofexoplanetsdiscovered,searchingforlow-massexoplanetshasbeengainingincreasingattention.Aplanetofgivenmassandsemi-majoraxiswouldproduce 24

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largerRVsignalaroundaMdwarfthanaroundasolar-typestar.ThefactmakesMdwarfsinterestingtargetsinRVsurveysforlow-massexoplanets. 1.5.1EssentialFactsAboutM-dwarfInastronomy,moststarsareclassiedasO,B,A,F,G,KandMaccordingtotheirspectralfeaturessuchasspectralenergydistributionandspectrallinecharacteristics.F,GandKstarsareusuallyreferredtoassolar-typestars.Mstarshavetheloweststellarmass(M0.45M)andthelowesteffectivetemperature(Te3700K)amongallclassiedstars.BecauseoflowTeandthuslowluminosity(L0.08L),MstarsareusuallycalledMdwarfsbecauseoftheirpositionsonaH-Rdiagram(lowerstellarmassend).Mdwarfsspendaverylongtimeonthemainsequence,whichisevenlongercomparedtotheageoftheuniverse.Therefore,Mstarsonthemainsequencehavenotevolvedtoanadvancedevolutionarystages.Accordingtothelawofblackbodyradiation,Mdwarfsaregenerallyfaintandemitthebulkofenergyinthenearinfrared(NIR).TherearefewMdwarfsthatcanbeseenbynakedeyesdespitethefactthatmorethan70%solarneighborhoodstarsareMdwarfs( Henry 1998 ). 1.5.2M-dwarfPlanetsRVsignalincreaseswithplanetmassandinverselywithsquarerootofstellarmassandsemi-majoraxis.Theequationisgivenby Zechmeisteretal. ( 2009 ):K=r G 1)]TJ /F4 11.955 Tf 10.26 0 Td[(e2msini p (M+m)a=28.4kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1msini MJup M MAU a!1=2, (1)whereKisRVamplitude,Gistheuniversalgravitationalconstant,eiseccentricity,misplanetmass,iisorbitalinclinationwithi=90whenseenedge-on,Misstellarmassandaissemi-majoraxis.Asconsiderableinteresthasbeenfocusedonsearchingforplanetsinthehabitablezone(HZ),Mdwarfsbecomepromisingtargetsforseveralreasons: 25

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AccordingtoEquation 1 ,forgivemanda,KislargerbecauseoflowstellarmassofaMdwarf BecauseoflowluminosityofMdwarfs,HZ,denedasaregionaroundastarwhereliquidwatermayexist,iscloserincomparedtosolar-typestarswhosestrongerradi-ationpushesHZfurtheraway.Forexample,HZaroundtheSunis1AU,wheretheEarthorbitis.Incomparison,HZistypically0.03-0.4AUforaMdwarfs( Zechmeisteretal. 2009 ) Theoreticalworkhasshownthatlow-massplanetsshouldbecommonaroundMdwarfs( Ida&Lin 2005 ) 1.5.3SurveysandResultsThereareseveralRVsurveystargetingM-dwarfplanets( Beanetal. 2010 ; Blakeetal. 2010 ; Bonlsetal. 2011b ; Clubbetal. 2009 ; Endletal. 2006 ; Zechmeisteretal. 2009 ).Theresultsofthesesurveysindicatethat,whilegasgiantsarerare( Endletal. 2006 ; Zechmeisteretal. 2009 ),low-massplanetsmaybeabundantaroundMdwarfs( Bonlsetal. 2011b ).21planetsaround15MdwarfshavebeendetectedbytheRVtechnique.TherstexoplanetaroundaMdwarfwasGJ876bdiscoveredin1998by Delfosseetal. ( 1998 ); Marcyetal. ( 1998 ).ThesecondplanetGJ876caroundthestarwasan-nouncedby Marcyetal. ( 2001 ),whichisanothergasgiantinresonancewiththeonepreviouslydiscovered.Thistypeofresonance,commonlyfoundforsmallobjectssuchasmoonsandastroids,isknownforgasgiantsforthersttime.In2004,anothergasgiantHD41004Bbwasdiscoveredby( Zuckeretal. 2004 ).TheHD41004binarystarsystemisuniquebecauseithasabrowndwarforbitingthefaintcompanionandaplanetorbitingthebrightcompanion.Thediscoveriesofgasgiantsaroundlow-massMdwarfsposechallengestoplanetformationandevolutiontheoryandmotivatetheoriststoupgradeandrenemodels.TherstNeptune-sizeexoplanetwasfoundby Butleretal. ( 2004 )withaminimummassmsiniof21M.ThethirdplanetaroundGJ876d,whichisa7.5Msuper-Earth,wasdiscoveredby Riveraetal. ( 2005 )in2005.Withthediscoveryofthefourthplanet,GJ876e( Riveraetal. 2010 ),whichisNeptune-sized,GJ876systembecomestherstknownSun-analogwithtwooutsidegasgiantsshepherdinganinsidelow-massplanetandthe 26

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outermostonetobeaNeptune-sizeplanet.GJ581bwasfoundby Bonlsetal. ( 2005 )inthesameyear,whichisthepreludeforaseriesofsubsequentdiscoveriesofothermem-bersc,d( Udryetal. 2007 )ande( Mayoretal. 2009b )inthesystem. Bonlsetal. ( 2007 )detectedaplanetwithaminimummassof11M,GJ674b.Accordingtotheiranalysis,theyfoundevidenceoftheexistenceofplanet-metallicitycorrelationforMdwarfs.Inthesameyear, Johnsonetal. ( 2007a )foundthethirdM-dwarfgiantplanetwithalongperiod(P=692.9d).Theyfoundgiantplanetoccurrencerateiscorrelatedwithhoststarmassevenaftercor-rectingformetallicity. Forveilleetal. ( 2009 )addedonesuper-Earthinthediscoverylistaftercorrectingtheclaimmadeby Endletal. ( 2008 ).Theplanetfoundby Baileyetal. ( 2009 ),GJ832b,holdtherecordforthelongestorbitalperiodandthelowesthoststarmetallicityamongcurrentcensusofM-dwarfplanets.GJ1214bistherstplanetaroundaMdwarfdiscoveredwiththetransitmethodandlaterconrmedwiththeRVtechnique( Charbonneauetal. 2009 ). Haghighipouretal. ( 2010 )foundasaturn-massplanet,HIP57050b.In2010,threemoregiantplanetdiscoveries,HIP79431b( Appsetal. 2010 ),GJ649b( Johnsonetal. 2010b )andGJ179b( Howardetal. 2010a ),wereannounced. Bonlsetal. ( 2011a )detectedanothershort-periodsuper-EarthGJ3634b.Thediscoverylistwillkeepbeingreplenishedasexoplanetscientistscontinuetopush-ingthedetectionlimitforsurveysusingavarietyofmethods.Forexample,1235planetarycandidateswereannouncedafterKeplerreleasesitsrst4monthsofdata( Boruckietal. 2011c ).ThereareM-dwarfplanetcandidateswaitingtobeconrmedbyothermethodssuchastheRVtechnique. 27

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CHAPTER2FUNDAMENTALPERFORMANCEOFTHEDFDIMETHOD 2.1IntroductionThepopularDopplerinstrumentsarebasedonthecross-dispersedechellespectrogaphdesign,whichwecalledthedirectechelle(DE)method.Inthismethod,theRVsignalsareextractedbydirectlymeasuringthecentroidshiftofstellarabsorptionlines.Thefundamentalphoton-limitedRVuncertaintyusingtheDEmethodhasbeenstudiedandreportedbyseveralresearchgroups(e.g., Bouchyetal. ( 2001 ); Butleretal. ( 1996 )).WhileDEisthemostwidelyadoptedmethodinprecisionRVmeasurements,atotallydifferentRVmethodusingadispersedxeddelayinterferometer(DFDI)hasalsodemonstrateditscapabilityindiscoveringexoplanets( Flemingetal. 2010 ; Geetal. 2006b ; Leeetal. 2011 ).Inthismethod,theRVsignalsarederivedfromphaseshiftoftheinterferencefringescreatedbypassingstellarabsorptionspectrathroughaMichelsontypeinterferometerwithxedopticalpathdifference(OPD)betweenthetwointerferometerarms( Erskine 2003 ; Erskine&Ge 2000 ; Ge 2002 ; Geetal. 2002 ).Thestellarfringesareseparatedbyapost-disperser,whichistypicallyamedium-resolutionspectrograph.DopplersensitivityofDFDIcanbeoptimizedbycarefullychoosingtheopticalpathdifferenceoftheinterferometer.TheDFDImethodispromisingforitslowcost,compactsizeandpotentialformulti-objectcapability( Ge 2002 ). vanEykenetal. ( 2010 )discussedthetheoryandapplicationofDFDIindetails.However,itsfundamentallimitforDopplermeasurementshasnotbeenwellstudiedbefore.Inthischapter,Iwillintroduceamethodtocalculatephoton-noiselimitedDopplermeasurementuncertaintyinthenearinfrared(NIR)wavelengthregion,whereweplantoapplytheDFDImethodforlaunchingaDopplerplanetsurveyaroundMdwarfs.NIRDopplerplanetsurveysareveryimportanttoaddressplanetcharacteristicsaroundlowmassstars,especiallyMdwarfs.MdwarfsemitmostoftheirphotonsintheNIRregion.DuetothelackofNIRDopplertechniques,onlyafewhundredsbrightMdwarfshavebeensearchedforexoplanetsusingopticalDEinstruments( Blakeetal. 2010 ; Clubbetal. 2009 ; 28

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Endletal. 2006 ; Zechmeisteretal. 2009 ).Todate,onlyabout20exoplanetsaroundMdwarfshavebeendiscoveredcomparedtomorethan700exoplanetsdiscoveredaroundsolartypestars(i.e.,FGKstars)despiteofthefactthatMdwarfsaccountfor70%starsinlocaluniverse.Nonetheless,searchingforplanetsaroundMdwarfsisessentialtoanswerquestionssuchasthedependenceofplanetarypropertiesonthespectraltypeofhoststars.Inaddition,thesmallerstellarmassofMdwarfsfavorsdetectionofrockyplanetsinhabitablezone(HZ)usingtheRVtechnique.However,thestellarabsorptionlinesinNIRarenotassharpasthoseinthevisibleband.Recentstudyby Reinersetal. ( 2010 )showsthatprecisionRVmeasurementscanonlyreachbetterDopplerprecisionintheNIRthaninvisiblewavelengthforMdwarfswithstellartypeslaterthanM4.Inthischapter,IwillreportresultsfromourstudyonfundamentallimitswiththeNIRDopplertechniqueusingtheDFDImethod.ThetheoryofDFDIhasbeendiscussedbyseveralpapers( Erskine 2003 ; Ge 2002 ; vanEykenetal. 2010 ; Wangetal. 2011 ).Readersmayrefertopreviousreferencesformoredetaileddiscussion.DFDIisrealizedbycouplingaxeddelayinterferometerwithapost-disperser(Fig. 5-1 ).Theresultingfringingspectrumisrecordedona2-Ddetector.TheformationofthenalfringingspectrumisillustratedinFig. 2-2 .B(,y)isamathematicalrepresentationofthenalimageformedatthe2-Ddetectoranditisdescribedbythefollowingequation: B(,y)="S0() hIT(,y)#LSF(,R),(2)whereS0()istheintrinsicstellarspectrumandisopticalfrequency.S0isdividedbyhtoconvertenergyuxintophotonux.ITistheintensitytransmissionfunction(Equation 2 ),yisthecoordinatealongtheslitdirectionwhichistransversetodispersiondirection,representsconvolutionandLSFisthelinespreadfunctionofthepost-disperserwhichisafunctionofandspectralresolutionR.InEquation 2 :isvisibilityforagivenfrequencychannel,theratioofhalfofthepeak-valleyamplitudeandtheDCoffset,whichisdeterminedbystellaruxS0();cisthespeedoflight;andistheopticalpathdifference(OPD)ofthe 29

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Figure2-1. AschematiclayoutofanRVinstrumentusingtheDFDImethod. Figure2-2. DFDIIllustration.S0()isastellarspectrum;IT(,y)isinterferometertransmission;B(,y)istheimagetakenata2-Ddetector.isopticalfrequencyandyiscoordinateofslitdirection.DFDImeasuresRVbymonitoringphaseshiftofstellarabsorptionlinefringesintheydirection(theslitdirection). 30

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interferometerwhichisdesignedtobetiltedalongtheslitdirectionsuchthatseveralfringesareformedalongeachchannel(Middle,Fig. 2-2 ).WeassumetheLSFisagaussianfunction(Equation 2 ),==R=2.35becauseweassumethatoneresolutionelementisequaltotheFWHMofaspectralline. IT=1+()cos"2(,y) c#,(2) LSF(0,)=1 22exp")]TJ /F5 11.955 Tf 8.8 8.09 Td[(()]TJ /F8 11.955 Tf 10.27 0 Td[(0)2 22#.(2) Figure2-3. ExamplesofsyntheticMdwarfstellarspectra(Vsini=0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),whicharegeneratedbyPHOENIX( Allardetal. 2001 ; Hauschildtetal. 1999 ).Thetoppanelshowsthespectrabetween0.8-1.35m,thebottompanelshowsanenlargedspectralregionaround1177nmshowingstellarlineproles. Figure 2-3 showshigh-resolution(0.005spacing)syntheticspectraofMdwarfswithsolarmetallicity( Allardetal. 2001 ; Hauschildtetal. 1999 ).Terangesfrom2400Kto 31

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Figure2-4. Top:powerspectrumofderivativesofstellarspectrum,F[dS0=d],withSRF()=F[LSF(,R)]fordifferentRoverplotted,whereLSFislinespreadfunction;Bottom:F[dS0=d]shiftedby=20mmusingaxed-delayMichelsoninterferometer.TheSRFsfordifferentRareoverplotted. 3100K,andloggis4.5.Norotationalbroadeningisaddedinthespectrum.MostabsorptionlinesareshallowwithFWHMsofseveraltenthsofan.SinceRVinformationisembeddedintheslopeofanabsorptionline,sharpanddeeplinescontainmoreRVinformationthanbroadandshallowlines.Mathematically,theslopeisthederivativeofuxasafunctionofopticalfrequency,i.e.,dS0=d.ThepowerspectrumofdS0=disobtainedbyFouriertransform.AccordingtopropertiesofFouriertransform,F[dS0=d]=(i)F[S0],whereFmanifestsFouriertransform,iistheunitofimaginarynumberandistherepresentationof=cinFourierspace.WeplotF([dS0=d]inFig. 2-4 aswellasthespectralresponsefunction(SRF),whichisF[LSF].SRFatR=5,000dropsdrasticallytowardhighspatial 32

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frequency(highvalue)suchthatitmissesmostoftheRVinformationcontainedinstellarspectrum.AsRincreases,SRFgraduallyincreasestowardhighwherethebulkofRVinformationisstored.AspectrographwithRof100,000iscapableofnearlycompletelyextractingRVinformation.UnlikeDE,DFDIcanshiftF[dS0=d]byanamountdeterminedbytheOPDoftheinterferometer( Erskine 2003 ).Forexample,Fig. 2-4 alsoshowsthepowerspectrumofF([dS0=d]ofafringingspectrumobtainedwithaDFDIinstrumentwitha20mmopticaldelay,whichshiftsF[dS0=]by20mm.Inthiscase,RVinformationhasbeenshiftedfromtheoriginalhighspatialfrequenciestolowspatialfrequencieswhichcanberesolvedbyaspectrographwithalowormediumRinDFDI. 2.2MethodologyofCalculatingPhoton-limitedRVMeasurementUncertaintyIntheDEmethod,anefcientwaybasedonaspectralqualityfactor(Q)wasintroducedby Bouchyetal. ( 2001 )tocalculatethefundamentaluncertaintyintheDopplermeasure-ments.TheQfactorisameasureofspectralproleinformationwithinagivenwavelengthregionconsideredforDopplermeasurements.HerewedevelopasimilarmethodtocalculateQvaluesfortheDFDImethod.InsteadofrepresentingthespetrallineproleinformationintheDEmethod,theQfactorinourDFDImethodrepresentsstellarfringeproleinformation.Weusehighresolution(0.005spacing)syntheticstellarspectrageneratedbyPHOENIXcode( Allardetal. 2001 ; Hauschildtetal. 1999 )becauseobservedspectraoflowmassstarsdonothavehighenoughresolutionandbroadeffectivetemperaturecoverage. Reinersetal. ( 2010 )haveconductedseveralcomparisonsbetweensyntheticspectrageneratedbyPHOENIXandtheobservedspectra.TheyconcludedthatthesyntheticspectraareaccurateenoughforRVmeasurementuncertaintycalculation.WeusedsyntheticstellarspectraofsolarabundancewithTerangingfrom2400Kto3100K(correspondingspectraltypefromM9VtoM4V)andasurfacegravityloggof4.5.TheQfactoriscalculatedforaseriesof10nmspectralslicesfrom800nmto1350nm.WearticiallybroadenspectrawithVsinifrom0kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1to10kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1assumingalimbdarkeningindexof0.6,whichisatypicalvalueforanMdwarf.Weconvolvetherotationalbroadeningprolewitheachspectralsliceof 33

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10nmtoobtainarotationally-broadenedspectrum.WeassumeaGaussianLSFwhichisdeterminedbyspectralresolutionR(Equation 2 ).AfterarticialrotationalbroadeningandLSFconvolution,werebineachspectralsliceaccordingto4.2pixelsperresolutionelement(accordingtotheopticaldesignofIRETby Zhaoetal. ( 2010 ))togeneratethenal2DimageonadetectorbasedonwhichwecomputetheQfactor. 2.2.1Photon-limitedRVUncertaintyofDE Bouchyetal. ( 2001 )describedamethodofcalculatingtheQfactorfortheDEmethod.Webrieyintroducethemethodhereandthereadercanreferto Bouchyetal. ( 2001 )formoredetails.LetS0()designateanintrinsicstellarspectrum.A0,adigitalizedandcalibratedspectrum,isconsideredasanoise-freetemplatespectrumfordifferentialRVmeasurement,whichisrelatedtoS0()viathefollowingequation: A0(i)=S0() hLSF(),(2)whereiispixelnumberandS0isdividedbyhtoconvertenergyuxintophotonux.AnotherspectrumAistakenatadifferenttimewithatinyDopplershift,whichissmallrelativetothetypicallinewidthofanintrinsicstellarabsorption.Assumingthatthetwospectrahavethesamecontinuumlevel,Dopplershiftisgivenby: v c= ,(2)wherecisspeedoflightandisopticalfrequency.TheoverallRVuncertaintyfortheentirespectralrangeisgivenby( Bouchyetal. 2001 ): vrms c=Q)]TJ /F5 7.97 Tf 5.07 0 Td[(12666664XiA0(i)3777775)]TJ /F5 7.97 Tf 5.07 0 Td[(1=2=1 Qp Ne)]TJ /F7 11.955 Tf 5.99 9.42 Td[(, (2) whereQisdenedas: Q266666666666664XiW(i) XiA0(i)3777777777777751=2,(2) 34

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andW(i)isexpressedas: W(i)=h@A0(i) @(i)i2(i)2 A(i).(2)TheQfactorisindependentofphotonuxandrepresentsextractableDopplerinformationgivenanintrinsicstellarspectrumandinstrumentspectralresolutionR.AccordingtoEquation 2 ,wecancalculatephoton-limitedRVuncertaintygiventheQfactorandphotonuxNe)]TJ /F6 11.955 Tf 8.12 -0.51 Td[(=XiA0(i)withinthespectralrange. 2.2.2Photon-limitedRVUncertaintyofDFDIAnewmethodofcalculatingtheQfactorforDFDIisdevelopedanddiscussedhere.Afteradigitalizationprocess,a2-DuxdistributionexpressedbyEquation 2 isrecordedona2-DdetectorinDFDI.Thedigitalizationprocessinvolvesdistributingphotonuxintoeachpixelaccordingto:1)pixelsperresolutionelement(RE);2)spectralresolution;3)numberoffringesalongslit.B0(i,j),whichisanoise-freetemplate,isthencalculated.B(i,j)isaframetakenatadifferenttimewithatinyDopplershift.iisthepixelnumberalongthedispersiondirection,andjisthepixelnumberalongtheslitdirection.Theobservableintensitychangeatagivenpixel(i,j)inDFDIisexpressedby: B(i,j))]TJ /F4 11.955 Tf 10.26 0 Td[(B0(i,j)=@B0(i,j) @(i)(i)=@B0(i,j) @(i)v c(i). (2) TheDopplershiftismeasuredbymonitoringtheintensitychangeatagivenpixelintheequation: v c=B(i,j))]TJ /F4 11.955 Tf 10.26 0 Td[(B0(i,j) @B0(i,j) @(i)(i).(2)FrameB0isassumedtobeanoise-freetemplateandthenoiseofframeBisthequadraticsumofthephotonnoiseandthedetectornoiseD: Brms(i,j)=q B(i,j)+2D.(2) 35

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Equation 2 isapproximatedunderphoton-limitedconditionsasBrms(i,j)=p B(i,j).Therefore,theRVuncertaintyatpixel(i,j)isgivenby: vrms(i,j) c=p B(i,j) @B0(i,j) @(i)(i).(2)TheoverallRVuncertaintyfortheentirespectralrangeisgivenby: vrms c=8>><>>:Xi,j"vrms(i,j) c#)]TJ /F5 7.97 Tf 5.07 0 Td[(29>>=>>;)]TJ /F5 7.97 Tf 5.07 0 Td[(1=226666664Xi,jW(i,j)37777775)]TJ /F5 7.97 Tf 5.07 0 Td[(1=2Q)]TJ /F5 7.97 Tf 5.07 0 Td[(126666664Xi,jB0(i,j)37777775)]TJ /F5 7.97 Tf 5.07 0 Td[(1=2=1 Qp Ne)]TJ /F7 11.955 Tf 5.99 9.41 Td[(, (2) where W(i,j)@B0(i,j) @(i)2(i)2 B(i,j),(2)and Q"Xi,jW(i,j) Xi,jB0(i,j)#1=2.(2)Equation 2 calculatestheQfactorfortheDFDImethod,whichisalsoindependentofuxandrepresentstheDopplerinformationthatcanbeextractedwiththeDFDImethod.AccordingtoEquation 2 ,wecancalculatephoton-limitedRVuncertaintygiventheQfactorandphotonuxNe)]TJ /F6 11.955 Tf 8.12 -0.51 Td[(=Xi,jB0(i,j)withinthespectralrange. 2.3ComparisonbetweenDEandOptimizedDFDI 2.3.1OptimizedDFDIOpticalPathDifference(OPD)ofaxeddelayinterferometerisacrucialparameterthataffectstheDopplersensitivityofaDFDIinstruments( Ge 2002 ).AnoptimizedOPDcan 36

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helpincreasetheinstrumentDopplersensitivity.WecalculatetheoptimalOPDforspectraofvariousTeandVsiniatdifferentspectralresolutions(Table 2-1 ).Weassumeawavelengthrangefrom800nmto1350nmandanOPDrangefrom10mmto41mmwithastepsizeof1mminthecalculationasdescribedinx 2.2.2 .OptimalOPDisselectedastheonewhichresultsinthehighestQfactorvalue.IncreasingVsiniordecreasingRnaturallybroadensabsorptionlines,decreasingthecoherencelengthofeachstellarabsorptionline( Ge 2002 ).Consequently,oursimulationsshowingeneralthattheoptimalOPDdecreaseswithincreasingVsiniordecreasingRvalues(Fig. 2-5 ).WealsonotethatTeinuenceonoptimalOPDisnotsignicant. Figure2-5. OptimalOPDcorrelationwithVsini(left)andspectralresolutionR(right).OptimalOPDforR=20,000(solid),50,000(dotted),80,000(dashed)areusedontheleftpanel.Vsini=2(solid),5(dotted),10(dashed)kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1areassumedontherightpanel.CompleteresultsofoptimalOPDcanbefoundinTable 2-1 .TeinuenceonoptimalOPDisnotsignicant,Te=2800Kisadoptedintheplot. 37

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Table2-1. OPDchoiceasafunctionofRandVsiniatdifferentTe RVsini[kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1]012345678910 5,00019,27,2919,19,2819,19,1915,15,1715,15,1513,13,1311,12,1210,11,1110,10,1010,10,1010,10,1010,00021,23,2719,21,2619,19,2017,17,1715,15,1513,13,1311,12,1211,11,1110,10,1010,10,1010,10,1015,00021,23,2621,22,2319,20,2017,17,1715,15,1514,14,1413,13,1312,11,1111,11,1111,10,1010,10,1020,00022,23,2621,23,2420,21,2119,19,1917,17,1715,15,1514,14,1413,13,1312,12,1211,11,1111,11,1125,00023,24,2623,24,2521,22,2219,20,2019,18,1817,17,1616,15,1514,14,1414,13,1314,13,1212,12,1230,00024,26,2624,25,2623,23,2421,21,2119,19,1918,17,1717,16,1616,15,1514,14,1414,14,1414,14,1435,00026,26,2826,26,2624,24,2522,22,2321,21,2019,19,1919,18,1717,17,1716,16,1616,16,1616,16,1540,00027,28,2827,27,2826,26,2624,24,2422,22,2221,20,2019,19,1919,18,1819,18,1819,18,1819,18,1645,00029,29,3028,28,2927,27,2825,25,2624,23,2322,22,2122,20,2022,20,2022,20,1822,18,1822,18,1850,00030,30,3129,30,3028,28,2827,26,2624,24,2424,22,2322,22,2222,22,2222,22,2222,22,2224,22,2255,00032,32,3231,31,3230,30,3027,28,2827,26,2624,24,2424,24,2424,24,2224,24,2224,24,2224,24,2460,00032,32,3432,32,3332,31,3229,28,2827,27,2627,26,2624,24,2424,24,2424,24,2424,24,2424,24,2465,00034,34,3534,34,3432,32,3232,30,3029,28,2827,27,2627,26,2627,24,2624,24,2624,24,2424,24,2470,00035,35,3635,35,3634,32,3432,32,3232,30,3032,28,2827,28,2827,28,2627,24,2624,24,2624,24,2675,00037,36,3737,36,3735,35,3532,32,3232,32,3232,32,3232,32,2832,32,2832,32,3232,32,3232,24,3280,00037,37,3937,37,3737,36,3635,34,3535,32,3232,32,3232,32,3232,32,3235,32,3235,32,3235,32,3285,00040,39,4140,39,3937,37,3737,36,3635,32,3235,32,3235,32,3235,32,3235,32,3235,32,3235,32,3290,00040,41,4140,41,4140,37,3937,36,3637,36,3637,36,3635,32,3635,32,3637,37,3637,37,3637,37,3695,00041,41,4141,41,4140,41,4140,37,3737,37,3637,37,3637,37,3637,37,3637,37,3737,37,3737,37,37100,00041,41,4141,41,4140,41,4140,40,4137,37,3737,37,3637,37,3637,37,3737,37,3737,37,3737,37,37105,00041,41,4141,41,4141,41,4140,41,4140,40,4137,37,3737,37,3737,37,3737,37,3737,37,3737,37,37110,00041,41,4141,41,4140,41,4140,41,4140,41,4137,41,4137,37,4137,37,3737,37,3737,37,3737,37,37115,00041,41,4141,41,4141,41,4140,41,4140,41,4137,41,4137,41,4137,37,4137,37,3737,37,3737,37,37120,00041,41,4141,41,4141,41,4140,41,4140,41,4137,41,4137,41,4137,37,4137,37,3737,37,3737,37,37125,00041,41,4141,41,4141,41,4140,41,4140,41,4137,41,4137,41,4137,37,4137,37,3737,37,3737,37,37130,00041,41,4141,41,4140,41,4140,41,4140,41,4137,41,4137,41,4137,37,4137,37,3737,37,3737,37,37135,00041,41,4141,41,4140,41,4140,41,4140,41,4137,41,4137,41,4137,37,4137,37,3737,37,3737,37,37140,00041,41,4141,41,4140,41,4140,41,4140,41,4137,41,4137,41,4137,41,4137,37,4137,37,3737,37,37145,00041,41,4141,41,4140,41,4140,41,4140,41,4140,41,4137,41,4137,41,4137,41,4137,41,4137,41,41150,00041,41,4141,41,4140,41,4140,41,4140,41,4140,41,4137,41,4137,41,4137,41,4137,41,4137,41,41 38

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WeinvestigatehowtheQfactorisaffectedifOPDisdeviatedfromtheoptimalvalue.WecalculatetheQfactorwhentheactualOPDisdeviatedfromtheoptimalOPDby5mm.WechoosethelowervalueofthetwoQsfromthe5mmdeviationfromtheoptimaldelay(bothpositiveandnegativesides)asQdeviated.WeplottheratioofQoptimalandQdeviatedasafunctionofspectralresolutionRinFig. 2-6 .WefoundthatdeviatingOPDby5mmdoesnotresultinseveredegradationoftheQfactor.Themaximumdegradationis1.115andoccursatRof5,000andVsiniof5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forastarwithTeof2800K.Thedegradationcanbecompensatedbyincreasingtheintegrationtimeby24%(1.1152)toreachthesamephoton-limitedDopplerprecisionaccordingtoEquation 2 .ThedegradationbecomessmallerasRincreases.AsshowninFig. 2-4 ,DFDIshiftsthepowerspectrumofdS0=dbyanamountdeterminedbytheinterferometerOPDsothattheSRFhasareasonableresponseataregionwheremostoftheRVinformationiscontained.TheSRFbroadensasRincreases.Therefore,itcanstillrecovermostoftheRVinformationinastellarspectrumevenifOPDisdeviatedfromtheoptimalvalue.Atlowandmediumresolutions(5,000to20,000),QfactordegradationbecomeslargerasVsiniincreases.Thisisbecauserotationalbroadeningremovesthehighfrequencysignalfromastellarspectrum,whichmakestheregioncontainingmostoftheRVinformationmoresensitivetothechoiceofOPDasthepowerspectrumdistributionbecomesnarrowerduelossofhighcomponent. 2.3.2InuenceofSpectralResolutionRIntheory,aspectrographwithaninnitelyhighresolutionwouldbeabletoextractalltheRVinformationcontainedinastellarspectrum.However,inpractice,itisimpossibletocompletelyrecovertheRVinformationwithaspectrographwithanitespectralresolutionwhosespectralresponsefunctiondropsatthehighspatialfrequencyend.AlthoughthepowerspectrumofthederivativeofthestellarspectrumisshiftedtothelowfrequencyregionwheremostoftheRVinformationiscarried,thepowerspectrumisstillbroadinthespatialfrequency()domain(seeFig. 2-4 ).Therefore,highRcanhelptoextractmoreRVinformation.Inawavelengthcoveragefrom800nmto1350nm,wecalculateQvaluesfor 39

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Figure2-6. RatioofQoptimalandQdeviatedasafunctionofspectralresolution,whereQoptimalisQfactoratoptimalOPDandQdeviatedisQfactorwhenOPDisdeviatedfromQoptimalby5mm.DifferentlinestylesrepresentdifferentVsiniwhilecolorsindicatedifferentTe. stellarspectrawithVsiniof0,2,5and10kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1atdifferentR(5,000to150,000withastepof5,000)inordertoinvestigatethedependenceofQonR(Fig. 2-7 ).WendthatmoreRVinformation(higherQfactor)canbeextractedasRincreases.QfactorsforDFDIandDEconvergeathighRbecausethespectralresponsefunctioniswideenoughinthedomaintocovertheregionrichinRVinformation,notaffectedbythepowerspectrumshiftinginvolvedinDFDI.Inaddition,theQfactoratagivenRincreasesasTedropsfrom3100Kto2400K,whichislargelyduetostrongermolecularabsorptionfeaturesintheI,YandJbands(seeFig. 2-3 ). 40

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Table2-2. PowerLawIndexasafunctionofSpectralResolutionR(Te=2400K) DFDIDE VsiniR5,00020,00050,0005,00020,00050,000[kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1]-20,000-50,000-150,000-20,000-50,000-150,00000.620.590.311.080.930.4420.630.560.271.070.890.3850.620.450.161.010.690.21100.580.280.090.870.390.10 WedivideRintothreeregions,lowresolution(5,000to20,000),mediumresolution(20,000to50,000)andhighresolution(50,000to150,000).WeuseapowerlawtotQforbothDFDIandDEasafunctionofR.ThepowerindicesofthreeregionsforTe=2400KarepresentedinTable 2-2 .AtlowRregion,remainsroughlyaconstantfor0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Vsini5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,butitdropsforstarswithVsiniof10kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1indicatingstellarabsorptionlinesbegintoberesolvedevenatlowR.AthigherRregions,decreasesasVsiniincreases,areducedvalueofimpliesdiminishingbenetbroughtbyincreasingR.Stellarabsorptionlinesarebroadenedbystellarrotation,andtheyareresolvedatacertainRbeyondwhichincreasingRdoesnotsignicantlygainDopplersensitivity.Overall,forDEislargerthanthatofDFDI,especiallyforlowandmediumR.Inotherwords,QDFDIislesssensitivetoachangeofR,andtheDFDIinstrumentcanextractrelativelymoreDopplerinformationatlowormediumspectralresolutionthantheDEmethod.Forexample,forslowrotators(Vsini=2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)atthelowRregion(R=5,000-20,000),QDFDI/R0.63.Dopplersensitivityvrmsisinverselyproportionaltotwofactors:Qandp Ne)]TJ /F1 11.357 Tf 7.96 -0.51 Td[(accordingtoEquation 2 and 2 ,whereNe)]TJ /F1 11.357 Tf 7.95 -0.51 Td[(isthetotalphotoncountcollectedbytheCCDdetector.Ne)]TJ /F3 11.955 Tf 9.28 -0.51 Td[(/(S=N)2Npixel,whereS=Nistheaveragesignaltonoiseratioperpixel,andNpixelistotalnumberofpixels.NotethatNe)]TJ /F3 11.955 Tf 9.02 -0.51 Td[(/Rifthewavelengthcoverage,S/Nperpixelandtheresolutionsamplingarexed.Therefore,vrms/R)]TJ /F5 7.97 Tf 5.06 0 Td[(0.63)]TJ /F5 7.97 Tf 5.07 0 Td[(0.5=R)]TJ /F5 7.97 Tf 5.07 0 Td[(1.13forDFDI.Incomparison,vrms/R)]TJ /F5 7.97 Tf 5.07 0 Td[(1.57forDEgiventhesamewavelengthcoverageandS/Nperpixel.Thepowerlawisconsistentwiththeprevioustheoreticalworkby Ge ( 2002 )and Erskine ( 2003 ). 41

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Figure2-7. Qfactorasafunctionofspectralresolution.(left:Te=2400K;right:Te=3100K.OpencirclesrepresentQDFDI;crossesrepresentQDE;solidlinesarebestpower-lawtsforQDFDI;dashedlinesarebestpower-lawtsforQDE) WecompareQfactorsforbothDFDIandDEatgivenRvalues,andtheresultsareshowninFig. 2-8 .Forveryslowrotators(0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Vsini2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),theadvantageofDFDIoverDEisobviousatlowandmediumR(5,000to20,000)becausethecenterofthepowerspectrumofthederivativeofthestellarspectrumisatahighfrequencydomainwhichcannotbecoveredinDEduetothelimitedfrequencyresponserangeofitsSRFatlowandmediumR.TheimprovementofDFDIis3.1times(R=5,000),2.4times(R=10,000)and1.7times(R=20,000)respectively.Inotherwords,optimizedDFDIwithRof5,000,10,000and20,000isequivalenttoDEwithRof16,000,24,000and34,000respectivelyintermsofDopplersensitivityforthesamewavelengthcoverage,S/Nperpixelandspectralsampling(otherwise,seemorediscussionsinx 2.3.3 ,thegainwiththeDFDIwouldbemore 42

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signicantforaxeddetectorsizeandexposuretime).Overall,DEwiththesamespectralresolutionasDFDIatR=5,000-20,000requires3-9timeslongerexposuretimetoreachthesameDopplersensitivityasDFDIifbothinstrumentshavethesamewavelengthcoverageandsamedetectionefciency(i.e.,Ne)]TJ /F1 11.357 Tf 7.95 -0.51 Td[(isthesame).TheimprovementofDFDIatR=20,000-50,000isnotasnoticeableasatthelowRrange.ThedifferencebetweenDFDIandDEbecomesnegligiblewhenRisover100,000.Inotherwords,theadvantageoftheDFDIoverDEgraduallydisappearsasRreacheshighresolutiondomain(R>50,000).Inaddition,theimprovementforrelativelyfasterrotators(5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Vsini10kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1)withDFDIislesssignicantthanitisforslowrotators. Figure2-8. ImprovementofQDFDIoverQDEasafunctionofspectralresolution. 43

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2.3.3InuenceofDetectorPixelNumbersIntheNIR,detectorpixelnumberistypicallysmallerthantheopticaldetector.Further-more,thetotalcostforanNIRarrayismuchhigherthananopticaldetectorwiththesamepixelnumber.Intheforeseeablefuture,detectorsizemaybeoneofthemajorlimitationsforDopplersensitivityimprovement.WestudytheimpactofthelimiteddetectorresourceontheDopplermeasurementsensitivity.Usingthesamedetectorresource,wendthatitisfairtocomparetheirDopplerperformanceforthesametargetwiththesameexposuretounderstandstrengthandweaknessforeachmethodalthoughDFDIandDEaretotallydifferentDopplertechniques.AccordingtoEquation 2 and 2 ,wedeneanewmeritfunction, Q0=Qp Ne)]TJ /F7 11.955 Tf 4.8 -0.51 Td[(,(2)tostudyphoton-limitedDopplerperformanceforbothmethodswiththesamedetectorsize.Notethatthenewlydenedmeritfunctionisdirectlyrelatedtophoton-limitedRVuncertainty,i.e.,inverseproportionality.Ne)]TJ /F1 11.357 Tf 7.95 -0.51 Td[(iscalculatedbyEquation 2 : Ne)]TJ /F6 11.955 Tf 8.12 -0.51 Td[(=FSteltexp 2.512mJ,(2)inwhichFisthephotonuxinthewavelengthcoverageregionofanmJ=0starwiththeunitofphotonss)]TJ /F5 7.97 Tf 5.07 0 Td[(1cm)]TJ /F5 7.97 Tf 5.07 0 Td[(2;isinstrumenttotalthroughput;Stelistheeffectivesurfaceareaofthetelescope;texpisthetimeofexposure;andmJistheJbandmagnitude.HereweuseIRET( Zhaoetal. 2010 )asanexampletoillustratestrengthsoftheDFDImethodforDopplermeasurements.IRETadoptstheDFDImethodandhasawavelengthcoveragefrom800nmto1350nmandaspectralresolutionof22,000.Foraxeddetec-torsize(i.e.,totalnumberofdetectorpixels)andxednumberofpixelstosampleeachresolutionelement,thetotalwavelengthcoverageofaDopplerinstrument,,is =Npix Porderc RNS,(2) 44

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whereNpixistotalnumberofpixelsavailableonaCCDdetectorandNSisthenumberofpixelsperresolutionelement,cisthecentralwavelengthandPorderisthenumberofpixelssamplingeachpixelchannelbetweenspectralorders1.Equation 2 showsthatisinverselyproportionaltoR.Table 2-3 givestherelationofRandassumingNpixel,PorderandNSasconstants.cissettobe1000nmbecauseitisapproximatelythecenteroftheYband.WecalculatedtheratioofQ0DFDIandQ0DEinwhichweusethephotonuxofastarwithaTeof2400K(Fig. 2-9 ).Q0IRETisconsistentlyhigherthanQ0DEregardlessofRoftheDEinstrument.Inotherwords,IRETisabletoachievelowerphoton-limitedRVuncertaintycomparedtoaDEinstrumentwiththesamedetector.Theresultseemstobedifferentfromtheconclusionwedrewinx 2.3.2 ,inwhichwecompareQfactorsofthesameRandandreachedaconclusionthatDFDIwithanRof22,000isequivalenttoDEwithanRof35,000(afactorof1.6gain)forthesamewavelengthcoverageandthesamedetectionefciency(Fig. 2-7 ).Thekeydifferencebetweenthiscaseandtheearliercaseisthexeddetectorresourceinsteadofxedtotalcollectedphotonnumbers.Sincelowerspectralresolutionallowstocovermorewavelengths,morephotonswillbecollectedforthesameinstrumentdetectionefciencyforbothDFDIandDEM.NotethatQ0consistsoftwocomponents,QandNe)]TJ /F1 11.357 Tf 4.8 -0.51 Td[(.Foragivennumberofpixelsonthedetector,Ne)]TJ /F10 7.97 Tf 4.3 -2.3 Td[(,DFDIishigherthanNe)]TJ /F10 7.97 Tf 4.3 -2.3 Td[(,DEduetothelargerwavelengthcoverage.Inaddition,QDFDI(DFDI)ismorethanQDE(DE).Consequently,weseeinFig. 2-9 thatQ0IRETishigherthanQ0DEatallRofaDEinstrument.Figure 2-9 alsoshowsthattheminimumofQ0DFDI=Q0DEisdependentofVsini.TheratioofQ0reachesaminimum(Q0DEreachesamaximum)aroundanRof50,000forslowrotators(Vsini5kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1).ItincreasesatthelowRendbecausethespectrographhasnotyet 1Inprinciple,eachfrequencychannelforbothDFDIandDEinstrumentscanbedesignedidentical.Inpractice,theDFDIinstrumenttendstouse20pixelstosamplefringesintheslitdirection,whichisonlyformeasurementconvenience,notarequirement.Infact, Muir-headetal. ( 2011 )hasdemonstratedaphase-steppingmethodwhichdoesnotrequiresam-plefringesintheslitdirection.) 45

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resolvedstellarabsorptionlines.Ontheotherhand,theratioincreasesatthehighRendbecauseoffewerphotons(seeTable 2-3 ).Forfastrotators(Vsini=10kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),theratioreachesaminimumaroundRof30,000. Table2-3. SpectralResolutionandwavelengthcoverageonagivendetector Rmin)]TJ /F8 11.955 Tf 10.26 0 Td[(max(nm)(nm) 25,000480800)]TJ /F1 11.357 Tf 7.6 0 Td[(128030,000400800)]TJ /F1 11.357 Tf 7.6 0 Td[(120040,000300850)]TJ /F1 11.357 Tf 7.6 0 Td[(115050,000240880)]TJ /F1 11.357 Tf 7.6 0 Td[(112060,000200900)]TJ /F1 11.357 Tf 7.6 0 Td[(111070,000170910)]TJ /F1 11.357 Tf 7.6 0 Td[(108080,000150920)]TJ /F1 11.357 Tf 7.6 0 Td[(1070 Figure2-9. ComparisonofQ0IRETandQ0DEatdifferentR.NotethatQ0=Qp Ne)]TJ /F1 11.357 Tf 4.79 -0.51 Td[(.Differentcolorrepresentsdifferentrotationalvelocity. 46

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Foraslowrotator(Vsini=2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)atlowRregion(R=5,000-20,000),QDFDI/R0.63.SinceDopplersensitivityvrmsisinverselyproportionaltotwofactors:Qandp Ne)]TJ /F1 11.357 Tf 7.95 -0.51 Td[(accordingtoEquation 2 and 2 ,theDopplersensitivitybecomesnearlyindependentofspectralresolutionfortheDFDImethod(/R)]TJ /F5 7.97 Tf 5.07 0 Td[(0.13)ifthedetectionsize(ortotalnumberofpixels)isxed.ThisindicatesthatwecanusequitemoderateresolutionspectrographtodispersethestellarfringesproducedbytheinterferometerinaDFDIinstrumentwhilemaintaininghighDopplersensitivity.Thisopensamajordoorformulti-objectDopplermeasurementsusingtheDFDImethodasproposedby Ge ( 2002 ).Incomparison,theDopplersensitivityfortheDEmethodstillstronglydependsonspectralresolutionforaxednumberofdetectorpixels(/R)]TJ /F5 7.97 Tf 5.07 0 Td[(0.57),indicatingthathigherspectralresolutionwillofferbetterDopplersensitivity. 2.3.4InuenceofMulti-ObjectObservationsAsdiscussedinx 2.3.2 and 2.3.3 ,theDFDIinstrumentcanbedesignedtohaveamoderateresolutionspectrographcoupledwithaMichelsontypeinterferometer.ModeratespectralresolutionallowsasingleorderspectrumorafeworderspectratocoverabroadwavelengthregionintheNIRregionwhilekeepingtheDopplersensivitiysimilartoahighresolutionDEdesignwhichrequiresalargedetectorarraytocoverspectrafromasingletarget.ThisindicatesthattheDFDImethodhasmuchgreaterpotentialforaccommodatingmultipletargetsonthesamedetectorasproposedby Ge ( 2002 )thanaDEinstrument.Inordertoevaluatethepotentialimpactofmulti-objectDFDIinstruments,weredenethemeritfunctionQ00as: Q00=Qp Ne)]TJ /F3 11.955 Tf 7.45 -0.51 Td[(Nobj,(2)whereNobjisthenumberofobjectsthatcanbemonitoredsimultaneously,andistheindexofimportanceformulti-objectobservations.FromtheperspectiveofphotoncountandS/N,multi-objectobservationsareequivalenttoanincreaseofNe)]TJ /F1 11.357 Tf 4.8 -0.51 Td[(,andthusis0.5.However,fromanobservationalefciencypointofview,Q00shouldbeproportionaltoNobjbecausethemoreobjectsareobservedsimultaneously,thequickerthesurveyisaccomplished,andisthereforeequalto1. 47

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Figure2-10. ComparisonofQ00DFDIandQ00DFDI,R=100,000atdifferentR.NotethatQ00=Qp Ne)]TJ /F3 11.955 Tf 7.46 -0.51 Td[(Nobj.Themaximumofeachcurveisindicatedbylledcircle.Differentcolorrepresentsdifferentrotationalvelocity,thesameasFig. 2-9 Weassumeadetectorthatcoversfrom800nmto1350nmatR=100,000sothatwecanusetheQfactorsobtainedinx 2.3.2 .Ne)]TJ /F1 11.357 Tf 7.96 -0.51 Td[(isaconstantsinceweassumeidentical.NobjisinverselyproportionaltothenumberofpixelsperobjectwhichisproportionaltospectralresolutionR(Equation 2 ).NotethatwedonotrequireNobjtobeanintegerbecausewecan,inprinciple,tafractionofspectrumonadetectortomakefulluseofthedetector.Q00sforbothDFDIandDEarecalculated.Figure 2-10 showstheratioofQ00andQ00R=100,000forDFDIundertwodifferentassumptionsof.For=0.5,i.e.,increaseofNobjisequivalenttophotongain,onlyaslightimprovementisachievedifthedetectorisusedformulti-objectobservationsatlowerresolutionthan100,000.Incomparison,fromasurveyefciencypointofview(i.e.,=1),weseeafactorof4-6timesboostofQ00inmulti-object 48

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Figure2-11. ComparisonofQ00DEandQ00DE,R=100,000atdifferentR.NotethatQ00=Qp Ne)]TJ /F3 11.955 Tf 7.46 -0.51 Td[(Nobj.Themaximumofeachcurveisindicatedbylledcircle.Differentcolorrepresentsdifferentrotationalvelocity,thesameasFig. 2-9 observations.ThetruncationatR=5,000isduetoapracticalreasonthatalowerresolutionthan5,000israrelyusedinplanetsurveyusingRVtechniques.Ontheotherhand,similarcalculationisalsoconductedfortheDEmethod(Fig. 2-11 ),inwhichwendthathighresolutionsingleobjectobservationisanoptimaloperationmodeforDEfromaperspectiveofphotongain(=0.5).At=1,theincreaseofQ00isafactorof3atthemost.WecomparethemaximumofQ00forbothDFDIandDEatdifferentVsiniinTable 2-4 .At=0.5,theadvantageofDFDIoverDEis1.1forawiderangeofVsini.Inotherwords,fromthephotongainpointofview,thereisnosignicantdifferencebetweenDFDIandDEinmulti-objectRVinstruments.However,fromthesurveyefciencypointofview(=1),weseeafactorof3boostofQ00inDFDIforslowrotators(Vsini2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),suggesting9 49

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Table2-4. Q00comparisonofDFDIandDEasafunctionofVsini DFDIDEVsini[kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1]RoptimalQ00DFDIRoptimalQ00DEQ00DFDI/Q00DE =0.5050,0007502110,00066231.065250,000680675,00060011.134525,000497950,00044241.1251015,000339425,00029961.133=1.005,0002638430,00087053.03125,0002429725,00084902.86255,0001892915,00078842.401105,0001287710,00072391.779 timesfasterintermsofsurveyspeed.Forfastrotators(i.e.,Vsini=10kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),theboostdropsto1.78.OurstudyconrmsthattheDFDImethodhasanadvantageformulti-objectRVmeasurementsovertheDEmethodassuggestedby Ge ( 2002 ). 2.3.5InuenceofProjectedRotationalVelocityVsiniProjectedrotationalvelocityVsinibroadensstellarabsorptionlinesandthusreducestheQfactor.WecarryoutsimulationscalculatingQfactorsofdifferentVsini(0kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1Vsini10kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1)atR=150,000andvariousTe.Weassumeawavelengthrangefrom800nmto1350nm.TheresultsareshowninFig. 2-12 .TheQfactordecreasesasVsiniincreases.Itisclearthatslowrotatorswouldbebettertargetstoreachhigherphoton-limitedRVprecisionbecausethespectrumofaslowrotatorcontainsmoreDopplerinformation. 2.4SummaryandDiscussion 2.4.1QFactorsforDFDI,DEandFTSWedevelopanewmethodofcalculatingphoton-limitedDopplersensitivityofaninstrumentadoptingtheDFDImethod.WeconductaseriesofsimulationsbasedonhighresolutionsyntheticstellarspectrageneratedbyPHOENIXcode( Allardetal. 2001 ; Hauschildtetal. 1999 ).Insimulations,weinvestigatethecorrelationsofQandotherparameterssuchasOPDoftheinterferometer,spectralresolutionRandstellarprojectedrotationalvelocityVsini.WendthatoptimalOPDincreaseswithincreasingRanddecreasingVsini.Empirically,theoptimalOPDischosensuchthatthedensityofthe 50

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Figure2-12. QDFDIasafunctionofVsini. interferencecombsmatcheswiththelinedensityofthestellarspectrum.Basedonthesimulationresults,theoptimalOPDisdeterminedastheonethatmaximizestheQfactor.Infact,optimalOPDsfoundfromempiricalwayandfromnumericalsimulationareconsistentwitheachother.Forexample,forVsini=0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1andR=50,000,simulationgivesanoptimalOPDof30mm.TheinterferencecombdensityofaninterferometerwithOPDof30mmis0.3at1000nm,whichindeedmatchesthewidthofatypicalabsorptionlineafterspectralblurringwithRof50,000.Anindependentmethodtocalculatephoton-noiselimitedDopplermeasurementuncertaintyintheopticalisbeingdeveloped,andtheresultswillbereportedinaseparatepaper( Jiangetal. 2011 ).WehavecomparedresultsfrombothmethodsandconrmedthatbothindependentmethodsproduceessentiallythesameresultsforbothopticalandNIRDopplermeasurements. 51

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WeinvestigatehowtheQfactorisaffectedifOPDisdeviatedfromtheoptimalvalueandndthatadeviatedOPD(5mm)doesnotresultinasignicantQfactordegradation,whichismitigatedasRincreases.WendthattheQfactorincreaseswithRforbothDFDIandDE,andeventuallyconvergeatveryhighR(R100,000).TheconvergenceofDFDIandDEmethodsisanaturalconsequencebecausethemeasurementmethoddoesnotmakeadifferenceafterthespectralresolutionbecomesextremelyhigh.Inaddition,QfactorsatagivenRincreaseasTedropsfrom3100Kto2400K,whichisduetostrongermolecularabsorptionfeaturesinNIR(seeFig. 2-3 ).TheQfactordecreasesasVsiniincreasesbecausestellarrotationbroadenstheabsorptionlines,leadingtolesssensitivemeasurement.WecompareQfactorsforbothDFDIandDEatagivenR.Forslowrotators(0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Vsini2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),DFDIismoreadvantageousoverDEatlowandmediumR(5,000to20,000)forthesamewavelengthcoverage.TheimprovementofDFDIcomparedtoDEis3.1(R=5,000),2.4(R=10,000)and1.7(R=20,000),respectively.Inotherwords,optimizedDFDIwithRof5,000,10,000and20,000areequivalentinDopplersensitiv-itytoDEwithRof16,000,24,000and34,000,respectively.TheimprovementofDFDIatR20,000to50,000isnotasnoticeableasatlowRrange.ThedifferencebetweenDFDIandDEbecomesnegligiblewhenRisover100,000.Forrelativelyfasterrotators(5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Vsini10kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1),theimprovementwithDFDIislessobviousthanitisforveryslowrotators.DFDIhasstrengthwhenthespectrallinesinastellarspectrumarenotresolvedbyaspectrograph,whichisthecaseforlowandmediumresolutionspectrograph.Undersuchconditions,thexeddelayinterferometerprovidesadditionalresolvingpowersforthesystem.Afterthelinesarefullyresolvedbythespectrographitself,theinterferometerinthesystembecomesdispensable,whichisthereasonwhyweseetheconvergenceofDFDIandDEatveryhighspectralresolution.FundamentalperformanceofaFourier-transformspectrometer(FTS)intheapplicationofDopplermeasurementshasbeendiscussedby Maillard ( 1996 ).Therearesimilarities 52

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betweentheFTSandtheDFDImethod,forexample:1,bothmethodsusetheinterferom-eterasanespectralresolvingelement;2,RVismeasuredbymonitoringthetemporalphasechangeataxedOPDoftheinterferometer.InDFDImethod,OPDisscannedineachfrequencychannelbecauseoftworelativelytiltedmirrors,andtheresolutionofthepost-disperserinDFDIischosentoensureareasonablefringevisibility.Therefore,theDFDImethodisaextendedversionoftheFTSmethodwithalow-mediumresolutionpost-disperser.However,onemajordifferencebetweenthesetwomethodsisthatthein-terferometeritselfisusedasaspectrometerbyOPDscanningintheFTSmethodwhileanadditionalspectrographisemployedintheDFDImethod.Theadvantageofintroducinganadditionalspectrographintothesystemisthatthevisibility(orfringecontrast)isnolongerlimitedbythebandpassasintheFTScase,whichisthereasonthattheDFDImethodcanbeappliedinbroad-bandDopplermeasurements. Mosseretal. ( 2003 )discussedthepossibilityofanFTSworkinginbroadbandbyintroducingalowresolutionpost-disperserandconcludedthattheFTSmethodisinferior(byafactorbetween1and2)toDEmethodevenafteremployinganpost-disperser.ThisconclusionshouldbeacceptedwithcautionsbecausetheycomparedanFTSwithapost-disperser(R=1200)withaDEinstrumentwithamuchhigherspectralresolution(R=84,000),whichisnotnecessarilyafaircomparison.Wedenenewmeritfunctions(Equations 2 and 2 )toobjectivelyevaluateDopplerperformanceforbothDFDIandDEmethods.ForQ0,themeritfunctionforsingleobjectobservation,wendthatQ0DFDIisconsistentlyhigherthanQ0DEregardlessoftheRoftheDEinstrumentundertheconstraintoftotalnumberofpixels,i.e.,boththeDFDIandDEinstrumentadoptthesameNIRdetector.TheDEinstrumentrequiresusingalargerdetectorinordertoreachthesamewavelengthcoverageastheDFDIinstrument.NotethattheaboveconclusionisbasedontheassumptionthatthenumberofpixelsperspectralorderarethesameforDFDIandDE.Inpractice,aDFDIinstrumentuses20pixelstosamplespatialdirection,i.e.,thedirectiontransversetodispersiondirection,while5pixelsareusuallyusedtosamplespatialdirectioninaDEinstrument.However,the20pixelssamplingis 53

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notarequirementforDFDIbutratherfortheconvenienceofdatareduction.Normally,7pixelssampleonespatialperiodofastellarfringe,whichinprincipleareadequatebasedonaphase-steppingalgorithmprovidedby Erskine ( 2003 ).Ifthesamedetectorisused,thesparepartofthedetectorinDFDIcanbeusedformulti-objectobservations.Consequently,inadditiontosingle-objectinstrument,wealsoinvestigateQ00,ameritfunctionformulti-objectRVmeasurementforbothDFDIandDE.Differentconclusionsarereacheddependingondifferentvalueof,anindexoftheimportanceofmulti-objectobservation.Fromapurephotongainpointofview,DFDIandDEinstrumentshavesimilarQ00valueswithQ00DFDIslightlybetterthanQ00DE(afactorof1.1).Fromasurveyefciencypointofview,aDFDImulti-objectinstrumentis9timesfasterthanitscounterpartusingDEforslowrotatingstars(0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Vsini2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)and4timesfasterforfastrotators(Vsini10kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1). 2.4.2ApplicationofDFDITheremaybeotherpracticalconcernsabouttheinstrumentusingtheDFDImethod,mostofthemareduetotherelativelowspectralresolutioncomparedtocurrentDEinstru-ments.Firstofall,anabsolutewavelengthcalibrationforaDFDIinstrumentisnotaspreciseasaDEinstrumentwithahigherspectralresolution.Forexample,ataspectralresolutionof22,000forIRET,alineprolewithaFWHMof0.45inYbandisexpected.Followingthemethoddescribedin Butleretal. ( 1996 ),itcorrespondsto136.4ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RVuncertaintyataS/Nof100ifonlyonespectrallineisused. Ramseyetal. ( 2010 )proposedtouseaU-Neemissionlampasawavelengthcalibrationsourceandithasapproximately500linesinYbandaccordingtotheirmeasurement.Therefore,afterallthelinesinYbandareconsidered,6ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RVuncertaintyisintroducedintheprocessofabsolutewavelengthcalibration.Incomparison,aDEinstrumentatRof110,000causes1.2ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RVuncer-taintyinanabsolutewavelengthcalibration.However,anabsolutewavelengthsolutionisonlyrequiredfortheDEmethodinordertomeasureRVdriftduetoinstrumentinstability,whichismeasuredinadifferentmethodinaDFDIinstrument.ItissimilartoastellarRV 54

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measurement,thedifferenceisthattheobjectisswitchedfromastartoanwavelengthcali-brationsource.VerticalfringemovementofabsorptionoremissionlinesofanRVcalibrationsourceismeasuredinsteadofcentroidmovementmeasurementinaDEinstrument.Inthiscase,aDFDIinstrument(e.g.,IRET,R=22,000)isequivalenttoaDEinstrumentwithRof37,000intermsofDopplermeasurementprecision(seex 2.3.2 ).Therefore,instrumentRVdriftcalibrationprocessintroducesanRVuncertaintyof3.5ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forIRETintheexampleofaU-Nelampcalibrationsource.Inaddition,theRVuncertaintycanbefurtherreducedbyincreasingS/Nandnumberofmeasurement.Secondly,atalowspectralresolution,itischallengingtoperformspectrallineproleanalysisandthusitrequireshigh-resolutionfollow-upinordertoconrmorexcludeapossibledetection.Lastbutnotleast,inabinarycaseinwhichtheobservedspectrumisblended,twoapproachescanbeusedforidentication:1,frommeasuredRV,iftheuxratioissmall,similartotheplanetcompanioncase,thenthelowermasscompanioncanbeidentiedinthemeasuredRVcurveeventhoughsmalluxcontaminationexists;iftheuxratioisaboutunity,indicatingstronguxcontamination,alargeRVscatteringisexpectedbecausethiscaseisnotconsideredandmodeledinthecurrentdatareductionpipeline;2,frommeasuredspectrum,eventheobservedspectrumisa2-DfringingspectruminDFDI,wecanstillde-fringethespectrumintoa1-Dtraditionalspectrum,onwhichspecialtreatmentcanbeperformedtoquantifytheblendingsuchasTODCOR( Zuckeretal. 2003 ). 55

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CHAPTER3COMPREHENSIVESIMULATIONSFORHABITABLEPLANETSEARCHINTHENIRDiscoveringanEarth-likeexoplanetinhabitablezoneisanimportantmilestoneforastronomersinsearchofextra-terrestriallife.Whiletheradialvelocity(RV)techniquere-mainsonethemostpowerfultoolsindetectingandcharacterizingexo-planetarysystems,wecalculatetheuncertaintiesinprecisionRVmeasurementsconsideringstellarspectralqualityfactors,RVcalibrationsources,stellarnoiseandtelluriccontaminationindifferentob-servationalbandpassesandfordifferentspectraltypes.WepredicttheoptimalobservationalbandpassfordifferentspectraltypesusingtheRVtechniqueunderavarietyofconditions.WecomparetheRVsignalofanEarth-likeplanetinthehabitablezone(HZ)tothenearfuturestateoftheartRVprecisionandattempttoanswerthequestion:HowclosearewetodetectingEarth-likeplanetintheHZusingtheRVtechnique? 3.1IntroductionThefundamentalphoton-noiseRVuncertaintieshavebeendiscussedinseveralpre-viouspapers( Bouchyetal. 2001 ; Butleretal. 1996 ).However,onlyintrinsicpropertiesofstellarspectraarediscussedintheirworkswhilenodetailedcalculationofRVuncertaintiesintroducedbythecalibrationsources.Inarecentpaper, Reinersetal. ( 2010 )consideredtheuncertaintiesintheNIRcausedbyRVcalibrationsources,i.e.,aTh-ArlampandanAmmoniagasabsorptioncell.However,thesetwoRVcalibrationsourcescannotbecom-pletelyrepresentativeofthecalibrationsourcesusedandproposedincurrentandplannedDopplerplanetsurveyintheNIR.Forexample,thereareotheremissionlampsavailableintheNIRforRVcalibrationsuchasaU-Nelampasproposedby Mahadevanetal. ( 2010 ).Inaddition,othergasabsorptioncellsbesidestheAmmoniacellhavebeenproposedintheNIR( Mahadevan&Ge 2009 ; Valdivielsoetal. 2010 ).Futhermore,in Reinersetal. ( 2010 ),thecalculationofRVcalibrationuncertaintyofagasabsorptioncellassumesa50nmbandwidthinKband,andthentheuncertaintywasappliedtootherNIRbands,whichispurelyhypothetical.Therefore,amorecomprehensiveanddetailedstudyofRVcalibration 56

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uncertaintiesisnecessaryatdifferentobservationalbandpassesintheNIRinthesearchofplanetsaroundcoolstars.Ontheotherhand,inthevisible,eventhoughthecurrentRVprecisionisnotlimitedbytheRVcalibrationsourcesuchasaTh-ArlamporanIodineab-sorptioncell,abetterunderstandingoftheirperformancesunderthephoton-limitedconditionhelpsusdiscernastageinwhichtheRVcalibrationsourcebecomesthebottleneckasRVprecisionkeepsimproving. Rodleretal. ( 2011 )recentlyinvestigatedRVprecisionachiev-ableforMandLdwarfs,butdidnotquantitativelydiscussedtheinuenceofRVcalibrationsourcesandstellarnoiseonprecisionDopplermeasurement.StellarnoiseisasignicantcontributortoRVuncertaintybudget,whichfallsintothreecategories:p-modeoscillation,spotsandplagues,andgranulations.P-modeoscillationusuallyproducesanRVsignaturewithaperiodofseveralminutes.Theoscillationmodehasbeenrelativelywellstudiedbypreviouswork(e.g., Carrier&Bourban ( 2003 ); Kjeldsenetal. ( 2005 )).Exposuretimeof10-15minisproposedinordertosmooththeRVsignatureinducedbyp-modeoscillation( Dumusqueetal. 2011 ).SpotsandplaguesinducedRVsignalhasbeendiscussedbyseveralpapers(e.g., Desortetal. ( 2007 ); Lagrangeetal. ( 2010 ); Meunieretal. ( 2010 ); Reinersetal. ( 2010 )). Meunieretal. ( 2010 )concludedthatthephotometriccontributionofplagesandspotsshouldnotpreventdetectionofEarth-massplanetsintheHZgivenaverygoodtemporalsamplingandsignal-to-noiseratio.GranulationisconsideredtobethemajorobstacleindetectionofEarthplanetsintheHZbecauseitproducesanRVsignalwithanamplitudeof810ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1basedonobservationontheSun( Meunieretal. 2010 ).Inaddition,thereisbyfarnogoodmethodofremovingtheRVnoisefromthisphenomenon. Dumusqueetal. ( 2011 )providedamodelofnoisecontributioninRVmeasurementsbasedonprecisionRVobservationonstarsofdifferentspectraltypeandatdifferentevolutionstages.ThetelluriclinesfromtheEarth'satmosphereareusuallymaskedoutincalculationsofthephoton-limitedRVuncertaintiesinNIR( Reinersetal. 2010 ; Rodleretal. 2011 ).Although Wangetal. ( 2011 )proposedamethodtoquantitativelyestimatetheinuenceof 57

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atmosphereremovalresidualonprecisionDopplermeasurementusingtheDispersedFixedDelayInterferometer(DFDI)method( Erskine 2003 ; Ge 2002 ; vanEykenetal. 2010 ),noattempthaseverbeenmadeforprecisionDopplermeasurementsusingahigh-resolutionEchellespectrograph.Inpractice,thetelluriclinesarenotmaskedout,butinsteadmodeledandremoved.Therefore,aquantitativewayofestimatingtheRVuncertaintiesproducedbytheresidualoftelluriclineremovalisnecessarybeforewefullyunderstandtheperformanceofanRVinstrument.Inthevisibleband,theestimationoftelluriclinecontaminationisequallyimportantashigherRVprecisionisrequiredinthesearchoflower-massplanetsaroundsolartypestars.Weaddresstwobasicquestionsinthischaperafterconsideringavarietyoffactorsincludingstellarspectrumquality,RVcalibrationprecision,stellarnoiseandatmospherecontamination:1,whichobservationalbandpassisoptimaltoconductprecisionDopplermeasurementsforstarsofdifferentspectraltypes;2,iscurrentRVprecisionadequatefordetectingEarth-likeplanetsintheHZinthemostoptimisticscenario,inwhichthestaristheleastactiveandtelluriclinesareperfectlymodeledandremoved.ThemethodsandndingsofthisstudywillprovideinsightstothedesignandoptimizationofaplannedorongoingprecisionDopplerplanetsurvey.Inaddition,italsohelpsustoaccessatwhatstageweareinthesearchofEarth-likeplanetsintheHZ. 3.2SimulationMethodology 3.2.1HighResolutionSyntheticSpectraBecauseobservedstellarspectradonothavehighenoughspectralresolutionandbroadeffectivetemperaturecoverage,wedecidetousehighresolutionsyntheticstellarspectrainthecalculationofphoton-limitedRVuncertainty.Forsolartypestars,i.e.,FGKtypestars(3750KTe7000K),weadoptthespectrawitha0.02samplingfrom Coelhoetal. ( 2005 ).ForMdwarfs,(2400KTe3500K),weusehigh-resolution(0.005sam-pling)syntheticstellarspectrageneratedbyPHOENIXcode( Allardetal. 2001 ; Hauschildt 58

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etal. 1999 ). Reinersetal. ( 2010 )conductedseveralcomparisonsbetweensyntheticspec-trageneratedbyPHOENIXandobservedspectrainNIR.Theyconcludedthatthesyntheticspectraareaccurateenoughforthepurposeofsimulations.Formoremassivestars(7000K
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Figure3-1. Comparisonsbetweensyntheticandobservedspectra.BlacklinesrepresentobservedspectraandredlinesaresyntheticspectraafterrotationallinebroadeningandLSFconvolutionatR=80,000.TheTeandVsiniarechosenaccordingtothespectraltypeandlinewidthempirically,theyarenotnecessarilythebest-tparametersfortheobservedspectra.ThechosenTeandVsiniare,fromtoptobottom,9000Kand80.0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forHD39060(A5V),6250Kand4.5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forHD30562(F8V),5750Kand6.0kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1forHD14802(G2V),4750Kand4.0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forHD10361(K5V),2900Kand10.0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forHD34055(M6V).ThedifferencebetweenobservedandsyntheticspectrumisalsoplottedatthebottomofeachpanelwithRMSofdifference. 60

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ofRVuncertainty.Photon-limitedRVmeasurementuncertaintyiscalculatedbasedonthemethoddiscussedinx 2.2.1 3.2.2RVCalibrationSourcesRVcalibrationsourcesareimportantinprecisionDopplermeasurementsbecausetheynotonlyprovidewavelengthsolutionsbutalsohelptrackdriftduetoinstrumentin-stabilities.TheRVuncertaintiesduetocalibrationsmustbeconsideredifwewanttofullyunderstandtheperformanceofanRVinstrument.Weconsiderthephoton-limiteduncer-taintiesintroducedbyRVcalibrationsourcesbasedontheirspectralqualityfactors.TwotypesofcalibrationsourceshavebeensuccessfullyappliedinRVmeasurementsinthevisiblebands:1),aTh-Aremissionlamp( Lovis&Pepe 2007 );2),anIodinegasabsorptioncell( Butleretal. 1996 ).SearchingforplanetsinNIRusingtheRVtechniquehasalreadybeenconductedbyseveralgroups( Beanetal. 2010 ; Blakeetal. 2010 ; Figueiraetal. 2010b ; Mahadevanetal. 2010 ; Muirheadetal. 2011 )andseveralhighresolutionNIRspec-trographswillbeputintouseintheforeseeablefuture( Geetal. 2006a ; Quirrenbachetal. 2010 ).WelimitthediscussionsintheemissionlampsandgasabsorptioncellsalthoughthereareothercandidatesforRVcalibrationsources,forexamples,lasercombs( Lietal. 2008 ; Steinmetzetal. 2008 ),whichareunfortunatelyveryexpensiveandnotyetreadilyavailable,andinterferometercalibrationsourcesasproposedby Wildietal. ( 2010 )and Wan&Ge ( 2010 )Inthefollowingdiscussions,theRVcalibrationsourcesarecategorizedbytheobser-vationalbandpassinwhichtheyareapplied.ThecorrespondingwavelengthrangeforeachobservationalbandpassisgiveninTable 3-1 .InBband,aTh-Arlampisasuitablecalibrationsource.ThelineslistofaTh-Arlampfrom Lovis&Pepe ( 2007 )isadoptedinmywork.OnlyThoriumlinesareusedinthecalculationbecausetheinstabilityofArgonlinesisattheorderof10ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,whichisnotstableenoughforhighprecisionDopplermeasurements.AIodineabsorptioncellisassumedinVbandforRVcalibration,aTh-Arlampisalsoconsideredinthisband 61

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Table3-1. Denitionofobservationalbandpassesusedinthisstudy:centerwavelengthandwavelengthrange Band0min)]TJ /F8 11.955 Tf 10.26 0 Td[(max(nm)(nm) B450400)]TJ /F1 11.357 Tf 7.61 0 Td[(500V545500)]TJ /F1 11.357 Tf 7.61 0 Td[(590R660590)]TJ /F1 11.357 Tf 7.61 0 Td[(730Y1020960)]TJ /F1 11.357 Tf 7.6 0 Td[(1080J12201110)]TJ /F1 11.357 Tf 7.61 0 Td[(1330H15801480)]TJ /F1 11.357 Tf 7.61 0 Td[(1680K22752170)]TJ /F1 11.357 Tf 7.61 0 Td[(2380 forcomparison.Weobtainedahighresolutionspectrum(R200,000)usingtheCoudeSpectrographatKittpeakforaIodinecellwitha6-inchlightpathat60C.Notethataniodinecellspectrumissuperimposedonastellarspectrum( Butleretal. 1996 ),theS/NofRVcalibrationisthusdeterminedbytheS/Nofthecontinuumofastellarspectrum.ThiscaseiscalledSuperimposinginthischapter.Howerver,forverystableinstruments,thereareotherwaysofcalibratingthenon-stellardriftincludingspatial( Mayoretal. 2003 )andtemporalapproaches( Leeetal. 2011 ).Inaspatialapproach,thelightfromastarandaTh-ArlampisfedontonearbybutdifferentpartsofCCDbytwoseparatebers(WecallthiscaseNon-CommonPathinthechapter).InBracketingmethod,ontheotherhand,RVcalibrationsareconductedrightbeforeandafterastellarexposureinatemporalapproach.Inbothcases,theS/NofRVcalibrationisnotdependentonstellarux.Thedisadvantageis,however,thelightfromacalibrationsourcedoesnotpassthroughtheinstrumentinexactlythesamepathoratthesametimeasthelightfromastar.WechooseaTh-ArlampastheRVcalibrationsourceinRband,wherestrongArgonlinesexistthatsaturatetheCCD.SinceweexcludeArgonlinesinRVcalibrationuncertaintycalculation,amorepracticalresultwhenArgonlinesareconsideredisexpectedtobeworseunlessaCCDwithhigherdynamicrangeisused.InYandJband,aU-Neemissionlampisproposedby Mahadevanetal. ( 2010 ),weusealineslistofUraniumprovidedbyStephenRedman( Redmanetal. 2011 ).InHband, 62

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aseriesofabsorptioncellsisproposedby Mahadevan&Ge ( 2009 ),inwhichamixtureofgascellsincludingH13C14N,12C2H2,12CO,and13COcreatesaseriesofabsorptionlinesthatspansover120nmoftheHband. Beanetal. ( 2010 )demonstratedthatanAmmoniaabsorptioncellisagoodcandidateforcalibrationsourceinKband.Therefore,weassumeanAmmoniacellinthecalculationofRVcalibrationuncertaintyintheKband. Valdivielsoetal. ( 2010 )proposedagasabsorptioncellwiththemixtureofacetylene,nitrousoxide,ammonia,chloromethanes,andhydrocarbonscoveringmostoftheHandKbands.Wedonotconsiderthiscellinthisstudysinceadetailedlineslistofthecellisnotavailable. 3.2.3StellarNoiseStellarnoiseisasignicantcontributortoRVuncertaintybudget,thereforewedevotethefollowingparttodiscussamethodofquantifyingitsinuenceonprecisionDopplermeasurement.GranulationisconsideredtobethemajorobstacleindetectionofEarthplanetsintheHZbecauseitproducesanRVsignalwithanamplitudeof810ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1basedonobservationontheSun( Meunieretal. 2010 ).Inaddition,thereisbyfarnogoodmethodofremovingtheRVnoisefromthisphenomenon. Dumusqueetal. ( 2011 )providedamodelofnoisecontributioninRVmeasurementsbasedonprecisionRVobservationonstarsofdifferentspectraltypeandatdifferentevolutionstages.WeadoptthismodelandquantifytheRVuncertaintycontributionofgranulationbasedtheirmeasurementofthreestars,i.e.,CenA(G2V),Ceti(G8V),andCenB(K1V).Thesumofthreeexponentiallydecayingfunctionsrepresentsapowerspectrumdensityfunctionwithcontributionsfromgranulation,meso-granulationandsuper-granulation,usingthevaluesgiveninTable2from Dumusqueetal. ( 2011 ).AnRVRMSerrorduetogranulationisthencalculationbasedonEquation(6)intheirpaperassuminga100-day(300-day)consecutiveobservationforK(G)typestarwithanoptimalstrategyfoundinthepaper,i.e.,threemeasurementspernightof10minexposureeach,2hapart.ThetotallengthofconsecutiveobservationisroughlyinaccordancewiththeorbitalperiodofaplanetintheHZ.WendthattheRVRMSerrordue 63

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togranulationis0.55,1.05and1.05ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1foraK1V,G8VandG2Vstarrespectively.ThesenumberaregoingtobeusedlaterinthisstudytoestimateatotalRVuncertainty.DetailedstudyofRVuncertaintyinducedbystellarnoisehassofarbeenlimitedinKandGtypestarsduetopracticalconcernssuchasstellarphotonuxandstellaractivity.Despitetheirintrinsicfaintnessandrelativehigherlevelofstellaractivityduetofastrotationanddeepconvectionzone,MdwarfsareamongprimarytargetsinsearchofplanetsintheHZ.TheRMSttingerroroforbitofGJ674b( Bonlsetal. 2007 )is0.82ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1afterRVnoiseduetoastellarspotismodeledandremoved,providingangoodtargetforEarth-likeplanetsearchwithanupperlimitofotherstellarnoisecontributionof0.82ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,ifweinterprettheRMSttingerrorisduetoancombinationofinstrumentinstability,photon-noiseandothersourceofstellarnoise.Weadoptthemodelproposedby Dumusqueetal. ( 2011 )toestimatetheRVRMSerrorduegranulationphenomenonforanMdwarfusingtheparametersforaKorGstar.Awareofthecaveatofdifferentstellartype,wendtheRMSerroris0.52,1.07and1.04ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1usingtheparametersforaK1V,G8VorG2Vstar.50-dayconsecutiveobservationisassumedwithanoptimalstrategydescribedin Dumusqueetal. ( 2011 ).Thetheoreticalcalculationofgranulation-inducedRVRMSerrorisworsethanobservationofGJ674busingparametersforGstars,suggestingtheparametersforGstarsarenotrepresentativeofoptimisticscenarioinobservationofMdwarfs.Therefore,weuse0.52ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,whichisaresultofusingtheparametersforKstars,asanestimationofgranulation-inducedRVRMSerrorforanMdwarfinanoptimisticcase. 3.2.4TelluricLinesContaminationGround-basedobservationsarepronetocontaminationbytelluriclines.PrecisionDopplermeasurementsintheNIRrequiresasignicantlevelofdisentanglementofstellarabsorptionlinesandtelluriclines.AsRVprecisionkeepsimproving,DopplermeasurementsinthevisiblebandsuchasB,VandRbandshouldalsoconsidertelluriclines,becausethecontaminationofthemwillnolongerbenegligible.Thequanticationoftelluriclinecontaminationhasbeendiscussedby Wangetal. ( 2011 )inthecontextofDispersedFixed 64

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DelayInterferometermethod( Erskine 2003 ; Ge 2002 ; vanEykenetal. 2010 ),herewepresentageneralizationofthemethodforthecaseofEchellespectrograph,whichisamoreconventionalapplication.Atmospherictransmission(AT)iscalculatedbyaserviceprovidedbyspectralcalc.combasedonamethoddescribedin Gordleyetal. ( 1994 ).ThefollowingequationdescribestheuxdistributiononaCCDdetectoriftelluricabsorptionlinesareconsidered: F()=S0() hAT()LSF=S0() h(1)]TJ /F8 11.955 Tf 10.26 0 Td[(AA())LSF=S0() hLSF+)]TJ /F4 11.955 Tf 11.45 8.09 Td[(S0() hAA()LSF=FS()+FN(), (3) whereS0()isstellarenergyux,whichisconvertedintophotonuxbybeingdividedbyh,ATistheatmospherictransmissionfunction,AAistheatmosphericabsorptionfunction,andisaparameterdescribingtheleveloftelluriclineremovalasarst-orderestimation.InEquation 3 ,photonuxdistributiononthedetector,F,iscomprisedofasignalcomponentFSandanoisecomponentFN.Ideally,werequirethatthedetectoruxchange,F,isentirelyduetothestellarRVchangevS.However,FisalsopartlyinducedbytelluriclineshiftvNresultingfromrandomatmosphericmotions.Therefore,bothvSandvNcontributetoF.WehavetwosetsofRVmeasurements,vS+(0,vrms,S)forstellarRVandvN+(0,vrms,N)forRVinducedbytheEarth'satmosphere,where(0,)representsrandomnumbersfollowingagaussiandistributionwithameanof0andastandarddeviationof.vrms,Sisthephoton-limitedmeasurementerrorforcomponentFSandvrms,Nisthephoton-limitedmeasurementerrorforcomponentFN.WeweighthenalRVmeasurementwiththeinversesquareofphoton-limitedRVuncertaintiesofthesetwocomponents,whichisexpressedbythefollowingequation: v=(vS+(0,vrms,S))v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,S+(vN+(0,vrms,N))v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,N v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,S+v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,N,(3) 65

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InpracticalDopplermeasurements,vSconsiststwocomponents,stellarRVandEarth'sbarycentricRV.DependingonthepositionoftheEarthinitsorbit,thereisanoffsetbetweenvSandvN,whichistheEarth'sbarycentricvelocity.TheEarth'sbarycentricmotionhasasemi-amplitudeof30kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1.Statistically,observedstarhasanannually-varyingRVwithasemi-amplitudeofon-average21.21kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1.Wearticiallyshiftastellarspectrumbyanamountlessthan21.21kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1inordertogenerateaoffsetbetweenstellarspectrumandAAspectrum.vrms,Sandvrms,NarethencalculatedforFSandFN.Wechoosethemedianofvrms,Ntorepresentatypicalvrms,Nvaluefromcalculationsbasedondifferentinputbarycentricvelocities.WefurtherassumethatobservedstarhasaconstantRV(i.e.,nodifferentialRV),andFNhasanRVuctuationwithanRMSofvN,ATMbecauseoftheEarth'sturbulentatmosphere.ThemeasuredRVuncertaintyvisequalto: vrms=(vrms,S)v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,S+(v2N,ATM+v2rms,N)1=2v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,N v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,S+v)]TJ /F5 7.97 Tf 5.07 0 Td[(2rms,N,(3)Inreality,RVuncertaintyofFNisnotdominatedbyphoton-noise,instead,itisdominatedbyatmosphericbehaviorssuchaswind,molecularcolumndensitychange,etc. Figueiraetal. ( 2010a )usedHARPSarchivedataandfoundthatO2linesarestabletoa10ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1levelover6years.However,longtermstabilityoftelluriclines(overyears)becomesworseifwetakeintoconsiderationothergasmoleculessuchasH2OandCO2.TheuncertaintyinducedbyatmospherictelluriclinesistransferredtovrmsviaEquation 3 .InordertocalculatethenalRVuncertainty,vrms,weneedtocalculatephoton-limitedRVuncertaintyvS,rmsandvN,rmsaccordingtoEquation 2 ,inwhichtwotermsneedtobecalculated:QandNe)]TJ /F1 11.357 Tf 4.8 -0.51 Td[(.Thespectralqualityfactors(QSandQN)forthetwocomponents(FSandFN)fromEquation 3 arecalculatedbasedonEquation 2 .Ne)]TJ /F10 7.97 Tf 4.3 -2.3 Td[(,SandNe)]TJ /F10 7.97 Tf 4.3 -2.3 Td[(,N,thephotonuxofFSandFNarecalculatedbasedonstellartype,magnitude,exposuretime,instrumentspecicationsandtelluricabsorptionproperties.NotetheratioofNe)]TJ /F10 7.97 Tf 4.3 -2.3 Td[(,SandNe)]TJ /F10 7.97 Tf 4.3 -2.3 Td[(,Nremainsconstantaslongasatmosphericabsorptionstaysunchangedbecausetelluriclineabsorptionisimprintedonthestellarspectrum. 66

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Themethoddescribedaboveprovidesaquantitativewayofansweringthequestionssuchas:1),howtheRVuncertaintyiscorrelatedwithdifferentlevelsofresidualoftelluriclineremoval;2),whatthecontributionofRVuncertaintyduetotelluriccontaminationisinthenalRVerrorbudgetineachdifferentobservationalbandpass. 3.3Results 3.3.1RVCalibrationUncertaintyRVcalibrationsourcesareusedtotrackthedriftthatisnotcausedbythestellarreexmotionduetoanunseencompanion.Aemissionlamporagasabsorptioncellisusuallyusedforsuchpurpose.WecalculatetheRVuncertaintiesbroughtbythecalibrationsourcesthemselvesbasedontheirspectralproperties.TheRVcalibrationsourcesindifferentobservationalbandpassesarediscussedinx 3.2.2 .Forgasabsorptioncells,weassumeacontinuumlevelof30,000ADU(withinthetypicallinearrange)onaCCDwitha16-bitdynamicrange,whichcorrespondstoaS/Nof425ifthegainisat6electron/ADU.Foraemissionlamp,weassumethatthestrongestlineinthespectralregionhasapeakuxof30,000ADU.Notethat4pixelsperresolutionelementisassumedthroughoutthepaper.Figure 3-2 showstheRVcalibrationuncertaintiesasafunctionofobservationalbandpassatdifferentspectralresolutions.NotethattherearecurrentlytwosuccessfulcalibrationsourcesinVband,i.e.,aTh-Arlamp(asterisk)andanIodinecell(square).Therefore,bothareconsideredandplottedforVbandinFig. 3-2 .TheresultsshowninFig. 3-2 arealsosummarizedinTable 3-2 .Gasabsorptioncellsusuallyofferhighercalibrationprecisionthanemissionlampsbecauseofdenserlinesdistributionandon-averagehigherS/N.However,thisconclusiondependsonthecalibrationmethods,itisusuallythecaseforNon-CommonPathandBracketingmethodwhileitisnotalwaystrueinSuperimposingscheme,whichwillbediscussedlaterinthissection. 3.3.2OptimalSpectralBandForRVMeasurementsTheoptimalobservationalbandpassforprecisionRVmeasurementsdependsonthequalityofastellarspectrum(Qfactor),photonux(S/N),RVcalibrationuncertainty,the 67

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Figure3-2. RVcalibrationuncertaintiesasafunctionofobservationalbandpassatdifferentspectralresolutions(color-coded).SquaresinVbandrepresentIodinecellmethodandasterisksinVbandrepresentTh-Arlampmethod.RefertoTable 3-2 forRVcalibrationsourcesindifferentobservationalbandpasses. Table3-2. RVuncertaintiescausedbycalibrationsourcesatdifferentspectralresolutions. RBVRYJHKTh-AraTh-Ar,IodinebTh-ArU-NecU-NeMixedcelldAmmoniae(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.06 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1) 20,0000.780.90,0.331.03.02.41.40.5740,0000.650.74,0.210.812.01.60.750.3060,0000.540.60,0.150.631.41.10.500.2180,0000.460.50,0.120.511.10.870.380.17100,0000.380.41,0.100.430.900.700.310.15120,0000.330.36,0.080.360.760.590.270.14 Note.a: Lovis&Pepe ( 2007 );b: Butleretal. ( 1996 );c: Redmanetal. ( 2011 );d: Mahadevan&Ge ( 2009 );e: Beanetal. ( 2010 ). 68

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severityoftelluriclinecontaminationandotherfactors.Wewillconsiderdifferentsituationsinthefollowingdiscussion.WeassumeaS/N(perpixel)of100atthecenterofYband(i.e.,=1020nm)atR=60,000,theS/Ninotherobservationalbandpassvarieswithstellarspectralenergydistribution(SED)andspectralresolutionaccordingly.TheS/Nreportedinthispaperisatthecenterofeachobservationalbandpass(seeTable 3-1 )unlessotherwisespecied.WewillinvestigatetheoptimalobservationalbandpassforprecisionDopplermeasurementsgiventhesameexposuretime,thesametelescopeapertureandthesameinstrumentthroughput(independentofwavelength). 3.3.2.1StellarSpectralQualityWestartwiththesimplestcaseinwhichtheRVuncertaintyisonlydeterminedbythestellarspectralqualityfactorandtheSEDofastar.Inotherwords,theRVcalibrationsourceisperfectandnouncertaintyisintroducedwhencalibratingoutthenon-stellardrift.Inaddition,telluriclinesareperfectlyremovedfromtheobservedstellarspectrum.Table 3-3 summarizestheobtainableRVprecisionsandtheS/Nsatthreedifferentspectralresolutions,i.e.,20,000,60,000and120,000.AnexampleofR=120,000isplottedinFig. 3-3 .WendtheoptimalobservationalbandpassisBbandforawiderangeofspectraltypesfromKtoA.TheoptimalobservationalbandpassforanMdwarfiseitherinRbandorinKband.Morespecically,Rbandisoptimalforanearly-to-mid-typeMdwarfwhileKbandforanlate-typeMdwarf.Thendingremainsthesameforawiderangeofspectralresolutionsfrom20,000to120,000.TheRVuncertaintyforanotherspectraltypeoratadifferentS/NcanbeobtainedbyeitherinterpolatingorscalingbasedontheresultsinTable 3-3 3.3.2.2StellarSpectralQuality+StellarRotationStellarrotationbroadenstheabsorptionlinesinastellarspectrum,resultinginlessDopplersensitivity.Itisthereforenecessarytoconsiderthestellarrotationinthediscussionofphoton-limitedRVuncertainty.Typicalvaluesofrotationvelocitiesofdifferentspectraltypesareobtainedbasedonthemeasurementresultsfrom Jenkinsetal. ( 2009 )forMdwarfsand Valenti&Fischer ( 2005 )forFGKstars.Inaddition,typicalrotationalvelocities 69

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Table3-3. Photon-limitedRVuncertaintiesbasedonstellarspectralqualityatdifferentspectralresolutionsfordifferentspectraltypes,averageS/Nperpixelisreportedinperentheses Spec.TypeBVRYJHK(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1) R=20,000A5V4.7(211.7)8.3(209.0)21.5(198.3)23.9(173.2)22.6(155.1)62.9(126.8)...(...)F5V3.8(166.9)6.3(177.1)12.8(182.1)23.3(173.2)20.9(164.2)27.2(148.9)...(...)G5V3.3(126.4)5.1(145.6)8.6(163.3)20.8(173.2)15.9(171.9)14.9(168.0)...(...)K5V3.7(88.8)4.8(110.9)7.5(141.0)17.3(173.2)13.5(181.8)11.2(199.8)...(...)M5V14.7(26.8)11.7(44.0)9.0(66.2)12.8(173.2)12.0(203.0)9.5(205.6)6.8(191.6)M9V28.2(9.3)18.4(17.9)13.8(26.6)8.2(173.2)4.6(250.5)5.4(246.3)2.7(250.9)R=60,000A5V1.5(122.2)2.7(120.7)6.9(114.5)10.0(100.0)9.6(89.5)24.6(73.2)...(...)F5V1.1(96.4)1.8(102.3)3.6(105.1)8.8(100.0)7.9(94.8)9.4(86.0)...(...)G5V1.0(73.0)1.6(84.0)2.5(94.3)6.9(100.0)5.7(99.3)5.6(97.0)...(...)K5V1.2(51.3)1.5(64.0)2.3(81.4)5.9(100.0)5.1(105.0)4.6(115.3)...(...)M5V4.6(15.5)3.5(25.4)2.5(38.2)4.8(100.0)4.4(117.2)3.8(118.7)2.4(110.6)M9V9.3(5.4)5.9(10.4)4.2(15.3)3.3(100.0)1.8(144.6)2.0(142.2)1.0(144.9)R=120,000A5V1.1(86.4)2.0(85.3)5.1(81.0)8.0(70.7)7.9(63.3)18.9(51.8)...(...)F5V0.7(68.2)1.2(72.3)2.4(74.3)6.4(70.7)5.9(67.0)6.5(60.8)...(...)G5V0.6(51.6)1.1(59.4)1.6(66.7)4.6(70.7)3.9(70.2)4.0(68.6)...(...)K5V0.8(36.3)1.0(45.3)1.5(57.6)4.0(70.7)3.5(74.2)3.0(81.6)...(...)M5V3.1(11.0)2.3(18.0)1.5(27.0)3.3(70.7)3.1(82.9)2.8(83.9)1.8(78.2)M9V6.4(3.8)4.1(7.3)2.8(10.8)2.3(70.7)1.3(102.3)1.5(100.6)0.8(102.4) 70

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Figure3-3. RVprecision(R=120,000)basedonspectralqualityfactorasafunctionofobservationalbandpassfordifferentspectraltypesfromM9VtoA5V(color-coded).AverageS/Nperpixelisalsoshownintheplot,seeTable 3-3 forresultsatotherspectralresolutions.KbandRVuncertaintiesarenotcalculatedforstarswithTehigherthan3500Kbecausetheyareusuallyobservedinthevisiblebandatcurrentstage. forearlytypestarssuchasAstarsareextrapolatedfromvaluesofsolartypestars.Table 3-4 summarizesthespectraltypesandthecorrespondingTeandVsiniusedinthepaper.AtrendofincreasingVsiniisseenasspectraltypemoveseithertoearlytypeend(FandA)orlatetypeend(M).Afterconsideringtypicalstellarrotationvelocitiesfordifferentspectraltypes(asshowninFig. 3-4 ,R=120,000),FandAstarsarenotsuitabletargetsforprecisionDopplermeasurementsbecauseoftheirintrinsichighstellarrotation.MdwarfsRVuncertaintiesaregettingworsethannon-rotatingcase,but2)]TJ /F1 11.357 Tf 7.6 0 Td[(5ms)]TJ /F5 7.97 Tf 5.06 0 Td[(1RVprecisionareexpectedinoptimalcases,i.e.,RbandforM5VandKbandforM9V.ForKandGstars,sub 71

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ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1precisionisreachedunderphoton-limitedconditionevenafterconsideringtypicalstellarrotation. Figure3-4. RVprecision(R=120,000)basedonspectralqualityfactorandtypicalstellarrotationasafunctionofobservationalbandpassfordifferentspectraltypesfromM9VtoA5V(color-coded).AverageS/NperpixelisthesameasshowninFig. 3-3 3.3.2.3StellarSpectralQuality+RVCalibrationSourceTheaboveRVprecisionsconsideringtypicalstellarrotationbroadeningaregoodindicatorsforRVplanetsurveysinwhichapopulationofstarsisobservedwithadistributionofstellarrotations.However,inthesearchforanEarth-likeplanet,adifferentapproachistakeninwhichstarswithfavorablepropertiesforDopplermeasurementsareassignedhigherpriorityinobservation.Thepropertiesusuallyincludeslowstellarrotationandlowstellaractivity.Therefore,wewillreducestellarrotationinthefollowingdiscussionsincewe 72

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Table3-4. SpectralType,correspondingTe,andtypicalstellarrotationVsini SpectralTypeTeVsini(K)(kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1) A0V9600131.0A2V9000108.0A5V840085.5A8V780062.5F0V740047.5F2V700032.0F5V675023.0F8V62506.5G0V60003.5G2V57502.2G5V55001.7G8V52501.8K0V50001.9K2V47501.8K5V45002.0K8V40002.5M0V37502.8M2V35002.8M5V31003.9M8V26006.8M9V24008.0 emphasizediscoveryofanEarth-likeplanet.AfterconsideringtheuncertaintiesbroughtbyanRVcalibrationsource,RVuncertaintiesinFig. 3-3 degradetothoseinFig. 3-5 (R=120,000).Twoscenariosofcalibrationareconsidered:Superimposing(Dotted),andNon-CommonPathandBracketing(Solid).ThedifferencebetweenthesetwoiswhethertheS/Ndependsonthestellarux.IntheSuperimposingcase,becausetheabsorptioncellisinthelightpathofthestellarux,thecontinuumoftheresultingIodineabsorptionspectrumisdeterminedbythethecontinuumuxofastar.Consequently,theRVcalibrationuncertaintyisstronglydependentontheincomingstellarux.InthecomparisonofthetwocasesinFig. 3-5 ,weseetheNon-CommonPathandtheBracketingmethodsalwaysintroducelessuncertaintyinRVcalibrationthantheSuperimposingmethod.ThemajorreasonforthatistheS/Nintheformercasemaybeoptimizedbyadjustingthesourceintensity(Non-Common 73

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Path)ortheexposuretime(Bracketing).Themainconclusionaboutoptimalobservationalbandpassfromx 3.3.2.1 remainsunchanged. Figure3-5. RVprecision(R=120,000)basedonspectralqualityfactorandRVcalibrationuncertaintiesasafunctionofobservationalbandpassfordifferentspectraltypesfromM9VtoA5V(color-coded).AverageS/NperpixelisthesameasshowninFig. 3-3 .SquaresinVbandrepresentIodinecellmethodandasterisksinVbandrepresentTh-Arlampmethod.DottedlinesshowresultfromSuperimposingcasesandsolidlinesforNon-CommonPathandBracketingcases.RefertoTable 3-2 forRVcalibrationsourcesindifferentobservationalbandpasses. 3.3.2.4StellarSpectralQuality+RVCalibrationSource+AtmosphereTheoptimalbandforDopplermeasurementsisintheNIR(Kbandinparticular)forstarswithspectraltypeslaterthanM5frompreviousdiscussionsinthispaper.However,oneimportantelementismissinginthediscussion,whichisthecontaminationfromthetelluriclinesintheEarth'satmosphere,whichisasevereproblemintheNIRobservation.The 74

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quantitativeanalysisoftelluriclinecontaminationisintroducedinx 3.2.4 andweapplythatmethodinestimatingtheRVuncertaintybroughtbythetelluriccontamination.Weconneourdiscussionsforlate-typeMdwarfssinceNIRobservationdoesnotgainadvantageforotherspectraltypesearlierthanM5V.Figure 3-6 showsanexampleofhowRVuncertaintyforanM9Vstarchangeswithobservationalbandpassunderdifferentvaluesof(i.e.,leveloftelluriclineremoval,seeEquation 3 ).1indicatesnotelluriclineremovaland0indicatescompleteremovaloftelluriclines(seeEquation 3 ).RVuctuationof10ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1duetorandomatmosphericmovementisassumedinthecalculation.BracketingRVcalibrationisassumedinthecalculation.Thereareseveralpointsworthnotinginthisplot:1),differentobservationalbandpassesareaffecteddifferentlybytelluriclines,thesignicanceoftelluriclinecontam-inationisindicatedbythespanofRVuncertaintiesatdifferentvalues.Forexample,BbandistheleastaffectedbytelluriclinesbecausetheRVuncertaintiesinBbandatdifferentlevelsoftelluriclineremovalremainroughlythesame,whileJ,HandKbandssufferseveretelluriclinecontaminationbecauseanysmallchangeofresultsinsignicantchangeofRVuncertainty.2),Ifthereisnoattemptofremovingtelluriclinesfromobservedstellarspectrum(purpleinFig. 3-6 ),thereisnoadvantageinobservinglate-typeMdwarfsinNIR,RVuncer-taintyisdominatedbyEarth'satmospherebehaviorintheNIR.Inthiscase,theoptimalbandisVandRband.Onlywhen0.01,i.e.,morethan99%telluriclinestrengthisremoved,theadvantageofobservinglate-typeMdwarfsintheNIRbecomesobvious,atafactorof3improvement.Inpractice,therehavebeenseveralexamplesinwhichtelluriclinemodelingandremovalisdemonstratedtobesuccessful. Vaccaetal. ( 2003 )achievedmaximumdeviationsoflessthan1.5%andRMSdeviationsoflessthan0.75%withR=2000andS/N100usingatelluricstandardstarnearbythesciencetargetstar. Beanetal. ( 2010 )hasshownthattheRMSdeviationisaslowas0.7%afterusinga3-componentmodel(Stellarspectrum,telluricabsorptionandAmmoniaabsorption)totanobservedNIRspectrum.Inbothcases, 75

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Figure3-6. RVprecision(R=120,000)consideringspectralqualityfactor,RVcalibrationuncertaintiesandtelluriccontaminationforanM9Vstarasafunctionof,i.e.,telluriclineremovallevel(color-coded).1indicatesnotelluriclineremovaland0indicatescompleteremovaloftelluriclines.BracketingRVcalibrationisassumedfortheresultsshownintheplot.SquaresinVbandrepresentIodinecellmethodandasterisksinVbandrepresentTh-Arlampmethod.RefertoTable 3-2 forRVcalibrationsourcesindifferentobservationalbandpasses. anvalueofbetterthan0.01hasbeendemonstratedshowinggreatpotentialofprecisionDopplermeasurementintheNIRband.Figure 3-7 showsthepercentagecontributionofRVuncertaintyintroducedbytelluriccontaminationatdifferentvalues.Ifnotelluriclineremovalisperformed,theRVuncer-taintyintheNIRisdominatedbythosecausedbytelluriccontamination,i.e.,thepercentagecontributionsaremorethan87.9%inY,J,HandKband.Incomparison,thepercentagecontributionoftelluriccontaminationinducedRVuncertaintyis2.9%,3.1%and53.0%inB, 76

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YandRbandrespectively.Asdecreases,i.e.,morestrengthoftelluriclinesisremoved,lessRVuncertaintyiscontributedtothenalRVuncertaintybudget.However,thereisstillasignicantfraction(morethan70%)ofRVuncertaintycontributedbytelluriccontaminationinJ,HandKbandevenafter90%oftelluriclinestrengthisremoved.Thepercentagecontributiondropsbelow10%throughoutconsideredobservationalbandpasseswhenmorethan99.9%strengthisremoved.Tosumupthediscussion,RVuncertaintyisdominatedbytelluriccontaminationintheNIRband.Therefore,telluriclineremovalintheNIRisanecessarysteptoreducethetelluriccontaminationandextractmoreofDopplerinformationintrinsicallycarriedbyastellarspectrum. Figure3-7. ThepercentagecontributionofRVuncertaintyinducedbytelluriccontaminationasafunctionofobservationalbandpass.Differentvaluesareindicatedbycolors.1indicatesnotelluriclineremovaland0.001indicates99.9%effectiveremovaloftelluriclines. 77

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3.3.2.5ComparisonstoPreviousWorkThereareworksthathavebeenpreviouslydoneinattemptstounderstandthefunda-mentalphoton-limitedRVuncertaintiesbasedonhighresolutionsyntheticstellarspectra. Bouchyetal. ( 2001 )calculatedQfactorsforasetofsyntheticstellarspectraforsolartypedwarfstars.Werestrictthecomparisontospectrawiththesameturbulencevelocity(Vt).SincethespectraforsolartypestarsinourstudyhaveaVtof1.0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,weonlycomparetheresultsfromspectrawithVtof1.0kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1in Bouchyetal. ( 2001 ).Table 3-5 summarizesacomparisonofourresultstothosefrom Bouchyetal. ( 2001 ).TheQfactorsfromourstudyaregenerally10)]TJ /F1 11.357 Tf 7.6 0 Td[(15%lowerifnostellarrotationisconsidered,i.e.,Vsini=0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1.Itmaybedueadifferentsamplingrateinthesyntheticspectra,0.005in Bouchyetal. ( 2001 )and0.02inourpaper.MorenefeaturesareseeninaspectrumwithhighersamplingrateandthusmoreDopplerinformationiscontained.Atlowstellarrotationrate(Vsini=4and8kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1),ourresultsagreewiththeirswithin6%,whichisimprovedcomparedtonon-rotatingcasebecausethenefeaturesaresmoothedoutbystellarrotation.Forfastro-tators,i.e.,Vsini=12kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1,10%differenceisseenintheworstcase,forwhichadifferentlimb-darkeningvaluemightberesponsible. Table3-5. ComparisonofQfactorsfromourresultsto Bouchyetal. ( 2001 )(numbersinparenthathes). TeloggVtVsini(K)(cms)]TJ /F5 7.97 Tf 5.06 0 Td[(1)(kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(14kms)]TJ /F5 7.97 Tf 5.07 0 Td[(18kms)]TJ /F5 7.97 Tf 5.06 0 Td[(112kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1 45004.51.030238(34940)17235(17080)8700(8440)5793(5380)50004.51.030001(33405)16607(16140)8305(7815)5432(4930)55004.51.026892(30375)14858(14700)7397(7020)4700(4385) Reinersetal. ( 2010 )investigatedtheprecisionthatcanbereachedinRVmeasure-mentsforstellarobjectscoolerthansolartypestarsintheNIR.Thetreatmentoftelluriclinesintheircalculationswastoblocktheregionswherethetelluricabsorptionisover2%and30kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1inthevicinity.Followingthemethoddescribedintheirpaper,wecalculatedthefractionofthewavelengthrangeaffectedbytelluriccontaminationinV,Y,JandHband,theresultsare2.4%,22.7%,60.0%and50.6%.Incomparison,theresultsare2%,19%, 78

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55%and46%forV,Y,JandHbandintheirpaper.Thedifferencemaybecausedbyadifferentatmosphericabsorptionusedincalculation.Both Reinersetal. ( 2010 )andwereachthesameconclusionthatNIRRVmeasure-mentsstarttogainadvantageovervisiblebandsformid-to-late-typeMdwarfs.However,wepredictthatYandHbandaresimilarintermsofgivingthehighestRVprecisionamongV,Y,JandHbands,whileitisfoundintheirpaperthatYbandistheoptimalbandconsider-ingstellarspectrumqualityandtelluriclinemasking.NotethatweadoptthedenitionofV,Y,JandHbandsaccordingto Reinersetal. ( 2010 )incomparisons.Table 3-6 summarizesourcalculationsofRVprecisioncanbereachedforanM9Vstar(Te=2400K)incompari-sontotheresultsintheirpaperaswellastheS/Nobtainedineachobservationalbandpass.Infurtherexamination,wecompareourresultsinJandHbandandndthatRVprecisionsinHbandareingeneralbetterthanthoseinJband.ItisexplainedbythewiderwavelengthcoverageandricherabsorptionfeaturesinHbandforanM9Vstar.Onthecontrast,theimprovementofRVprecisioninHbandisnotseeninthecomparisonofJandHbandintheresultsfrom Reinersetal. ( 2010 ).Inaddition,wereportbetterRVprecisionsinVbandbyafactorof1.5. Table3-6. ComparisonofpredictedRVprecision(intheunitofms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)betweenourresultsto Reinersetal. ( 2010 )foranM9dwarf(Te=2400K,Vsini=0kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1).TheFollowingresultsarecalculatedbasedonatelluricmaskingtreatmentinwhichtelluriclineswithmorethan2%absorptiondepthand30kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1withinitsvicinityaremaskedout. RS/NThisstudy Reinersetal. ( 2010 )VYJHVYJHVYJH 60000121001341285.13.04.92.78.02.24.64.08000010861161114.22.54.12.26.21.73.53.5100000977104993.92.43.82.05.31.52.93.3 Wehavealsoconductedsimilarcalculationto Rodleretal. ( 2011 )foranM9.5dwarf(Te=2200K,Vsini=5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)andthecomparisonispresentedinTable 3-7 .InsteadofndingYband,wendKbandgivesthehighestRVprecisionintelluric-contamination-freecase,whileHbandgivesthehighestRVprecisioninthecasewheretelluriclineswithan 79

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absorptiondepthmorethan3%aremaskedoutinRVcalculation.WealsonoticethatourpredictedRVprecisionsarelesssensitivetospectralresolution.Notethatstellarabsorptionlinestypicallybecomeresolvedbyspectrographafterspectralresolutiongoesover50,000,wedonotexpectRVprecisionincreasessteeplyasspectralresolutiongoeswellbeyond50,000. Table3-7. ComparisonofpredictedRVprecision(intheunitofms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)betweenourresultsto Rodleretal. ( 2011 )foranM9.5dwarf(Te=2200K,Vsini=5kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1).CaseAisforcompleteandperfectremovaloftelluriccontamination;CaseBisforthecaseinwhichtelluriclineswithabsorptiondepthof3%weremaskedout. RS/NThisstudy Rodleretal. ( 2011 )YJHKYJHKYJHK CaseA2000013918017115216.412.28.75.122.225.522.827.940000981271211089.46.95.43.26.98.77.810.8600008010499887.75.54.62.84.25.75.13.880000709085766.85.14.42.63.34.03.85.1CaseB2000013918017115218.819.813.916.424.229.729.139.3400009812712110810.510.98.59.58.712.211.917.3600008010499888.68.77.07.85.47.16.810.780000709085767.68.06.77.23.85.25.17.9 3.3.3CurrentPrecisionvs.SignalofanEarth-likePlanetinHabitableZoneOneofthemostintriguingtasksinexoplanetscienceistosearchandcharacterizeEarth-likeplanetsintheHZ.Overthepasttwodecades,greatadvanceshavebeenseenbutwehavenotyetdiscoveredanotherEarth.Wearetryingtoanswerseveralquestionsinthefollowingdiscussion:1),isitpossibletodetectanEarth-likeplanetintheHZusingtheRVtechnique?2),Ifso,atwhatS/Ninwhichobservationalbandpassandforwhichspectraltype?3),BasedonthecurrentavailableRVcalibrationsourcesandknowledgeofstellarnoise,isitpracticaltodetectanEarth-likeplanetintheHZinthemostoptimisticcase? 80

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3.3.3.1StellarSpectralQualityWerstconsideranidealsituationinwhichtheRVprecisionisonlydeterminedbytheQfactorofastellarspectrum.ThehighestS/Nperpixelobtainableforasingleexposureis425(assuming30,000ADUandagainof6electron/ADU).Figure 3-8 showstheRVprecisionsobtainableatspectralresolutionof120,000fordifferentspectraltypesindifferentobservationalbandpasses.OverplottedareRVsignalsofanEarth-likeplanetintheinner(dashed)andouteredge(dash-dotted)oftheHZofastarwithacertainspectraltype(colorcoded).ThepositionoftheHZiscalculatedbasedon Kastingetal. ( 1993 ).ThepositionoftheHZgetsclosertothehoststarasstellartemperatureandluminositydrops.TheRVsignalisenhancedbyboththedecreasingdistancetothestarandthedecreasingstellarmass.Earth-likeplanetisdetectableineveryobservationalbandpassataS/Nashighas425forMdwarfs.BandVbandbearthehighestprobabilityforKstarsandBbandisthesweetsoptforGstars.PredictedRVprecisionsarenotadequatetodetectthesignalofanEarth-likeplanetintheHZaroundFandAstarswithsingleexposureonacurrenttypicalCCDwith16-bitdynamicrange.Table 3-8 summarizestheS/NrequiredfordetectionofanEarth-likeplanetintheHZasafunctionofspectraltype,inwhichweassumethatadetectionispossiblewhentheRVprecisionisequaltothesignal.EventhoughitonlyrequireaS/Nof17foranM9VstarinBbandtodetectanhabitableEarth-likeplanet,theexposuretimecouldbeaslongas1hourevenattheKecktelescopeforaJ=6M9VstarandthereisnosuchbrightlateMtypestarinthesky.Inaddition,only25MstarsareavailablewithJbandmagnitudelessthan6( Lepine&Shara 2005 ).Incomparison,1minexposuretimeatKeckwillobtainaS/Nof175foraB=8star,whichisadequatefordetectinghabitableEarth-likeplanetaroundaK5Vstar.10%instrumentthroughputisassumedintheabovecalculations. 3.3.3.2StellarSpectralQuality+RVCalibrationSource+AtmosphereCurrentRVprecisionisnotonlyrestrictedbythephoton-limiteduncertaintydeterminedbyastellarspectrum,butalsobytheuncertaintiesbroughtbyanRVcalibrationsourceandthetelluriccontaminationfromtheEarth'satmosphere.Figure 3-9 showstheRVprecisions 81

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Figure3-8. RVprecisions(R=120,000)consideringspectralqualityfactorataS/Nof425asafunctionofobservationalbandpassfordifferentspectraltypesfromM9VtoA5V(color-coded).OverplottedareRVsignalsofanEarth-likeplanetintheinner(dashed)andouteredge(dash-dotted)oftheHZ. Table3-8. RequiredS/NfordetectionofanEarth-likePlanetintheHZasafunctionofspectraltype SpectralBandHZpropertiesBVRYJHKmainaoutvinvout(M)(AU)(AU)(ms)]TJ /F5 6.974 Tf 4.43 0 Td[(1)(ms)]TJ /F5 6.974 Tf 4.44 0 Td[(1) A5V3483608714724203891782735053...1.822.8934.1670.030.03F5V83715473088804270036976...1.301.6142.3250.060.05G5V3186251053314926602648...0.870.7201.0370.110.09K5V175280535171015701496...0.640.3870.5580.180.15M5V4758583283593301930.190.0680.0980.780.65M9V17222211899108580.100.0340.0491.511.26 82

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takingintoconsiderationofQfactors,RVcalibrationuncertaintiesandtelluriccontamination.Bracketingcalibration(ataS/Nof425)isconsideredinthecalculationsinwhichtheS/Nofcalibrationisnotdeterminedbythecontinuumoftheobservedstar.Twocasesarediscussedfortelluriccontamination,inonecasenotelluricremovalisattempted(solid)whileintheothercase99.9%oftelluriclinestrengthisremoved(dotted).Forthevisiblebands(i.e.,B,VandRband),RVprecisioninBbandisbarelyaffectedbytelluriccontamination(10ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1randomRVoftelluriclinesisassumedinthecalculations)butlimitedbytheRVcalibrationuncertaintyduetoaTh-Arlamp.TherearetwoRVcalibrationsourcesconsideredintheVband,aTh-ArlampandaIodineabsorptioncell,thelatteroneprovideshighercalibrationprecisionintheBracketingcase.Eventhoughonly2.4%ofthewavelengthrangeisaffectedbytelluriclinecontamination(x 3.3.2.5 ),handlingtelluriclinesisstillveryimportant.TheRVuncertaintybudgetofaK5VstarinTable 3-9 showsanexampleinwhichtheRVprecisionisworsethandetectionlimitifnotelluriclineremovalisinvolvedwhileitisbelowthedetectionlimitinthe99.9%removalcase(=0.001).ThisexampleaddresstheimportanceoftelluriclineremovalinthesearchofanEarth-likeplaneteveninthevisiblebandwheretelluriccontaminationislessseverethantheNIRband.After99.9%oftelluriclinestrengthisremoved,theRVuncertaintyofaK5VstarisdominatedbythespectralqualityofaK5VstarandtheRVcalibrationsource. Table3-9. Twoexamplesoftelluriccontamination SpectralTypeBandpassvS,rmsvATM,rmsvcalvrmsvHZ(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1) K5VV1.00.110.190.080.240.180.0010.010.14M5VK1.00.337.970.147.980.780.0010.170.39 Incomparison,intheNIR(Y,J,HandKband),theRVuncertaintiesaredominatedbytelluriccontamination,resultingRVprecisionsat5)]TJ /F1 11.357 Tf 7.6 0 Td[(10ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1thatarenotadequateinhabitableEarth-likeplanetdetections.Asimilarexampleofhowtelluriccontamination 83

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Figure3-9. RVprecision(R=120,000)consideringspectralqualityfactor(S/N=425),RVcalibrationuncertaintiesandtelluriccontaminationasafunctionofobservationalbandpassfordifferentspectraltypesfromM9VtoG5V(color-coded).OverplottedareRVsignalsofanEarth-likeplanetintheinner(dashed)andouteredge(dash-dotted)oftheHZ.Solidlinesrepresentnon-telluric-removalcaseswhiledottedlinesrepresentcasesinwhich99.9%ofthestrengthoftelluriclinesisremoved.SquaresinVbandrepresentIodinecellmethodandasterisksinVbandrepresentTh-Arlampmethod.RefertoTable 3-2 forRVcalibrationsourcesindifferentobservationalbandpasses. raisestheoorofRVuncertaintyisalsogivenforanM5VstarinKbandinTable 3-9 .Inthe99.9%removalcase,RVuncertaintiesintheNIRarenolongermainlydominatedbytelluriccontamination,butbyspectralqualityfactor.Tosumup,telluriclineremovalisanimportantandindispensablesteptowardthediscoveryofanEarth-likeplaneteveninthevisibleband.Aftertelluriclinesaresuccessfullyremovedfromobservedstellarspectrum,theRVprecisionislimitedbytheuncertaintycausedbystellarspectralqualityandRVcalibrationsources. 84

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Aftercompletelyremovingthetelluriccontamination,wecompareourpredictionofRVuncertaintiesandwhatisreportedfromHARPSinstrument( Mayoretal. 2003 ).AnexampleofHD47186( Bouchyetal. 2009 )isgiveninTable 3-10 .HD47186isaG5VstarwithaVsiniof2.2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,thebestachievableRVprecisionforthisstaris0.3ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1ataS/Nof250accordingto Bouchyetal. ( 2009 ).OurpredictionindicatesthatanRVprecisionof0.24ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1ispossibletoachieveatthesameS/Nforthesamewavelengthcoverage.Thedifferencemaycomefromthoseuncountedfactorsinourcalculation,forexample,stellarnoise.However,ourpredictionofRVprecisioniswithin20%toprecisionfromrealobservation. Table3-10. Predictionvs.HARPSobservation.a:HARPSobservationofHD47186from Bouchyetal. ( 2009 ),thebestachievableRVprecisionisataS/Nof250forthisG5VstarwithVsiniof2.2kms)]TJ /F5 7.97 Tf 5.07 0 Td[(1;b:ourRVuncertaintiespredictionforthisstarassumingthesameS/N,spectraltype,observationbandpassesandstellarrotation. BandpassvS,rmsvcalvrms(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)(ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1) HD47186aB+V+R......0.30G5VbB0.140.330.36V0.290.360.46R0.480.360.60B+V+R0.120.200.24 3.3.3.3StellarSpectralQuality+RVCalibrationSource+StellarNoiseAssumingthetelluriclinesareperfectlymeasuredandremoved,weconsidertheobtainableRVprecisionbasedonstellarspectralquality,RVcalibrationprecisionandstellarnoise.Stellarnoiseofdifferentspectraltypeisestimatedinx 3.2.3 basedon Dumusqueetal. ( 2011 ).Figure 3-10 showspredictedRVprecision,itisclearthattheRVprecisionforGandKtypestarsisnotadequatefordetectingEarth-likeplanetintheHZafterstellarnoiseistakenintoconsideration.However,itisstillpossibletodetectEarth-likeplanetsaroundMdwarfsbecauseofrelativelylargerRVsignal.ForanM5Vstar,visiblebandandKband 85

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provideadequateprecisionforanEarth-likeplanetdetectionwhileallbandpassesallowanEarth-likeplanetdetectionforanM9Vstar. Figure3-10. RVprecision(R=120,000)consideringspectralqualityfactor(S/N=425),RVcalibrationuncertaintiesandstellarnoiseasafunctionofobservationalbandpassfordifferentspectraltypesfromM9VtoG5V(color-coded).OverplottedareRVsignalsofanEarth-likeplanetintheinner(dashed)andouteredge(dash-dotted)oftheHZ.HZsofGandKtypestarsarenotplottedbecausetheyareoutofreachbasedonthepredictedRVprecision.SquaresinVbandrepresentIodinecellmethodandasterisksinVbandrepresentTh-Arlampmethod.RefertoTable 3-2 forRVcalibrationsourcesindifferentobservationalbandpasses. InordertocompareourpredictedRVprecisiontoobservations,wechoose69planetsdetectedbyHARPSsince2004afteraninstrumentupgrade( Mayoretal. 2003 )andplottheRMSerrorsofKeplerianorbitttingasafunctionTe(Fig. 3-11 ).Theminimaofthreesubsets(correspondingtoG,KandMstypestars)arefoundbasedonTe.Incomparison, 86

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thepredictedRVpredictions(aftercombiningresultsfromB,VandRbands)foraG5V,K5VandM5Vstarareplottedasopendiamonds.SincethepredictedRVprecisionisbasedonanoptimisticcase,wecomparethepredictionswiththeRMSminimawendintheobservation.ForMdwarfs,theminimumofRMSerrorsisfoundat0.8ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1withGJ674b( Bonlsetal. 2007 ).Incomparison,ourpredictionis0.62ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1consideringstellarspectralqualityfactor,RVcalibrationerrorandstellarnoise.If0.5ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1instrumentaluncertaintyasmentionedin Bonlsetal. ( 2007 )isaddedinquadrature,ourpredictioniswellmatchedwithHARPSMdwarfsobservationinthebestcasescenario. Lovisetal. ( 2006 )reported0.64ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RMSerrorsforaplanetsystemofaK0Vstar(i.e.,HD69830),whichisconsistentwithourpredictionof0.65ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1.ItisplottedinthebinwithTebetween5000Kand6000KbecausereportedTeof5385K.ForGtypestars, Bouchyetal. ( 2009 )reported0.91ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RMSerrorforHD47186bandc,aplanetarysystemaroundaG5Vstar.Incomparison,wepredictatotalRVuncertaintyof1.1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1.TheoverestimationofRVmeasurementuncertaintyispossiblyduetoanoverestimationofstellarnoiseoranincreasingS/Nbecauseofmultiplemeasurementsinrealobservation.WepredictatotalRVmeasurementuncertaintyof0.62,0.65and1.1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forspectraltypeM5V,K5VandG5Vconsideringstellarspectralquality,RVcalibrationandstellarnoise.Accordingtothecalculationsinx 3.2.3 ,RVuncertaintyduetostellarnoiseis0.52,0.55and1.05ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1fortheabovethreetypesofstars,accountingfor70.3%,71.6%and91.1%oftotalRVmeasurementuncertainty.Basedoncomparisonsofourpredictionsandobservation,wethereforeconcludethatstellarnoiseisonemajorcontributorinerrorbudgetofprecisionDopplermeasurement.MdwarfsshouldbetheprimarytargetsinsearchofEarth-likeplanetsintheHZ.UnlikeGandKstars,theRVsignalofEarth-likeplanetsintheHZisnotoverwhelmedbystellarnoiseforMdwarfsinthemostoptimisticcase. 3.4SummaryandDiscussionWeprovideamethodofpracticallyestimatingthephoton-limitedRVprecisionbasedonthespectralqualityfactor,stellarrotation,RVcalibrationuncertainty,stellarnoiseand 87

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Figure3-11. RMSerrorofKeplerianorbitttingforplanetsdetectedbyHARPSsince2004.PredictedRVprecisionsconsideringstellarnoisefordifferentstellartypesareoverplottedasopendiamonds. telluriclinecontamination.ThemethodologydescribedandtheresultspresentedinthispapercanbeusedfordesignandoptimizationofplannedandongoingprecisionDopplerplanetsurveys.Forpureconsiderationofstellarspectralqualitywithoutarticialrotationallybroadeningtheabsorptionlineprole,theoptimalbandforRVplanetsearchisBbandforawiderangeofspectraltypesfromKtoA,whileitisRorKbandformid-latetypeMdwarfs.Nevertheless,theaboveconclusionremainsunchangedafterconsideringtypicalstellarrotationofeachspectraltype.However,FandAstarsbecomeunsuitableforprecisionRVmeasurementsbecauseoftypicallyfaststellarrotation.Weconrmthendingin Reinersetal. ( 2010 )thattheNIRDopplermeasurementsgainadvantageformid-lateMdwarfs.However,insteadofndingYbandastheoptimalbandconsideringstellarspectrumquality 88

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andtelluricmasking,wendthatbothYandHbandsgivethehighestRVprecisionamongV,Y,JandHbands.Inacomparisonto Rodleretal. ( 2011 ),wendKbandistheoptimalbandforprecisionDopplermeasurementinatelluric-freecaseandHbandisoptimalinatelluric-maskingcase,whiletheyfoundYbandgivesthehighestRVprecisioninbothcases.Fundamentalphoton-limitedRVprecisionforevolvedstarshasbeendiscussedby Jiangetal. ( 2011 ),whichisvaluableforongoingRVplanetsearcharoundretiredstarsdiscussedin Johnsonetal. ( 2007b ).WealsoconsidertheuncertaintiesbroughtbycurrentavailableRVcalibrationsourcesatdifferentspectralresolutions(Fig. 3-2 ).Subms)]TJ /F5 7.97 Tf 5.07 0 Td[(1calibrationprecisioncanbereachedforeachobservationalbandpass.NotethattheQfactorsmaychangeasgaspressure,lengthoflightpathandtemperaturechanges.TheprecisionalsodependsonthemethodsusedintheRVcalibration.Wecategorizedthecalibrationmethodsintoseveralcases:Superimposing,inwhichthecalibrationspectrumisimprintedontoastellarspectrum;Non-CommonPathandBracketing,inwhichthecalibrationisconductedeitherspatiallyortemporally.Theformermethoddependsonstellaruxwhilethelatteronecanonlybeapplicableforverystableinstruments.Thereareothercalibrationsourceswehavenotincludedintothediscussionsinthisstudy,forexample,lasercombs( Lietal. 2008 ; Steinmetzetal. 2008 ),theFabry-Perotcalibrator( Wildietal. 2010 )andtheMonolithicMichelsonInterferometer( Wan&Ge 2010 ).Oncetheybecomemoreeconomicallyaffordableormoretechnicallyready,theRVprecisionwillbegreatlyimprovedinthefuture.Forthersttimewehavequantitativelyestimatedtheuncertaintycausedbytheresidualoftelluriccontaminationremovalforhighresolutionechellespectroscopymethod.Dependingonthetelluricabsorption,differentobservationalbandpassesareaffecteddifferently.BbandistheleastsensitivetotelluriccontaminationbecausetherearebarelyanytelluricabsorptionfeaturesinBband.However,theNIRbandssufferthemostinprecisionRVmeasurementsbecausethestellarabsorptionlinesandtelluriclinesaremixedtogetherseverelyinthisspectralregion.Onlywhen0.01,i.e.,morethan99%ofstrengthoftelluric 89

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linesisremoved,theadvantageofNIRobservationofmid-latetypeMdwarfsbeginstoshow,whichisafactorof3improvement.ThisquantitativemethodinestimatingtheRVuncertaintyinducedbytelluriccontaminationcanbeeasilyadaptedtootherproblems,forexample,estimatingthemoonlightcontamination.Besidestelluriclineremoval,telluriclinemaskinghasalsobeendiscussedinseveralofpreviousstudies( Reinersetal. 2010 ; Rodleretal. 2011 ; Wangetal. 2011 ).In Reinersetal. ( 2010 ),telluricabsorptionwithdepthmorethan2%and30kms)]TJ /F5 7.97 Tf 5.06 0 Td[(1inthevicinityisblockedoutwhenmeasuringRV.Basedonthisblockingcriterium,thephoton-limitedRVuncertainty,vrms,S(refertoEquation 3 ),foranM9VstaratR=100,000is3.9,2.2,3.9,2.2ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1inV,Y,JandHbandrespectively(seeTable 3-6 ).Incomprison,vrms,Nis71.3,5.8,6.5,3.7ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1inV,Y,JandHbandrespectively.ExceptforVband,vrms,Sandvrms,Nareatthesameorderofmagnitude,andtheuncertaintycausedbytelluricabsorptioncannotbeneglectedeventhoughthatthespectralregionwithanytelluricabsorptionofmorethan2%isblocked.Ifmorestrictcriteriumoftelluriclinemaskingisapplied,fewerphotonsareconsideredinmeasuringtheRV,whicheffectivelyincreasesthephoton-limitedRVuncertainty.Inordertoreachphoton-limitedRVprecisionpredictedbypureconsiderationofspectralQfactor,telluricremovalshouldbeappliedinwhichtelluriccontaminationismeasuredormodeledandthenremovedfrommeasuredstellarspectrum.RVuncertaintyduetostellargranulationistakenintoconsiderationinthispaper.Highfrequency(min)stellarnoisesuchasp-modeoscillationsusuallyhaveaRVamplitudeof0.1to4.0ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1( Schrijver&Zwaan 2000 )andtheycanbeaveragedoutwithintypical10)]TJ /F1 11.357 Tf 7.6 0 Td[(15exposuretime.RVuncertaintiesduetolowfrequency(10)]TJ /F5 11.955 Tf 9.05 0 Td[(100day)stellarnoisesuchasstellarspotshavebeendiscussedinrecentpapers,forexample, Desortetal. ( 2007 )and Reinersetal. ( 2010 ).Theamplitudesofspot-inducedRVrangefromonetoseveralhundredms)]TJ /F5 7.97 Tf 5.06 0 Td[(1.SincestellarspotinducedRVuncertaintiesareperiodicandthereforecanbemodeledandremoved,however,theamplitudeofresidualisunknownatthisstage. 90

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WecomparetheRVprecisionbasedonstellarspectralqualityandthesignalofanEarth-likeplanetintheHZofastarwithacertainspectraltype.WendthatitislikelytodetectahabitableEarth-likeplanetaroundG,KandMstarswhileitistoodemandingtodetectonearoundFandAstars.BbandistheoptimalbandforGandKstarsandKbandforMdwarfs.Afterconsideringpracticalissuessuchastelluriccontamination,wendthat,exceptforBband,everyobservationalbandpassisaffectedbytelluriccontaminationtosomeextent.ThemajorRVmeasurementuncertaintycomesfromtelluriccontamination,whichoverwhelmstheRVsignalofanhabitableEarth-likeplanetaroundGandKstars.Surprisingly,telluriccontaminationbecomesanissueinVbandeventhereisonly2.4%ofspectralregionaffectedbytelluriclines.Aftertelluriclinesareremovedataveryhighlevel,i.e.,0.001,theerrorfromRVcalibrationbecomesthemajorcontributorofDopplermeasurementuncertainty.Afterstellarnoise(granulationonly)istakenintoconsideration,whichisdominantcontributortoRVuncertainty,MdwarfsbecometheonlytypeofstarthatissuitableforthesearchforEarth-likeplanetsintheHZ.TheRVprecisioninthediscussionofhabitableEarthdetectabilityconsidersfourfactors:stellarspectralquality,RVcalibrationuncertainty,stellarnoiseandtelluriccontamination.However,thediscussionofstellarnoiseshouldbetreatedwithgreatcautionforseveralreasons:1),stellarnoiseisnotverywellunderstoodandcharacterizedatthisstage;2),itisdifferentfromcasetocaseandthereforeitisdifculttodrawageneralconclusion;3),ahabitableplanetsearchisdifferentfromaplanetsurvey,thetargetsarechoseninfavorofdetectionatthebestcasescenario,forexample,highstellarux,slowstellarrotation,lowstellaractivityandlowstellarnoiseandsoon.Therefore,thestellarnoiseassumedinthisstudyisinthebestcasescenarioaccordingtocurrenttheoryandobservation.Inaddition,weassumethehighestsignalwithinlinearrange(30,000ADU)forcurrenttypicalCCD(16-bitdynamicrange)insingleexposureinthediscussion,notethattheS/Ncanalsobeimprovedbymultipleindependentmeasurements.TheS/NsrequiredforEarth-likeplanetdetectionsareprovidedinTable 3-8 basedonstellarspectralquality.PleasenotethattheHZ 91

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changesovertimeastheluminosityofthehoststarchanges.Italsodependsonpropertiesofaplanetsuchasatmospherecomposition,albedoandorbit.ThepurposeofdiscussioninthispaperistoprovideabasicideaofthecomparisonofcurrentbestobtainableRVprecisiontoatypicalRVsignalofanhabitableEarth-likeplanet. 92

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CHAPTER4PLANETSEARCHAROUNDMDWARFS 4.1Introduction 4.1.1CurrentStatusAsofMay2011,thereare35planetsin28planetarysystemsofMdwarfs.RadialvelocitytechniqueisthemostproductivemethodinM-dwarfplanetsearchwithdiscoveriesof21planetsin15systems.Microlensingranksthesecondwith12planetdetectionsin11systems.Transitingmethodhasbyfardetected2planetsaroundMdwarfs.GiantplanetoccurrencerateforMdwarfsisgenerallythoughttobelowerthanthatforsolar-typestars. Bonlsetal. ( 2011b )foundalowfrequency(f)ofgiantplanetaroundMdwarfs,f1%forP=1-10dayandf=2+3)]TJ /F5 7.97 Tf 5.07 0 Td[(1%forP=10-100day,Pdenotesorbitperiod.Incomparison, Cummingetal. ( 2008 )foundthatthefrequencyis10%forsolar-typestars.Ontheotherhand,low-massplanetsarefrequentlydetecteddespiteofanadversedetectionbias. Bonlsetal. ( 2011b )foundthatsuper-Earths(msini=1)]TJ /F5 11.955 Tf 10.32 0 Td[(10M)areabundantaroundMdwarfswithafrequencyof35%.Giventhefrequencyofsuper-EartharoundMdwarfs,therearemanyplanetsawaitingfordiscoveries. 4.1.2ChallengesMdwarfsemitthebulktheirenergyinthenearinfrared(NIR),theyarethusmuchmorebrighterintheNIRthanintheopticalwavelengths.NIRobservationprovidesapromisingwayofdetectingplanetsaroundMdwarfs.However,thereareseveralobstaclesthatpreventusfrommakingdiscoveries. 4.1.2.1AtmophsereGround-basedNIRRVmeasurementisseverelyaffectedbytheEarth'satmosphere,whichconsistsabsorptionlinesofmanyspeciesincludingH2O,O2,CO,CO2andsoon(telluriclines).TheRVofthesespeciesiscorrelatedwiththemotionoftheatmospheresuchaswindandturbulence.MeasuredRVofanMdwarfisthereforeaffectediftelluriclinesarenotproperlyremovedfromobservedspectrum. 93

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4.1.2.2WavelengthCalibrationSourcesAnotherfactthathaslimitedtheimprovementofRVprecisionintheNIRisthelackofastableandprecisewavelengthcalibrationsource.Awavelengthcalibrationsourceprovideacaliber(absolutewavelengthcalibration)forthoseobtainedstellarspectraandenablesexclusionofinstrumentinstabilityandmeasurementofstellarRV.UnlikethosematuredwavelengthcalibrationsourcesinthevisiblebandssuchasanIodinecell( Butleretal. 1996 )andaTh-Aremissionlamp( Lovis&Pepe 2007 ),thequestforasuitablewavelengthcalibrationsourceintheNIRremainsawideopenquestion.Pleaserefertox 3.2.2 foramoredetailedandcompletereviewoftheeldofwavelengthcalibrationsources. 4.2TacklingAdversitiesinNIRRVMeasurement 4.2.1SoftwareAdvancement 4.2.1.1PreciseTelluricLinesRemovalTelluriclinesexitandsometimespopulateintheNIRpartofanobservedstellarspectrum.StellarRVwillnotbepreciselymeasuredunlesstelluriclinesarecarefullyremovedfromanobservedstellarspectrum.Severalwaysoftelluricremovingschemehavebeenproposedandpracticedincludingtelluriclineforwardmodeling( Beanetal. 2010 )andobservingatelluricstandardstar( Vaccaetal. 2003 ).Theformermethodreliesonasynthetictelluricabsorptionspectrumtoforwardmodelanobservedspectrumtogetherwithspectraofanstellartemplateandanabsorptioncellthatprovidesanabsolutewavelengthsolution.Itiscomputationallyintenseandthecrosstalkbetweendifferentcomponentsinthemodelisdifculttofullyunderstand.Thelattermethodreliesonanobservationofatelluricstandardstar,usuallyafast-rotatingearlytypestarwhoseintrinsicspectrumisalmostfeatureless.Telluriclinespectrumcanbeobtainedbasedontheobservedspectrumofantelluricstandardstar.Thismethodrequiresmoreobservationaltimebutreducesthecomplexityindatareductionprocess.Iadoptedthetelluricstandardstarmethodtoremovetelluriclinesfromanobservedspectrum. 94

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Figure 4-1 showanexampleofanobservedstellarspectrum(GJ411)contaminatedwithtelluriclines(mostlywaterlines)intheNIRregionbetween8130and8270.MostoftheobservedlinesarenotassociatedwiththestarbutformedbytheEarth'satmosphere.Thecomparisonbetweenatelluric-line-removedstellarspectrumandasyntheticstellarspectrumcanbeseeninFig. 4-2 .Atthisstage,aprecisemeasurementofstellarRVbecomespossible.Tofurtherinvestigatethetelluricremovallevel,Idividethesetwospectraandremoveoutliersduetomismatchesbetweenanobservedspectrumandasyntheticspectrum.Ifoundtheresidualaftertelluriclineremovalhasanrmsof0.027indicating97%oftelluriclinestrengthhasbeensuccessfullyremoved. Figure4-1. Comparisonbetweentwospectrabefore(black)andafter(red)removingtelluriclines. 95

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Figure4-2. Comparisonbetweenanobservedstellarspectra(blackGJ411,telluriclinesremoved)andasyntheticspectrum(red). 4.2.1.2BinaryMaskCrossCorrelationStellarRVcanbemeasuredwiththeso-calledcrosscorrelationfunction(CCF)methodinwhichafullywavelength-calibratedstellarspectrumismultipliedwithaseriesofDoppler-shiftedtemplatespectra.ThemaximumoftheCCFcorrespondsthemostlikelystellarRV.Thetemplatespectracanbeobtainedeitherfromobservationorfromtheoreticalhigh-resolutionsyntheticspectra.Iadoptedthebinarymaskcrosscorrelationtechnique( Baranneetal. 1996 ; Pepeetal. 2002 ; Queloz 1995 ).Thetemplatespectraaregeneratedfromsyntheticstellarspectraandeachabsorptionlinehasaboxedshapewhosewidthanddeptharedeterminedbytheactuallinewidthandlinedepth.Theadvantageofthebinarymaskcrosscorrelationtechniqueisthatitmakesitpossibletoselectspeciclinesofinterests 96

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Figure4-3. Telluriclineremovalresidualis2.7%afterejectingpointsthatarecausedbymismatchesbetweenanobservedspectrumandasyntheticspectrum(markedbyredasteriks). whilediscardingpotentialcontaminatingspectralregion.ItisparticularlyusefulwhendealingwithspectratakenintheNIRwhichisseverelycontaminatedbythetelluriclines.ThistechniqueisalsousefultoeliminatenoisesourcecontributedbyweakabsorptionlinesinthelowS/Ncase.Figure 4-4 showsanexampleofawavelength-calibratedstellarspectrumintheNIR(black)andabinarymasktemplate(red)usedinthecrosscorrelationprocess. 4.2.2HardwareAdvancementThereisalackofpreciseandstablewavelengthcalibrationsourceintheNIRalthoughmanycandidateshavebeenproposed( Beanetal. 2010 ; Lietal. 2008 ; Mahadevan&Ge 2009 ; Redmanetal. 2011 ; Steinmetzetal. 2008 ; Wildietal. 2010 ).SuitablegascellssuchasanIodinecellintheopticalwavelengthsarehardtondintheNIRandtheprecisionis 97

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Figure4-4. Red:anexampleofbinarymasktemplate.Black:wavelength-calibratedstellarspectrum. limitedbytemperatureandspeciescontamination.Sourcessuchasemissionlampsarefacingagingproblem.What'smore,theirspectraareirregularandnon-uniform,whichposeschallengesindatareductionprocess.Etaloncanbeusedforprecisewavelengthcalibration( Wildietal. 2010 ).However,thehighnessemirrorlimitsitsoperatingbandtoberelativelynarrow.Laserfrequencycombstechnique( Lietal. 2008 ; Steinmetzetal. 2008 )iswidelybelievedtobethenextgenerationwavelengthcalibrationsourceintheNIR,butitisstilltooexpensiveandinmaturedtobeincorporatedwithanastronomicalDopplerinstrument. 98

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WearedevelopingaMichelsoninterferometer(namelythesinesource)thatcanbeusedasapreciseandstablewavelengthcalibrationsource.Whencomparedtopreviously-mentionedcandidatesources,ithasmanyadvantages: 1. Itispreciseenoughtoprovideacalibrationprecisionofbetterthan10cms)]TJ /F5 7.97 Tf 5.07 0 Td[(1whichwouldenabledetectionofanEarth-likeplanetinthehabitablezoneofasolar-typestar. 2. Itsspectralstabilityissolelydependentuponitsthermalstability.Thisfeaturereducesthecomplexitywhendesigningtheinterferometeranditsenclosure. 3. IthasaverybroadwavelengthcoveragefromopticalwavelengthstotheNIRwithaoperationbandwidthofmorethan800nm.ThewidewavelengthcoverageallowsustosimultaneouslyincludemorespectralregiontoincreaseS/N. 4. Thespectralfeatureisuniformedandperiodicalwhichenablesfastandprecisedataprocessing. 5. Itiscompactwithadimensionof222inchforitsopticscomponents.Whenequippedwithathermalenclosure,itsdimensionis444inch.Theapplicationsofthesinesourceisexible.Itcanbeusedeitherasanabsorptioncell(Fig.??)orasanemissionlampsource(Fig. 4-6 ).IntheformercasethestarlightgoesthroughthesinesourcebeforerecordedbyaCCD.ThisapplicationissimilartotheIodineabsorptioncellmethodcurrentlybeingusedattheHIRESfortheKecktelescope,butthewavelengthregionwithcalibrationisdramaticallyincreasedbecauseofthebroadwavelengthcoverageofthesinesource.However,theEarthstelluriclineshavetobeconsideredintheNIR.Inthelattercasewherethesinesourceisusedasanemissionlampsource,starlightandtheemissionspectrumofthesinesourcearedirectedtothedetectorbytwonearbybutseparatedbers.ThiscaseissimilartothecalibrationschemeadoptedbyHARPSusingaTh-Aremissionlamp,butdatareductionisexpectedtobesimpliedbecauseofthesimplespectrumoutputofthesinesource.WehaveconducteddemonstrationexperimentsbothinthelabandattheobservatoryusingtheEXPERTinstrumentattheKPNO2.1mtelescope.Figure 4-7 showsthecompar-isonbetweentwodifferencewavelengthcalibrationsources.Oneisthesinesource,theotheroneisaTh-Arlamp.Ina2-dayexperiment,wehavedemonstratedthatthemeasured 99

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Figure4-5. Applicationofthesinesourceasanabsorptioncell.Obtainedspectrumisamultiplicationofstellarspectrum,atmosphereabsorptionandsinesourcespectrum. instrumentdrift(inms)]TJ /F5 7.97 Tf 5.06 0 Td[(1)fromthetwosourcesovertimeisconsistentwitheachotherat10.7ms)]TJ /F5 7.97 Tf 5.06 0 Td[(1levelwhichisatthelevelofpredictedphoton-noiselimitedmeasurementerror.Thesinesourceprovidesanoverallmoreprecisemeasurementandthusabetterwavelengthcalibrator. 4.3M-dwarfPlanetSearchandCharacterization-Results 4.3.1TelluricLineRVStabilityIthaslongbeenwonderedhowstabletheEarth'stelluriclinesareanddifferentRVstabilitiesofdifferentspecies. Figueiraetal. ( 2010a )hasstudiedO2linesRVstabilityusingHARPSdataandconcludedthatO2linesareasstableas10ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1over6yearsandtheintrinsicstabilityofO2linesisevenhigher(2-3ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)whenasimplephysicalatmosphere 100

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Figure4-6. Applicationofthesinesourceasanemissionlampforsimultaneouswavelengthcalibration.Stellarspectrumandsinesourcespectrumarearrangednexttoeachother. modelisconsidered. Beanetal. ( 2010 )mentionedthatH2Olinesisasstableas20ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1overhalfyear.Thesmallwavelengthcoverage(onespectralorder,8130-8270forH2Oand6870-6935forO2)andunknownsystematicerrorpreventusfromstudyingtheabsoluteRVstabilityofthesetwospecies.However,therelativestabilitycanbestudiedbysubtractingoneRVmeasurementresultfromtheother.Figure 4-8 showstherelativeRVstabilitybetweenH2OandO2linesandtheRVscatterrmsis18.3ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1over10days.Theresultisconsistentwithpreviousstudiesby Figueiraetal. ( 2010a )and Beanetal. ( 2010 ).Theimplicationisthattelluriclinescanbeusedasawavelengthcalibrationsourceat10-20ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1precisionlevel. 101

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Figure4-7. Top:MonitoredRVdriftover2-dayperiod.ResultsfromSinSourceareplottedinblackwitherrorbars,andtheresultsfromTh-Arlampareinred.Themedianofmeasurementerroris3.8ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forSinSourceand18.4ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1forTh-Arlamp.Bottom:ThedifferencebetweenresultsfromSinSourceandTh-Arlamp.ThetwomethodstrackeachotherwithanRVRMSof10.7ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1. 4.3.2RVMeasurementsofaReferenceStar-GJ411GJ411isaknownRVstablestarwithanRVscatterrmslessthan7ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1accordingto Endletal. ( 2006 ).WehaveshowninFig. 4-9 thattheRVrmsis24.7ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1withaTh-Arlampwavelengthcalibrationand40.6ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1withtelluriclinesasawavelengthcalibrationsource.Bothresultsare1.5timesworsethanphoton-noiselimitedprediction.Theresultsareexpectedtobeimprovedsignicantlywithlargerwavelengthcoverage. 102

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Figure4-8. TelluricwaterlinesRVstabilitycomparedtoO2linesRVstability. 4.4M-dwarfPlanetSearchandCharacterization-FutureWorks 4.4.1SearchingForPlanetsAroundMDwarfswithEXPERTTheendeavorofsearchingforexoplanetshasledtodiscoveriesofover400planetsaroundstarsofspectraltypeA-M1.Only16(4.0%)planetshasbeenfoundaroundMtypestarsdespiteofthefactthattheymakeupmorethan70%ofthegalaxyincludingsolarneighbors.Asoftoday,planetssearchprogramsaroundMtypestarshaveresultedinrelativelowdetectionratecomparedtosolartypestars( Cummingetal. 2008 ; Endletal. 2006 ; Zechmeisteretal. 2009 ).PartofthereasonisthatRVmeasurementprecisionis 1http://exoplanet.eu/ 103

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Figure4-9. RVmeasurementsforGJ411.Black:resultsusingaTh-Arlampasawavelengthcalibrationsource.Red:resultsusingtelluriclinesasawavelengthcalibrationsource. relativelyworseforMtypestars,makingitdifculttodetectNeptune-likeorlowermassplanets.Ontheotherhand,theRVmeasurementispreciseenoughtodetectclose-inJupiter-likeplanets.TherelativelowdetectionrateindicatesthelowfrequencyofgasgiantaroundMtypestarscomparedtosolartypestars. Endletal. ( 2006 )gives1upperlimitonfrequencyofgasgiantaroundMtypestarsof<1.27%. Butleretal. ( 2006 )estimatesplanetsfractionof1.8%1.0%forplanetsmassesover0.4MJ. Johnsonetal. ( 2007a )ndsthisfraction1.8%forstellarmassrangefrom0.1-0.7M,and Cummingetal. ( 2008 )foundthatMdwarfsare10timesunlikelytoharboragasgiantwithina2000-dayorbitcomparedtosolartypestars.Incontrast, Bonlsetal. ( 2007 )foundthatplanetslessmassivethan25MaresignicantlymorefrequentaroundMdwarfswhichsupportsthepredictionthatthe 104

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frequencyofNeptune-likeplanetsarehigheraroundMtypestarsthanGtypestars( Ida&Lin 2005 ; Kennedy&Kenyon 2008 ; Kennedyetal. 2006 ).Among16planetsdiscoveredaroundMtypestars,3(18.8%)hasbeenfoundmoremassivethan1MJandanother3(18.8%)planetsbetween0.6MJand1MJ.TheseJupiter-likeexoplanetsposechallengestotraditionalcore-accretionmodelinthewaythatclassicalcoreaccretionmodelhassevereproblemwithforminggasgiantplanetsduetolessmassiveprotoplanetarydisk( Laughlinetal. 2004 ),whilecompetinggravitationalinstabilitymodelcaneffectivelyformJupiter-likeplanetsaroundMtypestars( Boss 2006 ).MoreobservationsareneededtoconstrainplanetaryformationtheoryanddiscoveriesofplanetaroundMtypestarswillhelpusaddressthequestionfromstatisticalperspective.WeareproposingtouseEXPERT(EXtremelyhighPrecisionExtrasolaRplanetTracker)DEM(DirectEchelleMode)atKPNO2.1mtelescopetoconductpreciseRV(radialVelocity)measurementsof41mid-lateMtypestarsofspectralrangebetweenM3.5andM6.Wehaveincreasedthemid-lateMtypestarssamplebyafactorof1.5comparedtosimilarRVsurveyscombinedinthepast( Cummingetal. 2008 ; Endletal. 2006 ; Zechmeisteretal. 2009 ).Itwillproviderobuststatisticalconstraintsonthefrequencyofclose-inJupiter-likeplanetsandNeptune-likeplanetsaroundmid-lateMtypestars.Simulationsshowthatwewillbeabletoobtainbetterthan3m/sRVprecisioninphoton-noiselimitcasein10minexposureforJ=7target.Incomparison,a20Mplanetinaedge-onorbitwithaof0.1AU(p20day)aroundastarof0.3MproducesRVsignalwithsemi-amplitudeof10m/s.Mtypestarsemitthebulkofenergyaround1m,soitmaybeadvantageousobservingtheminNIR(nearinfrared).Practically,weareinevitablyfacingtwomajorobstaclesgoingintoNIR:(1)telluriclinescontamination;(2)lackofsourceforabsolutewavelengthcali-bration.Wewillobservebrightfastrotatingearlytypestars(mostlyAtype)astelluriclinesstandardstarandtrytoremovethecontaminationoftelluriclinesfromMtypestarspectrum.Telluriclinescontaminatingportioninspectrumwillbemaskedduringdatareductionifnotelluricstandardstarisavailable. 105

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4.4.2Multi-BandStudyofRadialVelocityInducedbyStellarActivitywithEXPERTSearchingforearth-likehabitableexoplanethaslongbeenpursuedbyplanethunters.However,itisextremelydifculttoachieveduetothelowRVsignal(0.1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1)forsolartypestarincontrastwithcurrentRVprecisioninvisibleband(1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1).Ontheotherhand,planetsaroundlate-typestars(i.e.Mdwarfs)inducerelativelylargerstellarreexmotionduetolowerstellarmass,thushigherRVsignal.Therefore,Mdwarfsbecomefavorabletargetsinthesearchofhabitableplanets.However,visiblebandobservationofMdwarfsisdifcultduetotheintrinsicfaintnessoftheobjects.Therefore,RVmeasurementsinnearinfrared(NIR)isbecominganincreasinglyinterestingeld.Meanwhile,itisclaimedthatRVjittersduetostellaractivitiesarereducedinNIR,whichbecomesantheradvantageofNIRRVmeasurements.Theargumentappearstobeobservationallyconrmedinanumberofcases( Beanetal. 2010 ; Huelamoetal. 2008 ; Martnetal. 2006 ; Pratoetal. 2008 ).However,inaboveobservations,themeasurementsinvisiblebandandinNIRwerenotconductedsimultaneously,inwhichwecannotruleoutthepossibilitythatthestarsobservedwereexperiencingalessactiveperiodduringtheNIRobservation.Therefore,simultaneousmeasurementsarerequiredtoconrmthetrendofdecreasingRVjittertowardlongerwavelength,i.e.,NIR. Desortetal. ( 2007 )studiedtheRVinducedbyastarspotandgaveanempiricalcorrelationbetweenRVamplitudeinvisibleandotherparameterssuchasspectraltype,spotsizeandVsini. Reinersetal. ( 2010 )carrieditfurthertocomparetheRVamplitudeinducedbyastarspotinthevisibleandNIRband.Accordingtothesimulation,theyfoundthattheRVamplitudeinYbandisatleasttwicesmallerthanthatinVbandforhotstarspot.ItisantheoreticalsupportforthetrendofdecreasingRVinNIRwhichneedsconrmationfrommulti-bandsimultaneousobservation.Wehavedemonstratedtheshort-termRVprecisioninIband(0=806nm,=149nm)byobservingaphotometricstablestar,KEP11859158.TheRMSis57.6ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1for7days'continuousobservation(Fig. 4-10 ).WeusedTelluriclinesaswavelengthandRVcalibration 106

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referencesincethelinesofTh-Arlampareverysparseinthesamewavelengthregion.WealsoremovedthetelluriclinesfromstellarspectrausinganearbytelluriclinestandardstarwhichisobservedatalmostthesametimeastheobservationofKEP11859158.WesuspectthatthescatteringoftheRVmeasurementsisduetotheintrinsicinstabilityoftelluriclines(i.e.,wind,watervapordensity,etc.).WehavedevelopedanewRVcalibrationsource,i.e.,anRV-calibrationinterferometerwhichproducesdenselinesoverlargewavelengthcov-erageincludingB,V,R,IandYband.In-labdemonstrationhasshownthatRVcalibrationsinVbandusingTh-ArlampandwiththeRV-calibrationinterferometertrackeachotherandtheRVcalibrationwiththeinterferometershowsmuchsmallerRMSscatteringthanTh-Arlamp(3.8ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1VS.18.4ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,Fig. 4-7 ).AftertherecentinstallationoftheRV-calibrationinter-ferometerinFeb2011,theRVprecisionbeyond0.7misexpectedtobegreatlyimproved.21.9ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1RVRMSscatterwiththesamestarhasbeenreachedintheVbandusingDEMofEXPERT(Fig. 4-11 ).MostobjectswithRVmeasuredbothinVandNIRbandshowRVjitteroramplitudemorethan300ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1inVband.Plus,spectraltypesoftheobjectsspanfrombrowndwarfs( Martnetal. 2006 )toactiveyoungstars( Pratoetal. 2008 ).IftheRVjitterisindeedsmallerinNIRthaninVbandbyafactorofatleassttwoaspredictedby Reinersetal. ( 2010 ),thenitisobservableusingEXPERT.Furthermore,thequestionofmulti-bandRVjitterdifferencedependenceonspectraltypewillforthersttimebeansweredifthesampleoftargetsiscarefullychosen.Thendingoftheproposalwillprovideinsightsforfuturehabitableearth-likeexoplanetsearchingmissioninNIR,helpingunderstandtheRVjitterofstarsofdifferentspectraltypes. 4.4.3Mid-LateTypeMDwarfPlanetSurveyUsingFIRSTWeproposestoconductapilotsurveyof50J8nearbyMdwarfsforexoplanetswiththe2-mAutomaticSpectroscopicTelescope(AST)in2013-2015withanewgenerationcryogenichigh-resolution(R=60,000,0.8-1.35micron)nearIR(NIR)cross-dispersedechellespectrograph.Thisinstrument,calledFIRST,isscheduledtoseerstlightattheAST2m 107

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Figure4-10. RVmeasuredinIbandusingDEMofEXPERTforKEP11859158,aphotometricstablestar.TheRVRMSis57.6ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1. telescopeinJune2013.ThispilotsurveyisdesignedtotunethenewinstrumentandthedatapipelinetogetreadyforlaunchingaNIRhighprecisionDopplerexoplanetsurveyof215Mdwarfs. 4.4.3.1ScienceJusticationHistorically,exoplanetsearcheshavefocusedonstarswithmassessimilartothatofourSun.Lowermassstarshavereceivedlessattentioninlargepartbecausetheyareintrinsicallyfaintandcool,emittingmostoftheirlightatNIRwavelengthswhereourobservationaltechniquesarelesswelldeveloped.RadialvelocitysearcheshaveprobedtheearlyMdwarfs(mass0.4Solarmass)andhaveclearlyindicatedthatshort-periodgiantplanetcompanionstoearlyMdwarfsarerare( Endletal. 2006 ; Johnsonetal. 108

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Figure4-11. RVmeasuredinVbandusingDEMofEXPERTforKEP11859158,aphotometricstablestar.TheRVRMSis21.9ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1. 2007a ).However,planetarycompanionstostarsatthepeakofthestellarmassfunctionandbelow(mass0.4Solarmass)remainessentiallyunexplored.ThesesmallstarsrepresentperhapsourbestopportunitytodetectEarth-massplanets,includingthoseorbitingintheHabitableZone(HZ),givencurrentlevelsofRVprecision.Atthesametime,theseplanetswillnecessarilybeinthesolarneighborhood,makingthemamongstthemostimportanttargetsforfuturespace-basedeffortstodirectlyimageEarth-likeplanetsandtostudytheiratmospheres.Thereismountingevidencethatsub-Neptunemassplanets,includingtheSuper-Earths,maybeverycommoninorbitaroundlow-massstars.WhileKeplerobservesonlyasmallnumberoflow-massstars,itprovidesevidencethattheplanetmassfunctionincreases 109

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towardsmallerplanetaryandstellarmass.Ananalysisby Howardetal. ( 2011 )indicatesthatupto30%ofearlyMdwarfshaveSuper-Earthsizedplanetswithorbitalperiodslessthan50days.AnalysisoftheHARPSRVdataby Bonlsetal. ( 2011b )ndssimilarlyhighvaluesfortherateofoccurrenceofsuper-EarthsorbitingearlyMdwarfswithperiodsbetween10and100days(35%)aswellevidenceforasignicantpopulationwithorbitalperiodslessthan10days(36%).ThedetectionofasystemofMars-sizecompanionsorbitingalateMdwarfinKeplerdataby Muirheadetal. ( 2012 ),oneofonlyahandfuloflate-MdwarfsintheKeplereld,providesfurthercircumstantialevidencethatthesetypesofcompanionsareverycommon.MdwarfslaterthanM4areofgreatscienticinterest.Forthesestars,themass,size,andtemperatureofthestarsbegintorapidlydecrease.Todate,mostexoplanetsearchestargetingMdwarfshavebeenconductedatvisiblewavelengths( Bonlsetal. 2011b ; Endletal. 2006 ; Johnsonetal. 2007a )forstarswithspectraltypeearlierthanM4becauseofintrinsicfaintnessofmid-latetypeMdwarfs.Thereareonly12M4orlatertypestarswithV12northof-30degrees( Reid&Gizis 1997 ).Forcomparison,thereareabout300nearbystarsM4orlaterwithJ9( Lepine&Shara 2005 ).Therefore,forthelatesttypesofstars,anobservingprogrammustoperateintheNIR.Thecurrentstate-of-the-artforNIRRVdetectionofplanetsaroundlateMdwarfshasbeendemonstratedwiththeVLTsCRIRESwithmoderatesimultaneouswavelengthcoverage(364)usinganammoniagascellforcalibration( Beanetal. 2010 ).Long-term(6months)RVprecisionsof5m/shavebeendemonstratedwiththissystem.Withthisprecision,andtheobservingtimeavailableatthisfacility,searchesforgiantplanetsorbitinglate-Mdwarfscanbecarriedout.TheprecisionthatwillbedeliveredbyFIRST(betterthan4m/s)andthecadenceenabledbyTSUsASTmakethissystemalogicalnextstepforsurveyoflow-massplanetsaroundmid-latetypeMdwarfs. 110

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4.4.3.2TargetSelectionOurtargetswereselectedfromthefollowingcatalogs:GlieseCatalogofNearbyStars;GlieseCatalogofNearbyStarscrossidentiedwith2MASS( Staufferetal. 2010 );ROSATAll-SkySurvey:NearbyStars( Hunschetal. 1999 ).Theselectionwasbasedonthefollowingcriteria: J10anddec-20 MV8.7andV-K3.5 RatiobetweenXrayluminosityandbolometricluminosity,RX-3.0215Mdwarfswereselectedwiththeabovecriteria(Note:beforewelaunchthesurvey,wewillusethe2MASScatalogtorejectadditionalMdwarfswithaJ14stellarcompanionwithin5arcsecandreplacethemwithslightlyfainterMdwarfs.Duringthesurvey,wewillrejectspectroscopicbinariesafter3RVmeasurementsfromourtargetsandreplacethemwithnewsurveytargets).Basedontheempiricalequationofrotationvelocityvs.RXin Kiraga&Stepien ( 2007 ),weexpect87%ofourMdwarfswithrotationalvelocitylessthan5km/s.Therefore,mostofthemareinactivestars,whichcanhelptominimizetheRVjitterscausedbystellaractivitiesalthoughthejitterlevelissignicantlyreducedinNIR( Ma&Ge 2012 ; Reinersetal. 2010 ).Figure 4-12 andFigure 4-13 showthenumberdistributionintheJandVbandsoftheFIRSTMdwarfsurveytargetsandtheeffectivetemperaturedistributionofthesurveytargets. 4.4.3.3PlanetYieldPredictionAMonte-CarlosimulationisusedtoestimateplanetyieldofthesurveywithFIRST.Foreachselectedstar,anRVmeasurementprecisioniscalculatedbasedonitsapparentmagnitudeandspectraltype.Figure 4-14 showsRVmeasurementprecisionintwocases.Inthebaselinecase,weassumetotalmeasurementerrorconsistsof1.5timesphoton-noisemeasurementuncertainty,0.5m/swavelengthcalibrationerror,98%telluriclinemaskingerror.ThebaselinecaserepresentsareasonablescenariofortheFIRSTsurvey.Inthepessimisticcase,weassumetotalmeasurementerrorconsistsofa3m/sunknown 111

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Figure4-12. VandJbanddistributionforFIRSTsurveytargets. systematicerrorinadditiontoerrorsourcesassumedinthebaselinecase.ThepessimisticcaserepresentsaworstcasescenariofortheFIRSTsurvey.Stellarmassisestimatedfromitsabsolutemagnitude.OnceRVprecisionandstellarmassisknown,wecangenerateadetectabilityplotonmass-periodspace.Morespecically,foragivenplanetmassandorbitalperiod,wegenerateaRVcurveof100daysfromwhich24RVpointsarerandomlydrawntoformaRVdataset,eccentricitydistributionfollowsthatfrom( Wang&Ford 2011 ).Thedatasetisthenanalyzedbyadetectioncodebasedonperiodogram,ifthepeakoftheperiodogramagreeswiththeinputperiodandthefalsealarmprobability(FAP)islessthan1/1000,thenwemarkitasadetection.Thistestisrepeated100timesforeachgivenplanetmassandperiod.Therefore,thereisadetectability/completenessplotforeachstar.The 112

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Figure4-13. TedistributionforFIRSTsurveytargets.3180Kisusedtodivideearlyandmid-latetypeMdwarfsinthesample. surveycompletenessplot(Fig. 4-15 andFig. 4-16 )istheaverageofcompletenessplotsofallselectedstars.Toestimateplanetyield,forexample,estimatednumberofdetectedsuperEarths(1-10M),theplanetyieldiscalculatedbyxN,wherexisthemedianofcompletenessintheregionwith1Msini10Mand1Period100day,=0.35( Bonlsetal. 2011b )isthefrequencyofSuper-EartharoundMdwarfswith1-100dayperiod,andNisthesamplesize.Intotal,23planetsareexpectedtobedetectedforthepessimisticcaseincluding5super-Earths,2giantplanet(msini100M)and16intermediate-massplanets(10-100M).Forthebaselinecase,30planetsareexpectedtobedetectedincluding10super-Earths,2giantplanetsand18intermediate-massplanets. 113

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Figure4-14. PredictedRVmeasurementprecisionfortheFIRSTsurvey.Blackdotsrepresentbaselinecase(1.5timesphoton-noise+calibrationerror+98%telluricmaskingerror)andreddotsrepresentpessimisticcase(1.5timesphoton-noise+calibrationerror+98%telluricmaskingerror+3.0m/sunknownsystematicerror).DashedlinerepresentsthebestprecisionachievedbyHARPSMdwarfplanetsurvey. 114

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Figure4-15. ThepredictedsurveycompletenesscontoursbasedonobservationstrategyandRVprecisionforthepessimisticcase. 115

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Figure4-16. ThepredictedsurveycompletenesscontoursbasedonobservationstrategyandRVprecisionforthebaselinecase. 116

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CHAPTER5ACCURATEGROUPDELAYMEASUREMENTFORRVINSTRUMENTSUSINGTHEDFDIMETHOD 5.1IntroductionAsofApr2012,thereareover700discoveredexoplanets,andmostofthemaredetectedbytheradialvelocity(RV)technique1.RVprecisionof1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1hasbeenrou-tinelyachieved( Bouchyetal. 2009 ; Howardetal. 2010b )withinstrumentssuchasHARPS( Mayoretal. 2003 )andHIRES( Vogtetal. 1994 ),whicharecross-dispersedechellespectrographs.Whilecross-dispersedechellespectrographsarecommonlyusedininstrumentsforprecisionRVmeasurements,amethodusingadispersedxeddelayinter-ferometer(DFDI)hasofferedanalternativemethod( Flemingetal. 2010 ; Geetal. 2006b ; Leeetal. 2011 ).Inthismethod,aMichelson-typeinterferometerisusedincombinationwithamoderateresolutionspectrograph,RVsignalsarethenextractedfromphaseshiftofinter-ferencefringesofstellarabsorptionlines( Erskine 2003 ; Erskine&Ge 2000 ; Ge 2002 ; Geetal. 2002 ).ThedetailsabouttheDFDItheoryandapplicationsarediscussedin vanEykenetal. ( 2010 )and Wangetal. ( 2011 ).InstrumentadoptingtheDFDImethodhasdemon-stratedadvantagessuchaslowcost,compactsizeandmulti-objectcapability( Flemingetal. 2010 ; Ge 2002 ; Geetal. 2006b ; Leeetal. 2011 ; Wisniewskietal. 2012 ).TheMARVELS(Multi-objectAPORadialVelocityExoplanetLarge-areaSurvey)( Geetal. 2009 )isaground-basedDopplersurveywiththemaingoalofobtainingalarge-scale,statisticallywell-denedsampleofgiantplanets.Ithasoperatedsince2008until2014.Itwillsearchforgaseousplanetsaround11,000starsthathaveorbitalperiodsrangingfromhoursto2years,andarebetween0.5and10Jupitermasses.Ithascompletedobservationof3,300starswithover94,000RVdatapoints,i.e.,onaverage28datapointsperstar.Over250binariesandadozenofbrowndwarfshavebeendetectedfromthesurvey. 1http://exoplanet.eu/;http://exoplanets.org/ 117

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IntheDFDImethod,axeddelayinterferometer( Wanetal. 2011 2009 )playsacrucialroleincreatingstellarspectralfringesforhighprecisionRVmeasurements( Erskine 2003 ; Ge 2002 ).TheDopplersensitivitycanbeoptimizedbycarefullychoosingthegroupdelay(GD)oftheinterferometer( Wangetal. 2011 ).Morespecically,GDofaninterferom-etershouldbechosensuchthatthespatialfrequencyofwhitelightcombs(WLCs)matcheswiththatofastellarspectrumafterrotationalbroadening.GDisdenedbythefollowingequation: GD()=)]TJ /F5 11.955 Tf 11.93 8.09 Td[(1 2d d,(5)whereisphaseshiftandisopticalfrequency.TheinterferometerinaDFDIinstrumentisusuallydesignedtobeeld-compensatedtominimizetheinuenceofinputbeaminstabil-ity( Wanetal. 2009 ; Wangetal. 2010 ).Itisrealizedbycarefullyselectingglassmaterialsandthicknessesoftwosecondsurfacemirrorssuchthattheirvirtualimagesareoverlapped.Becauseglassesareusedintheopticalpaths,doesnolongerlinearlychangewithfre-quency,thereforeGDisdependentonopticalfrequency.AninaccurateGDmeasurementmaysignicantlylimittheRVmeasurementaccuracy( Barker&Schuler 1974 ; vanEykenetal. 2010 ).Inpractice,theremaybeseveralmethodsofmeasuringGD: 1. CalculateGDbasedonglassrefractiveindicesusingSellmeierequationandthick-nessesfrommanufacturerspecication. 2. Forwardmodelthespectrumofaknownspectralsource,suchasanIodinecelloraTh-Arlamp. 3. Measurephaseandfrequencyusingawhilelightsource,suchasatungstenlamp. 4. CalibrateGDusingasourcewithknownvelocity.Method1isstraightforwardbutmaylackofadequateprecisionbecauseofuncertaintyinparametersinSellmeierequationandmanufacturertoleranceforglassthickness.Method2holdsgreatpromiseforaccuratelydeterminingGDbuttheresomecurrentpracticalissues 118

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preventingusfromadoptingthismethod(seemoredetaileddiscussioninx 5.5.2.1 ).WewilluseMethod3and4tomeasureGDofaninterferometerinthispaper. Figure5-1. IllustrationoftheDFDImethod.Tiltedlinesrepresentinterferencecombsgeneratedbyaninterferometer.Verticallinerepresentsanstellarabsorptionline(solid:originalpositionwithafrequencyof0;dashed:shiftedpositionwithafrequencyof). IntheDFDImethod,GDdeterminesthephase-to-velocity(PV)scale,theproportion-alitybetweenthemeasuredphaseshiftandthevelocityshift.SincetheDFDImethodisrealizedbycouplingaxeddelayinterferometerwithapost-disperser,theresultingfring-ingspectrum)]TJ /F1 11.357 Tf 7.6 0 Td[(stellarabsorptionlinessuperimposingontheWLCs)]TJ /F1 11.357 Tf 7.61 0 Td[(isrecordedonaCCDdetector(illustratedinFig. 5-1 ).Thefringephaseisexpressedbythefollowingequation: (,y)=2(,y) c,(5) 119

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whereyisthecoordinatealongslitdirection,whichistransversetodispersiondirection,istheopticalpathdifference(OPD)ofaninterferometerandcisthespeedoflight.Twomirrors(arms)oftheinterferometeraredesignedtobetiltedtowardseachotheralongtheslitdirectionsuchthatseveralfringesareformedalongeachchannel.TheintersectionofastellarabsorptionlineandaWLCmoves(fromPotoPinFig. 5-1 )ifthereisashiftofanabsorptionlineduetoachangeofstellarRV.Consequently,asmallchangeofinthedispersiondirection,x,isinduced: x=d dvv=d dd dvv=d d cv=)]TJ /F3 11.955 Tf 9.43 0 Td[(v, (5) where)]TJ /F1 11.357 Tf 9.94 0 Td[(isdenedasphase-to-velocityscale(PVscale).ItisdeterminedbytheGDofaninterferometer,whichbecomesexplicitifEquation 5 and 5 arecombined: )]TJ /F6 11.955 Tf 10.1 0 Td[(=)]TJ /F5 11.955 Tf 7.61 0 Td[(2GD c.(5)AtresolutionstypicallyadoptedbytheDFDImethod(5,000R20,000),stellarlines(linewidth0.1)arenotresolvedandameasurementofxisextremelydifcult.Instead,y,phaseshiftalongydirectioncanbemeasured,whichisequaltoxifthecombsgeneratedbyaninterferometerareparalleltoeachother.Thisisagoodapproximationatveryhighordersofinterference.TheadvantageofmeasuringyinsteadofxisseenfromFig. 5-1 ,inwhichthephysicalshiftinthedirectionisampliedinydirection,theamplicationrateisdeterminedbytherelativeanglebetweentheinterferometercombsandastellarabsorptionline.Therefore,yisrelativelyeasiertomeasurecomparedtoxanditismeasuredbyttingawell-sampledperiodicaluxsignalalongtheydirectionintheDFDImethod.Comparedtoconventionalhigh-resolutionEchellemethod,thenumberoffreedomfortheDFDImethodinthettingprocessismuchlessandsmallDopplerphaseshiftcanberelativelyeasierdetectedwithasimplefunctionalform,i.e.,asinusoidalfunction.However,wewanttopointoutthatwhiletheDFDImethodprovidesaboostininstrumentDoppler 120

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sensitivity,theDopplersensitivityisnotstronglydependentontheamplicationratebecauseuxslopedecreasesasamplicationrateincreases,whichnegatesthegainofphaseslope. 5.2GDMeasurementUsingWhiteLightCombs 5.2.1MethodMARVELS(Multi-objectApachePointObservatoryRadialVelocityExoplanetLarge-areaSurvey)instrumentcoversawavelengthrangefrom500nmto570nmandusesapost-dispersivegratingwithaspectralresolutionof11,000afteraxeddelayinterfer-ometer( Geetal. 2009 ).ATh-Aremissionlampandaniodineabsorptioncellserveaswavelengthcalibrationsources.TheinstrumentsetupofMARVELS( Geetal. 2009 ; Wanetal. 2009 )issimilartotheequipmentsthatmeasureGDasdescribedin Kovacsetal. ( 1995 )and Amotchkinaetal. ( 2009 ),inwhichawhitelightinterferometer(WLI)iscombinedwithapost-disperser.However,theOPDisscannedbyamovingpicomotorin Amotchk-inaetal. ( 2009 )whileitisrealizedbytworelativelytiltedarmsintheWLCmethodusingMARVELSinstrument.WLCsaregeneratedbytheinterferometerwhenfedbyawhitelightsource(e.g.,atungstenlamp).(),thephaseofeachfrequencychannel,ismeasuredandthenunwrappedtoremoveambiguityof2.GDisthenderivedbytakingthederivativeof()accordingtoEquation 5 .Fig. 5-2 showsanexampleWLCscreatedbyaninterferometerwithaxeddelayof4mm.Thecombsiscreatedbyaninputcontinuummodulatedwithfrequencyduetocon-structiveanddestructiveinterference.ThephaseisdeterminedbyEquation 5 .ThephasecanbemeasuredwithaFourier-transform-basedalgorithmdescribedby( Rochford&Dyer 1999 ):thesignalH()isobtainedbyrstlyremovingthenegativeFouriercomponentsofF())]TJ /F1 11.357 Tf 7.6 0 Td[(theuxdistributionwithfrequency)]TJ /F1 11.357 Tf 7.6 0 Td[(andthenconductinganinverseFouriertransform.Thephases()areobtainedbycalculatingandunwrappingtheargumentsofH().Fig. 5-3 showstheunwrappedphasemeasuredfromuxdistributioninFig. 5-2 .GDcanbedeter-minedbymeasuringthederivativeofunwrappedphasewithrespecttofrequencyaccordingtoGDdenition(Equation 5 ). 121

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Figure5-2. SimulatedWLCsofaninterferometerwithaxeddelayof4mm.Fluxismodulatedwithfrequencyduetoconstructiveanddestructiveinterference. 5.2.2DataReductionStandardspectroscopyreductionproceduresareperformedwithanIDLdatareductionpipelinededicatedtoMARVELS.Figure 5-4 showsanexampleofnormalizeduxasafunctionoffrequencyforaprocessedspectrum.Azoom-insub-plotshowstheWLCsproducedbyfrequencymodulationoftheinterferometer.Visibility,denedastheratioofhalfofpeak-valleyvaluetotheDCoffset,increaseswithfrequencyintheredpartofthespectrum.Theincreasingvisibilityintheblueendofthespectrumisnotphysicalbutcausedbyanincreasingphotonnoiseandouralgorithmofvisibilitycalculation.ThefringephasesasafunctionofarecalculatedbytheHilberttransformtech-nique( Rochford&Dyer 1999 )describedinx 5.2.1 .Wendthatthephasechangebetween 122

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Figure5-3. PhaseofsimulatedWLCs.PhasecanbecalculatedbyFourier-transform-basedalgorithmdescribedinx 5.2.2 pixelsexceedsinthebluepartandthereforethephaseunwrapcannotbesuccessfullyapplied,sowedecidetouseonlypartofthespectrumwithapixelrangefrom1800to3800forphaseunwrapping.Athird-orderpolynomialisusedtotasafunction.GDisthencalculatedaccordingtoitsdenition(Equation 5 ). 5.2.3GDMeasurementResultsThetopviewandsideviewoftheMARVELSinterferometerareshowninFig. 5-5 .60bersaremountedandeachcreatestwospectra,oneispickedfromtheforwardingbeamandtheotheroneisfromthereturningbeam(seeFig. 5-5 ).Intotal,120spectraareformed,allowingustomeasureGDat60positionsontheinterferometeralongvertical(slit)direction.Eachpositioncorrespondstoabernumber.Thereare24pixelsalongtheslitdirection 123

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Figure5-4. Thenormalizeduxandvisibility()asafunctionoffrequencyofatungstenspectrumtakenwithMARVELS.Thesolidlineisthenormalizeduxandlledcirclesrepresentvisibilitiesindifferentfrequencychannels. foreachspectrum.Wechose15rowsinthemiddletomeasureGDbecauseofrelativelyhigherphotonux,andthussmallerphotonnoiseinthemiddleregionofthespectrum.ThetoppanelofFig. 5-6 showsphasemeasurementresultsforcenterrowasafunctionoffrequencyatdifferentbernumbers.Phasettingresidual(shownonthebottompanelofFig. 5-6 ,RMS=0.9rad)isconsistentwithphoton-noiselimitedmeasurementerror(seex 5.2.4 fordetails).GDforaparticularbernumberisobtainedbyaveragingtheresultsofGDmeasurementsforthoserowsassociatedwiththeber.Figure 5-7 showstheresultsat=550THzasafunctionofbernumber.Notethatthetwoarmsoftheinterferometerareintentionallytiltedtoeachotherandthe60bersareevenlymountedalongtheslit 124

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direction.ThemeasuredGDsshouldgraduallyvarywithbernumber.Weuseasecond-orderpolynomialtottheGDvariationwiththebernumber.ThettingresidualhasanRMSof0.0046ps.Figure 5-8 showsttedGDasafunctionoffrequencyfordifferentbers.GDvaries0.15ps(0.6%)acrossmeasurementrangefrom540to565THz.IgnoringGDdependenceoffrequencywouldresultin180ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1measurementoffsetbetweentwoendsofmeasurementrange(assumingatrueRVof30,000ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,whichisatypicalstellarRVvalueduetotheEarth'sbarycentricmotion).Table 5-1 providesthepolynomialttingcoefcientsofGDvs.bernumberatdifferentfrequencieswithinmeasurementrange. Figure5-5. Top:topviewofanindividualberbeamfeedingoftheMARVELSinterferometer.Twospectraareformedbyoneber.One(SlitA)isfromthereturningbeamarmwhiletheotherone(SlitB)isfromtheforwardingbeamarm.Bottom:sideviewoftheberarraybeamfeedingoftheMARVELSinterferometer.Thereare60bersyielding120spectra.Notetheexaggeratedwedgeangleoftheshownsecondsurfacemirror,GDgraduallychangesalongtheverticaldirection. 125

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Figure5-6. Top:whitelightcombsphaseasafunctionoffrequencyatdifferentberlocations.Bottom:phaseresidualafterthird-orderpolynomialtting. 5.2.4GDMeasurementErrorAnalysisTwophysicalparameters,and,aremeasuredintheexperiment.Theuncertaintyofthemeasurementis0.8radunderphoton-noiselimitedconditionassumingaS/Nof120andatypicalfringevisibilityof1.5%.Theuncertaintyduetothewavelengthcalibrationis0.002THz(0.02).WeconductabootstrappingprocesstoinvestigatetheuncertaintyofGDcausedbythemeasurementuncertaintiesofand.Weaddgaussiannoiseswithstandarddeviationofmeasurementerrorstobothandandcalculatethegroupdelay.Werun1000iterationsforbootstrappinginordertoestimatetheuncertaintyofGD.ThemedianoftherelativeerrorofGDmeasurements,GD=GD,is4.410)]TJ /F5 7.97 Tf 5.07 0 Td[(5.Incomparison,themedianoftherelativeGDmeasurementerroris1.810)]TJ /F5 7.97 Tf 5.07 0 Td[(4aftersmoothingbytting 126

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Figure5-7. Measuredgroupdelayasafunctionbernumber.Filledcirclesaremeasuredresults,solidlinerepresentsthebestsecond-orderpolynomialttingwithanRMSttingerrorof0.0046ps.MeasurementresultscanbefoundinTable 5-1 atotherfrequencies. apolynomialtoGDvariationwiththebernumber.ThisnumberdoesnotagreewiththerelativeGDerrorpredictedbythebootstrappingexperiment.Wesuspectthattheuncountederrorinthebootstrappingsimulationcomesfromimagedistortionduetoopticswhichthedatapipelinehasnotfullycorrectedfor,e.g.,spectrumcurvature,spectrallineslant,etc.Ina2-DspectrumasillustratedinFig. 5-1 ,thephaseshiftbetweenadjacentpixelsalongslitdirectionis0.6rad,andthephaseshiftbetweeneachwavelengthchanelis2.5rad.Animperfectspectrumcurvaturetracingtendstoshiftpixelintheslitdirectionwhileanimperfectslantcorrectioncanaffectpixelshiftinginbothslitanddispersiondirections.Therangeofunwrappedphaseis4000rad.Foroneber,ifagradually-changingphaseerror 127

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Figure5-8. GDasafunctionoffrequencyatdifferentbernumbers.GDmeasurementresultsvs.frequencyandbernumberscanbefoundinTable 5-1 isintroducedbythedatapipelinewhencorrectingfortheopticaldistortion,forexample,0.4raddeviationfromtruevalueatoneendwhilenodeviationattheotherend,arelativeerrorofGDwouldbecausedwithanestimationof0.4=4000=110)]TJ /F5 7.97 Tf 5.07 0 Td[(4.Ifdifferentbersaretreatedindependently,whichisthecasefortheMARVELSdatareductionpipeline,thenthisgradually-changingphaseerror,introducedbyimperfectopticaldistortioncorrection,mayexplainthestandarddeviationerrorweseeafterthepolynomialttingforGDsasafunctionofbernumber. 128

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Table5-1. GDmeasurementresultsasafunctionofspectrumnumber(GD(#)=C0+C1#+C2#2)andstandarddeviation(GD)atdifferentfrequencies() [THz]C0C1C2GD[ps] 540.0000-2.5195975233e+014.1224184409e-03-1.3537701123e-050.0066542.0000-2.5216854974e+014.8064043246e-03-2.3438315106e-050.0070544.0000-2.5219400268e+013.8524896642e-03-8.1355543995e-060.0057546.0000-2.5232735338e+013.5346857088e-03-2.6247780042e-060.0055548.0000-2.5246276599e+013.5171101494e-03-3.3394045215e-060.0047550.0000-2.5259435353e+013.3877963986e-03-1.7477210448e-060.0046552.0000-2.5270775315e+012.8029235840e-039.6338400722e-060.0066554.0000-2.5286851225e+013.5335302125e-03-4.0412480867e-060.0093556.0000-2.5295872014e+013.4000221425e-03-2.5813725398e-070.0067558.0000-2.5303654859e+012.3432563708e-031.8389109388e-050.0087560.0000-2.5319347252e+012.6464319719e-031.2493738694e-050.0075 5.3GDCalibration:ObservinganRVReferenceStar 5.3.1MethodAdeviatedPVscalewouldresultinaninaccuratevelocitymeasurementgiventhesameamountoffringephaseshift: =v0)]TJ /F3 7.97 Tf 6.78 4.94 Td[(0=v,(5)wherev0representsameasuredvelocityshiftwhilevrepresentsatruevelocityshift.CombiningEquation 5 and 5 ,weobtainthefollowingequationfromwhichGDcanbecalculatedbyusingthemeasuredvelocityshiftofanobjectwithaknownvelocity. GD=GD0v0 v.(5)Thisapproachissimilartothatof Barker&Schuler ( 1974 ),butthedifferenceisthatthelatterappliedcorrectionfordiscretelaserfrequencieswhileweseekcorrectionsforacontinuousfrequencydistribution.InordertorealizethemethodofGDcalibrationusinganRVreferencestar,weneedto:1),assumeaGD0thatisclosetothetruevalueofGD;2),measurevelocityshiftv0basedonanassumedGD0;3),knowthetruevalueofthevelocityshiftofanRVreferencestar. 129

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Table5-2. MARVELSpredictedRVuncertainty(atanaverageS/Nof100)vs.Te Te[K]450050005500600065007000 v0[ms)]TJ /F5 7.97 Tf 5.06 0 Td[(1]1.92.32.73.13.54.0 5.3.2GDCalibrationPrecisionThecalibrationprecisionusinganRVreferencestarisdeterminedbymeasurementerrorofv0: GD=GD0v0 v,(5)wherev0isRVmeasurementuncertainty. Wangetal. ( 2011 )providedamethodofcalculat-ingphoton-limitedRVuncertaintyfortheDFDImethod.Underphoton-noiselimitedcondition,weexpecttheGDcalibrationerrortobedeterminedbythephoton-limitedRVuncertaintywithintheinstrumentbandwidth,whichisprovidedinTable 5-2 .InEquation 5 ,GD0isusu-allyestimatedtobewithinafewpercentoftrueGD,visstatistically10,000ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1givenauniformreferencestardistributionandaquarteryearobservationalavailability.Therefore,relativeerrorofGDmeasurementis210)]TJ /F5 7.97 Tf 5.07 0 Td[(4foranRVreferencestarwithaTeof4500K. 5.4ImplementationofMeasuredGDinAstronomicalObservationsWeuseanRVreferencestar,HIP14810(V=8.5),asanexampletoshowtheRVmeasurementresultsafterimplementationofthenewlymeasuredGDusingtheWLCmethod.HIP14810isastarknowntoharbor3planetsanditsRVjitterisestimatedtobe2ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1( Wrightetal. 2009 ).AfterRVchangesduetoinstrumentdrift,theEarth'sbarycentricmotionandorbitingplanetsareremoved,RVRMSerroris17.13ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1buthasnotreachedthepredictedphoton-limitedRVuncertainty(4.8ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1,S/N=80withahalfwavelengthcoveragefrom535to565nm).RVRMSerrorisexpectedtobefurtherreducedafterthedatapipelineisimprovedinthefuture.WealsoexaminethereferencestarGDcalibrationmethod.Weuseonespectralblockwithinmeasurementrangecenteringat550THz(540-560THz)andsetGD0tobeanarbitraryvalueof-23.873ps.ThemeasuredRVs(barycentricvelocitynotcorrected)areshowninFig. 5-9 .AfterapplyingcorrectionaccordingtoEquation 5 ,wendthatthe 130

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GDis-25.1070.027ps.Incomparison,GDmeasurementresultofbernumber51(theberforHIP14810)usingtheWLCmethodgives-25.0910.005ps(refertoTable 5-1 ).WeconrmthattheGDsmeasuredbythesetwomethodsareconsistentwitheachotherat68%signicancelevel. Figure5-9. Top:measured(v0)andtrue(v)RVsofHIP14810(barycentricvelocitynotcorrected)overaperiodof70days.Bottom:theratioofv0andvasafunctionoftime. ManybinaryandbrowndwarfdiscoveriesaremadeowingtothewellcalibratedinterferometerGD.AsofJune2012,morethan250binarieshavebeendiscoveredandadozenofbrowndwarfshavebeendiscoveredbyMARVELSandlaterconrmedbyfollow-upobservationsconductedatotherobservatories( Flemingetal. 2010 ; Leeetal. 2011 ; Wisniewskietal. 2012 ). 131

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Table5-3. ComparisonbetweentwomethodsofGDmeasurementandcalibration WLCRS SpectrumCoverageHalfFullS/N15100forV8aCurrentprecision4.610)]TJ /F5 7.97 Tf 5.07 0 Td[(39.310)]TJ /F5 7.97 Tf 5.07 0 Td[(3ps0.027psCurrentRVerrorb2ms)]TJ /F5 7.97 Tf 5.07 0 Td[(110.8ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1Potentialprecision3.110)]TJ /F5 7.97 Tf 5.07 0 Td[(4ps510)]TJ /F5 7.97 Tf 5.07 0 Td[(3psPotentialRVerrorb0.1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(12ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1DependenceonobservationXDependenceonpipelineXX Note.a:assumingMARVELSthroughput;b:RVerroriscalculatedassumingatruevelocityshift(v)of10,000ms)]TJ /F5 6.974 Tf 4.44 0 Td[(1accordingtoEquation 5 5.5SummariesandDiscussions 5.5.1SummariesThePVscaleisanimportantparameterintheDFDImethodthattranslatesameasuredphaseshifttoanRVshift,andisdeterminedbythegroupdelay(GD)ofaninterferometer.WehaveprovidedanddiscussedtwomethodsofGDmeasurementandcalibration:1),GDmeasurementusingwhitelightcombs(WLCs)generatedbytheinterferometerinaDFDIDopplerinstrument;2),GDcalibrationusinganRVreferencestar(RS).Table 5-3 summarizesthemainresultsandthecomparisonbetweenthesetwomethods.TheaccuracyofGDmeasurementissufcientforcurrentRVprecisionachievedwithinstrumentsusingtheDFDImethod( Flemingetal. 2010 ; Leeetal. 2011 ; Muirheadetal. 2011 ).However,highermeasurementandcalibrationprecisionisrequiredinthenearfutureashigherRVprecisionisachievedbyDFDIinstrumentsinsearchforexoplanets.RSandWLCmethodscanserveascomplementarymethodsofGDmeasurementandcalibrationforDFDIinstruments. 5.5.2Discussions 5.5.2.1WhiteLightComb(WLC)MethodTheGDmeasurementusingWLCscreatedbytheinterferometerprovidesadirectwayofcalibratingthePVscale.Intheregionwherecombsarevisible,effectiveS/Nisrelatively 132

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low(15)becauseoflowcombvisibility(1.5%).Inaddition,GDcannotbemeasuredintheregionwherecombsarenotvisible,whichlimitstheapplicationofthismethod.WeareabletomeasureGDinaregionthataccountsforhalfofthespectrumcoverage.Extrapolationbeyondthemeasurementrangemayresultinlargeuncertainties.Themajorissuefacingthemethodisthatthedatareductionpipelinemayhaveintroducedunknownerrorswhilecorrectingopticaldistortionssuchasspectrumcurvatureandslant.Inprinciple,wecanuseatungstenlampwithaniodinecelloraTh-Arlampinsteadofatungstenlampinordertoincreasethefringevisibility.However,therearesomepracticalconcernsthathinderusfromapplyingtheabovesolutions:1)lineblending,becauseoflowspectralresolution,manyspectrallinescannotberesolvedanditisnotcertainatthisstagehowlineblendingaffectsphasemeasurement;2)illuminationcorrection,whichisrequiredtocorrectforilluminationproleinslitdirectioninordertoproperlymeasurethephase.Weadoptaself-illuminationcorrectionprocedureinthepipelinewhichrequiresacertaincontin-uumleveltobesuccessfullyachieved.Th-Arislessaffectedbylineblendingifacarefullineselectionprocessisinvolved,butitdoesnothaveenoughcontinuumlevelforself-illuminationcorrection.AnexperimentisbeingconductedinwhichasecondinterferometerisusedtoimprovethevisibilityofWLCssothatGDismorepreciselymeasuredatahighereffectiveS/Nforawiderfrequencycoverage.WhencomparedtopreviousworkintheeldofGDmeasurement, Amotchkinaetal. ( 2009 )achievedameasurementprecisionof110)]TJ /F5 7.97 Tf 5.06 0 Td[(4ps.OurmeasurementofGDhasatypicalaccuracyof610)]TJ /F5 7.97 Tf 5.07 0 Td[(3ps(limitedbyeffectiveS/Nandsystematicerrors),whichismorethananorderofmagnitudelower.However,itisshowninx 5.2.4 thatsubstantialimprovementwouldbeabletobeachievedoncethedatareductionpipelinehasabetterhandleofopticaldistortion. Wanetal. ( 2010 )measuredGDfortheMARVELSinterferometerusingascanningWLImethodandachievedaprecisionof0.610)]TJ /F5 7.97 Tf 5.07 0 Td[(52.410)]TJ /F5 7.97 Tf 5.07 0 Td[(5ps,whichismorethantwoorderofmagnitudebetterthantheresultsinthispaper.However,therearepracticalconcernsusingGDmeasurementresultsfromthescanningWLImethod 133

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becausetheyarenotmeasuredinsitu,thereforeitisnoteasytoassociateapositioninaWLImeasurementtoaberposition. 5.5.2.2ReferenceStar(RS)MethodTheGDcalibrationusinganRVreferencestar(RS)isaself-calibratingprocessandhasthepotentialofachievingahighcalibrationprecisionifthefollowingrequirementsaremet:1),theRVreferencestarhasalargevelocityshiftduringaobservationwindow;2),theRVreferencestarisbright;3),thedatareductionpipelineisabletoproducethephoton-limitedRVprecision.Inaddition,GDispracticallymeasuredwithinacertainbandwidth: GD()=RGD()!()d R!()d,(5)where!()isweightfunction.Thebandwidth,,shouldbesmallsuchthatthedispersioneffectisnegligible.ThelimitationsstatedabovepreventusfrompreciselydeterminingthePVscaleatthepositionofeachberbecausenoteveryberhasacontinuousobservationonabrightknownRVreferencestar.ForMARVELS,thebrightestRVreferencestaravailablehasVmagof8andtheresultingS/Nis100perpixel.However,theRSmethodisaverypromisingapproachforasingle-objectDFDIinstrumentbecauseonlyonebrightreferencestarisrequiredintheeld.WeareplanningtoapplythismethodincalibratingGDofanotherDFDIinstrument(EXPERT)atKPNO2.1mtelescope( Geetal. 2010 ).NotethattheS/Ncanbefurtherincreasedbyconductingmultipleindependentmeasurementsandincreasinginstrumentthroughput. 5.5.2.3AFutureM-DwarfSurveyWiththeDFDIMethodTheMARVELSsurveyconceptcanbeadoptedbyafutureM-dwarfplanetsurveyusingtheDFDImethod.AstheplanetsaroundMdwarfsaregainingincreasinglymoreattention,asurveyofnearbyMdwarfsisimportantinordertocompletelyunderstandplanetoccurrencerateforMdwarfsanditsdependencyonstellarmetallicity,stellarmassandactivityandsoon.ThistypesurveyisidealforlargeareaspectroscopictelescopessuchtheSDSS2.5mtelescopeandtheLAMOSTtelescope.Thereareabout15,000mid-latetypeMdwarfs 134

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withJmagnitudelessthan12inthenorthernsky( Wangetal. 2011 ).Consideringabout6squaredegreeeldofviewforthesetelescopes,thereare5suchstarsintheeldofview.Accordingtopreviousstudyonthemulti-objectcase,aresolutionof5,000wouldbeoptimalforamulti-objectDFDIsurvey(seex 2.3.4 ).Atsuchresolution,weexpect125m/sRVprecisionforaJ=12starand8m/sforaJ=6star.Withthisdesignconceptandinstrumentspecication,a1Kby1Knearinfrareddetectorifsufcientfor800to1350nmwavelengthcoverage. 135

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CHAPTER6ECCENTRICITYDISTRIBUTIONFORSHORT-PERIODEXOPLANETS 6.1IntroductionThediscoveryofexoplanetshassignicantlyadvancedourunderstandingofformationandevolutionofplanetarysystem( Marcy&Butler 1996 ; Mayor&Queloz 1995 ; Wolszczan 1994 ).AsofFebruary2011,over500exoplanetshavebeendiscoveredincluding410sys-temsdetectedbyradialvelocity(RV)technique1.Theeccentricitydistributionofexoplanetsisverydifferentfromthatofsolarsystem.Forsufcientlyshort-periodplanets,itisexpectedthattidalcircularizationwouldleadtonearlycircularorbits.Yet,severalshort-periodplanetsappeartohaveeccentricorbits.Severalmechanisms(e.g.planetscattering,Kozaieffect)havebeenproposedtoexplaintheobservedeccentricitydistribution( Ford&Rasio 2008 ; Juric&Tremaine 2008 ; Takeda&Rasio 2005 ; Zhou&Lin 2007 ).Thischapteraimstoimproveourunderstandingofthetrueeccentricitydistributionanditsimplicationsfororbitalevolution.TheBayesianapproachoffersarigorousbasisfordeterminingtheposterioreccentricitydistributionforindividualsystem.TheBayesianmethodisparticularlyadvantageousrelativetotraditionalbootstrapmethodwhentheorbitaleccentricityispoorlyconstrainedbyRVdata( Ford 2006 ). Ford ( 2006 )discussedeccentricityestimationusingMarkovChainMonteCarlo(MCMC)simulationintheframeworkofBayesianinferencetheoryandfoundaparametersetthatacceleratesconvergenceofMCMCforloweccentricityorbit.Forapopulationofplanetsonnearlycircularorbits,eccentricityestimatesforplanetsoncircularorbitarebiasedresultinginoverestimationoforbitaleccentricities( Zakamskaetal. 2011 ).Furthercomplicatingmatters,thepopulationofknownexoplanetsisnothomogeneous,andtheobservedeccentricitydistributionisaffectedbythediscoverymethod,selectioneffectsanddataanalysistechnique. 1http://exoplanet.eu/;http://exoplanets.org/ 136

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6.2MethodWeselectallthesystemswith:1)asingleknownplanetdiscoveredwiththeradialvelocitytechniqueasofApril2010;2)anorbitalperiodoflessthan50days;and3)apubliclyavailableradialvelocitydataset.Weexcludeplanetsdiscoveredbythetransittechniqueinordertoavoidcomplicationsduetoselectioneffects( Gaudietal. 2005 ).Weperformanorbitalanalysisoneachsysteminoursampleusing:1)astandardMCMCanalysis(x2.1)and2)anewmethod,)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis(describedinx2.2).Wefocusontheeccentricityestimationforeachplanetsincetheeccentricityisanimportantindicationoforbitalevolutionandtidalinteraction. 6.2.1BayesianOrbitalAnalysisofIndividualPlanetWeperformedaBayesiananalysisofthepublishedradialvelocityobservationusingamodelconsistingofonelow-masscompanionfollowingaKeplerianorbit.Ifalong-termRVtrendisincludedintheoriginalpaperreportingtheRVdataorifalineartrendofmorethan1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1yr)]TJ /F5 7.97 Tf 5.07 0 Td[(1isapparent,thenweaddtothemodelaconstantlong-termaccelerationduetodistantplanetaryorstellarcompanion.WecalculateaposteriorsampleusingtheMarkovChainMonteCarlo(MCMC)tech-niqueasdescribedin Ford ( 2006 ).EachstateintheMarkovchainisdescribedbytheparameterset~=fP,K,e,!,M0,Ci,D,jg,wherePisorbitalperiod,Kisthevelocitysemi-amplitude,eistheorbitaleccentricity,!istheargumentofperiastron,M0isthemeananomalyatchosenepoch,Ciisconstantvelocityoffset(subscriptiindicatesconstantfordifferentobservatory),Distheslopeofalong-termlinearvelocitytrend,andjisthejitterparameter.Thejitterparameterdescribesanyadditionalnoiseincludingbothastrophysicalnoises,e.g.,stellaroscillation,stellarspots( Wrightetal. 2005 )andanyinstrumentnoisenotaccountedforinthereportedmeasurementuncertainties.TheRVperturbationofahoststarattimetkduetoaplanetonKeplerianorbitandpossibleperturbationisgivenby vk,~=K[cos(!+T)+ecos(!)]+D(tk)]TJ /F8 11.955 Tf 10.26 0 Td[(),(6) 137

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whereTisthetrueanomalywhichisrelatedtoeccentricanomalyEviatherelation, tan T 2!=r 1+e 1)]TJ /F4 11.955 Tf 10.26 0 Td[(etan E 2!.(6)TheeccentricanomalyisrelatedtothemeananomalyMviaKepler'sequation, E(t))]TJ /F4 11.955 Tf 10.26 0 Td[(esin[E(t)]=M(t))]TJ /F4 11.955 Tf 10.26 0 Td[(M0=2 P(t)]TJ /F8 11.955 Tf 10.26 0 Td[().(6)Wechoosepriorsofeachparameterasdescribedin Ford&Holman ( 2007 ).Thepriorisuniforminlogarithmoforbitalperiod.ForKandjweuseamodiedJefferyspriorintheformofp(x)/(x+xo))]TJ /F5 7.97 Tf 5.06 0 Td[(1( Gregory 2005 )withKmin=j,min=0.1ms)]TJ /F5 7.97 Tf 5.06 0 Td[(1.Priorsareuniformfor:e(0e1),!andM(0!,M<2),CiandD.Weveriedthattheparametersin~didnotapproachthelimitingvalues.Weassumeeachobservationresultsinameasurementdrawnfromnormaldistributioncenteredatthetruevelocity,resultinginalikelihood(i.e.,conditionalprobabilityofmakingthespeciedmeasurementsgivenaparticularsetofmodelparameters)of p(~vj~,M)/Ykexp[)]TJ /F5 11.955 Tf 7.6 0 Td[((vk,~)]TJ /F4 11.955 Tf 10.26 0 Td[(vk)2=22k] k,(6)wherevkisradialvelocityattimetk,andvk,isthemodelvelocityattimetkgiventhemodelparameters~.Noisekconsistsoftwoparts.Onecomponentisfromtheobservationuncertaintyk,obsreportedintheradialvelocitydata,andtheotheristhejitter,j,whichaccountsforanyunforseenadditionalnoiseincludinginstrumentinstabilityandstellarjitter.Thetwopartsareaddedinquadratureinordertogeneratek.WecalculatetheGelman-Rubinstatistic,R,totestfornonconvergenceofMarkovchains.WeperformaMCMCanalysisforRVdatasetofeachsysteminthesampleandobtainposteriorsamplesofhandk,whereh=ecos!andk=esin!.Thisparameterizationhasbeenshowntobemoreeffectiveindescriptionoftheeccentricitydistributionforloweccentricorbits( Ford 2006 ).Wetakestepsinhandkandadjusttheacceptancerate 138

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accordingtotheJacobianofthecoordiantetransformation,soastomaintainapriorthatisuniformineand!.Meanvalues,handk,fromposteriorsamplesofhandkareadoptedtocalculateeMCMCusingtheequationeMCMC=p h2+k2.TheposteriordistributionofeisnotalwaysGaussiandistributionespeciallyneare0.Therefore,itisnotappropriatetocalculatetheuncertaintyofeusingtheequationoferrorpropagationinwhichgaussiannoiseisassumed.Weuseposteriordistributionofetoinferthecredibleintervalofe.Theboundariesoftheregionwhere68%posteriorsamplespopulateareadoptedaselowerandeupper(Table 6-1 ).Figure 6-1 illustratestwoexamplesofhowthecredibleintervalsareinferredforHD68988(eccentricorbit)andHD330075(circularorbit). Figure6-1. ExamplesofhowcredibleintervalsofstandardMCMCanalysisarecalculatedusingposteriordistributionofe.Greyregioncontains68%oftotalnumberofposteriorsamplesofe. 139

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6.2.2)]TJ /F12 11.357 Tf 9.93 0 Td[(AnalysisofIndividualSystemsAfullyBayesiananalysisofthepopulationofexoplaneteccentricitieswouldbecom-putationallyprohibitiveduetothelargenumberofdimensions.Therefore,wedevelopahybridBayesian-frequentistmethodtoevaluatethesignicanceofanon-zeroeccentricitymeasurement.WecombineabootstrapstyleapproachofgeneratingandanalyzingsyntheticdatasetswithMCMCanalysisofeachsyntheticdatasettoobtainafrequentistcondencelevelforeacheccentricitythataccountsforbiases.First,weperformthestandardMCMCanalysisdescribedinx2.1ontherealRVdatasetandadoptthemeanvalueofeachorbitalparameterin~excepte.Wegenerateaseriesofsimulatedradialvelocitydatasetsatdif-ferentvaluesofe.TheadoptedKisscaledaccordinglytoK/(1)]TJ /F4 11.955 Tf 10.69 0 Td[(e2))]TJ /F5 7.97 Tf 5.07 0 Td[(0.5.Thesimulatedradialvelocitydatahasthesamenumberofobservations,andeachsimulatedobservationtakesplaceatexactlythesametimeandthesamemeananomalyastherealobservation.Gaussiannoisewithstandarddeviationofk(x2.1)areaddedtosimulatedradialvelocitydatasetsatdifferenteccentricities.EachsimulatedRVdatasethasthesamereportedRVmeasurementuncertaintiesastherealRVobservations.StandardMCMCanalysisisthenperformedoneachofthesimulatedRVdatasets.Forbothrealandsimulateddatasets,weconstructatwo-dimensionalhistogramusingtheposteriorsamplesin(h,k)spacetoapproximateatwo-dimensionalposteriordistributionforhandk,dr(hi,kj)andds(hi,kj),whereiandjdenotebinindices,andthesubscriptsrandsdenotetherealandsimulateddataset.Wecomparethedistributionforeachsimulateddatasetds(hi,kj)tothedistributionforrealradialvelocitydatasetdr(hi,kj).Toquantifythesimilaritybetweends(hi,kj)anddr(hi,kj),wecalculatethestatisticdenedas)]TJ /F6 11.955 Tf 12.11 0 Td[(=[NXi=1NXj=1(ds(hi,kj))]TJ /F8 11.955 Tf 10.26 0 Td[(s)(dr(hi,kj))]TJ /F8 11.955 Tf 10.26 0 Td[(r)]=[sr(N2)]TJ /F5 11.955 Tf 10.26 0 Td[(1)],whereNisthenumberofbinsinhorkdimension,andrepresentmeanandstandarddeviation.Inotherwords,the)]TJ /F1 11.357 Tf 9.94 0 Td[(statisticisobtainedbycross-correlatingtwoposteriordistributionsinhandkspace.Figure 6-2 illustratestheprocessbywhichweobtain)]TJ /F1 11.357 Tf 9.93 0 Td[(forthecaseofHD68988.Ifds(hi,kj)matchesdr(hi,kj),weexpecttoobtaina)]TJ /F1 11.357 Tf 9.93 0 Td[(valuethatapproachesunity(blue 140

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andredcontours).Ifthesamplesdiffersignicantlythen)]TJ /F1 11.357 Tf 9.94 0 Td[(decreasestowardszero(redandgreencontours).Foreacheccentricity,wesimulated21radialvelocitydatasetsandcompareds(hi,kj)withdr(hi,kj)toobtain21)]TJ /F1 11.357 Tf 9.93 0 Td[(statisticsbetweensimulatedandrealRVdata.Wechoosethemedianvalue)]TJ /F1 11.357 Tf 9.93 0 Td[(asanindicatorofoverallsimilarityatgiveneccentricity. Figure6-2. ContoursofposteriordistributioninhandkspaceforHD68988(Solid-68%ofsamplepointsincluded;dashed-95%ofsamplepointsincluded;dotted-99%ofsamplepointsincluded).Redcontoursareposteriordistributionforrealobservation,greencontoursareforsimulatedRVdatasetwithe=0.00,andbluecontoursareforsimulatedRVdatasetwithe=0.13.EccentricityofHD68988is0.12500.0087accordingto Butleretal. ( 2006 ). Basedonaboveanalysisofsimulatedradialvelocitydatawithdifferentinputeccentric-itiese,weobtainarelationshipbetween)]TJ /F1 11.357 Tf 9.93 0 Td[(ande,i.e.)]TJ /F5 11.955 Tf 6.77 0 Td[((e).Weuseahigh-orderpolynomialtointerpolatefor)]TJ /F5 11.955 Tf 6.77 0 Td[((e).Wedeneetobetheeccentricityatwhich)]TJ /F5 11.955 Tf 6.78 0 Td[((e)reachesmaximumandweinterpreteasanestimatorofeccentricity.ForHD68988(Fig. 6-3 ),inputeccentricity 141

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rangesfrom0.00to0.29withstepsizeof0.01.)]TJ /F5 11.955 Tf 6.78 0 Td[((e)reachesmaximumate=0.134.Weestimatestatisticalcondencelevelofeusingeverypairofposterioreccentricitysamplescalculatedfromthedatasetsthataregeneratedassumingthesameeccentricity.ConsidertheexampleofHD68988again:21setsofposteriordistributioninhandkspace,ds(hi,kj),areobtained.Comparisonbetweeneachpairgivesa)]TJ /F1 11.357 Tf 9.93 0 Td[(statisticbetweensimulatedRVdata.20i=1=210)]TJ /F1 11.357 Tf 9.93 0 Td[(statisticsintotalforsimulatedRVdatasetsarecalculatedateccentricityof0.13and68.1%ofpairs(143outof210)havea)]TJ /F1 11.357 Tf 9.93 0 Td[(statisticgreaterthan0.3714.Wedenethisvalue,)]TJ /F11 7.97 Tf 6.78 -1.79 Td[(c,0.68,asthecritical)]TJ /F1 11.357 Tf 9.93 0 Td[(valueforHD68988at68%condencelevelfore=0.13.Therefore,if)]TJ /F1 11.357 Tf 9.94 0 Td[(statisticobtainedincomparisonbetweends(hi,kj)anddr(hi,kj)islessthan)]TJ /F11 7.97 Tf 6.77 -1.79 Td[(c,0.68,wearguethattheeccentricityinferredfromsimulatedRVdatasetisnotconsistentwiththeobservedeccentricityofthesystemat68%condencelevel.Inthecasewhereeislocatedbetweengridsofsimulatedevalues,wecalculate)]TJ /F11 7.97 Tf 6.78 -1.79 Td[(c,0.68ateusinginterpolationof)]TJ /F11 7.97 Tf 6.77 -1.79 Td[(c,0.68atnearbyevalues.Weuseahigh-orderpolynomialtoapproximatethediscretedata)]TJ /F5 11.955 Tf 6.77 0 Td[((e).Thepolynomialislaterusedtoinfere,lowerandupperlimitofeccentricity.ForHD68988,)]TJ /F11 7.97 Tf 6.77 -1.8 Td[(c,0.68is0.3663ate=0.134afterinterpolation.Usingtherelationshipbetween)]TJ /F1 11.357 Tf 9.93 0 Td[(ande(Fig. 6-3 ),welookfortheevaluescorrespondingto)]TJ /F11 7.97 Tf 6.77 -1.8 Td[(c,0.68asestimatorsofthelowerandupperlimitforeccentricityoftheplanetsystemata68%condencelevel.InHD68988,weobtainede=0.134+0.040)]TJ /F5 7.97 Tf 5.07 0 Td[(0.040using)]TJ /F1 11.357 Tf 9.94 0 Td[(analysis.Incomparison,wehaveobtainede=0.119+0.025)]TJ /F5 7.97 Tf 5.07 0 Td[(0.022usingastandardMCMCanalysisand Butleretal. ( 2006 )reportede=0.1250.009. 6.3ResultsforIndividualPlanetsInoursampleof50short-periodsingle-planetsystems,wesuccessfullyanalyzed42systemsusing)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis,and46systemsusingstandardMCMCanalysis(Table 6-1 ).Alltheerrorarebasedona68%condencelevel()]TJ /F1 11.357 Tf 9.94 0 Td[(analysis)ora68%credibleinterval(MCMC).TheunsuccessfulcasesinstandardMCMCanalysisareHD189733,HD219828,HD102195andGJ176.InHD189733,mostoftheRVdatapointsweretakenduringobservationofRossiter-McLaughlineffect,whichisnotmodeledhere.MCMCanalysisfailsforHD219828andGJ176becauseoflowsignaltonoiseratio(K=7ms)]TJ /F5 7.97 Tf 5.06 0 Td[(1andnobs=20 142

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Figure6-3. )]TJ /F1 11.357 Tf 9.93 0 Td[(asafunctionofeccentricityeforHD68988.Opencirclesareresultsfromsimulations,solidlineistheresultofpolynomialtting.Thelongdashedlineisthecriticalthreshold,)]TJ /F11 7.97 Tf 6.78 -1.8 Td[(c,0.68atthe68%condencelevel. forHD219828andK=4.1ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1andnobs=57forGJ176).RVdatapointsofHD102195weretakenat3observatoriesandMCMCanalysisiscomplicatedusingdifferentobservatoryoffsets.Inadditiontotheabove,)]TJ /F1 11.357 Tf 9.94 0 Td[(analysiswasunsuccessfulforHD162020,GJ86,HD17156andHD6434.Since)]TJ /F1 11.357 Tf 9.94 0 Td[(analysisinvolvesgeneratingsimulatedRVdata,thelimitednumberofobservationsandpartialphasecoverageforthesesystemscancausepoorconvergenceforsomesimulateddatasets.Theselimitationsbecomemoresevereforsystemswithhigheccentricity(e.g.HD17156,e=0.684)sincephasecoverageismoreimportantforeccentricorbits. 143

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Figure6-4. Top:comparisonbetweenstandardMCMCanalysis(blue)andpreviousreferences(red);Bottom:comparisonbetweenstandardMCMCanalysis(blue)and)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis(green).Thesystemswheredisagreementstakeplacearemarkedwithanumber:1,HD149026;2,Boo;3,HD195019. 6.3.1Comparison:StandardMCMCandReferencesInFig. 6-4 (top),wecompareresultsfromtwosources,standardMCMCanalysisandpreviousreferences.Inmostcasesthetwomethodsprovidesimilarresults.Wefoundthereare3systemsforwhichtheeccentricityestimatesarenotconsistent,i.e.,thepublishedeccentricityerrorbardoesnotoverlapthe68%credibleintervalfromourMCMCanalysis.TheyareHD149026,Boo,HD195019.ForHD149026, Satoetal. ( 2005 )settheeccentricitytobezerowhenttingtheorbit.Incontrast,wetreateccentricityasavariableandthestandardMCMCmethodfoundthatthe68%credibleintervalforthesystemsmentionedabovedoesnotincludezero.Inaddition,HD149026bisaknowntransiting 144

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planetandtransitphotometryprovidesadditionalconstraintsoneccentricitywhichwehavenotincluded( Charbonneau 2003 ). Knutsonetal. ( 2009 )measuredt=20.9+7.2)]TJ /F5 7.97 Tf 5.07 0 Td[(6.2minute(2.9)forHD149026whichisinconsistentwithzeroeccentricity,becauseeecos!' 2Pt,wherePisperiodandtisthedeviationofsecondaryeclipsefrommidpointofprimarytransits.StandardMCMCresultsforothertwosystems(BooandHD195019)arenotconsistentwiththosefrompreviousreferenceseventhougheccentricitywastreatedasvariableinpreviousreferences. Butleretal. ( 2006 )reporte=0.0230.015forBooande=0.0140.004forHD195019.Onthecontrary,standardMCMCanalysisgivese=0.0787+0.0382)]TJ /F5 7.97 Tf 5.07 0 Td[(0.0246forBooande=0.0017+0.0049)]TJ /F5 7.97 Tf 5.07 0 Td[(0.0017forHD195019(SeeTable 6-1 ). 6.3.2Comparison:StandardMCMCand)]TJ /F12 11.357 Tf 9.93 0 Td[(AnalysisFigure 6-4 (bottom)comparestheresultsfromstandardMCMCanalysisand)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis.Wendthatthe68%credible/condenceintervalsforthetwomethodsoverlapinthecaseswheretherearediscrepanciesbetweenstandardMCMCanalysisandpreviousreferences(seex 6.3.1 ).Thecondenceintervalfromthe)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisisgenerallylargerthanthecredibleintervalfromastandardMCMCanalysis.Thelargeruncertaintyfrom)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisislikelyduetotheanalysisaccountingfortheuncertaintyineachvelocityobservationtwice,rstwhengeneratingsyntheticdatasetsandasecondtimewhenanalyzingthesimulateddata.Thus,the)]TJ /F1 11.357 Tf 9.94 0 Td[(analysisresultsinslightlylargeruncertaintyineccentricityestimation.InordertounderstandthebehaviorofstandardMCMCanalysisand)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisforplanetsonnearlycircularorbits,weconductanadditionalexperimentgeneratingmanysyntheticdatasetswhereeachsystemhasasingleplanetonacircularorbit.WeassumethattheyareobservedatthesametimesandwiththesameRVmeasurementprecisionsasactualRVdatasets.Inordertounderstandthebiasofeachmethodfornearlycircularsystems,wecomparetheoutputeccentricitiesandtheiruncertainties.UsingstandardMCMCanalysis,wendthat76.42.9%ofthesimulateddatasetsareconsistentwithzerousinga68%credibleinterval,and23.61.6%ofthesimulateddatasetshave68%credibleintervalsthatdonotincludezero.Incontrast,for16.81.4%ofthesimulateddatasets,the 145

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)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisdoesnotresultina68%condenceintervalthatincludeszero.Inbothcases,morethan68%ofdatasetsareconsistentwithacircularorbitata68%levelusingeithermethod.Usingthe)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis,6.83.0%moresimulateddatasetsareconsistentwithacircularorbitthanbasedonthestandardMCMCanalysis.Thisconrmsourintuitionthatthe)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisisalessbiasedmethodforanalyzingsystemsatverysmalleccentricity.Thus,the)]TJ /F1 11.357 Tf 9.93 0 Td[(analysismaybeausefultoolinassessingthesignicanceofameasurementofasmallnon-zeroeccentricity.Inparticular,wend5cases(11.4%)inwhichelower=0for)]TJ /F1 11.357 Tf 9.93 0 Td[(analysiseventhoughelowerforMCMCisgreaterthanzero(e.g.,HD46375,HD76700,HD7924,HD168746,HD102117).Asimilarexperimentisconductedexceptthataneccentricityof0.2isassignedtoeachsysteminsteadofzeroeccentricity.Again,weassesstheaccuracyofthetwomethodsbycomparingtheinputandoutputeccentricities.WhenusingtheMCMCmethod,wendthatthe68%credibleintervalfortheeccentricitydoesnotincludetheinputeccentricityfor26.02.1%ofsimulateddatasets.Whenusingthe)]TJ /F1 11.357 Tf 9.93 0 Td[(method,wendthatthe68%condenceintervalfortheeccentricitydoesnotincludetheinputeccentricityfor18.91.8%ofsimulateddatasets.Again,thereisalargerfractionofresultsfromtheMCMCmethodthatarenotconsistentwiththeinputatasizableeccentricity,indicating)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisislesslikelytorejectthecorrecteccentricity.Wealsoinvestigatethebiasofthetwomethodsatasignicanteccentricity(i.e.e=0.2).Inthecaseswheretheoutputeccentricitiesareconsistentwiththeinput,wendthat47.93.3%oftheoutputeccentricitiesarebelow0.2while52.13.5%ofoutputsareover0.2forMCMCmethodindicatingtheMCMCmethodisnotbiasedatasizableeccentricity,whichagreeswiththendingfrom Zakamskaetal. ( 2011 ).Incomparison,)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisisalsoanunbiasedanalysiswith49.93.3%belowinputand50.13.3%exceedinginput.Therefore,wendnoevidenceforsignicantbiasofeithermethodfordatasetswithasignicanteccentricity. 146

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6.3.3Discussionof)]TJ /F12 11.357 Tf 9.94 0 Td[(AnalysisThedifferentmethodsforanalyzingDopplerobservationsarecomplimentary.BayesianmethodsandMCMCinparticularareroutinelyusedtosamplefromtheposteriordistributionfortheKeplerianorbitalparametersforagivensystem.However,theanalysisofanexo-planetpopulationismorecomplicatedthansimplyperformingaBayesiananalysisofeachsystem.Toillustratethispoint,considerapopulationofplanetsthatallhaveexactlycircularorbits.Duetomeasurementuncertaintiesandnitesampling,thebest-teccentricityforeachsystemwillbenon-zero.Similarly,sinceeccentricityisapositive-denitequantity,theanalysisofeachsystemwillresultinaposteriordistributionthathassignicantsupportfore>0.Thispropertyremainsevenifonecombinesmanypointestimates(e.g.,best-t),frequentistcondenceintervalsorBayesianposteriordistributions.Whiletheposteriordistributionfortheorbitalparametersrepresentsthebestpossibleanalysisofanindividualsystem,theinevitablebiasfornearlycircularorbitsisapotentialconcernforpopulationanalyses.Therefore,itisimportanttoapplydifferentmethodsforpopulationanalyses(e.g., Hoggetal. ( 2010 ); Zakamskaetal. ( 2011 )).Sinceweintendtoinvestigatethepotentialroleoftidaleffectsontheeccentricitydistributionofshort-periodplanets,wedevelopedahybridtechniquetoassessthesensitivityofourresultstobiasintheposteriordistributionforplanetswithnearlycircularorbits.Thishybridtechnique()]TJ /F1 11.357 Tf 9.93 0 Td[(analysis)involvesperformingBayesiananalysesofeachindividualplanetarysystemalongwithseveralsimulateddatasets,eachofvaryingeccentricity.TheMCMCanalysisofeachdatasetallowsustoaccountforthevaryingprecisionofeccentricitymeasurementsdependingonthevelocityamplitude,measurementprecision,numberofobservationsandphasecoverage.Weassesstheextentoftheeccentricitybiasesbyperformingthesameanalysisonsimulateddatasetswithknowneccentricity.Wecomparetheposteriordensitiesfortheactualdatasettotheposteriordensityforeachofthesimulateddatasetstodeterminewhichinputmodelparametersareconsistentwiththeobservations.WecanconstructfrequentistcondenceintervalsbasedonMonteCarlo 147

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simulations(i.e.,comparingtheposteriordistributionsforthesyntheticdatasetstoeachother).Thebasicapproachofthe)]TJ /F1 11.357 Tf 9.94 0 Td[(analysisissimilartolikelihood-freemethodsmorecom-monlyusedinapproximateBayesiancomputation.Inthiscase,wedohavealikelihoodwhichallowsustosamplefromtheposteriorprobabilitydistributionusingstandardMCMC.Wecomparetheposteriordensitiescalculatedforseveralsimulateddatasetstotheposteriordensityoftheactualobservations,soastoassesstheaccuracyandbiasofthestandardMCMCanalysis. Figure6-5. Cumulativedistributionsfunctions(CDFs)ofeccentricitiesfromdifferentmethods.ThesolidredlineisforeMCMC,adoptedfromMCMCmethod(Table 6-1 ).ThedottedredlineissimilartoMCMC,butanyeccentricitywithelowerof0fora95%credibleintervalisassignedto0.Thebluelinesarefor)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis,wherethesolidlineisfore)]TJ /F1 11.357 Tf 8.48 1.79 Td[(fromthe)]TJ /F1 11.357 Tf 9.94 0 Td[(analysis(Table 6-1 )andthedottedlineissimilar,butanyeccentricitywithelowerof0fora95%condenceintervalisassignedto0. 148

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Theproblemofbiasedeccentricityestimatorsfornearlycircularorbitsisfamiliarfrompreviousstudiesofbinarystars.Inparticular, Lucy&Sweeney ( 1971 )investigatedthepossibilityofmistakenlyassigninganeccentricorbittoacircularspectroscopicbinaryduetoinevitablemeasurementerrors.Asmanyspectroscopicbinariesmayhavebeenaffectedbytidalcircularization,theysuggestedassigningacircularorbittoanysystemforwhichtheeccentricitycredibleintervalcontained0.Whenstudyingapopulationofsystemsforwhichcircularorbitsarecommon,thisapproachsignicantlyreducesthechanceoferroneouslyconcludingthesystemhasanon-zeroeccentricity.Oneobviousdisadvantageofthisapproachisthatitwouldleadtoanegativebiasforsystemswheretheeccentricityisoforder=K,whereisthetypicalmeasurementprecisionandKisthevelocityamplitude.Forbinarystars,=Kmaybesmallenoughthatthisisnotasignicantconcern.Forexoplanets,where=Kmaybeassmallas2)]TJ /F5 11.955 Tf 11.16 0 Td[(3,suchaprocedurewouldresultinasignicantnegativebiasformanysystems.The)]TJ /F1 11.357 Tf 9.94 0 Td[(analysisoffersanalternativeapproach,whichmaybeparticularlyusefulwhenanalyzingtheeccentricitydistributionofapopulationofplanetarysystems.Forthesakeofcomparison,weconsideramodernizedversionofthe Lucy&Sweeney ( 1971 )approachwhichisbasedontheposteriordistributionfromastandardMCMCanal-ysisorthecondenceintervalfromour)]TJ /F1 11.357 Tf 9.94 0 Td[(analysis.Weconstructahistogramorcumulativedistributionoftheeccentricitiesforapopulationofsystems,usingasinglesummarystatisticforeachsystem:theposteriormeanforthestandardMCMCanalysisortheeccentricitythatmaximizesthe)]TJ /F1 11.357 Tf 9.93 0 Td[(statistic.Following Lucy&Sweeney ( 1971 ),weadoptaneccentricityofzeroforanysystemforwhichthe95%signicancelevel()]TJ /F1 11.357 Tf 9.93 0 Td[(method)orthe95%credibleinterval(MCMC)includese=0.ThecumulativedistributionfunctionsoftheeccentricitiesusingdifferentmethodsareplottedinFig. 6-5 .Basedonthegeneralized Lucy&Sweeney ( 1971 )approach,81%(70%)oftheshort-periodplanetsystemsinoursampleareconsis-tentwithcircularorbitsusingthe)]TJ /F1 11.357 Tf 9.94 0 Td[(analysis(standardMCMCanalysis).Clearly,the Lucy&Sweeney ( 1971 )approachresultsinalargefractionbeingassignedacircularorbit,largely 149

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duetothechoiceofa95%threshold.Thefractionassignedacircularorbitissensitivetothesizeofthecredibleintervalusedwhendecidingwhethertoseteacheccentricitytozero.Thereisnostrongjusticationforthechoiceofthe95%threshold(asopposedto68%or99.9%threshold)andtuningthethresholdtoagreewithothermethodsnegatestheprimaryadvantageofthe Lucy&Sweeney ( 1971 )method,thatitrequiresnoadditionalcomputa-tions.Therefore,wedonotrecommendusingthe Lucy&Sweeney ( 1971 )approachtolearnabouttheeccentricitydistributionforapopulationwhen=Kisnotlarge. 6.4TidalInteractionBetweenStarandPlanetSeveralfactorsaffecttheeccentricitydistributionofshort-periodplanetsincludingtidalinteractionbetweenhoststarandplanetandpossibleperturbationofanundetectedcompanion.Wewilldiscusshowthesefactorsaffecttheeccentricitydistributionandwhethertheeffectisobservable.Inordertounderstandtheinuenceoftidalinteractiononeccentricitydistribution,werstdivideoursampleintotwosubsets,onesubsetcontainssystemswithalong-termRVtrendwhiletheothersubsetcontainssystemsthatdonotshowalong-termRVtrend(Fig. 6-6 ).Thesystemsthatarenotedwithalong-termvelocitytrendinclude51PEG,BD-103166,GJ436,GJ86,HD107148,HD118203,HD149143,HD68988,HD7924,HD99492andBoo.Wefurtherdividetheno-trendsubsetintotwogroups,onegroupisdistinguishedbyage=circ1,andtheothergroupisdistinguishedbyage=circ<1,wherecircistidalcircularizationtimescaleandageistheageofthehoststar.Weinvestigatewhetherthereisasignicantdifferenceintheeccentricitydistributionbetweenthesetwogroupsasexpectediftidalinteractionisanimportantfactorinshapingeccentricitydistribution.Following Matsumuraetal. ( 2008 ),weestimatecircusing: circ=2 81Q0p nMp Ma Rp5Q0p Q0Mp M2R Rp5F+Fp)]TJ /F5 7.97 Tf 5.07 0 Td[(1,(6)wherethesubscriptspanddenoteplanetandstar,Mismass,Risradius,aissemi-majoraxis,Q0ismodiedtidalqualityfactorandn=[G(M+Mp)=a3]1=2isthemeanmotion. 150

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Table6-1. ComparisonofEccentricitiesCalculatedFromDifferentMethods NameRef.MCMC)]TJ /F4 11.955 Tf -252.45 -14.44 Td[(ereferefeMCMCelowereuppere)]TJ /F4 11.955 Tf 27.46 1.8 Td[(elowereupper HD41004B0.0810.0120.0580.0000.1090.0000.0000.085HD860810.0080.0040.0580.0080.1660.0130.0010.098HD1897330.0000.000..................HD2123010.000...0.0150.0000.0510.0340.0000.091GJ4360.1590.0520.1910.1460.2480.2230.1280.295HD634540.000...0.0180.0000.0380.0030.0000.017HD1490260.000...0.1920.1180.2700.1820.0500.344HD834430.0120.0230.0070.0000.0200.0060.0000.037HD463750.0630.0260.0520.0300.0840.0650.0000.121HD1799490.0220.0150.0140.0000.0240.0130.0000.042Boo0.0230.0150.0790.0540.1170.0860.0340.139HD3300750.000...0.0190.0000.0870.0080.0000.095HD881330.1330.0720.0760.0000.1270.0860.0000.175HD26380.000...0.0410.0000.0760.0050.0000.042BD-1031660.0190.0230.0100.0000.0300.0190.0000.064HD752890.0340.0290.0210.0000.0430.0000.0000.063HD2094580.000...0.0080.0000.0160.0050.0000.018HD21982810.000.....................HD767000.0950.0750.0620.0030.1040.0450.0000.104HD1491430.000...0.0120.0000.0220.0090.0000.014HD1021950.000.....................51Peg0.0130.0120.0070.0000.0140.0060.0000.020GJ67420.1000.0200.0700.0000.1470.0470.0000.131HD496740.0870.0950.0500.0000.1190.0000.0000.073HD1097490.000...0.0450.0000.0700.0420.0000.143HD79240.1700.1600.1190.0220.2240.0540.0000.297HD1182030.3090.0140.2930.2640.3280.2970.2180.367HD689880.1250.0090.1180.0960.1430.1340.0940.174HD1687460.1070.0800.0790.0250.1390.0860.0000.155HD18526930.2760.0370.2760.2420.3140.2790.1750.389HD1620200.2770.0020.2770.2740.279.........GJ17640.000.....................HD1303220.0110.0200.0070.0000.0520.0310.0000.086HD1081470.5300.1200.5260.4290.6240.5560.3020.698HD43080.0000.0100.0680.0000.1230.0600.0000.111GJ860.0420.0070.0420.0340.051.........HD994920.0500.1200.0360.0000.1250.0560.0000.115HD278940.0490.0080.0240.0000.0450.0290.0000.093HD332830.4800.0500.4580.4010.5230.4830.3860.544HD1950190.0140.0040.0020.0000.0070.0060.0000.014HD1021170.1210.0820.0680.0380.1210.0520.0000.119 151

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Table 6-1 .Continued NameRef.MCMC)]TJ /F4 11.955 Tf -252.45 -14.44 Td[(ereferefeMCMCelowereuppere)]TJ /F4 11.955 Tf 27.46 1.8 Td[(elowereupper HD1715650.6840.0130.6830.6720.691.........HD64340.1700.0300.1590.1240.202.........HD1922630.0550.0390.0260.0000.0550.0150.0000.118HD1176180.4200.1700.3520.2330.5110.3810.2120.538HD2246930.0500.0300.0310.0000.0550.0190.0000.089HD4369160.1400.0200.0900.0550.1330.1040.0210.182Crb0.0570.0280.0480.0000.0690.0550.0000.123HD4565270.3800.0600.4340.3710.4960.4430.2990.588HD1071480.0500.1700.0280.0000.1410.0730.0000.221 References:1 Meloetal. ( 2007 );2 Bonlsetal. ( 2007 );3 Johnsonetal. ( 2006 );4 Forveilleetal. ( 2009 );5 Barbierietal. ( 2009 );6 daSilvaetal. ( 2007 );7 Santosetal. ( 2008 );erefanderefarefrom Butleretal. ( 2006 )ifotherwisenoted. 152

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Figure6-6. Distributionofshort-periodsingle-planetsystemsin(e,age=circ)space.FilledcirclesaresystemsshowingnolinearRVtrendandopencirclesaresystemsshowinglong-termlinearRVtrends.DifferentcolorsindicatedifferentcombinationsofQ0andQ0p Matsumuraetal. ( 2008 )adopted106asatypicalvalueforQ0forshortperiodplanetarysystemhoststarsandconsideredQ0prangingfrom105to109.WeuseQ0=[106,107]andQ0p=[105,107,109]inouranalysis.ThefactorsFandFparedenedinthefollowingtwoequations: F=f1(e2))]TJ /F5 11.955 Tf 11.46 8.09 Td[(11 18f2(e2),rot n,(6) Fp=f1(e2))]TJ /F5 11.955 Tf 11.46 8.1 Td[(11 18f2(e2)p,rot n,(6) 153

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whereisrotationalfrequency.Forshort-periodplanetsonecouldsetp,rot=n=1basedontheassumptionthatalltheplanetsinoursamplehavebeensynchronizedsincesynch10)]TJ /F5 7.97 Tf 5.07 0 Td[(3circ( Rasioetal. 1996 ).Inordertocheckwhetherourconclusionissensitivetothechoiceof,rot=n,weconductcalculationswithother,rot=nvaluesinwhichwechoosestellarrotationperiodtobe3,30,and70daysforallthestars.Wendthatthisrangefor,rot=ndoesnotchangetheconclusionsinthechapter.Therefore,forfuturediscussion,weadopt,rot=n=0.67,whichresultsinstellarrotationperiodsconsistentwithtypicalvaluesfrom3to70days( Matsumuraetal. 2008 ).Theuncertaintiesin,rot=nareaccountedforbyoursubsequentdataanalysis(Equation. 6 ).Andf1andf2areapproximatedbytheequations: f1(e2)=(1+15 4e2+15 8e4+5 64e6)=(1)]TJ /F4 11.955 Tf 10.26 0 Td[(e2)13=2,(6) f2(e2)=(1+3 2e2+1 8e4)=(1)]TJ /F4 11.955 Tf 10.26 0 Td[(e2)5.(6)PlanetradiusRpisestimatedbasedon Fortneyetal. ( 2007 ).Weassumethattheplanetandhoststarareformedatthesameepoch.WeassumethatplanetstructureissimilartoJupiterwithacoremassfractionof25M=MJ=7.86%.RadiiofGJ436bandHD149026bareadoptedfromreferencepapersbecausethereisafactorof2differencebetweenobservedvalues( Torresetal. 2008 )andtheoreticallycalculatedvalues.Stellarradiusandageestimationsareobtainedfromthefollowingsourceswithdescendingpriority:1, Takedaetal. ( 2007 );2,nsted.ipac.caltech.edu;3,exoplanet.eu.ThecalculatedcircvaluesarepresentedinTable 6-4 inadditiontotheresultsofMCMCanalysisofindividualsystemandotherstellarandplanetaryproperties.WeusetheeccentricityposteriorsamplesforeachsystemforwhichthestandardMCMCanalysiswassuccessful(i.e.,resultsofx2.1)toconstructtheeccentricitysamplesoftwogroupsseparatedbyage=circ.Wenotethatthereareconsiderableuncertaintiesintheestimationofageandcirc,soage=circ>1doesnotnecessarilymeanthattheactual 154

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systemageislargerthantheactualcircularizationtime.Weconsiderthesensitivityofourresultstotheseuncertaintiesbyadoptingaprobabilityfunction: (age circ)=8>>>>><>>>>>:1)]TJ /F5 11.955 Tf 10.26 0 Td[(0.5exp)]TJ /F8 11.955 Tf 10.26 0 Td[((age circ)]TJ /F5 11.955 Tf 10.26 0 Td[(1)ifage circ10.5exp)]TJ /F8 11.955 Tf 10.26 0 Td[((circ age)]TJ /F5 11.955 Tf 10.26 0 Td[(1)ifage circ<1(6)whereisaparametertuningthecondenceofageandcircestimation.Forexample,ifage=circ=2and=1,then(age=circ)=0.816,meaningthereis81.6%chancethatthesystemisfromthegroupofage=circ1becauseoftheuncertaintiesinageandcircestimation.Therefore,wetake81.6%oftheeccentricityposteriorsamplesofthesystemtoconstructeccentricitysampleforthegroupofage=circ1andtheremaining18.4%eccentricityposteriorsamplestoconstructeccentricitysampleforgroupofage=circ<1.Theparameterreectsourcondenceinageandcircestimation.Ifwearenotverycondentintheestimationofageandcirc,thenwesettoasmallvalueapproachingzero,sohalfoftheeccentricityposteriorsamplesforeachsystemareassignedtothegroupwithage=circ1andthetheotherhalfareassignedtothegroupwithage=circ<1.Afterconstructingtheeccentricitysampleforthetwogroups,weusetwo-sampleK-Stesttotestthenullhypothesisthatthesetwosamplesfromtwogroupsweredrawnfromthesameparentdistribution.Theresults(Table 6-2 )showthatweareunabletoexcludethenullhypnosisatalowpvalue(statisticoftwo-sampleK-Stest)becauseofthesmalleffectivesamplesize(N08).Ifmax=0.2,wheremaxisthemaximumdifferencebetweencumulativedistributionfunctionsoftwogroups,wecanexcludethenullhypnosisatp=0.05onlyifN0ismorethan44.Incomparison,ourcurrentsamplesizeisinadequatetodrawastatisticallysignicantconclusiononwhetherornotthethegroupsarefromthesameparentdistribution.However,wedoseeahintofadifferencebetweencumulativedistributionfunctionsoftwogroups(Fig. 6-7 left),therearemoresystemswithlow-eccentricityforthegroupwithage=circ<1,whichisaconsequenceoftidalcircularization.Wealsondthattheconclusionisunchangedforawiderangeof,Q0,andQ0pvalues. 155

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Figure6-7. Theleftpanelcomparesthecumulativedistributionfunctionofeccentricityfortwogroupsofplanets:1)age=circ1(solidline)and2)age=circ<1(dottedline).Therightpanelcomparescumulativedistributionfunctionsofeccentricityfortwosubsetsofplanets:planetswithoutalong-termRVslope(dashedline)andplanetswithalong-termRVslope(dash-dottedline). Weconductsimilartestfortwosubsetsdistinguishedbywhetherornotalong-termvelocitytrendisrecognizedandndthesimilarresultthatourcurrentsamplesizeisinade-quatetodrawastatisticallysignicantconclusiononwhetherornotthethegroupsarefromthesameparentdistribution.Again,weseeahintofadifferencebetweencumulativedistri-butionfunctionsoftwosubsets(Fig. 6-7 right)althoughitisnotstatisticallysignicant.Therearemoresystemswithlow-eccentricityforsubsetsshowingnosignofexternalperturbation.Themaximumdifferencebetweenthecumulativedistributionfunctionsmaxis0.123,N0is8.37andK-Sstatisticis0.999.Inthatcase,weneedaneffectivesamplesizeof119inorder 156

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Table6-2. Two-sampleK-Stestresult.maxisthemaximumdifferencebetweencumulativedistributionfunctionsofeccentricityfortwogroupsseparatedbyage=circ=1.N0istheeffectivesamplesize,calculatedby(N1N2)=(N1+N2),pisthesignicancelevelatwhichtwo-sampleK-Stestrejectsthenullhypothesisthatthetwoeccentricitysamplesarefromthesameparentdistribution;isaparametertuningthecondenceofageandcircestimation. Q0p105107109 Q0=106=1000max0.0880.1750.086N07.578.767.57p1.000.931.00=1max0.1030.1600.093N07.728.727.02p1.000.971.00=0.001max0.0430.0270.061N08.598.508.08p1.001.001.00Q0=107=1000max0.0880.2860.190N07.578.422.80p1.000.431.00=1max0.1030.2590.168N07.728.362.81p1.000.561.00=0.001max0.0400.0270.065N08.618.397.67p1.001.001.00 tomakeastatisticallysignicantconclusion(p=0.05).Inothercases,largersamplesizeisrequiredsincemaxisless.Wehaveshownthatanydifferenceineccentricitydistributiondependingonexpectedtimescalefortidalcircularizationorthepresenceofadditionalbodiescapableofexcitinginnerplanet'seccentricityisnotstatisticallysignicant,althoughthismaybeaconsequenceofsmalleffectivesamplesize.Thedataarealsoconsistentwiththeargumentthatbothfactorsplayrolesinaffectingtheeccentricitydistribution. 157

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6.5EccentricityDistributionWeseekananalyticalfunctionthatisabletoapproximatetheobservedeccentricitydistributionforshortperiodsingleplanetarysystemsintheframeworkofBayesianinference.Forthispurpose,werstexcludesystemsshowinglong-termRVtrendstoreducetheeffectofperturbationontheestimatedeccentricitydistribution.Wealsoassumethatthedistributionofage=circinoursampleisrepresentativeofshort-periodsingle-planetsystems.UsingtheposteriorsamplesofeccentricityfromstandardMCMCanalysis,weobtainanobservedeccentricityprobabilitydensityfunction(pdf)f(e)bysummingtheposteriordistributionstogether.Whilenotstatisticallyrigorous,thisprovidesasimplesummaryofourresults.Logarithmicbinningisadoptedbecausetheshapeoff(e)atloweccentricityisofparticularinterest.Theuncertainty(e)foreachbinissetbyassumingaPoissondistribution.Then,weuseabrute-forceBayesiananalysistondthemostprobablevaluesofparametersforthecandidateeccentricitypdff0(e)thatapproximatestheobservedeccentricitypdf.Intheobservedeccentricitypdf,thereisapile-upinsmalleccentricitiesnearzeroandascatterofnonzeroeccentricitylessthan0.8.Therefore,weuseamixtureoftwodistributions:anexponentialpdffexpo(e,)=(1=)exp()]TJ /F4 11.955 Tf 7.61 0 Td[(e=)torepresentthepile-upofsmalleccentricitynearzeroandeitherauniformdistributionoraRayleighdistribution( Juric&Tremaine 2008 )torepresentthepopulationwithsizableeccentricities.Weassumeauniformdistributionforparametersinprior(~),where~isvectorcontainingtheparametersforf0(e).Ourresultsarenotsensitivetothechoiceofpriors.WeadoptPoissonlikelihoodforeachbinintheformoffPoisson(n;)=(nexp()]TJ /F8 11.955 Tf 7.6 0 Td[())=n!,i.e.,L(eij~)=fPoisson)]TJ /F4 11.955 Tf 4.1 -8.21 Td[(f0(eij~)N;f(ei)N,whereNistotalnumberofposteriorsamples.Thevalueoff0(eij~)Nisroundedifitisnotaninteger.Theposteriordistributionof~iscalculatedasp(~j~e)=MYi(~)L(eij~)1 MZMYi(~)L(eij~)1 Md~,whereMisthenumberofbins.Wehaveexploredarangeofbinsizefrom10to40binsinlogarithmalspaceandconcludethattheresultsofBayesiananalysisdonotchangesignicantlywithchoiceofbinsize.Since 158

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theplausiblevaluesofparametersforf0(e)arelimited,wedonothavetoexplorealargeparameterspace.Therefore,abrute-forceBayesiananalysisispractical.WeapplyBayesiananalysistothreedifferentplanetpopulationsfordifferentvalues(=0.001,1,1000)assumingQ0=107andQ0P=107:1)systemswithoutalong-termRVslopeandage=circ1;2)systemswithoutlong-termRVslopeandage=circ<1;3)theunionof1)and2).TheresultsofBayesiananalysisarepresentedinTable 6-3 Figure6-8. MarginalizedprobabilitydensityfunctionsofparametersforanalyticaleccentricitydistributionwithamixtureofexponentialandRayleighpdfs.Uncircularizedsystems(group2)arerepresentedbycoloredsolidlineswhilecircularizedsystems(group1)arerepresentedbycoloreddashedlines.Differentcolorsindicatedifferentvalues,red:=0.001;green:=1;blue:=1000.Solidblacklinesareforunionofgroup1and2. Figure 6-8 showsmarginalizedprobabilitydensityofdifferentparametersforanalyticaleccentricitydistributionwithamixtureofexponentialandRayleighdistributions(f0(e)= 159

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fexpo(e,)+(1)]TJ /F8 11.955 Tf 10.33 0 Td[()frayl(e,)).isthefractionofexponentialdistributioncomponentandfrayl(e,)representsRayleighdistributionwiththeformoffrayl(e,)=(e=2)exp()]TJ /F4 11.955 Tf 7.61 0 Td[(e2=22).Group1(potentiallycircularizedsystem)isrepresentedbydashedlinesofdifferentcolorsindicatingdifferentvaluesofwhilegroup2(systemsunlikelytohavebeencircularized)isrepresentedbycoloredsolidlines.Group1andgroup2arewellmixedif=0.001,i.e.,alooseconstraintisappliedontheboundaryofage=circ=1.Whenadopting=0.001,themarginalizedposteriorpdfsforbothgroups1and2approachthemarginalizedpdfforgroup3(solidblacklines;theunionofgroup1and2,includingallplanetswithoutavelocityslope).Whenweadoptlargervalues(=1or1000),,thefractionoftheexponentialcomponentofthepdfisgreaterforgroup1(Fig. 6-8 top:blueandgreendashedlines)thangroup2(Fig 6-8 top:blueandgreensolidlines).Thisisconsistentwiththehypothesisthatsignicanttidalcircularizationaffectedgroup1.Forgroup1,thefractionoftheexponentialcomponentofthepdfisconsistentwithunity,implyingthattheeccentricitydistributionforplanetsfromgroup1canbedescribedbyanexponentialpdf.Incomparison,thereisasubstantialfractionofRayleighcomponent(40%)fortheanalyticalfunctiondescribingeccentricitydistributionforuncircularizedsystems(i.e.,group2).Similarconclusionsarealsodrawnfortheanalyticaleccentricitydistributionwithamixtureofexponentialanduniformdistribution(Fig. 6-9 )intheformoff0(e)=fexpo(e,)+(1)]TJ /F8 11.955 Tf 10.43 0 Td[()funif(e,),wherefunif(e,)isuniformdistributionwithlowerboundaryof0andupperboundaryof.Figure 6-10 showsthecumulativedistributionoftheeccentricitiessamplebasedonsummingtheposterioreccentricitysamplesofeachsystem(greensolidline)andtheCDFsofthetwoanalyticfunctions(bluedottedanddashed)withtheparametersthatmaximizetheposteriorprobability(seeTable 6-3 ,subset3).Forcomparison,thecumulativedistributionofeMCMCfromstandardMCMCanalysis(Table 6-1 )isshowninred.Thedifferencebetweentheredandthegreenlinecannotbedistinguishedat0.05signicantlevelinaK-Stest(N0=46). 160

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Figure6-9. Marginalizedprobabilitydensityfunctionsofparametersforanalyticaleccentricitydistributionwithamixtureofexponentialanduniformpdf.Uncircularizedsystems(group2)arerepresentedbycoloredsolidlineswhilecircularizedsystems(group1)arerepresentedbycoloreddashedlines,red:=0.001;green:=1;blue:=1000.Differentcolorsindicatedifferentvalues.Solidblacklinesareforunionofgroup1and2. 161

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Figure6-10. Cumulativedistributionsfunctions(cdf)ofeccentricitiesfromdifferentmethods.MCMC:mostprobableeccentricities,eMCMC,adoptedfromMCMCmethod(Table 6-1 );MCMC,sum:eccentricitiesbysummingupposteriordistributionsamplesofeachsystem;AF1:cdfoftheanalyticalfunctionwiththemostprobableparameters,f0(e)=fexpo(e,)+(1)]TJ /F8 11.955 Tf 10.26 0 Td[()frayl(e,);AF2:cdfoftheanalyticalfunctionwiththemostprobableparameters,f0(e)=fexpo(e,)+(1)]TJ /F8 11.955 Tf 10.26 0 Td[()funif(e,). 162

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Table6-3. Bayesiananalysisresults.Group1:systemswithoutlong-termRVslopeandage=circ1;group2:systemswithoutlong-termRVslopeandage=circ<1;group3:unionof1and2.Numbersinbracketareuncertaintiesofthelasttwodigits.Thefractioncolumnreportsthepercentageoftrialsinwhicheccentricitytestingsamplesgeneratedbyanalyticalfunctioncannotbedifferentiatedfromtheobservedeccentricitysampleat0.05condencelevel. subsetf0(e)=fexpo(e,)+(1)]TJ /F8 11.955 Tf 10.26 0 Td[()frayl(e,)f0(e)=fexpo(e,)+(1)]TJ /F8 11.955 Tf 10.26 0 Td[()funif(e,)[10)]TJ /F5 7.97 Tf 5.07 0 Td[(2][10)]TJ /F5 7.97 Tf 5.07 0 Td[(1][10)]TJ /F5 7.97 Tf 5.07 0 Td[(1]fraction[10)]TJ /F5 7.97 Tf 5.07 0 Td[(2][10)]TJ /F5 7.97 Tf 5.07 0 Td[(1][10)]TJ /F5 7.97 Tf 5.07 0 Td[(1]fraction 0.00116.83(84)7.61(40)3.38(36)100.0%7.08(82)7.35(38)7.56(41)100.0%26.67(59)7.56(32)3.15(22)100.0%6.86(59)7.32(33)7.06(23)100.0%1.00017.71(58)9.78(20)0.83(54)99.6%7.96(59)10.0(12)1.8(11)99.9%25.55(53)6.10(31)3.19(16)100.0%5.55(57)5.66(34)7.12(18)100.0%1000.17.41(72)8.9(12)0.95(24)99.9%7.62(56)9.2(11)2.10(57)100.0%25.24(49)5.95(30)3.27(16)100.0%5.11(53)5.44(33)7.20(19)100.0%36.67(47)7.54(25)3.22(19)100.0%6.89(47)7.33(26)7.22(20)100.0% 163

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Inordertocheckwhethertheanalyticalfunctionwiththemostprobableparametersisanacceptableapproximationtotheeccentricitydistributionforshort-periodsingle-planetsystems,wegeneratetestsamplesfromtheanalyticalfunctionsandcomparetheresultingeccentricitysamplestoobservations.ForeachtestsamplethereareN0eccentricitiesfollowingthedistributionoftheanalyticalfunctionf0(ej),whereN0istheeffectivesamplesize.Eacheccentricityisperturbedbyansimulatedmeasurementerrorthatfollowsthedistributionofposteriorsamplesforouranalysisoftheactualobservations.ThetestsampleisthencomparedtotheobservedeccentricitysamplesobtainedbystandardMCMCanalysisusingtwo-sampleK-Stest.WereportinTable 6-3 thepercentageoftrialsinwhicheccentricitysamplesgeneratedbyouranalyticalfunctioncannotbedifferentiatedfromtheobservedeccentricitysampleat0.05condencelevel.Allthecandidateanalyticalfunctionswehavetestedareabletoreproducetheobservedeccentricitydistributioninmorethan99.6%ofthetrialsat0.05condencelevel.Weconcludethatthebest-tanalyticalfunctionisanadequateapproximationtoobservedeccentricitydistribution.Wealsocomparetoanalyticaleccentricitydistributionusedin Shen&Turner ( 2008 )f0(e)[1=(1+e)a)]TJ /F4 11.955 Tf 10.57 0 Td[(e=2a]inwhicha=4,althoughitisnotspecicallyforshortperiodsingleplanetarysystems.Similartowhatwedidinprevioustest,wefoundthatin39.0%ofthetests,theeccentricitysamplesgeneratedbyanalyticalfunctionscannotbedifferentiatedfromtheobservedeccentricitysampleat0.05condencelevel.FromtheresultsofBayesiananalysis,thereisacleardifferencein,thefractionofexponentialdistribution,betweengroup1and2,suggestingtheroleplayedbytidalcircularization.Group1withage=circ1showsmoreplanetswithnear-zeroeccentricities(75%)ascomparedtogroup2(55%)withage=circ<1(Table 6-3 ).Sincetheeccentricitysamplestestedwereperturbedbymeasurementerrors,theanalyticalfunctionwefoundcanbeinterpretedasanapproximationoftheunderlyingeccentricitydistributionofshort-periodsingle-planetsystems.However,theactualparametervaluesanduncertaintiesintheanalyticalfunctionaredependentuponthequalityofobservationandthenumberof 164

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systemsinthesampleofshort-periodsingleplanets.Asthemeasurementprecisionandthesamplesizeimprove,wewillbeabletobetterconstrainthevaluesofparametersintheanalyticalfunctionwhichapproximatestheunderlyingeccentricitydistribution. 6.6Discussion Figure6-11. Distributionofshort-periodsingle-planetsystemsinoursampleinperiod-eccentricityspace(top:p10day;bottom:p>10day).Systemsthatarenotconsistentwithzeroeccentricityaccordingto)]TJ /F1 11.357 Tf 9.93 0 Td[(analysis(lledcircles)orMCMCanalysis(opencircles)aremarkedwithcorrespondingnames.Wealsoincludetransitingplanets(markedascross)forcomparison.Transitingsystemsnotconsistentwithzeroeccentricityaremarkedwithnumbers:1,WASP-18;2,WASP-12;3,WASP-14;4,HAT-P-13;5,WASP-10;6,XO-3;7,WASP-6;8,WASP-17;9,CoRoT-5;10,HAT-P-11;11,HAT-P-2. Themedianeccentricityofshort-periodsingle-planetsystemsinoursampleis0.088.Whencomparedtomedianeccentricityforallthedetectedexoplanets0.15,itsuggeststhatthepopulationmaybeaffectedbytidalcircularization.Weuseeccentricityestimatedby 165

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)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisinthefollowingdiscussionsinceitisalessbiasedmethodforaccessingsmalleccentricity.Figure 6-11 showstheperiod-eccentricitycorrelation.Wewouldexpectplanetswithsufcientshortperiodtobetidallycircularized.Whilethisisgenerallytrue,thereare3(17.6%)planetswithP<4dandnon-circularorbits:GJ436,HD149026andBoo.ForGJ436,itissuspectedthatanoutercompanionmaybepumpingtheeccentricity( Manessetal. 2007 ; Ribasetal. 2008 ).Similarly,observationsofBoobareinconsistentwithcircularorbit,butmightbeexplainedbytheperturbationofanunseencompanionindicatedbyalong-termRVlineartrend( Butleretal. 2006 ).SecondaryeclipsetimingindicatesthattheeccentricityofHD149026isquitesmallbutinconsistentwithzero( Knutsonetal. 2009 ).Consideringplanetswithorbitalperiodsupto10days,thereare4additionalsystemsthatarenotcircularized.HD118203,HD68988andHD185269mightbeduetoperturbationsbyadditionalbodiesinthesystem( Butleretal. 2006 )whilethenon-zeroeccentricityofHD162020bmaybeattributedtoadifferentformationmechanism( Udryetal. 2002 ).Withperiodlongerthan10days,thereare8(44.4%)planetswithage=circ>1whichhavenon-zeroeccentricityandnodetectedlong-termlineartrend.Incomparison,thereare8(28.6%)eccentricplanetswithorbitalperiodslessthan10days.Theincreasingfractionofrecognizablyeccentricorbitsasperiodincreasesissuggestiveofdecreasingtidalcircularizationeffect,butlargesampleofplanetsisrequiredtodrawrmconclusion.Thediscoveryofover705planetscandidatesbytheKeplermissionpresentsanexcellentopportunitytoanalyzetheeccentricitydistributionofshort-periodplanets( Boruckietal. 2011b ; Fordetal. 2008 ).Webrieyconsidertransitingplanets.Wenotethat11of58(19.0%)transitingsystemsasofJune2010arenotconsistentwithzeroeccentricity,andtheorbitalperiodsforalltransitingplanetsbut3(i.e.CoRoT-9b,HAT-P-13cHD80606b)arelessthan10days.Weinferthatthetidalcircularizationprocessmightbeeffectiveforisolatedplanetswithorbitalperiodoflessthan10days.Alternatively,theplanetformationandmigrationprocessesforshort-periodgiantplanetmaynaturallyleadtoasignicantfractionofnearlycircularorbits,evenbeforetidaleffecttakesplace. 166

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ItisworthnotingthatHD17156b( Fischeretal. 2007 )hasoneofthemosteccentricorbitsamongshort-periodplanetsystemsinspitethefactthatage=circcouldbewellover10(Fig. 6-6 ,red).However, Barbierietal. ( 2009 )foundnoindicationsofadditionalcompanionsbasedonobservationsofdirectingimaging,RVandastrometrymeasurement. Anglada-Escudeetal. ( 2010 )investigatedthepossibilitythata2:1resonantorbitcanbehiddenbyaneccentricorbitalsolution.Itisinterestingtoexploresuchpossibilityonthisparticularsystemtosolvethediscrepancyofhigheccentricityandagetocircratio.Anotherpossibilityisthatthesystemisintheprocessofcircularizationthatbeganwellafterthestarandplanetformed(e.g.,duetoplanetsscattering).However,wearecautiousindrawingconclusionssinceage=circcouldbelessthanunity(Fig. 6-6 ,greenandblue). 6.7ConclusionWeapplystandardMCMCanalysisfor50short-periodsingle-planetsystemsandconstructacatalogoforbitalsolutionsfortheseplanetarysystems(Table 6-4 ).WendgeneralagreementbetweenMCMCanalysisandpreviousreferenceswiththeprimaryexceptionbeingcaseswhereeccentricitywasheldxedinpreviousanalysis.Wedevelopanewmethodtotestthesignicanceofnon-circularorbits()]TJ /F1 11.357 Tf 9.93 0 Td[(analysis),whichisbettersuitedtoperformingpopulationanalysis.Wendtheeccentricityestimationsfrom)]TJ /F1 11.357 Tf 9.93 0 Td[(analysisareconsistentwithresultsfrombothstandardMCMCanalysisandpreviousreferences.Ourresultssuggestthatbothtidalinteractionsandexternalperturbationsmayplayrolesinshapingtheeccentricitydistributionofshort-periodsingle-planetsystemsbutlargesamplesizesareneededtoprovidesufcientsensitivitytomakethesetrendsstatisticallysignicant.Weidentifytwoanalyticalfunctionsthatapproximatetheunderlyingeccentricitydistribution:1)mixtureofanexponentialdistributionandauniformdistributionand2)amixtureofanexponentialdistributionandaRayleighdistribution.WeuseBayesiananalysistondthemostprobablevaluesofparametersfortheanalyticalfunctionsgiventheobservedeccentricities(Table 6-3 ).Theanalyticalfunctionscanbeinterpretedastheunderlying 167

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distributionofeccentricitiesforshort-periodsingle-planetsystems.Thus,theanalyticalfunctionscanbeusedinthefuturetheoreticalworksoraspriorsforeccentricitydistribution. 168

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Table6-4. CatalogofShort-PeriodSingle-PlanetSystems NamePKe!M0(day)(ms)]TJ /F5 6.974 Tf 4.44 0 Td[(1)(deg)(deg) HD41004B1.323638.91594e-054599.21337.570.0580+0.0511)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0580149.072.2119.572.1HD860811.998090.00677822189.6512.090.0575+0.1080)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0501-27.471.564.874.5HD18973312.21860.0005205600.000290270HD2123012.245710.00014769957.263.010.0147+0.0365)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0147172.885.656.085.6GJ4362.643949.85041e-0518.071.030.1912+0.0571)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0449-5.615.3-66.614.5HD634542.817470.00038224763.191.820.0177+0.0203)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0177-122.976.225.776.3HD1490262.878070.0014657154.6311.900.1918+0.0777)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0743114.225.910.922.5HD834432.985725.30373e-0556.001.050.0070+0.0135)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0070117.382.2134.082.0HD463753.023586.44902e-0533.670.810.0524+0.0320)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0229113.734.3-53.234.3HD1799493.09253.30046e-05112.621.770.0104+0.0099)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0104-170.763.442.463.3Boo3.312493.12595e-05469.5914.860.0787+0.0382)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0246-141.625.024.424.9HD881333.415660.00084113434.133.570.0761+0.0508)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0761-2.860.4-39.560.1HD26383.437520.0082387666.262.830.0407+0.0351)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0407126.978.6-123.477.7BD-1031663.48780.00010485860.531.440.0104+0.0192)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0104-14.683.936.683.7HD752893.509287.2946e-0554.841.870.0211+0.0217)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0211136.373.9-162.274.1HD2094583.524722.81699e-0584.300.880.0082+0.0078)]TJ /F5 6.974 Tf 4.43 0 Td[(0.008243.868.492.568.5HD3300753.64130.0018711197.848.790.0187+0.0684)]TJ /F5 6.974 Tf 4.43 0 Td[(0.018738.291.635.291.7HD21982823.8330.001370.5000HD767003.971010.00020319427.241.310.0616+0.0426)]TJ /F5 6.974 Tf 4.43 0 Td[(0.058712.354.345.653.8HD1491434.072060.000320041149.281.650.0123+0.0093)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0115-155.255.9-150.955.9HD10219534.11380.00055763200051PEG4.23083.72905e-0555.650.530.0069+0.0066)]TJ /F5 6.974 Tf 4.43 0 Td[(0.006954.172.385.172.3GJ6744.69440.001825919.461.090.0700+0.0766)]TJ /F5 6.974 Tf 4.43 0 Td[(0.07000.571.277.671.1HD496744.947390.00097492511.781.180.0495+0.0691)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0495-96.179.382.479.3HD1097495.239210.00093553328.491.120.0451+0.0250)]TJ /F5 6.974 Tf 4.43 0 Td[(0.045172.355.8-155.155.8HD79245.397850.000966973.740.440.1186+0.1050)]TJ /F5 6.974 Tf 4.43 0 Td[(0.097025.055.662.154.8HD1182036.133220.00129898213.756.670.2943+0.0342)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0298-27.35.5-2.94.8HD689886.276990.0002195184.634.690.1187+0.0246)]TJ /F5 6.974 Tf 4.43 0 Td[(0.021632.411.261.310.8HD1687466.403980.00097946128.411.380.0791+0.0595)]TJ /F5 6.974 Tf 4.43 0 Td[(0.054114.847.2-147.247.3HD1852696.837960.0011914689.394.200.2758+0.0386)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0334173.36.4-99.45.9HD1620208.428267.76104e-051808.845.260.2765+0.0023)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0025-151.20.368.31.0GJ17648.7830.00544.10.52000HD13032210.70860.00184045108.107.310.0068+0.0455)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0068128.393.9-10.493.8HD10814710.89840.0031676724.603.620.5161+0.0979)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0966-54.414.0-68.09.9HD430815.56460.02135564.200.340.0682+0.0547)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0682-166.960.6-13.059.5GJ8615.7650.000382114376.642.790.0416+0.0092)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0073-93.712.4-76.812.0HD9949217.04950.005250918.390.980.0364+0.0891)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0364-138.693.6-137.693.5HD2789418.00590.016324856.881.750.0240+0.0211)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0240131.366.5-60.028866.5HD3328318.18010.0083058424.482.410.4576+0.0656)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0569156.09.6-60.07.4HD19501918.20180.000595221269.701.600.0017+0.0049)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0017-127.447.4-175.547.3HD10211720.82100.0100646510.200.910.0685+0.0529)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0647137.672.0-9.572.5HD1715621.21780.00371004279.888.430.6829+0.0080)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0106121.31.1-156.71.2HD643421.99750.012760434.301.520.1589+0.0434)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0345155.415.8-14.215.5HD19226324.35460.0050767551.132.620.0256+0.0297)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0256-147.078.9-56.778.9HD11761825.82210.015504512.251.700.3524+0.1583)]TJ /F5 6.974 Tf 4.43 0 Td[(0.1192-105.622.9-123.321.1HD22469326.7320.024593439.731.540.0313+0.0236)]TJ /F5 6.974 Tf 4.43 0 Td[(0.031311.668.6162.568.9HD4369136.99160.0350715125.984.060.0897+0.0432)]TJ /F5 6.974 Tf 4.43 0 Td[(0.034691.326.216.724.9Crb39.84590.0091786565.252.200.0476+0.0218)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0476-62.142.2-172.642.5HD4565243.68960.10582535.762.840.4339+0.0625)]TJ /F5 6.974 Tf 4.43 0 Td[(0.063283.212.877.710.2HD10714848.61683.5920410.014.150.0279+0.1135)]TJ /F5 6.974 Tf 4.43 0 Td[(0.0279126.590.8-50.188.8 169

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Table 6-4 .Continued NametrendJitterNobsMR(day)(ms)]TJ /F5 6.974 Tf 4.44 0 Td[(1d)]TJ /F5 6.974 Tf 4.44 0 Td[(1)(ms)]TJ /F5 6.974 Tf 4.43 0 Td[(1)(MS)(RS) HD41004B2452532.699......2761.45168.901490.400.40HD860812453753.2......32.135.74261.211.22HD18973312454037.612......1586......HD2123012453388.9......8.531.78231.051.19GJ4362452992.10.00370.00150.411.46550.410.46HD634542453238.057......5.701.22260.800.78HD1490262453545.35......2.413.26171.301.50HD834432452248.9......3.121.67511.001.02HD463752451920.7......3.280.60500.920.94HD1799492452419.1......9.441.06881.211.22Boo2450529.2-0.0510.003694.308.13981.351.33HD881332453180.0......5.671.61211.201.93HD26382453323.282......5.484.87280.931.01BD-1031662451844.70.0050.00264.001.74311.010.84HD752892452593.9......4.731.73301.211.28HD2094582452499.3......3.270.86641.141.14HD3300752452968.399......24.595.00210.700.90HD21982822453898.63......1.727......HD767002452655.1......1.354.21351.131.34HD1491432453413.10.0270.00560.481.96171.201.61HD10219532453895.96......6.159......51PEG2450404.4-0.00450.000460.270.912561.091.18GJ6742453823.784......3.550.55320.350.46HD496742452308.9......3.560.82391.060.95HD1097492453426.3......0.311.10201.211.28HD79242454096.650.350.072.590.22930.830.78HD1182032453351.20.140.016623.083.96431.232.15HD689882452441.3-0.0650.005613.302.34281.181.14HD1687462452510.8......0.411.62160.931.04HD1852692453795.0......7.702.10301.281.88HD1620202451672.02......11.023.43460.780.74GJ17642454399.8......2.557......HD1303222452430.2......10.253.79120.880.85HD1081472452407.0......8.651.49541.191.25HD43082453338.121......0.580.78410.900.92GJ862452199.4-0.2600.002910.671.62420.770.80HD994922452523.850.00350.000663.960.51860.860.76HD278942453344.278......4.481.07200.750.90HD332832453560.0......0.270.94241.241.20HD1950192451844.0......10.501.221541.071.38HD1021172452931.7......0.571.73441.111.26HD171562454111.21......3.680.64341.241.63HD64342451753.933......7.391.151300.790.57HD1922632451867.6......6.991.31310.810.77HD1176182452838.0......3.301.20571.091.20HD2246932453607.2......1.002.78231.331.70HD436912454046.7......10.352.58361.381.92Crb2451181.1......0.652.79261.001.28HD456522453692.66......8.382.20450.831.04HD1071482452799.90.0030.00123.531.04351.141.12 170

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Table 6-4 .Continued NameMPRPcircageRV(MJ)(RJ)(Gyr)(Gyr)ref. HD41004B18.401.060.266.327HD860811.501.080.566.218HD1897331............9HD2123010.401.070.285.9011GJ4360.070.385.576.006HD634540.391.060.671.0012HD1490260.360.653.462.006HD834430.401.041.1011.686HD463750.231.020.6711.886HD1799490.921.052.902.566Boo4.131.064.121.646HD881330.301.001.499.566HD26380.481.042.383.0012BD-1031660.461.032.701.846HD752890.471.033.033.286HD2094580.691.054.072.446HD3300750.621.061.976.2113HD2198282............2HD767000.230.992.809.846HD1491431.331.057.017.6014HD1021953............351PEG0.471.036.616.766GJ6740.041.130.210.5515HD496740.100.983.253.566HD1097490.280.9912.0410.3014HD79240.031.050.610.8816HD1182032.141.053.684.6017HD689881.861.0580.153.406HD1687460.250.9920.3112.406HD1852690.941.0419.344.208HD16202015.000.98148.000.7619GJ1764............4HD1303221.091.04670.0810.806HD1081470.260.988.433.206HD43080.051.00169.248.6820GJ863.911.056701.628.486HD994920.110.96692.891.8021HD278940.621.023620.333.9012HD332830.330.99203.123.208HD1950193.691.054051.079.326HD1021170.170.953205.499.406HD171563.201.0419.108.0010HD64340.401.004393.146.8522HD1922630.641.0214630.882.566HD1176180.180.951435.345.686HD2246930.711.0323506.412.008HD436912.491.0442743.172.8018Crb1.091.03166705.717.646HD456520.471.0112199.365.0023HD1071480.210.96122900.605.606 171

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Table 6-4 .Continued References:1 Bouchyetal. ( 2005 );2 Meloetal. ( 2007 );3 Geetal. ( 2006b );4 Forveilleetal. ( 2009 );5 Manessetal. ( 2007 );6 Butleretal. ( 2006 );7 Zuckeretal. ( 2004 );8 Johnsonetal. ( 2006 );9 Winnetal. ( 2006 );10 Winnetal. ( 2009 );11 LoCurtoetal. ( 2006 );12 Moutouetal. ( 2005 );13 Pepeetal. ( 2004 );14 Fischeretal. ( 2006 );15 Geetal. ( 2006b );16 Bonlsetal. ( 2007 );17 Howardetal. ( 2009 );18 daSilvaetal. ( 2006 );19 daSilvaetal. ( 2007 );20 Udryetal. ( 2002 );21 Udryetal. ( 2006 );22 Marcyetal. ( 2005 );23 Mayoretal. ( 2004 );24 Santosetal. ( 2008 )Stellarradiusandageestimationsareobtainedfromthefollowingsourceswithdescendingpriority:1, Takedaetal. ( 2007 );2, nsted.ipac.caltech.edu ;3, exoplanet.eu .circiscalculatedassumingQ0=107andQ0p=107 172

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CHAPTER7SUMMARY,CONCLUSIONANDCONTRIBUTIONMydissertationworkspavethewayformassivedetectionsofM-dwarfplanets.Morespecically,IcomparetwoexistingRVtechniques,i.e.,theDEmethodandtheDFDImethod,andndtheirownstrengthinexoplanetsearch.IanswerthequestionwhetherandhowtodetectahabitableEarth-likeplanet.ItakeefforttoovercomechallengesinNIRRVprecisionmeasurement.IhelpdesignandunderstandthescienticdeliverableofaM-dwarfplanetsurvey.Itakepartintoamulti-objectplanetsurveyprojectusingtheDFDImethodandhelpsolveinstrumentationproblemwhichleadstofruitfuldiscoveries.Ienvisionamulti-objectM-dwarfplanetsurveytocompleteourunderstandofplanetformationaroundstarswithabroadmassspectrum.InadditiontomymainworkonMdwarfplanet,IpresentmycollaborationworkwithDr.EricFordontheeccentricitydistributionofshort-periodplanets. 7.1Chapter2IconductedcomprehensivecomparisonbetweentheDEandtheDFDImethod.Inordertoensureafairandquantitativecomparison,IinventedamethodofcalculatingtheQfactorfortheDFDImethod.Thismethodisanaturalextensionoftheworkof Bouchyetal. ( 2001 )inwhichtheydevelopedamethodofcalculatingtheQfactorfortheDEmethod.ComparisonbetweentheDEmethodandtheDFDImethodhasbeendonebyseveralpreviousworksuchas Erskine ( 2003 ); Erskine&Ge ( 2000 ); Ge ( 2002 ); Geetal. ( 2002 ); vanEykenetal. ( 2010 ),butithasneverbeendoneassystematicandcompleteasmyworkdescribedinthisdissertation.Ihaveconsideredmanycasessuchasthesamewavelengthcoveragecase,thesamedetectorcaseandthemulti-objectcase.Myworkprovidesaguidanceforfutureexoplanetsurvey:1,asurveyofalargesampleofstarsshouldadopttheDFDImethod,whichenablesbothadequateRVprecisionandhighsurveyefciency;2,highprecisionlow-massexoplanetsearchshouldadopttheDEmethodwithahighresolutionspectrograph. 173

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7.2Chapter3FollowingtheconclusionofChapter2,IinvestigatedintothehighprecisioncaseforwhichtheDEmethodwithahighresolutionspectrographissuitable.IwastryingtoanswerthequestionifweareabletodetectahabitableEarth-likeplanetgivenspectralpropertiesandstate-of-the-artRVtechnique.Thetheoreticalframeworkisgivenby Bouchyetal. ( 2001 )thatprovidesthemethodtoestimateRVprecisionforagivenspectrumatagivenS/N.Thereareseveralpreviouspapersonthistopicsuchas Reinersetal. ( 2010 ); Rodleretal. ( 2011 ),butmyworkistherstofitskindinwhichIhaveconsideredallthedominantfactorsthatcontributetoRVprecisionerrors.TheyincludestellarpropertiessuchasspectralSED,linepropertiesandrotationalvelocity,wavelengthcalibrationerror,stellarnoiseandtelluriccontamination.IconcludedthatNIRobservationofmid-latetypeMdwarfsisthemostrealisticandlikelyapproachinthesearchforahabitableEarth-likeplanet. 7.3Chapter4ThereareseveralchallengesinNIRprecisionRVmeasurement.IhaveconductedpioneeringworkusingtheEXPERTspectrograph(R=27,000)attheKNNO2.1mtelescope.Myworkincludesobservation,dataanalysisandhardwaredevelopment.Forobservation,IhasworkedwithothermembersintheETgroupforproposalwritingandnightlyobservation.IhavetwoproposalsasPIacceptedforwhichatotalof14nightswereawarded.Fordataanalysis,Ihaveusedatelluricstandardstarmethoddescribedby Vaccaetal. ( 2003 )toremovetelluriccontaminationfromobservedstellarspectrumanddemonstrated2.7%removalresidual.IhaveforthersttimeintroducedthebinarymaskcrosscorrelationtechniqueintotheNIRRVmeasurementdataanalysis.Thistechniquehasonlybeenusedfortheopticalwavelengthsbeforeby Baranneetal. ( 1996 ).Forhardwaredevelopment,IwasmainlyworkingwithDr.XiaokeWanonthedevelopmentofthesinesource.WehavetogetherconductedfeasibilitydemonstrationinlabandattheKPNO2.1mtelescope,Iwaspartiallyinvolvedinexperimentanddataacquisitionandmainlyinchargeofdataanalysis.Thesinesourcehasbeendemonstratedastableandprecisewavelengthcalibrationsource 174

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andfurtherdemonstrationexperimentisrequired.IwasheavilyinvolvedintheFIRSTsurveyproposal.Mycontributionismainlyontargetselectionandplanetyieldsimulation.Ipredictedthat30planetswillbedetectedincluding10super-Earths,2giantplanetsand18intermediate-massplanets. 7.4Chapter5TheothermajorconclusionfromChapter2isthatasurveyoflargesamplestarsshouldadopttheDFDImethodatalow-mediumspectralresolution.TheMARVELSprojectisplanetsurveythatfollowsthisconclusion.Thisprojecttakesenormouseffortindesign,instrumentbuilding,commissioning,operatinganddataanalysis.Mymajorcontributiontothisprojectisinstrumentationandinterferometercalibration.TheRVerrorduetointerferometerislimitedwithin2ms)]TJ /F5 7.97 Tf 5.07 0 Td[(1.OwingtomyworkinMARVELSinterferometergroupdelaycalibration,over250binaryandadozenofbrowndwarfshavebeendiscoveredandtheyprovideavaluableinsightofformationandevolutionoflow-massstellarcompanionandbrowndwarf.IwasawardedtheArchitectstatusfortheMARVELSprojectin2012formycontributiontothisprojectandwillbeguaranteeddataaccess.IhaveenvisionedanM-dwarfplanetsurveyandprovidedaconceptstudyofsuchsurvey. 7.5Chapter6ThischapterreportsmycollaborationworkwithDr.EricFordontheeccentricitydis-tributionofshort-periodplanet.WeapplystandardMCMCanalysisfor50short-periodsingle-planetsystemsandconstructacatalogoforbitalsolutionsfortheseplanetarysys-tems.Wedevelopanewmethodtotestthesignicanceofnon-circularorbits()]TJ /F1 11.357 Tf 9.93 0 Td[(analysis),whichisbettersuitedtoperformingpopulationanalysis.Ourresultssuggestthatbothtidalinteractionsandexternalperturbationsmayplayrolesinshapingtheeccentricitydistributionofshort-periodsingle-planetsystemsbutlargesamplesizesareneededtoprovidesufcientsensitivitytomakethesetrendsstatisticallysignicant. 175

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BIOGRAPHICALSKETCH JiWangwasborninGuilin,abeautifulcitylocatedinthemountainousareainSouthernChina.18yearsafterhappilylivinginGuilin,hemovedtotheUniversityofScienceandTechnologyofChina(USTC)inHefeitopursuehisdreamofbeingascientistandstudyingabroad.AtUSTC,thebestuniversityinChina,hestartedtolearnandtoappreciatetheconceptofhardworkandsmartthinking.HebuiltasolidfoundationforstudyandresearchatUSTCforthecomingyearsbeforebeingadmittedintotheAstronomyprogramattheUniversityofFlorida.Hespent6goodyearsinFlorida,wherehestudiedhardbutexercisedharder.Thankstoallthesportfacilitiesoncampus,hebecameanall-aroundathleticwhoisgoodatbasketball,tennis,golf,swimming,longdistancerunning.Nowheisreadyforhisnextstop,YaleUniversitywherehewillworkasapost-docintheDepartmentofAstronomy.HopefullyinafewyearshewillndatenuretrackpositionsomewhereeitherintheUSorbackinChina. 185