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Understanding and Reducing Dispersed Fixed-Delay Interferometric Data for Extrasolar Planet Searches

Permanent Link: http://ufdc.ufl.edu/UFE0021717/00001

Material Information

Title: Understanding and Reducing Dispersed Fixed-Delay Interferometric Data for Extrasolar Planet Searches
Physical Description: 1 online resource (191 p.)
Language: english
Creator: Van Eyken, Julian C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: 51peg, dfdi, doppler, et, exoplanets, extrasolar, hd102195, instrumentation, interferometer, michelson, pipeline, planets, radial, reduction, rotation, spectrograph, spectroscopy, velocity
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: The dispersed fixed-delay interferometer represents a new instrument concept for high-precision radial velocity surveys for extrasolar planets. A combination of an interferometer in series with a medium resolution spectrograph, it has the potential for performing multi-object surveying down to faint magnitude limits, where previous radial velocity techniques have been limited to observing only one target at at time. The sample of relatively bright stars that can quickly be surveyed using traditional radial velocity techniques with current technology is becoming exhausted, and the radial-velocity planet discovery rate is beginning to level out. Because of the large sample of extrasolar planets needed to perform statistical analyses to be able to understand aspects of planetary formation and evolution, such a multi-object instrument represents the next logical step in instrumentation for extrasolar planet research. As a useful by-product, the instrument has the potential to provide simple measurements of the projected rotational velocities of stars. The development of this instrument has necessitated the development of new data reduction procedures to efficiently and reliably turn the raw data into meaningful results, and a careful consideration of precision levels and sources of error. This in turn has required fleshing out a detailed understanding of the physical principles of the instrument. The single-object Exoplanet Tracker installed at Kitt Peak National Observatory and the multi-object W. M. Keck Exoplanet-Tracker at Apache Point Observatory are the first two fully-fledged astronomical radial velocity instruments to have been built based on this new technique. The former served to prove the concept with a successful confirmation of the known planet, 51 Peg b. We were later able to use it to discover a new planet, HD 102195 b, or ET-1. The latter has also demonstrated detection of known planets, and has been the workhorse for a pilot survey in preparation for a planned major wide-field survey over the next decade.
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 Julian C Van Eyken.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Ge, Jian.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021717:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021717/00001

Material Information

Title: Understanding and Reducing Dispersed Fixed-Delay Interferometric Data for Extrasolar Planet Searches
Physical Description: 1 online resource (191 p.)
Language: english
Creator: Van Eyken, Julian C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: 51peg, dfdi, doppler, et, exoplanets, extrasolar, hd102195, instrumentation, interferometer, michelson, pipeline, planets, radial, reduction, rotation, spectrograph, spectroscopy, velocity
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: The dispersed fixed-delay interferometer represents a new instrument concept for high-precision radial velocity surveys for extrasolar planets. A combination of an interferometer in series with a medium resolution spectrograph, it has the potential for performing multi-object surveying down to faint magnitude limits, where previous radial velocity techniques have been limited to observing only one target at at time. The sample of relatively bright stars that can quickly be surveyed using traditional radial velocity techniques with current technology is becoming exhausted, and the radial-velocity planet discovery rate is beginning to level out. Because of the large sample of extrasolar planets needed to perform statistical analyses to be able to understand aspects of planetary formation and evolution, such a multi-object instrument represents the next logical step in instrumentation for extrasolar planet research. As a useful by-product, the instrument has the potential to provide simple measurements of the projected rotational velocities of stars. The development of this instrument has necessitated the development of new data reduction procedures to efficiently and reliably turn the raw data into meaningful results, and a careful consideration of precision levels and sources of error. This in turn has required fleshing out a detailed understanding of the physical principles of the instrument. The single-object Exoplanet Tracker installed at Kitt Peak National Observatory and the multi-object W. M. Keck Exoplanet-Tracker at Apache Point Observatory are the first two fully-fledged astronomical radial velocity instruments to have been built based on this new technique. The former served to prove the concept with a successful confirmation of the known planet, 51 Peg b. We were later able to use it to discover a new planet, HD 102195 b, or ET-1. The latter has also demonstrated detection of known planets, and has been the workhorse for a pilot survey in preparation for a planned major wide-field survey over the next decade.
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 Julian C Van Eyken.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Ge, Jian.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021717:00001


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IwouldliketoexpressmygratitudetoallthoseatPennState,theUniversityofFlorida,andelsewherewhohavehelpedmeinputtingthisworktogether,andwhohavehelpedtokeepmesane:toJianandthewholeETinstrumentteamforkeepingETaoatandgivingmetheopportunitytoworkonagreatproject;tomycommitteemembers,RobertBuchler,SteveEikenberry,ElizabethLada,andAtaSarajedini,fortheirhelpfulinput;toDimitri,Brian,andSuvrath,mypartnerincrime,forproofreading,andmuchhelpfuldiscussionandexchangeofideas;tothestaoftheKittPeakandApachePointobservatories,fortheirpatienceandwillingnesstohelp;toLarryRamsey,foragooddoseofsanityandperspective;toSteinnSigurdsson,forbeingsoencouraging{andgettingmeintothiswholemessintherstplace;toTamaraand(again)Suvrath,theothertwomusketeers,fortheirpatientacceptancewhileIooadedallmycultureshockandstress;toallthoseatPennStatewhobroughtmeoverhereandturnedmeintoaproto-astronomer;andtothegradstudentsatUFastroandmycontemporariesatPennState:therecanbefewnercrowds,andIwishtherewasroomtonameeveryeverysingleoneofyou.(Ally'all.Andthankyou,incidentally,forclearingallmyforgottenbiologyexperimentsoutofthefridgeasIbecameincreasinglyoutoftouchwiththeoutsideworldduringthelastfewmonths.Ididnotice.)IamalsotremendouslygratefulforallthenancialsupportIhavereceived,fromthegenerousPennStatefellowship;fromtheSPIEscholarshipprogram;andfromtheNASAJPLMichelsonFellowshipprogram.Finally,IwouldespeciallyliketothankNoraandFr.Guy,forhelpingkeepmyfeetonthegroundwhileIwastryingtoseebeyondtheclouds.IamgenuinelynotsureIcouldhavedonethiswithoutyou.ToGod:foradventuresindistantlandsthatIwouldonceneverhavedreamedof;forsurroundingmewithsuchwonderfulfriends;andforaUniversevaster,strangerandmorefantasticalthananyhumanction. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 LISTOFSYMBOLSANDABBREVIATIONS ..................... 12 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1Background ................................... 15 1.2ExoplanetResearch ............................... 16 1.3TheSearchforExoplanets ........................... 18 1.4TheRadialVelocityTechnique ......................... 20 1.5TheETProgramandtheDFDIConcept ................... 23 1.5.1TheNeedforaNewInstrument .................... 23 1.5.2TheDFDIPrinciple ........................... 24 1.5.3ABriefHistory ............................. 25 1.5.4TheETproject ............................. 27 2INSTRUMENTPRINCIPLESANDTHEORY .................. 28 2.1FormationofaFringingSpectrum ....................... 28 2.2FringePhaseandVisibility ........................... 31 2.3TheInterferometerComb ........................... 35 2.4FromPhasetoVelocity ............................. 38 2.5CalculatingtheInterferometerDelay ..................... 40 2.6HandlingaFiducialReferenceSpectrum ................... 43 2.6.1TheAdditionApproximation ...................... 45 2.6.2AnAlternative:Combined-BeamReference .............. 48 2.7PhotonErrorPropagation ........................... 49 2.7.1PhotonErrorforMultipliedReference ................. 49 2.7.2PhotonErrorforReferenceSpectruminAddition .......... 52 3INSTRUMENTHARDWARE ............................ 55 3.1SubsystemOverview .............................. 55 3.2TheSingle-ObjectETatKPNO ........................ 58 3.2.1TestRun,2002 ............................. 59 3.2.2Upgrades,2004 ............................. 59 3.2.3ThecurrentKPNOET ......................... 60 5

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...................... 61 4THEDATAREDUCTIONPIPELINE ....................... 65 4.1ObservingProcedureandCalibrations .................... 66 4.2UserFront-End ................................. 67 4.3FibreMapping ................................. 70 4.4DataReduction ................................. 71 4.4.1Preprocessing .............................. 72 4.4.1.1Bias/darksubtractionandatelding ............ 72 4.4.1.2Badpixelmasksandcosmicrayrejection .......... 72 4.4.1.3Trimming ........................... 73 4.4.1.4Slantcorrection ........................ 73 4.4.1.5Illuminationcorrection .................... 76 4.4.1.6Filtering ............................ 77 4.4.2FringeFitting .............................. 78 4.4.3RadialVelocityMeasurement ...................... 80 4.4.3.1Referenceextraction ..................... 80 4.4.3.2Bulkshiftcompensation ................... 84 4.4.3.3Animprovedapproach .................... 85 4.4.4BarycentricCorrection ......................... 86 5SOURCESOFERROR ............................... 88 5.1ErrorBars .................................... 88 5.2FringeFitting .................................. 91 5.3IlluminationCorrection ............................. 91 5.4SlantCorrection ................................. 95 5.5FluxCentroiding ................................ 97 5.6BarycentricCorrection ............................. 98 5.7ContaminatingSpectra ............................. 98 5.8MoonlightContamination ........................... 102 5.9ResidualInterferometerComb ......................... 103 5.10TheAdditionApproximation ......................... 104 5.11OtherSourcesofError ............................. 108 5.12PlanetDetectionLimits ............................ 110 6RESULTSFROMTHEKPNOET ......................... 112 6.1Conrmationof51Pegb ............................ 112 6.1.1Observations ............................... 112 6.1.2DataAnalysis .............................. 112 6.1.3Results .................................. 114 6.1.4Discussion ................................ 116 6.2DiscoveryofET-1 ................................ 119 6.2.1EarlyMeasurements ........................... 120 6.2.1.1Observationsanddataanalysis ............... 120 6

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........................... 122 6.2.1.3Photometry .......................... 126 6.2.2FollowupandConrmation ....................... 128 6.2.3Discussion ................................ 134 6.3Addendum:51Pegb,ET-1,andtheAdditionApproximation ....... 134 7RESULTSFROMTHEMULTI-OBJECTKECKETATAPO ......... 138 7.1Prototype .................................... 139 7.2May2006Results ................................ 140 7.3November2006Results ............................. 142 7.4Discussion .................................... 146 8MEASURINGSTELLARROTATIONWITHADFDI .............. 150 8.1Background ................................... 150 8.2TheoreticalPredictions ............................. 152 8.2.1TheBroadenedLineProle ...................... 152 8.2.2FringeVisibility ............................. 153 8.2.3PredictedRelation ........................... 156 8.3Simulations ................................... 159 8.3.1Description ................................ 159 8.3.2ResultsandDependenceofvsinionStellarParameters ....... 160 8.3.3ComparisonwithTheoreticalPredictions ............... 164 8.4Observations .................................. 165 8.5Conclusions ................................... 169 9CONCLUSIONSANDCONTRIBUTIONS ..................... 171 9.1WorkRemaining ................................ 171 9.1.1RVmeasurement ............................ 171 9.1.2Stellarrotation ............................. 173 9.2FuturePossibilities ............................... 173 9.3SummaryandCurrentStatus ......................... 174 9.4Contributions .................................. 176 9.5Credits ...................................... 178 APPENDIX ATHESPECTROGRAPHRESPONSEFUNCTIONANDTHELSF ....... 180 BFRINGEFORMATION:ANALTERNATIVEVIEWPOINT .......... 182 REFERENCES ....................................... 184 BIOGRAPHICALSKETCH ................................ 191 7

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Table page 3-1KeckETinstrumentspecications ......................... 63 6-1RVmeasurementsfor51Peg ............................. 116 6-2RVmeasurementsforCas ............................. 118 6-3Meanphotonlimitingerrorestimationfor51PegandCasobservations .... 118 6-4OrbitalparametersforHD102195 ......................... 130 6-5CompleteradialvelocitiesforHD102195 ...................... 131 6-6StellarparametersforHD102195 .......................... 133 8-1Dependenciesofmeanvisibility, ....... 163 8

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Figure page 2-1Dispersedinterferometerschematic ......................... 29 2-2Arepresentationofthesummation(thickline)offringesduetoeachindividualwavelengthofwhitelight(thinlines) ........................ 34 2-3Interferogramshowingthecoherenceenvelopeduetoarectangularbandpass 36 2-4Simulatedinterferometercomb ........................... 37 3-1OpticaldesignofETattheKPNO2.1mtelescope ................. 61 3-2OpticaldesignoftheKeckETinterferometer ................... 63 3-3OpticaldesignofthespectrographfortheKeckET ................ 64 4-1ExamplescreenshotfromtheETpipelinegraphicaluserinterface ........ 69 4-2ExamplescreenshotfromtheETelectronicobservinglog ............. 70 4-3Examplenon-fringingThArframefromthemulti-objectKeckET ........ 74 4-4Slantcorrectionofanon-fringingThArspectrum ................. 75 4-5Pre-processingsteps ................................. 77 5-1Phaseandvisibilityerrorsduetocurve-ttingalone ................ 92 5-2Fringettingerrorsforpoorilluminationcorrection ................ 94 5-3Eectofuniformarticialspectrumshiftinthedispersiondirection ....... 96 5-4Fringealongonechannelduetotarget(uppercurve),andcontaminatinglowuxfringe(lowercurve) ............................... 99 5-5Vectorrepresentationofthesummationofthefringesduetothetargetsourceandbackgroundcontamination ........................... 100 5-6Simulationsofmoonlightcontamination ...................... 103 5-7Vectorrepresentationofthesummationofthetruecomplexvisibilityandtheerrortermduetotheadditionapproximation ................... 106 5-8Analyticallycalculatedexpectederrorduetotheadditionapproximation .... 108 5-9Simulationsshowingtheadditionapproximationerror ............... 109 5-10Radialvelocitysemi-amplitude,K,fordierentminimumplanetmasses ..... 111 6-1Smallsectionofrawfringingspectrumof51Peg .................. 114 9

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........................ 115 6-3PlotofRVmeasurementsforCas ......................... 117 6-451PegRVmeasurementsinDecember2004withtheupgradedET ....... 120 6-536UMa(knownRVstablestar)short-termprecisionmeasurementswiththeupgradedET ..................................... 121 6-6ExamplesectionofrawspectrumtakenwithupgradedET ............ 122 6-7Knownplanet-bearingstar,55Cnc,measuredwithcurrentKPNOET ..... 123 6-8RVstablestar,36UMa,measuredoverafewdayswithcurrentKPNOET ... 124 6-9BesttKeplerianorbitforearlyET-1RVmeasurements ............. 125 6-10Asforgure 6-9 ,butphasefoldedonthebest-tperiod ............. 126 6-11Lomb-ScargleperiodogramforearlyET-1data .................. 127 6-12EarlyphotometryofHD102195 ........................... 128 6-13FoldedcombinedradialvelocitiesforHD102195 .................. 129 6-14UpdatedperiodogramforHD102195 ........................ 132 6-15Dierentialbarycentriccorrectionsforearly51Pegmeasurements ........ 136 6-16DierentialbarycentriccorrectionsforHD102195measurements ......... 137 7-1Resultsfromthe20-objectprototype ........................ 141 7-2ExamplerawdataframefromearlyKeckET ................... 143 7-3ExamplesolardatafromearlyKeckET ...................... 144 7-4Knownplanet-bearingstarsfromearlyKeckETdata ............... 145 7-5SmallselectionofdierentexamplesearchstarresultsfromearlyKeckETdata 146 7-6KeckETmeasurementsofHD209458fromNovember2006 ........... 147 7-7KeckETmeasurementsofHIP14810(TYC123117271)fromNovember2006 147 7-8Representativeselectionofday-skyRVdatafromKeckET,November2006 .. 148 8-1Top-hatresponsefunctioncentredonasingleGaussianabsorptionline ..... 155 8-2Gaussianbest-tapproximationtoanormalisedrotationalbroadeningprole 157 8-3Analyticallypredictedvisibilityvs.vsinicurve .................. 159 10

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.. 161 8-5Contoursofvisibilityvs.vsinifromsimulations .................. 162 8-6Simulateddataoverplottedonthetheoreticalcurvefromgure 8-3 ....... 164 8-7Realdataandoverplottedsimulationsforawell-matchedsubsetofobservedtargets ......................................... 167 8-8Alleighttargetswithoverplottedvisibilitycontoursfromsimulations ...... 168 11

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Wolszczan&Frail ( 1992 ).Usingexquisitelyprecisepulsetimingmeasurements,theywereabletodetectthetinyshiftinthedistanceofthepulsarfromtheEarthduetothepulsar'sreexmotionasitsplanetsorbited.Therstunambiguous`traditional'planetarycompaniontoamain-sequencesolar-typestarwasdetectedin1995,orbiting51Pegasi.Usingtheradialvelocity(RV)technique, Mayor&Queloz ( 1995 )detectedthetinyDopplershiftinthespectrumofthestarduetoitsreex`wobble'causedbyitsorbitingcompanion.Moreplanetdetectionsquicklyfollowed(e.g., Marcy&Butler 1996 ; Butler&Marcy 1996 ; Butleretal. 1997 ),andtheeldofexoplanetresearchwasborn.Arecentcompilationoftheknownexoplanetswasgivenin Butleretal. ( 2006 ).

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Ida&Lin 2004a b 2005 ; Armitage 2007 ),whichinturncanbegintotellusabouttheformationofourownSolarSystem,ourownEarth,andultimatelyouroriginsandplacewithintheUniverse.Thediscoveryofexoplanetshasprovidedanimportantstepinthequesttondlife{ifitexists{elsewhereintheUniverse(e.g. Seageretal. 2005 ; Seguraetal. 2005 ; Raymondetal. 2006 ),andhasbeenanimportantcatalystforthebirthoftheeldofastrobiology( Morrison 2001 ).Thenowapparentubiquityofplanetsfullsamajorrequirementfortheexistenceoflife,asfarasweunderstandit,beyondourownworld. Perryman 2000 ; Marcyetal. 2005 ; Udryetal. 2007 ,forreviewsoftheeld).Around1%ofthestarssurveyed,forexample,harbour`hotJupiters',veryclose-ingas-giantplanetswithminimummassesontheorderofaJupitermassormore,andorbitalperiodsofonlyafewdays(<10d)( Udryetal. 2007 ).Withtheexceptionoftheextremelyclose-inplanetswheretidalinteractionwiththehoststarisexpectedtohavecircularisedtheorbits,themajorityofplanetsarealsoonveryeccentricorbits,withamedianeccentricityof0.26( Udryetal. 2007 ),unlikeourownsystemwheretheyareallonhighlycircularisedorbits.Averywiderangeofminimumplanetmasseshasalsobeenfound,fromthelowerlimitofdetection{aroundaNeptune-massandrecentlyevenseveral{Earth-masses{uptotherareinstanceswheretheminimummassis10MJ,rightontheboundarywherethedistinctionbetween 16

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Kleyetal. 2004 ,andreferencestherein),othersinteractingmorestronglyinchaoticsystems(e.g. Correiaetal. 2005 ).Finally,assurveytimebaselineslengthenandsensitivityimproves,systemssimilartoourownSolarSystem(`SolarSystemanalogues')arejustbecomingaccessible( Marcyetal. 2002 ).Inordertomakeasoliddetectionofanexoplanet,itisgenerallynecessarytoobservethecandidatetargetforthedurationofatleastoneorbitalperiod.Largelyforthisreason,SolarSystemanaloguesareonlynowbecomingaccessibletocurrentdetectiontechniques:Jupiterwouldpresentthemosteasilydetectableofourplanetswerewetoviewitfromanotherstellarsystem,butthebaselineofobservationssincethestartofmajorplanetsurveysisonlynowbecominglongenoughtomatchits12-yearorbitalperiod.Thisdiversityhasraisedmanyquestionsinexoplanetresearch.Someofthecurrentmajorquestionsinclude:Howubiquitousareexoplanets?Fromcurrentsurveys,6%ofsolar-type(i.e.generallylateF-toearlyM-typemainsequence)starshavebeenfoundtoharbourplanetsdowntoacompletenesslimitof0:5MJ,andaround1%harbourhotJupiters.Oftheknownplanetbearingstars,12%haveknownmultipleplanetsystems,andthismayinfactbethenorm.Assurveysincreaseinsensitivity,itisbeginningtoappearthateventhemajorityofsolar-typestarsmayharbourplanets( Udryetal. 2007 ).Howdopropertiesofthehoststaraectthoseofcompanionplanets?Planetpresenceisfoundtobestronglycorrelatedwithhoststarmetallicity( Fischer&Valenti 2005 );itisalsostartingtoappearthatmoremassivestarsmaypossiblyleadtomoremassiveplanets( Satoetal. 2007 ; Ida&Lin 2005 ). 17

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Marcyetal. 2005 )uptothebrowndwarfmassboundaryat13MJ.Thereisapronouncedabsenceoflowmasscompanionsinthemassrange20{60MJ,knownasthe`browndwarfdesert',beyondwhichpointhydrogenburningstarsbegintotakeover.Thisistakentosuggestdierentformationmechanismsforstarsandplanets.Whatistheorbitalperioddistribution?Thereisfoundtobeapileupintheperioddistributionofplanetsatperiodsofabout3d.Veryfewplanetsarefoundinorbitsshorterthanthisperiod.Goingouttolongerperiods,thereisaslowincreaseinoccurrenceagainwithincreasingperiod.Insingle-planetsystemswithperiodslessthan100d,onlyplanetsofmasslessthan2MJhavebeenfound.Whatistheeectofstellarbinarityonplanetformation?Owingtotheaddeddicultyinvolvedinndingplanetsaroundmultiplestarsystems,theytendtohavebeenexcludedfromsurveys,andsothisremainssomewhatofanopenquestion( Udryetal. 2007 )Whatgivesrisetothehigheccentricitiesofexoplanetorbits?Ifplanetsforminprotostellardebrisdiscs,onemightnaivelyexpectsimplecircularorbitstoresult,andyetmostplanetsareonsurprisinglyeccentricorbits.( Tremaine&Zakamska 2004 ).Whatdothesequestions,alongwithothercorrelationsbetweenplanetparameters,tellusabouttheformationandevolutionofplanets?Eachpieceofstatisticalinformationgatheredfromplanetsurveysultimatelyprovidescluestoguideandconstrainthetheoreticalmodelsthattellusaboutplanetaryformation(e.g. Ida&Lin 2004a b 2005 ; Armitage 2007 ). Beuzitetal. 2007 ,andreferencestherein).Mosttechniques,however,dependonindirectobservationsofthehoststartoinferthepresenceofacompanionplanet.Oneoftheearliestclaimstothedetectionofaplanetwasmadeby vandeKamp ( 1963 )usingastrometry,orbitingBarnard'sstar,althoughthisclaimwaslaterdiscreditedby Gatewood&Eichhorn ( 1973 ).Aplanet-bearingstarexecutesaverysmallreex 18

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Perryman 2000 ).Todate,thetechniquehasnotseenanysuccessindiscoveringplanets(see Sozzetti ( 2005 )forafullreviewoftheeld).Evenmoreindirect,buttothispointmoresuccessful,isthegravitationallensingtechnique(e.g. Gouldetal. 2006 ),wherethepropermotionofafaintplanet-bearingstarcausesittopassinfrontofanothermuchbrighterstar,gravitationallylensingthebackgroundstar'slighttowardstheEarthandcausingabriefbrighteninginmagnitude.Smallexcursionsintheexpectedshapeofthelightcurveofthebackgroundstarduringtheeventbetraytheexistenceofaplanetarycompanion.Theeventsareonlyone-timedetectionswhichcannotberevisited,however,andthehoststarsaretypicallyextremelyfaint,makingthemveryhardtofollowupwithothertechniquesandlimitingtheinformationthatcanbegainedabouttheplanet.Asearlyasthe1950s, Struve ( 1952 )pointedoutthatafractionofexoplanets,thosewhichtransittheirparentstars,shouldbedetectablethroughphotometryofthehoststarduringthetransit.Thetransitmethodhastheadvantagethatitisrelativelyeasytotargetmanystarsatonceinawideeld,andsowhatitlosesintermsofnumbersofplanetswhichtransit,itgainsinsurveynumbers;italsoisuniquelyabletodeterminetheradiusoftheplanet,anditsorbitalinclination.Thetransitmethodhasbeenslowtotakeobuthasrecentlybeenrapidlygainingground( Charbonneauetal. 2007 ).TheoriginalpulsartimingmethodemployedbyWolszczanhassincebeenrelativelyunproductive.However,thesametechniquehasbeenappliedtosimilarlyexoticpulsatingwhitedwarfs,andisshowingsomepromise( Mullally&Winget 2006 ; Mullallyetal. 19

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).Evenmorerecently,arstplanethasbeenclaimedusingtimingofpulsationsofapost{red-giantstar,adramaticnewextensiontotherangeofstartypeswhichhavebeenfoundtohostplanets( Silvottietal. 2007 ).Ofallthetechniquesforndingexoplanets,however,byfarthehighestyieldhascomefromtheradialvelocitytechnique,anditisthisthattheETinstrumentsdependon. 20

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Gray 1992 ).Whentheradialvelocityofthestarchanges,abulkDopplershiftinthewavelengthoftheabsorptionlinesresults,anditisthisthatenablesthemeasurementofchangesinradialvelocity(specically,theinformationontheDopplershiftisintheslopesofthelineproles).Alongwiththetransitmethod,theradialvelocitytechniquewasproposedby Struve ( 1952 )astheprimarymethodforndingexoplanets.Exoplanetradialvelocitysurveyshavetraditionallydependedonrecordingveryhighresolutionspectratoobtainwellresolvedabsorptionlines,andeithercrosscorrelatingthespectrawithreferencetemplatespectra,orttingfunctionstothelineprolesthemselvestomeasurethepositionsofthecentroids.Bestinternalprecisionshavetypicallyreacheddowntothe3ms1level( Butleretal. 1996 ; Vogtetal. 2000 ),andarenowreachingaslowas1ms1orbetter( Pepeetal. 2005 ).(Forcomparison,aJupiteranalogueinacircularorbitaroundasolar-typestarwouldcausesinusoidalradialvelocityvariationswithanamplitudeofabout12:5ms1.)Theveryhighlevelsofprecisionrequiredforplanetdetectionandthedicultyofdirectlymeasuringabsolutewavelengthsmeansthatsomekindofstationaryreferencespectrumisinvariablyusedasacalibration.EarlyRVmeasurementswerelimitedintheirprecisionlargelybecauseofdierencesbetweenthesourceandreferenceilluminationprolesenteringtheinstrument( Grin&Grin 1973 ).Furthermore,thereferencewasofteneitherpassedalongaseparatebeampaththroughtheinstrument,ortakenatadierenttime:inbothcases,dierentialinstrumentdrifts(e.g.duetothermalexureorchangesinairpressure)wouldaectsourceandcalibrationdierently. Grin&Grin ( 1973 )madetherstattemptstoovercometheseproblemsbyusingasimultaneoussuperposedreference:inthiscase,theyproposedusingthetelluriclinesimposedbytheEarth'satmosphere.Inthisway,thereferencespectrumpassedthroughtheinstrument 21

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Campbell&Walker ( 1979 )overcametheseproblemsbyinsertingaglasscelllledwithhydrogenuoride(HF)gasintotheopticalpathoftheinstrumentasaducialreference,superimposingawellcontrolledsetoflinesintherange8670{8770Aandachievingaprecisionof15ms1.Ofthe21starstheyhadobservedover12years,nonewerefoundtoharbouranyplanetsattheirdetectionlimits( Walkeretal. 1995 ),thoughthisresultisconsistentwiththenow-knownfractionofplanet-bearingstars( Udryetal. 2007 ).Hydrogenuoridealsopresentedanumberofproblemsasareference,however,nottheleastofwhichwasthatthegasiscorrosiveandhighlytoxic.Thenow-favourediodinereferencewasrstadoptedby Marcy&Butler ( 1992 ).SubstitutingaglasscelllledwithheatediodinevapourratherthanHFhadtheadvantageofallowingforamuchsmallercelllength,andcreatedareferencespectrumthatcoveredthemuchbroaderwavelengthrangeof5000{6300A.Sincebroaderwavelengthcoverageamountstomoreux,onecouldsearchmuchfaintertargets,wherepreviousmagnitudelimitshadsignicantlylimitedthenumberofstarsavailableforsearch.Theyearssurroundingthesedevelopmentsalsosawthebirthoftheopticalbreinastronomy.Opticalbresallowedtheinstrumenttobecompletelyseparatedfromthetelescope,sothatitcouldbesituatedinabettercontrolledandmorestableenvironment.Furthermore,theopticalscramblingpropertiesofbrescouldbeusedtoerasethechangingspatialstructureofthebeamprolebeforeenteringtheinstrument,sothataconsistentilluminationwasalwayspresented( Heacox 1986 ).Itthusbecamepossibleforthersttimetoconsiderusingaparallelreferencebeamrunningalongsidethemainsciencebeamforplanethunting.ThiswastheapproachadoptedfortheELODIE 22

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Baranneetal. 1996 ),usingaThoriumArgon(ThAr)emissionspectrumasareference,anditwasELODIEthatwasusedtorstdiscover51Pegb( Mayor&Queloz 1995 ).IntimeMarcy'sgroupwasabletopushtheirRVprecisiondowntothe3ms1level( Butleretal. 1996 )usingtheiodinetechnique.TherearenowanumberofotheractiveplanetsurveysusingtheRVmethod,includingtheLick,KeckandAATprogram( Marcyetal. 2005 );theCORALIE( Udryetal. 2000 ,andreferencestherein),ELODIE( Baranneetal. 1996 ; daSilvaetal. 2006 ),andHARPS( Pepeetal. 2000 2005 )programs;andtheMcDonaldObservatory( Cochran&Hatzes 1993 )andAFOE( Brownetal. 1994 )programs,amongothers.Precisionsdowntobetterthan1ms1arenowbeingreachedbytheHARPSgroup,usingparallelThAr(withtheoptionofusingiodineasanalternative).Areviewofradialvelocitydiscoveriesisgivenin Udryetal. ( 2007 ). 1.5.1TheNeedforaNewInstrumentDespitetheachievementsinplanetdetection,moreplanetsarestillneededtoconstrainformationandevolutionarymodels.Thisisinpartbecauseofthediversityofplanetpropertiesthathasbeenfound;butitisalsobecause,todate,themajorityofsurveyshavenotusedwell-denedunbiasedtargetlists,makingitdiculttoperformrobuststatisticalanalyses.Tothispoint,theprimaryconcernhasbeenndingplanetsintherstplace,andmostsurveyssuerfromcompletenessissuesorbiasestowardplanetdetection(e.g. daSilvaetal. 2006 ). Armitage ( 2007 )concludesthatthereisstillastrongneedforlargeuniformsurveystoenlargethestatisticalsampleavailable:ofalltheplanetsknown,hewasonlyabletondauniformsubsampleof22thatsatisedtherequirementsneededforastatisticalcomparisonwithmodels.AfewthousandstarshavebeensearchedbetweenthevariousRVsurveys,includingmoststarsdowntovisualmagnitude8,butinstrumentlightthroughputissueshavemadeitlesseasytogomuchfainterthanthis,andRVdetectionrateshavebeenlevelling 23

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Kaneetal. 2004 ).Furthermore,thecomplementaryinformationgainedfromRVdetectionsremainsofgreatvalue.ThereisthereforeastrongcaseforndingatechniquewhichenablesRVsurveyingdowntofaintermagnitudesandatconsiderablyfasterspeedsthanhavebeenachievedoverthelastdecade.ThegoaloftheETprogramdescribedinthisworkistosatisfythisrequirement. Erskineetal. ( 2006 )asanexternallydispersedinterferometer,or`EDI').Theeectiveresolutionoftheinstrumentisdeterminedprimarilybytheinterferometer,sothepost-dispersingspectrographcanbeofmuchlowerresolutionthanintraditionaltechniques,andconsequentlycanhavehigherthroughput( Ge 2002 ; Geetal. 2003b a ).ThetechniqueiscloselyrelatedtoFouriertransformspectroscopy:thepost-dispersereectivelycreatesacontinuumofverynarrowbandpassesfortheinterferometer,increasingtheinterferencefringecontrast.Alltheinformationneedediscontainedinthefringephaseandvisibility(seechapter 2 ).ItturnsoutthatsinceweareonlyinterestedintheDopplershiftofthelines,measurementsarerequiredatonlyonevalueofinterferometerdelay(hence`xeddelay').Thecostoftheinstrumentiscomparativelylow,andmostimportantly,itoperatesinasingle-ordermode.WheretraditionalechellespectrographtechniquesoperatebyspreadingasinglestellarspectrumoveranentireCCDdetectorinmultipleorders,herethespectrumonlytakesuponestripalongthedetector.Spectrafrommultiplestarscanthereforebelinedupatonceonasingledetector( Ge 2002 ).Incombinationwithawide 24

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Mahadevanetal. 2003 ). Erskine&Ge 2000 ; Ge 2002 ; Geetal. 2002 ; Erskine 2003 ).Thesameapproachisbeingfollowedby Erskineetal. ( 2006 )intheinfra-red,inanattempttondplanetsaroundlate-typestars.Asimilarapproachisdiscussedby Mosseretal. ( 2003 )forasteroseismologyandthemeasurementofstellaroscillations;morerecentlythetechniquehasalsobeenadoptedfortheUSNOdFTSinstrument( Hajianetal. 2007 )(inthislastcase,theinterferometerdelaycanalsobevariedsothathighresolutionspectracanbereconstructed).Theideaofdispersedinterferometryisbynomeansnew.Michelsonhimselfrecognisedtheuseofinterferometersforspectroscopy( Michelson 1903 ),andevenproposedcombiningadisperserinserieswithaMichelsoninterferometer.Inthiscasethedisperser,aprism,wasplacedbeforetheinterferometer,allowingonlyanarrowbandwidthoflighttoentertheinterferometerintherstplace,buttheunderlyingphysicsisessentiallyequivalent.InwhatwasprobablytherstrealisationofaDFDI, Edser&Butler ( 1898 )placedaFabry-Perottypeinterferometerinfrontofaspectrographtoproducedispersedfringes(eectivelyaninterferometercomb-seechapter 2 ),whichtheyusedasaducialreferenceformeasuringthewavelengthsofspectrallines.Suchdispersedfringeswerelatertobecomeknownas`Edser-Butlerfringes'( Lawson 2000 ).Thevariouscombinationsofinterferometerswithdisperserscameintousesomewhatlaterintheeldofastronomy. Geakeetal. ( 1959 )placedaFabry-Perotinterferometerbeforeaspectrographtoallowthespectrographslittobewidenedandhenceincreasethethroughput.P.ConnesdevelopedtheSISAMtechniqueforspectroscopy,replacingthemirrorsinaMichelsoninterferometerwithgratingstoselectasmallbandpassandso 25

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Jacquinot 1960 ).ThelaterSHS( Harlanderetal. 1992 )andHHS( Frandsenetal. 1993 ; Douglas 1997 )techniquessharedtheuseofinternallydispersedMichelsoninterferometers,butusedthedispersioninsteadforthepurposeofscanningarangeofinterferometerdelays,obviatingtheneedformovingparts. Barker&Hollenbach ( 1972 )wereabletomeasurethevelocityhistoryofaprojectileinthelaboratoryinreectedlaserlight,inanearlyexampleoftheuseoftruexed-delayinterferometryforvelocimetry.TheuseofaMichelsoninterferometerforastronomicalRVmeasurementswasproposedshortlyafterwardsby Gorskii&Lebedev ( 1977 )and Beckers&Brown ( 1978 ). Forrest&Ring ( 1978 )alsoproposedusingaMichelsoninterferometerwithaxeddelayforhigh-precisionDopplermeasurementsofsinglespectrallinesforthedetectionofstellaroscillations.Duringthe80s, Connes ( 1985 )proposedanoveltechniqueusinglasertrackingofaFabry-Perotinterferometer,whichinturntrackstheDopplershiftsofstellarspectrallinesforhighprecisionmeasurements.Morerecently, McMillanetal. ( 1993 )haveusedaFabry-Perotinterferometercombinedwithacross-dispersedechellespectrographforprecisionDopplermeasurements(seealso McMillanetal. 1994 ).Fixed-delayMichelsoninterferometerswithverynarrowbandpasseshavealsobeenusedforproducingDopplerimagesby Shepherdetal. ( 1985 )(theWAMDIIinstrument)formeasuringupperatmosphericwinds,and Harvey&TheGONGInstrumentTeam ( 1995 )(theGONGproject)forDopplermeasurementsacrossthesolardisc.Here,fringephasemeasurementsweremadeovertheeldofanimagetogiveDopplermeasurementsateachpointintheeldofview.Manyoftheseinterferometricinstrumentssueredfromthelimitationofhavinganextremelynarrowbandpass,tendingtolimittheirapplicationtoonlybrighttargets.TheDFDItechniqueusedintheETinstrumentsallowsforanarbitrarilywidebandpass,limitedonlybythespectrographcapabilities,whilestillretainingthehighresolution 26

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Erskine&Ge ( 2000 )and Geetal. ( 2002 ).TheETprojectbeganshortlyafter. Geetal. 2003b ; Mahadevan 2006 ).TwoETinstrumentshavenowbeenbuilt:thesingle-objectprototypeET,permanentlyinstalledattheKPNO2.1mtelescopein2003afteratemporarytestruninAugust2002,andthemulti-objectKeckET,installedattheAPOSloan2.5mtelescopeinMarch2005,andthenupgradedandmovedtoamorestablelocationlaterthatyear(seechapter 3 ).ProofofconceptwasachievedusingtheKPNOETwiththerstDFDIplanetdetection,aconrmationofthecompanionto51Pegasi( vanEykenetal. 2004a ).Ourrstplanetdiscovery,HD102195b(ET-1)wasalsolatermadeusingthisinstrument( Geetal. 2006a ).Themulti-objectKeckETisafullscaleinstrumentbeingdevelopedtosatisfythesurveyrequirementslaidoutinsection 1.5.1 ,anditishopedthatitwillhaveadramaticimpactontheplanetdiscoveryrateoverthecomingdecade. 27

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Goodman ( 1985 ); Erskine&Ge ( 2000 ); Lawson ( 2000 ); Erskine ( 2003 ); Ge ( 2002 ); Geetal. ( 2002 ); Mosseretal. ( 2003 ); vanEykenetal. ( 2003 ).Anattemptismadeheretodrawtogether,expandon,andmorepreciselystatethetheoreticalmaterialneededfordatareduction,andtoprovideasomewhatcompleteoverviewofthephysicsunderlyingtheinstrument'sworking.Thisprovidesuswiththefundamentalmathematicsnecessaryforwritingadatareductionpipeline. 2-1 showsahighlysimpliedschematicofadispersedxed-delayinterferometer(DFDI),consistingofthetwomaincomponents,abre-fedMichelsoninterferometerandadisperser,followedbyadetector.Lightinputfromthebreissplitintotwopathsalongthearmsoftheinterferometerandthenrecombinedatthebeamsplitter.Theoutputisfedtothedisperser,representedforconvenienceasaprism,thoughgenerallythiswillbeaspectrograph.Anetalonisplacedinoneoftheinterferometerarmstocreateaxedopticalpathdierence(or`delay'),d=d0,betweenthetwoarms,whileallowingforadequateeldwidening( Hilliard&Shepherd 1966 ).d0istypicallyontheorderofmm.Inpractice,aniodinevapourcellcanalsobeplacedintheopticalpathbeforetheinterferometertoactasaducialreference(section 2.6 ).Inputtingawidebeamofmonochromaticlightintotheinstrumentwithbothinterferometermirrorsexactlyperpendiculartothelighttravelpathwillgiveeitherabrightoradarkfringeattheoutputoftheinterferometer,asshowningure 2-1 A, 28

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Dispersedinterferometerschematic.ycorrespondstopositionintheslitdirection,andindicateswavelengthinthedispersiondirection.A)Outputfrominterferometeralonewithmonochromaticlightinput,andmirror2untilted.B)Thesamewithmirror2tiltedalongtheaxisintheplaneofthepage,asshown.C)Imageondetectorwithmonochromaticlight.D)Detectorimagewithwhitelightinput.E)Imagewithstellarspectruminput.F)AsforEbutatlowresolution. 29

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2-1 C.Switchingtheinputspectrumtowhitelight,whichcanbethoughtofasacontinuumofneighbouringdeltafunctions,leadstoasimilarfringepatternonthedetectorateverywavelengthchannel.Duetothefactthat,intermsofnumberofwavelengths,theopticalpathdierenceisdierentfordierentwavelengths,eachfringeisslightlyosetinphasefromitsneighbours(andveryslightlydierentinperiod).Thisgivesrisetotheseriesofparallellinesknownastheinterferometer`comb',showningure 2-1 D.Goingfurtherandinputtingastellarspectrumintotheinstrumentwouldsimplygivetheproductofthestellarspectrumandthecomb,asingure 2-1 E.Finally,changingtotherealcaseofalowormediumresolutionspectrographasforanET-typeinstrument,thecombisnolonger(orbarely)resolved,andweseeaspectrumlikethatingure 2-1 F.Suchaspectrumissometimesreferredtoasaspectrum\channelledwithfringes",alsoknownasEdser-Butlerfringes( Edser&Butler 1898 ; Lawson 2000 ; Ge 2002 ).TheremainingfringescontainhighspatialfrequencyDopplerinformationthathasbeenheterodyneddowntolowerspatialfrequenciesbytheinterferometercomb( Mahadevan 2006 ; Erskine 2003 ).Itisthisheterodyningthatallowsfortheuseofalow-resolutionspectrographatlowdispersion,andisthekeytotheDFDItechnique. 30

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Michelson 1903 ).Hereweintroducetheconceptofa`whirl'( Erskine&Ge 2000 ).Thephaseandvelocityforafringecantogetherbethoughtofasrepresentingavector,withthevisibilityrepresentingthemagnitude.Anarrayoftheserepresentingafullspectrumofwavelengthchannelsiscalledawhirl.By`wavelengthchannel',wemeanspecicallyaninnitesimallywidestripalongtheslitdirectionatpixelpositionj,wherejisnotnecessarilyaninteger.Thewhirliswhatwemeasuredirectlyfromafringingspectrumandcontainstheinformationrelevanttovelocitydetermination.Vectoroperationssuchasaddition,subtractionandscalarproductscanbeperformedonthesewhirlsjustasfortheindividualvectors( Erskine&Ge 2000 ).Tounderstandwhatdeterminesthevaluesofthephaseandvisibilityforafringe,wecanconsiderthecontributionfromeachwavelengthoflighttoaparticularwavelengthchannelonthedetector.Eachcontributingwavelengthhaspassedthroughtheinterferometer,andforanidealinterferometer,willcontributeasinusoidof100%visibilitylikethatingure 2-1 C.Theintensityofthesesinusoidsonthedetectorcaneachbedescribedby
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Jacquinot 1960 ).Normalisingbydividingthroughbythetotalux,wecandenethecomplexquantitysuchthat Goodman 1985 ),anddescribesthephase,,andamplitude,ofthenormalisedfringes(i.e.thevisibility),asafunctionofd.isreferredtohereasthecomplexvisibility.Morerigorousderivationsofthiscanbefoundin Goodman ( 1985 ,ch.5)and Lawson ( 2000 ),butthissimplicationisadequatefortheexplanationhere.InordertounderstandtheactualformofthefringesseeninDFDI,itisimportanttorealisethatthenatureoftheinstrumentissuchthatforanygivenwavelengthchannel,thecontributingspectrumQhasaverynarrowpassband(forourinstruments,=1A=5000A=2104).WeimagineQasbeingequaltoafunctionQ0ofcharacteristicwidthandcentredatzerowavenumber,whichhasbeenshiftedinwavenumbersothatitscentrefallsatwavenumber= F[Q0]d:(2{4) 32

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.BythereciprocalscalingpropertyofFouriertransforms,thesecondterm,theFouriertransform,canbeexpectedtovaryonscalesofthereciprocalofthewidthofQ0,thatis,onscalesof1=.Since1=1= ,therealpartoftherighthandsiderepresentsasinusoidalfringeoffrequency 2{2 wecanwrite: F[Q0]d RQ()d(2{5)Ifwedene: (d);(2{6)wecanrewriteequation 2{5 as: =1+V(d)cos(2d +V(d)): Thisclearlyshowstheformofthefringe.Overlargerangesofd,thefringeappearslikea`carrierwave',givenbythecosineterm,thatisslowlymodulatedinphaseandamplitudebyanenvelopeV(alsoknownasa`coherenceenvelope', Lawson 2000 ).Overthelengthoftheslitdirectiononthedetector,wesampleonlyaverysmallrangeofdelays,sothatd0d=2dd0+d=2,whered0isdeterminedbytheinterferometeretalon,asbefore.Overthisrange,thevariationinVisnegligible,soweseeonlyauniformsinusoid.Inmeasuringthephaseandvisibilityofthefringe,weessentiallymakeameasurementofVatthexeddelayd=d0.ThephaseosetofthesinusoidisdeterminedbytheargumentofV,V.Themeasured(absolute)fringevisibilityissimplytheamplitudeofthenormalisedfringe,V.Wenotethat,asshownbyequation 2{6 ,Vandareverycloselyrelated,theonlydierencebeingaphaseoset,which,foragivenwavelengthchanneljatwavenumber

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2-2 demonstratesconceptuallyhowthesummationof100%visibilityfringesduetoanarrowbandofwavelengthssumstogiveamodulatedsinusoid.Aplotisshownofthemeasuredfringeintensityagainstdelay,aplotknownasaninterferogram.Inthiscasethecontributingspectrumiswhitelightthrougharectangularbandpass,i.e.asimpletop-hatfunction.TheFouriertransformofatop-hatfunctioncentredat=0isasincfunction,andthisistheshapeofthemodulationseen. Figure2-2. Arepresentationofthesummation(thickline)offringesduetoeachindividualwavelengthofwhitelight(thinlines)passingthroughanarrowrectangularbandpass(thinlines),resultinginasinc-modulatedsinusoid.(AfterLawson,P.R.2000,inPrinciplesofLongBaselineStellarInterferometry,ed.P.R.Lawson(Pasadena:NASAJPL),ch.8,g.8.2.) 34

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2-3 ,theamplitudeofthemodulatingcoherenceenvelope,V,isshownmoreexplicitly,andweseehowmeasuringthefringeoveranarrowrangeofdelaysarounddaroundd0givesanapproximatelyuniformsinusoid.Thiscorrespondsdirectlytotheimageseenalongthelengthoftheslitdirectioninagivenwavelengthchannelonthedetector(seee.g. 2-1 F).Again,herethecaseisshownforwhitelightthrougharectangularbandpass,sothatV(d)isasincfunction,withzeroesatd=n=(n2Z+),whichmodulatesasinusoidofperiod1= .SincetheLSFisatleasttheoreticallyanimageoftheslit,atop-hatisareasonablygoodrepresentationoftheLSF,andthereforealsooftheresponsefunctionofawavelengthchannel(seeappendix A ).Inpracticethepassband,= ,willbemuchnarrowerthanindicatedinthegure,sothatthevariationofVwillbemuchslowercomparedtothesinusoid,andthesinusoiditselfhighlyuniformoverd.Foramorecomplicatedinputspectrum,suchasthatfromastar,thecoherenceenvelopewillgenerallyalsohaveamorecomplicatedshape,thoughthevariationswillstillbeslowind.Eachwavelengthchannelwillhaveitsownuniquepieceofspectrumcontributingtoit,andthereforeeachwillhaveitsownparticularphaseandvisibility.Itisthisthatgivesrisetothevariedpatternsoffringesthatareseeninthenalfringingstellarspectra(e.g.gure 2-1 F).Analternativeapproachtounderstandingtheformationofthefringingspectrumistothinkofitasaninnite-resolutionfringingspectrum,givenbytheproductoftheinputspectrumandaninterferometertransmissionfunction,convolvedwiththeLSFofthespectrograph.Thisistheapproachadoptedby Erskine ( 2003 )and Mahadevan ( 2006 ).Aderivationrelatingthetwoapproachesisoutlinedinappendix B 35

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Interferogramshowingthecoherenceenvelopeduetoarectangularbandpassmodulatingthesinusoidalfringe.Alongtheslitdirectionofafringingspectrum,averysmallpartoftheinterferogramissampledovertheranged0d=2. anidealisedinniteresolutionspectrograph,wjisadeltafunction.Shiningwhitelightintotheinstrument,sothatP()=1,leadstoQjalsobeingadeltafunctionforallj.Byequation 2{6 ,thecoherenceenvelope,V(d)isthenormalisedFouriertransformofthisdeltafunctionshiftedtod=0,sothatV(d)=1atalldelays.Equation 2{7 thengivestheverysimpleformoftheresultinginterferogram: 2-4 ,plottingcontoursofintensityonthedetectorasafunctionofwavelength=1=onthex-axisversusdelayintheydirection.Sincemapslinearlytoxpositiononthedetector(atleastforanidealspectrograph),anddelaymapslinearly 36

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Figure2-4. Simulatedinterferometercomb.Settingalargeinterferometerdelayandchoosingthewavelengthrangeoverwhichthespectrumisobservedselectsa`window'inthecomb(shownschematically)wherethefringesareapproximatelyparallel.Theordersofsomeofthefringes,n,areshowndowntherighthandside.(Inpracticethe`window'chosenisatmuchlongerwavelengthandmuchhigherorder.) Inreality,ofcourse,thespectrographdoesnothaveinniteresolution.Ifwewidenthedelta-functionresponsefunctionsothatwisnowatop-hatfunction,weapproachthe 37

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2-3 ,wherethefringesaremodulatedbyaslowsincenvelope.Inthesegmentoftheinterferogramimagedbythedetector,theenvelopeisclosetoconstantinvalueoverthelengthoftheslit.Sincethetop-hatfunctionissymmetricandreal,itssinc-shapedFouriertransformisalsoreal,sothereisnochangeinphaseofourinterferogram.Thevisibilityissimplyreducedaccordingtothewidthofw,,andaccordingtowheretheinterferometerdelayd0isset.Wecanseefromthisthatbyappropriatelychoosingthedelayandspectrographslitwidthwecannullouttheinterferometercombbyndingaminimumintheenvelope.EarlyexperimentschangingtheslitwidthanddelaywithETprototypesdidindeedshowthiskindofsinc-typevariationinthecombvisibility. 2{3 (orsee Goodman 1985 ,ch.5)by: 38

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2.6 )derivefromthisformula.ThekeytotheDFDIradialvelocitytechniqueisthefactthatDopplershiftsofthespectrumresultindirectlyproportionatephaseshiftsofthefringes.ThisisadirectconsequenceoftheFouriershifttheorem(D.J.Erskine,privatecommunication).IfthespectrumshiftssuchthatP()!P0()=P(+),andwecorrectlyfollowtheshiftinthedispersiondirectionsothatwenowcomparetothewavelengthchannelcorrespondingtowj+j=wj(+)(assumingthattheresponsefunctionmaintainsthesameforminnearbychannels,andnotingthatjisnotnecessarilyaninteger),thentheshifttheoremgives: =2d0=2d0v c=2d0 39

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c Ge 2002 ).Thisissetatdesigntime,andremainsxedfortheinstrument.AnnualvariationsintheRVofastarduetotheEarth'smotioncanbeaslargeas60kms1,evenforanRV-stabletarget. 2{12 ),determiningissynonymouswithmeasuringthedelay.Toarstapproximation,thedelaycanbecalculatedfromthepropertiesoftheetalon.Theetalonhastwoeects:rst,itproducesapathlengthdierenceduetothefactthatthewavelengthofthelighttravellingthroughitisreducedbyafactor1=,where 40

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2t2t;(2{13)wheretisthethicknessoftheetalon.Thefactoroftwoisintroducedsincethelightmustpassthroughtheetalontwiceasittravelstowardandthenawayfromthemirror.Thesecondeectoftheetalonistoreducetheapparentdistancebetweenthemirrorandtheetalon,byanamountt(11=)wheretisthethicknessoftheetalon.Sincethevirtualimageformedmustcoincidewiththeimageoftheunimpededmirrorintheotherinterferometerarm,themirrorintheetalonarmmustbemovedbackbythisdistancetocompensate.Hencethepathlengthinwavelengthsisincreasedbyanamount: 2t(11=);(2{14)whereagainthefactoroftwoisduetopassingtwicethroughtheetalon.Addingthesetwoeects( 2{13 and 2{14 )weobtainthetotalpathdierenceduetotheetalon: Barker&Schuler 1974 ,D.J.Erskine,privatecommunication),enoughforaninitialestimate.Amoreprecisemeasureofthedelaycanbedeterminedsimplybycountingfringesintheinterferometercomb.Weknowfromequation 2{8 thatthephaseofthecombvariesas 41

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@=1 2@n @=2d 2;(2{16)wherenisthefringeorder,giving: @:(2{17)Inotherwords,bycountingthefringedensityoverwavelength,wecanimmediatelycalculated0,andhence.Sincethereisa2dependencein@n=@,careneedstobetakentoaccountforthedependenceproperlywhendeterminingthefringedensityatagivenwavelength.Thismaybemoreeasilydoneinwavenumberspaceinstead,sincethefringedensityisuniformwithwavenumber,andd=@n=@.Inpractice,countingfringesisoftennoteasy,sincethecombisoftenbarelyresolved(usuallybydesign).Aslongasthecombisnotunder-sampledonthedetector,thiscanbeovercomebytemporarilyusinganarrowerslitinthespectrograph,sinceinprinciplethedelayshouldonlyneedtobedeterminedonce.Evenso,itisusuallypossibleinpracticeonlytocountoverafewhundredstoathousandortwofringes,givingatbestanaccuracyontheorderofonepartin1000.Overa60kms1variation,thisisstillonlygoodtothe60ms1level.Othermethodsofmeasuringthedelayareunderinvestigation,buttodate,themethodofchoicehasbeensimplytoobserveknownstablereferencestarsoverthetimebaselineofinterestandusetheirknownapparentchangesinvelocityduetotheEarth'smotiontocalibrate.Providedthereferencestarsaregenuinelystable,andtheyarepositionedintheskysuchthattheirbarycentricmotionsarelarge,thistechniquewill 42

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2{11 ,achangeind0convenientlyhasmathematicallyexactlythesameeectasachangeinvelocity,v.)Theimagecanalsoshiftonthedetectorinboththeslitandthedispersiondirections.Toaccountfortheseinstrumentalartifacts,anabsorptionreferenceisinsertedintotheopticalpath{inthecaseoftheETinstruments,aglasscelllledwithiodinevapourmaintainedataxedtemperature.Inthiswayaniodinespectrumcanbemultipliedwiththestellarspectrum.Sincetheiodineisstationarywithrespecttotheinstrument,itsspectrumwilltrackinstrumentshifts,whichcanthenbesubtractedfromthemeasuredstellarshifttorevealthestar'sintrinsicmotion.Todothis,foreachtargettobeobserved,twofringing`template'spectraaretaken,onebeingpurestarwithnoreferenceinthebeampath,andtheotherpurereference{i.e.apureiodinespectrumtakenbyshiningatungstencontinuumlampthroughthecell.Thesetemplatesarethenusedtoseparateoutthestellarandiodinecomponentsofthecombinedstar-iodinedata(referredtohereas`data'or`measurement'frames,asdistinctfrom`template'frames).Aformalismisrequiredtoextracttheiodineandstellarspectrafromthecombinedspectrum.Inordertoproceed,wedenethefollowingsymbols:j{thepixelnumberinthedispersiondirectionwhichidentiesthecolumnalongwhichafringeismeasuredintheslitdirection,correspondingtoasinglewavelengthchannel.Strictlyspeaking,thewavelengthchannelisinnitesimallywideonthe 2.6.1 43

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A ).M(j){thecomplexvisibilityvector(ie.phaseandabsolutevisibility)forafringeatwavelengthchanneljinasingleDopplermeasurementframeofcombinedstar-iodinedata,anensembleofsuchvaluesforaspectrumacrossalljcomprisinga`whirl.'S(j){themeasuredcomplexvisibilityforthestartemplateatchannelj.I(j){themeasuredcomplexvisibilityfortheiodinetemplateatchannelj.M()Cm()M(){theinputspectrumforacombinedstar/iodinedataframe,whereCmisthecontinuumfunctionandincludesanoverallnormalisationfactorforM,whichisthecontinuum-correctednormalisedspectraldensity,equaltoonewherevertherearenoabsorptionlines.Cmisassumedconstanttoagoodapproximationoverthescaleofthewidthoftheresponsefunctionw(seebelow)andinstrumentLSF,and0M1.S()Cs()S(){thesameforthestartemplatespectrum.I()Ci()I(){thesamefortheiodinetemplatespectrum.s();i(),suchthatS1s;I1i;0(s;i)1:w(j;){theresponsefunctionatpositionjonthedetector,bywhichismeantthespectrumthatcontributestoaninnitesimallywidewavelengthchannelatthedetectorplaneifperfectcontinuumlightispassedthroughtheinstrument.(NotethatwisverycloselyrelatedtotheinstrumentLSF{seeappendix A )d{theinterferometerdelay,xedtoavalueofd=d0,asusual.{phase/velocityscalingconstant,alsoasusual.WeassumefornowthecasewherethereisnoDopplerorinstrumentshiftineitherphaseorinthedispersiondirection,forbothstarandiodinecomponents,andnophotonshotnoise.Heretheaimissimplytoreconstructthedatawhirlfromthetwotemplatewhirls.Oncethisisachieved,itisconceptuallyarelativelytrivialsteptoallowforshiftedandnoisydata:thetemplatewhirlsneedonlytobeshiftediterativelyinphaseandinthe 44

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2{9 ,thecomplexvisibilitymeasuredatdetectorchanneljforthetemplatesandcombinedstar-iodinedatacanbewrittenexactlyas: 2{20 intermsof 2{18 and 2{19 .Thisismadedicultbytheconvolutions,whichappeartorequireknowledgeofthetemplatespectraatallpossiblevaluesofthedelaydinordertobeevaluated.ThenatureofETissuchthatwemeasureitonlyatonevalue,d0.Anadditiveapproximationiscurrentlyusedtoaddressthisproblem,whichisdescribedinsection 2.6.1 whereAisascalingconstanttoallowfordierenceintotaluxlevelbetweenthetemplatesanddata,andC0ACsCiisaconstantoverthewidthoftheresponsefunction.Ifweassumethateithersoriorboth1,thenthe`crosstalk'term,si,canbeneglected.Sinceiandsessentiallyrepresentlinedepths,thismeansthatweareassumingeithervery 45

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2{21 inequation 2{20 : 2{18 and 2{19 into 2{22 wecanwrite: 2{24 containstwotermsinthenumerator,thecombterm,bwjd0,andacrosstalkterm,dsiwjd0.Itisinprinciplepossibletoarrangetheinstrumentsuchthatatdelayd=d0theinterferometercombhaszerovisibility,bychoosingthedelayandslitwidthsothatMcontinuumisatazeropointofbw.Alternatively,itispossibletolow-passFourierlterthedataimagebeforemeasuringthewhirls, 46

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2{21 ,wenallyhavethewhirladditionapproximation,whichwecanwrite: 2{25 showsthatwewouldexpectKs;Ki1.Asfarasthisapproximationholdsgood,andtotheextentthatKsandKiareapproximatelyconstantacrossallchannelsj,itisthenasimplemattertoallowforDopplerandinstrumentdriftbyallowingthetemplatewhirlstorotateinphaseandshiftinthedispersiondirectionasafunctionofj,andminimising2intheresidualstondthebesttsolutionforthemeasureddataM.However,wenotethatinfacttheassumptionsmaderegardingKsandKiareactuallynotlikelytobeterriblygood{particularlygiventhattheiodineducialcelltypicallyabsorbsatotalof40%oftheincidentlight.Furthermore,thereislittlereasontoassumethatKsandKishouldbeconstantfromchanneltochannel.Wecanrecastthem,rewritingequation 2{25 as: 47

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5.10 ).Ifanexactsolutioncanbefound,itwouldsolvetheadditionapproximationproblem.Findingabetterapproximationmayalsobeapossibility,butwehavebeenunabletondeitheratthetimeofwriting.OnepossiblesolutiontotheproblemistoactuallyphysicallysuperposeaducialThArortungsten-illuminatediodinespectrumontopofthestellartargetspectrum,forexamplebysplicingtwobresintoone,onecomingfromthetelescopeandonefromthereferencelamp.Inthiscase,thetwospectranowcombineadditivelyinsteadofmultiplicatively.Wecanthenwrite: 2{9 wecannowwrite: 48

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2{28 bythedenominator,dMwj0(whichrepresentsthetotaluxalongthechannelinthecombineddata),weessentiallyndwehaveanexpressionwhichisasummationofuxvisibilityterms.Sincevisibilityisdenedas(ImaxImin)=(Imax+Imin),whereImaxandIminarethemaximumandminimumfringeintensities,thenmultiplyingbytotaluxinthechannelgivesaquantityequaltotheamplitudeofthefringe.Henceequation 2{28 isreallysimplysummingfringeamplitudes,andisexactlywhatweexpectwhenthetwoinputspectraarecombinedadditively:theresultingimageonthedetectorshouldsimplybeadirectintensitysummationoftherespectiveimagesthatwouldbeobtainedindividually. 2.7.1PhotonErrorforMultipliedReferenceItisusefultohaveanestimateoftheleveloferrorexpectedpurelyfromphotonshotnoisetobeabletoassessinstrumentperformance.Thephotonerrorinthephasemeasurement(andhencevelocitymeasurement)fromasinglewavelengthchannelcanbeestimatedfollowing Ge ( 2002 ).Thisgivesessentially: djp 49

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=hsm;jst;jihim;jit;ji;(2{32)whereh:::irepresentsaweightedmeanoverj,sm;jandim;jrepresentthephasesforthestarandiodinecomponentsofthecombinedstar/iodinedata(`measurement')frameandst;jandit;jarethephasesmeasuredintheseparatepurestarandiodinetemplates.Forconvenience,weimmediatelymapthesephasestocorresponding`velocity'measurementsbymultiplyingbothsidesby(thoughwiththecaveatthatavelocitymeasurementofasinglechannelinasinglespectrumhasnophysicalmeaninginitselfuntilitisdierencedwithanotherspectrum): v=hvsm;jvst;jihvim;jvit;ji;(2{33)Usingwithcorrespondingsubscriptstorepresentthevariouserrorsinthisequation,wemightexpectatotalphotonerrorinvtobegivenby: Ge ( 2002 ).MonteCarlosimulationsofsinusoidtssuggestthattheRMSslopegivesmoreaccurateresults. 50

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Pj1=2j:(2{35)Inpractice,thetwotemplatetermsinequation 2{34 areneglected,fortworeasons.Therstissimplybecauseingeneralthetemplateswillhavesignicantlyhigheruxthanthedataframe:theiodinetemplatecanbetakenwitharbitrarilyhighuxsinceitisobtainedwithaquartzlampasasource;andthestellartemplateisusuallydeliberatelytakenwithhigheruxthanthedatasothatitdoesnotcompromisetheentiredataset.Thesecondreasonisalittlemoresubtle.AllRVmeasurementswiththiskindofinstrumentaredierential,measuredrelativetothetwotemplateswhicheectivelysetthezeropointofthemeasurementsforthestarandiodine,asseeninequation 2{33 .Sincethis`zeropoint'isthesameforeveryRVmeasurement,anyerrorinthezeropointwillnotcontributetotheRMSscatterinasetofmeasurementswhichusesthesametemplates.Sincephotonerrorsgoas1=p where 2{35 51

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2{36 : 2{31 .Itshouldbenoted,however,thattheseformulaeforthephotonlimitareforthevaluesexpectedgiventhefringevisibilitythatwasobtained.Variousinstrumenteects{forexampledefocus{canreducethevisibilityfromitsoptimumandhencereducethephotonlimitingprecisionfromitsoptimum.Asanexample,atypicaldatapointtakenfromaKPNO2.1mETrun(single-object)inJanuary2006for51Peg(V=5:49mag,10minexposure)withgooduxgivesmeansignal/noiseratios(S/N)perpixelforstartemplate,iodinetemplate,anddataframeof106.6,129.0and86.6respectively.Thesevaluesgivephotonerrorsforthestarandiodinecomponentsof5.6and4:5ms1respectively,whichwhenaddedinquadraturegiveatotalphotonerrorof7:2ms1.ThisisforonlyoneofthetwooutputbeamsoftheKPNOinstrument:aweightedaverageoverthetwobeamsgivesanalphotonerrorof4:8ms1.Itisinterestingtonotethattheerrorduetotheiodinereferenceisinfactcomparabletothatduetothestar,sincethesignalintheiodinecomponentofthedataframeisintrinsicallylimitedbythemagnitudeofthetargetbeingobserved.Thisispossiblyanadditionalargumentforpursuingthemethodofcombiningstarandreferencebeamsadditivelyratherthanbythenormalinsertionofanabsorptioncellintothestellarbeampath. 52

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2{36 : 2{35 .Theindividualcomponentssm;jandim;jmustbereevaluated,however,sincethephotonnoisefromthetwoseparatesourceswillnowcombineadditively(forexample,ifoneofthesourcesisconsiderablybrighterthanthesecond,itsphotonnoisewilldominateoverthesignalinthesecond).Wecanthinkofaneectivevisibilityforthetwocomponentsinthecombineddata,sm;jandim;j.Rememberingthatfringeamplitudeisgivenbytheproductofthevisibilityandthemeanuxinthefringe,wecanwrite: 2.6.2 (wetakethesetobeindependentofchannel).Substitutingtheseeectivevisibilitiesinequation 2{31 gives: Usingthesewecannowevaluateequation 2{38 toobtainanestimateofthephotonlimitingerror,sothat: dvuut "Ejp 53

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54

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2 ,thereareseveraladditionalsubsystems: 2.6 ).Thecellisactivelyheatedtomaintainaxedtemperatureof600:1Csothattheabsorptionspectrumremainsstable. 55

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2-1 B).TheinterferometermirrorinthearmwithouttheetalonissetonaPZTactuatedmount.UsingsoftwarewritteninLabviewbyCurtisDeWittandlaterupgradedbyPengchengGuo,thecomputerfeedsbackcontinuouslytothePZTactuators,adjustingthemirrorpositiontokeepthelaserfringeslockedatthesamephase,sothattheinterferometerpathdierenceinprincipleremainsconstantunderanyenvironmentalchangesorinstrumentdrifts.ThefacilityalsoexistsfortheinstrumentoperatortodialthephaseofthefringesmanuallyfromthePZTcontrolsoftware,whichappliespistonmotiontothemirrorandchangesthedelaybytinyamounts,eitherforcalibrationpurposes,orsothattheoriginaldelaycanberecoveredintheeventthatitislost.Utilisingthefactthatthereisasmallwavelengthdependencyinthephasechangeasafunctionofdelay(equation 2{11 ),itispossibletocompare`before'and`after'measurementsofpureiodinespectraintheeventofphaselossbyusingthedatareductionsoftwaretoestablishthenumberofphasewrapsthatmayhaveoccurred,andtherebybreakthemodulo2phasedegeneracy.Inthiswayexactlythesameinterferometerdelaycanbere-established,towithinaverysmallfractionofawavelength.Thesamesoftwarecanalsobeusedto`jitter'themirror,scanningitrapidlybackandforthoverarangeofsometensoffringes.Overanyexposurelongerthanafewseconds,thiswashesoutthefringessucientlytoproduceastandardnon-fringingspectrum,andisusedforcreatingvariouscalibrationframesforthepipeline(seesection 4.1 ). 56

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Mahadevan 2006 ). 5.5 .) 58

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Mahadevan ( 2006 ).Abriefsummaryofthesetupatvariouspointsisdescribedhere. Geetal. ( 2003b ).Thiswasthesetuprstusedtoconrmtheknownplanetaround51Peg(section 6.1 ).AMichelson-typeinterferometerwasemployed,withonemirrortiltedbyafewwavelengths,andwitha4mmBK7glassplateinsertedinonearm,givingatotalopticalpathdierencebetweenthearmsof7mm.TheinterferometeroutputwasfedintoaspectrographofCzernyTurnerdesignwithtwoparabolicmirrorsandarst-orderreectiongrating,andanadjustableentranceslit.Thespectrographoperatedatf=7:5andthenaloperatingresolutionwasmeasuredatR=4540.UsingaKPNO1k3kback-illuminatedCCD,weobtainedawavelengthcoverageof270Acentredaround5445A.Theimagewasspreadover300pixelsintheslitdirection,givingatotalofaround12fringeperiods.Thef=8telescopebeamwasfedintoa200mbre,matchinga2:500stellarimage.Duetofocalratiodegradation,theoutputfocalratioofthebrewasf=6,whichwasconvertedtof=7:5tofeedthespectrograph.Thisetenduemismatchledto40%photonlossattheslit. vanEykenetal. 2004b ),withthefollowingimprovements:Thespectrographwascompletelyreplacedwithacustomdesignedandbuiltcollimatorandcamera,allowingforafasterfocalratio(f=5collimatorandf=2camera)withlessaberration.Thegratingwasreplacedwithanoptimisedvolumephaseholographic(VPH)transmissiongratingwithsubstantiallyhigherthroughput,operatingataroundR5000anddesignedtopeakintransmissionat5300A( Mahadevan 2006 ). 59

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2-1 ),andalsoasecondwhichinafullyon-axisdesignisactuallyreectedbackdowntheinputpath.Afterthenewupgrades,bothoutputbeamsfromtheinterferometerwereutilisedbypickingothesecondbeamwhereitisslightlydisplacedfromtheinputbeamowingtothetiltedmirrorintheinterferometer.Thesecondoutputbeamwasthenfedalongapathparalleltotheprimarybeam,imagingasecondidentical(thoughphase-inverted)spectrumontothedetector.Anapproximately1=p2improvementinprecisionisobtainedbyaveragingtheresultsfromthetwospectra,equivalenttodoublingthethroughput.Anewcryotiger-cooled4kx4kback-illuminatedFairchildCCDdetectorwasinstalled,allowing600Awavelengthcoverage,centredaroundroughly5510A.Athirdbeamwasalsopassedthroughtheinstrumentforcalibrationsources,allowingforsimultaneousparallelThArducialmeasurements.Thephaselockingsoftwarewasenhancedtolockfringeperiodandangle,inadditiontothephasewhichwasallthatwasmeasuredpreviously.Theinstrumentwasinstalledinitscurrentfullthermalenclosure. 6.2 ),andisdetailedfullyin Mahadevan ( 2006 ).Tosummarise,a200mbre(2:500onsky)feedsthef=8beamfromeitherthe2:1morthe0:9mcoudefeedtelescopeintotheinstrumentatf=6.ThespectrographgratingisaVPHDickson-typegrating,with92%eciencyovera600Abandwidth,designedtocreateaspectrumwithresolutionR5100onthedetectorwithacoverageof5000{5640A.Thespectrumhasa6.7pixelresolutionelement(FWHM)andcovers58pixelsintheslitdirectionfortypically4{5fringeperiods.Theimageisnowrecordedonacryotiger-cooledback-illuminated4k4kCCDdetectormanufacturedbySpectralInstrumentsInc.,with15mpixels,90%quantumeciency,andlinearitybetterthan1%upto80%offullwellcapacity(wherefullwellisat99,000e).Thecontinuous-cyclecryo-coolingprovidedbythecryotigerallowedforasignicantimprovementinimagestability.(Previouslynitrogencooled,thedetector'spositionwoulddriftbyseveralpixelsintheslitdirectionasthenitrogenevaporatedandthedewar's 60

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OpticaldesignofETattheKPNO2.1mtelescope,byJianGeandDeqingRen.DesignedtooperateatresolutionR=5100,withawavelengthcoverageof640A. momentchanged;manualrellingwouldagainnudgeitbyseveralpixels.)AMelles-GriotHeNelaserat0:6238misusedfortheinterferometerphaselockingsystem.Theglassiodinecelliscylindricalwithalengthof150mmanddiameterof50mm,andisstabilisedtoatemperatureof600:1C.Themainspecicationsaresummarisedandcomparedwiththemulti-objectKeckETintable 3-1 ,andthelayoutoftheopticaldesignisshowningure 3-1 61

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Wanetal. 2006 ).Atthetelescopeendthebresarepluggedintoacouplerintheplatecartridge,whichcontainsametalplatethattsatthefocalplaneofthetelescope.Short180mbresinsidethecartridgerunfromthecouplertoholesdrilledintheplateatpositionscorrespondingtothoseofthetargetstarsintheeldthattheparticularplateisdrilledfor.Thediameteroftheseshortbrescorrespondsto300onthesky,feedingatf=5.Attheotherend,thelongbresarefedtotheinstrumentatf=4inthreeverticalgroupsof20bres,runningalongsideandslightlyosetrelativetoeachother.Afterpassingthroughtheinterferometertheseareimagedontothreeslitsintheinstrument,sothatthespectracyclethroughthethreeslitsinsequenceasaswestepspectrumbyspectrumdownthedetector(withsomeexceptionsintheorderingtowardstherstandlastspectraforpracticalreasons.)Thedetectorisagainacryotigercooled4k4kSpectralInstrumentsCCD,with15mpixels,92%quantumeciency,andlinearitybetterthan1%upto80%offullwellcapacity(wherefullwelloccursat94,000e).TheVPHgratinginthespectrograph(non-Dicksontype)( Zhao&Ge 2006 )operatesatR5100,withspectracoveringalargerrangethantheKPNOETat5000{5900A.59ofthe60spectrafallonthedetectorinthecurrentalignment(thelastdoesnotquitet),andeachcoversabout409650pixels,againwithtypicallyontheorderof4{5fringesintheslitdirection.Thisinformationisalsosummarisedintable 3-1 .Theopticallayoutfortheinterferometerisshowningure 3-2 ,andthatforthespectrographingure 3-3 .Afulldetailedinstrumentdescriptioncanbefoundin Geetal. ( 2006b ); Wanetal. ( 2006 ); Zhao&Ge ( 2006 ). 62

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KeckETinstrumentspecications KPNOETAPOKeckET #targets1+calibration59#interferometeroutputs21Fibrediameter200m/2:500180m/300Telescopef=#f=8f=5Instrumentinputf=#f=6f=4ResolutionR51005100Resolutionelement6.7pix5.0pixDetectorS.I.4k4kS.I.4k4kSpectralcoverage5000{5640A5000{5900ASizeofspectrum409658pix409650pix Figure3-2. OpticaldesignoftheKeckETinterferometer,byBoZhaoandJianGe,showingallsixtybeams. 63

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OpticaldesignofthespectrographfortheKeckET,whichfollowsimmediatelyaftertheinterferometershowningure 3-2 ,matchingattheslitplane.(ByJianGe.) 64

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3.1 ),wetake:Tungsten-illuminatediodine,nofringes:severaltakenatthebeginningandendofanobservingrun.Providesalowresolutioniodinespectrumforreference,andinsomecircumstancesthesecan,ifnecessary,beusedasatelds.(Althoughdividingoutaniodinespectrumwillremovetheintensityvariationsduetotheiodineabsorptionlines,itwillnotactuallychangefringevisibilityorphase,andithastheadvantagethattheilluminationfunctionislikelytobettermatchthatofdatawiththeiodinecellinserted,givingabetterheadstartonilluminationcorrection.Thedownsideisthatanyhintofresidualfringinginthiscalibrationwillintroducesystematicphaseerrorsinthedata.)Puretungsten,nofringes:severaltakenatthebeginningandendofeachobservingrun,providingasimplecontinuumillumination.Anaverageovertheseframesisusedasamasterateldimage.Thorium-Argon,nofringes:Severaltakenatthebeginningandendofeachrun.UsedtocreateanaveragedtemplateforslantcorrectionofspectrallinestomatchtheCCDaxes.Thorium-Argonwithiodine,nofringes:Thesecanbeusedforslantcorrectionofspectrawheretheiodinecellisswitchedinifitissuspectedthatinsertionoftheiodinecellisaectingtheslantoftheslitimageinthespectra.Thesecalibrationsaretakenatthebeginningandendofeachrun.Oncejitterisswitchedo,theinterferometerphaselockerisrestarted,andthephaserestoredtothesamevalue,modulo2.Beforeandafterjittering,asingletungsten-illuminatediodineframeisalwaystaken.Anon-the-yroughreductionoftheseframesisperformed,andthefringephasesdierencedbetweenthetwoframesacrossthelengthofthespectra:anyphasewrapsthatmayhaveoccurredareeasilyidentiedbecauseoftheinversedependencyofphaseshiftonwavelengthwhentheinterferometerdelaychanges(equation 2{11 ).Ifanyhaveoccurred,thephaseissteppedbacktoitsoriginalvalueusingthephase-lockersoftware,andanotheriodineframeistakentoverifythatanear-zeroradian 66

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2.5 ).Tungsten-illuminatediodine:severaltakenatthebeginning,middleandendofthenight,usedfordeterminingacalibrationofthefringeperiodintheslitdirectionasafunctionofwavelength,averagedoverthenight.Thiscalibrationisthenappliedtoallfringettingforthewholenight'sdata.Anadditionaliodineframeistakenconsecutivelywithanypurestartemplatespectrumtoprovideacorrespondingreferencetemplatespectrum.Theseframescanalsobeusedperiodicallytocheckthatthephaselockerhasnotskippedanylaserfringes,followingtheprocedureoutlinedaboveforphase-wraprestoration.Purestartemplates:takenatleastonceforeachtargetortargeteldtobeobserved,toprovidethetemplateforseparatingoutstarandreferencespectruminformationinthecombinedstar/referencedata.Starwithiodine:theactualdataframesusedtomakeRVmeasurements.Biasframes:severalzero-lengthexposurestakeneachnightandaveragedforstandarddetectorbiassubtraction.Darkframes:usuallyseveraltakeneachnight,matchingthelengthofthelongestexposureofthenight,withnolightinputintotheinstrument.Averagedtogetheroverthewholerunandusedforstandardsubtractionofanystraybackgroundlightsourcesthatmaybepresentintheinstrument. 67

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4-1 ).Severalsubsidiarysimple-textparameterlescontaintheremaininginformationneededforthepipelineconguration.ThepipelineiswrittenintheIDLdataanalysislanguagebyResearchSystemsInc.,andisdesignedtobelaunchedeitherfromtheIDLcommandline,orfromtheGUI.Thesoftwareisdesignedsothatdierentdefaultparameterlescanbechosenfordierentinstrumentsandcongurations,whilsteasilyallowingfornetuningforparticularunanticipateddatasituations.Foreveryparticulardatareductionrun,itshouldbepossibletosavethefullsetofparametersasarecordoftheparticularprocessingthatwasdone.Inaddition,recordsofallprocessingperformedonanimagearestoredinthecorrespondingheaderinformationforthatimage.TheearlypartsofthedatareductionprocedurethatfollowedstandardastronomicaldatareductionmethodswereoriginallyperformedusingtheIRAF Tody 1986 1993 )softwarepackage.Toenableincorporationoftheentireprocedureintoasinglepipeline,theinitialIRAFpartsofthedataprocessingwerelaterreplacedwithIDLcodewrittenbyCraigWarner,alongwiththeparameter-editor/launchGUI.Atthistime,theGUIonlyincorporatesthoseparameterswhichaectthepartsofthedatareductionprocedurethatwereoriginallyperformedusingIRAF,andthesubsequentdatareductionmustbelaunchedindependentlyfromtheIDLcommandline.AddingintherequisiteparameteroptionstotheGUIandincorporatinglaunchofthefullpipelinefromtherearerelativelysimpletasks,however,whichareexpectedtobecompletedsoon.TherawdataarestoredasFITSimages( Hanischetal. 2001 ).AutomationofthepipelineisfacilitatedinpartbyreadinginwhatinformationisavailablefromtheFITS

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ExamplescreenshotfromtheETpipelinegraphicaluserinterface,writtenbyCraigWarner,showingoptionsforsomeofthepreliminarypre-processingsteps.Theinterfaceisusedtoedittheparameterleforthepipeline. headerinformation(epochofobservation,exposurelength,etc.),andinpartbyreadingininformationfromelectronicloglescreatedbytheobserver,whichcontaininformationontheimagetype(i.e.,targetorcalibrationframeandwhatkindofcalibration,whethertheiodinecellisswitchedin,whethertheinterferometerPZTisbeingjitteredtowashoutfringes,etc.),andthenameofthetargetortargeteld.TheelectroniclogsarecreatedduringobservationusinggraphicalsoftwarewrittenintheJavalanguage( Goslingetal. 1996 ) 4-2 ),basedonanoriginalMicrosoftExcelspreadsheettemplatebyAndrewVandenHeuvel.Itisanticipatedthatinthefuture,moreinformationwillautomaticallybeincludedintheFITSheadersduringobservation,includingtelescopepointinginformation,observatory

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ExamplescreenshotfromtheETelectronicobservinglog,writtenbyCraigWarner.Thelogproducesdatalesthatareautomaticallyreadablebythedatareductionpipeline. information,instrumentstatus(iodinecellin/outetc.),sothattherecanbelessroomforhumanerrorbetweenskyandRVresults. 70

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4.4.4 ).Intheelectroniclogsforthemulti-objectinstrument,the`targetname'recordedistheplateidentierfortheeld.Usingthemastermaps,theidentierisexpandedoutintoalistofstarswiththeirassociatedspectrumindexnumbers,andthenthedatacanbeprocessedbythestandardsingle-objectpipeline. 71

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4.4.1.1Bias/darksubtractionandateldingBiasanddark-subtractionandateldingwereoriginallyperformedinusingstandardIRAFroutines( Tody 1986 1993 )inearlyversionsofthedatareductionprocedure.TheseroutineswerelaterreplacedwithWarner'sIDLcode,andareperformedasforstandardastronomicaldataprocessing.Amasterbiasframeiscreatedbyaveraging(bymeanormedian)allthebiasframestakenoverarun,withoutlierrejection,andsubtractedfromalldataframes.Asimilarlyaveragedbias-subtractedmasterdarkisalsocreatedforanentirerun,usingoutlierrejectionacrosstheframesforeachpixeltoreliablyremovecosmicrays.Themasterdarkisscaledbyappropriateexposuretimesandalsothensubtractedfromalldataframes.Theinitialateldingisalsoperformedfollowingstandardastronomicalimageprocessingprocedures.Amasterateldiscreatedfromaveragedjitteredpure-tungstencontinuumcalibrationframeswithoutfringes,againwithoutlierrejectiontoremovecosmicrays.Theaverage(eitherameanoramedian)canbeweightedbytheexposuretimeorbyvariousmeasuresoftheaverageuxintheateldframes(mean,median,meanwithoutlierclipping,etc.),toaccountfoructuationinthebrightnessofthetungstenlamp.Theateldingtakescareofpixel-pixelgainvariationsintheCCDchipwhichcanotherwiseappearasnoisesimilartophotonshotnoise.Italsousuallygoesalongway{althoughnotalltheway{towardsatteningtheilluminationproleofthespectra. 72

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4-3 ).Thisresultsinbothslantandsomedegreeofcurvatureoftheslitimage(referredtocollectivelyas`slant'forconvenience),andhenceofthespectrallines.Inordertoallowfortheasymmetry,weapplyanalgorithmtocorrecttheeectandalignthespectrallinesexactlywiththeCCDcolumns,sothatwecaneasilytsinusoidstoeachcolumnduringthefringettinglateron.Theslantismeasuredusingthenon-fringingThArcalibrationframes,whichprovidesharpemissionlinesthathelptoprovideanaccuratedetermination.Thespectrumischoppedintotypically25segmentsinthedispersiondirection.Foreach 73

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Examplenon-fringingThArframefromthemulti-objectKeckET,showingall59spectra.Distortionoftheeldthatleadstoslantintheslitimageisclear.Wavelengthscalesareosetincyclesofthreespectraduetothearrangementofthebreimagesonthespectrographslits.GreyscaleindicatesADUperpixel. 74

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Slantcorrectionofanon-fringingThArspectrum.A)Beforecorrection.B)Aftercorrection.Imageisfromaportionofthefthspectrumonthedetector,takenfromthesameimageasgure 4-3 .GreyscaleindicatesADUperpixel. segment,therelativeosetbetweeneachpixelrowandthecentralrowisdeterminedbycross-correlationafterapplyingalow-passlter( Verschueren&David 1999 ).Thecrosscorrelationfunctionisevaluatedateachintegerlagvalue,andanimplementationoftheinterpolationschemeproposedby David&Verschueren ( 1995 )usedtomeasurethepeakpositiontosub-pixelaccuracy.Thisgivesusameasurementofthemeanslantforeachsegment.Beforeapplyinganycorrections,themeasuredslantsforeachsegmentaremedianlteredandthensmoothedbyboxcaraveragingoverthe25segments,toreducenoiseandtoremoveanyoutliersthatmaybecausedbybadsegments(e.g.anywithverylowux,fewornospectrallines,orstraycosmicrays).Correctionsarethenappliedtoallthedataspectrabyinterpolatingthemeasuredslantsfromsegmenttosegmentforeachpixel,andthenremappingeachpixelofthespectrumtoitsnewposition,alsobyinterpolation.TestsoftheinternalprecisionofthealgorithmaremadebymeasuringtheslantusingtheThArcalibrationframes,applyingthatcorrectiontothecalibrationframesthemselves,andthenre-measuringtheslantonthecorrectedcalibrationframes.DoingthisgivestypicalRMSdeviationsfromzeroalongtheslittypicallyontheorderof0.01pixels.Anexampleoftheeectsoftheslantcorrectioncanbeseeningure 4-4 75

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4-5 showstheeectofthisatelding.Arawdataleandtheattenedandpixel-pixelcorrectedversionareshownalongwiththeextractedilluminationfunction,and 76

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Pre-processingsteps.A)Rawiodine+brominefringingspectrumfromtheearlyprototypesingle-objectET.Interferometercombisclearlyresolved.B)Imageafterpixel-pixelateldingandilluminationcorrection,andC)afterlowpasslteringtoremovetheinterferometercomb.D)Illuminationfunctionextractedbyself-illuminationcorrectionalgorithm.E)Pixel-pixelgainvariationseenbyself-illuminationcorrectionofaateldframe.Notethatthespectrallinesareclose-toverticalintherawdata,soslant-correctionisdiculttosee. thepixel-pixelatobtainedbyremovingtheilluminationfunctionfromoneoftheatles,whereimperfectionsintheCCDresponsecanclearlybeseen. 2.6.1 ,itisimportantthatthereisnointerferometercombpresenceinthedata.Thisisensuredbyasimpleonedimensionallow-passspatialFourierlterof 77

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4-5 C.Afterltering,theimagepreprocessingiscompleteandthedataarereadyforfringemeasurement. 78

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79

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5.2 ,gure 5-1 ).Theresultsfromthefringettingaresavedas`whirl'lesforeachspectrum. 4.4.3.1ReferenceextractionInthesimpliedcaseofaperfectlystableinstrumentanddatawithnooverlaidreferencespectrum,calculatingthevelocityshiftisrelativelysimple:thephaseshiftinagivenwavelengthchannelisdirectlyproportionaltothedierentialRVshiftofthetarget,scaledbythephase-velocityfactor(section 2.4 ).Forsmallshifts,itisadequatetoperformaweightedmeanofthephaseshiftsoverallthechannelstoobtainahighprecisionDopplershiftthatusestheinformationacrossthewholespectrum.Forlargershifts,thephaseshiftcannolongerbeassumedtobeindependentofwavelength,,anda`twist'isintroducedintothephaseshiftasafunctionofwavelength.Becausethewavelengthscaleisnotlinearonthedetector,thefunctionalformofthisshift(asafunctionofdetectorcoordinate)isnotsimple,butinprinciplecouldbeestablishedbydeliberatelysteppingtheinterferometercavity,andempiricallyttingapolynomialfunctiontothetwistedfunction().Onceestablished,thefunctioncouldbescaledandttoany():atachosenreferencepointinthespectrum(usuallythecentre),thephaseshiftofthefunctionwouldcontaintheinformationofanaveragedvalue,andcouldeasilybetranslatedtoahighprecisionvelocityshift.Thisistheapproachthatwouldbeusedforaparallelsimultaneousreferencespectrum(iodineorThAr),orforbracketingpure-stardataexposureswithpurereferenceframes:anyapparentdriftinthereferencewouldsimplybesubtracted,leavinguswiththetruestellarDopplershift.Inpractice,however,asuperposediodinereferencespectrumhasalwaysbeenusedtoaccountforinstrumentdrift,duetovariousinstrumentinstabilitiesthatrenderotherapproachesimpractical.Inordertoextractthestellarandiodinewhirlsfromthecombinedwhirl,Mj,twotemplatesaretakenateachobservation,givingapureiodinewhirlIj(takenwithatungstenlampasalightsource),andapurestarwhirl,Sj.In 80

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2.6.1 .Essentially,weareaimingtosolveequation 2{26 ,whichwerestateas: 2{27 hasonlyrecentlybeenappreciated,andhasnotbeenincorporatedintothepipelineyet.WeexpressageneralvectorforthejthwavelengthchannelofawhirlinpolarcoordinatesusingthenotationMj=[Mj;Mj]withMjrepresentingthevectorlength(i.e.thevisibility)andMjthephase.UsingsimilarnotationforSjandIj,wecanthenwrite: 81

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Lookingattherighthandsideofthexcomponentandusingtrigonometricidentities,wecanrewritetherstterm: 4{4 ).Henceweendupwithtwolinearequationsinfourunknowns: OnechannelisthusinsucienttodetermineKsandKicompletely.However,bycombiningalltheavailablewavelengthchannelsj,weeectivelyhave2000equationsacrossa4096pixel-widedetector.Thesystemisthenoverdetermined:inanidealworldonlyfourequationswouldberequiredandallothersetsoffourwouldyieldthesameresult,butintherealworld,allsetsoffourwillgiveslightlydierentresults.Thesystem 82

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Pressetal. 1992 ,section2.6).Thisyieldswhatisessentiallya`bestt'result.OnceKsandKiaredetermined,theycanbeconvertedbacktotheirpolarequivalents,andthevaluesKsandKiextracted,henceyieldingandnallytherelativevelocityshiftviaequation 2{11 .Thisnewapproachyieldsresultsnumericallyextremelyclosetothepreviousapproachemployedby Erskine&Ge ( 2000 ),buthastheadvantagethatitmakesnoassumptionsaboutthevaluesofthescalingconstantsKsandKibeyondtheirwavelengthindependence.Italsoallowsforweightingoftheequationstoaccountfordierentmeasurementaccuraciesintheindividualchannels(althoughinpracticewegenerallyassumeuniformweighting{seesection 5.1 ).TheSVDsolutionprovidestheinnermostlayerofnestediterationintheradialvelocitydetermination.Wehavenotyetaccountedforshiftofthespectrallinesthemselvesinthedispersiondirection.ThiscanbecausedbothbynaturalDopplershift(byasmallbutsignicantamount)andbyimageshiftonthedetectorbecauseofslightosetofthebeampathwhentheiodineabsorptioncellisinsertedandbecauseofpossibleinstrumentinstability.Itisimportanttocomparethephasesandvisibilitiesofexactlythesamepartofthespectrumfromdataframetodataframeforameaningfulshiftcomparison.Toaccomplishthis,thetemplatewhirlsareshiftedbysub-pixelamountsusinginterpolation(eitherlinearorspline,dependingonspeedandaccuracyrequired).Theshiftsareallowedtooatasafreeparameters,andthe2valuefortheresidualsbetweenthebest-tsolutionandthecombinedstar/iodinedataframeisminimisedusingtheIDL`AMOEBA'algorithm(basedontheroutineofthesamenamein Pressetal. 1992 ).Finally,whererequired,athirdlayerofiterationcanbeusedtoiterativelyrejectoutlierchannelsinthebest-tsolution,toeliminatetheeectsofanyremainingcosmic 83

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84

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4.4.3.1 ,wesimplyadduptosecond-orderpolynomialstoboththestarandiodinetemplates,withoatingparameters.Twovisibilityscalingparametersarealsoallowedfor,asbefore(thefullchanneldependencyaccordingtoequation 2{27 hasnotyetbeenincorporated,thoughthiscanbeeasilydone).Freeparametersarealsoallowedtoaccountforthedispersiondirectionshiftaswell:inthiscase,againuptoasecond-orderpolynomialwavelength-dependentshiftcanbeallowedfor,insteadofsimplyassumingauniformshift.Apolynomialshiftallowsforbothuniformimageshiftscausedbyimageshiftonthedetector,andwavelength-dependentDopplershift.Wenowhaveatotalofupto14free 85

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McCarthy 1995 ,foradetaileddiscussion).OurbarycentriccorrectioncodeisbasedonaslightlymodiedroutineoriginallywrittenbySuvrathMahadevan.Ifnecessary,thetargetstellarcoordinatesarerst 86

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Landsman 1993 ). McCarthy 1995 ).InthecaseoftheKeckETinstrumenttheprecessionisnotneededsincethecoordinatesareobtaineddirectlyfromtheplatedrillingles,whichalreadyincludeprecessionandpropermotioneects.Forthesingle-objectKPNOET,coordinatesareautomaticallyobtainedfromtheonlineSIMBADdatabase Stump ( 1980 ),whichtakesintoaccountEarth-moonmotionandaccountsfortheperturbationsoftheSolarSystemplanets,claiminga1ms1precisionlevel.Thecoordinatesandaltitudeoftheobservatoryareconvertedfromgeodetictogeocentriccoordinates,andusedtocalculatetheobservatoryvelocityduetothediurnalrotationoftheEarth,followingthecalculationsoutlinedinthedocumentationfortheIRAFroutine`rvcorrect'.Theorbitalanddiurnalvelocitiesaresummedtogivethefullbarycentriccorrection.Relativisticeects(gravitationalandtransverseDopplershifts)areneglected,asisparallacticmotion.Overallprecisionsarefoundtobegoodtoapproximatelythe1ms1level(seesection 5.6 ). 87

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2.7 .Forthemostpart,wehavesettledonusingthephotonlimitingnoiseforourerrorbars,calculatedindividuallyforeachRVdatapoint.Short-term(few-hour)experimentshaveshownagoodmatchwiththeseresults,andmoreinternally-consistenterrorestimateshaveusuallyyieldedfairlysimilar-sizederrorbars.Asecond,andprobablymoreinclusiveestimatefortheerrorbarscouldinprinciplebeobtainedbylookingatthescatterintheresidualsinphaseandvisibilitybetweenthedataframeandthebest-solutionsuperpositionofthetransformedstarandiodinetemplates.SincesucharesidualismeasuredateachCCDcolumninthespectrum,itiseasytocalculateanRMSvaluefortheresiduals,anditshouldbepossibletotranslate 88

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5.10 ),innon-trivialways.Sincetheadditionapproximationcausessystematicerrorsthatappearonlyoverlargevelocityshifts,usingthismethodtorepresenterrorbarswouldcurrentlynotbehelpfulonthescaleswherewearetryingtomakemeasurements:withinsmallvelocityshiftregimes(i.e.overtheshortterm),theerrorbarscouldappearmuchlargerthantheactualRMSscatterintheRVresults.Analternativewaytoestimateerrorbarsistopropagatethestandarderrorsfromthefringetting(section 5.2 )throughtheiodineextractiontothenalresult.ThiswastheapproachusedearlyintheETproject,andtoproduceerrorbarsforthe51Pegresultsinsection 6.1 .Toconvertthestandarderrorsinthevisibility-phasevectorforagivenwavelengthchannel,and,toerrorsinCartesiancoordinates,xandy,asnecessaryforthelinearisationinequations 4{6 and 4{7 ,weusethestandardformalismforcombiningindependenterrors: @22+@x @22(5{1) y2=@y @22+@y @22;(5{2)wheretherepresentstherespectivemeasurementerrors.Givenx=cosandy=sin,wend: x2=2cos2+22sin2 y2=2sin2+22cos2: 89

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4.4.3.1 )canbeweightedaccordingtotheseerrors,anderrorbarsforthenallydeterminedvelocityshiftcanbeextractedfromtheresultsusingstandardstatisticsforsingularvaluedecomposition( Pressetal. 1992 ).Equations 5{3 and 5{4 willingeneralbeusefulinthattheymaypartiallycatcherrorsfromthefringettingduetounexpectedartifactsinthedatareduction,suchasnon-uniformfringesandsoforth.However,itisinterestingtonotethatinthecaseofwellbehaved,evenly-distributedindependentrandomerrors,assuming,itcanbeshownthat: ==:(5{5)Thisrelationfollowssimplyfromthedimensionalityofand.Ifwerepresentthepairasavector,,itisreasonabletoexpecttheerrorinthisvectormeasurementtobeanerrorcircleintwodimensionalspace.Theradiusofthiscircleisequalto,sointhedirectionparalleltotheerrorissimply.Intheperpendiculardirection,theerroristan,butsincetheerroriscircular,itmustalsoequalandhenceweendupwithequation 5{5 .InCartesians,theerrorinbothaxesislikewisegivenbytheradiusoftheerrorcircle,sothat x=y=:(5{6)Followingatreatmentsimilartothatin Ge ( 2002 )forthemeasurementerrorinphase,itcanalsoeasilybeshownthatforanidealsinusoidalfringetheerrorinthevisibilitymeasurementisgivensimplyby =r 90

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5-1 .100simulationswererunforeachpointontheplot.Thedottedlinewithdiamondpointsrepresentsthemeanofthestandarderrorsreturnedbythecurvettingroutine,withtheerrorbarsrepresentingthestandarddeviationinthatmeanoverthe100iterations;thethinsolidlinewiththecrossesrepresentstheRMSdierencebetweentheinputphaseinthesimulationandthephasemeasuredbythecurve-t,withtheerrorbarsrepresentingtheexpectedstatisticaldeviationinthatRMSduetothenitenumberofsimulations;andthethicksolidlinerepresentsthetheoreticalphotonlimitingcurvepredictedbyequation 2{31 .Itcanclearlybeseenthatallthreecurvesmatchverywell.Itisthereforenotthoughtthatthereisanysignicanterrorduetothecurve-ttingroutinesthemselves. 4.4.1.5 ).Tothecurve-troutine,anyilluminationcorrectiondeciencylookslikeadditionalnoiseinthefringe.Wemightthereforeexpecttheateldingerrorstobecomesignicant 91

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Phaseandvisibilityerrorsduetocurve-ttingalone,forasinglewavelengthchannel.Thinsolidlinewithcrosses:RMSerrorfromMonte-Carlosimulations;dottedlinewithdiamondpoints:meanstandarderrorestimatefromcurve-tting;thicksolidline:theoreticalcurveduetophotonlimit. 92

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5-1 ,thistimeagainstS/N,(gure 5-2 )showsthattheerrorsbegintodivergefromthepredictedphotonerrorsatS/Nratiosaboveabout20,justaswewouldexpectfora5%(i.e.S/N=20)illuminationcorrectionerror.(Itisalsointerestingtonotethatthestandardcurve-treturnederrorsapparentlydonotcapturethiseectatall.)ItthereforeseemsreasonabletosuggestthatilluminationcorrectionneedstobegoodtobetterthanthephotonS/Nforthespectrum;otherwiseprecisionsarelikelytoreachaoorattheequivalentS/Noftheilluminationcorrection.Inpractice,thepipelinecanallowextraparametersinttingthesinusoids,however,toatleastpartiallycompensateforpooratelding,andsotheerrorsareactuallylikelytobelessseverethanthisestimate.Therearetwoparticulareectswhichcantripuptheilluminationcorrectionalgorithm,requiringthatsomecarebetaken.Therstiswhere,bypurecoincidence,alargeportionofaspectrumcontainsfringesthatallhappentohavesimilarphases,whichweterm`fringeconspiracy'.Inthiscase,theilluminationcorrectionalgorithmwilltendtointerpretthealignedfringesasbackgroundilluminationfeatures,anddividethemout,subtlyaectingthephasesofthoseandneighbouringfringes.Giventhesheernumberofspectraweobtain,thisinevitablyhappensonoccasion,andcancausesignicantsystematicerrorsinthesamewaythatacontaminatingspectrumwould(seesection 5.7 ).Theoptionexistsinthepipelinesoftwaretotryandovercomefringeconspiracybyperformingadditionalsmoothingoftheextractedilluminationfunctionintheslitdirection,inadditiontothatinthedispersiondirection,toremoveanyresidualfringes.Unfortunatelythistransversesmoothingalsohastheeectofremovinganygenuinefeaturesintheilluminationfunctionthathavesimilarspatialscalestothefringes,whichcanthemselvescontributesystematicerrorsifnotsuccessfullyremoved.Boththeseeectsareveryhardfortheilluminationcorrectionalgorithmtodistinguishfromgenuinefringes.Theycanbestbeconsideredascontaminatingspectra,ratherlikeresidualcomb(section 5.9 ):forfeaturesatthe0.5%fractionallevelin 93

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Fringettingerrorsforpoorilluminationcorrection.Thinsolidlinewithcrosses:actualRMSerrorfromMonte-Carlosimulations;dottedlinewithdiamondpoints:meanstandarderrorestimatefromcurve-tting;thicksolidline:theoreticalcurveduetophotonlimit.1000MonteCarlosimulationsareperformedperpoint,witha5%modulationofthefringesalongtheslitlength.TheactualRMSerrorsbegintodeviatefromthephotonlimitatS/N>100%=5%=20. 94

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4.4.1.4 )isanimportantstepforachievingthebestpossibleprecision.Spectraarecorrectedforslantbyshiftingeachrowofthespectruminthedispersiondirectionuntilalltherowsarealigned,withtheslitimagefollowingexactlyalongtheCCDcolumndirection.Wecangainacrudehandleontherequiredprecisionofthecorrectionbyconsideringtheeectsofshiftinganentirespectruminthedispersiondirectionbysub-pixelamounts.Wetakeafullyprocesseddatawhirl(fromrealdatathathavebeenslantcorrectedaswellaspossible),shiftitbysmallamountsinthedispersiondirectionusinginterpolation,andthenmeasuretheapparentensemblephaseshiftacrossallwavelengthchannelsbetweentheshiftedandunshiftedframes.Theresultsofsuchanexperimentfromanearly51Pegfringingspectrumtakenwiththesingle-objectETareshowningure 5-3 95

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Eectofuniformarticialspectrumshiftinthedispersiondirection. Themagnitudeoftheeectdependsontheparticularspectrumused,andthewaythefringesforthatspectrumhappentoalignoverthefullwavelengthcoverageofthespectrum.Figure 5-3 showsoneofthemoreextremecases,showinganerrorof200ms1perpixelofshift.Foranacceptablysmallerrorof,say,1ms1(smallerthanthephotonnoiseinalmostallcases),wewouldthereforerequireanimageshiftoflessthan0.005pixels.Wecanthinkofthisasaworst-caselimitfortheprecisionofthedispersionshifttobeappliedtoeachsinglerowcutalongthespectrum.Sinceinrealitytheslantcorrectionisnotapplieduniformlytoeachrow,butratherwitharoughlymonotonicvariationfrompositivetonegativealongtheslitdirection,theactualprecisionrequirementislikelytobesomewhatlessstringent.InpracticetheslantcorrectionshowsanRMSprecisionoftheorderof0:01pixels,andthisisdeemedsucientlyaccurateforthetimebeing.(Thisgureisdeterminedbytakingslant-correctedimages,re-measuringtheslantasafunctionofrownumber,andcalculatingtheRMSaboutzero.) 96

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3.1 )showthat,exceptincasesofextremevaryingcloudcover(wheredataislikelytoberejectedanyway),uxweightingusuallymodiesthetime-of-exposureby1{10%oftheexposurelength.Eventakingtheextremeexampleofcompletecloudcoverageforjustthesecondhalfofanexposurewouldmovethetime-of-exposureforwardbyaonlyaquarteroftheexposuretime.Foraonehourexposure,wemightthereforeexpect,eveninsuchaparticularlybadcase,anerrorof15min,or15ms1inbarycentriccorrection.Forthemulti-objectKeckET,targetsthatrequiresuchlongexposuresusuallyhavephotonerrorsatthe30ms1levelormore,sotheuxcentroidingerrorisnotdominant.ForshortexposuresatKittPeakonbrighttargets,wemighttake,say,5minexposuresattheshortest.Inaverybadcase,thismightleadtoa1:3ms1error:again,thisissucientlybelowthephotonlimit(usually2{3ms1atbest)thatuxcentroiding,whilesignicant,isnotaprimaryconcern.TheseerrorsarementionedheresincetheyarerelevanttotheresultsreportedsofarwithET.Nonetheless,ux-centroidingiseasytoimplement,thehardwareisalreadyin 97

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4.4.4 )canbeaverysignicantsourceofsystematicerrorifnotdoneaccurately.Fortunatelytheproblemhasalreadybeenfairlythoroughlyaddressedintheeldofastronomy.AcomparisonofourbarycentriccorrectionroutinetoheliocentricvelocitiesgivenbytheIRAFroutine,`rvcorrect'( Tody 1986 1993 ), 2{24 asbackgroundspectra,wecanalsotrytoassesstheirrelativesignicance.Figure 5-4 showsafringealongonedetectorcolumnduetothetargetsourcealone,withfringeamplitudeas,meanuxFs,andphases.Forsimplicityweassumeno 98

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Fringealongonechannelduetotarget(uppercurve),andcontaminatinglowuxfringe(lowercurve).Measuredfringeisasummationofthesetwofringes. iodineducialreference,sinceweareonlyaimingforanorder-of-magnitudeestimate.AsecondcontaminatingfringeofloweramplitudeacandmeanuxFcduetobackgroundcontaminationisalsoshown,withphasec.Ifthespatialfrequencyofthefringesisf,thenthesummationofthesetwofringeswillgivethetotalmeasuredfringe: wherexidentiespositionalongtheslit.Fs+Fcrepresentsthemeanvalueofthemeasuredux.Thelasttermrepresentsthevaryingsinusoidalfringe.Weareinterestedinthephaseerror,",introducedintothemeasuredfringebythecontaminatingspectrum.Sinceweareonlyinterestedinthephaseinformation,weignoretheosettermFs+Fc,andrepresentthevaryingtermasavectorsummation,asshowningure 5-5 ,whereasandacrepresentthesourceandcontaminantfringeamplitudesasbefore.Theangleisthedierencebetweenthesourceandcontaminantfringephases,=cs.Usingthesinandcosinerulesfortriangleswecanshow: sin" 99

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Vectorrepresentationofthesummationofthefringesduetothetargetsourceandbackgroundcontamination. Inthelimitthatthetwospectraareofsimilarformandverycloseinvelocity,sothatissmall,then sin"ac as+ac:(5{10)Ifweassumethesourceandcontaminantfringevisibilitiesareapproximatelyequal,sothatas=Fsac=Fc,thenac=asFc=Fs,whichisequaltotheuxratioofthetwofringes.Ifthecontaminatingfringeismuchfainterthanthesource,sothatFsFcandasac,and"isthereforesmall,then sin""ac 100

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5-5 itcanbeseenthataworstcasescenarioiswherethecontaminantinallwavelengthchannelsissystematicallyosetbyanamountsuchthatthebackgroundcontaminantvectorisperpendiculartothemeasuredvector(or,approximately,where==2).Inthiscase,"ac=asFc=Fs,sothat: 5{9 ,againassumingFsFcandas=Fsac=Fc)ac=asFc=Fs,butnowtakingasuniformlyrandomlydistributed,wecanndtheRMSvalueforthephaseerrorinonechannelas: rms(sin")rms(")rms(ac 101

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5{12 to 5{15 canbeapplieddirectlytoestimatethemagnitudeoftheerrorsintroducedbybackgroundscatteredmoonlightcontamination.A300brewithabright-timeskybackgroundof19magarcsec2duetoscatteredmoonlightfromtheatmospheregivesatotalof16:9magofskybackground.Foramagnitude12star,thisgivesasource-to-contaminationuxratioofabout90.Intheworstcasescenario,fromequation 5{13 ,andassuming3300ms1rad1(roughlycorrectfortheKPNOET),wend"v37ms1.Thiswillapplywherethestellarspectrumissimilartothemoonlightspectrum(notuncommon,sincemosttargetsaresun-like),andinthecasewherethevelocitydierencebetweenstarandmoonlight,v,iscoincidentallyaround5kms1.Atsmallervelocities,theerrorwillscaleroughlylinearlyas"v=v=90uptothispoint.Afterthat,itwillimproveagainasincreasesto,wherethephaseerroronceagainapproacheszero.Asincreases,thebehaviourislikelytobesomewhatoscillatory,untilvislargeenoughthatthetwospectraarecompletelyuncorrelated.Forn=1000independentwavelengthchannels(ie.4000pixelchannelswithanLSF4pixelswide),theerrorshouldthenapproachequation 5{15 ,withavalueofaround"v0:8ms1.Thisshouldalsobethetypicalerrorsizewhenthestarandmoonspectraareverydierentinform.SimulationsoftheeectofmoonlightcontaminationbyMahadevanshowreasonableagreement.Figure 5-6 showstheRVdeviationcausedbysyntheticmoonlightcontaminationaddedtoasyntheticstellarspectrum,andthenmultipliedbytheinterferometerresponsefunctionanddegradedinresolutiontosimulaterealinstrumentspectra.Theresultingspectraareputthroughthestandardpipeline(ignoringtheiodinereference)toassesstheeectsofthecontamination. 102

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Simulationsofmoonlightcontamination,showingthesystematicerrorintroducedbycontaminatingmoonlightat19magarcsec2foraV=12F9Vstarona300bre,asafunctionofvelocitydierencebetweentargetstarandmoonspectrum.[CourtesyofSuvrathMahadevan,privatecommunication.] 5{9 throughby 103

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5{15 andwrite: 2.6.1 ,wecanalsofollowasimilarapproach,treatingthecrosstermfromequation 2{24 whichisignoredintheapproximation(orrather,treatingthelackofcrossterm)asifitwereacontaminatingspectrum.First,weconsiderthesimpliedcaseoftwodiscreteoverlappingGaussianabsorptionlines,fromtemplatespectralabelledAandB(e.g.aniodineandastellarline),combinedbymultiplicationtogivethemeasuredspectrum,labelledM.Bothlinecentresareexactlycoincident.ThefractionallinedepthsarerepresentedbyD(0D1),withcorrespondingsubscriptsa,bandm.From Ge ( 2002 ),wehavethat: 104

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Intheadditionapproximation,thecomplexvisibilitiesofthetemplatespectraareaddedtogether.Inthissimplecase,thetwolinesarecentredatthesamewavelengthandbothareGaussian,sothatonelineissimplyascaledversionoftheother.BythelinearityofFouriertransforms,thismeansthatthephasesofthetwocomplexvisibilitiesmustbeidentical,sothatintheadditionapproximation,thetwoabsolutevisibilitiesaddtogivema+b.Theremainingterminequation 5{18 ,DaDbK=",isthereforeapproximatelytheerror,thedierencebetweentheaddedtemplatesandtheactualmeasuredvisibility.Inthemoregeneralcasethatthetwolinecentresarenotexactlycoincidentorthesameshape,sothattherespectivetemplatefringesarenotinphase,theerrortermwillalsoincludeaphasedierence,becomingatwodimensionalvector,"ei".Takingtheerrortermaboveasareasonableestimateofthelengthofthisvectorandassuming"isuniformlyrandomlydistributed,wecancalculateacorrespondingrepresentativeerrorinphaseofthesummationapproximation.Figure 5-7 showstheadditionofthe\true"(measured)complexvisibilityandtheerrortermtogivethesolutionaccordingtothesummationapproximation(comparetogure 5-5 ).representsthephaseofthetruecomplexvisibility,and"representstheerrorinthemeasurementofthatphase.Ifweassumetheresultingmeasuredvisibilityvectorsandtheerrortermsareuncorrelatedfromchanneltochannel,andifwetakeDaandDbtobesomekindofrepresentativeaveragelinedepthforthetwospectraacrossallj,wecanderivethetypicalexpectedvelocity 105

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Vectorrepresentationofthesummationofthetruecomplexvisibilityandtheerrortermduetotheadditionapproximation. errorfollowingthesamereasoningasforequation 5{16 andwrite: wherenisagainthenumberofindependentwavelengthchannels.Figure 5-8 showstheexpectedtypicalerrorasafunctionofaveragelinedepthforthesimpliedcasewherethetypicaldepthsofthetwospectraareequal.Foraveragelinedepthsof,say,80%forbothstarandiodine,andagaintaking3300ms1rad1andn=1000thisgivesatypicalerrorduetotheadditionapproximationof50ms1,whichisclearlyverysignicant.Theerrorwillmanifestasasystematicerrorinthevelocityresponseoftheinstrument,essentiallyaddingnoisewhichvariesasafunctionofthespecicoverlappingofthelinesbetweentargetstarandreferencespectrum.It 5-7 andingure 5-5 aremeasuredfromdierentorigins,theyareinbothcasestakentobeuniformlyrandomlydistributedbetween0and2,sothatthesamereasoningappliesforboth. 106

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5-9 showstheresultsofsimulatedfringingspectrarunthroughthepipelinebyMahadevantoseetheeectofnon-linearityduetotheadditionapproximation,andshowsbroadagreementwiththeseexpectations.Wenoteherethat,alongthesamelines,usingthecorrect,ux-dependent,andhencechannel-dependent,visibilityscalingfactors,KsandKi(equation 2{27 )isalsoofsignicance.Sincethesehaveonlyrecentlybeenfullyunderstood,andhadbeenpreviouslybeenallowedtobefreeparametersthatareconstantoverallchannels,thefullformhasnotyetbeenincludedinthepipeline.Thisomissioncanitselfbeexpectedtocontributesomesystematicerror.Mahadevanhasperformedtestswithsimulationstomodelthecross-talktermintheadditionapproximation,andfoundthatincludingthefulluxdependencyalongwiththecross-talktermisindeedessentialtoavoidsystematicerrors.However,includingtheux-dependencyalonewithoutincludingthecross-talktermleftremainingsystematicerrorsonasimilarscaletothosefoundwithoutincludingeither.Clearlytheadditionapproximationisaverysignicantsourceofsystematicerror,anditwillbeessentialtosolvetheproblem.Itcan,however,bemitigatedinthemeantimebyjudiciousselectionofobservationtimesandpositionsoftargetsonthe 107

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Analyticallycalculatedexpectederrorduetotheadditionapproximation,assumingapproximatelyequallinedepthsforbothstarandreferencespectra. sky,sothatthebarycentricmotionoftheEarth{usuallythedominanteectthatcausesthenon-linearitytobecomesignicant{isminimised.Suchobservationsareoftennothardtoachieveatleastoverperiodsofafewdays,andbecauseofthisfact,wehavebeenstillbeenabletomakemanyusefuldetectionsofgenuineastrophysicalRVsignals. 4.4.3.2 )correctsthiseect,andhowmuchthesamealgorithmintroduceserrorsofitsown.(Althoughfutureversionsofthesoftwareshouldrenderthisproblemirrelevantbynolongermakingsuchanapproximation.)TheeectofLSFvariationalongthelengthoftheslit,causedbynon-uniformslitillumination,alsohasyettobequantied,ashastheeectofdistortionalongthelengthoftheslitgivingrisetonon-uniformspatialfrequencyofthefringes.Likewise,itwouldbegoodtoestablish 108

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Simulationsshowingtheadditionapproximationerror,byMahadevan.Thenon-linearityintheRVresponsehasthesameorderofmagnitudeandoccursonthesameinputvelocityscaleasexpectedfromtheoreticalpredictions.[CourtesyofSuvrathMahadevan,privatecommunication.] atwhatloweruxlimitthedatareductionalgorithmsbegintobreakdownsothattheresultsnolongerfollowsimplephotonnoisepredictions,andtobetterestablishthenoiseooratwhichpointthereisnogaintobehadbyincreasingthephotonux.ManyoftheseissueswillbeeasiertoaddressonceMahadevan'scodeforproducingrawdatasimulationsisfullyoperational:itshouldtheninprinciplebepossibletosimulatevariouseectsoneatatimeinacontrolledmanner,runthesimulationsthroughthepipeline,andestablishsemi-empiricallythesizeoftheerrorcausedbyeachone. 109

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5-10 showsthevelocitysemi-amplitude,K,expectedinthereexmotionofastarduetoaplanetarycompanionasafunctionoforbitalperiod,atdierentminimumplanetmasses(msini).Thiscanbeusedasaroughguidetothemassesofdetectableplanetcompanionsatagivenerrorlevel,atleastinthecaseofrandomuncorrelatederrors.Kscalesrelativelyweaklyasthe2=3powerofstellarmass,wheremassesaregenerallyintherange0:7{1:6MfortheETsurveys.AtthecurrentbestETprecisionof3ms1short-termforbrightSun-likestars,wendadetectionlimitofaboutmsini>0:1MJforhot-Jupiters(period<10d),atthe3-sigmacondencelevel.AttypicalKeckETprecisionsof30{40ms1,wemightexpectsomethingmorelikea1:0MJdetectionlimitatperiodslessthan10d. 110

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Radialvelocitysemi-amplitude,K,fordierentminimumplanetmasses(msini),assuminga1Mprimaryandzero-eccentricityorbits. 111

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vanEykenetal. 2003 2004a ).ItwasalsousedtomaketherstactualdiscoveryofanexoplanetusingtheDFDItechnique( Geetal. 2006a ),thendbeingdesignatedET-1(HD102195b).Anoverviewoftheinstrumentitself,alongwithreferencestomoredetaileddescriptions,isgiveninchapter 3 3.2.1 ),showingaroot-mean-square(RMS)scatterof11:5ms1inthemeasurementsovervedays.Theresultswerereportedin vanEykenetal. ( 2004a ),wherewealsoshowedcomparisonmeasurementsoftheRV-stablestar,Cas,whichdemonstratedanRMSscatterof7:9ms1oversevendays,startingtoapproachtheprecisionlevelsobtainedwithtraditionalRVtechniquesbasedoncross-dispersedechelles.Theseresultsaresummarisedhere. 4 .RawspectrawerersttrimmedanddarksubtractedusingstandardIRAFroutines,withbiasbeingsubtractedalongwiththedarksinonestep.Pixel-pixelateldingwasperformedusingnon-fringing 112

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3.1 ).TherestofthedatareductionwasthenperformedusingcustomsoftwarewrittenintheIDLdataanalysislanguage.Theimageswere`selfilluminationcorrected'usinganalgorithmtoextracttheunderlyingcontinuumilluminationfunctionwhichisdividedoutfromeachimage.Thisavoidsproblemswithchangesintheilluminationovertime.ThespectrawerethencorrectedforslantsothattheslitdirectionwasexactlyalignedwiththeCCDpixelaxes.Theywerethenlow-passFourierlteredinordertoremovetheinterferometercomb.Afterthesepre-processingsteps,thephaseandvisibilityweredeterminedforeachwavelengthchannelbyttingasinusoidtoeachcolumnoftheCCDimage,eachpixelbeingweightedaccordingtothenumberofcountsintheoriginalnon-ateldeddata,ontheassumptionofphotonnoisedominatederror.Sincefringespatialfrequencyvariesonlyslowlyasafunctionofwavelength,wetasmoothfunctiontothefrequenciesobtainedfromthesinusoidts,andthenperformedasecondpasswiththefrequenciesxedtomatchthisfunction,helpingtoreducerandomerrors.Purestellarandpureiodinetemplatespectraweretakenatthebeginningofeachobservation,andthesewereusedtomathematicallyextractthephaseshiftsofthestarandtheiodineindividually,andhencecalculatetheintrinsicstellarvelocityshiftcorrectedforinstrumentaldrifts(seesection 4.4.3.1 ).Finally,theRVduetothemotionoftheEarthwassubtractedtoleaveanintrinsicstellarrelativevelocitycurve.Theexposuretimewastakentobethecentreoftheexposure.Errorbarswerebasedonthestandardstatisticalcurve-ttingerrorsdeterminedduringmeasurementofphaseandvisibility.Theerrorsweretranslatedtoerrorbarsthroughcalculationsappropriatetothealgorithmsusedtoextractthenalintrinsicstellar 113

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Smallsectionofrawfringingspectrumof51Pegwithiodine,obtainedatKPNOonthenightofAugust14,2002(V=5:5mag,S/N50perpixel).(Theinterferometercombcanbeveryfaintlyseennearthetoppartoftheimage.)[Reproducedfrom vanEykenetal. ( 2004a ).] RV(section 5.1 ).Theyareexpectedtogiveareasonableguidetotherandomscatterexpectedinthedata,althoughtheymaynotcatchallsystematicerrors.Oncloserinspection,thedatafromthistrialrunwerefoundtoshowsignicantvariationinthefringephaseandvisibilityalongthelengthoftheslit,perhapsduetonon-uniformslitillumination,ortoaberrationanddistortionintheoptics.Wethereforecutthespectraintothreeslicesalongthedispersiondirectionandtreatedeachsliceseparately,inordertoobtainsinusoidaltslessaectedbythissystematicerror.AweightedaverageofthethreeresultswasusedtogivethenalRVplot. 6-1 ,obtainedin25minutesatvisualmagnitude5.5withS/Nperpixelinthecentralstripofaround50.TypicalexposuretimesforCas(mag3.5)were30minatanS/Nof80(includingiodinecelllosses). 114

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PlotofRVmeasurementsfor51Peg,withthepredictedcurveover-plotted.RMSresidualsare11:5ms1.[Reproducedfrom vanEykenetal. ( 2004a ).] Figure 6-2 showstheradialvelocityvariationmeasuredfor51Pegafterdiurnalmotionwassubtracted.Thezeropointwaschosenarbitrarily.Over-plottedistheexpectedcurveextrapolatedfromorbitalparametersdeterminedby Naefetal. ( 2004 ).Thesamedataarelistedintable 6-1 .S/Nperpixelratiosobtainedwereintherange40{60forstar+iodinespectra.ThetemplatesusedfortheprocessingwerefromthenightofAugust16,2002(August17UT),andS/Nfortheiodineandstartemplateswereapproximately300and70perpixelrespectively(forthecentralofthethreestrips).AveragingoverthethreedetectorstripsgaveanRMSdeviationfromthepredictedcurveof11:5ms1.Thevalueofthereduced2is2.70. 1 vanEykenetal. 2003 ),dueinparttousingallthreedetectorstripsandalsotoseveralimprovementsinthereductionsoftware. 115

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RVmeasurementsfor51Peg[Reproducedfrom vanEykenetal. ( 2004a ).] JuliandateRadialvelocityError2450000(ms1)(ms1) 2500:866535.47.12500:887946.86.82500:909429.46.82501:907718.16.82501:926330.87.02501:944129.97.02502:939278.18.42502:957361.68.32502:975157.97.82503:925050.96.72503:948057.96.82503:969227.66.62504:896044.26.52504:917728.16.62504:939215.46.8 ResidualsafterdiurnalcorrectionforthestarCasareshowningure 6-3 andtable 6-2 ,usingtemplatesfromthenightofAugust15,2002(Aug16UT).CasisaknownRVstablestar(W.D.Cochran2002,privatecommunication)andisthereforeexpectedtoshowzeroshiftatourcurrentlevelofprecision.Thethreeimagestripsareaveraged,weightedaccordingtoux.TheRMSscatteris7:9ms1,withareduced2of2.03.TypicalS/Nperpixelinthecentralstripisaround70{90forstar+iodinespectra,270fortheiodinetemplate,and100forthestartemplate.Under1.5arc-secseeingconditions,weobtainedatotalinstrumentthroughputof4%,fromabovetheatmospheretothedetector,includingsky,telescopetransmission,breloss,instrumentandiodinecelltransmissionanddetectorquantumeciency.Thisthroughputwasobtainedusingonlyoneinterferometeroutput.Excludingslitloss,thetransmissionoftheinstrumentitselffrombretodetectorwas19%. 2.7.1 ,usingequation 2{37 .Forthe51Peg 116

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PlotofRVmeasurementsforCas,anRVstablestar,expectedtoshowzeroshift.RMSresidualsare7:9ms1.[Reproducedfrom vanEykenetal. ( 2004a ).] measurements,averagingoverallthedatapointsgivesameanphotonerrorduetothestarcomponentof6:7ms1,andameanerrorduetotheiodinecomponentof7:2ms1.DoingthesamefortheCasmeasurementsyieldsmeanerrorsof5:3ms1and4:3ms1forstarandiodinecomponents,respectively.Addingthecomponenterrorsinquadraturegivesatotalmeanphotonerrorof9:8ms1for51Peg,and6:8ms1forCas,assummarisedintable 6-3 vanEykenetal. ( 2004a ),wheretheerrortermsduetothetemplatesthemselvesshouldhavebeenneglected,asdiscussedinsection 2.7.1 .Theguresgivenherearemorerealistic,buttheconclusionsremainessentiallythesame. 117

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RVmeasurementsforCas.[Reproducedfrom vanEykenetal. ( 2004a ).] JuliandateRadialvelocityError2450000(ms1)(ms1) 2498:83186.45.72498:85458.35.12498:882311.25.42498:893610.95.32499:78396.86.32499:79522.96.02499:80643.65.82499:81741.75.82499:82841.45.92501:87015.95.32501:87779.65.42501:885015.35.52501:89267.75.42502:85280.04.92502:86431.74.92502:87525.45.12502:886010.85.42504:80931.45.72504:820110.16.42504:83112.55.42504:840111.75.9 Table6-3. Meanphotonlimitingerrorestimationfor51PegandCasobservations,comparedwithRMSresidualsfromthedata. TargetStarcomponentIodinecomponentTotalphotonerr.RMSresiduals(ms1)(ms1)(ms1)(ms1) 51Peg6.77.29.811.5Cas5.34.36.87.9 wecanseethattheinstrumentwasalreadyperformingclosetothephotonlimit:thereductionsoftwarewassuccessfullyextractingalmostthemaximumpossibleinformationfromthedata.Forcomparison,RMSscattersobtainedpreviouslyfor51Peghadbeen13ms1( Mayor&Queloz 1995 ),5:2ms1( Marcyetal. 1997 ),and11:8ms1( Naefetal. 2004 ).Giventhephotonlimit,theerrorbarsinthedataappeartobesomewhatunderestimated(leadingtothelargevaluesforthereduced2).Apossiblecauseofthiswasthelowpass 118

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3.2.2 ).Alongwithcontinuingrenementstothesoftware,theupgradesenabledustomakesubstantiallyimprovedmeasurementsof51PeginDecember2004,showingRMSresidualsnowaslowas7:9ms1over6datapointsandanaveragephotonlimitingerrorof9:4ms1usingonlythe0:9mcoudetelescope,ratherthanthe2:1m(seegure 6-4 ).Earlier,inMarch2004,wehadbeenabletoobtainashort-termRMSof3:6ms1over2hronanotherknownstablereferencestar,36UMa(visualmagnitude4.83),usingthe2:1mtelescope,withmeanphotonlimitingerrorsof4:0ms1(seegure 6-5 ).Inboththesecases,theRMSwasagainconsistentwiththephotonlimit,giventheuncertaintyintheRMSduetothesmallnumberofdatapoints.Theerrorbarsrepresentthecalculatedphotonlimitingerrorinallofthefollowingresultsinthissection.Beginninginthewinterof2004,webeganasmall-scalesurveywiththeinstrumenttosearchforshortperiod(<10d)planets,andwewereeventuallyabletouncoverourrst 119

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51PegRVmeasurementsinDecember2004withtheupgradedET,withpredictedRVcurveoverplotted.7:9ms1RMSresidualswereobtainedwiththe0:9mcoudetelescope{animprovementoverpreviousthe11:5ms1obtainedwiththe2:1min2002(gure 6-2 ). newplanet,ET-1,orbitingthestarHD102195.Theinitialtentativendings,discussedhere,werereportedin vanEykenetal. ( 2005 );theocialdiscoverywasannouncedin Geetal. ( 2006a ),representingthersttimeanexoplanethasbeendiscoveredbyRVmeasurementsaroundastarfainterthanmagnitudeV=8usingasub-metersizedtelescope. 6.2.1.1ObservationsanddataanalysisObservationswereconductedofaround90previouslyunsearchedstarsduringtheperiodfromDecember2004{May2005usingtheKPNO0.9mcoudefeedtelescopefromDecembertoMarchandthe2:1mtelescopeinMay.WechosemosttargetsprimarilyfromtheN-starcatalogue( Grayetal. 2003 ),selectingdwarfstarsoftypeF{KwithvisualmagnitudeV=8{9,highmetallicity,andwhereknown,slowrotationandlowactivity 120

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36UMa(knownRVstablestar)shorttermprecisionmeasurementswiththeupgradedETinMarch2004,showingphoton-limitingperformanceat3:6ms1RMSresiduals. indicators(targetselectionisdiscussedinmoredetailin Mahadevan 2006 ).Anyknownvisualdoublesorvariablestarswererejectedfromthelist.Wherepossible,candidatesshowingsignicantRVvariationearlyinthesurveywerefollowedupusingthe2:1mtelescopeinMay2005.Thesurveywasdividedintoveobservingrunsofdurationsvaryingfrom8{21nights,therstfourrunsusingthecoudefeedtelescope,andthelastswitchingtothe2:1mpartwaythroughtherun.Atotalof59nightswereallocatedforthecoudefeed,and7nightsonthe2:1m.Observationsweremadeon43ofthenights,theremainderbeinglostprimarilyduetobadweather.Starandiodinetemplatesweretakenatthebeginningandendofeachobservingrun,andtypically5{6RVmeasurementswiththeiodinecellinthebeampathwereacquiredpersurveystarforeachrun.Typicalexposuretimesforthecoudefeedwere25minforstarsatmagnitudeV8and40minforV9.Onthe2:1m,exposuresweregenerallykeptto10min.Atypicalrawspectrumfromthecurrentupgradedinstrumentisshown 121

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ExamplesectionofrawspectrumtakenwithupgradedET(55Cncwithoutiodine,May2005). ingure 6-6 .Thedatawerereducedusingessentiallythesamepipelineasisnowused,describedinchapter 4 ,althoughtheprocesswasthennotasstreamlined,andlackedthefullyautomatedmulti-objectmultiplexingcapability. vanEykenetal. 2005 ).Inordertoestablishanindependentestimateofoursurveyprecision,wedenedstarswhichappearedRVstableatourprecisionleveltobethoseshowingareduced2<2overtheperiodofobservation.Atthe0.9mcoudeduringonetypicalobservingblock,7outof15searchstarssatisedthiscriterion.Forthose,wefoundameanRMSscatterof18:9ms1comparedtoameanphotonlimitingerrorof21:2ms1,atameanvisualmagnitudeof8.26andtypicalexposuretimesof20-30min.(ThelowervalueoftheRMSiscomfortablywithintherangeexpectedduetostatisticalvariation).Fromthe2:1mobservations,usingthesamecriterion,wefound12stablestarsoutof24searchstars,forwhichwefoundameanRMSof17:6ms1,comparedtoameanphotonlimitingerrorof17:4ms1.Inthiscase,themeanVmagnitudewas8.48,andtypicalexposureswere10min.Inbothcasestheerrorswereconsistentwiththeexpectedphotonlimit.Ofthe90starssurveyed,weselectedthosethatshowedanRMSdeviationaboutaconstantRVgreaterthan2.5timesthemeanphotonerroraspossiblecandidates.Weruledoutanythatshowedvariationgreaterthan1000ms1ontheassumptionthatthesewerelikelytobebinarystars.10candidatesremained,andHD102195appearedtobethemostpromising. 122

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Knownplanet-bearingstar,55Cnc,measuredwithcurrentKPNOET.PredictedRVcurvedueto55Cncbisoverplotted.Additionalscattermaybeduetoacloser-inplanetnotincludedinmodel. Figure 6-7 showsresultsthenobtainedatthe2:1mfortheknownplanetbearingstar,55Cnc,usedasacontrol,withthepredictedRVcurveduetoits14:7dcompanion55Cncbover-plotted.Theotherknowncompanions( Butleretal. 2006 )areneglectedhere,sincetheyhavemuchlongerperiodsandwouldhaveverylittleeectoverthistimescale,withoneexception:partofthereasonforthesomewhatlargereduced2valuemaywellbebecauseofaninnerplanet,55Cnce,reportedby McArthuretal. ( 2004 )tohavea2:8dperiodandvelocitysemi-amplitudeof6:7ms1.(Modellingthisplanetintothepredictedcurveisdicultbecauseitislikelytobegravitationallyinteractingwith55Cncb.)RMSdeviationaboutthepredictedcurvewas7:8ms1,comparedtoameanphotonlimitingerrorof4:2ms1.Figure 6-8 similarlyshowstheperformanceovertheperiodofafewdaysagainusingthebrightRV-stablereferencestar36UMa. 123

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RVstablestar,36UMa,measuredoverafewdayswithcurrentKPNOETin2005.Smallesterrorbaris4:1ms1. Figures 6-9 and 6-10 showapreliminarybest-tKeplerianorbitcurvetotherstsetsofdataforHD102195,showingtheplanetarycompanionRVsignal.Dataforthiscandidatewereobtainedinthreesetsofobservations,eachofaboutoneweek,inJanuary,March(0.9m)andMay2005(2:1m).Theosetsbetweenthedatasetswerearbitrary,eachobservingrunbeingreducedseparately:theosetswerethereforechosentogiveagoodmatchtoasingleKepleriansignalwithnolong-termtrend.Azero-eccentricityRVtatthemostlikelyperiodgiventhedata,4:85d,wasconsistentwithaplanetarycompanionofminimummassMsini=0:41MJorbitingat0:052AUfromthehoststar,withanRVsemi-amplitudeof58:1ms1,RMSresidualsof20:2ms1,andreduced2of1.49(wheretheerrorbarsshownareagainthephotonlimitingerrors).ThesevaluesgaveamasslimitofMsini=0:41MJ,consistentwithotherknownhot-Jupiterexoplanetsinthesameperiodrange.Figure 6-11 showsaLomb-Scargleperiodogram( Lomb 1976 ; Scargle 1982 )forthedatathentakentothatpoint,showingthepoweratvariousperiodsintheRV 124

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BesttKeplerianorbitforearlyET-1RVmeasurements,assumingazeroeccentricityplanetarycompanion.Osetsbetweenthethreeobservingrunswerearbitrary,andchosenassumingnolongtermRVdriftbetweentheruns.

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Asforgure 6-9 ,butphasefoldedonthebest-tperiodof4:85d. signal.Themostsignicantperiodicityof4:85dwasusedasaninitialguessforthe2minimisationinthecurvetting.Itcanbeseen,however,thatthesamplingofthedatagaveanumberofotherperiodsofsimilarlikelihood.Moredatawereneededtonarrowdownthepossibilities:laterfollowupdata(section 6.2.2 )weretoshowthatinfactthepeakataround4:1dwasthecorrectone. 126

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Lomb-ScargleperiodogramforearlyET-1data.Themostprobableperiodismarkedwithanasteriskat4:85d,showingasignalabovethebackgroundatthe99.7%signicancelevel. comparabletoaplanetarysignature.Asstellarrotationbringsthespotinandoutofview,coveringdierentpartsofthedisc,theRVsignalcanbemodulated.TheeectcanbedistinguishedfromtrueDopplermodulation,however,becauseinthecaseofsurfacefeatures,theapparentmagnitudeofthestarwillvaryinsynchronywiththeRVsignal.Itisthereforefairlystandardproceduretotakephotometricmeasurementsofsuspectedplanetbearingstarstoruleoutsuchproblems.Figure 6-12 showspreliminaryphotometricmeasurementsofthesametargettakenafewweekslaterbyGregoryHenryatTennesseeStateUniversity,usingtheAutomaticPhotometryTelescopeatFairbornObservatory.Somevariationwasseenatthe0.01maglevel,withsomewhatofaperiodicityaroundroughly12days,showingthattherewasindeedsomestellarvariation.The12-dayperiod,however,wouldlikelycorrespondtotherotationalperiodofthestar,andwasfarenoughremovedfromtheperiodoftheRVcurvenottoruleoutthepossibilityofaplanetarycompanion. 127

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EarlyphotometryofHD102195,theparentstarforET-1(combinedStromgrenbandydierentialmagnitudes).TakenbyGregoryHenryusingtheAPTatFairbornObservatory.A12dperiodicityisevident,atthe0:01maglevel.Errorbarsare0:001mag.[Reproducedfrom Geetal. ( 2006a ).] Geetal. ( 2006a ),butarebrieysummarisedhereforcompleteness.InDecember2005,afurther21datapointsweremadeusingETandthe2:1mtelescopetotryandweedouttheharmonicsintheperiodogramandpindownthecorrectperiod.FromNovember2005{January2006,afurther10RVmeasurementswereindependentlymadewithconventionalRVmethodsusingtheMcDonaldObservatory9mHobby-Eberlytelescope(HET),inaneortledbyDonaldSchneideratPennStateUniversity.UsingtheHighResolutionSpectrograph(HRS),themuchlargertelescopeapertureallowedforinternalerrorsofonly2ms1,andthemeasurementsconvincinglyconrmedtheRVsignaldetectedwithET.CombiningtheHETmeasurementswiththeETmeasurementsallowedforamuchmorepreciseconstraintontheorbitalparameters 128

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FoldedcombinedradialvelocitiesforHD102195,combiningETmeasurementsfromtheKPNOCoudeFeedtelescope(lledcircles),2:1m(lledtriangles),andHRSmeasurementsfromtheHET(opencircles).Twoorbitalcyclesareshownandeachobservationisplottedastwopoints,foldedonthebest-torbitalperiod.ThebestKeplerianttothedata(overplotted)yieldsanorbitalperiodof4:11434d.[PlotproducedbyRobertWittenmyer,UniversityofTexas;reproducedfrom Geetal. ( 2006a ).] ofET-1.ThecombinedRVdatasetsareplottedingure 6-13 ,alongwiththebest-tKeplerianorbitsolutioncalculatedbyRobertWittenmyerattheuniversityofTexas. 3 Mahadevan ( 2006 ). 129

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OrbitalparametersforHD102195. ParameterValue Geetal. ( 2006a );anerrorinthedependenceoftheminimumplanetmassonthestellarmasshasbeencorrectedhere.] TheperiodogramwasupdatedbyStephenKaneusingthefulldataset,andisshowningure 6-14 forcomparisonwithgure 6-11 .The4:11dpeakisnowclearlythestrongest.EricFord,thenatBerkeley,performedafullBayesiananalysisofthedatatoobtainthebestpossibledeterminationoftheorbitalparameters,andfoundfullyconsistentresults,attributingthepreviouslyused4:8dperiodtoaliasingbythelunarcycle(sinceobservationsweregenerallytakenonlyduringbrighttime,aroundthetimeoffull-moon).Asbefore,theosetsbetweenthedierentrunsandinstrumentswerearbitrary,andsowereallowedtooatasfreeparametersintheanalysis.Theorbitalparametersforthebestsolutionarelistedintable 6-4 .Thefullsetofradialvelocitymeasurementsfromalltheobservingrunsislistedintable 6-5 .AmeasurementofthestellarparametersforHD102195,listedintable 6-6 ,wasleadbyEduardoMartnattheInstitutodeAstrosicadeCanarias,usingthehighresolutionSARGspectrographonthe3:5mTelescopioNazionaleGalileoatLaPalmainJune2005.ThismeasurementwasalsousedtocheckforthepresenceofanyfaintstellarspectroscopicbinarycompanionwhichmightbeosettingthelinecentroidpositionsandmasqueradingasaDopplershift;nonewasfound.TheSARGspectrawerealsousedbyMartn'steamtoperformalinebisectoranalysis.A`linebisector'tracesthecentreofanabsorptionlineasafunctionofdepth 130

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CompleteradialvelocitiesforHD102195 JDInstrumentaRVErrorsJDInstrumentaRVErrors2450000ms1ms12450000ms1ms1 Geetal. ( 2006a )] 131

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UpdatedperiodogramforHD102195,byStephenKane,usingthefullcombineddatasets.Dierentfalsealarmprobabilitiesareindicatedbythedashedlines.Thepeakpowerisnowveryclearat4:11d,withafalsealarmprobabilityof106.[Reproducedfrom Geetal. ( 2006a ).] intheline,givinganindicationofvariationinthesymmetryoftheline.Analysisofthevariationofthelinebisectorcangiveinformationaboutstellarvariability,particularlythepresenceofstarspotsorchangesinthegranulationpattern,whichcancreatelineasymmetriesthatlooklikeRVvariations( MartnezFiorenzanoetal. 2005 ).Overthe3daysofSARGobservations,nosuchbisectorvariationwasfound.Furtherhigh-precisionphotometricanalysisbyGregoryHenryupuntilFebruary2006showedthesameperiodicityasbeforeat12:30:3d,butnowatmuchloweramplitude.Thisbehaviourisconsistentwithslowstarspotorplagevariationonastarrotatingata12:3dayperiod.Theperiodisalsoconsistentwiththeprojectedrotationalvelocity 132

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StellarparametersforHD102195 ParameterValue [Reproducedfrom Geetal. ( 2006a )] (vsini)withtheSARGmeasurementsiftheplanet'sorbitisclosetoedge-on,althoughthephotometryalsoruledoutanytransitsignaturestoahighdegreeofcondence.Finally,measurementsandanalysisofemissionlinecoresatthebottomoftheCaIIHandKlines,createdinthestellarchromosphere,wereundertakenbySuvrathMahadevan( Mahadevan 2006 ),usingtheKPNO0:9mcoudespectrograph.Thesecanbeusedasanindicatorofstellaractivity,andtheresultswereconsistentwiththeinterpretationofHD102195asamildlyactivestar.SimilarmeasurementsweremadeusingaspectrumobtainedwiththeFOCEShighresolutionechellespectrographattheGerman-SpanishAstronomicalObservatory(CAHA,Almera,Spain),ledbyDavidMontesattheUniversidadComplutensedeMadrid,leadingtosimilarconclusions.VariationinthechromosphericactivitymeasuredbyMahadevanshowednosignalattheorbitalperiodoftheplanetthatwouldcauseconcernfortheplanetaryinterpretation,althoughsomevariationwasseenthatcouldhavebeencorrelatedwiththe12:3dstellarrotationperiod. 133

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2.6.1 and 5.10 ,whichwehadnotatthetimeappreciated.Theadditionapproximationappearsasanon-linearityintheRVresponsethatsetsinwhenthescaleofRVvariationbecomeslargerthanoforderafewkms1.SinceexoplanetaryRVsignaturesareonmuchsmallerscalesthanthis,itisprimarilythemotionoftheEarthrelativetotheSolarSystembarycentre(whichcanbeaslargeas60kms1)thatdetermineswhentheadditionapproximationbecomesaproblem.Thepositionsontheskyandtimesofobservationof51PegandHD102195weresuchthatthedierentialbarycentriccorrections,showninguresgures 6-15 and 6-16 ,wererelativelysmalloverthelengthsoftheruns. 134

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6-16 wecansee,however,thatinattemptingtojoinrunstogether,theiodineapproximationwouldcontributeverysignicantrandomrun-to-runoseterrors,onthescaleofmanytensofmeters(seesection 5.10 ).ThiserrorsourcethereforelikelyexplainswhyattemptstolinkthesetsofobservationsbetweenthevariousETrunswerelargelyunsuccessful,forcingustoallowoatingosetsbetweeneachrun. 135

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Dierentialbarycentriccorrectionsforearly51PegmeasurementswiththeKPNOET.Thecontinuouslineshowsthebarycentriccorrectionasafunctionoftime.Thepointsmarktheactualtimesofobservation.ThearbitraryRVzero-pointischosenforclaritytoindicatethescaleofvariation. 136

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DierentialbarycentriccorrectionsforHD102195measurementswiththeKPNOET.Asingure 6-15 ,thecontinuouslineshowsthebarycentriccorrection,andthepointsmarkactualtimesofobservation.Insetsshowzoomedinplotsforeachobservingrun.They-axesallhavearbitraryzero-pointschosenforclaritytoindicatethescaleofvariation.

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Ge ( 2002 ); Geetal. ( 2002 2003a ); Mahadevanetal. ( 2003 );preliminaryresultswerereportedin Geetal. ( 2006b ); Mahadevanetal. ( 2005 ); vanEykenetal. ( 2007 ).Majorengineeringupgradesarealsocurrentlyunderway,includingreplicatingtheKeckETtodoublethesimultaneoustargetcapacityto120objects,inanticipationofa`Multi-objectAPORadialVelocityExoplanetLarge-areaSurvey' Geetal. 2007 ).Herewereporttheearlyresultsfromtheprototype,andfromobservationswiththefull60-objectKeckETupuntilNovember2006whichformedthetrialsurveyfortheproposedpilotprogramthatwouldrunfrom2006{2008.Abriefdescriptionoftheinstrumentisgiveninsection 3.3 .Themostup-to-dateinstrumentdescriptioncanbefoundindetailin Geetal. ( 2006b ); Wanetal. ( 2006 ); 138

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( 2006 ).Datawerereducedusingtheproceduresoutlinedinchapter 4 ,thefullmulti-objectpipelinebeingbuiltupontheoriginalKPNOETpipelineduringthecourseoftheobservations.TargetselectionwasledbySuvrathMahadevanandRogerCohen,andisdiscussedindetailin Mahadevan ( 2006 ).Earlyeldsweremainlychosenaroundknownplanet-bearingorRV-stablereferencestars.Asidefromthesereferencestars,surveystarsintheeldswerechoseninthemagnituderange7:6
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Geetal. ( 2006b ).Alongwiththeotherdataobtained,theyindicatedthatalthoughtherewereproblemstobeironedout,performancewassucienttoproceedwithafull60-objectinstrument. 7-2 showsanexampleraw-dataframe,showingafullsetofsimultaneousstellarspectra.Dataweretakentomeasuretheapparentsolarvelocityreectedothedaytimesky(gure 7-3 ),providingazeroreferenceforwhichtypicalRMSscatterswerearound18{27ms1overthedurationoftherun,and12{16ms1(matchingthephotonlimit)overperiodsofafewhours. 7-4 showsexampleresultsobtainedforthreeknownplanet-bearingstarsusedasreferences,includingET-1,showingperformanceclosetothephotonlimit.Fromtheknownhot-Jupiteroccurrencerateof1%amongsolar-typestars,weexpected4hot-Jupitertypeplanetstobepresentinthetotalsample.Amongthesearchtargetsfromtherun(examplesof 140

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Resultsfromthe20-objectprototypeinstrumentinApril2005,forRV-stablestar36UMa,knownplanet-bearingstar55Cnc,andanexamplesurveystar,TYC49854851.ExpectedRVcurvesareoverplottedfor36UMaand55Cnc.Errorbarsrepresentphotonerrors. 141

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7-5 ),around15werechosenasinterestingcandidatesforfurtherfollowup.Thesepreliminaryresultswerereportedin vanEykenetal. ( 2007 ).Althoughwehadobtainedreasonableresults,therewasstillclearlyroomforimprovementatthispoint.Instrumentthroughputwasaroundathirdofthedesignspecications,increasingthephotonerrorscorrespondinglybyafactorofp 2{31 ),thisalsocouldcauseafurtherfactoroftwolossinprecision.The12-daynoiseoorof20ms1seeninthesolardataalsosuggestedfurthersourcesofsystematicerror.Theseissuesweretobetargetedinthefollowingengineeringruns. 7-6 and 7-7 showmeasurementsoftwoknownplanet-bearingreferencestars,HD209458(magnitudeV=7:70)andHIP14810(V=8:59),alongwiththeirpredictedRVcurves.Bothshowagoodmatchtothepredictions,withsignicantlysmallererrorsthaninMay,HD209458showingRMSresidualsof14:1ms1andameanphotonlimitingerrorof8:5ms1.AlthoughtherewereonlytwodatapointsforHIP14810,thedierenceinRVbetweenthe 142

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ExamplerawdataframefromearlyKeckET,May2006,showing54usablesimultaneousstellarspectra,withoutiodine.A)Fulldataframe.B)Magniedportion.Interferometerfringescanfaintlybeseen.

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ExamplesolardatafromearlyKeckET,May2006,obtainedbypointingbreendsdirectlytodaytimesky,showingtheexpectedzerovelocitydrift.Shortterm(fewhour)RMSmatchesthephotonnoise.Resultsarefromsimultaneousmeasurementswithdierentbres(`#'indicatesspectrumnumberonthedetector). twopointsiscloseenoughtothepredictiontobeausefulconrmationofperformance,withaphotonlimitingerrorof16:4ms1.Thegeneralimprovementininstrumentprecisioncanclearlybeseenintheday-skydatatakenduringtherun(gure 7-8 ).Mostofthe59spectraonthedetectorshowedRMSscattersintherange7{10ms1overthevedaysoftherun,downfrom18{27ms1intheMaydata.WhileafewhadlargerRMSscatters,someapproachedtheirphotonlimit,typically6ms1(downfrom12{16ms1).Themeasurementofskydatasimultaneouslywithallthebresprovidesausefultest:inprincipleallthebresshouldbemeasuringexactlythesameresults,withinthenoise.Anysystematicdierences 144

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Knownplanet-bearingstarsfromearlyKeckETdata,May2006,usedasreferences:HD178911B,(msini=6:3MJ,period=71:5d),HD102195(ET-1,msini=0:48MJ,period=4:1d),andHD118203(msini=2:13MJ,period=6:1d).PredictedRVcurvesfrompreviouslypublishedmeasurementsareoverplotted(parametersfrom Butleretal. ( 2006 )andtheonlineExtrasolarPlanetsEncyclopaedia, 145

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SmallselectionofdierentexamplesearchstarresultsfromearlyKeckETdata,May2006.OneshowsanapparentsinusoidalRVsignal.Ifnotduetosystematicerrors,thiscouldbeaplanet,ormorelikely,aspectroscopicbinary.AKeplerianttothedatasuggestsacompanionofmsini11MJ,period12d. mustthereforebeduetoinstrumentorpipelineissues.Thecausesofthesedierencesseenarestillunderinvestigation. 3.1 ),leadingtopoorcorrectionformanyofthe 146

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KeckETmeasurementsofHD209458fromNovember2006.Errorbarsrepresentphotonlimitingerrors.PredictedRVcurveisoverplotted(orbitalparameterstakenfrom Naefetal. 2004 ). Figure7-7. KeckETmeasurementsofHD14810(TYC123117271)fromNovember2006.Errorbarsrepresentphotonlimitingerrors.PredictedRVcurveisoverplotted(orbitalparameterstakenfrom Butleretal. 2006 ). 147

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Representativeselectionofday-skyRVdatafromKeckET,November2006,fromdierentbres,showingexpectedzerovelocity-shift.Resultsareforfoursimultaneousspectranearthecentreofthedetector,numbers27{30(indicatedby`#'attopofplots). spectra.Uncorrectedinstrumentdriftfortheseobservationswithouttheiodinereferencewasstillattensofkms1,andasignicantreductioninfringevisibilityformanyofthespectrawasseenduethelargephasedriftsoccurringduringthelengthofexposures.Thesystemhasnowbeenupgradedtotwo-beamphaselocking,andfringesforallofthe59objectsarestableto1=300waves,or60ms1,over5hours.Systematicerrorswerestillseeninmanyinstances.Sourcesoferror,stillunderinvestigation,mayhaveincludedmoonlightcontamination,scatteredlightcross-talkbetweenspectra,andeectsofdistortionintheopticaldesign.Mostsignicantly,theadditionapproximationerror(seesections 2.6.1 and 5.10 )hadnotyetbeenaddressed.Itislikelythatthislasteectexplainedwhywewereabletogetgoodresultsinsomecases,butsawsystematicerrorsinmanyothers. 148

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2.4 thatthevisibilityofafringeisgivenbythenormalisedabsolutevalueofthecomplexFouriertransformoftheinputspectrum,evaluatedattheinterferometerdelayd=d0.Forasinglespectralline,thereciprocalscalingprincipleofFouriertransformsimpliesthatasthelinegetsbroader,itsFouriertransformasafunctionofdwillbecomenarrower.Atthexeddelayvalued0,wecanthereforeexpectthefringevisibilityduetothelinetodecreaseasthelinewidthincreases.Stellarrotationhastheeectofbroadeningspectrallines:itisthereforeinterestingtoask,canmeasurementsoffringevisibilitybeusedtoinfertheprojectedstellarrotationvelocityusingaDFDIinstrument?Thebroadeningofastellarspectrallineaboutitscentralwavelengthiscausedbyacombinationofseveraleects( Carroll&Ostlie 1996 ).`Naturalbroadening',duetotheHeisenberguncertaintyprinciple,representsafundamentalphysicallimittohownarrowalinecanbe:theabsorptioniscausedbyphotonsexcitingelectronstohigherorbitals,andthelimitedtimethatanelectronspendsinitsexcitedstatemeansthatthereisacorrespondinguncertaintyintheenergyofthatorbital,andhenceintheexactwavelengthatwhichabsorptionoccurs.`Doppler',or`thermal'broadening,iscausedbyaspreadintheDopplershiftsoftheabsorbingatomsinthegasofthephotosphereduetotheirthermalmotion.`Pressure'and`collisional'broadeningarecausedbyperturbationsintheorbitalsoftheabsorbingatomsduetoclosepassageandcollisionswithotherneutralatomsandions,andarethustiedwiththepressureofthegasinthephotosphere.Otherbroadeningeectsincludemicro-andmacro-turbulence,Dopplerbroadeningcausedbylarge-scaleconvectivemotionsofthegasinthephotosphere.Oftheseeects,thecoreofthelineproleisdominatedbythermalbroadeningandmicro-andmacro-turbulenceinsolartypestars.Thecombinationofalltheseeectsgiveswhatwetermthe`intrinsic' 150

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Gray 1992 ).Theobservedspectrumisasummationofthespectracontributedbyeachpointonthestellardisc,eachwithitsownprojectedline-of-sightvelocityduetotherotationofthestar,andweightedbythelimbdarkeningproleacrossthediscofthestar.Theline-of-sightvelocityvariesfromzeroatthecentreofthedisktoamaximumvsiniatthelimbontheequator,wherevistheequatorialrotationvelocity,andiistheangleofinclinationbetweenthelineofsightandtherotationaxisofthestar.Thereisadegeneracybetweenvandiwhichmeansthatonlythecombinationvsiniismeasurableusingspectroscopictechniques.Thetotalobservedlineproleisfoundtobeaconvolutionoftheintrinsiclineproleandarotationprolewhosewidthisdeterminedbythevalueofvsini,andwhosepreciseshapeisdeterminedbythestellarlimb-darkeningprole( Gray 1992 ).Weaddressthequestionofwhetherwecanmeasurevsinibymeasuringsimplythemeanvisibilityofthefringesacrossafringingspectrum.Theexactformofaspectrumdependsoneectivetemperature,Te,andmetallicity,[M=H], 151

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Cohenetal. 2006 ).Ifthestellarparametersarealreadyknown,avsinimeasurementwouldbeverysimpletoachieve,andcouldpotentiallybeausefulbyproductofaDFDIsurvey.Wediscussheretheresultsfromapreliminaryinvestigationintothefeasibilityofsuchmeasurements. 8.2.1TheBroadenedLineProleTounderstandthebehaviourofthefringevisibilityasvsinichanges,itishelpfultotryandperformaveryroughtheoreticalanalysis.Webeginbyconsideringasinglespectralline,modellingitforsimplicityastheconvolutionoftwoGaussianfunctions,onerepresentingtheintrinsiclineprole,andtheotherapproximatingtherotationalbroadeningkernel.Workinginwavenumberspace,,weexpresstheresultingspectrum,P,as: 152

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8{1 : Theproleofthebroadenedlineisthereforejustg=gigr,whichwecancalculatebytakingtheFouriertransformover: wheredistheconjugatevariableto,whichwelaterassociatewiththeinterferometerdelay.NowreversingtheFouriertransformtondg,wend: 8{3 ,forasinglerotationallybroadenedline,wecanwrite: 153

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2 (equation 2{9 ): 8{6 intothecomplexvisibilityequation 8{7 ,wecanwritetheFouriertransforminthenumeratorofthecomplexvisibilityequation 8{7 as 8-1 weseethatwecanassumeIw(1g)I(wg)sothat: A ,wisessentiallyjustthereverseoftheLSF,sothatthewidthsofwandtheLSFarethesame:forthepurposesofthediscussioninthischapter,theconceptsofresponsefunctionandLSF(orresolutionelement)canbeusedinterchangeably. 154

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Top-hatresponsefunctioncentredonasingleGaussianabsorptionline. Thebwtermvanishesbecauseitisapurecombterm,andtheinterferometerdelayd0(and/orthelowpasslteringinthedatareduction)isalwayschosensuchthatthecombisinvisibleandbw!0atd=d0(seesection 2.6.1 ).Usingequation 8{4 wecansubstitutebgintoequation 8{9 sothat: 8{7 ,whichisthetotaluxunderP()w().Referringagaintogure 8-1 ,andusingthestandardresultfortheareaunderaGaussian,wecanseethat: Finallysubstitutingthenumeratoranddenominator(equations 8{10 and 8{11 )intothecomplexvisibilityequation 8{7 andtakingtheabsolutevaluetodeterminethe 155

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wi!ip 2:(8{13)Wecanthereforewriteforthevisibility: 2p Gray ( 1992 ,equation17.2)(seegure 8-2 )gives FWHMr1:30L=1:30 cvsini(8{16)whereListhemaximalDopplershiftatthelimbofthestellardisc,relateddirectlytovsiniviatheDopplershiftequation,andcisthespeedoflight. 8{14 to 8{16 ,wetakeapproximatelysolarparameters,withinstrumentparameterscorrespondingtothoseassumedinthesimulationsdiscussedlater(section 8.3 ):wetakeFWHMi=0:1Aforthetypicalstellarintrinsiclinewidth;w=0:97A 156

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Gaussianapproximationtoanormalisedrotationalbroadeningprole.Solidline:broadeningprolefollowing Gray ( 1992 ,equation17.2,withlimb-darkeningcoecient"=0:6).Dashedline:best-tGaussian.They-axisisinunitsofthemaximalstellarDopplershiftL,correspondingtotheprojectedequatorialrotationvelocity,vsini. forthewidthoftheLSF(sincethisisthewidthofthelow-passlterappliedinthepre-processingforthesimulatedimages){thesameasthewidthofthespectrographresponsefunction;d0=7mmfortheinterferometerdelay;and 8.3 ),andchopitinto0:97Asegments,correspondingtothewidthoftheLSF.Wendaminimumuxlevelforeachsegment,andtakeameanoverallthesevalues,givingavalueforiof0:40.Wheretherearetwolineswithinaresolutionelement,thelinearityofFouriertransformssuggeststhattheresultingcomplexvisibilitywillbeasummationofthecomplexvisibilitiesduetothetwoindividuallines,scaleddownbytheadditionaluxlossbelowthecontinuumlevel.Ifonelineismuchdeeperthantheother,itwilldominatethevisibility;wherethetwoareofsimilar 157

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8-3 showsthepredictedvisibilityasafunctionofvsinifortheseparameters,usingtheanalyticalvisibilityequation(equation 8{14 ).Lookingattheformoftheequation,weseethatitisitselfaGaussianfunctionoftherotationalbroadening,andhenceofvsini,scaledapproximatelybytheratiooftheareaenclosedbytheintrinsiclineshapetotheareaunderthespectrographresponsefunction.ChangingeithertheintrinsiclinedepthortheintrinsiclinewidthhasonlytheeectofscalingtheGaussian{neitherchangestheshapeofthefunction.Thisfortunatecircumstancemeansthatallfringevisibilitiesacrossallchannelswillchangebythesamefractionalamountunderachangeofvsini,sothatitshouldbeentirelymeaningfultotakeameanvisibilityoveracompletespectrumtoimproveS/N.Thismeanvalueshoulditselfalsoscaleinexactlythesameway. 8{4 ,whichleadstothenumeratorofthevisibilityequation(equation 8{14 ):thebroadeningkernelmustdependonlyonvsini,andtheintrinsicprolecannothaveanyvsinidependencybydenition.Thedenominatoroftheequation,thetotaluxwithintheresponsefunction,mustalsoremainapproximatelyindependentoftherotationalbroadeningbecausethebroadeningisuxconserving. 158

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Analyticallypredictedvisibilityvs.vsinicurveforaSun-likestarobservedwithanET-typeinstrumentoperatingat5000{5800Awithapost-lteringresolutionelementofwidth0:97A. 8.3.1DescriptionAgridofsimulatedDFDIspectrawithdierentstellarparameterswascreatedbySuvrathMahadevanusinghigh-resolutionsyntheticmodelspectraprovidedbyRogerCohen,forthepurposesofinvestigatingwhetheritispossibletorecoverthevariousstellarparametersfromETspectra.Thehigh-resolutionsyntheticspectrawerecreatedusingthesoftware`SPECTRUM'( Gray&Corbally 1994 )whichcombinesaKuruczmodelatmosphere( Castelli&Kurucz 2003 )withitsownspectrallinelisttocalculateanormalisedspectrum.ThesewerethenprocessedthroughMahadevan'sinstrumentsimulationcodetoproducefringingspectrasimilartothosethatwouldbeseenbytheETinstruments.ThegridcoveredtherangesTe=4000{6500Kinstepsof250K;logg=2:0{5:0dex(cgsunits,logcms2),instepsof0:5dex;[M=H]=2:5{+0:5dexinstepsof0:5dex,andadditionally[M=H]=+0:2dex;andvsini=0;2;5and10kms1.RandomGaussian 159

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8-4 showsplotsofmeanvisibilityagainstthefourstellarparameters,representingcutsacrossthesimulationgridallcentredonthegridpointthatcorrespondstotheclosestmatchtosolarvalues.Foreachfreeparameteronthex-axis,theremainingparametersarexedatTe=5750K,logg=4:5,[M=H]=0:0,andvsini=2:0kms1(cf.truesolarvalues,Te=5770K,logg=4:44,[M=H]=0:00,andvsini=1:7kms1,from Valenti&Fischer 2005 ).Figure 8-5 showssimilarplotsofmeanvisibilityagainstvsini,thistimeshowingcontoursofTe,logg,and[M=H].Ineachcase,theunplottedparametersareagainheldatthenearest-to-solarvaluesinthegrid.Sincethesimulationshavephotonnoiseadded,itisnecessarytoestimatetheresultingerrorsinthemeanvisibilitymeasurements, j 160

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Fringevisibilitydependencyondierentstellarparametersfromsimulations.Diamondmarksnearest-to-solargridpoint.Foreachfreeparameter,theremainingparametersareheldxedatthevaluesnearesttosolarinthesimulationgrid.Errorbarsareontheorderof0.016%invisibility. pixel-widespectrumand5.0pixelresolutionelement).Thisequationgivestypicalerrorsforallthesimulateddataof =@(vsini),andacrosseachoftheotherthreeparameters,@ =@Xi,whereXirepresentstheparameterswiththesubscriptitakingvalues1;2;3,correspondingtoeachparameter(nottobeconfusedwiththeorbitalinclinationinvsini).Wecanthencalculate: @ @Xi(8{18) 161

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Contoursofvisibilityvs.vsinifromsimulationsfordierentstellarparameters.Contoursareshownforeectivetemperature,surfacegravityandmetallicity(inK,dexanddex,respectively).Eachcontourislabelledwithitsrespectivevaluedownthelefthandside.Errorbarsareontheorderof0.016%invisibility. 162

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Dependenciesofmeanvisibility, Parameter,Unitsof@ =@X@(vsini)=@XXX(%/unitX)(kms1/unitX) Wecalculatethesegradientsbytakingcutsacrossthegridinallfourparameterdirections,centredatthenearest-to-solargridpoint:themeangradientistakentobethatofastraightlinepassingthroughthevisibilitiesatthesmallestandlargestparametervaluesineachdimension(i.e.,equivalently,themeangradientofeachoftheplotsingure 8-4 ).Theresultsofthesecalculationsaresummarisedintable 8-1 .Wenotethat,apartfromadierenceininterferometerdelay,mostinstrumentaleects{whichwillmostlikelyaectthespectrographresolutionandthereforethewidthoftheresponsefunctionw{willenduponlyscalingthevisibilitymeasurements(asseeninequation 8{14 ).Suchascalinginvisibilitywillcanceloutinequation 8{18 ,sothatthevaluesfor@(vsini)=@Xshouldstaybroadlythesame.Thesedependencymeasuresareofcourseonlymeansacrossthegrid,however,andthereforeonlyprovideroughestimatesofthedependencies.Amoreexacttreatmentwouldconsiderthepartialdierentialsateachpointinthegrid.Finally,wenotethatwecanestimatetheeectofuncertaintyinthevisibilitymeasurementsduetophotonnoisebytakingthereciprocalofthevsinigradientfromtable 8-1 ,@(vsini)=@ =4:59kms1perpercentvisibility.Forthepreviouslycalculatederrorlevelof 8.5 ). 163

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Simulateddataoverplottedonthetheoreticalcurvefromgure 8-3 .Solidlinewithcrosses:simulateddata.Dashedline:analyticalprediction.Stellarparametersforthesimulationsarefornear-solarvalues,aslistedinthelegend. 8-6 showsthetheoreticallypredictedvisibility{rotation-raterelationfromsection 8.2 overplottedonthecorrespondingdatafromthesimulationsforthenearest-to-solarparameterset.Followingthepreviousarguments,wewouldprobablyexpectsomescalingdierencebetweenthetwocurvesduetodierencesinshapeofthespectrographresponseprole,theapproximatenatureofthechoiceofrepresentativelinedepth,andsoforth.Infact,aremarkablygoodagreementisseenwithoutanychangeinscaling.Thismayinfactsomewhatcoincidentalgiventheapproximatevaluesusedintheanalyticalcalculations,butitneverthelessshowsthatagoodestimateofthebehaviourcanbeobtainedanalytically.SinceatruerotationalbroadeningproleisratherdierentinshapefromaGaussian,weexpectsomedeviationintheanalyticalmodelfromreality.Atsmallvsini,where 164

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8-2 wecanseethatamoreaccuratebroadeningproledropsomoresteeplyandisnarrowerthantheGaussianapproximation:wethereforecanexpectthetransformtodropomoreslowly,sothatatlargevsini,weshouldseeavisibilityfor`real'spectralargerthanpredictedbytheanalyticalmodel.Thisbehaviourisindeedseenintheoverplottedcurvesingure 8-6 .Amoreaccurateanalyticalcurveshould,inprinciple,bereasonablyeasytoobtainbyperforminganumericalFouriertransformoftherealrotationalbroadeningproleratherthanusingtheGaussianapproximation. Valenti&Fischer ( 2005 )(hereafter`VF05').Oftheeight,asubsetoffour(HD134044,HD3861,HD3674,andHD17037)wereparticularlycloselymatchedinTe,loggand[M=H],dieringonlyinvsini.Weusethesetargetsheretoperformameanvisibilityanalysisandcomparewiththesimulationsinsection 8.3 .Simplemeanfringevisibilitieswereobtainedinthesamewayasforthesimulations,aswereerrorbarsonthemeanvisibilities.Toobtainagoodmatchwiththeknownstellar 165

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logg=4:27dex(range4.23{4.32dex);and [M=H]=0:0525dex(range0.04{0.07dex).Figure 8-7 showsaplotofmeanvisibilityagainstvsiniforthesubsetoftargets,wherethevaluesofvsiniandthecorrespondingerrorsarealsotakenfromVF05.Overplottedisthecurveexpectedfromaninterpolationofthesimulationstomatchthemeanvaluesforthisdataset.Clearlythetruevisibilityisover-predicted;asecondoverplottedcurveshowsthesamesimulationsscaledinvisibilitybyafactorof0.59togiveabettermatchtothedata:agoodtisnowseen.Thereasonforthemismatchmaybeduetoseveraleects.Theremaybeadierenceineectiveresolutionbetweenthesimulationsandthedata{althoughtheresolutionelements(andhencetheresponsefunction)shouldmatchwellinwidthafterltering,theLSFisGaussianinthesimulations,butmaybeclosertoatop-hatintherealdata.Itisalsolikelythatthefringesintherealdataaresomewhatundersampledbytheinstrumentpointspreadfunction(PSF)sincethefringedensityisratherhighinthisdataset:undersamplingcanleadtoadistinctreductioninvisibility.Finally,theremaybeacertainamountofdefocusoraberrationintheinstrument,whichwouldagainblurthefringesout.Alloftheseeectscanbeexpectedtoscaledownthevisibility,anditisthereforenotunreasonabletohavetoallowaconsequentscalingfactor.Figure 8-8 showsthefulldataset,includingthosefromgure 8-7 ,overplottedwithcontoursfromthesimulationgridinterpolatedtomatchtheparametersforthevariousothertargets.Thistimeallthecontoursarescaledbythesamefactorof0.59asbefore.Agoodmatchisseeninallcases,consistentwiththeerrorbars;HD12414fallsjustoutsideitsvsinierrorbarbutonlybyastatisticallyreasonableamountgiventhetotalnumberofdatapoints. 166

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Realdataandoverplottedsimulationsforawell-matchedsubsetofobservedtargets.Thesimulateddata(dashed/solidlinewithcrosses)arefromgridinterpolationsatstellarparametersmatchingthemeanparametersofthefourtargets(seelegend).Thedottedlinerepresentstheun-scaledsimulations;thesolidlinerepresentsthesamesimulationsscaledinvisibilitybyafactorof0.59. Theonlyexceptionistheday-skydatapoint,whichseemstoliearatherlongwayothesolarvsinicontour.Thereasonforthisdiscrepancyisnotyetclear,althoughitisnotunreasonabletoexpectthatthecharacterofthesolarspectrumreectedotheEarth'satmospheremayberatherdierentfromadirectlyobservedsolarspectrum.Ifthespectrumisintegratedoverlayersoftheskywhicharemovingatdierentvelocitiesduetovaryingwindspeeds,forexample,thespectrallinesmaybecomebroadened.Infactthespectrumwasalsoactuallytakenduringtwilight:itisknownthatwhentheSunislowinthesky,dierentialskytransmissionacrossthesolardisccanaectthespectrallinesquite 167

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Alleighttargetswithoverplottedvisibilitycontoursfromsimulations.Contoursarechosentomatchthetargetstellarparametersaslabelledinitalics.`Subset'indicatescontourforthemeanparametervaluesusedforthewell-matchedtargetsubsetasshowningure 8-7

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Demingetal. 1987 ).Overthe5minutelengthoftheexposure,thereisagoodchancethatthedierentialextinctionwasalsochangingratherrapidly,soitisperhapsnotsurprisingthatthisdatapointdoesnotmatchterriblywell. [(vsini)]2=Xi[Xi]2@(vsini) 169

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9.1.1RVmeasurementOneofthemostsignicantproblemsintheETprojectremainstheadditionapproximationnon-linearity.Withsystematicerrorsonthescaleofupto100ms1dominating,itishardtotryandtackleothersmallersourcesoferror.Theidealanswerwouldbetondamathematicallyexactsolution,oratleastamoreaccurateapproximation.Anotherverygoodsolutionwouldbe`combinedbeam'superpositionofthereferencespectrum,whereaThArortungsten-illuminatediodinespectrumisliterallyaddedtothestellarspectrum,forexamplebysplicingtwoinputbrestogetherintoone.Inthiswaytheadditionapproximationisnolongeranapproximation{itisexact.Combinedbeamsuperpositionwouldaddphotonnoisetothestellarspectrum,althoughprovidedthetwospectraarewellbalancedinux,theadditionalnoisemaynotbemuchworsethanthatduetouxlosswhenusinganiodineabsorptioncellinthebeampath.Balancinguxesbetweenstarandreferenceover60simultaneoustargets,however,maypresentachallenge.CurrenteortsbySuvrathMahadevantomodeltheerrortermusinghigh-resolutioniodineandsyntheticstellarspectrashowsomepromise.Inthisapproach,agridofcorrectionsacrossvelocityandstellarparameterspacecouldinprinciplebeapplied.Ifinstrumentstabilitycanbecontrolledwellenough,simplyrunningparallelreferencespectraalongsidethestellarspectra,oralternatively,bracketingstellarexposuresintimewithreferenceexposuresmayprovideathirdsolution.Anotherusefulbenetofcorrectingtheadditionapproximationerrorterm,ifitcanbedone,wouldbetoallowonetomeaningfullychopupthespectraintosmall,essentiallywavelength-independentsections.Atthispoint,theerrorsthatwoulddominateeachsectionbecauseoftheadditionapproximationarelikelytobesolargethatsuchchoppingwouldbelargelyfruitless.However,ifchoppingcanbedone,itwouldallow 171

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Gallandetal. ( 2005 )andsubsequentpapers).Furthermore,thesestarsgenerallyhavehigherrotationalvelocities,onthescaleof10{100kms1( Gray 1992 ),againreducingtheDopplerinformationcontent.TheexceptionallystrongBalmerlinesinthesespectraaretoodeepandintrinsicallybroadenedtoobeofmuchuseforprecisionmeasurementswithaDFDIinstrument.Nonetheless,therestillremainmanyveryshallowmetallines,anditisinprinciplepossibletotuneaDFDIinstrumenttobesensitivetotheserotationallybroadenedlines.EarlyexperimentswithsimulateddatausingaspectrumofVega(anextremeA0Vcase)suggestedthatforarotationalvelocity,vsini,of25kms1,aprecisionof50ms1mightbeachievable.Atamoreextremevsiniof150kms1,theachievableprecisionmightbearound340ms1.Althoughtheseareverymuchlowerprecisionsthanforlater-typestars,itisstillavelocityspacethathasbeenlargelyunexplored:several{Jupiter-massplanetsinfew-dayorbitsshouldstillbedetectable,asshouldbrowndwarfs.Sincetherearehintsthatmoremassivestarsmayharbourmoremassiveplanets( Satoetal. 2007 ; Ida&Lin 2005 ),thiswouldbeaninterestingareaofinvestigation. 174

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Geetal. 2007 ).TheKeckETinstrumentisstillbeingupgraded:theinterferometerandinterferometermountshaverecentlybeenreplacedtoimprovestability;theopticshavebeenrealignedtoimprovethroughput,aberrationandslit-illuminationhomogeneity;andtheslitwidthhasbeenslightlyreducedtotryandmitigatetheproblemswithaliasingoftheinterferometercomb.Themeasurementofprojectedstellarrotation(vsini)bymeasuringtheaveragefringevisibilityforastellarspectrumalsoappearspromising.Evenatthesuggestedprecisionlevelsof3:4kms1fromthepreliminaryinvestigations,thiscanstillbeausefulbyproductoftheETsurveys.Itisreasonabletoexpectthatthereisroomforimprovementinthisprecisionlevel.AstheETinstruments'overallprecisionandreliabilityimprovesinpreparationfortheplannedMARVELSsurvey( Geetal. 2007 ),wehopethattheETinstrumentswillbe 175

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2{31 )takenfrom Ge ( 2002 ),thediscussionofanalyticalerroranalysis,includingthepreciseformoftheadditionapproximationcross-talkterms,ismyown,andtomyknowledge,new.TheinvestigationintothemeasurementofstellarrotationvelocitiesrepresentsastartonanewprojectthatIhavetakenon.Thesemi-empiricalinvestigationintothefringevisibilitydependenceonstellarparametersis,tomyknowledge,newwork.Therelevantmathematicalanalysispresentedhereisalsomyownindependentwork,thoughIunderstandthatDr.JianGehaspreviouslydonesomesomewhatsimilarcalculations,andhadalsopreviouslyproposedtheideaofusingfringemeasurementsformeasuringstellarlinewidthsandabundancestudies.ItshouldalsobenotedthatRogerCohenalsoperformedsomeearlycloselyrelatedworkonmeasuringstellarrotations(alongwithotherstellarparameters)withtheETinstruments,usingasomewhatdierentapproach( Cohenetal. 2006 ).IamindebtedtoSuvrathMahadevanandRogerfortheirgridof 176

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vanEykenetal. 2004a );andFigures5,8,and9(top),andtables1,3and5,from\TheFirstExtrasolarPlanetDiscoveredwithaNew-GenerationHigh-ThroughputDopplerInstrument"( Geetal. 2006a )(c2004and2006:TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A.).TheauthorhasbeenaVisitingAstronomerattheKittPeakNationalObservatory,NationalOpticalAstronomyObservatory,whichisoperatedbytheAssociationofUniversitiesforResearchinAstronomy,Inc.(AURA)undercooperativeagreementwiththeNationalScienceFoundation.ThisresearchhasmadeuseoftheSIMBADdatabase,operatedatCDS,Strasbourg,France.TheHobby-EberlyTelescope(HET)isajointprojectoftheUniversityofTexasatAustin,PennsylvaniaStateUniversity,StanfordUniversity,Ludwig-Maximillians-UniversitatMunchen,andGeorg-August-UniversitatGottingen.TheHETisnamedinhonourofitsprincipalbenefactors,WilliamP.HobbyandRobertE.Eberly.IRAFisdistributedbytheNationalOpticalAstronomyObservatories,whichareoperatedbytheAssociationofUniversitiesforResearchinAstronomy,Inc., 178

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Landsman 1993 ), 179

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2{9 inanotherway.IfwewriteouttheFouriertransformsexplicitlyasintegrals,weobtain: A ,weknowthatthespectrographresponsefunctionisjustthereverseoftheLSF,sowecanwrite: B{1 : [P()L0()](j):(B{4)Ifjisthecomplexvisibility,thentheintensityasafunctionofdelayandwavenumberjmustbegivenby: whereItrepresentsthetotaluxcontributingtothechannel,andwehavedroppedtheexplicitnotationofthe(j)functionaldependencefortheconvolutions.Sincethe 182

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Finally,becauseconvolutionisdistributive,wecanwrite: wherewedeneT()1+cos(2jd0),whichcanbethoughtofastheinterferometertransmissionfunction,equivalenttotheinterferogramthatwouldbeobtainedforpurewhitelightandaninniteresolutionspectrograph(exactlyasinequation 2{8 ).Inotherwords,wehavesimplytheinputspectrummultipliedwiththeinterferometertransmissionfunction,andthenconvolvedwiththeLSFduetothespectrograph.Thinkingintwodimensions,tomatchthewide-slitformatoftheactualETspectra,wecanreplacetheLSFtoits(insomesenses)twodimensionalequivalent,theinstrumentpointspreadfunction(PSF).Exactlythesameresultscanbederivedinfrequencyspace,simplybysubstitutingfor.Thiswayoflookingatfringeformationistheapproachusedby Erskine ( 2003 )andfollowedby Mahadevan ( 2006 ).ThesameformalismisalsousedbyMahadevantoproducesimulatedDFDIspectra. 183

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Afternarrowlyavoidingbeingsoldinatwo-for-onebundlealongwithhismothertoaMasaiwarriorinAfrica(fortherespectablesumofsevengoats),JulianvanEykenwaseventuallybornin1976inruralSomerset,England,wherehegrewupunderthecareoftwoofthebestparentsintheworld.AsateenagerheattendedWellsCathedralSchool,wherehespecialisedinphysics,mathematicsandmusic.HegraduatedfromCambridgeUniversitywithadegreeinnaturalsciencesin1998,andwentontopursuehisPhDinastronomyonafellowshipatPennStateUniversity,wherehewaslaterawardedtwoSPIEscholarships,andaMichelsonFellowshipbyNASAJPLin2004,forhisworkontheETinstruments.LaterthatyearhemovedtotheUniversityofFloridatocompletehisPhDworkingonthesameproject.HecandotheVulcansalutewithbothhands,knowshisAfricanbushcreaturenoisesaswellashisfarmanimalsounds,andisalwaysalittlesuspiciousaboutwhatturnlifemighttakenext. 191