UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
REREDUCING THE SOUTHERN POLAR ZONE OF THE YALE PHOTOGRAPHIC STAR CATALOGUE By JANE E. MORRISON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995 To Abby, Ben, Taylor and Atticus. ACKNOWLEDGMENTS This project would never have been possible with the support and advice from many members of the astronomical community, my family and friends. First of all I would like thank my thesis advisor, Dr. Eichhor for his patience, advice, insight and colorful conversations. By his own example I have developed a passion for scientific research. I would like to thank the members of my committee, Haywood Smith, John Oliver, Ralph Selfridge, and KwanYu Chen for their counsel and support. I would like to thank my parents for their moral support and encouragement. I would especially like to thank them for instilling in me the desire to always find the best in every situation and for sharing with me their belief that I can achieve any goal I set my mind to. Both of these qualities have enabled me to finish this project while enjoying nearly every moment. An extra special thanks goes to my friend and colleague, Ricky Smart, for his enduring patience and unending support, without which this project may not have been possible. I would also like to thank him for never a dull moment. For fun, food and frolicking I would like to send hugs and kisses to Jaydeep Mukherjee, Sumita Jayaraman, Caroline Simpson, Ron Drimmel, Chuck Higgins, Leonard Garcia, Stephen Kortenkamp, Elaine Mahon and cast of thousands. Thanks to Lenny and Croaker for ghostwriting my biographic sketch. iii For providing the raw measurement of the Southern Polar Zone of the Yale Photographic Star catalog and an early release of the International Reference Star catalog, I would like to thank Tom Corbin and Sean Urban from the United States Naval Observatory. For providing background information concerning this project I would like to thanks Dorrit Hoffleit, Fred Fallon, and Bill Van Altena. The bulk of the computing has been carried out on the Northeast Regional Data Centers IBM computer under their Research Computing Initiative program. This computing time has saved me many hours. TABLE OF CONTENTS ACKNOWLEDGMENTS .............................. iii LIST OF TABLES ........ .............. ......... vii LIST OF FIGURES ...... ........................ viii ABSTRACT ........... ... .................. .....xi CHAPTERS 1. INTRODUCTION ...... ......... ............... 1 Original Reduction ................ ........... 7 Why Do a ReReduction? ............................ 8 2. PHOTOGRAPHIC ASTROMETRY .................... 14 Standard Coordinates and their Relationship to the Equatorial C oordinates . .. . 15 Determining the Equatorial Coordinates from the Measured Coordinates ....... .. ............... ...... .. 17 The Relationship Between the Standard and Measured Coordinates 20 Some Commonly Used Models ......... ............. 28 4Constant M odel ..... ....... ..... .......... 29 6Constant M odel ....... ...... .. .......... 31 12Constant M odel ........................... 32 Spherical Correction .. .......................... 34 3. LEAST SQUARES ............. ................. 45 Theory of Least Squares ................... ....... 45 Traditional (Linear Least Squares) .................. 47 Nonlinear Least Squares ........... ............... 49 4. METHOD OF PLATE REDUCTIONS .................... 56 Single Plate Reduction: Using Least Squares to Determine the Plate Param eters. ........... ..................... 57 Final Positions from a Single Plate Reduction, .............. 68 Overlap Plate Reduction .......................... 69 Example of Overlap Reduction ................. ...... 83 Set up the equations of condition ................. 86 Determine J' ................................ 90 Determine a .............................. 93 Determine 3 ........ 5. OBSERVATIONS ......... Observations for the Southern I Measurement of Stars ..... Magnitude System . 6. PLATE REDUCTIONS ...... Reference Catalog ........ External Catalog ........ Determining the Plate Model . Single Plate Reduction ..... Final Positions from the Single Overlapping Plate Reduction . Conclusions ........... Future Work ....... . BIBLIOGRAPHY . . BIOGRAPHICAL SKETCH .... 'olar Zone . Plate . .. . . . .. .. 94 . .. 96 . 96 . 97 . 10 1 . 102 . 102 . 104 . 107 . .. 108 . 114 . 119 . .. .. 126 ...... 129 . 168 . 172 Solution LIST OF TABLES Yale Zones. ................................... 6 Refraction Information for plates 132 . ..... 43 Refraction Information for plates 3364 . ..... 44 The Yale Plates ..... . ........... 98 Single Plate RMS Results ....... .............. 110 Overlap Plate RMS Results . ... 125 RMS Single Plate Comparisons: 15constant model ... 127 RMS Overlap Plate Comparisons: 18constant model ....... .128 Table 1: Table 2: Table 3: Table 4: Table 5: Table 6: Table 7: Table 8: LIST OF FIGURES Figure 1: The plates for the Southern Polar Zone of the Yale Catalogue .12 Figure 2: The Stars of the Southern Polar Yale Zone of the Yale Photographic Star Catalogue . . 13 Figure 3: Gnomonic Projection . ..... ........ 15 Figure 4: Origin Shift ......... .... .... .. ......... 22 Figure 5: Rotation of xy axis with respect to the rq axis . 23 Figure 6: Nonperpendicularity of axes . . 26 Figure 7: 4Constant Model: origin shift and rotation of axes .. 29 Figure 8: The effects of refraction on plate 1. . 39 Figure 9: Example of two overlapping plates . ... 84 Figure 10: Distribution of Magnitudes . ... 105 Figure 11: The different in positions found from the single plate adjustment for plates I and 24 and plates 1 and 25. . ... 115 Figure 12: The different in positions found from the single plate adjustment for plates 61 and 64 and plates 62 and 63 . .... 116 Figure 13: Magnitude Dependent Measuring Error, the units of the variances are arcseconds squared ................... ...... 123 Figure 14: 6Constant Single Plate Residuals . ..... 131 Figure 15: 6Constant Single Plate Residuals . ..... 132 Viii Figure 16: 6Constant Single Plate Residuals . Figure 17: 6Constant Single Plate Figure 18: 6Constant Single Plate Figure 19: 8Constant Single Plate Figure 20: 8Constant Single Plate Figure 21: 8Constant Single Plate Figure 22: 8Constant Single Plate Figure 23: 8Constant Single Plate Residuals Residuals Residuals Residuals Residuals Residuals Residuals Figure 24: 13Constant Single Plate Residuals Figure Figure Figure Figure Figure Figure Figure Figure 13Constant Single Plate 13Constant Single Plate 13Constant Single Plate 13Constant Single Plate 15Constant Single Plate 15Constant Single Plate 15Constant Single Plate 15Constant Single Plate Residuals Residuals Residuals Residuals Residuals Residuals Residuals Residuals Figure 33: 15Constant Single Plate Residuals ix ..... . 134 . . 135 . . 136 . . . 137 . . . 138 . . . 139 . . . 140 . . 14 1 . . 142 . . 143 . . 144 . . 145 . . 146 . . 147 . 148 . . 149 . . 150 . . . 1 3 3 Figure 34: 20Constant Single Plate Residuals Figure 35: 20Constant Single Plate Residuals ... .. 152 Figure 36: 20Constant Single Plate Residuals . .... 153 Figure 37: 20Constant Single Plate Residuals . .... 154 Figure 38: 20Constant Single Plate Residuals . ..... 155 Figure 39: 13Constant Overlap Residuals . ..... 156 Figure 40: 13Constant Overlap Residuals . 157 Figure 41: 13Constant Overlap Residuals . . ... 158 Figure 42: 13Constant Overlap Residuals . ..... 159 Figure 43: 15Constant Overlap Residuals . . ... 160 Figure 44: 15Constant Overlap Residuals. . . 161 Figure 45: 15Constant Overlap Residuals. . . 162 Figure 46: 15Constant Overlap Residuals . . ... 163 Figure 47: 18Constant Overlap Residuals. . . 164 Figure 48: 18Constant Overlap Residuals . . 165 Figure 49: 18Constant Overlap Plate Residuals . .... 166 Figure 50: 18Constant Overlap Residuals . ..... . 151 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REREDUCING THE SOUTHERN POLAR ZONE OF THE YALE PHOTOGRAPHIC STAR CATALOGUE By Jane E. Morrison August, 1995 Chairman: Heinrich Eichhorn Major Department: Astronomy We have rereduced the Southern Polar Zone of the Yale Photographic Star Catalogue (70 to 90 declination) using an overlapping plate technique. This region was photographed in 195556 with 64 overlapping plates and the reduction of these plates (i.e. the determination of the equatorial coordinates of the star images on the plates) was completed in 1971. Because of the scarcity of observations from the Southern Hemisphere prior to the 1970's, this data set is particularly valuable. The plates where photographed so that they overlapped 50% in right ascension and 50% in declination. Thus every region of the sky was covered by at least 2 plates and sometimes as many as 20 different plates. Originally these plates were individually reduced. Thus after the reduction, some of the stars had many as 20 different position estimates (i.e. right ascension and declination). Clearly a star occupies one position at one time, but this can not be obtained without mathematically enforcing this constraint on a multiple plate reduction process. Instead, the separately computed values were averaged to give the "best" (in the mathematical statistics sense) estimate for the stellar position. Taking advantage of the high degree of overlap of these plates, we used a more powerful reduction method for overlapping plates called the overlapping plate technique. This method enforces the simple fact that, at a given instant of time, a star is at only one position. When this constraint is enforced, all the reduction parameters on all plates are obtained simultaneously by solving one large system of equations. The application of these improved reduction procedures and resources to the original measurements have yielded star positions which are a 20% improvement over those originally published. These new estimates of star positions are capable of yielding improved proper motions, which are invaluable data for many aspects of astronomy. CHAPTER 1 INTRODUCTION Astrometry, one of the oldest branches of astronomy, deals with the determi nation of locations and the positional changes of celestial bodies. The compilation of star catalogs is the domain of astrometry devoted to the determination of ac curate and precise positions of celestial bodies. One of the most important tasks of astrometry is to establish a reference frame and maintain it by determining the motions of the reference stars which it represents. Usually these reference frames are defined by a star catalog (i.e. FK4 (Fricke et al., 1963), FK5 (Fricke et al. 1988) and soon the HIPPARCOS catalog). The compilation of star catalogs has been a continuing processes over the centuries. One of the first recorded star catalogs was compiled by Hipparchus (128 B.C.) and contained 850 stars (Abell et al., 1987). Throughout the centuries astronomers have devised and improved upon the methods to compute accurate and precise positions for as many stars as possible. Today the largest star catalog is the Guide Star Catalog (GSC, Lasker et al., 1990) containing roughly 20 million objects with a limiting magnitude of 15m and positional standard errors from the plate centers to the edges which vary from 0".5 to 1".1 in the northern celestial hemisphere and from 1".0 to 1."6 in the southern celestial hemisphere (Taff et al., 1990). Containing fewer stars, but with a much higher expected positional accuracy, the HIPPARCOS catalog is the first 2 star catalog to contain observations solely made in space. This catalog is being produced by the European Space Agency from observations taken by the satel lite HIPPARCOS and will contain roughly 120,000 star positions with limiting visual magnitudes between 7.7m and 8.7m depending upon the galactic latitude. The individual rmserrors (root mean squareerrors) are expected to be 1.3 milli arcseconds (mas) per coordinate in position (Kovalevsky, 1995). Star positions at only one epoch are of limited use, but they become of enor mous value when combined with several epochs and changes in the positions are observed. From studying stellar positional changes several important astronomi cal parameters can be determined; probably the most important of these are stellar parallaxes and proper motions. Trigonometric parallaxes are the only method for determining stellar distances based only on geometry. Once the distances of stars are known, then a host of other quantities can be found, for example, absolute magnitudes and in double star systems, actual orbit semimajor axes and the sum of the masses of the stars can be determined. Proper motions in combination with radial velocities allow one to study the motion of stars in clusters, to identify stellar associations, to determine the orbital motion of double and multiple star systems and to study the motions within the Galaxy which leads to an understanding of the evolution, structure and rotational curve of the Galaxy. The considerable distances to stars reduce their proper motion to very small amounts; for most stars these are considered negligible within the current proce 3 dures of measuring precision. It is almost always an angle too small to measure with precision in a single year. The majority of stellar proper motions can only be determined after a cumulative effect (as a rule) over many decades produces a measurable change in the stars' position. Most of the detectable proper motions are for nearby stars. Though there are several hundred stars with proper motions greater than 1."O per year (the largest motion being that of Barnard's Star, 10."25 per year), the mean proper motion of all nakedeye stars is less than 0."1 (Abell et al., 1987). Therefore, in order to detect accurately proper motions we must have extremely accurate and precise positions for the stars with preferably decades having elapsed between observations. Star catalogs, in particular old star catalogs, are extremely useful for the determination of proper motions for a large number of stars. For example, stellar proper motions can be detected by comparing catalogs containing the same stars with epochs several decades apart or by combining the positions found in a single catalog with new observations of the same stars. Thus, one way to improve upon the current proper motion estimates is to improve upon the accuracy of the star positions found in the old star catalogs by using improved reduction techniques and better reference material than was originally available. This is one of the justifications for the work reported on in this paper. For this project we have rereduced (redetermined) the star positions found in the Southern Polar Zone of the Yale Photographic Star Catalogue (700 to 900 declination, Lii 1971). Because of the scarcity of southern hemisphere 4 observatories in the first half of this century, there are few accurate old (by this we mean 50 years or more) star catalogs containing southern declination star positions. Thus, this region of the Yale Zone Catalogues is particularly interesting for it contains some of the oldest accurate observations (epoch 19551956) taken with photographic plates of the southern polar zone. In addition, the photographic plates that were used to create this catalog were photographed so that they heavily overlapped (50% in right ascension and 50% in declination); thus, many of the same stars are on several plates. This heavily overlapped pattern is particularly well suited to our reduction technique which has improved the accuracy of the stellar positions and will lead to more accurate and precise proper motions when calculated from the positions found in the improved catalog. The Southern Polar Zone of the Yale Photographic Star Catalogue is one of many zones which make up the Yale Zone Catalogues. In 1913 the director of the Yale Observatory, Frank Schlesinger ("father of modern astrometry") initiated a project to photograph the entire sky. He planned to reobserve by photography the positions of the Astronomische Gesellschaft Catalogue (AGK series) stars and then by comparing the older AGK positions with the new Yale determinations to detect the proper motions of stars to the ninth magnitude. Ida Barney collaborated with Schlesinger on this project and after Schlesinger's death in 1942, continued on her own until 1959. Between 1925 and 1959 they (with very few other colleagues) published some 20 volumes of accurate positions and attainable proper motions 5 for nearly 146.000 stars in the zone from 300 to +300, +500 to +600 and +850 to +900 (Hoffleit. 1962). Dirk Brouwer, director of the Yale Observatory from 1941 to until his death in January 1966, continued Schlesinger's project and extended it to more southern declinations: 300 to 900. Early in 1960, Brouwer obtained a 2year contract from the U.S. Army Map Service to reobserve all the stars in the Cape Zone Catalogues for the zones 300 to 500 and 600 to 90o, involving some 70,000 stars, in order to improve the positions and particularly the proper motions in those zones. Dorrit Hoffleit was appointed to supervise the project, starting at 300 and working southward (Hoffleit, 1983). The southern catalogs, like the northern ones, were divided into zones (ex tended regions of the sky bounded by parallels of declination). Except for the polar caps, each zone was covered by photographic plates centered at the same declination, with the right ascensions of the centers spaced such that the vertical center line of a plate almost coincided with the vertical edges of one of its neigh bors (i.e. overlap 50% in right ascension). This produced a double coverage of the sky. The overlap in declination was at most only a few degrees and often none. The Southern Polar Zone has a much stronger overlap pattern than the other zones. This region consists of 4 belts of zones with a declination overlap of 50% between the zones and an overlap of 50% in right ascension within the zones. The Southern Polar zone was photographed according the scheme given in Table 1. 6 Table 1: Yale Zones Number of Plates Center Declination Separation in R.A. 24 75 1h 00m 20 80 1 12 16 85 1 30 4 90 600 The strong overlap pattern is shown the polar plots in Figure 1. The first plot shows the different declination zones the center representing the Southern Celestial Pole (900) and the other zones are represented by concentric circles, moving outward from the pole: 850, 800, 750, 700 and 650. The first plot is also only 3 plates, the next 4 plots are graphs for the plates in each zone and the last plot shows all the plates for the region. The number of plates contained is each zone given in Table 1. The Yale Zone Catalogues have been of enormous importance to positional astronomy and in particular the determination of stellar proper motions. Together they contain hundreds of thousands of stars. Schlesinger had planned to eventually cover the whole sky; however, this aim was not quite achieved, the zones +600 to +850, +300 to +500 and 500 to 600 are still lacking. The region we are concerned with is the 700 to 900 zone. This was the last Yale zone to be photographed (19551956) and has the largest star density and magnitude range of all the Yale zones. 7 Original Reduction The plate material for the Yale Southern Polar Zone Catalogue consists of 64 photographic plates with a sky coverage of 1 l x 110 each. These plates were exposed between September 15, 1955 to August 7, 1956 at Sydney, Australia, measured at the Yale Observatory between 1963 and 1968 and the reduction was completed in 1971. The reduction of the plates (i.e. the determination of the equatorial positions of the images on the plates) was completed in 1971 (Hoffleit, 1971). Figure 2 (top) shows the stars for this catalog on a polar plot. The bottom graph in Figure 2 shows the references used to rereduce the catalog. In order to explain the reasoning behind our desire to reduce this catalog, we will briefly outline the procedure for reducing plates. The details of this are given in Chapter 4. The reduction of photographic plates requires the use of a set of reference stars, whose stellar images are on the plate and whose equatorial coordinates are known by previous investigation. In the traditional method of reducing plates, estimates of certain plate parameters, which are numerically different and characteristic for each plate, are determined from least squares reductions of the reference stars' equatorial coordinates with respect to the rectangular coordinate on the plate. These plate parameters are then used to convert field stars images (i.e., nonreference stars) to equatorial coordinates. This method of reducing each plate separately has become known as the "single plate method." The original reduction of the Southern Polar Zone was performed using a single plate method by Phillip Li and Dorrit Hoffleit. The reference catalog used was the Second Cape Catalog for 1950. The Yale plates were taken between 1955 and 1956, so the reference star positions taken from the Cape were corrected to the epochs of the Yale plates by applying proper motions for intervals of three to nine years. On the average there are 150 to 200 reference stars per plate. After the reduction, the resulting positions of stars that had been recorded on more than one plate were averaged together. The resulting catalog consists of 18,702 stars with their positions and proper motions on the FK3 system (Third Fundamental Katalog, Astronomisches RechenInstitut, 1957). "The tabulated probable errors of the positions represent simply the internal consistency between different Yale plates used for each star. They imply that the average probable error of each coordinate determined from a single plate amounts to 0."33. This is consistent with the average formal probable error of a plate solution, namely +0."35" (Lii, 1971, p. 29). Note that the (usually quoted) standard errors are about 1.5 times this amount or 0."53. Why Do a ReReduction? A major drawback of the single plate method is that it produces a different position estimate (right ascension, declination) for each star on each plate, even if 9 that star also appears on more than one plate. Clearly the star can occupy only one position at one time, but this cannot be obtained without mathematically enforcing this constraint on a multiple plate process. Instead, the separately computed values are averaged to give the "best" (in the mathematical statistics sense) estimate for the stellar position. As just mentioned, the most important constraint, which was never enforced, is the simple fact that any one star, at a given instant of time, occupies only one position. In the case of the Yale Southern Polar Zone, the plates were so strongly overlapped that most stars occurred on 2 plates, some even on 20 plates. Thus after performing the single plate reduction there were as many as 20 different position estimates for some of these stars. Taking advantage of the rich overlap we have rereduced the plates using a more powerful reduction technique, the overlapping plate technique (Eichhorn, 1960; for review Eichhorn, 1985). In this method the critical constraint that a star can only have one position at one time is enforced. Then all the reduction parameters on all plates are obtained simultaneously by solving one large system of equations (i.e., by a block adjustment). Whereas the single plate is limited to individual least square reductions of just the reference stars on each plate, the overlapping plate method reduces all the plates at the same time using all the stars (not just the reference stars). More accurate (because the systematic errors on individual plates are diminished as a consequence of the forced platetoplate agreement which must now occur) and thus also more precise 10 positions than those obtained from the single plate reduction are generated by the overlap method. This method also has many capabilities that are not possible with a single plate reduction, such as reducing systematic errors between neighboring plates and the estimation of parameters common to certain data sets at varying levels. Though the theoretical formation of the overlap method was known when the original reduction was done, it was not used. Originally it had been planned to use an overlap reduction, but as Hoffleit explained in the preface of this catalog, "the planned more elaborate reductions may be a long time forthcoming because of a present shortage of both funds and personnel" (Hoffleit, 1971, p. 3). In fact, when the computations for the Yale Catalogue were carried out, it would have been very difficult to take advantage of the huge number of existing constraints, because enforcing them requires a computing effort that, while almost modest by contemporary standards, was totally prohibitive at the time. Many of the existing geometric and physical constraints could not be enforced and were therefore wasted. The solution using all the available information has become possible only through the emergence of fast computers with large memory capacities. In addition, we were able to use a more precise and accurate reference star catalog, the International Reference Stars (IRS, Corbin, 1991) which was made available to us by the US Naval Observatory (see Figure 2, bottom). Thus by adjusting the originally measured position of the stars' images on the plates on 11 the basis of better reference star position estimates (which were not available when the original reductions were carried out) and using a superior adjustment algorithm, the early positions of these stars have been improved by roughly 20%. The improvement in the positions found in this old star catalog will improve the proper motion estimates calculated with them. For example the ARCS (Astro graphic Catalogue Reference Stars, Corbin and Urban, 1991) and PPM (Positions and Proper Motion Catalog, Bastian et al., 1993) catalogs (both key reference catalogs) used the Yale Southern Polar Zone Catalogue for their determination of proper motions. Thus by improving the positions in the Yale Catalogue we can improve other catalogs that use the positions from the Yale Catalogue. This is especially important in the Southern Celestial Hemisphere, where there are few old reference catalogs. The wholesale improvement of the precision and accuracy of proper motion estimates is one of the keys to improving the precision and accuracy of a large variety of astronomical data. This is one of the reasons why the measurements in the Yale Catalogue, which is based on plates taken several decades ago, are so important. 75" Zone Plates 1, 3 and 25 80 Zone 85 Zone 90' Zone 70 to 90 Zone Figure 1: The plates for the Southern Polar Zone of the Yale Catalogue I Star Images for the Yale Southern Zone IRS Reference Stars Figure 2: The Stars of the Southern Polar Yale Zone of the Yale Photographic Star Catalogue CHAPTER 2 PHOTOGRAPHIC ASTROMETRY One method of creating a star catalog (a list of right ascension and declination of a group of stars) is to photograph a region of the sky with photographic plates. Once the plates are developed, the positions of the star images on the plate in a certain rectangular coordinate system (x,y) are determined with a measuring machine. The positions of the measurements on the photographic plate can be derived only with the use of a set of known positions of stars whose images are among those recorded on the plate. Their equatorial coordinates are usually found from existing star catalogs. We call such stars reference stars and all other stars for which we do not have equatorial coordinates, field stars. The goal of catalog compilation is to determine accurately the spherical coordinates (for our purposes, equatorial coordinates) of all the stars imaged on the plate (field stars as well as reference stars). In the ideal case, one assumes that the optical system (telescope + plate holder) is equivalent to a pinhole camera, in which is governed by the laws of the gnomonic projection. A gnomonic projection is the projection of a spherical surface onto a plane through a point. For descriptive purposes we will assume the existence of the fictitious "celestial sphere" and we will further assume that the center of the celestial sphere lies at the optical center of the objective system. 15 In this perfect case the portion of the "celestial sphere" where the telescope is pointing is projected onto a plane through the focal point of the telescope. The resulting rectangular coordinates projected onto the plane are known as the standard coordinates ((,,I). The usefulness of photography for astrometric positional work is connected with the fact that there exists a simple, rigorous and parameterfree geometrical relationship between positions of objects on the sky and their standard coordinates projected onto the plate. Standard Coordinates and their Relationship to the Equatorial Coordinates 7 tangent plane T(a0 ,8.) celestial sphere objective = center of celestial sphere \ photographic plate Figure 3: Gnomonic Projection 16 Imagine a plane which is tangential to the celestial sphere at the point ,o, 6,. In the plane define a left handed coordinate system ((,r) (as seen from the center of the sphere), such that its origin is at the tangential point. The positive tr axis is tangential to the hour circle through co and points northward. The ( axis is tangential to the parallel of declination at 6o and points east (in the direction of increasing right ascension). This plane coordinate system is thus, by definition, parallel to the focal plane of the telescope and therefore parallel to the plane of the photographic plate being used to record the images. The tangential point can also be described as the point in which a line normal to the plate coincides with the optical axis of the telescope and penetrates the plate close to its geometrical center. Thus star positions on the celestial sphere are projected in straight lines through the center of the sphere (which is the also the focal point of the telescope) and onto the tangent plate. This geometrical situation is illustrated in figure 3. If we further assume that the radius of the sphere is equal to 1, the well known relationships between a point on the celestial sphere (a, 6) and the gnomonic projection ((, 7) of this point onto a plane which is tangent to the unit sphere at (ao,6o) is given by cos 6 sin (a ao) sin 6 sin 60 + cos 6 cos 6o cos (a oo) sin 6 cos 60 cos 6 sin 6, cos (a ao) sin 6 sin 60 + cos 6 cos 6b cos (a ro). These equations are rigorous and free of estimated parameters, and can be inverted so that a and 6 can be found from ( and rf. Note that ( and 77 are in units of the telescope's focal length. However, in practice no real optical device produces images in the same way as a pinhole camera. One needs to take into account such effects as coma, radial distortion, rotation and tilt of the plate when photographed or measured. The measured coordinates (x,y) are actually approximations to the standard coordinates (Jr). Determining the Equatorial Coordinates from the Measured Coordinates The goal of a plate reduction is the derivation of the spherical equatorial coordinates from the objects' measured positions on the plate. Usually this process is achieved in two steps. First the measured coordinates are transformed to the standard coordinates, then these are converted to equatorial coordinates. The second step has already been dealt with (i.e., equation 1), so only the first remains. The establishment of the relationship between the standard coordinates (,,r7) and the measured plate coordinates (x,y) of the star's images is the fundamental problem of determining the equatorial coordinates. In the ideal case (assuming a pinhole camera), even though the precise position of the image of the tangential point (0o, o) on the plate is not known, it is reasonable to assume that it is near the geometric center of the plate. Plates are usually exposed in such a manner that it is also reasonable to assume that the edges of the plate (i.e. the x and y axes) are closely parallel to the ( and 77 axes, respectively. If the origin of the 18 measured coordinates is chosen to lie at the geometric center of the plate and the axes are properly oriented, to first approximation the standard coordinates are related to the measured coordinates by a factor of the telescope's focal length, s: )= + +0(1). (2) The unit of length on the plate in the (r system is such that the standard coordinates are equal to 1 at a distance of 1 focal length from the tangential point. Each departure from the ideal imaging process, a gnomonic projection, results in deviations from the perfect relationship in the equation (x c S cos 6 sin (3) y d sin p cos )\ 6 s where c, d and 6 are all small quantities of the first order. The end result is a mapping between the standard coordinates and the measured coordinates. Tradi tionally, this mapping is called a plate model. The plate parameters have tradition ally been called plate constants. We prefer the terminology "plate parameters" which indicates that these are adjustment parameters to be estimated by some statistical process. In the traditional method of reducing plates one assumes there exists for each plate the following relationships f(xi,yi; mi, c; i, 6i; a, a2,,...,an) = 0 19 between the measured coordinates xi, yi of the image of the ith star, its magnitude mi and color index ci, and its coordinates ai, Sii, on the one hand and a set of plate parameters {ak ) on the other hand. If all the plates had been taken with the same telescope, the plate parameters will belong to groups such that some elements vary from plate to plate, others vary from plate to plate but are constrained, and finally some are the same for all plates obtained on the same telescope. (Eichhorn, 1974). After the functional form of the plate model is established, the plate constants (or estimates thereof) are determined from a statistical adjustment (usually a least squares adjustment), by fitting the measured coordinates to the corresponding spherical coordinates of the reference stars (a, 6). In this case, equations (4) then become the equations of condition for a least squares adjustment in which the xi, yi, ai and 6i are regarded as observations and the at, a2, ..., a, as the unknown parameters. The mi and ci are assumed to be known for every star. Of course, there must be enough reference stars to make the system overdetermined (i.e., the number of reference stars must be greater than the number of parameters). A good rule of thumb is that there should be at least 3 times as many reference stars as plate parameters. Once the plate parameters are determined the best estimates of the standard coordinates for all the stars (field stars as well are reference stars) can be obtained. Then, after routinely inverting the gnomonic projection (equation 1) the equatorial coordinates for all the stars are computed. 20 The Relationship Between the Standard and Measured Coordinates Even under the most favorable conditions the assumption of a perfect gnomonic projection can only approximate the actual circumstances. It is clear that neither telescope nor measuring machine can be constructed with perfect ac curacy. First of all, there is no optical system in practical use which unites in a single point all light rays that enter the front component of the objective from the same direction. In reality there are many deviations from the ideal situation which require small corrections to be added to equation (2). These departures from the perfect gnomonic projection are "instrumental." Others, which rotate and distort the coordinate system with respect to which the position estimates of the reference stars are catalogued, may be termed astronomical effects and include refraction, aberration, precession and nutation. We will deal separately with these two types of "corrections." The "instrumental" corrections will be characterized by the plate model and the "spherical" corrections will be discussed in the next section. Below is a list of some of the more common departures from the ideal gnomonic projection; this list is by no means exhaustive: 1. Shift in the origin of the xy coordinate system with respect to the (r7 coordinate system (c and d in equation 3 2. Rotation of the rq axis with respect to the xy axis. 21 3. The (r; scale may be different from the xy scale, due to a poorly known focal length. 4. Tilt of the xy focal plane with respect to the (q focal plane, this will cause an incorrect tangential point. 5. Different scales on each axis, due to different measuring screws or otherwise introduced by the measuring device. 6. Lens aberrations (distortion, spherical aberration, coma, astigmatism, chro matic aberration) affect the position of the stellar centroids either by enlarging them (thus making it more difficult to determine the center) or by an actual shift in positions, both of which vary depending on the location on the plate. These effects plus peculiarities of the process of photographic image forma tion render the positions of the star's images on the plate dependent on their magnitudes and spectra expressed in terms of color indices. These conditions lead to the inclusion of the mi and ci terms in the mapping between xy and (7] coordinate systems. 7. Besides lens aberration "smeared out" or "blow up images can be produced by guiding errors, developing errors (e.g. emulsion shifts), and bad seeing. These can and do affect the position of the stellar centroids. Each departure from the ideal imaging process results in deviations from the relationship x=s<; y=srj. Given below are the relationships between the standard 22 coordinates and the measured coordinates to correct for many of the commonly experienced errors. 1. Translation error: If the photographic plate is not centered correctly on the measuring machine, there will be a systematic difference between the measured coordinates and ideal coordinates. Ar (c,d) Figure 4: Origin Shift x s = c y sl = d 4 4 23 2. Rotation: If the photographic plate is correctly centered, but rotated relative to the measuring machine axes by an angle 0 (0 > 0 for counterclockwise). x = s( cos 6 + / sin 0) (6) v = .( rcos 6 ( sin o). A Figure 5: Rotation of xy axis with respect to the &7 axis 3. Imprecise focal length: An incorrect focal length can cause differences in the scales between the ,rj system and x,y system. This can be corrected by making small changes in ( and r: x s = (7) y sq = er. 24 4. Center error (tangent point error): For wide fields ( say over 50 x 50) the extraneous methods for the determination of no and 60 yield approximate values of these equations and thus they must be corrected differentially. If the actual position of the tangential point is (oo+dco), (6o+db0) instead of oo,60, values of jq will be incorrect if calculated only from to,So. It can be shown to first order in doo and d6o (Eichhorn, 1971) that the centering error can be corrected (to first order) by x s, = (cos Sodo),2 + d6,r (8) y srs =(cos bodoao)( + d6orl2 provided that the plate was oriented during measurement so that the x and ( axis as well as the y and q axes are nearly parallel and (, r1) are in radians If dao and dbo turn out to be large, it is recommended that the calculation be repeated with the newly calculated ,r7 now referred to the tangential point ao+dao and 60+d60. This may be repeated until further repetitions result in no change in the assumed ao, 6o (Eichhorn, 1974). 5. Tilt: If the photographic plate is not perpendicular to the optical axis of the telescope but titled by an angle w relative to the focal plane, the errors introduced will be x s = (pg2 + q@) tanw (9) y sr = (p + qri2) tanw where p and q are constants (Taff 1981). One can see the terms for the corrections for zero point error and tilt have the same form. This is because if 25 we have a situation where there is a tangential point error this is the equivalent of having an imaginary plate whose true tangent point coincides with our assumed one, but is inclined to the actual plate. Since these two effects can be modeled in a similar manner, they are sometimes combined under that same name. If construction and proper alignment of the telescope has been carefully executed such second order errors may be neglected. However, in the case of large plates (over 50 x 50), it is important that these higher order terms be included (K6nig, 1962). 6. Radial distortion: If the image process produces a net radial distortion, it can be modeled. Radial distortion originates at the intersection point of the optical axis with the focal plane and its origin will, therefore, also be very close to the tangential point. This aberration is radially symmetric and usually modeled in the form n x S= E a(k + 772)1 k=l1 1 (10) y s''= 1 Rk(W + q2)k k=1 where R is a constant (Eichhorn 1974). Unless the radial distortion is very strong, and for the field sizes used on photographic catalog astrometry, only the first terms of equations (9) are necessary for appropriate modeling of radial distortion. Thus with n = 1, x s=R~((2 + rl) (11) y si==Rr(r2 + r2). 26 7. Nonperpendicularity of axes: If the axes of the measuring machine are not perpendicular to each other (in other words x is not perpendicular to y), let be the angle between the q and y axes (McNally, 1975) x s=qr tan u' (12) v sq=(1 sec ,)r. y rl A 4 X4 Figure 6: Nonperpendicularity of axes 8. Coma: The effective focal length of the instrument may depend on the apparent magnitude m of the object that is being imaged. This is coma; it is radial and accounted for by x s=tmx (13) y srt=tmy. 27 Experience shows that the coefficient s is rather sensitive to the rate of change of the ambient temperature during the exposure. Thus it can not be regarded as a constant for any one objective, although the values obtained in practice will have a tendency to cluster around a certain value (see Eichhorn & Gatewood, 1967). 9. Decentering distortion: If all the components of the objective are not prop erly aligned, the resulting imperfection is known as decentering distortion. Brown (1966) found that the appropriate corrections to the measured x and y coordinates are =1 (14) ys,=={2[P,,7+P&,r]+P,( 2 +i2)} 1+ Pk+2(&2 +2,)k k=l again, under the conditions imposed by catalog photographic astrometry, the terms with Pk+2 may usually all be assumed to equal zero, this results in a second order effect x s&=2[P,& +P297] + P, (2 +r2) (15) y sq=2[Plq+P2,2] + P, (2+2). One can see by comparison, the first brackets are exactly the same structure as those that model the correction to the tangential point error (number 4) and tilt (number 5). If the model includes the provisions for correcting the tangential point (or tilt) anyway, the above decentering terms in the brackets can be may "lumped" with them. Then by just adding the Pi (&2 + q72); P.(2( + q2)terms one can account for the corrections due to decentering distortion. 28 As just indicated the sources of errors are not independent of each other and there is much crossover in effects but they can be modeled in a first order (sometimes second order) approximation that will more than suffice considering their magnitude and the magnitude of the measuring errors. It is worth noting that the coefficients which describe the distortions are very insensitive to changes in the ambient temperature and may thus be regarded as more or less constant for any one objective. This statement is not true for coma, which is, in effect, a dependence of the effective focal length (scale) on magnitude. Some Commonly Used Models The difficulty in establishing the relationship between the standard and mea sured coordinate system is in determining which terms to include in the model. Eichhom (1985) writes the relationship (2) in the form (x) = s(/ +Ea (16) where E, the model matrix, is a function of (, r7, m and c, and a is the vector of plate parameters. The form of the model matrix is determined by probing possible deviations from the gnomonic projection, first adding appropriate terms to a very basic model, and then investigating the adjustment residuals produced by additional terms. When significant terms are found they are added to the model. This trial and error process continues until one is satisfied that one has found the 29 "best" possible model which approximates the "true" system. We will now look at some of the models which have been employed in the past. 4Constant Model The first model we will look at is the simplest and therefore the most basic model; it corrects for errors in the measurements. We will assume the tangential (ao 60) to be known or determinable by extraneous methods and thus not subject to the adjustment. The scale and orientation of the standard coordinates are fixed by their definition. However, the system of the measured coordinates is dependent on the measuring machine and the accidental orientation of the photographic plate on it. Consequently, the two systems may not agree because the origins of each system might be different and the axes of one system may be rotated with respect to the ones of the other. This situation is demonstrated in figure 7. Y A S    T(C,D) X Figure 7: 4Constant Model: origin shift and rotation of axes x = s( cos o + sin o) + c (17) y = s( coso sin o) + d. where o is the angle of rotation, s the focal length and c and d the difference in the origins. Combining these effects results in the following model. If we let a = s coso and b = s sino, then the equation becomes, x =a + br + c (18) y = a7l b + d. Frequently, equation (18) is written in the form (x) = 5 r( ) + ( qrl r, l 0 a =) s ( 0 )( (19) In this case plate model, E, is the 2x4 matrix and the vector of plate constants a is (a, b, c, and d). There are four plate constants, so this model is called the 4 constant model. Usually, the plates are measured in such a way that the xaxis is almost parallel to the Iaxis which will (if x and y are strictly rectangular) render the yaxis almost parallel to the qraxis. This will make b very small compared to a. So, when O 0, a = pcos o cos 0 m 1, (20) thus, a p and b= psin sin s 0, (21) therefore, b 0. 31 Thus the term a is the correction to the assumed focal length. If the xy and rq axes are tilted with respect to each other, this term may have a small effect due to this rotation, but the dominant effect is the correction to the focal length. The inversion of the above equation is (' = 1 (a b f (x c (22) q ) + b b a bd or after introducing certain quantities [()=(B AB)()+(C) (23) where A, B, C, D make up the matrix of plate constants. Thus for the 4constant model a rigorous equation exists for and 77 to be solved in terms of the plate constants and x and y. 6Constant Model The 4constant model does not correct for different scales in the x and y measurements. Correcting for this effect, equation (17) becomes (x ( cos sinm (') + (c ( (24) Y ) ( sin cos 0 ) d e () or after multiplying out x = s cos Of + s sin rl + c a( (25) y = s sin O+ + s cos Or + d er. 32 After substituting the following values, a = cos o e cos o b =sin o, f = sino, c= and d= d S S a b (X+ 0 0 0 i (26) f \d 1. a & e: correction to assumed focal length. 2. b & f: correction for rotation of xy system with respect to (r system. This also corrects for nonperpendicularity of the x and y axes. These should be approximately equal and opposite in sign. 3. c & d: differences in origin of the xy system and (r1 system. For the 6constant model there exists an equation analogous to equation (23) for the 4constant model. In other words ( and r can be rigorously solved for from the plate constants and x and y. This is not true for the other models we will be using. In those cases, an initial approximation is used and the solution then iterated until a set tolerance is met. 12Constant Model So far our models have only dealt with terms linear in the standard coordinates ( and r7. However, there are many "second order" effects that need to be considered, especially when dealing with large plates like the Yale plates (11 x 110). A standard model for the conversion of the measured coordinates to the 33 standard coordinates considers corrections for 1) rotation, 2) translational error, 3) tangential point correction, 4) radial distortion, 5) tilt and 6) coma ; it is the 12constant model: a b C e 0 0 0 q 1 (q q2 0 m mo qm q1(2 + r12) p q 9 i \h 1. a & e: correction to assumed focal length (different in x and y). 2. b & f: correction for rotation of xy system with q system. This also corrects for nonperpendicularity of the x and y axes 3. c & d: differences in origin of the xy system and (77 system 4. p & q: Tangential point correction and tilt 5. i & j: magnitude "equations" 6. g: coma 7. h: radial distortion 34 All terms of a are at least one order of magnitude smaller than x, y or (, rl and s is, of course, an approximation for the focal length. This clearly shows that the rigorous formulas give (, r/ not in radians, but rather in units of the focal length; they are indeed dimensionless quantities. This model is very similar to one we chose to best represent the spherical coordinate system. The actual model and method of determining the model for the Yale plates is given in Chapter 6. Spherical Correction We stated earlier that we split the deviations from the perfect gnomonic projection into instrumental and astronomical effects. The instrumental effects are corrected by the plate model. The astronomical deviations arise because the image formed on the photographic plate represents the refracted apparent topocentric positions of the objects. The standard coordinates are computed from the coordinates listed in star catalogs (e.g. mean positions). To account for the difference between the refracted topocentric apparent positions (x, y) and the mean positions (, rt), corrections for refraction precession, and for extreme accuracy, diurnal aberration, should be applied. We will not consider diurnal aberration, since its effect is smaller than our measuring error. The effects of precession are accounted for by correcting the reference catalog and influence the results only slightly through refraction. 35 Atmospheric refraction causes light from a celestial body to be bent as it passes through the Earth's atmosphere. Assuming a simple model for the atmosphere as a plane atmosphere and for small zenith distances, total refraction in the zenith distance can be calculated by z z = Az = 3 tan z (28) where z and zo are the topocentric apparent (i.e., unaffected by refraction) and observed refracted zenith (i.e., effected by refraction) distance of the object, respectively. 3 is the constant of refraction and depends weakly on the temperature and pressure at the observing station (usually this value is about 1 minute of arc). Since Az > 0, astronomical refraction raises celestial objects toward the zenith. Astronomical refraction ideally does not affect the azimuth. For large zenith distances, allowance has to be made for the variation of 3 within the field of the plate; in practice, it suffices to put 3 = 3o + 3' tan2 z, where 3' and 3o are refraction constants and can be found in refraction tables (KOnig, 1933). Thus equation (28) becomes Az = /o tan z + /' tan3 z. (29) We have neglected higher order terms for zenith distance less than 700. We will split the effects of atmospheric refraction into those of "absolute" and "differential." By "absolute" we mean the amount of refraction affecting the 36 center of plate or more precisely the tangential point. We can think of refraction as shifting all the positions on the plate by this constant amount plus a differential amount which varies across the plate. For example in Figure 8 (top) we have plotted the effects of refraction for plate 1. As one can see the positions are shifted mostly in the y direction (an average of about 49"). In Figure 8 (bottom) we have shown the effect across the plate by subtracting the tangential point refraction. Note that the scales for the two graphs are not the same; in the bottom left hand corner of each graph we have plotted the scale: 50" for the top graph and 5" for the bottom one. We will apply a correction for differential refraction, and since our plate model already has a linear term in it we will allow the linear effect of absolute refraction to be taken care of by our plate model. It has been a frequent practice to account for differential refraction by adding higher order terms to the model. This is usually achieved by adopting a full second and third order polynomial expansion in the coordinates. There are two major disadvantages to using this approach. Firstly, if the number of parameters is large an unfavorable error propagation of the resulting star positions in the plate field due to the random errors in the xy measurements and the reference star positions is introduced (Eichhorn and Williams, 1963); in particular when the number of reference stars per plate is small. Secondly, all effects are mixed up in the model and therefore it is impossible to separate different contributions and evaluate the physical meaning of the plate parameters in detail. As we mentioned above we will precorrect the x,y measurements for the differential atmospheric refraction. These values are used as observations in the plate reduction. In this way, the number of parameters in the model can be kept to a minium which will give us a better chance to disentangle the complicated origins of many of the plate parameters (Zacharias et al., 1992). The usual approach in plate reduction calculations is to transform the equa torial system to the horizon system and make refraction corrections in a simple way in the horizon system. In this manner the effect of refraction is added to the reference positions. However, this leads to the situation where the same star imaged on different plates has different positions due to refraction. From Chapter 1 we stated that the power of the reduction technique that we use, the overlap ping plate method (Chapter 4), is that it enforces the constraint that a single star photographed on many plates must have only one position in the sky. Therefore correcting the stars' equatorial coordinates for refraction leads to a different posi tion depending on what plate it was imaged, thereby invalidating our constraint. We have chosen, instead, to precorrect the measured star images so that they represent the position on a plate if the plate had been observed outside of the atmosphere. We have used the equations developed by K6nig (1933) for calcu lating the corrections for refraction, Ax and Ay, which are to be applied to the apparent topocentric coordinates, x and y. After calculating the zenith distance of 38 the plates at the time of exposure and then allowing for temperature and pressure we can determine 30 from the refraction Tables 713 in K6nig (1933) and let 3' = 0.082". Table 2 and Table 3 contain all the necessary information to determine refraction with the aid of Konig's refraction tables. In addition, the latitude at the Sydney observatory is 330 51' 41."1, the longitude is 10h 4m 49.s06, and we have assumed an altitude at sea level and a barometer reading of 750 millibars. Refraction Correction: Absolute 0.037 0.034 3.50318 YRMS: Refraction Correction: Relative 0 % %( v 41 11 Hi It, , It% .. .. C. . *  ~N \  <* I '*;. ^< * * / ,,, ",', ,, '' ,,, : ,^ '^ ;n^'. n 1i n 0.037 0.034 3.10016 YRMS: Figure 8: The effects of refraction on plate 1 0.108 i 0.109 X RMS: 0.106 49.2025 0.108 0.036 0.036 0.109 X RMS: 0.106 5.58124 i i I r i i I i I 40 The tangential point is taken to be the refracted ao, o0. The rigorous mathe matical relations between the refracted positions and the topocentric positions are somewhat complicated. For practical application, only development into series can be considered; it is therefore necessary to state which terms must be retained and which can be neglected. In order to obtain the simple expressions for the coefficients of the series, Kbnig introduces the following auxiliary quantities k = tan z sin y k2 = tan z cos X (30) k3 = 1 + k k4 = 1 + k2 where y is the (refracted) parallactic angle at the tangent point (taken to be the refracted tangential point) and ki and k2 have a simple geometrical meaning; they are the tangential coordinates of the zenith in the plane of the plate. The corrections for the measured values are (to the third order): Ax = [3k3 +20/'(1+ kl + k )k ]x [3k(k2 + tan bo) + 23'(1 + + k + )kk2]y + 2[3 + 20'(1 + k + k2)]klk2y klk3x2 (31) 23k k2xy /3kk4y2 + /k 2x3 + 30klk2k3X2y + ,(kak4 + 2k k2)xy2 + /kik2k4y3 41 Ay = [3k(k2 + tan o) + 23'(1 + k2 + k )kik2]x + [3k3 + 23'(1 + k2 + k )k ]y + [3 + 23'(1 + kf + k:)] (k2 k )y (32) 3k2kax2 23kik xy 3kk4y2 + 3kik2k3x3 + 3(k3k4 + 2k2 k)x2y + 33kik2k4xy2 + 3k y3. Let XR = 3lkk3x2 23kfk2xy 3klk4y2 + 3k x3 (33) +33klk2k3x2y + 3(k3k4 + 2k k )xy2 + 3klk2k4y3 and YR 3k2kax2 20kik xy 3k2k4y2 + 3klk2k3x3 (34) +i3(k3k4 + 2kI k)x2y + 3/3klk2k4xy2 + 3k y3 also define A = 3ka + 2'(1 + k + k2)kl B = [Oki(k2 + tan) + 2/'(1 + k2 + k )kik2] (35) C = 2[3+20'(1 +k k) + k i]k2y D = [ + 2/'(1 + ki + k2)] (k2 k). Substituting these variables into equation we can rewrite equations (31) and (32) as Ax = Ax + By + C + XR (36) Ay = Bx + Ay + D + YR. Written in this form the refraction terms which cause a change in the scale (A), rotation (B), linear refraction terms (C and D), residual refraction (XRand YR) cab be separated. Generally, for small fields and for zenith distance less than 70 the 42 terms A and B can be neglected. However, this is not so for plates taken at very high declinations. We have included the terms for change of scale and residual refraction, but not rotation. Rotation was not included because the rotation caused by refraction cannot be separated from that caused by the position of the plates in the measuring machine. Table 2: Refraction Information for plates 132 PL RA0 LMST MON DAY 1 0.0 0.108 10 15 2 1.0 1.042 11 11 3 2.0 2.058 12 5 4 3.0 2.75 11 20 5 4.0 4.342 1 6 6 5.0 5.408 1 13 7 6.0 6.158 1 8 8 7.0 7.092 1 8 9 8.0 8.092 1 18 10 9.0 9.108 1 18 11 10.0 10.063 2 7 12 11.0 11.258 4 12 13 12.0 11.958 4 11 14 13.0 13.083 7 3 15 14.0 14.008 4 10 16 15.0 15.233 6 12 17 16.0 16.250 6 12 18 17.0 17.150 6 5 19 18.0 18.150 6 5 20 19.0 18.883 6 5 21 20.0 20.238 9 19 22 21.0 21.175 10 13 23 22.0 22.208 10 15 24 23.0 23.275 10 15 25 0.0 0.096 11 20 26 1.2 1.413 10 13 27 2.4 2.496 11 20 28 3.6 3.713 11 20 29 4.8 4.629 1 13 30 6.0 6.213 1 9 31 7.2 7.046 1 13 32 8.4 8.579 1 18 TEMP HA Z D REFR 68 0.108 41.15 48.25 70 0.042 41.14 48.05 70 0.058 41.14 48.06 70 23.975 41.14 48.05 70 0.342 41.21 48.18 75 0.408 41.26 47.78 75 0.158 41.16 47.63 75 0.092 41.14 47.61 77 0.092 41.14 47.43 77 0.108 41.15 47.43 75 0.063 41.14 47.60 65 0.258 41.18 48.58 65 23.958 41.14 48.51 60 0.083 41.14 48.99 65 0.008 41.14 48.51 60 0.233 41.17 49.04 60 0.250 41.18 49.05 60 0.150 41.15 49.00 60 0.150 41.15 49.00 60 23.883 41.15 48.99 60 0.237 41.18 49.04 68 0.175 41.16 48.27 68 0.208 41.17 48.28 68 0.275 41.19 48.32 70 0.096 46.14 57.22 68 0.212 46.16 57.47 70 0.096 46.14 57.22 70 0.113 46.14 57.23 75 23.829 46.15 56.70 75 0.213 46.16 56.71 75 23.846 46.15 56.70 77 0.179 46.15 56.49 Table 3: Refraction Information for plates 3364 PL RAO LMST MON DAY TEMP HA Z D REFR 33 9.6 9.717 34 10.8 10.967 35 12.0 12.258 36 13.2 13.404 37 14.4 14.729 38 15.6 15.792 39 16.8 16.971 40 18.0 18.088 41 19.2 19.433 42 20.4 20.567 43 21.6 21.746 44 22.8 22.875 45 0.0 0.088 46 1.5 1.583 47 3.0 3.250 48 4.5 4.550 49 6.0 5.900 50 7.5 7.854 51 9.0 9.325 52 10.5 10.575 53 12.0 12.308 54 13.5 13.783 55 15.0 15.183 56 16.5 16.750 57 18.0 18.000 58 19.5 19.771 59 21.0 21.154 60 22.5 22.550 61 0.0 0.133 62 6.0 6.350 63 12.0 12.083 64 18.0 18.263 77 0.117 46.14 56.48 65 0.167 46.15 57.79 65 0.258 46.17 57.82 65 0.204 46.16 57.80 60 0.329 46.18 58.41 60 0.192 46.15 58.35 65 0.171 46.15 57.79 60 0.088 46.14 58.33 60 0.233 46.16 58.36 65 0.167 46.15 57.78 68 0.146 46.15 57.45 68 0.075 46.14157.44 68 0.088 51.14 68.46 70 0.083 51.14 68.20 70 0.250 51.15 68.23 70 0.050 51.14 68.20 75 23.90 51.14 67.56 75 0.354 51.16 67.61 75 0.325 51.16 67.61 75 0.075 51.14 68.90 65 0.308 51.16 68.90 65 0.283 51.15 68.89 65 0.183 51.15 68.87 60 0.250 51.15 69.55 60 0.000 51.14 69.52 60 0.271 51.15 69.55 60 0.154 51.14 69.53 68 0.050 51.14 68.46 68 0.133 56.14 82.15 77 0.350 56.14 80.76 65 0.083 56.14 82.62 60 0.262 56.14 83.42 CHAPTER 3 LEAST SQUARES Since we will use the method of least squares to determine our plate parame ters for both the single plate overlapping plate reduction techniques, we will first give an outline of this data analysis method. Theory of Least Squares Consider a set of n observations represented in vector form by xo, and assume these observations are unbiased (i.e. without systematic error). The vector of true values is represented by x and v = x xo is the vector of errors in the observations. From the assumptions that the observations are unbiased we can assume that the errors will have a Gaussian distribution (multivariate normal distribution) and we will regard their covariance matrix, 0r, as being known. Thus we can express the probability density distribution function of the errors as the multivariate gaussian k(v) = CevT V. (37) We further assume that the observations satisfy a set of equations of condition F(xo + v, a) = 0 (38) 46 where a is a vector of q parameters. Some of the components of the vector equations F=O may not explicitly contain any of the components of x; such equations would be condition equations involving parameters only. In this analysis we will distinguish these from the general condition equations by calling them parameter constraints. If the q parameters are not mutually independent, the relations existing be tween them must be included among the parameter constraint equations. The observations must be functionally independent, that is, not one of the n observa tions can be derived from any or all of the remaining (n1) observations. There must be at least n observations; in other words, n>q. Otherwise there will be a deficiency in the model. If q = n then such a circumstance will lead to a unique solution. When n > q, redundancy is said to exist and adjustment is needed in order to obtain a the best set of estimates for the model variables. Adjustment is meaningful only in those cases in which the data available exceed the minimum necessary for a unique determination. Since the data points are usually obtained from observations, which contain errors, redundant data are usually inconsistent in the sense that each sufficient subset yields different results from another subset. No unique result (no one vector a) will satisfy all the equations, but will give rise to (at least some nonzero) residuals, v. Making the basic assumption, called the principle of maximum likelihood, we assume that the set of measurements which we obtain is actually the most probable 47 set of measurements. Thus the best estimates of the coefficients a are the ones that maximize the probability of obtaining the particular set of measurements which we actually obtained. Clearly, the way to maximize 0 is to minimize the value of the exponent. The principle of maximum likelihood thus leads to the conclusion that we should minimize vTav. The principle of least squares consists in finding estimates of v and a which minimize the quadratic form vT0rlv while simultaneously satisfying F(xo + v. a) = 0. Thus if the random variables to which the observations re fer are normally distributed, the least squares method will give identical results to those from the maximum likelihood method. Traditional (Linear Least Squares) Traditional least squares is when we assume that the observation errors are of the same precision and uncorrelated and that exactly one observation occurs in each equation of condition. Since the observations are uncorrelated the covariance matrix a becomes diagonal. Furthermore, if the observations are of the same precision the covariance matrix will be of the form o = ooI, where I is the identity matrix. The minimum principle of least squares becomes : 1n (39) vT1V = V 2 v 1 v = v  minimum. i=1 48 n Thus in this case the minimum of v2 = vTv and vTalv coincide. This i=l last case is the oldest and the most classical, and possibly the one that gave rise to the name "least squares" since in this case we seek the "least" of the sum of the squares of the residuals. In traditional least squares the condition equations F(x,a) = 0 usually have the form Aa + Xo = v. (40) where A = . In this case the "adjustment unknowns" or adjustment parameters", a, occur linearly in the condition equation, and each equation contains exactly one observation. The "normal equations" for finding the estimates of a are easily found for this case.Multiplying the above equation by its transpose gives: (aTAT + x)(Aa + Xo) = (vT)(v) (41) aTATAa + 2aTATxo + x0xo = vT for matrix A and vectors a and x. Also, note that vT v is not a vector or a matrix, but a number and we may therefore transpose any of the terms without knowing its value. To find the minimum of vT v, we differentiate with respect to each component of a and set the derivative equal to zero (vTv) a(aTATAa + 2aTATxo + xoxo) Oa 9a (42) =2(aTATA+x A) =0 yielding aTATA = xTA. (43) After transposing and rearranging, we get the traditional least squares solution in its simplest form: a = (ATA)fATxo (44) the solution of the normal equations. Nonlinear Least Squares The conditional and the constraint equations involved in an adjustment prob lem can, in general, be nonlinear. However, least squares adjustments are gener ally performed with linear functions, since it is rather difficult and often imprac tical, at least at present, to seek a least squares solution of nonlinear equations. Consequently, whenever the equations in the model are originally nonlinear, some means of linearization must be performed on the equations. Series expansions, a Taylor's series in particular, are often used for this purpose, where only the zero and firstorder terms are used and all other higherorder terms are neglected. When applying a series expansion, an initial set of approximate values for the unknowns in the equations must be chosen. The choice of those initial approx imations is an important aspect of solving the problem at hand. Unfortunately there is no concrete and unique way of choosing approximations that can be ap plied to all adjustment problems. In all cases an attempt should be made to obtain 50 the closest approximations that can be obtained by using relatively simple and uninvolved techniques. We will now use the more generalized least squares treatment to drop some of the restrictions found in the traditional least squares method. In the following development 1. Observations may be correlated. 2. More than one observation may appear in an equation of condition. 3. The equations of condition are assumed nonlinear in both parameters and observations. 4. Nonlinear constraints among the parameters are allowed. As before the vector Xo is the vector of observations, x is the vector of true values x=xo + v where v is the vector of corrections. In order to find estimates of the vector of parameters a and x in the generalized least squares we will use the method of Laplace multipliers. At the solution the numerical value of vTcrv = S. These conditions are the same as S* = vTolv 2FT(a, xo + v)A, where A is a Lagrange multiplier. This means that the values of S and S* reach their minimum at the same values of a and v. As before, a has q components, v has n components and F has p components. Thus there are only p+nq free components. Differentiating S* with respect to 51 the components of a and v gives = lv 2XTA 0v OV (45) OS* T  = 2ATA da where ofi af_ X _v av2 * Xpxn (46) S x=xo,a=ao (46) Of af af OF 3aal 5a, Apxq = (47) Ba /a 2 x= xo=xa=oaa=ao da1 ea2 iaq x=xo,a=ao The following equations must, therefore, be rigorously satisfied at the solution v = XTA n equations ATA = 0 q equations (48) F = 0 p equations. This means that A and X must be evaluated at the solution for a and for x. When one considers that A has p unknowns, then one can see that there are exactly the same number of unknowns as there are equations. The equations are in general nonlinear and therefore must be solved by successive approximations. A general rigorous and noniterative algorithm for the 52 solution exists only for the case that the elements of x and a occur linearly in the function F. When F is nonlinear in the elements of either x or a, or both, equations which are practically equivalent to the F and which are linear in the pertinent variables can be derived, e.g., in the following way: assume that an approximate solution of the above equations has been obtained somehow, linearize these equations about the approximate solution and by iterations determine corrections to it. Start by taking an approximation ao for a and using the observations xo as approximations for x. If we write a = ao + a, with a being small corrections to ao, the equation F=0 can be developed as a Taylor series ,F(xo, ao) 0F(xo, ao) F(x,a) = F(xo.ao) + (x Xo) + (a ao) + 0(2)= 0 xr Ba F(x, a) = F(xo, ao) + Xov + Aoa + 0(2)= 0 (49) where Xo and Ao are evaluated at x0 and ao; and also, let F(xo, ao) = F0. From equation (48) we have v = aXTA; so, by substituting into the above equations we get Fo + XooXoTA + Aoa = 0. (50) Combining this with ATA = 0 yields the Generalized Normal equations Xo 7XT Ao[ A] [Fo] [01 AT 0 0 0 (51) Note that A is needed for computing the correction v. If XaXT is nonsingular, we can eliminate A from the equation by multiplying the equation (50) by AT(XoXT) 1 yielding (~1 A (Xo X0) Fo + AOA + Ao(XoTX)1Aoa = 0. (52) Since, ATA = 0, rearranging gives us AT(XoaXor)IAa = AT(XoCaXT)Fo (53) a = [AT(XoaXT)I A]'AT(XoaX;)^Fo. Premultiplying equation (50) by (Xoo'XT) and solving for A gives (XoaXo)1Fo + A + (XoXoX)Aa = 0 (54) A = (Xoc'Xo)1Aa (XoaXT)'Fo. A = W(Aa + Fo) (55) where W = (XoaXT)1. W is frequently called the weight matrix. Since we know ATA = 0, we can premultiply equation (55) by AT, and get AT = ATWAa ATWFo = 0. (56) Solving for a gives a = [ATWA] ATWFo (57) It can be shown (Brown 1955; Jefferys 1980, 1981) that (ATWA)1 is the covariance matrix of a, so that the square roots of its diagonal terms are the standard deviations of the corresponding terms of a. Note that XoXT is not always nonsingular; for instance, when F=O contains equations in which none of 54 the components of x occur explicitly i.e., parameter constraints. This, however, produces in general no essential singularities. It is now possible to solve for v. From v = oXTA we get v = X T[WA[ArWA]ATW W}Fo. (58) Then anew = ao + a (59) Xnew = Xo + V define the improved solution. If the improved solution is insufficiently accurate, the process should be repeated until satisfactory convergence is attained. The natural starting point for this scheme is to set xo= x as the initial approximation, so that v = 0, and to use a vector a of initial approximations for a for ao. Obtaining a suitable a is sometimes a difficult question, it must be close enough to the solution that the process given above converges. Comparison with Traditional Least Squares It is informative to compare this general method with the results from the traditional least squares method. In the traditional case we are dealing with the a) noncorrelated observations of equal precision (i.e. ar = aoloI = OTolm) of which b) exactly one observation occurs in each equation of condition, and that c) the equation of condition is solved with respect to it, we have X = I and therefore (XoXT)' = ~Im, so that in this very specialized case a = (ATA) ATFo 55 Also, the observation errors reduce using this simplification and X = Im a = (o7Im (60) W = Im to v = (XT[WA[ATWA]lATW W}Fo v = aolm(Im)[ImA(AT ImA)AT Im Im]Fo (61) 70o 00 070 (0" = [Im A(ATA) AT Fo. This formula is well known from classical leastsquares theory and evidently independent of a0. CHAPTER 4 METHOD OF PLATE REDUCTIONS We will now use the theory of least squares for the determination of the plate parameters. Recall that from chapter 2 the determination of the equatorial coordinates is a twostep process. First the reference stars' equatorial coordinates are converted to standard coordinates. Second, after the functional form of a plate model is established between the measured coordinates and the standard coordinates a least squares reduction is performed on the measured and standard coordinates of the reference stars to determine the plate parameters. Usually a great deal of analysis is involved in determining the functional form of the plate model. First a simple 4 or 6constant model is used and approximate values are obtained for some of the plate parameters. A least squares reduction using only the reference stars on the plate (single plate reduction) is done to find the best values for the parameters. After the reduction, the residuals from the actual measurements (x, y) and those calculated using these plate parameters and the cataloged equatorial coordinates are studied. From the residual plots it is decided which new terms, if any, need to be added to the plate model. Once the "best" plate model has been established, the plate parameters corresponding to this model are used to determine the equatorial coordinates of all the stars imaged on the plate (not just the reference stars). After position estimates for all the stars have 56 57 been determined an overlap solution can be performed. This is because in an overlap solution all the images of stars that occur on at least two plates or are reference stars are considered in the normal equations. Single Plate Reduction: Using Least Squares to Determine the Plate Parameters. We will first reduce each plate separately. In the next section we will develop a multiple plate reduction technique. We will use the formulation developed by Eichhom (1985) to set up the equation for a single reduction and also an overlap reduction. In a least squares adjustment we regard the measured relative coordinates of the images and the position estimates from the reference star catalogue as observables. These two types of observables give rise to two types of condition equations. Consider data from only one plate on which there are n stars, m of which are reference stars. Without restricting generality, we may assume that the measured image coordinates of the vyth star, x/, yp are related to their standard coordinates ip, t77 by the equation =( s(} + Z=a. (62) ,, is the model matrix, which depends on 7, rt, and possibly other data pertaining to the star in question, such as magnitude or color index, a is the vector of the plate parameters, which constitutes a subset of the adjustment unknowns and is a vector of dimension p. In what follows, we shall keep the model matrix E 58 general and hold to the assumption that Ea is at least one order of magnitude smaller than s( and sj. Using this relation we establish the first set of condition equations (yv (,i ,a F =\ y2(a( 662)) (2 ) X) o. (63) (Xm ( m(Om, m m)\ \ mm (m(am, 6m) / In our approach, the observables are xu and yp and we regard the adjustment parameters to be not only the plate constants a, but also the spherical coordinates (aCp,6p) of the stars. ( and i7 are used only as auxiliary quantities. We consider the tangential point, ao and bo, as being known. As stated in Chapter 2 if the assumed tangential point is not the actual one, then this is can be accounted for by an appropriate form of E and additional components of a. The second type of observations (estimates of the reference star's spherical coordinates from an existing catalogue) states that the actual spherical coordinates (c 6/) are equal to those estimated in the catalogue plus corrections. To take care of convergence of the meridians, we introduce the factor cosb in the right ascension equations yielding /(Oc ao) cos \, 61c 61 (a2C 0 ) cos 6, G(2mx) = 6= 0 (64) (Omc Om) COS 6m \ m m / where aic...bmc are the catalogue positions and ol...bm are the adjustment un knowns which occur both in F=0 and G =0. We will now combine the 4m condition equations F and G into one set H. HT = (FT GT). (65) Each equation, regardless of whether it belongs to F=0 or G0 contains exactly one observable, therefore ( X = I4m. (66) We assume that the measurements of all the coordinates are free from bias and not correlated. The variance (in units of squares of the units in which xy and yp were measured) of x; is vp and yp is Op. The covariance matrix of the rectangular coordinate measurements is Orx = diag(/iv, &,2, v2, 02,..., Vm, and in most cases, it will suffice to write Ox = mI'2m (68) 60 where arn is the common variance of all coordinate measurements and 12m is, of course, the identity matrix of order 2m. Also let's assume that all the reference star coordinate's are uncorrelated, so that the covariance matrix a,, of the catalogues reference star coordinate estimates acos6 and 6 will be Fyav = diag(pl, 7,..., pn, OIm). (69) The factors cos2 6 are introduced in order to make vp/v the dispersion of cutcos(y6), that is the dispersion on the great circle. Not only will this simplify some of the formulas we need in our calculation, but it compensates for the convergence of the meridians. Thus, the observations are XO = (xx,yX2 z,Y2,...,Xm, Ym; OYlcCOS 61, 1l,...QamcCOS bm, m) (70) and their covariance matrix is a = diag(vl, 0l, V2, 12, ...Vm, ;m ; pI, a 2, 2, ...P,, &m) (71) or o 0 xv = (V (72) 0 ) (0 : 61 The vector a of adjustment parameters consists of the corrections to the reference star positions and the initial plate parameters. As approximations for the position we choose their cataloged values which renders Go = 0. As initial approximations to the vector of the plate parameters we choose 0. Thus we have Fo = F(a,,,,0) = d Ho = d (73) Go The vector of the adjustment parameters a = with a 3T = (cos 6dda1, db6, ..., cos 6,dam, d6m). (74) From our least squares analysis in Chapter 3, the corrections to the parameters a are given by a = [ATWA]'ATWHo (75) where W = (XcrXT) Since all the equations have only one observation, X = I and our weight matrix becomes W = r1Im so we can write a = [ATor A]'ATcrHo. (76) In order to solve the normal equations, we have to evaluate explicitly the terms of the matrix A and since a = ( we can rewrite the above equation as A A() = A oT1 (). (77) 62 A is the Jacobian matrix of the condition equations with respect to the adjustment parameters, so we can generally write, OH O(F ( ) G) (78) A 8a) 0(3.a) G) (dG)) ) If Icl << 1 and if the initial parameter vector is a = 0 then, the values we obtain for a from equation (77) will be the actual parameter values rather than the corrections. Let's take a closer look at what these partial derivatives equal: 3d, (04 cos6b,6 b) (79) sB., where B = ( ) 0 F I When v does not equal = 0. Therefore (a) is block diagonal, consisting of m blocks of dimension 2 x 2 on the main diagonal; these blocks are sBy. The factor matrix is required because we regard cos6 do, and not do as the correction, thus d( = ( o) ) cos da + . (80) \ \ / cos 6 \d6 The other elements of A are a= = S (81) OG 9G &9G G I 9G 2G 1G =I2n and 0G a = 0. (83) Therefore, A=(B ) (84) We will substitute this into equation (77) and examine the remaining terms. A T o ;7 sBlT 1 (85) ( s nT o ,x I o ra 1 ATo.1A= T 1 0 SSBnTox'ln BToxl (86) SaBT 1BS 1 01 ( , .sZTr B ST1 } It is informative to look at the dimensions of A and the elements that make up A. For m reference stars and p plate parameters we have * B: block diagonal (2m x 2m), whose elementary matrix is a (2 x 2). * 1 : block diagonal (2m x 2m), whose elementary matrix is a (2 x 2 ). * 1 : block diagonal (2m x 2m), whose elementary matrix is a (2 x 2). * : (2m x p) Therefore the dimensions of the 2 x 2 blocks of A are: * (1,1) position : (2m x 2m) * (1,2) position : (2m x p) * (2,1) position : (m x 2p) * (2.2) position : (m x m) After substituting in equation (86) into equation (77) we get (.2BTcx IB + a sB r ' J\ sBTx1 1 d s )\1B ) = S1O ) )1 O 0 (87) and expanding this gives two equation vectors (s2BTo'r B + 3a') + sBTo a = sB Tad (88) and T xlB + T 1a= OTi ld. (89) Solving the first for 3 gives 3 = (s2BToB + a 1) sB T l(d Za). (90) Substituting in 3 to the second equation and letting Y =s2l 1B(s2BTr lB + 1)BT x, (91) we (after some algebra) get the following .T(y a~ )d = T(Y_rl).a. (92) Let J = o Y or J= ('s 2 1B (s2B Trx B+ or BTr1 (93) J= 65 using the inversion lemma (Henderson and Searle, 1981) we get J= (x + sBoB') ) (94) Substituting this equation for J into equation (92) yielding ETJEa = WTJd. (95) We can solve this equation for a: a = (ETJ) 'TJd. (96) Note that J is also block diagonal, consisting of 2 x 2 blocks on the main diagonal. We can therefore write more explicitly a 6= ( VT) J .. (97) V=l V=l Next, we have to find explicit expression for the terms of J. These depend on ax, oa and the terms of B. The terms of B are essentially the terms of ( ) Since we have good approximations of the values of (, j which are approximately (x/s) and (y/s) respectively, we will determine J from these quantities. Following the procedure suggested by Eichhorn (1985) we will define R2 = 1 + 2 + 72 T2 = 2 + (cos 60 r sin 60)2 S = R2 sin 60/T (98) U = ,R(sin bo + cos bo)/T V = R2(cos 60 7 sin o)/T W = (2Rcos 6o/T + V/R and note that R SU + VW. (99) B can now be written as B = (100) From our earlier discussion we wrote J = (ox + s2BaaBT) However with the formulas above the elements of B can be easily calculated without the explicit knowledge of the star's spherical coordinates. If we further introduce the variables, QS = pUW + OSV Y = pU + ci V (101) Z = pW2 + oS2 and introduce (P 0 (S2 )0 (= )aS= (S 2& (102) the weight matrix J can be written in the form J= (v+Z Q (103) Q 0 + Y) Note that for any nonsingular matrix D the following is true l{d2 d12 D 2= d dl (104) A dai d ). 67 Therefore the determinant of J is A = vo + Zo + Yv + R'ap and we get the weight matrix belonging to the vth star as J,= I 1 Ov +=Y, Q, (105) V",vO + Yvv + ZVov + R6ap v Qv V + Z We now need to explore more fully the computation for the corrections to the star positions. We now assume that we have calculated the vector of plate parameters, a, from equation (97), and henceforth we consider that the vector d Ea is known. As mentioned before, we have m different independent sets of equations for calculating the corrections3T = (cos 6Sda, d6b) for each star. From equation (90) we have the formula for the corrections to the star parameters, expanding this equations out (and substituting &a'1 = s2a1) we have 1 T T 3= B B + a ) sB (d a) 1 [( 1 0 U 10 W _S 1 S 1 S V 1 V (W U U)((1 0td V 1 0)(d Ea) (106) Remember that 3 is different for each star and that the symbols W, U, V, S above are simple numbers and not matrices. To calculate the inverse of the 2 x 2 matrix in this equation, we need its determinant. After some algebra we get the determinant of 3 to be S= p(v + Yv + Z + R6p). (107) vapa 68 Except for the factor. I, this is the reciprocal of the determinant of J. After multiplying the equation for 3 (equation 106) by its determinant we get 1 .s( o 4 Zo + Y "' + R6pa) 3( [S: Ioor 0 V (108) Note that 3 is in radians. Its is (if one investigates more closely ) a sophisticated weighted average of the catalogue positions and that which follows from the plates. Final Positions from a Single Plate Reduction, Once we obtain, by an adjustment procedure, estimates for the plate constants, we use them with an inverse of the plate model to determine estimates for the standard coordinates for all the measured images (reference and nonreference). Then using the inverse gnomonic projection, the equatorial coordinates are de termined. One of the drawback of the single plate solution is that if the star occurred on more than one plate, the estimates for the same star's coordinates will, of course, be different depending on which plate they originate from. Thus, the single plate solution does not use the simple fact that a star can at one time oc cupy only one position. Since all the plates are reduced individually, a star that is found on n plates will have n different values for its coordinates. Clearly the star only has one position at one time, but this cannot be obtained without mathemat ically enforcing this constraint on a multiple plate reduction process. Instead, the 69 separately computed values are averaged to give the "best" (in the mathematical statistics sense) estimate for the stellar position. In addition, the random errors are reduced by a factor of 1 but this does not reduce the systematic errors. Another drawback of the single plate solution is that when using only the reference stars in the reduction we would more than likely be extrapolating the magnitude terms to the field stars which are typically fainter than the reference stars. It was considerations of this kind which led to the method of overlapping plates, in its full form first published by Eichhorn (1960). The method of overlapping plates avoids the multiplicity of several "best" star positions and increases considerably the accuracy of at least some of the plate constants, thus reducing the systematic errors (i.e., increasing the accuracy) of the final positions. However, its implementation requires an arithmetic effort exceeding that which is necessary for traditional plate reductions by a factor of at least 2 orders of magnitude. This task has become possible only within the last few years, because cheap, powerful computers have become easily available, removing the label of "major job" from a complete overlap solution. Overlap Plate Reduction The basic principle of the overlapping plate method is that all the star coordinates, not just the reference stars, are considered together with the plate parameters, as adjustment parameters (unknowns). Since the overlap solution 70 regards the coordinates of each star as adjustment parameters as long as images of the star were measured on more than one plate, the number of parameters increases in proportion to the ratio of the field stars to the reference stars. Obviously  since the number of unknowns has increased dramatically the analytical and computational problem is exponentially more complex and time consuming than the single plate reduction. In addition, the adjustments pertaining to individual plates also lose their mutual independence because any two plates which have a star in common can now no longer be reduced individually. Another unavoidable defect of the single plate reduction is a large parameter variance (Eichhorn and Williams, 1963). Even if the geometry of the imaging sys tem was known precisely and the errors of the measuring machine were modeled exactly, there will still exist unavoidable random errors in the xy measurements and reference star positions. This leads to an unfavorable error propagation of the resulting star positions, in particular when the number of parameters is large and the number of reference stars per plate is small. The accidental errors of these plate parameter estimates thus reappear as systematic errors in the positions that were calculated with them. Such systematic differences even showing up as functions of model terms are unavoidable (and their magnitudes can even be estimated) for those stars (field stars) which were not involved in the derivation of the singleplate parameter estimates. An overlap solution enforces the constraint that a star can only have one position at one time, producing more accurate plate 71 parameters, thus inevitably leading to smaller systematic errors in the star posi tions. Also, since the positions themselves are regarded as adjustment unknowns, there can not be any systematic residuals between plates (except those introduced by model deficiencies). In addition, when there is a strong overlap between plates, fewer reference stars are needed per square degree to obtain the same accuracy since, in effect, the total area covered by the overlapping plates is treated as one large plate. We assume that there are altogether m stars involved in the adjustment, including both reference and field stars found on more than one plate. Each star number: yp = 1, 2,..., m. The numbers t are assigned to the stars in some organized fashion. There are n plates and each plate is assigned a number v = 1, 2,..., n. On the vth plate there are my stars. For example, (xyv, ypv) are the measured rectangular coordinates of the images of the lth star on the vth plate. mr of the m stars are reference stars and the reference stars are assigned a number Pri, PTr2, ..., Prm,  The stars used in a overlap reduction are either field stars which occur on at least two plates or reference stars. The estimates for the field stars' equatorial coordinates are found from the single plate solution. When the star appears on more than one plate the average of the single plate solution is used. Thus the averaged ay, 6b of the field stars are used to find the standard coordinates /vy, rJuv on all plates on which the star appears. However, for the reference stars 72 the catalogued values ac/, b6c are used as initial approximations to these stars' spherical coordinates. For the establishment of the condition equations, it is important to fix the order of the equations and the order of the adjustment parameters. The condition equations are arranged in the following order. Plate by plate we establish the Ftype equations for each star on that plate which either occurs on at least on other plate or as a reference star in order of increasing star number. These are followed by the Gtype equations which are produced by the estimates of the spherical coordinates of the reference stars, also in the order of increasing star numbers. The order of the observations is the same as the order of the equations which they generate. We chose the order of the parameters as follows: star by star, in numerical order, corrections to the coordinates, i.e. cos 6Sdca, and d6, with p = 1, 2,... m are followed by the frame parameters al, ..., an. The equations of condition are identical to those in the single plate solution. As before the total vector of conditions equations H=0 consists of "plate equa tions", F=0 and "catalogue equations", G=0. First let's discuss the F=0 equations. Plate by plate we establish the Ftype equations for each star on that plate which either occurs on at least one other plate or as a reference star. The observations are xpand y,,, the measured coordinates of the image of the /th star on the Vth plate. Exactly one observation occurs in each equation of condition, as before. 73 The frame equations for the v plate are F, = (v v a,v. (109) There are my such equations for each plate, for a total of 2my equations. (uP and rytv are the standard coordinates of the yth star with respect to the assumed tangential point of the vth plate. S, is the model matrix for the /th star on the v th plate and av is the vector of plate parameters on the vth plate. Expanding the above equation, we get x/A S4 \ y a YI V s 7 av (XIVL/ S^i.f " F. = ,vY s% a (110) ACvm  SAVmV V vav , v s ) IsnI ^ At first glance the above notation looks rather confusing. However, x,,, Y is the first star on the v th plate. Since all the stars are given a particular number designation numbered 1, 2...., m. There has to be a way to keep up with the running star number p and the star number on each plate, which we will refer to as vk. The number of stars on the vth plate run from p to pu,, These subscripts are useful in designating the total running star number and the star number on a particular plate. See the example at end of this chapter on how to assign the running star number. The entire vector F is F, F, F= = 0. (111) Each vector F; has the dimension 2mv x 1 and the dimension of the total vector is E (2m, x 1). Now on to the reference star condition equations. There are a total of mr reference stars, numbered Pri, 1r2,, ,rm,. For each reference star we have an estimated value of its position (acy, 6cp ) from the catalog. The G0 equations. look like this: (ac',. C, ,) cos S 6r + Erl ScLrl 6rL, + Elrl (ac,r2 ac2,2) cos 6 r2 + Er2 G = bcr2 6r2 + Lr2 .(112) (aC/rmr arm,r cos 6/im, + ,rm, 6Crmr 6 rMr, + E,,rmr Since only a small fraction of all stars are reference stars, it follows that Pfk and pLrk+1 of two neighboring reference stars normally do not have the same number. That means that the subscript r is just to keep track of reference stars, but just because a star has a subscript r it does not mean that it is given a dif ferent number from that of a field star. G does not consist of a separate equations for each plate, as they do for the equations F, it has a dimension of 2mr x 1. 75 As before, the total vector H consists of the plate equations F and the catalog equations G. H= =0. (113) \G/ Likewise the vector of residuals is H = (dT,0) (114) The covariance matrix is similar to that in a single plate setup, except that ox is broken down into blocks belonging to the various frames. Again we are assuming no correlation between observations, so the covariance matrix is diagonal. It consists of V v, lUpv, the variances of Xjv and ypv, and the pp ,ap which are cos261P times the variance of ap and the variance of 6y, respectively, for those Cp which belong to a reference star a = diag(o, 7x2, ,... Ox.; 7a) = (x 0 (115) The dimensions of each a,, : 2m, x 2m,, so the dimension of the total matrix n ax : E (2m, x 2m,), while the dimension of ea : 2m, x 2m,. V=1 As with the single plate each equation contains only one observation so, X () = I, where x = (xl, yl, X2, Y2, ..., Xm, Ym; alc Cos 6, 1, ...amc COS 6m, m), so XaXT = ) = x 0 (0 0a The mathematical development of an overlap solution is very similar to the single plate method discussed in the last section. The major difference is the 76 increase in the observations (all the stars), which enormously increase, the size and structure of the matrices. We will present the mathematical development similar to the single plate development, noting the changing the structure of the matrices. As before, the correction to the parameters is given by a= AT,1A 'A THo (116) where a = (3) and 3 is the vector of corrections to the spherical coordinates (a,, cos 6,6, 6b). The total vector of plate parameters for all the plates is a, which is made up of the individual vectors for each plate /a, \ a2 as \an, Our primary concern is the establishment of the matrix A f( = (OH = (F, G) ( ) (a A = =(,a) (9F ( Similar to the single plate reduction we have for each star (117) or the overlap case: S(118) .) (118) Ga ) / (119) OFpAV, O(a ( cos k.6) sB k 0, a (a',, Cos 61, k, 1,) vk) 77 where each BLk is a 2 x 2 matrix. For each plate we get OF. 03 S= o B 0 i 0) (120) 0 ... 0 B ... ... ... ... 0O 0 0 ... 0 B ... ... ... 020) where the dimension of each By matrix is (2mv x 2m). Thus each line pair contains exactly one Bp,, matrix (of dimension 2 x 2) but only in those column pairs which belong to a star whose image was measured on the vth plate. In other words, the rows correspond to the stars on plate v, while the columns are for all the stars, therefore B ,, is not the null matrix only when the pth star is on the Vth plate. The entire matrix B for all the plates is written as sB, sB, sB= (121) * \sB,, n and this matrix is of dimension E 2m, x 2m. v=l Furthermore, we have 8F, S= =, (122) Oa, where ,, is of dimension 2my x ly, ly being the number of components of the vector a on the vth plate. In E,, each star with number p that occurs on the vth 78 frame generates a 2 x l; matrix 2,,. Thus .= (123) and the total matrix E is of the form OF , (124) Oa n n and has dimension E 2mE, x Z l,. v=l v= Furthermore, 0G,, 0G_, OG, G = G OG= K. (125) d3 cos 6,,VOv, O6v,, All nonzero elements of the matrix K are 1. However, since not all stars are reference stars, K is not the identity matrix. K has the dimension of 2mrx 2m. If the pthstar is also a reference star then the row pair corresponding to this reference star and the column pair for this pth star will contain the 12 identity matrix. Since G is independent of the plate parameters, we have (as in the single plate result) 0G = 0. (126) da Combining all these results (as in the single plate reduction), the matrix A can be written as A = o (127) and has dimension 2 i m, + 1m, x 2m + E 1,. Now that we have looked at the various matrices that make up the matrix A, let's look at the structure of the normal equations. The structure of the normal equations looks the same as in the single plate solution: (AT A) = ATaol (128) However, these equations are much more complicated than in the single plate case. In detail AT sBTox1 KTOa1 B 2 A T A= a a A=1 0 \K 0 x 0 \ (129) fsBr B + KTK s2Bo .' S .T.x lB / ) ,T =Tx which can also be written as \ s.< erBn 0 n ,Bn / S L sBTKT K BTO (130) Q :B a 1 0n 'A= (1 T O1 1BT, i 1B) ... ( with L = KT rK + s2BTc,' B. Now let's take a closer look at structure of the matrices that make up equation (130). The example given later on will help to clarify these matrices, so the reader may want to refer to this example. Because ax is a diagonal matrix, 80 multiplication by this matrix leaves the structure of the matrix by which it is multiplied unchanged. We noted from the discussion above that the By (2mv x 2m) are very sparse matrices and have a structure similar to that of K. Each row pair contains exactly one nonzero 2 x 2 matrix, however not every column pair has one of these blocks, (i.e. some of the column pairs contain all zero elements). Nonzero elements occur in a column pair of By only if the corresponding star occurs on the plate corresponding to the row for that plate. Compact blocks with nonzero elements in BTv will interact with compact blocks in By, wherever the pth star's image was measured on the vth plate. The result of the interaction will result in a 2 x 2 block in the Pth row pair and column pair of L. The structure of BT aB is thus similar to that of BT, except that the vth column pair would be "replaced" by a matrix of the width Iv. The matrix KTo, aK will be a diagonal matrix, however some of the elements on the diagonal may be zero. The result of adding this matrix to s2BTc, 'B is that the diagonal blocks (pp, a ) will add to the diagonal blocks in the pth double row and column. L is thus a blockdiagonal. Since the dimensions of the matrices BTv are 2m x 2my and those of =,V are 2my x ly, the product sBf'aV1, is a sparse matrix of dimensions 2mxly, whose nonzero 2 x 2 elements sB a ,,1VS occur only on those of its line pairs whose numbers correspond to a star which occurs on the vth plate. 81 7 xL is blockdiagonal, the individual blocks being ,, the matri ces of the product sums of each plate's model matrix with the covariance matrix. Finally, consider the right hand side of the normal equations (equation 128), n Arl1H. The resulting vector has the dimension (2rn + F l,) x 1 which is a v=1 vector of the order of twice the number of stars plus the total number of plate parameters. A0TdsB T K T OW 1 0 ) \ /d A7H= =T S0 (131) (.BT) sB T = T )7x 1d Consider the 2 x I vector: s E B wildly,. Nonzero contributions to these v=1 sums are made only when images of the uth star were measured on the vth plate. Substituting equation (130) and (131) in the normal equations (128) yields ( L sBTar'1 ( lBT gS 1B T (a) = ( BT) d. (132) The first row gives L3 + sBTx la= sBTxld and the second row gives sTxlBi3 + I Txl'Sa = Scrd. As before, the solution to the normal equa tions is done in steps. First we eliminate the star parameters, 3, and solve for the plate parameters a. Solving the first equation for 0 gives : 3 = sL 1BTr (d a). (133) Substituting 3 into the second row of the normal equations yields ('1 xI s 'x1BL1B T )(d Sa) = 0. (134) (a / x Let J' = x s2a1BLBTI1 (135) Then T (J')(d Ea) = 0 TYJ'd = eTJ'Sa (136) a = (TJ' ) TJId. The inversion lemma cannot be used to simplify J' as it was used to simplify J because the 2m x 2m matrix KTr1K term in L is a diagonal matrix with nonzero elements on the diagonal only in those double blocks whose ordinal numbers correspond to reference stars, thus it is singular. Unlike the matrix J', J is not blockdiagonal. Since we have established above that L is blockdiagonal with blocks of dimension 2 x 2, the structure of J' is the same as that of BBT. Thus the resulting T J'S is "bandedbordered" (De Vegt and Ebner, 1972) and thus there are simplified routines to invert it. Now that we have determined (i.e., estimated) the plate parameters, we can use them to find the star parameters by equation (133). In terms of the individual stars n n = s ,.,,,,.L1 T1 SI = s 2 + s B T B B lT o(dr ,,a,). (137) V=1 V=1 v=l v=l The symbol bvy is of Kronecker type, it equals zero if the pth star is not a reference star, and it equals 1 for reference stars. The 2 x 1 vector B/ = (da, cos 6P, d6)T 83 is a submatrix of the previously encountered 2my x 1 vector 3/ which stands for the pairs (da cos 6, d6) of all the stars which occur on the vth frame. Thus if a star occurs on n plates, there will be n different 3's after the reduction, whereas in the case 3p, all the star parameters for one star are calculated after all the plate parameters have been calculated. This substantially amounts to taking a weighted mean for all the contributions of all frames on which images of the star were measured. Example of Overlap Reduction. Although the solution of the normal equations in the overlap case is very similar tot that of the single plate case, the interaction of the stars imaged on more than one plate complicates the reduction. It is therefore extremely useful to illustrate the method with an example. We will use a very simple example. Consider the case where we have 8 stars imaged on 2 plates, 6 stars on the first plate and 6 stars on the second. Let three of the stars be reference stars. 84 plate 1 2 * 5 3 5 4 1 6 plate 2 S : field star reference star Figure 9: Example of two overlapping plates One can see that each plate has a star in the corer to which no number has been assigned. Each of these two stars occurs on only one plate, and none is a reference star, thus in the overlap reduction they are ignored. After the reduction has been performed, the plate parameters are used to calculate the equatorial coordinates of these isolated stars. Using our notation established earlier we have m=6 n=2 m = 3 m = 5 m2 = 5 Indexing for the stars on plate I is PLV, the first star on plate 1, pi, 1 = 11 = 1.1 the second star on plate 2, p1, = 221 = 2, 1 the third star on plate 1, P 1, = 331 = 3, 1 the forth star on plate 1, i1 = 441 = 4, 1 the fifth star on plate 1, /1il1 = 551 = 5, 1 or t r, = 2 or /r2 = 4 Indexing for the stars on plate 2 is * Star 1: the first star on plate 2, P2'22 = 112 = 1,2 * Star 3 : the second star on the second plate, P222 = 322 = 3, 2 * Star 4 : the third star on the second plate, p2,2 = 432 = 4.2 P12 = 4 * Star 5 : the forth star on the second plate, 2, 2 = 542 = 5, 2 * Star 6 : the fifth star on the second plate25,2 = 652 = 6, 2 or 11r3 = 6 We have four prime goals: 1. Set up the equations of condition. 2. Use these equations to find J'. 3. Use J', to determine a. 4. Use a, to find 3. * Star 1 * Star 2 * Star 3 * Star 4 * Star 5 Set up the equations of condition The equations F have the form F ( ) = S, Ev,a, 0 (138) and the equations G become G ((CArk 6rk) COS /irk =0, (139) (bcMk rk) where k goes from 1 to mr, the total number of reference stars. The index ac r is the pth number of the k'h reference star. Therefore in our example: * Pr1 = 2 * Pr, = 4 * lrT3 = 6 It is important to fix the order of the equations. In the last section we stated that the equations were ordered plate by plate and within a plate, in order of increasing star number. However, from a programming point of view we found it simpler to order the equations star by star. In other words we group all the stars together that occur on the separate plates. This is equivalent to ordering the equations plate by plate (the row and columns of the matrices are interchanged). Therefore the equations are ordered star by star in order of increasing star number. The total vector H becomes I I , ,I a Yii Stil (xii s) \ Y12 s712 / S i _2a  Y12 srh 2 Xz2 SJ21 Y21 Sqz21 /X31 31 d 31 S731 / d12 (3'2 S<32 \ d32 Y32 S32 32a (41 41 d32 S d41 H4= S42 42a d42 (140) Y42 S42 d, X5 S51 d5 Y51 S7751 ad, (x2 s52 d52 S52 s<522 Y.52 S752 52a2 (X62 Sf62 \ , =62a2 \ 0 / \(62 S7762 ) (aC2 a2) cos 6, (bC2 62) (ac4 04) cos S4 (b4 64) (6c6 66) cos 66 (6c6 6 6) The (total) covariance matrix is a = ) where the covariance matrix of 0 0a the measurements is 0r11 0 0 0 (7,, 0 0 0 0', 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o0 00 o0 00 o0 o00 Each submatrix in x is a 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (731 0 0 0 0 0 0 o (732 0 0 0 0 0 0 0 o41 0 0 0 0 o 0 0 (742 0 0 0 0 0 0 0 (751 0 0 0 0 0 0 0 0o6, 0 0 0 0 0 0 0 62, x 2 diagonal matrix of the form a,, = P(P (0 (141) 0) a~/ n n The matrix ox then has dimensions E 2mg, x E 2m,. The covariance matrix v=l v=1 of the reference stars has the form (a, (a = 0 0 0 0 o"6 (142) where the diagonal elements are 2 x 2 block matrices of the form S(a cos2 6S a = 0 (143) so the dimension of 0a is 2mr x 2mr. By has dimension 2my x 2m, so in this example the total matrix B is a 01 = 
Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ET9Q47DJG_PCG0P6 INGEST_TIME 20120229T17:06:27Z PACKAGE AA00009035_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 