AfiALYSES OF THE LIGHT CURVES 0F TEN ECLIPSENG BINARY SYSTEMS Thesis for the Degree of Ph. D. MTCHIGAN STATE UNIVERSITY DEANNE DOROTHY PROCTOR 1971 ITTTTWTUTTIITT'TITITT'TTIITTITTTTTTI 31293 01713 6627 LIBRARY Michigan State University This is to certify that the thesis entitled ANALYSIS OF THE LIGHT CURVES OF TEN ECLIPSING BINARY SYSTEMS presented by Deanne Dorothy Proctor has been accepted towards fulfillment of the requirements for Ph. D. degreein Physics Date Sept ember 1971 0-7639 ‘— ? emome av ' T HOAG & sous :. . T BUUK amnm ma. LIBRARY amoens it a m SPRIHGPOR’. HICHIGA! q ABSTRACT ANALYSIS OF THE LIGHT CURVES OF TEN ECLIPSING BINARY SYSTEMS By Deanne Dorothy Proctor Computer programs for the Fourier analysis, rectification, and solution of eclipsing binary light curves have been written. Both Kopal's method and the method of differential corrections have been generalized to include third light. The method of differential cor- rections has been further generalized to include orbital eccentricity directly. Synthetic light curves were used to validate the computer pro- grams, as well as to determine the effect of dispersion and number of observations on the ability to extract the desired parameters. Analy- sis of synthetic data indicated limb-darkening coefficients may be extracted from observations of sufficient accuracy and density. This conclusion was found to hold for partial as well as completely eclips- ing systems. In addition, it has been found possible to extract values of third light. In some cases, however, correlation between parameters, combined with observations of insufficient quality or quantity, may prevent convergence. The data from 10 eclipsing binary systems have been rectified and subsequently analyzed using differential corrections. The systems are C0 Lacertae, CM Lacertae, RX Arietis, V338 Herculis, Y Leonis, RW Mono- cerotis, BR Cygni, BV 430, BV 412, and SW Lyncis. It was often found necessary to solve the light curves for each combination of assumptions as to type of primary minimum (occultation Deanne Dorothy Proctor or transit) and possible presence of third light. Calculation and com- parison of o(est.) and o(ca1.), the estimated and calculated standard deviations, proved valuable in the determination of convergence. Equality of the standard deviation of the Fourier analysis and the standard deviation of the entire light curve, to within their probable errors, indicated the adequacy of the fit for each curve. For those light curves for which b was varied, choice of b, the exponent of the light in the weight, did not seem to cause significant change in the parameters obtained. Convergence of the iterative procedure was obtained for the sys- tems C0 Lacertae, CM Lacertae, RX Arietis, and Y Leonis. Convergence for the V curves of BV 412 and BV 430 was also satisfactory. However, convergence of the B curves of these two systems was obtained only if the number of variables included in the differential corrections was limited to six. V338 Herculis and RW Monocerotis exhibited satisfactory convergence; however, the standard deviations of the Fourier analyses for these light curves were not in good agreement with the respective standard deviations obtained from the differential corrections analysis. The V light curves of BR Cygni and SW Lyncis exhibited satisfactory convergence. The B light curves did not. Further observation of V338 Herculis, RW Monocerotis, BR Cygni, and SW Lyncis is recommended. 0f the ten systems studied, two (C0 Lacertae and BR Cygni) showed evidence of third light. Limb-darkening coefficients resulting from the analyses are com- pared to the theoretical values. Results for limb-darkening coef- ficients in V show satisfactory agreement with theory, while limb- darkening coefficients in 8 show more scatter. ANALYSIS OF THE LIGHT CURVES OF TEN ECLIPSING BINARY SYSTEMS BY Deanne Dorothy Proctor A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics ‘1’ ‘fi f f? ‘ ”Z; ’7 In memory of my father Charles Albert Blake ii ACKNOWLEDGMENTS I wish to express my deepest gratitude to Professor A. P. Linnell for suggesting this problem and for his guidance and many helpful sug- gestions in the preparation of this thesis. I would also like to acknowledge the financial support of the National Science Foundation and Michigan State University throughout my graduate work. Special thanks go to my husband for his encouragement and support throughout our graduate studies. iii TABLE OF CONTENTS LIST OF TABLES. LIST OF FIGURES Chapter I. II. III. INTRODUCTION . A. Historical Aspects. B. Discussion of Model . C. Review of Notation and Units D. Statement of Problem . E. Previous Work F. Purpose METHOD . A. Theoretical Light Intensities . . l. Oblate star, uniform brightness . 2. Oblate star, complete darkening . 3 Oblate star, intermediate darkening. 4 Oblate star, gravity darkening 5. Reflection . . 6. Combined effects . B. Rectification . l Rectification for reflection . 2 Rectification for oblateness . 3 Light rectification formulas . 4. Phase rectification . C. Fourier Analysis . D. Rectification Procedure . . B. Effect of Third Light on Rectification . F. Differential Corrections. . . G. KOpal's Method . . . . DESCRIPTION OF COMPUTER PROGRAMS U100“) Fourier Analysis Program. Rectification Program. . . Differential Corrections Program . Kopal's Method Program . . . Program Accuracy iv Page vi . viii H 17 17 18 20 20 21 22 24 25 25 27 28 30 36 38 40 43 52 55 SS 59 61 66 67 IV. SOLUTION OF SYNTHETIC LIGHT CURVES V. ANALYSIS OF PUBLISHED DATA . A. CO Lacertae. B. CM Lacertae. C. RX Arietis . D. V338 Herculis . E. Y Leonis. . F. RW Monocerotis. G. BR Cygni. H. BV 430 I. BV 412 J. SW Lyncis VI. SUMMARY AND CONCLUSIONS . APPENDICES A. DISTRIBUTION OF APPARENT BRIGHTNESS 0N STELLAR DISK. B. VON ZEIPEL'S THEOREM . C. LUMINOUS EFFICIENCY CALCULATIONS . D. THE FUNCTIONS xf°° AND xftr. E. PARTIAL DERIVATIVES OF xf°° AND xf”. F. ERRORS IN THE PUBLISHED DATA BIBLIOGRAPHY . Page 69 79 84 89 94 97 . 101 . 104 . 108 . 112 . 116 . 120 . 124 . 129 . 133 . 138 . 146 . 148 . 156 . 158 LIST OF TABLES Table Page 1. Differential Correction Terms. . . . . . . . . . 48 2. Iterative Solution of Synthetic S Cancri (b=1) . . . . 72 3. Iterative Solution of Synthetic Light Curve (b=0) . . . 73 4. Iterative Solution of Synthetic BR Cygni (b=%) . . . . 74 S. Iterative Solution of Synthetic BV 412 (0:0.0) . . . . 7S 6. Iterative Solution of Synthetic BV 412 (o=0.074,b=%) . . 76 7. Iterative Solution of Synthetic BV 412 (o=0.074,b=%) . . 76 8. Iterative Solution of Synthetic Eccentric Light Curve (e=.15,w=45.0°) for i, e, w, and to. . . . . . . . 77 9. Iterative Solution of Synthetic Eccentric Light Curve (e=.15,w=45.0°) . . . . . . . . . . . . . . 78 10. Iterative Solution of Synthetic Eccentric Light Curve (e=.15,w=45.0°) excluding Lg . . . . . . . . . . 78 11. Parameters of CO Lacertae . . . . . . . . . . . 87 12. Parameters of CM Lacertae . . . . . . . . . . . 91 13. Parameters of RX Arietis . . . . . . . . . . . 95 14. Parameters of V338 Herculis . . . . . . . . . . 99 15. Parameters of Y Leonis . . . . . . . . . . . . 102 16. Parameters of RW Monocerotis . . . . . . . . . . 106 17. Parameters of BR Cygni . . . . . . . . . . . . 110 18. Parameters of BV 430. . . . . . . . . . . . . 114 19. Parameters of BV 412. . . . . . . . . . . . . 118 20. Parameters of SW Lyncis. . . . . . . . . . . . 122 vi Table Page 21. Gravity Darkening as a Function of Limb-Darkening. . . I37 22. Errors in the Published Data. . . . . . . . . . 156 vii LIST OF FIGURES Figure Page 1. Orbital parameters of the true ellipse in space. . . . 11 2. Orbital parameters for the true orbit . . . . . . . 11 3. Orbital geometry . . . . . . . . . . . . . . 18 4. Projection of Components . . . . . . . . . . . l8 5. Schematic light curve with influence of oblateness. . . 21 6. Schematic light curve with influence of reflection. . . 24 7. Orbital geometry for projected distance of centers. . . 31 8. Projection of components against the sky . . . . . . 31 9. Summary of transformation from tri-axial ellipsoid with reflection, gravity darkening, and limb darkening to sphere with limb darkening . . . . . . . . . . . . . 35 10. Flow chart of FOURIER . . . . . . . . . . . . 58 11. Flow chart of RRECK . . . . . . . . . . . . . 60 12. Flow chart of DIFCORT . . . . . . . . . . . . 65 13. Limb-darkening coefficients in V. . . . . . . . . 127 14. Limb-darkening coefficients in B. . . . . . . . . 128 15. Effective temperature and color temperature . . . . . 145 viii I. INTRODUCTION A. Historical Aspects The term double star is generally applied to pairs of stars that appear single to the unaided eye, but are resolvable with the aid of a telescope. The discovery of the first double star, around 1650, was made by Riccioli (Aitken 1964, p. 1). Several dozen double stars were discovered in the following century and in 1767 it was first suggested that the phenomenon was due to something other than chance projection. John Michell (1767) suggested actual physical association of the members of some double stars. This was confirmed 36 years later by William Herschel. Herschel (1803) presented results of measurements and analyses of the relative position of the components of six double stars. He concluded that certain double stars are true binary systems, that is, systems of physically associated stars. At present we have calculations for the orbits of over 500 such systems. (The catalogue of Baize (1950) contains calculated orbits for 252 visual binaries.) Stars exhibiting variation in brightness had been observed for hundreds of years when, in 1783, John Goodricke suggested an eclipsing nature for some of the variables. In that year Goodricke reported observing periodic minima in the light of Algol and suggested that the cause of the loss of light was the interposition of a body revolv- ing around Algol. Proof of the binary nature of Algol came with spectroscopic investigations of radial velocities. Dappler in 1842 presented his formula for the shift in the wavelength of light as a function of relative velocity of the source and observer. Then, Vogel (1889) observed the periodic shifting of the radial velocity of Algol and noted that the times of conjunction obtained from the radial velocity curve coincided with the minima of light. Estimates for the percentage of double or multiple stars in the vicinity of the solar system range from 30 to 50% of the total popula- tion (Kuiper 1935). No preferential orientation has been found for the inclination of the orbital planes for binary stars (Chang 1929, Finsen 1933, Huang and Wade 1966). Based on this assumption, the prob- ability' P of a system having an inclination between 11 and 12 is P - cos 11 - cos 12 (1.1) (Binnendijk 1970). Eclipsing binaries should not be uncommon. As of 1968, over 20,000 variable stars had been catalogued. Of those stars, 4062 have been classified as eclipsing binaries (Kukarkin et. a1. 1969). (The catalogue of Koch, Plavec, and Wood (1970) contains the results of the analysis of 216 eclipsing binary systems.) From observations of the light of an eclipsing binary as a function of time it is possible to extract information regarding the physical and geometrical prOperties of the system. This is done by adopting a physically reasonable model and expressing the theoretical value of the light as a function of time in terms of the related para- meters. The parameters of the model are then adjusted to obtain the best fit between the theoretical light curve and the observed light curve. The model is discussed in the next section. B. Discussion of Model The members of a binary system are distorted by tidal action and rotation. As a first approximation, each component may be regarded as a tri-axial ellipsoid (Jeans 1928, p. 225). The adopted model for the binary system consists of a pair of similar tri-axial ellipsoids. The errors introduced by the assumption of similarity will be discussed later in this section. Let a8, b8' and c8 be the axes of the larger star and as, b8 and c8 be the axes of the smaller star, each expressed in terms of the sep- aration of the components as unit of length. The axes a8 and as are taken along the line joining the centers of the components, the axes c and c are taken as the polar axes or axes of rotation, and the axes g 3 b8 and be are the remaining axes. The axes c8 and c8 are assumed par- allel to the orbital angular momentum vector. Assuming the axes of rotation constant in magnitude and direction, their projections, as viewed by the observer, are constant. It has been shown by Russell (1945) that, for a particular form of the surface brightness, the light observed for a pair of similar tri-axial ellipsoids (axes as, be, c8 and as, bs’ c3) with orbital inclination i is the same as would be observed for a pair of similar oblate spheroids (axes as, b8, b8 and as, ha, be) with inclination j, where 2 2 c c tanzj = -37 tanzi = J: tanzi . (1'2) b8 b8 The form of the apparent surface brightness assumed is n va) .. (99%1] . (1.3) where J(y) is the apparent surface brightness, y is the angle of fore- shortening (the angle between the normal to the surface of the star and the line of sight), H is the perpendicular distance from the center of the star to the tangent plane of the point under consideration, and n is unrestricted. This form of apparent surface brightness is con- sistent with the theoretical form described later. Thus there is no loss of generality in replacing the model of similar tri-axial ellip- soids with the model of similar oblate spheroids of axes a8, b8, b8 and as, bs’ b8 and orbital inclination j, where j is given by equation (1.2). The assumption of similarity (equal oblateness) of the equatorial forms is not an unreasonable first approximation. The dynamical theory of equilibrium gives (1.4) where m8 is the mass of the star of larger radius, m8 is the mass of the star of smaller radius, and Kg is a function of the variation of density with radius that does not exceed 0.02 in any well-determined case (Russell and Merrill 1952, p. 40). The quantity is is defined by E83 - a b c . (1.5) 8 8 8 Corresponding expressions hold for the smaller star. Defining the oblateness of the equatorial shape as 01 we have a m 2 i 3 65 mg is . (1.7) For main-sequence stars the mass-radius relation (Russell and Moore 1940, p. 112) gives 50 e .1 g mg '6'” ii" (1.9) 5 S and the oblateness ratio is a weak function of the mass ratio. The assumption of similarity of the polar flattening is perhaps less justified. Russell and Merrill (1952, p. 40) give _£_£_ = _£__$_ {.83 (1+2Kg)wg2 , (1.10) where mg is the ratio of angular velocity of rotation to angular veloc- ity of revolution for the larger component. A corresponding expression holds for the smaller star. Define the oblateness for the polar flat- tening as b-c n - —b— o (1011) Then, if the stars are taken to have synchronous rotation and revolu— tion (Koch, Olson, and Yoss 1965 and Olson 1968) (1.12) The assumption of similarity of shape is seen to be best for components of nearly equal mass. If the masses are reasonably similar and the radii are less than one third of the separation of centers of the components, the departure from spherical shape will not be more than a few percent. Consider the errors resulting from the assumption of similarity of shape. From equation (1.2) 3 - tan'1( E-tan i), (1.13) so _£§L_- tan i d(%) 1 + [gjztanzi ’ (1'14) Thus 1 - A. . T HE) , (1.15) 5) tan 1 where Aj is the error in j resulting from an error of A[§) in (E). Assuming the error in (g) resulting from the assumption of similarity of shape is no greater than the difference of the values of (g) for each star we have (1.16) or A(%) < Ing-nsl . (1.17) Thus for the error in the inclination 1 Aj < ———-—c 2 . Ing'nsl (5) tan 1 . (1.18) Next, considering the error in b we have, from equation (1.6), that b - a(l-s) , (1.19) so Ab - aAs , (1.20) where Ab is the error in b resulting from an error As in the equatorial oblateness. Again assuming that the error in the oblateness caused by assuming similarity of shape is less than the difference between the values of oblateness for each star, we have Ab < slag-68' , (1.21) or 0m m1... Ab < as 1 (1.22) As an example consider a system with components ag - .25, a8 - .20, and i . 76'. From equation (1.4) and equation (1.10) values of the oblateness for this system are as - es - .017, n - .014, ns - .010. 8 Then from equations (1.8), (1.19), and (1.22) we have Aj < .06' and Ab < .0002. These errors in the inclination and radii are to be compared with the corresponding errors resulting from observational error in the light. For a synthetic light curve of approximately 800 points and a standard deviation in the light values of $2, the standard devi- ation in the inclination is approximately 0.1° and the standard deviation in the radii is approximately .0006 (Linnell and Proctor 1970a). Light curves discussed in Chapter V of this work typically have 400 obser- vations and standard deviations in the light of 22. The errors in the inclination and radii due to observational errors are correspondingly greater for these curves. Typically the standard deviation in the inclination is 0.2' and the standard deviation in the radii is .004. Comparing these values with the values Aj < .06° and b < .0002, we see that errors in assuming similarity of shape are less than the errors resulting from observational scatter in the light. C. Review of Notation and Units Based on the discussion of the previous section, we substitute for the similar tri-axial ellipsoids with inclination i, the mathe- matically equivalent prolate spheroids with inclination j. The latter form is called the Russell Model. Reflection from each star will also be included. Providing for the possibility of excess or uneclipsed third light, light curves based on this model may then be considered a function of fourteen parameters and the time. These parameters are as follows: rg - semi-major axis, larger star r - semi-major axis, smaller star j - inclination of plane of orbit of equivalent oblate spheroids L - light of larger star L - light of smaller star L3 - excess uneclipsed light, third light x - limb-darkening coefficient, larger star 8 x8 - limb-darkening coefficient, smaller star to - time of minimum projected distance of centers during primary minimum e - eccentricity of orbit w - longitude of periastron e - oblateness of ellipsoids S S - parameters related to the light reflected from the cooler and hotter stars. We note that for orbits of small eccentricity (e s .02) the change in the oblateness due to the variation in distance between the com- ponents is at most of the same order of magnitude as the error occuring due to the assumption of similarity of the components. We thus take the unit of length to be constant and equal to so, the semi-major axis of the orbit. The unit of light intensity, U, is defined initially as U . Lg+Ls+L3 (1.23) It is customary to normalize the light such that U - 1, so u n Lg +Ls 1 . (1.24) 10 where L n - L /(U-L ) (1.25) s s 3 and L8“ - Ls/(U-L3) . (1.26) Third light may arise from an unresolved third companion, from gas streams, or from gas shells in the system. Alternatively, the source of excess light may not be physically associated with the binary system. A field star may be of such small angular displacement from the system that its light cannot be eliminated from the measurements. Koch (1970) gives a discussion of sources of third light. Diagrams of the orbital parameters are given in Figure l and Figure 2. The angle 0 is the position angle of the nodal point between 0' and 180°. It cannot be determined from the light curve. Note that for the orbital parameters the convention used is that of spectroscopic notation. The primary is moving about the secondary and w is measured from the ascending node, the node at which the star is moving away from the observer (Aitken 1964, p. 154). Observations do not give the sign of the inclination and therefore do not tell the quadrant. The angle 6 is the phase angle measured in the plane of the orbit from the time of conjunction (primary minimum). The angle 0, called the true anomaly, is measured from periastron in the plane of the orbit and in the direc- tion of motion of the primary. Thus we have the relation 0 e - 0+w-90 . (1.27) The limb-darkening coefficients are parameters in an expression giving the distribution of brightness over the apparent projected 11 Observer Al Plane of true orbit Periastron Figure l. Orbital parameters of the true ellipse in space. I Plane through primary I perpendicular to line e : of nodes I ‘ Primary ”0831 Periastron p01nt Secondary ’ line of nodes (11 Figure 2. Orbital parameters for the true orbit. 12 stellar disks. This variation is due to both the finite optical depths of the atmospheres and the variation of temperature with depth in the atmosphere. Thus the apparent surface brightness depends on the angle of foreshortening. For a given wavelength the adapted form of the expression for apparent surface brightness J(y) is J(y) I J(0)(l - x + x cos y) , (1.28) where y is the angle of foreshortening, J(0) is the surface intensity at the center of the projected disk, and x is the limb-darkening coef- ficient. The values of x are restricted such that - 1 s x 5 1 . (1.29) The expression for apparent surface brightness is an approximation linear in the limb-darkening coefficient. Comparison of the first order theoretical values of x (Munch and Chandrasekhar 1949) with those produced by the third order theory of Kopal (1959 p. 160) indicate that over the wavelengths covered by the UBV system, the maximum dis- crepancy in the values of x is about 0.03. This error is comparable with the probable error in x resulting from observational dispersion in a light curve of 800 points and observational scatter of £2 (Linnell and Proctor 1970a). However, for the observed light curves treated in Chapter V of this work the probable errors in x are typically three to four times as great. Until more accurate curves containing greater numbers of observations are available, use of equation (1.28) is an adequate approximation. In general, x is expected to be a function of the wavelength of observation, atmospheric absorption coefficients, and the effective I” (a) in: If Vii ref Fri Ia; 13 temperature of the star (Kopal 1959, p. 159). (A derivation of the limb-darkening law, equation (1.28), is given in Appendix A.) A com- pletely theoretical determination of x involves knowledge of the chemical and physical processes occuring in the atmosphere. Model stellar atmospheres and their effective linear limb-darkening coef- ficients are discussed by Grygar (1965), Gingerich (1966), Margrave (1969), and Parsons (1971). These theoretical values of limb darken- ing can be compared with observed values. D. Statement of Problem The continual change in the size of the apparent projected area of the spheroidal stars as a function of phase angle results in a variation of the light received by an observer. The amount of light reflected from each star in the direction of the observer also changes as a function of phase angle. Schematic light curves showing the effects of oblateness and reflection are shown in Figure 5 and Figure 6. Estimates of the oblateness and ratio of reflected lights can be obtained from information resulting from Fourier analysis of the non- eclipse variation combined with knowledge of the spectral type of the primary and the ratio of the depths of eclipse. The oblateness and ratio of reflected lights are used to transform the curve from that of similar spheroids to that of certain equivalent spheres, in a manner to be described later. This transformation process is called recti- fication. The underlying reason for rectification is that this trans- formation eliminates the necessity to tabulate or calculate special functions for every value of oblateness and ratio of reflected light. Each light curve can be transformed to its equivalent Spherical Model 14 light curve and functions for the Spherical Model can then be used. We also note that the Spherical Model is the limiting case of zero "interaction" between the components. An initial value of to is generally available in the literature, along with P, the period of the orbit. Observations of minima over several years allow very accurate period determinations. The geometric parameters, lights of each star, and limb-darkening coefficients remain to be determined. Various methods have been devised for the analysis of the Spherical Model light curve, the method com- monly employed being the graphical one of Russell and Merrill (1952). The method of Russell and Merrill was initially designed to provide preliminary estimates of the parameters (Russell and Merrill 1952, p. 27). Subsequent modifications of the method can be applied to pro- duce parameters of greater weight (Russell and Merrill 1952 p. 58). Certain specifically chosen points are taken from a free-hand curve drawn through the observations. Thus each observation is not, in general, given equal weight in the solution. However, for visual, photographic, and photometric observations with probable errors of a single observation commonly 42, this method produces parameters that satisfactorily fit the light curves. Values of limb-darkening coef- ficients are assumed in this method of solution. For modern photo~ electric light curves, probable errors of a single observation of 82 are not uncommon. It is likely that an analytic method of solution can extract more information from the data. In particular, limb- darkening coefficients and probable errors of the parameters are desired. Solution by computer is desirable to handle the large amounts of data used in the more rigorous methods. 15 E. Previous Work Some of the earliest attempts at utilization of a computer includ- ed those of Hamid, Buffer and Kapal (1951) on RZ Cassiopeiae and Buffer and Collins (1962) on S Cancri and AR Cassiopeiae. Kopal's Second Method (1959) was used. A maximum of six parameters were determined. These were r8, r8, 1, A (depth of eclipse), x (limb-darkening coef- ficient of eclipsed star), and U (unit of light). In the Huffer and Collins analysis of S Cancri and AR Cassiopeiae corrections to the limb-darkening coefficient and its probable error were suspiciously small. Jurkevich (1964) and West (1965) also programmed Kopal's Second Method. Neither included corrections to the depths of eclipse, unit of light, or limb-darkening coefficients, although West did examine variance as a function of successive values of limb-darkening. Tabachnik (1969) programmed Kapal's First Method (1959, p. 319). Tabachnik minimizes a variance which is a function of x, the appro- priate limb-darkening coefficient, and k, the ratio of radii. In principle the method allows for corrections to the depths of eclipse and unit of light, but, in the published results not all of the variables were included simultaneously. Wilson (1969) presented an ingenious method for finding limb- darkening coefficients by enforcing a condition between the coef- ficients for the larger and smaller components at corresponding phases (9,6+180‘). This method though is rather restricted. It requires completely eclipsing systems, small, well-known eccentricity and absence of third light. One of the eclipses must be represented piecewise by an analytic series. A computer is useful for the method. 16 Kitamura (1965) developed a procedure involving the Fourier trans- form of the light curve. It provides uniform treatment of partial, total, and annular eclipses. The method also determines whether the rectification process is satisfactory and whether the rectified light curve is acceptably represented by the eclipse effect alone. Kitamura ultimately resorts to the method of differential corrections for his final analysis. F. Purpose The purpose of this study is to extend and. apply the equations involved in the differential corrections method and Kopal's Second Method. Both approaches are generalized to allow for the possibility of excess or uneclipsed third light. In the case of differential corrections, the effects of orbital eccentricity are explicitly in- cluded. Previously, light curves were analyzed for "fictitious" circular elements. These fictitious elements could be transformed to the true elements if values of the eccentricity and longitude of periastron were available. The results were good to second or third order in e, the orbital eccentricity (Kopal 1950, p. 106ff). The requisite equations for rectification and analysis are de- scribed in Chapter 11. Chapter III provides a brief description of the programs and Chapter IV describes the validation of the programs using synthetic light curves. Finally, Chapter V contains the results of analysis of published observations for 10 eclipsing binary systems. The systems considered are relatively well-separated systems, thus the relations discussed in Part B of this chapter should provide good first approxflmations. II. METHOD It is necessary to calculate the theoretical light intensity seen by a distant observer for a spheroidal star as a function of phase angle 6. The effects of limb darkening, gravity darkening, and reflection will first be considered individually in Section A. In Section B of this chapter the rectification equations are discussed. Sections A and B thus relate to the formal properties of the Russell Model. Details of the analysis procedure begin in Section C. A. Theoretical Light Intensities The form of the limb-darkening law of apparent surface brightness allows calculation of xi, the light from a limb-darkened star, as a linear combination of 2”, the light of a uniformly bright star (x=0.0), and in, the light of a completely limb-darkened star (x=1.0). The apparent surface brightness J(y) at y is ch) = J(O)(l - x +'x cos y) , (2.1) where y is the angle between the normal to the surface of the star and the line of sight and J(0) is the central surface brightness. Thus x2 = I J(y) cos y do , (2.2) where do is the surface element, facing the observer, at angle y. So J(O)(l-x) ] cos y do + J(0) x [ coszy do (2.3a) D (2.3b) 2o II (1-x) in + x 2 where 17 18 10 ll J(0) f cos y do (2.4) and 2D J(O) f coszy do . (2.5) The derivations below follow the presentation of Binnendijk (1960, p. 290ff), with minor changes in notation. 1. Oblate star, uniform brightness For the uniformly bright star we may calculate t” by multiplying the surface brightness J(0) by the apparent projected area AP of the star. Thus z-o II J(0) f cos y do (2.6a) J(0) Ap . (2.61)) The orbital geometry is given in Figure 3. (In Figure 3 the plane containing the angle (90-j) is perpendicular to the line of nodes.) The ellipsoid has axes a, b, and b, with major axis along the line of centers. The projected ellipse has axes d and b. Figure 4 is a view of the components in the plane of the orbit. Primary . Primary Orbit A ¢ d Secondary 4 Secondary SQ Observer Observer Figure 3. Orbital geometry. Figure 4. Projection of components. .I ‘H- 19 The value of d is found by requiring that the equation for the inter- section of the line of sight and the ellipse have a single root. This gives a2 sinzo + b2 coszo (2.7a) a2 (l - ee2 coszo) . (2.7b) Q. ll Here ee is the eccentricity of the ellipsoid. From the cosine rule of spherical trigonometry cos ¢ = sin j cos 9 . (2.8) Thus we have AP = ndb (2.9a) = nab[l - ee2 sinzj c0526)l5 (2.9b) .2 . 2. 2 = nab(1 - gee SIn 3 cos 6) (2.9c) to first order in e62. For small ee we have for the oblateness s e = (a - b)/a = l - (l - eez)l5 (2.10a) (2.10b) Substituting equation (2.9c) in equation (2.6b) and using equation (2.10b) we have £U= J(0)rab(l-e sinzj c0526) (2.11s) = £U(90)[l-e sinzj c0526) . (2.11b) where £U(90), the light at quadrature, is 2U(9o) = J(0)nab . (2.12) c- Us ~‘.' 20 2. Oblate star, complete darkening Evaluation of the integral in equation (2.5) to first order in 5 results in D 2 1 3 . 2. 2 2 = 3-(1 + §-5)rabJ(O)(l - §-€ s1n 3 cos 8) (2-133) 2 1D(9o)(1 - g-e sinzj c0526) (2.13b) where D 2 l 2 (90) = 3-(1 + g-e)wabJ(O) (2.14) (Binnendijk 1960, p. 301). 3. Oblate star, intermediate darkening For intermediate limb darkening of an oblate star (1 - x)tU + xiD (2.15s) 10 II I2 (1 - x)gU(90)(1 - e sinzj c0526) +x20(90)(1 - g-s sinzj c0526) , (2.15b) 50 X x 2 = l - f(x) 6 sinzj c0526 , (2.16) 1(90) where (1-x)2”(90) + % x20(90) f(x) = u D (2.17) (l-x)£ (90) + xi (90) and x1(90) = (1 - x)1”(9o) + xtD(9o) . (2.18) 21 With equation (2.12) and equation (2.14) D 2 (90) = 3 (1 + .1. a) (2.19) U 3 5 , 2 (90) and to first order in 6 x2 15+ 2 2 x = l - --—--’-(—s sin j cos 6 (2.20) g(90) lS-Sx This variation is shown schematically in Figure 5. 270° 0° 90° 180° 2700 Figure 5. Schematic light curve with influence of oblateness. 4. Oblate star, gravity darkening Von Zeipel (1924) first demonstrated the prOportionality of the emergent flux and local gravity at a point on the surface of a star. This relation has also been derived by Chandrasekhar (1933). Let H be the intensity of the total radiation emergent normally from the atmosphere and let 3 be the local surface gravity. Then 22 Ill: =§—=1-(1-3—) o 0 go , (2.21) where go is the mean surface gravity and H0 is the correSponding inten- sity. Proof of Equation (2.21) along with the assumptions involved is given in Appendix B. Kopal (1959, p. 172) has shown that, assuming stars radiate like black bodies, the surface brightness at wavelength A will be 3'; u H l ‘< ”,3 I If (2.22) O 00 O U where y, the gravity darkening coefficient, is a function of wavelength and effective temperature. Integration of equation (2.22) over the surface of the star gives the light variation associated with local gravity, . 2. 2 y1(90) - l - (l + y) s 51n J cos 6 . (2.23) Again y11.(90) is the light at quadrature. 5. Reflection Let Lh and Lc be the intrinsic luminosities of the hotter and cooler stars re5pectively. Let 2 Sh be the total amount of light reflected from the hotter star and let 2 Sc be the total amount of light reflected from the cooler star. If stars reflected light like mirrors the light outside of eclipse would be 2 = Lh + Sh(l + cos e) + Lc + Sc(l - cos ¢) , (2.24) Where 9 is the angle between the line of sight and the line of centers 23 of the stars in space. The difference in sign for the cosine terms is a result of the difference in phase of r between the components. Thus we may write 1 = Lh + 2.5 Sh f(¢) + Lc + 2.5 Sc f(¢+n) , (2.25) where for "mirror-like" stars 2 f(¢) = g-(l + cos ¢) . (2.26) The normalization of the phase function, equation (2.26), is chosen so that the form for "mirror-like" stars may be compared to the form generally adOpted. Rigorous calculation of f(¢) for more physically realistic models of reflection is extremely complicated. The form generally adOpted is f(¢) = 0.2 + 0.4 cos o + 0.2 coszo , (2.27) where cos ¢ = sin j cos 6 . (2.28) (See Russell and Merrill 1952, p. 44.) Then t=(Lc.Lh)+%—(sc+sh) - (Sc-sh)cos¢ + é-(sc + sh) coszo . (2.29) This variation is shown schematically in Figure 6. 24 270 0 90 180 270 Figure 6. Schematic light curve with influence of reflection. 6. Combined effects The combined effects of reflection, oblateness, limb-darkening and gravity darkening produce for the light received from both stars outside eclipse, . 2. 2 1 L - 2(90)(1 - N e s1n 3 cos 6) + 2'(Sc + Sh) ' O O 1 O 2 O 2 - (Sc - Sh) 51n 3 cos 8 + -2--(SC + Sh) s1n J cos 8 , (2.30) where 15 + _ X N — ————15 _ 5X (1 + y) , (2.31) and 2(90) is the sum of two terms of the form of equation (2.18), one fer each star. It has been assumed for the purpose of rectification that thelimb-darkening and gravity darkening are the same for both stars . 25 B. Rectification The result desired from rectification is the elimination of the variation in the light curve due to reflection and oblateness in such a manner as to retain the physical meaning associated with the parameters of the model. If £(obs.) is the observed light value at phase angle 6 and 2 is the corresponding theoretical light value, then we may write £(obs.) = i + 62 (2.32) where 62 is the associated observational error. For points outside eclipse 2 is given by equation (2.30), so that to first order in e . 2. 2 1 £(obs.) = 2(90)[l - N e $1n 3 cos 8] + 2'(Sc + Sh) - (SC - Sh) $1n 3 cos 6 1 . 2. 2 + 2 (Sc + Sh) $1n 3 cos 8 + 62 . (2-33) 1. Rectification for reflection From equation (2.29) and equation (2.33) it is seen that the light from the cooler and hotter stars may be symmetrized by the addition of the quantities of light ARC and A2 where h, At = 1-S + S sin j cos 8 + 1-S sinzj c0526 (2 34a) c 2 c c 2 c ' and A2 = 2-S - S sin j cos 8 + 1-S sinzj c0526 (2 34b) h 2 h h 2 h ’ ' Thus if 2(obs.) is the observed light at phase angle 6,then firp’ the observed light partially rectified for reflection,may be defined as 26 __ 1 . . zrp - £(obs.) + 2'(Sc+sh) + (Sc-Sh) $1n 3 cos 6 1 . 2. 2 + §.(Sc+sh) $1n 3 cos 6 . (2.35) In effect, this partial rectification adds sufficient luminosity to the outer faces to bring them to equality with the illuminated sides. It thus provides completely illuminated stars at all phase angles. While the above rectification is exact for the non-eclipse portion of the light curve, no sensible error occurs by continuing its application right through eclipse (Russell and Merrill 1952, p. 48). Note that zrp still varies as a function of phase angle. This is due to the "non- mirror like" quality of the reflection. We can eliminate this variation and complete the rectification for reflection by division as follows: 2 2 = m . 2. 2 (Lg+Ls)+(Sc+Sh)+(Sc+Sh) $1n 3 cos 6 1 . . 1 . 2. 2 2(obs.)+2(Sc+8h)+(Sc~Sh)51n 3 cos 8 +§(Sc+Sh)s1n 3 cos 6 = (2.36) O 2 O 2 (Lg+LS)+(Sc+Sh)+(Sc+Sh)51n 3 cos 6 , where we have used the substitution 2(90) = Lg + Ls . (2.37) The denominator of the right-hand side of equation (2.36) has been determined by the combination of equation (2.33) and equation (2.35), 27 2. Rectification for oblateness To eliminate the non-eclipse variation due to oblateness, the light at 6 is divided by the appr0priate value of (l - N e sinzj c0526). The quantity N e sinzj is known as the photometric ellipticity. Thus if 2 and A c r h r are the lights of the cooler and hotter stars rectified 3 a for reflection and we define i c r 2 = ’ (2.38) c,rr l - N s sinzj c0526 ’ c c L h r g = ’ (2.39) h,rr . 21 2 1 - Nheh Sin 3 cos 6 a the total rectified light grr is 2rr = 2c,rr + 2h,rr (2'40) 2 A = C’rz 2 + h” 2 2 (2.41) 1 - Neec sin j cos 8 1 - Nheh sin j cos 6 zc r + 2h r .. 3 9 - -—- . 2. 2 (2'42) 1 - Ne s1n 3 cos 8 2r = 2.4 -—— . 2. 2 , ( 3) l - Ne s1n 3 cos 6 where ___ 2 N s + i N 6 N6 = CJr C C h,r h h (2.44) A + A 28 If it is assumed that s = e = e and N = N = N, then c h c h TE = Na . (2.45) 3. Light rectification formulas Combining equation (2.36) and equation (2.43), we have for the light rectification formula 1 . . l . 2. 2 2(obs.)+§(Sc+Sh)+(Sc-Sh)s1n J cos 6 +§(SC+Sh)51n 3 cos 6 (Sc+sh)sin?j c0526) Lg+LS+SC+Sh ] rr (Lg+Ls+Sc+Sh)(l-Ne sinzj cosze)[1 + (2.46) We note that, excluding observational error, if £(obs.) is the light of spheroidal stars with reflection and gravity darkening, then the rectified light is that which would be observed for spherical stars with Russell Model parameters rg, rs, xg, x5, Lg, L5’ and j. The variation of the non-eclipse portion of the light curve has been eliminated. (See Russell 1946, 1948.) To first order in small quantities (Sc’ Sh’ e) the order of the rectification for reflection and oblateness is immaterial. It is necessary to obtain an expression for the rectified light in terms of empirically determinable quantities. Define 1 DO - 2'(Sc + Sh) , (2.47) D1 = - (Sc - Sh) sin j , (2.48) _ 1 . 2. D2 - i-(Sc + Sh) 51n j . (2.49) From the theoretical expression for the observed light outside eclipse, equation (2.30), using equations (2.47), (2.48), and (2.49) we have the relation i = £(90)[1 - Ne sinzj cos2 29 2 6) + D0 + D1 cos 8 + D2 cos 6 A Fourier analysis of the non-eclipse variation produces where We then have cos 9 + A cos 29 2 cos 6 + A, c0526 , 2 = A0 ' A2 ’ = 2 A2 D1 = A1 , = - l_ S +Sh A1 2 S -S sin ' ’ h J = D0 Sin 3 (2.50) (2.51s) (2.515) (2.52) (2.53) (2.54) (2.55) (2.56) Thus we can empirically determine 0 D and D if we know Sc/Sh' The procedure for estimating SC/Sh is discussed in Appendix C. We have that l . . 5-(Sc+8h) +(Sc-Sh)51n J cos 8 0’ 1’ 2 1 . 2. 2 + E-(SC+Sh)51n 3 cos 6 2 D0 - 01 cos 8 + D2 cos 9 (2.57) 11d 1" -& ,1 4 CT 30 and (Sc+Sh)sin2j c0528 (SC+S h+Lg+Ls)(l - Ne sinzj c0526) 1 + L +L +8 +8 g s c h + 2 D c0529 (2.588) = 2(90)(1 - Ne sinzj cosze) + 2 D0 2 = (Ad’+ D0) + (A;'+ D2) c0526 , (2.58b) where we have neglected quantities of second order in Sc’ Sh’ and e, used equations (2.50) and (2.51b), and equation (2.37). Thus, substituting from equations(2.57) and (2.58b), equation (2.46) becomes £(obs.) + DO - D1 cos 9 + D2 c0526 = (2.59) (A6 + D0) + (A; + D2) c0526 rr We have in equation (2.59) a formula for the rectified light in terms of empirically determinable quantities. 4. Phase rectification It is possible to express the geometrical dependence of the theoretical value of light as a function of two dimensionless variables (Kopal 1946, p. 24ff). These variables are normally taken to be the ratio of radii k, where rs k = '1':— , (2.60) 8 and the geometrical depth of eclipse p, where 6 - d 9 =74 (2.61) 31 In the above formula <18 and d5 are the apparent projected axes along the projected line of centers and 6 is the projected distance of cen- ters. From Figure 7 and Figure 8 it can be seen that 62 = 6 2 + 6 2 (2.62) R2(sin26 sinzj + coszj) . (2.63) I I6 I l 2 _-..,--_.-1 Observer Figure 7. Orbital geometry for Figure 8. Projection of com- projected distance of centers. ponents against the sky. Here R is the separation of the centers of the components ao(l - e2) R - (2.64) 1 + e cos(6-w+90) and so is the semi-major axis of the components (a0 = 1). External contact of the apparent projected ellipsoids occurs at p - +1; internal contact occurs at p I -1. An eclipse is called total or complete if the minimum value (pmin) of the geometrical depth during eclipse satisfies 32 pmin s -1 - (2.65) An eclipse is called partial if 1 _ pmin s l . (2.66) For 1 < pmin (2.67) no eclipse occurs. We may fit the rectified light with the equation arr(e) = 2(k.p(e)) . (2.68) where 2(k,p(e)) is the theoretical light for spherical stars with Russell Model parameters rg, rs, L8, L , xg, x5, and j at phase angle 6 and 5 geometrical depth 8 - d . 9(6) = —-—-—ii . (2.69) d s The radii project in the same ratio so that .rs ds k =T=arr (2.70) 8 8 Using equation (2.63) and equation (2.70) sinze sinzi + coszj R 1.-zcos 6 1 9(9) = - E’ (2.71) 1‘ a s where z = 2 e sinzj = e 2 sinzj . (2.72) 33 However, with the substitutions . 2 sinzer = 51“ 9 2 (2.73) 1 - z cos 9 ’ 2 sinz' - z sin ir = l- (2.74) 1 - z ’ 2 c052' cos ir = -———4L (2.75) 1 - z ’ we may write the geometrical depth as RJ sinzer sinzir + coszir -r pcer) = EL- (2.76) r . s We note that this is the value of p that would be obtained for a pair of Spherical stars with paremeters rg, r , L , L , x , x , and ir at Dr. 8 5 8 5 Thus we observe that as an alternative to equation (2.68) we may fit S the rectified light with the equation zrrcer) = 2(k.p(er)) (2.77) where i(k,p(6r)) is the theoretical light for spherical stars with Spherical Model parameters rg, r , L , L x . s g s’ g’ xs, and 1r at phase angle or. This transformation procedure will be followed. A summary of the transformations from the tri-axial model to the spherical model is given in Figure 9. 34 Figure 9. Summary of transformation from tri-axial ellipsoid with reflection, gravity darkening and limb darkening to sphere with limb darkening. \ ‘x 71" (slur taxi-5 P ."/’A'— //I’A/Tb\\ In)! (I r ux i :5 Ia" axle-s 35 mGOMHwfl—HOM mafia. .nueomowm Hafium mewcoxumo xufl>ouw opoev oo>oaew woweoxwmo xuw>onm one :ofiuoofimou .mmoeoumfloo Hofinouosco on use sowuowhm> wefleoxnmo xufi>oaw a coauoeflaoefl m sowuoewfloefi x meweotho ost x wefieoxumv mafia meficoxnov xufi>mum o: .H noun Hcowuoemm vwonoemm confine o are \ mflxo wmaom a+a mfixn Hagen .m shaman m A no» u n an» N o N N eowuoEHOMmcmwh newuoofimos seapooamoh weweoxuoo xuw>oum _.. 53822: x wcweoxsmo mafia moweoxump xuw>oum a :owumeflaomfi x weweoxwmo mafia peoponmm Dungeon ofiommwa~o aefixnnwnu mass uoaom 36 C. Fourier Analysis As a preliminary to rectification, Fourier analysis of the non- eclipse variation is necessary. Data for light curves are commonly given in magnitude differences as a fUnction of time, such that Am = m - mc = -2.5 1og(£(obs.)/£c) , (2.78) where 111C and 2c are the magnitude and light of the comparison star and m and £(obs.) are the magnitude and light of the eclipsing system (which may include excess light from an uneclipsed third source). Thus -O.4(m-mc) = e-(m-mc)/1.08S73620 9.(obs.)/iC = 10 (2.79) The method of least squares is used to determine the Fourier coefficients in the equation 1 +2 0 A A 2(Obs.) _ v 3 _ 0 _l_ i - z - 2 + 1 cos 6 + 2 cos 26 + .... c c c c c B B + Eg-sin 6 + EZ-sin 26 + ---° . (2°80) c c 3 the phase angle from conjunction at primary minimum. The Fourier Here AV is the light of the variable, 2 is the excess light and 6 is analysis is normally carried to terms of order 26. The occurence of the odd harmonics will be discussed in Section D of this chapter. For non-eccentric orbits 9=£1(t-t) (281) P o ’ ' where P is the period of the orbit and t is the time of the observation. For eccentric orbits Kepler's equation is solved. (See, for example, 37 Kopal (1946, p. 94ff).) Taking 1c as unity and assuming that harmonic terms arise only from the variable we have t = A + A cos 6 + A cos 26 + ---- v 0 l 2 + B1 sin 6 + B2 sin 26 + .... . (2.82) Thus, A0 = so - 23 . (2.83) Merrill (1970) states that conventional least squares analysis, carried through terms of 26, gives no indication as to the presence or absence of higher order harmonics in the data. He also demonstrates that failure to include the cos 36 term can vitiate the resulting estimates of the reflection effect. On the other hand, inclusion of terms in 36 and higher may diminish the weights of all the coefficients (Russell and Merrill 1952, p. 53). For these reasons, at least two least squares Fourier analyses were carried out on the non-eclipse variation of each light curve studied. The first analysis was the conventional series carried to terms of order 26. The second analysis included higher order terms (normally cos 36 and sin 36). From an examination of the resulting residuals and standard deviations it could be determined whether or not the inclusion of higher order terms resulted in a sig- nificant deviation from a normal distribution. Choice of phase ranges for the non-eclipse variation will be discussed in Chapter V. 38 D. Rectification Procedure The rectification formula used to transform the light curve to the equivalent spherical model light curve was adopted from Jurkevich (1964) and Binnendijk (1960), ns £(6)-;;18nsin n6 - 2:3Ancos n6 + DO-Alcos 6 +D2c0526 2rr(6) = (2.84) (A6 + D0) + (A; + D2) c0526 where n5 is the number of significant sine terms and nc is the number of significant cosine terms from equation (2.82). The resultant £rr(6) is the observed intensity corrected for asymmetry, higher order cosine terms,.ref1ection, and ellipticity. The higher order cosine terms and the sine terms as yet have no generally accepted theoretical justifi- cation. Their presence in the rectification formula represents an empirical correction. The various constants.in the rectification formula are calculated from the Fourier coefficients, obtained from the outside-eclipse variation, and an estimate of the ratio of the reflected lights Sc/Sh° As discussed in Section B of this chapter I AO - A0 - A2 , (2.85) A2 = 2 A2 , (2.86) (S + S ) A _ 1 _ 1 c h 1 D0 - §.(s + Sh) - - 5- (2.87) (Sc - Sh) sin j ’ D = A1 = -(Sc - sh) sin j , (2.88) _ l . 2. _ . 2. D2 - -2--(Sc + Sh) s1n j - D0 s1n j . (2.89) we thus require an estimate for Sc/Sh' For bolometric observations we may make the approximation 39 S _c_ Sh where Ih/Ic is the ratio of surface luminosities (Binnendijk 1960, Ih Depth of Primary 1 - 2r(0) a __.= = ————————- (2.90) I c Depth of Secondary l - 2r(n) ’ p. 313). For observations taken at an effective wavelength A it is necessary to include the effect of luminous efficiency when calculating Sc/Sh° Russell and Merrill (1952) provide a method of graphically evaluating Sc/Sh' For computer reduction, the required equations are given by Jurkevich (1964). The discussion presented by Jurkevich contains errors of a typographical nature. For this reason and for completeness the deve10pment is reproduced in Appendix C. As discussed in Section 8.4 of this chapter, it is necessary to rectify the phase angles to complete the transformation from the ellip- soidal model to the spherical model light curve. From Jurkevich (1964, p. 139), correcting the typographical error sin 6r = Sln 6 (2.91) J1 - z c0526 and cos 6 =J 1 - z cos 6 (2.92) r 2 l - z cos 6 , where D AI 2 ' 2 2 —7w———- A0 ' 0 z I (2.93) 15+x ISZSR (1*Y) Here the limb-darkening coefficient x and the gravity darkening coef- ficient y have been taken as the same for both stars. For the initial rectification, theoretical values for the brighter component may be used. 40 For computational purposes equations (2.91) and (2.92) may be combined to give _1 sin 6r 9r = 2 ta“ more: ”-943 In this process the inclination has been transformed as well, so that cos i = 33-1-3- (2.95) 71 - z ’ sinzj - z Sln 11‘ = r—l—:'T . (2.96) The (irr’er) data may now be analyzed according to the Spherical Model. E. Effect of Third Light on Rectification It is necessary to consider the effect of third light in the rectification of the light curve. Let ns nc . 2 R(6) - -) Bn51n n6 - E An cos n6 + D0 - Al cos 6 + D2 cos 6 (2.97) n=l n=3 and I 2 E(6) — (Ao - A2 + D0) + (A2 + D2) cos 6 . (2.98) Then 2v + R(6) 2. =9. (6) =-——-—-— (2°99) And we have 41 2. 3 z + .2 + 1 + R(6) 3 rr E(6) ___= v a l + R(6) (2.100) 3 O ———-— E6 +9. E6 +2, E(6) () 3 c) 3 Define a: 2'3 £3 = -———— (2.101) E(6) and 2. z * = 1 + —--31—-—— (2.102) C E(6) Then * * 2 = 2rr+ £3 rr 2c ns nc 2 2-23 B»Sin n6 —I: A cos n6 + D - A cos 6 + D cos 6 n=1 n n=3 n O l 2 2 10 . 2 (° 3) (so - A2 + D0) + (A2 + D2) cos 6 Thus we see that use of the rectification formula, equation (2.84) produces rectified light where, if there is excess light, the rectified * * excess light 23 and the rectified scale factor 2c are slightly variable. For light curves of the "Algol" type this variation is in general less * may be than 1% during eclipse. Thus, for a first approximation, Arr * analyzed as though 23 and 2,: were constant, say by the method of iterative differential corrections. From this analysis we obtain an 'k estimate of 23 and we may solve for £3 42 i 2'3 t = , 2 (2.104) (80 - £3 - A2 + 00) + (A2 + D2) cos 6 , 2' I t = -———£L——-— (a - A + D ) + (A + D ) c0526 (2.105) * O 2 2 2 2 . (l + £3 ) With this estimate of 13 we may use equation (2.83) for A0 and eliminate the excess light during rectification. From equation (2.93) we have D2 - A2 A: - D s = ——£L——-11— (2.106) N sinzj and it can be seen that failure to exclude the excess light results in an under-estimation of the oblateness. Note, however, that for the systems discussed in Chapter V, that unless the excess light is a major fraction of the light of the system, failure to take it into account results in an error in the oblateness of approximately the same order as that caused by observational error in the Fourier coefficients. 43 F. Differential Corrections Wyse (1939), Irwin (1947), and K0pal (1959, p. 367ff) have deve10ped the initial equations necessary to determine the differential corrections to the initial parameters that describe the eclipsing binary system in the Spherical Model. The extended equations are described below. At this point it is customary to dr0p the subscripts on trr’ ir’ and er. This practice will be followed, keeping in mind that the quantities discussed are, in fact, the rectified values. We ad0pt the terminology of KOpal (1959, p. 307). The deeper minimum will be called the primary and the shallower minimum will be called the secondary. The eclipse of the smaller star by the larger will be referred to as an occultation eclipse and the eclipse of the larger star by the smaller will be referred to as a transit. The eclipsing binary light curve is to be fitted to the equation to = 2c (2.107s) u - xf(k,p)L (2.107b) where to is the observed light value, 1c is the calculated or theoretical light value, 0 is the unit of light, xf is the fractional light loss appr0priate to the type of eclipse (occultation or transit), and L is the total light of the eclipsed star. Further discussion of the function f is given in Appendix D. Each observed point provides an equation of condition of the form X I p 2.’ I I , I I to = U - f(rg, rs, cos 1, x, e, w, to, t)L , (2.108) 44 where'rg', rg', coszi’, x’(limb-darkening coefficient of eclipsed star), e'(orbital eccentricity), w’(longitude of periastron), t6’(time of primary minimum), L’, and U"are the true parameters. Assuming an initial approximate set of parameters rg, rs, coszi, x, e, m, L, and U (U=l), we have, expanding equation (2.108) to first order in the differential corrections to the parameters x x x l = l - fo + AU-foLaL fl—g-Ar + 34i- r + -3—£—- coszi 5 cos 1 x x x x 3 f a f 3 f 8 f + 3x Ax + Do As + 58—-Aw + 3‘0 AtO , (2.109) x axfoc axftr where 5;—-Ax 1s 5;———- x5 or 5§;_-'Axg as appropriate. Deflne 2c calculated with the current estimate of the true parameters (the initial estimate on the first iteration) as x 2. Ac - 1 - f(rg,rs,cos 1,x,e,w,to,t)L (2.110) and define x x x A2(o-c) = AU - foL - L 2—£-Ar + a—£-Ar +§-£;—-Acoszi 6r g 8r 5 . g 5 aces 1 ext axf axf axf + '5;— AX+ 5-6- Ae+ 5-w— AID-l- 5?; to . (2.111) The equation of condition, for a given iteration, is then to - 2c = A£(o-c) . (2.112) Note 20 - 2c is the light residual with AC calculated from current parameters and A£(o-c) is an estimator of 20 - 2c. Thus, equation (2.112) is an attempt to account for the residuals in terms of changes in the current system parameters. 45 Writing A£(o-c) explicitly for the various types of points we have (a) for*points outside eclipse A£(o-c) I AU , (2.113) (b) for points in transit eclipse X tr X tr X tr A2.(o-c) = AU - xftrAL - L 3;:— Ar + 5—5— Ar + f Acoszl g 8 g g ars secs 1 X tr X tr X tr X tr Bxg 8 36 3m 3to o (2.114) and (c) for points of occultation eclipse x oc x oc x oc A2(o-c) = AU - xf0c AL — L a f Ar +§—£——-Ar + a f Acoszi s 5 ar g 6r 5 2. g s Boos 1 x oc x oc x oc x oc a g 8 f a f 3 f I 3x5 Axs+ 8e Ae+ 3w Aw+ 3‘0 Ato (2.115) We have the fhrther condition that Lg, LS, L3, and U are related by Lg + LS + L3 = u . (2.116) Thus 8Ls = AU - ALg - 813 , (2.117) where L3 is the possible excess light. We may now write equation (2.115) in the form 46 A£(o-c) = (l - xfoc)AU + xfocALg + xfocAL 3 x oc x oc x oc - L a-£——-Ar + §—£I—-Ar + 3 f Acoszi 5 3r g 6r 5 . g s 3cos 1 x oc x oc x oc x 0c 3 f 6 f 3 f 8 f + —_6xs Axs+ 3e Ae+ _—8w Aw-I- -——-—ato Ato . (2.118) We have used coszi as a parameter rather than i, following the recom- mendation of Irwin (1947). The evaluation of the various partial derivatives of xf0c and xftr is discussed in Appendix E. The kth equation of condition is weighted according to wk1 7;;'= -;71;— , (2.119) k 1 is the observational weight of the kth point and b = 0, 2 where w a or kI 1 according to the scale on which random errors are assumed constant. (Linnell and Proctor 1970b). Given the apparent magnitude of the system and the aperture of the telescope used for the observations, Young's Table IV (Young 1967, p. 794) may be used to estimate the most appro- priate value for b. Let S be the weighted sum of squares of residuals of the equation of condition, equation (2.112). Then N S s kzl wk((£o-2c)k - A£(o-c)k)2 . (2.120) Define (£0 - £c)k = Yk (2.121) 47 and NP A£(o-c)k = 121 81k ci (2.122) Then g [ %F 2 S = w Y — B. C. (2.123) _ k=1 k k i=1 1k 1 , where N is the total number of observed points and i is summed over the differential corrections to be included, NP in all. The various C1 and Bik appear in Table l. The dual use of i for the orbital inclination in coszi and as a subscript should cause no confusion. Application of the least squares criterion results in the matrix of normal equations AS = 9. . (2.124) where N Amj = kél wk emk ejk (2.125) and i G = w 8 Y . (2.126) m k=1 k mk k Then -1 £3. = A 9. (2.127) produces the components of the 2 vector which are the differential 48 Table 1. Differential Correction Terms. ' Occultation Transit out§1de 1 C' B 8 Eclipse 1 11‘ i15— Bile x oc x tr 3 f a f 1 - — Arg L5 81‘ Lg 8r 0 8 6 f 3 f 2 - — ATS LS 8r Lg DIS 0 3 Acoszi .L5 3 f 2 -L a f 2 0 Boos 1 g 6cos i 5 AU +1 ' xfoc +1 +1 x oc 8 f 6 AXS -Ls ax 0 O x tr 8 f 7 AX 0 _ Lg 8x 0 x oc x tr 6 f a f 8 - — Ato L5 31: Lg at 0 o 9 ALS +xfoc O 0 X 0C X tr 10 e _ 3 f _ 3 f A L5 as Lg 3e 0 X 0C X tr 11 Am -L 8 f, -L 6 f 0 £011 The the The .1; 1:11; VA. IPQ‘ the 49 correction terms. The covariance matrix SC is given by S -1 (SC)ij = W (A )1]. (2.128) (Ostle 1964, eq. 8.69 and 8.70). The simple correlation coefficient between the ith and the jthvariables is defined by the covariance between the two variables divided by the product of their standard deviations. Thus the matrix of simple correlation coefficients Sc OI‘I' is defined by the elements (SC)jj (scorr)ij = 9C SC (2.129) ‘ ii 11 The partial correlation coefficients are defined by -1 C ) = (Scorr)ij corr ij _1 _1 (2.130) JEcorr)ii (Scorr)jj where (Sgirr) is the matrix inverse of S (Smillie 1966, eq. 3.7.1). corr' The values of the simple and partial correlation coefficients are limited to values in the range [-l,+l], with values near the end points indicating higher correlation. We have for the standard deviation of the weighted light obser- vations o(est.) ="N—:§-fil3- , (2.131) where S may be calculated from the individual residuals (equation (2.123)) or, alternatively, 50 N 2 NP S = 2 YR - .2 ci Gi , (2.132) =1 i=1 a form not requiring the calculation of the individual residuals. We note that o(est.) is an expression for the standard deviation of the observations from the spherical model light curve.that is first order in the differential corrections. Let 2 - t ) , (2.133) S = 1 wk (2k cn " k 11 M2 where 2c“ is calculated with the incremented parameters (rg+Arg, rs+Ars, etc.). Then Sn 2 o(cal.) = W (2.134) Equality of o(est.) and o(cal.) is a test of convergence, indicating A£(o-c) does not contain systematic errors that can be accounted for by a significant change in the parameters. Equality of o(est.) and o(cal.) will not occur unless higher order terms in the expansion of ac are negligable compared to first order terms. The probable errors of the parameters follow from the root of the appropriate covariance matrix element. For example, the probable error of Ar is 8 % P.E. Ar = 0.6745 (SC) (2.135) g 1,1 ° We have assumed that the uncertainty of the differential correction to a parameter is equal to the uncertainty of the respective parameter in the final iteration (Piotrowski 1948). 51 The method of differential corrections offers several advantages: (1) Each observed point is given proper weight in the solution. This is not true with Russell's graphi- cal solution. (2) The same set of equations apply to partial as well as completely eclipsing systems. (3). The effects of orbital eccentricity can be included directly. 52 G. KOPAL'S METHOD Kopal (1959, p. 321ff) has deve10ped an iterative method for the solution of eclipsing binary light curves that is suitable for adaption to a computer. The equation used in fitting the light curve follows from KOpal (1959, p. 332); but, applied to all parameters the equation becomes XOC - 2 a (l-o) /w(p -l)c1+2/G(p+1)c2+/503+ ”ZE'C4* Tag—cS (2.136) (U-Ao) axaoc xatr (U-lt) axatr . 2 b ax C6+ b C7+ b ax C8 ‘ JG 51“ 9 ’ 2 s 2 2 g where the intrinsic weight of a given point is given by 6 -(u-l)(59) w = b P. (2.137) 22 C2(l+kp) The a in the C5 term is xaoc or xatr as apprOpriate to the data point. Contributionstxlthe C4 and C6 terms occur only for points in an occul- tation eclipse and contributionstxlthe C7 and C8 terms occur only for transit eclipse. The variables 10 and At are the light values at in- ternal tangency of the occultation and transit eclipse respectively. Choice of b (the weighting condition) depends on observational circum- stances (Young 1967 and Linnell and Proctor 1970b). The regression equation, equation (2.136), must also be multiplied by the observational weight of the data point under consideration. The C1 in equations (2.136) and (2.137) are related to the system parameters as follows: 53 2 2. C1 - rS csc 1 , C - r r csczi 2 - g s ’ c - ‘ 26 3 ‘ 51“ int. ’ C4 = - AA , C5 = - AU , C6 = AxS , C7 = - AAt , C8 = Axg , where 6. 1nt. (2. (2. (2. (2. (2. (2. (2. (2. is the phase angle at internal tangency. l38a) 138b) 138C) l38d) 138e) l38f) 138g) l38h) A least squares fit of equation (2.136) to the data produces the various Ci’ which in turn can be solved for the system parameters. We note that the above equations apply only to completely eclipsing systems. A more detailed discussion of Kopal's method as used in computer solution of eclipsing binary light curves is given by Linnell and Proctor (1970a). In addition to the discussion by Linnell and Proctor, we note that A = U - L o s x tr At - U - f (k,-l) Lg Lg + LS + L3 = U , (2.139) , (2.140) , (2.141) (The function xftr is discussed in Appendix D.) Thus, with the values of Ag, At, and U obtained from the least squares solution we have L = U - A s o x tr Lg - (U-At)/ f (k,-l) and L = U - L - L 5 (2.142) (2.143) 9 (2.144) 54 Several problems arise with the use of K0pa1's method. The equations given above apply only to complete eclipses. Different equa- tions must be used for partial eclipses. Also Kopal's method requires inversion of the a fUnctions for the corresponding geometrical depths of eclipse at each point. This causes difficulty when the observed a values lie outside the theoretically permissible range. Further, ec- centric orbits can't be handled directly. Finally, the normal equations used in KOpal's method do not rigorously satisfy the least squares condition in that the weights are not independent of the parameters, though they are treated as such in calculating the error sum of squares of the residuals. Consistency of the results obtained by Kopal's method and the differential corrections method has been demonstrated for completely eclipsing systems (Linnell and Proctor 1971). However, because of the previously discussed limitations of Kopal's method, only the method of differential corrections was applied in the solutions of the systems discussed in Chapter V. fit (1‘ an: dal a 1 in pr: 11: 3y; III. DESCRIPTION OF COMPUTER PROGRAMS A. Fourier Analysis Program The program FOURIER calculates from one to ten Fourier coefficients for the non-eclipse portion of the light curve. The data points are fitted to an equation of the form N§ Ni 2(obs.) I a + A cos n 6 + B sin m 6 (3.1) ° iIl “i i i-l mi 1 by the method of least squares. The ni and 1111 are the integers desired in the harmonic expansion, NC is the number of cosine terms, and N8 is the number of sine terms. The various An1 and Bmi, t, and ab are expressed in units of Ag, the light of the comparison star. An abbre- viated flow chart of the program FOURIER is given in Figure 10. The program requires several control parameters to determine: (1) the form and order of the input data, (2) the number of data points, and (3) the number of Fourier analyses to be carried out for the current data set. Data may be in the form of phase or time units and light or magnitude units. For each Fourier analysis to be carried out, the program requires a set of integers to determine which harmonic terms are to be included in the solution. A maximum of ten coefficients may be included without program modification. Phase limits of the non-eclipse portion of the light curve may be read in directly; alternatively, for circular orbits system parameters may be read in and phase limits calculated. For each point the phase angle 6 is calculated. For circular orbits 55 56 6-211 (t-t) (3.2) -§- 0 where P is the period and to is the time of minimum projected distance of centers. For eccentric orbits 6 is calculated from Kepler's equa- tion (equation (E-9)). The point is classified according to its phase value. If the point is in the non-eclipse portion of the light curve, its contribution to the normal equation is calculated following the standard formulas for the method of least squares. (See for example Ostle 1963, equation 8.59.) If the point is outside the non-eclipse portion of the light curve, it is omitted from the calculation. After each point has been processed a check is made to determine if there are sufficient points for solution. (The number of points must be greater than the number of coefficients being determined.) If there are insuf- ficient points, solution for the present set of coefficients is termin- ated; otherwise, solution continues with the inversion of the matrix form of the normal equations and calculation of the Fourier coefficients. The matrices of simple and partial correlation coefficients are calcu- lated. Standard deviations of the light residuals and individual Fourier coefficients are calculated. The program also calculates the Fourier coefficients and standard deviations normalized to do, the constant in the Fourier expansion. Individual residuals are calculated and plotted in a histogram. The histogram for a normal distribution with the same standard deviation is superimposed for comparison. The Kolmogorov- Smirnov goodness of fit test (Ostle 1963, p. 471) is applied to deter- mine if the normal distribution satisfactorily fits the residuals. S7 Calculation of different sets of Fourier coefficients for the data is carried out as desired. The entire process is repeated for each set of data points. 58 Figure 10. Flow Chart of FOURIER 1 ' "Read control parameters for data input Read phase-magnitude-empirical weight data I ‘—-*“Read control parameters to determine inclusion of harmonic terms IDetermine phase limits for eclipse 'Fbr each point: Classify point according to phase range Calculate contribution to normal equations AbrSufficient points for solution? 1.1 I Yes Invert matrix of normal equations Obtain Fourier coefficients Calculate individual residuals from regression equation Calculate standard deviation of light residuals Calculate simple and partial correlation coefficients Calculate standard deviation of Fourier coefficients Plot histoggam of regression equation residuals e Another Fourier analysis to be performed on this ___dsta_asc1 Yes 1N° '-(——1 Another (Leta set agilable? End 59 B. Rectification Program The program RRECK transforms observational data to the equivalent spherical model data. An abbreviated flow chart of the program is given in Figure 11. The program requires values of control parameters that determine: (1) the form and order of input data, (2) number of rectifications for the current data set, and (3) the number of data points in the data set. The data may be in the form of phase or time units and light or magnitude units. Conversion of the data to phase and light units is carried out as necessary. The rectification formula is determined using the input values of Fourier coefficients, limb-darkening and gravity-darkening coefficients, angle of inclination and color temperature of the primary. Input control parameters allow three options for determining the ratio of reflected lights: (1) using an input value for the ratio of reflected lights, (2) using input depths of eclipse to calculate the ratio of reflected lights, or (3) using luminous efficiency calculations to find the ratio of reflected lights. Rectified values of the light and phase may be output on cards or magnetic tape. After each independent rectification of the data set has been performed, calculations continue on succeeding data sets. 60 Figure 11. Flow Chart of RRECK l 1 Prepare tape (if necessary) I Read control parameters for data input Read phase-magnitude-empirical weight data TIT I _1 J; Read control parameters for S /S option Read control parameters for output Options Read Fourier coefficients and constants used in calculation of rectification formula Determine S [Sh and oblateness Calculate constants used in rectification formulgm For each point: Calculate rectified light Calculate rectified phase Punch rectified data or transfer rectified data to tape (as desired), Wes ~—efAnother rectification setggg§113b1e1_ ‘JEEHfAEother data set avail ble? I7 hi 61 C. Differential Corrections Program The program DIFCORT produces from one to eleven differential correc- tions to spherical model parameters. An abbreviated flow chart of the program is given in Figure 12. The program requires values of control parameters that determine: (1) the form and order of the input data, (2) the number of initial parameter sets for which differential corrections are to be found, and (3) the number of data points in the data set. The data may be in the form of phase or time units and light or magnitude units. Conversion of the data to phase and light units is carried out as necessary. The data may be on cards or magnetic tape. The program requires initial values for the spherical model parameters rg, rg, Lg, L , i, e, w, tg, and RF, where P is the period, and RF 2.5 log U (3.3a) 1.0857362 ln 0 . (3.3b) RF is the reference magnitude corresponding to the unit of light. Also required is a set of integers to indicate which differential correc- tions are to be included in the solution. (A maximum of ten differential corrections may be included simultaneously.) Control parameters to determine the maximum number of iterations and the type of solution (occulation eclipse, transit eclipse, or both) are also required. Using the current values of the spherical model parameters, the minimum value of the geometrical depth for each eclipse is calculated along with 2min the corresponding value of the light. Thus 62 oc x oc oc 2min I l - f (k,pm1g) L8 (3.4) and tr _ _ x tr tr 2min 1 f (k’pmin) Lg , (3.5) where p::n and p;:g are the minimum values of geometrical depth for occulation and transit eclipse respectively. The primary (deeper) minimum is then associated with the type of eclipse having the smallest value of minimum light. The ranges of partial and total phase of each eclipse are calculated. Each data point is then classified according to its phase range as being: (1) outside eclipse, (2) in partial phase of occulation, (3) in total phase of occulation, (4) in partial phase of transit, or (5) in total phase of transit. Partial derivatives required for the regression equation, (2.112), are calculated for the point. The value of mg is calculated using current spherical model parameters. The point's contribution to the normal equations, equation (2.124), is included. At this point in the calculation a check is made to determine if there are sufficient points to obtain a solution. If there are suf- ficient points, calculation continues with the matrix inversion of the normal equations. Otherwise, solution of the present set of param- eters is terminated. The Gauss-Jordan method (Smillie 1966, p. 134) is used for solving the normal equations. As a check the program calculates the values of AAIl-g, where I‘is the unit matrix. Each matrix element should equal “H zero, within rounding errors. 63 Individual residuals of the regression equation are computed and used to calculate the standard deviation, o(est.), of the observed points from the calculated values. The standard deviation is also cal- culated using equation (2.132), a form not requiring calculation of individual residuals. A histogram of the residuals is plotted. The Kolmogorov-Smirnov test of goodness of fit is applied to check the residuals for conformity with a normal distribution with standard devi- ation o(est.). The simple and partial correlation coefficients are calculated along with probable errors of the parameters. The current values of the spherical model parameters are then incremented by the differential corrections. The values of limb-darkening coefficients, luminosities, radii, and eccentricity are restricted as follows: -1 S x S 1 , (3.5) 051.51 , (3.7) 051-51 , (3.8) rg 5 rs , (3.9) 0 S e f 0.999 . (3.10) Values of x less than zero are included to allow for the possibility of limb brightening. The theoretical values of light calculated with the incremented parameters are then used to calculate the standard deviation of the observed values. It is customary to normalize the light curve such that the non-eclipse portion is unity. Thus, the light values are normalized by the replacement _3__,, 1+AU . (3.11) 64 Calculation of differential corrections for the incremented param- eters is repeated for the maximum allowed number of iterations. Itera- tion on succeeding sets of initial parameter values is then carried out. The entire procedure is repeated for each data set. 65 Figure 12. Flow chart of DIFCORT J, :1 Read control parameters for data input , Read phase-mggnitude-empirical weight data . Read initial values of parameters Read control parameters for maximum number of iterations and type of solution ———9- With current parameters: Calculate minimum values of p and 2 for each eclipse Determine type of primary minimum Calculate ranges of partial and total phase for each eclipse 111 For each point: Classify point according to phase range Calculate partial derivatives required for regression equation 1 Determine 2 with current parameters Calculate contribution to normal equations N24 Number of points sufficient for solution? LYes Invert matrix of normal equations Obtain differential corrections Calculate‘Ae'lfi; Calculate individual residuals from regression equation Calculate standard deviation of light residuals Calculate simple and partial correlation coefficients Calculate probable errors of parameters Increment parameters (within allowable limits) Calculate residuals using incremented parameters Plot histogram of Eggression equation residuals ILJffiMaximum number of iterations reached? {Yes ‘ Another set of initial parameter values for 4 yes thig dggg get? :INO 1%other data set available? No 66 D. Kopal's Method Program The Kopal's Method program, called CFIT, is restricted to completely eclipsing systems and spherical orbits. The data are assumed rectified. The program produces the values of the parameters C CZ, and C3, which 1’ are functions that may be solved for rg, rg, and coszi. The program also allows for inclusion of differential corrections to U, Ag, At, xg, and xg, where Ag and At are the values of light at internal tangency (p I -l) of the occulation and transit eclipse respectively. From the equations At - U - x£“(k,-l) L , (3.14) s Lg + Lg + L3 - U , (3.15) where xftr(k,p) is defined in Appendix D, we see that we may solve for Lg, Lg, and L3 as follows Lg - u - Ag , (3.16) Lg - (u - At)/xftr(k,-l) , (3.17) L - U - L - L , (3.18) 3 g 3 Further description of the program is given by Linnell and Proctor (1970a, p. 1043). 67 E. Proggam Accuracy The programs FOURIER, RRECK, DIFCORT, and CFIT are written in the CDC 3600 FORTRAN language. The precision of the CDC 3600 in single precision is approximately 10 decimal digits. The programing of the direct eclipse functions (xa°c(k,p) and xaFr(k,p))used is described by Linnell (l965a,b; l966a,b,c). The stated programming objective for these functions was to obtain a frac- tional error of 10-6. This was obtained for most values of k and p. The maximum fractional error was given as 10’s. In most regions of the k-p plane for which eclipsing systems have meaning the absolute errors are less than 10-6. The matrix inversion in the programs FOURIER, DIFCORT, and CFIT was carried out in double precision with corresponding word length accuracy of approximately 25 decimal digits. The Gauss-Jordan method (Smillie 1966) is used for carrying out the matrix inversion. Further, to insure minimum rounding and truncation error, the matrix inversion routine chooses as pivot element, at each stage of the matrix inversion, the element largest in absolute value in the rows and columns not con- taining previous pivot elements. As a check the program calculates the 1 matrix‘AAI -5, where I is the unit matrix. In no case has an element of this matrix been found to be larger than 10-19. Typically the elements of this matrix are several orders of magnitude smaller. Further discussion of accuracy is given in Linnell and Proctor (1970a). Final validation of the programs rests in the solution of synthetic light curves with known parameters. Discussion of such solutions is given in the following chapter. 68 Complete program listings are on file in the Astronomy Department, Michigan State University. IV. SOLUTION OF SYNTHETIC LIGHT CURVES This chapter contains the results of the application of the method of differential corrections to synthetic light curves. The synthetic light curves are constructed on the Spherical Model and include random errors with normal distribution and assignable dispersion. Synthetic light curves are useful in the validation of the programs. Convergence on known parameters provides the most convincing test of program reliability. Synthetic light curves may also be used to eval- uate the effect of the dispersion and number of observations on ability to extract the desired parameters. Synthetic light curves may be based on parameters obtained from the results of actual light curve analysis. Subsequent solution of these curves may provide further confidence in the results; alternatively, the solution may indicate the need for more observations of greater accuracy. Synthetic light curves with zero dispersion were used to validate DIFCORT. Resulting light residuals were on the order of 10-6. Table 2 gives the results of analysis of a synthetic light curve similar to the light curve of the system S Cancri. The dispersion is comparable to that obtainable under optimum observational conditions. Primary minimum is a deep occulation eclipse, while secondary minimum is very shallow. Thus xg can be reliably determined, but the uncertainty in xg is rather large. Satisfactory convergence on the parameters is demonstrated. Table 3 shows the results of a test for separability of rg and xg. Irwin (1947) has shown that for certain values of parameters the ratio of the coefficients of xg and rg is nearly constant. There is 69 70 the possibility that the correlation is so great as to prevent separation. To test for this possibility, a synthetic light curve was constructed with 800 total points of intrinsic dispersion o I .005. The parameters closely accord with the parameters adopted by Irwin for his example. Convergence was not as good as for the previous example; however, result- ing parameter values were at most 235 standard deviations from the true values. As an example of a system with third light, a synthetic light curve corresponding to BR Cygni was constructed. The curve had a total of 430 points, 130 in occulation and 100 in transit. Convergence occured as shown in Table 4. Solution of actual data for BR Cygni is discussed in Chapter V. An illustration of the complications that occur due to correlation of the system parameters is given using synthetic BV 412 data. For the o=0.0 data with initial estimates of the parameters given in Table 5, convergence to the true parameters occurs in three iterations. The light curve with o I .0074 was solved twice, once allowing L3 to vary to its true value (Table 7). Convergence (Table 6) and once holding L3 was obtained only by holding L3 constant. It was noted that when L3 was allowed to vary, the absolute magnitude of the correlation coeffi- and r , L , and coszi were greater than 0.99. Also :ients between L3 10:3 that this is a partially eclipsing system. It has often been ssumed that it is not possible to determine limb-darkening coefficients at such systems (Wilson 1968). We conclude from these results that 12 determinability is based more on the accuracy of the light curve and e density of observations, and less on the geometrical depth of lipse. 71 Extensive tests on eccentric systems have not yet been carried out. However, the results of iterative solutions of a synthetic curve with Initial parameters zero dispersion are given in Tables 8, 9, and 10. In the first solution only the were the same in all three cases. Satis- differential corrections for i, e, w, and tg were calculated. When all parameters factory convergence occured in three iterations. except L3 were included, there was no indication of convergence after This is probably due to correlation of the variables three iterations. combined with sufficiently large error in the initial estimates of the Higher order differential corrections then become signif- parameters. (Compare o(est.) and o(cal.) for the first iteration in Tables 8 icant. and 9.) The absolute values of the simple correlation coefficients of L with r and coszi are large (greater than 0.97). Since for total 8 8 eclipse an estimate of L can be obtained from the light during total phase of occultation, a third solution, omitting differential corrections The results to L was carried out. Satisfactory convergence occurs. are given in Table 10. 72 r1 - .H. .mpmflom com mcwmucoo Eseficfle comm .mpewom oow meemueoo o>eso unwed “Romeo m oepoeuexm one 0000.0 00000. 0NN00. 00¢. 00¢. 0woso. H000~. 000.00 Human. onus hH000. H0000.H NHO.H 000.“ 00000.“ 00000.H o¢0.0 H H0000.“ .0.n mnm¢.0 nm0<.0 mHH.HI smooo. Nwm00. waumo. H00. H00. MONNO. whoom. 0H<.mm momwm. 0a 000¢.0 mmw¢.0 mHH.HI 00000. Nmm00. wHumo. N00. mm“. 00550. 0n00N. 0H¢.mw H0000. 0 0000.0 m0~0.0 0NH.HI mqooo. mwmo0. NHNmo. coo. 0N0. 0Hmmo. Nqoom. mmq.mm 00000. 0 nmNn.0 0¢¢o.0 NHH.HI N0000. M0500. NMNMO. own. 000.H mmowo. N0m0H. 0N0.nw mwmos. m 0000.0 Hcmm.m mma.Hl 00000. 00500. NHNMO. 0H0. HOH. momxo. hNO0H. 0mm.m0 m00mm. 0 00N0.H N00m.N NHH.HI «HH00. NH500. mwNmo. 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Selection, although somewhat subjective, was based on the following criteria: (1) Well-separated systems (2) Coverage of entire phase range (3) Number and quality of observations (4) Individual observations published (5) Lack of obvious complications. Before the analyses of the individual systems are discussed, it is necessary to consider the type of eclipse to be associated with the consecutive minima. If spectroscopic radial velocity curves are avail- able, it is possible, in principle, to determine whether the primary minimum is an occultation or a transit. Let L1 be the luminosity and J1 the mean surface brightness of the star of greater surface brightness (the star being eclipsed during primary minimum). Let L2 and J2 be the corresponding quantities for the star of lesser surface brightness. The star approaching the observer immediately before primary minimum is thus the star of luminosity L2 and mean surface brightness J For bolometric light we have 2. , 2 I1 - r1 J1 (5.1) where r1 is the radius of the star of greater surface brightness and r2 is the radius of the star of lesser surface brightness. Thus 79 80 2 r1 - L1/1'2 (5.2) r22 J1/J2 ‘where an estimate of L1/L2 is obtained from the spectroscopic observations and an estimate of J1/J2 is obtained from the depths of eclipse. In this way it can be determined if the star of greater surface brightness is the larger or smaller star and hence whether the primary minimum is an occultation or a transit. Unfortunately, spectroscopic information is not always available; or if it is, the errors associated with the estimates of L1/L2 and JllJ2 may prevent positive determination of the type of eclipse. Thus it is not always possible to make an "a priori" judgement as to the type of eclipse. Both possibilities must then be considered. The results of analysis of the light curves of 10 eclipsing binary systems are presented in the following sections. For each system a general discussion is presented and tabular data follow. In each table the source of the original photoelectric data is given, along with the spectral type of the primary. The spectral type of the secondary is given if available. The spectral type or range of spectral types of the secondary, as found by subsequent luminous efficiency calculations, is given in parentheses. The value of the period is followed by the adopted designation for the type of primary minimum. The data in each table are divided into three sections. Section A contains the results of the Fourier analysis. The phase ranges of the points included in the Fourier analysis are given in parentheses. For each light curve the results of two Fourier analyses are presented. Normally the first analysis for each color is the 81 analysis carried to terms of order 26, while the second analysis is to terms of order 36. The standard deviation (normalized to no) of the resulting residuals is presented in the second column of Section A. This is followed by so, the constant of the Fourier expansion, and the remaining Fourier coefficients (normalized to a (Note do - A0 if it O)' is assumed that there is no third light.) The Fourier coefficients are followed by NDF, the number of data points used in the Fourier analysis. The adopted values of the Fourier coefficients are followed by initial estimates of the ratio of surface brightnesses and the color temperatures of the primary components. The color temperature of the primary compo- nent is taken from Figure 15, using the known spectral type of the star. This procedure for estimation of the temperature follows the recommen- dation of Jurkevich (1964, p. 185). The temperature of the secondary and the ratio of reflected lights resulting from.the luminous efficiency calculations is given next, followed by the subsequent value of e, the oblateness of the equatorial cross section and N (given by the equation (2.31)). Section B of each table gives the coefficients used in the recti- .fication formula, equation (2.84). RFO, if given, is the value of the reference magnitude initially subtracted from observed magnitude dif- ferences in order to normalize the non-eclipse portion of the light curve to unity. Section C of each table contains the equivalent spherical model Parameters. The value of b, the exponent of the light in the weights of ‘the conditional equations, is given in parentheses. The parameters designated as "Initial" are those determined by the author publishing the original data. Often it was not clear whether the value of 82 inclination given by the author was if, 1, or i. The values of inclina- tion are simply included in the tables as they were given in the original paper. In addition to the geometric parameters and luminosities obtained with the differential corrections method, the reference magnitude RF is given, where RF - 2.5 log U (5.3) The geometric depth of eclipse po is also given. For eccentric orbits the value of po from the primary minimum is used. The last three columns present the various standard deviations of the rectified data. The standard deviation o(b - 0) of the light values is given, followed by the standard deviation of the weighted light values o(cal.) and its estimator o(est.). (See equations (2.131) and (2.134).) The number of observations used in the solution of the light curve is given beneath the tabular data. This information is followed by values of A0 and At. For partially eclipsing systems A0 and At are the calculated values of light for the occultation eclipse and the transit eclipse, respectively, at minimum geometrical depth. For completely eclipsing systems A0 and At are the calculated values of light for internal contact (p - -l) of the occultation and transit eclipse respec- tively. Also included are the ratio of the mean surface intensities Jg/Js and the ratio of the central surface intensities (Jg/J8)c, where 2 fa - rs L3 (5.4) J3 r 2 L 8 s and 83 J 3 - J {1% 3 x8._& (5.5) c Js (Kopal 1950, p. 53). 84 A. C0 Lacertae C0 Lacertae is a tenth magnitude system exhibiting a small value of orbital eccentricity. The system is also notable for the short period of its apsidal motion. Semeniuk (1967), from an analysis of 27 times of minima, obtained e - 0.027 and w - 6S.4° for the epoch of her observa- tions. Smak (1967), from his spectroscopic analysis of the system, classified the primary component as 38.51V and the secondary component as B9.5V. The recent photoelectric observations of Semeniuk (1967), have been chosen for analysis. Semeniuk, in her analysis of the data, reflected the descending branches of minima onto the respective ascending branches and grouped the observations into normal points. These normal points were rectified for ellipticity only. She assumed values for the limb-darkening coeffi- cients and made a preliminary analysis using the iterative method of Piotrowski (1948) and Kopal (1959). She reported lack of convergence for the primary minimum. Using the values obtained for the analysis of the secondary minimum, she made a single differential corrections solution for the geometric parameters and luminosities. The results of the indi- vidual B and V Semeniuk solutions are listed as the initial values of the parameters in Table 11C. Results of analysis of individual observations, using the programs FOURIER, RRECK, and DIFCORT, are given in Table 11. The values of e and u:given by Semeniuk were used. Inclusion of terms of order 36 decreased the standard deviation of residuals in both the B and V light curves. Thus the corresponding 85 coefficients were used in the subsequent rectification. Five errors were found in the published phase values. The Julian dates of these observations, along with the corrected phase values, are listed in Appendix F. Following rectification, a solution of the B curve was attempted assuming that the primary minimum was a transit and that there was no P“ excess light. Convergence did not occur. The results of the sixth j iteration are given in Table 110 under the designation ”Bl". Extreme divergence was exhibited in the seventh iteration. The 32 solution is F discussed below. b Iterative analysis including excess light did converge. The param- eters and their respective probable errors are given in Table 110 under the designation "Adopted". The V light curve also converged under the assumption of primary minimum a transit allowing excess light. Results are listed in the table. The geometric parameters from the B and V light curves are in good agreement. The difference in the standard deviation obtained from the Fourier analysis and the differential correc- tions analysis is comparable to the probable error of the standard devia- tion and can be accounted for by the error in the choice of b, the exponent of the light in the weights. To determine if the choice of b significantly affects resulting parameters, the V light curve was ana- lyzed again with b - 1. All resulting parameters were less than 8 standard deviation from the values obtained with b - k. In the process of analysis, a solution of the B curve was carried out on the assumption that primary minimum was an occultation. Surpri- singly, convergence was obtained in this case also. The resulting parameters are given in Table 116 under the designation "32". It 86 has commonly been assumed that iterations will converge only if the type of eclipse has been correctly identified (Kopal 1959, p. 334). Note the close correspondence of the geometrical parameters and luminosities of this solution with those of the solution assuming the primary minimum a transit. However, the resulting values of limb-darkening coefficients assuming primary minimum an occultation are not in good agreement with the theoretical values discussed later. Unfortunately, the spectroscopic data of Smak were not sufficiently accurate to determine the type of eclipse; however, it is felt that the assumption of primary minimum a transit and presence of third light provides the best solution. The standard deviation is a few percent smaller for this case. In addition, limb-darkening coefficients result- ing from this assumption are in good agreement with the theoretical values discussed later. 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N.mw mm. 0000030 m 0 -000 0-000 0 -000 o 0 0 .0 0 0- 0 00- 0000000 0000000 ou000 0 00 0 0 0 x x 0 0 0 0 0000 u 00 00000000 00002 000000000 0:000>0=00 .u 0.0.00000 00 00000 89 B. CM Lacertae CM Lacertae, an eighth magnitude double-line spectroscOpic binary, was observed by Alexander (1958) and later by Barnes, Hall, and Bardie (1968). The former investigator did not report individual observations. Thus, only the more recent photoelectric observations of Barnes, Hall, and Hardie were chosen for study. Spectroscopic observations were obtained by Sanford (1934) and re-examined by Popper (1967). Alexander, in the analysis of his data, concluded that the primary minimum was an occultation. Barnes, Hall, and Bardie felt the alterna- tive assumption was better for fitting their data. Consequently, the Barnes datanwerestudied under both assumptions. The results are pre- sented in Table 12C. For the U light curve convergence was obtained for both assumptions. However, the residuals assuming primary minimum an occultation have a significantly smaller standard deviation. It is interesting to note the correspondence of the values of the limb-darken- ing coefficients. For the two assumptions x - .33 and .24 for the brighter star and x - .87 and .78 for the cooler star. While the B and V light curves did not converge assuming primary minimum a transit, observe that there is agreement between the values of the radii of the hotter and cooler stars for each assumption. The V light curve was analyzed allow- ing inclusion of excess light. The resulting value of excess light was approximately two standard deviations from zero. Since the geometrical parameters of the V solution agreed well with the B and U solutions, it was not felt worthwhile to re-analyze the V data excluding the excess light. During the analysis four observations in the U light curve, four observations in the B light curve and one observation in the V light curve were found to have residuals greater than three standard deviations from 90 the calculated light curve. These points were omitted from subsequent iterations. They are listed in Appendix F. The distinction between occultation and transit eclipse diminishes as the ratio of radii approaches unity. However, it is seen from this analysis that the assmnption of primary minimum an occultation provides the more consistent results for CM Lacertae. Parameters based on this assumption have been adopted. 91 aooo.a woos.“ mace.“ Haoo. Haoo.fl Haoo.fi mmooo.n .o.a o.~ mmoo. mm.v ooem comma ca.~ «a mmoo.- oHoo.+ wooo.- maoo.+ owoo.- mmoo.+ Nvmmm. moo. > mace.“ mace.“ sage.“ oHoo.H emcee.“ .o.m em mfloo.+ aooo.- whoo.- Naoo.+ NmNNm. wmo. > case.“ odes.“ HHoo.H mace.“ mace.“ mace.“ emcee.“ .o.m o.~ mooo. mm.m ooooH oooma on.~ em Hmoo.- mooo.- Haoo.+ mmoo.- Hmoo.- mmoo.- memes. one. m Haoo.H HHoo.H mace.“ mace.“ wmooo.s .o.g em nooo.- aooo.+ maoo.- odoo.- amuse. own. m esoo.H vHoo.H mace.“ mace.“ mace.“ mace.“ “coco.“ .o.a so oHoo.+ mooo.- mooo.- Hooc.+ omoo.- aooo.+ maeme. Noo.H a waoo.a vHoo.a mace.“ oHoo.u ovooo.a .0.“ o.~ ovoo. ov.~ comm ooofla ve.~ em vooo.- Nooo.- mmoo.- mooo.+ mHva. vmm. : O O U I z m zm\ m fixes a fixovne a\na moz o<\mm o<\Nm o<\Hm o<\m< o<\~< o<\a< am-m«v an ass o<\o A em.o - om.o can ¢¢.o - co.o ".mm V mamxsma< amassed .< :Oflumuafiuuo .u moo.a .1: Am< - em< .>~< mwomav .ow .m.m.<.a .oauum: .m .m .Hamm .m .o .mocssm .u .m asawcwa thawnm moaned max» Haywoomm «was Hmcwmfiuo .3283 5 .3 2305.5 .2 03“ H 92 omom.w- maao.- oooo.a omoo.- mmoo.+ omoo.- > “Ham.w- mqfio.- mmoo.a wsoo.+ omoo.- waoo.+ m mamv.w- owfio.- weoo.a woeo.- oooo.+ mooo.- = one ~a+m< oo+m Na an on mucvfiowmmwou Nana—HOW fiOMHNUMWHHUfi“ .m fi.b_u=ouv NH magma 93 o e U m w a o m w a o p a o 0 one u A o\ oo oom moo u o\ o owes A ono u H coHpsHom > oooooom map you .sdeficfle thwaooom no moA was aseficfie meSHHm so ow .mcoflum>homno «mm mcflmucou o>h=o unmwA > och U m w o . m m a . H a . O .omm. u A o\ oo oco now u o\ o omen n H oaom u H :oHusHom H ooHoooH oHH Hem .ESEMAZME bmvfloomm Cw mom wad ESEMfiHE khmfiwhm HZ.” mm «mcowuma/Hmmno Ham mfiflmufioo Q>HSO ucwfla m 05H 0 m m m w u H o .oHH. u A o\ oo was .Hom. u o\ o .ooha. u H oNom. u H :oHoaHoH o oeHoooH was How .ASHEMAJE AHMVfioomm cw NOH wad EDEHCHME khmfifihm a“ em .mcoHumzmmDo HmN mcfimubou mafia Ham“: D 2.5. .GOMpmuAsooo cm ma asawafle unmeflum “any :oflumadmmm saw so momma mam meoHusAom moumovm age .uwmqmuu a ma azawcwa xnmeflsm was» cowumesmmm was no woman was mcoflusAom m> was .A> .Am .A: one mooo.H oomo.H moHo.H HoH.H moH.H omoo.H Nmoo.H 5N.o H ooHo.H .a.o oooo. oooo. HAmo. NNo.- Aooo. Hooo. Aamo. ooAm. one. omm. maoA. mHoA. oo.ao Homo. omHoooH > mooo.H Nvo.H omA.H ooN.H «moo.H NNoo.H ao.o H moHo.H .o.o oHAo. ooHn. omoo. HNB.- mooo. Avon. omom. moo. HHo.- wooA. omaA. oo.oo mooo. ooHooo< o aooo.H aon.H mHH.H omA.H oooo.H omoo.H NH.o H oAHo.H .o.o Homo. Homo. oomo. oHA.- oooo. ommn. meow. oom. man. woeH. mmaH. mo.eo omoo. oouooo< o mooo.H ommo.H oomo.H NoH.H moH.H Hmoo.H omoo.H mm.o H oHoo.H .o.o Hooo.- ooHH. NHom. oonm. woo. «on. HaoH. oooH. no.5o mooH.H N> mooo.H moHo.H omH.H oAH.H omoo.H mooo.H oA.o H ooHo.H .o.o Hooo.- oon. voo. owe. ooH. oooH. mooH. oo.oo movo.H H> mooo.H non.H NoN.H oHH.H oHoo.H avoo.H oo.o H HoHo.H .o.o Hooo. omoo. oon. ooH. NHm. mHaH. oNoH. Ho.oo NHHo.H Ho oooo.H oooo.H oHH.H moo.H oHoo.H Hmoo.H HH.o H omoo.H .o.o moNA.H moNH.H mooo.H Noa.- Hooo.- «Amm. coon. moo. omm. QNNH. mohA. mm.eo oHoo. Ho oom. o. o. HoA. n.5o oo. HHHHHHH > com. o. o. ooA. H.5o No. HHHHHHH o mom. o. o. HoA. o.ao oo. HHHHHHH o A~-oHo A~-oHo A -oHo co m H w m HM H m A.HmooonhcAHooo ouHoo mm H H H x x H H H H AN\H n Ho HHcoaon Heooz HHUHHoHom Hovo>H=om .u aw.u=oov NA oAQmR 94 C. RX Arietis RX Arietis is a ninth magnitude eclipsing binary system. The :solution obtained by McCluskey (1966) for his data indicated primary Ininimum was a transit. The light curves were analyzed using this zassumption. Both the U and V light curves were solved permitting and ‘then excluding excess light. Two errors were found in the published phase values for RX Arietis. 'There were four observations with residuals greater than six standard deviations from the calculated light curve. It is felt that these are the result of typographical errors. These four points were omitted for the final iterations. Each of these points is listed in Appendix F. The results of the iterative solutions are given in Table 13C. The solutions excluding excess light have been adOpted. Agreement between B and V solutions for this assumption is good. Also, the V solution converged to a value of L3 within its probable error of zero. It is felt that the L3 convergence to a non-zero value in the 8 solution allowing third light is due to correlation (as in the case of the synthetic BV 412 light curve). As seen in the table, limb-darkening coefficients for the primary component can be determined with reason- able accuracy; but, since the secondary eclipse is very shallow, the probable error in the limb-darkening coefficient for the smaller star is quite large. 95 Homo.- oHomm. oooo.+ Aooo.- omoo.+ > AANo.- Hmoom. voo.+ oooo.- Naoo.+ o No+m< oo+m< No Ho oo mucofiofimmoou «Assnom coflamofimfluoom .m voo.H oooo.H ono.H omooo.H .o.o New oHoo.- Homo.- mvo.- ovsmm. Amm. > mooo.H oooo.H oooo.H Aooo.H omooo.H .m.m ~.N Homo. m~.m oooo ooma Ho.o New ono.- mooo.- ommo.- ANAo.- NHANH. mom. > HAoo.H oooo.H voo.H Hmooo.H .o.m NHN aooo.+ Ammo.- oHAo.- AAANH. «mm. o mooo.H Hooo.H oooo.H oooo.H Amooo.H .o.o ~.N Aomo. mo.~ ooao ooma A.mA New oooo.- mooo.- mmmo.- oon.- AoHNm. mom. H 0 o o u 2 H Hm\ m Axoo A Axoooo oAHo moz o<\~o o oouooom HHH Hoo .esaflcfle Ahwwcooom :H NAA was adaficfle Aumenm :M nmA .mcofium>Homno woe mcfimucoo o>uso psmflA > use 0 m w m w . o .A.NN u A o\ oo nos o.mH u o\ o .omoo. u HA mono. u A coHHsHoH m oHHoooH HHH Hos .asaficfle xummcooom ca oAA new asaflcwa thafihm ca ovA .mcoflpm>homno Aom mcfimucoo o>~30 unmfla m och .uwmcmnu m mm ezeficfis Asmaaum wasp cofiumasmmm map co momma one mcoflpaAOm och 96 Nooo.H aooo.H Nom.AH Noo.H oooo.H omoo.H mH.o H mmoo.H .o.o Haom. Anew. HHmm. No.H- oooo. Homo. mooo. HAN.H- mom. HNAN. mmom. AH.oo momm. oopooo< > mooo.H Hmao.H «who.H NAm.AH ANA.H mooo.H oooo.H oo.o H omoo.H .o.o oAHm. oAHm. oAmm. No.A- oooo.- NNHo.- mono. AHoo. on.H- on. AHAN. omom. Hm.oo HAmm. A> mono. Aooo. H. H. mm. om. o.oA Am. HHHHHoA > Nooo.H HAoo.H oHH.NH Hoo.H oHoo.H Amoo.H mm.o H Hooo.H .H.o oomm. AHmm. Homo. oo.o- oooo. mHNo. mvo. Hom.m- How. HoHN. moan. AA.oA Ammo. ooHoooH o Nooo.H onNo.H AANo.H mNo.HH mAA.H Hmoo.H omoo.H Hm.o H Hooo.H .o.o movm. oon. oomm. oo.H- mooo.- oooA. Homo. coma. on.N- Noo.- AoNN. oAmm. oo.~o ommo. Hm NoHo. oooo. H. H. NH. on. o.oA am. AHHHHHA o A~-vo AN-oHV A~-oHo on ma mH HH .wH Hx .wx HH .ww H H A.Hmm~o A.Hmooo Aounoo A~\H u so msaoaHAm Hmooz HHHHHHHHH HomHH>H=om .o A.H.H=ouo MA HAHHA 97 D. V338 Herculis V338 Herculis, an eclipsing binary of approximately the tenth mmagnitude, has been observed independently by Vetesnik (1968) and Iialter (1969). Both investigators classified the primary minimum as a transit. The more numerous observations of Vetesnik were chosen for a tudy . The analysis presented here is based on the assumption that pri- mary minimum is a transit and that there is no third light. There is good agreement between the resulting geometrical parameters of the B and V solutions for this assumption. Iterative solution of the B curve allowing excess light converged to parameters that were within one standard deviation of those obtained for the B analysis excluding third light. It is seen in Table 140 that while the limb-darkening coefficient ‘of the larger star can be reasonably well determined, the secondary 'minimum is too shallow to permit reliable evaluation of the limb-darken- ing coefficient of the smaller star. One error was found in the published phase values for the V light curve. The Julian date of this observation is J.D.Hel 2439648.4767 and the corrected phase value is 0.9683. There is also evidence of systematic error in the data. There are runs of constant sign in the residuals. .Agreement between standard deviations of the Fourier analysis and the vNAo.u vmoo.A wvoo.+ wnoo.u wvoo.+ m No+m< oo+m< me An on mucofioflmmoou «Adamo» :owumoAmHuoom .m mooo.H mesa.“ coco.“ AAoo.H mooo.H oaoo.n meooo.w .o.m v.m mooo. vw.n oomv comm m.oA Nam Awoo.+ mooo.+ omoo.u nooo.n mmoo.u vao.u wnown. mAn. > mooo.H coco.“ mace.“ wooo.H mvooo.w .o.m mom Nooo.+ emoo.- v~oo.- nmoo.n vkon. emu. > coco.“ mooo.H mooo.u coco.“ woos.“ aces.“ mmooo.w .o.m o.N mwoo. mv.m comm oomoA w.mA omv Nooo.- mmoo.+ «Hoo.u vAoo.n owoo.n whoo.s mmmmm. nae. m mooo.w mooo.H mooo.H mooo.H mmooo.w .o.m omv wmoo.+ ono.u nwoo.u Nooo.n nmmmm. mmv. m U U . U I 2 H HHA m Aeoo e Aeoooe oAHo moz o<\mo o<\~o o<\Ho o<\m< o .o.o - o.o ooH H.o - H.o u no .NAo.o - oom.o ocH NHH.o - ooo.o u.$w v HHHAHHH< HoHHsom .< newsman "abeflcwa Ahmaflum .m oom.~ ” vowuom .II Aouxv m< " max» Amhooomm AoooHo mmH .oA .u.<.o .HHcmoHo> .2 u HHHo HaonHHo .HHAHUHHm omm> Ho HHoHoaHHHo .HH oHHHe 100 o m m m m . o .A.HA u A o\ oo oqo .HH.A u o\ o .HHNH. u HA Homo. u A :oHHHHoHH > HHHHH HHH Hoe .Eseficwe Aemmcooom so omA mam Edeflcfia Aumsflum a“ nmm .mcoHHm>Homno mom mawmucoo o>eso HawHA > 0:9 0 m m m m . . o .o.o~ u A o\ oo oco .o.HA u HA o omoo. u HH HHoo. u H :oHHHHoHH o AHHHH oHH Hos .ESEACAE Aummeooom :H wnA mam Edaficfis Ahmawum so mma .mcofium>homno own mcwmucoo o>hso szwA m och .mecmuu m ma esswcwe AHmSHHm use» :OMHQESHmm map :0 momma one meowpsAom 0:9 mooo. wmoown NNB. H Awo. fl w~oo.n A000.“ om.o H owes.“ .o.m omw. mew. mew. www.u mooo.u omno. memo. www.mn Amm. nwmm. omAm. ow.mw mmvn. Amman > mmo. mvm. m. m. BNN. com. mw.©w wmn. HmwpficH > NNoo.H HHH.AH HAH. H HAoo.H oaoo.H o~.o H HHHo.H .o.o Noe. ova. woo. Hoo.- Nooo. oeoo. Amoo. HHH.H- vo.A Homo. AAAH. oo.oo HooA. HHHHH o mmo. mom. 0. o. BNN. com. ww.ow emu. Amwuflcu m A -oHo A~-oHo A -oHo o .1» H .m H or. H -mx AHHHHHH Est mono: H s. H H H H H H H H AN\H u no HHaoaoAm Hoooz AHHHHHHHH HHHAH>H=om .o A.o.HHooo HA HHHHH 101 E. Y Leonis Y Leonis is a single-line spectroscopic binary of approximately tame tenth magnitude. Struve (1945) derived spectroscopic elements. The system is notable for its deep primary minimum. The photometric (Sbservations studied here are the broad band (3000A wide) infared (8000A) observations of Johnson (1960). The UBV observations of .Johnson covered essentially only one primary minimum. It was felt that they were not sufficiently numerous to warrent analysis. There was one error found in the published phase values. The Julian date of this observation along with the corrected phase value is given in Appendix F. Preliminary elements obtained by Johnson are given as the initial parameter values in Table 15C. The Johnson IR data were analyzed assuming primary minimum an occultation, both allowing and then excluding excess light. The results are given in Table 15C. The difference between the resulting parame- ters are negligible. Notice, however, exclusion of third light signif- icantly reduces the probable errors of the parameters. The values of the standard deviation of the residuals from the Fourier analysis and from the differential corrections analysis are in very good agreement. However, there are relatively few observations contributing to the determination of the limb-darkening coefficients (48 observations in the occultation eclipse and 51 observations in the transit.). It is felt that re-observation of Y Leonis in UBV covering the entire phase range would be worthwhile. 102 owmoo.n voovH. emaoo.+ mmNoo.u mmHoo.+ mm ~o+m< oo+m< No Ao oo mucoflofimmoou «Asanom cofiumofimfiuoom .m oooo.H AAoo.H oooo.H oooo.H Hooo.H oNoo.H omooo.H .H.o oA Hooo.- ANoo.+ Hooo.+ Hooo.- HHHo.- oAmo.- oooHA. one. HA mHoo.H oNoo.H Hmoo.H voo.H HAooo.H .H.o o.o ooHo. o.oo ooHH ooHo A.oA oA oNoo.+ Hooo.+ oHAo.- oHNo.- NHoHA. HNA. «H U U U a 2 H HH\ H Aeoo H AeooHH o\Ho Hoz o<\mo o mo muopoamumm .mA oanmh 103 o s U m w Q o W m A o p A e o Hooo u A o\ oo ego NHH u o\ o HoHo u H oHoH u H ooHHsHOH oHHoooH HHH HHH .aseficfla Ahmmcooom ea Am mam Edeficfla Ahmeflum so we .mcofium>nomno ewA mcfimucoo o>nso unmAA one .aoflumuasooo cm ma asaficfie Ahmeflnm Hosp cowum83mmm one no momma one mcofiusAom oak oooo. oooo. oHA. ooo. Hooo. voo. AH.o oHHo. o.o.o ooHA. ooNA. ooNA. ooo.- oHoo. NoHo. ooHH. moo. ooH.- Homm. oHoN. AA.Ho Home. H oz oooo. NHHo. oooo. HoH. moo. whoo. omoo. oo.A ooHo. H .HAA oHNA. oHNH. oANA. Aoo.- oooo. oNoo. HAHH. HoHA. moo. ooH.- ooNN. HHHN. o~.Ho Home. H HHHz ooH. ooo. oooN. ooow. AH.HH oooo. HHHHHHH A~-vo mm-oAM.r.-qu on o H HH H w H w A.HHHoo A.AHHoo mouowo mm H H H x H H H H H Ao n Ho HHHHaHAm Hoooz HHHHHHHHH HHHAH>Haom .o A.o.H=oHo HA HHHHH 104 F. Rw Monocerotis RH Monocerotis, a ninth magnitude system, has been classified as a single-spectrum binary by Heard and Newton (1969). The system has been studied photometrically in two series of infrared observations by Brukalska, Rucinski, Smak, and Stepien (1969). From their preliminary analysis Brukalska, et. a1., reported a negative limb-darkening coef- ficient for the secondary component. As the Brukalska Series I observations did not cover the non- eclipse portion of the light curve, only the Series II observations are discussed here. The Fourier analysis carried to terms of order 36 has a signif- icantly smaller standard deviation than the analysis carried to terms of order 26. The large sine terms are a preliminary indication of complications in the system. Although there are a large number of observations, a significant range of the non-eclipse portion of the light curve is not covered. Analysis has been carried out on the assumption that primary minimum is an occultation. Initial analysis indicated asymmetry in the residuals and absence of third light. Thus differential correc- tions to to were calculated and differential corrections L3 were excluded in succeeding iterations. Inclusion of to was accompanied by a significant reduction of the standard deviation. (It should be observed, however, that inclusion of sine terms iJI the rectification introduces systematic variation which may partially simulate a change in the reference time to.) Contrary to the analysis of Brukalska, et. al., the resulting 105 limb-darkening coefficients are in reasonable agreement with the theore- tical values. However, the standard deviation of the residuals from the entire light curve is not in good agreement with the standard deviation from the Fourier analysis. As shown in Table 160 this cannot be accounted for by change in the choice of b. The results of iterations with three different values of b show little variation. The observation on J°D'Hel. 2439454.8463, apparently containing a typographical error, was omitted from the solution. Fourteen observa- tions between phases 0.067 and 0.087 have systematically positive resid- uals between 28 and 6% standard deviations from the calculated curve. This phase range was covered on only one night during the photometric study. Thus the solution presented in Table 160 should be viewed with some reserve. Further observation of the system would be useful. 106 Aceo.u mmwm.~ smoo.+ nooo.u nmoo.+ «A ~o+m< oo+o< No to Ho oQ mucofiofimmoou «Asahom cowumoflmflpoom .m oHoo.H oNoo.H oHoo.H NAoo.H omoo.H ANoo.H oNoo.H .H.H ~.N AAHo. o.om oooo oooAA Hm.o woo oHHo.+ ooHo.+ oooo.+ oooo.- HoHo.- oHoo.- AHHH.A one. «A oHoo.H ono.H HAoo.H voo.H «Hoo.H .H.H Hon ANoo.+ HAoo.+ oomo.- omNo.- Homo.A moo. HA 0 U U I 2 H HHA H Aeoo H AeooHH HAHo Hoz o<\mo o<\No o<\Ao o<\o< oAomno oww mcfimpsoo o>H=o HamHA one .coAHmHAsooo cm HA aseflcfla Ahmefinm Hana :oAHmESHHm may no woman was HGOHHSAom Hooo.H ooooo.H HAoo.H Amo.H mHo.H HAoo.H Aooo.H omoo.H .H.H omo.o «mo.o mmo.o mm.A- mooo.- mmmoo. oomA. HoHN. mom. moH. HmNN. Hmom. oo.oo oomA. AHHAH ouo Hooo.H ooooo.H oooo.H HHo.H AHo.H NAoo.H mooo.H omoo.H .H.m vo.A omo.A oHH.o Hm.A- oooo.- Homoo. moon. HAHN. HNH. moH. HNNN. vom. oo.oo ooma. AHHAH.«H Hooo.H ooooo.H Hooo.H Hmo.H oHo.H AAoo.H Hooo.H ANoo.H .H.H oNA.A AoA.A HAo.o Hm.A- voo.- mmmoo. AHHA. HAHN. Hum. omH. omNN. AAom. oo.oo ooHA. AHHAH Auo ANA. mam. H. H. moon. ooom. .oo NH. AHAHAHA om-vo AN-vo A -vo o o H H H H1» H [Hm mfwmbvb mfnmobb OflBb am mm H 1— A K K .H H ..H X munoeoAm Homo: AHUAHosmm HeoAm>A=cm .u A.H.H:ooo oA HAHHH 108 G. BR Cygni BR Cygni, a ninth magnitude system,has been observed by Wehinger (1968). Wehinger presented a solution for the primary minimum of the V curve only. The V curve exhibits uniform light for phase values within approximately 0.011 days of to. On the strength of this feature Wehinger assumed primary minimum was a complete occultation eclipse, even though the B curve did not indicate a similar characteristic. The B light curve shows night-to-night variation of about 0.03 magnitudes. This variation was particularly apparent in the phase ranges 0.1 to 0.2 and 0.4 to 0.5. Solution of the B and V curves were attempted assuming primary minimum a transit and excluding third light. Apparent convergence on the parameters was obtained. Resulting parameters are given with the designation "Bl” and "V1" in Table 17C. Note, however, the values of o(est.) and o(cal.) are significantly different in both B and V solutions. Solution of the V light curve assuming primary minimum a transit and allowing excess light resulted in large negative values of excess light and hence was not considered further. Using Wehinger's results as initial parameter values, an iterative solution assuming primary minimum an occultation eclipse and excluding third light was attempted. The results of three iterations are given in Table 17C with the designation "V2". Note the negative values of the limb-darkening coefficients. In an attempt to find a more satisfactory fit, the solution was repeated using the same initial parameters, but in this case allowing third light. Convergence occured in three iterations. The resulting limb-darkening coefficient for the primary component is not unreasonable. Iterative solution of the B curve assuming primary 109 minimum an occultation, both allowing and excluding third light, were divergent. A solution of the B curve assuming geometric parameters of the V3 solution, but excluding differential corrections to rs was then attempted. Convergence occured. The results of this solution are given in Table 17C with the designation "82". Further iteration excluding differential corrections to L8 were divergent. Designation of the primary eclipse as an occultation eclipse seems to provide the most satisfactory results. It is felt that further obser- vation, especially in B, will be needed to determine the system parameters with greater reliability. 110 ommo.- HHHA.A AAoo.+ Hon.- NAoo.+ > ono.- AoNH.A mHoo.+ mvo.- mooo.+ o ~o+m< oo+o< No Ao oo \ \ mucowowmmooo «Aseuom cofiumoflmfluoom .m HAoo.H voo.H voo.H oNoo.H mmoo.H oHoo.H ono.H .H.e ~.~ AoNo. H.Am ovo oommA mA.m ooA AAoo.- HAoo.- mHoo.+ omoo.- Avo.- oAAo.- AANA.A HA.A > voo.H voo.H mNoo.H voo.H HAoo.H .H.H ooA HAoo.- mHoo.+ Hvo.- voo.- oANA.A HA.A > ono.H AAoo.H AAoo.H mmoo.H HNoo.H «Hoo.H HNoo.H .H.H ~.~ HANo. o.AH ooom ooHAA oH.AA ooA NAoo.- oooo.- mooo.+ oNoo.+ HANo.- Nooo.- mHAA.A AN.A H AAoo.H AAoo.H Hmoo.H voo.H Hmoo.H .H.o ooA oooo.+ Hooo.+ AANo.- ANAo.- omAA.A A~.A o 2 H HH\HH AeooHH AeooHH Ho\oo Hoz oHAmo oHANH o<\Ao o<\m< o<\~< o<\A< o< AmmeW A oo.o - Ao.o Ham om.o - AA.o u.$m v HAHHAHHH HHAHsoH .H new :ofiHmHAsooo "ESEACAE xemswnm .mmmn.A A moaned II. Aem . New mm " omxu Amuuoomm Amovo omA .mA .o.< .HHHHAHHz .< .o " HHHH AHHAHHHo Aamxo Ho Ho HHHHHHHHHH .AA HAHHH 111 s U W m G 0 [W m Q s p a o o .mmH u a 5\ 5V vcm MON u H\ H comm u 4 snow u & cowuaflom m> 0:» How .assfieAs Aummeooom :A em mes ESEACAB Aumsmpm em omA .mcoHum>homno Ame chmucoo o>Hso unmAA > one m w m m o .NAA. u HA HA Ho Hem .voo. u HA o .ooNo. u HA .AmNm. n A :oAHsAoH No HHH Ham .Essmcma Ahmmsooom :A om mam aseAcfia Ahmawnm :H NNA .mcoAHm>Homno Ame chmucoo o>eso H:MAA m och .eoAHmHAsooo so HA asamcfls mamswnm Hosp eoAHmasmHm on» so woman who HcoAHsAom m> was .N> .mm och .HAmemnu a HH asaAcwe AumEAum Hogu :oAHmasmmm map so momma ohm meoAHsAom A> was Am och Hooo.H HHAo.H AHAo.H omA.oH Hmm.oH oooo.H NNoo.H Na.o H NNNo.H .H.o HAm.A HAm.A NmN.A oo.A- Hooo.- HHoN. ANHH. oNoN. oom.o HAm.o- omAN. ommm. mm.mm ooAA. m> mNoo.H omA.oH ooN.oH voo.H HAoo.H oA.o H oNoo.H .H.H Aom.A moH.A HAm.A Ho.o- voo.- mHoH. AHoH. ooN.A- mmo.o- omoN. onm. oH.oA HNmm. N> Nooo.H oNoo.H ooN.oH ooo.oH ono.H Hmoo.H NNoo.H .H.H Amm.A AAH.A mom.A Hm.A- Hooo. AHoA. Hmoo. mHH.o AHA.o oAHN. oon. oo.oo oHHA. A> Noo. mom. H. H. HAN. oom. oo.oo mNN. AHAHAHA > oooo.H Hvo.H mvo.H omA.oH NoN.H ANoo.H HH.N H mHoo.H .H.H Hmm.A Hmm.A ooN.A om.o- Nooo.- HHHA. oooo. mmoA. moH.o Amo.A omAN. HAHm. Ho.mo Boom. No Nooo.H oNoo.H HAm.oH NmA.H HAoo.H ovo.H Nvo.H .H.H ooH.A HoA.A AAH.A Hm.A- oNoo. ommo. omHo. AoA.A ooo.o mooN. onm. oo.oo mNHA. Ao AN-vo AN-vo A -vo on HA mH HH wH Hx Haw H.H mm, A A AmHHHoo AAAHHoo ouooo AN\A u Ho HHHHaHAm AHooz AHHAHHHHH HHHAH>AHHH .o A.H.H=ouo NA HAHHH 112 H. BV 430 BV 430 (RS Cha), a sixth magnitude system, has been observed in- dependently by Chambliss (1967) and Schoffel and Mauder (1967). Since the latter's observations were not published, the Chambliss data were chosen for further study. Chambliss based his solution on the assumption that the primary minimum was a transit. Using this assumption, the B and V light curves were analyzed both allowing and then excluding third light. When third light was included, convergence to large negative values of excess light occured in both the B and V light curves. Results of the third itera- tions of B and V excluding third light are presented in Table 18C, under the designation ”Bl" and ”V1". Note the attempted "interchange" of the larger and smaller star, as indicated by values of k=>l.0. Solutions assuming primary minimum at occultation and excluding third light were then attempted. The results are given in Table 18C under the designations ”82” and "V2". Observe the close correspondence between the V2 and the B1 and V1 geometrical parameters and also between the limb-darkening coefficients for the brighter and less bright compo- nents for the V1 and V2 solutions. With the ratio of radii close to unity it is extremely difficult to distinguish between occultation and transit eclipse. The results of the B2 solution were somewhat puzzling considering the correspondence of the other three solutions. Examina- tion revealed correlation coefficient between rs and L8 was -0.98. A solution assuming the V2 results for rs and coszi and excluding differ- ential corrections to re and coszi was attempted. The luminosities of this solution were used and differential corrections to the remaining 113 parameters were calculated. The resulting values of the geometrical parameters were within 8 probable error of the input values. For the final three iterations only the differential corrections to r8 were excluded. The resulting parameters were virtually unchanged from the input values. These parameters are designated "B3" in Table 18C. Even though the procedure followed for the B3 solution is somewhat subjective, the resulting parameters and reduced standard deviation seem to justify the procedure. The B3 geometrical parameters are in good agreement with the V2 values. 114 mmHAo.- onHH. AoNAo.+ HHmoo.- NANAo.+ > oHNNo.- HHHmH. Hono.+ Hoooo.- Hono.+ H No+N oo+m< No Ao oo \ HHcvoHmmoou «Asshom :ofipmofimfiuoom .m Hooo.H Hooo.H HAoo.H voo.H HNoo.H oHooo.H .H.H o.N ooNo. mm.A ooHo ooooA HA.A HHN NAoo.- voo.- HNoo.- omNo.- Nooo.- oNHHH. mA.A > Hooo.H Hooo.H HAoo.H ono.H HHooo.H .H.H HHN voo.- oooo.- mmNo.- mmoo.- HmHHH. mA.A > voo.H Hooo.H oNoo.H HAoo.H AAoo.H oHooo.H .H.H o.N oNNo. Ao.A ooAAA ooomA Hm.A oHN oNoo.- Nmoo.+ voo.- NHoo.- mHNo.- HAHNH. NN.A H voo.H Hooo.H AAoo.H HAoo.H oHooo.H .H.H oHN voo.- omoo.+ NANo.- Nooo.- HHHNH. oN.A H U o o n 2 H HH\ H AHoo H AHHUHH H\Ho Hoz o<\NH oHAAH o<\m< o<\N< o<\A< o< ANowvw .F A Ao.o - Hm.o HHH AH.o - Ho.o H.Am v HAHHAHHH HHAHsoH .< :oAHmHAsooo "aseficfia Aumsahm .o HHo.A " HoAHHH AAH Ho HHHHHEHHHH .HA HAHHH o H w . . H m . . H . . o .HmH. u A o\ oo HHH ANH u o\ o Aooo u A HHHH u A HoAHsAoH N> HHH HHH .ESEAGAE Ahmecooom cw nmA mew Esawcfla Aumsflum :a AAA .mcofium>nomno mmm chmueou o>Hso szfiA > one o H w . H w . . u . . 0 .Ann. u A 5\ my mam men. n H\ a moon u A Aenm n A coflusAom mm 02H you .ESEA:AE Ahmecouom cw omA ecu ssefiefia Aymawnm :H omA .Hsoflum>homno emm Hewmueoo o>psu uzwflA m one .eoAHHHAsouo em HA asawcfia HAHSHHQ Hana :oflumesmmm ecu :0 women one HcoApsAoH N> new .mw .mm one .Hfimcmuu m HH adefiefis Anmswum page cowumasmmm onu no women mum HcoflusAOH A> wee Aw one 115 Hooo.H HHNo.H mAA.H mHA.H oNoo.H Aooo.H HN.o H AAoo.H .H.H oNN.A oNN.A AAA.A HoH.- Hooo.- HNHH. HAHH. HHA. HHA. ooHN. oHHN. NH.NH HoHo. N> Hooo.H ANmo.H omA.H mNA.H HAoo.H HHoo.H NN.o H oHoo.H .H.H mooo.u mmme. been. wew. wen. mmmm. eHmN. on.~w cnoo.H H> New. wmo. o. o. mom. eom. n.ew on. AmfluwcH > coco.“ Hooo.H ooA.H ewe.“ wNoo.H MN.o H memo.“ .o.m mme.H mme.~ nmN.A mom.s wmoo. wmem. Nome. 0N5. Nun. weem. ommm. me.mw mmhm. mw coco.“ neoo.H mm~.u mwo.H ANoo.H ono.H om.o H wees.“ .o.m owe.~ owe.~ mem.A me.u omoo. meme. mmnm. AMA. emo. mmom. mwom. Ao.ow comm. Nw coco.“ mama.“ mmH.H eNH.H nooo.H ewoo.H mm. H mmoc.fi .o.m oHoo. mace. momm. omo. owe. emmm. mwmm. wm.mw hmoc.a Aw Amm. ooo. o. o. HoN. HoN. A.HH HA. AHAHAHA H AN-vo AN-vo AN-vo on HH HH .HMH Hx ,Hm HH .1HH A H A.HHHoo A.AHHoo AouHoo AA u o .HN> .mH .NH .AH ANAA.H o HA>o HHHHaHAH AHHoz AHHAHHHHH HHHAH>AsoH .u A.H.H:ouo HA HAHHH 116 I. BV 412 BV 412, an eighth magnitude system, was observed spectrosc0pically by Mammano, Margoni, and Stagni and photoelectrically by Harris (1968). Harris states that the spectroscopic observations indicate primary eclipse is a transit. The Harris observations were analyzed under this assumption. Two errors were found in the published phase values. The Julian dates of these observations are listed in Appendix F. The V observations were first analyzed allowing third light. The iterations converged to a large negative value of excess light. It was assumed this was due to correlation between the parameters. The V data were subsequently re-analyzed excluding third light. Satisfactory convergence occured. The results of this analysis appear with the designation "AdOpted" in Table 19C. While the B curve iterative solution excluding third light was divergent, analysis allowing third light converged. The results of the convergent solution are given in Table 19C under the designation "Bl". The geometrical parameters of the Bl and the ad0pted V solutions are not in good agreement. Examination of the correlation coefficients of the Bl solution showed a correlation coefficient between rs and coszi of 0.99. A procedure similar to that used for BV 430 was used in an attempt to find accordant results for the B and V light curves. Geomet- rical parameters of the V solution were used as initial values of an iterative solution of the B light curve. Differential corrections to rs were excluded and it was assumed that there was no third light. Convergence was obtained in four iterations. The resulting parameters 117 were used in an iterative solution excluding differential corrections to Lg. The parameters changed by less than one standard deviation. There is a discrepency between the standard deviation of the Fourier analysis and the differential corrections analysis for the B light curve. Resid- uals between 2% and 6 standard deviations from the calculated curve were found for eighteen B observations. Fifteen of these observations are in the non-eclipse portion of the light curve. This accounts for the standard deviation of the Fourier analysis being 10% greater than that obtained by the differential corrections analysis. Resultant B parameters, designated "Adepted" in Table 19C, show good agreement with the V solution. 118 HeoH.- anew.~ owoo.+ omflo.- «Hoo.+ > HHHH.- nowH.~ NNHo.+ ammo.- amfio.+ m ~o+~< oo+m< No Ho co \ Hueowoflwmoou «Hashom :oflueoMMwuoom .m coco.“ Hooo.fi mace.“ Nfioo.fl coco.“ Face.“ mace.“ .o.a o.N oemo. a.H~ ooflw oomvfl Hw.e NHH Hfioo.+ Hooo.+ «Noo.+ mmoo.- aoeo.- moflo.- OHHH.H emu. > mace.“ Hose.“ coco.“ woos.“ Hoes.“ .o.a NHH Hooo.+ omoo.+ «Heo.- aHoo.- HNHH.H Han. > “coo.” mace.“ coco.“ vHoo.H mace.“ ANoo.H mace.“ .o.a o.N memo. N.HH coma oomofi HH.oH Ham Hooo.+ Hfioo.+ «Hoo.+ vmoo.- Hweo.- VHHo.- emao.~ oom. m coco.“ coco.“ NHoo.H Hoes.“ Hose.“ .o.a mam HHoo.+ Nfloo.+ omeo.- Heco.- o~ao.~ cam. m o o o .. z w ;H\ H Agog a fixooge n\:e aoz o<\mm o<\mm o<\Hm o<\H< o<\~< o<\H< o< fimcwvw A H.o - o.o egg «.0 - o.o ".mm W Homafiaq< “amused .< uflHeenu "seawafla xueaflum .v Hun.o u moaned mm.em . m.~mv o< " omxu Heywoomm " «poo Hecwmfiuo Awomfio veg .Ha .H.< .Hfiuuez .e .< Nae >m Ho Huouoaanma .HH «Home 119 o H m .HH.H n A H\ Hg Hem .HHH. u o e U m m A m m Q we m u A 6\ av one nn.m u m\ o mam“. u .Edsficws xuooeoooH :M moH woo assfiefia xhoeflum ea mew .Hcofluo>uomno Hmn .uwHeonu o HM onfiHoo xHHEHHQ pone :oflumssHHo HH\HH . .ssaficfle xuoocooom :« moH one ESEMGME xHHSMHm a“ eem .Hcowuo>homno Nmn amen. u u o< :owusaom > ooumooo onu pom choucoo o>n=o uanH > one H .HHHH. u H O 4 .enmm. u « cowusaoH m topmoeo onu you chouzoo o>uso unmfia m och onu :o ooHoo one Hcofluofiom Hooo.H o~oo.H HHH. H HHo.H HNoo.H HHHH.H ov.o H HHoo.H .H.HH HHHH. HHHH. HHHH. HHH.- Nooo. HHHo. HHHH. HHH. HHH. HNHN. HHHH. NH.BH HHHH. HHHH0H< > HHH. Hem. H. H. HHN. HHH. HH.HH HH. HHHHHHH > Hooo.H HHoo.H HHH. H HHH.H HHoo.H HHHH.H He.o H HHoo.H .H.m HoNH. HHNH. HHHH. HHH.- HHHH. HHHH. HHHH. HHH. Hue. HHHN. HoHH. HH.HH HHNH. HHHHOH< H Hooo.H HHHH.H HHHH.H NHH.HH HHH.H HHHH.H HHHH.H HN.H H HHHH.H .H.H HHNH. HHNH. HHoH. HHH.- Hooo. HHHH. NHHH. Hana. NHH.H- Hao. HHHN. NHHH. ~H.Ha HHHH. HH HNH. Ham. H. H. HHN. HHH. HH.HH HH. HHHHHsH H A -HHH fiN1HHH mm-oHH o H H .w, H mm, H w mmHHHHo @233 Saab a "a H .— H x x H H H s. nN\H n by mhmpOEdHNm H0602 HdUfiHOSQm HGQHN>MfiGm .U A.H.H:oov HH HHHHH 120 J. sw Lyncis SW Lyncis, a ninth magnitude system, has been observed by Gleim (1967) and Vetesnik (1968). Fourier analysis indicated the standard deviation of the Gleim data is approximately twice as large as the standard deviation of the Vetesnik data. Since the Gleim data are also less numerous, only the analysis of the Vetesnik data is discussed here. Both Gleim and Vetesnik concluded that primary minimum of SW Lyncis is a transit eclipse. Analysis was based on this assumption. Vetesnik indicated that some of his data apparently contained systematic errors. He omitted certain observations from his analysis. Accordingly, four- teen V observations and thirteen B observations designated by Vetesnik were thus excluded from the differential corrections analysis. One V observation felt to contain a typographical error was also excluded. Julian dates of these observations are given in Appendix F. The V light curve was first analyzed allowing third light. Conver- gence occured. The resulting parameters are designated as "V1" in Table 20C. Iterative solution of the V light curve excluding third light was not completely convergent. The parameters of the iteration having the smallest value of o(cal.) are given in Table 20C with the designation "V2". An attempt was made to improve convergence by omit- ting differential corrections to Lg. Although the resulting standard deviation is a few percent larger than the V1 solution, convergence was satisfactory. The results of this solution are designated "V3" in Table 20C. Iterative solution of the B light curve allowing third light was divergent. Iterative solution excluding third light did not exhibit 121 satisfactory convergence. The parameters of the iteration having the smallest value of o(cal.) are given in Table 20C with the designation "Bl". The partial correlation coefficient between rs and coszi for this iteration is 0.98. The geometrical parameters of the V2 solution were used as initial parameter values in an iterative solution excluding differential corrections to coszi. A small decrease in the standard deviation of the residuals was obtained. The results are designated "BZ" in Table 20C. Further iterations using the 82 solution and ex- cluding differential corrections to rS were divergent. The 82 and V3 solutions seem to provide the most consistent results. However, these parameters should be regarded with considerable reserve until they can be supplemented with the results of more numerous obser- vations of greater accuracy. 122 wmno.n Humm.o mNoo.u omoo.+ omoo.u > HHHH.- aHHo.H ooHo.- HHHo.+ ooHo.- H Ho+H< oo+m< No Ho oo HHHHHHHHHHoo HHHEHoH ooHHHonHHuHH .H emmummz H> mummumaz Hm HHoo.H Hooo.H HHoo.H Hooo.H HHoo.H Hmoo.H Haoo.H Heoo.H Hooo.H .H.o mHoo.+ Hooo.+ aHoo.+ mooo.+ HoHo.- HHoo.+ ooHo.- oaoo.+ NHHH. Hoa. > Hooo.H mooo.H oHoo.H oHoo.H Hooo.H .H.H H.~ Homo. HH.m ooao ooma HH.H Hooo.+ Hooo.+ a~oo.- oeoo.+ HHoo. Hoa. > mHoo.H HHoo.H aHoo.H Hooo.H aHoo.H HHoo.H oHoo.H omoo.H amoo.H .H.o mmoo.- oHoo.- «Hoo.+ «Hoo.- aooo.- HHoo.- HHHo.- HHHo.+ HaHo. omH. H Hooo.H Hooo.H HHoo.H HHoo.H HHoo.H .H.o H.H HaHo. oH.m ooHo oomH o.oH HHoo.+ Hooo.- HHHo.- oHHo.+ HHHH. HmH. H o muow, ”mo H“ o 2 H HH\ H HH oagoo <\ H o<\HH o<\HH o<\HH o<\e< o .2 " HHHH HHHHHHHo Hfioch 3m mo Huopoaonom .cm «Home 123 o H w H w o .H.NH u A o\ oo Hoe .HH.a u H\ o .Hoom. u HH .Nomo. u H ooHHoHoH Ho HHH HoH .Essfiefie xneecoooH a“ onH use aseflnfie muesfinm a“ NmH .Hcoflue>uomoo mmc Homeueoo o>u3o unmflfi > och . u H H . . H H . . H . . o HNH u A o\ no one H HH u o\ o HaHm u 4 «com u H ooHooHoH NH HHH Hoo .Essficme kneocooom HH emH use 5:5«cwe xuesfinm cm mmH .Hcomue>noHno Hmo Homeucoo o>u3o acuma m oak .uwmeehu e H“ ssefiewa Anesfium peg» eofiumssHHe ocu co ooHeo one Hcofiuofiom Hooo.H o.o H HHo.H oHoo.H Hooo.H aN.o H HHoo.H .H.o HHHH. moNH. NoHa. HH.H- Hooo.- Hoqo. Nomo. o.N - mHH. NNHN. HmHH. Na.NH HoHH. H> Hooo.H Hooo.H o.H H oHo.H HHoo.H mHoo.H HH.o H omoo.H .H.o mHHH. HHHH. NoHa. HN.H- aooo.- oomo. oomo. o.o - Hoa. HHaN. mHHH. aH.HH oHoH. N> Hooo.H mmHo.H H. H HHo.H mNoo.H Hmoo.H Hooo.H .H.o HooH. oNoH. NoHa. HH.H- oNoo.- NaoN. NNmo. ooHa. a.H - aoN. HaoH. ooNo. o.oo HoNa. H> oNHo. oHHo. H. H. aHN. HHH. H.Na NH. HHHHHHH > Hooo.H Hooo.H H.H H HHH.H aooo.H oeHo.H ooHo.H .H.H HHo.H oHo.H Hoao. HN.H- oooo. Hmoo. Homo. H.oH- oHH. oHaN. oooo. aH.HH NHNH. NH Hooo.H Hooo.H H.H H aHH.H aHoo.H oHHo.H aH.o H oaNo.H .H.o mHo.H NHo.H HmHo. HH.o- Hooo. Hmoo. HHHH. o.m - HHm. HHHN. mHNo. HH.oa oaoH. HH oHo. HHo. aHN. HHH. H.Na oNH. HHHHHHH H “N-oHH aN-oHH a -oHo oo H H ”w HH {HH HH HH H nHHHHo n. 28% mouBo HH H H H . H aNafl u no HHoHaHHH Hoooz HHHHHHHHH HHHHH>HooH .o a.o.H:ouo oN HHHHH VI. SUMMARY AND CONCLUSIONS To summarize, we have discussed the transformation from the model of similar tri-axial ellipsoids to the spherical model for an eclipsing binary system. KOpal's method and the method of differential corrections were discussed. Both methods were generalized to include third light. The method of differential corrections was further generalized to include orbital eccentricity directly. Synthetic light curves were used to validate the computer programs, as well as to determine the effect of dispersion and number of observations on the ability to extract the desired parameters. Analysis of synthetic data indicated limb-darkening coefficients may be extracted from observations of sufficient accuracy and density. This conclusion was found to hold for partial as well as completely eclipsing systems. In addition, it has been found possible to extract values of third light. In some cases, however, correlation between parameters, combined with observations of insufficient quality or quantity, may prevent convergence. The data from 10 eclipsing binary systems have been rectified and subsequently analyzed using differential corrections. The systems are CO Lacertae, CM Lacertae, RX Arietis, V338 Herculis, Y Leonis, RW Mono- cerotis, BR Cygni, BV 430, BV 412, and SW Lyncis. It was often necessary to solve the light curves for each combina- tion of assumptions as to type of primary minimum and possible presence of third light. Calculation and comparison of o(est.) and o(cal.), the estimated and calculated standard deviations, proved valuable in the determination of convergence. Equality of the standard deviations of the Fourier analysis and the standard deviation of the entire light 124 125 curve indicated the adequacy of the fit. For those systems in which b was varied, choice of b, the exponent of the light in the weight, did not seem to cause significant change in the parameters obtained. Excellent results were obtained for the systems CO Lacertae, CM Lacertae, RX Arietis, and Y Leonis. For these light curves convergence was obtained for the standard set of parameters (rg, rs, coszi, L , Ls, xg, xs, and, if necessary, L3). For each of these light curves the standard deviation for the entire light curve was in good agreement with the standard deviation obtained from the Fourier analysis of the non-eclipse variation. In addition, for the systems with multi-color observations the resulting geometric parameters from the separate curves showed good agreement. For the B light curves of BV 412 and BV 430 convergence was obtained only if the number of variables in the parameter set for a given iteration was limited to six. The V curves converged with a complete set of seven variables. Resulting geometric parameters and standard deviations showed satisfactory agreement. The iterative analysis of V338 Herculis and RW Monocerotis exhibited satisfactory convergence for the entire set of seven para- meters (again excluding e, w, to, and possibly L The agreement of 3)° the geometric parameters from the individual color curves is good for V338 Herculis. However, for each of the light curves of these two systems the standard deviations of the Fourier analysis is not in good agreement with the standard deviation obtained for the entire light curve. Further observation and analysis is indicated. The V light curve solution of BR Cygni was satisfactory; however complete convergence was not obtained for the B light curve. Similar 126 results were obtained for SW Lyncis. It is felt that the parameters for BR Cygni and SW Lyncis should be viewed with reserve until further observations are made. With the exception of V338 Herculis, the dispersion of the obser- vations was larger for the B light curve than for the V light curve of each system. It is interesting to note that for the systems where difficulty in the convergence of one curve occured, lack of convergence was in the B light curve. Of the ten systems studied, two (CO Lacertae and BR Cygni) showed evidence of third light. Values obtained for V and B limb-darkening coefficients and their probable errors are given in Figures 13 and 14. The theoretical results are given in the figures for comparison. The theoretical values for spectral types BO through A0 are from Grygar (1965). The remainder of the theoretical values result from least squares fits of the model stellar~atmospheres limb darkening given by Gingerich (1966) and Margrave (1969) to the linear limb-darkening law, equation (1.28). Results for the limb-darkening coefficients in V show reasonable agree— ment with theory, while limb-darkening coefficients in B show somewhat more scatter. In conclusion, it is suggested that greater numbers of high quality observations are needed to reduce the uncertainty of the limb- darkening coefficients. Of the 20 limb-darkening coefficients given in Figures 13 and 14, only four had more than 150 observations in the corresponding minimum. Light curves containing 300 observations per minimum should provide satisfactory determination of the corresponding limb-darkening coefficient. 127 .> cw Hucoflommmooo mcficoxnevunafiq oaks Heywoomm .HH HHHHHH HHHHaH 2H NHH >H one >H HHHao HH HHHooHo: HHH> HHHHHH< xH vduhm ow..— EU oeuhooeq ou ucoaomaoo HNMVBQO‘DO H xueoeoooH O unoeomaoo xueafium AU osfie> Heomuonoonu d J H.o N.o m.o e.o m.o o.o m.o o.o 128 L9 .m a“ Hpcomommmooo mcmcoxneounswq omxh Heywoomm .HH oHoHHH j HHuoaH 2H NHH >H omo >H HHHao HH HHHHHHH: HHH> HHHHHH< NH ONHHOUMA 2U omuhmuma 00 H HNMVNQO‘C neocomaoo xHeocoooH O unonomaoo Aneafinm AU ozHe> Heowpouoogu . »\ H.o N.o m.o e.o m.o o.o 5.9 m.o m.o o.H APPENDICES APPENDIX A. DISTRIBUTION OF APPARENT BRIGHTNESS ON THE STELLAR DISK The discussion of limb darkening presented here follows closely the discussion given by Kepal (1959, p. lSOff). However we have used x for the limb-darkening coefficient rather than u. Temperature variation in the semi-transparent stellar atmosphere results in an apparent surface brightness that is dependent on the angle of foreshortening. Radiation viewed normally originates, on the average, at greater stellar depth than that viewed tangentially. Assuming the semi-transparent atmosphere represents such a small fraction of the total stellar radius that it can be regarded as plane- parallel layers, the equation of transfer of the radiation is cos y g%-= Kp(B - I) , (A-l) where I(r) is the intensity of the radiation at a distance r from the center of the star, B is the source function (emissivity), v is the angle of foreshortening (angle between the radius vector and the line of sight),and K is the coefficient of Opacity and p is the density of the stellar material. (More complete discussion of the equation of transfer is given by Mihalis (1970, Ch. 1).) Define the Optical depth T such that d1 = -Kp dr ; (A-Z) then U—=I'B ) (A’s) where 129 130 u = cos y . (A-4) Assuming that the energy sources in the atmOSphere are neglibible, so that the atmospheric layers merely transmit radiation without gain or loss, the net flux of radiation is n 1 F = 2 f I sin y cos y do = 2 f In du . (A—S) 0 -1 F is thus constant and independent of T. Also under the assumption of negligible atmospheric energy generation, the source function (giving the radiation emitted at a point) will be 1 " 11 3(1) =§IOI51nvdy=§fl Idu . (A-6) The source function consists of incident light from all directions. Combining equations (A-3) and (A-6), we have T.‘ (LIO- H H 1 =I-%f1du. (A-7) -1 Equation (A-7) is an integro—differential equation for the intensity I(T,u) of radiation at any optical depth T in an arbitrary direction y. It describes the radiative transfer of energy which is absorbed and re- emitted (or isotropically scattered with unit albedo) in the plane- parallel atmosphere. The two boundary conditions are that the net flux is constant and independent of T and that no radiation is incident on the star, thus 1(0.u) = 0 (A-8) for 0 2 u 2 -1 . (A-9) 131 Equation (A-7) has no known closed form solution. But, for the case of interest (T = O) Wiener and HOpf (1931) have shown the solution to be TI/2 _1 I(O,u) = 1%: F exp %_ 6 tan (u tan 6) d6 (A-lO) Vl+p 0 I - O cot O ’ where F is given by equation (A-S). Equation (A-10) may be expanded in a Taylor series around u = 1. For the linear approximation Milne (1921) has given the result 1(0’”) = 1 - x + x cos y , (A-ll) where x = 0.6. The above result is valid for bolometric observations. Assuming local thermodynamic equilibrium, KOpal (1959, p. ISSff) has shown 2 3 IA(0,u) — BA(Te) (A0 + Alu + Azu + Asp + ) , (A-12) where B). is the Planck function and the Ai are functions of Te(the Optical depth at which the temperature equals the effective temperature), the Planck function and its derivatives evaluated at Te’ and the ratio of the mean absorption coefficient to the frequency dependent absorption coefficient. (The A1 discussed here are not tO be confused with the A1 used in the Fourier expansion of the non-eclipse variation Of the light curve.) In addition, KOpal (1959, p. 158) has shown that third-order theory may be approximated by the linear theory, so that in adOpting the linear limb-darkening law 132 J(v)/J(0) = (1 - x + x cos Y) (A-13) the coefficient x is given by 32(A + A ) + 30 A x = 1 2 3 (A-l4) 32(AO + A1) + 28 A2 + 25 A3 Explicit expressions for the A1 in terms Of the parameters discussed above are given by Kopal (1959, equations (1-24)). Thus, in general, we expect the limb-darkening coefficients to be a function Of effective wavelength and spectral type (effective temperature), as well as the absorption coefficient of the stellar atmosphere. APPENDIX B. VON ZEIPEL'S THEOREM H. von Zeipel (1924) proved the emergent radiation flux at the surface Of a rotationally or tidally distorted star in radiative equili- brium is prOportional to the local gravity. The version of the derivation of von Zeipel's theorum given here follows the derivations Of KOpal (1959, p. l70ff) and Chandrasekhar (1933, p. 539ff). If pr is the radiation pressure, Kv the frequency dependent ab- sorption coefficient and Fv the energy flux, then, as has been shown by Mihalas (1970, p. l3ff), the variation Of the radiation pressure with depth in the stellar atmosphere is ——=.3 K F dv (3—1) c v v , where c is the velocity of light, p is the density of stellar material and 2, measured normal to the surface, increases outward in the atmos- phere. Scattering has been excluded. Defining the mean absorption coefficient K as l K - E. J KV Fv dv , (B-2) 0 equation (B-l) becomes dp __r.=_E_F (8-3) d2 C s or more generally ‘* -_Ea+ _ Vpr - c F (B 4) + (Motz 1970, p. 101), where F is the total energy flux. 133 134 Assuming all energy is transported by radiation (the condition of radiative equilibrium), conservation Of energy requires -> + V ° F = so , (B-S) where e is the rate Of energy liberation per unit mass. Substituting .+ for F from equation (B-4), equation (B-S) becomes “*. _ ...-ae__ _ V ( Vpr ) . (B 6) Expressing pr as a function Of P, the total pressure (gas plus radia- tion), the left-hand side Of equation (B-6) may be written as 3 ‘* 1" - L _1._13.r _ v.($vpr)—W§3X[K3X:] (B7) 3 _ a l__P_1._P. ago-[.3 .31] 03-8) d 2 _SL _1__Er_ .1. 11’. - dP K d? p dn .l—LPI‘ $.[l‘6p] (8-9) KC") 0 9 where we have used 2 3 2 dP _ .32 [am] am.) W 1-1 1 and measured n normal to the surfaces of constant potential. If the force on a particle arises from a potential V, then we may write the equivalent potential for motion with respect to axes that are rotating with constant angular velocity o about a polar axis x3 as V = V +-l-w2(x 2 + x22) . (B-ll) 135 (See for example Danby (1962, p. 47).) Thus for the rotating star with gravitational potential V + + 2 V ° V? = - 4NGO + 2w . (B-12) If hydrostatic equilibrium is assumed then + -> VP = p VW (B-l3) and equation (B-12) becomes + 1 + 2 V - S—VP = -4nGp + 2w . (B-l4) Combining equations (B-6), (B-9), and (B-l4) we have 60 C 2 d [ dp ) = g 51‘ ( ZflGp — 012) — (3-15) Along an equipotential surface the right-hand side Of equation (B-lS) is constant (KOpal 1959, p. l70ff), thus dP ] l.[ dn = constant . (3-16) surface 0 ]2 surface If it is assumed the constant is non-zero, the equipotential surfaces must be equidistant. But, in a rotationally or tidally distorted star, this is not possible. Thus the constant is zero and for each equi- potential surface d l dPr _ 33(de )‘0 ’ (8‘17) since the pressure gradient is non-zero. Then Idpr_ E'HP' - constant . (B-18) 136 + Thus for the normal component of F we have Fn=-EB-Efi'=“:p—a$ a?!- , (3-19) and 1 dP dV FD '3 ; 'd—n- a]; (B-20) Thus we expect that at the boundary of the star the intensity H Of total radiation emerging normally from the atmosphere should vary as —— = —-———- (3-21) where g and go are the local and mean surface gravities and H and Ho are the corresponding intensities. Further, KOpal has shown that assuming black body radiation the surface brightness “A at a particular wavelength A may be expressed as “A _:1_y[1-§] (8-22) H gO : 0 where 1 T dB y - -4— [§ d—T] .1" (3-23) 8 . T is the actual temperature of the atmosphere and T6 is the effective temperature. B is the Planck function. The theory of stellar atmOSpheres thus indicates that the limb- darkening coefficient and the gravity~darkening coefficient are not independent. (See KOpal (1959, p. 159 and p. 172).) The adOpted 137 values of the gravity-darkening coefficient as a function of the limb- darkening coefficient are those suggested by Russell and Merrill (1952) and tabulated by Jurkevich (1964, p. 186). A tabulation of y as a function of x is given in Table 21. The values of N as calculated by equation (2.31) are also given in Table 21. Table 21. Gravity Darkening as a Function of Limb Darkening. x y N 0.4 0.08571 2.2 0.6 1.00000 2.6 0.8 1.2278 3.2 1.0 1.2500 3.6 APPENDIX C. LUMINOUS EFFICIENCY CALCULATIONS Portions of the following treatment have been adopted from Jurkevich (1964, p. l40ff) and Linnell (1971). Let Lh.band Lc be the intrinsic bolometric luminosities of the b hotter and cooler stars. Then L I Q ---1 h,b — h,b 4111'h (C-1) and 4 2 c,b O Tc,b 4111‘c , L (c-z) where rh and rc are the radii of the hotter and cooler stars expressed in physical units and T and TC are the corresponding effective h,b ,b temperatures. The total energy from the hotter star intercepted by the cooler star is 2 r ALh = L —C— (c-s) C h 432 , where a is the separation of the stars expressed in physical units and Lh/41rrh2 is the surface luminosity Of the inner hemisphere and includes heating by the radiation Of the cooler star. Similarly = L — . (c-4) In deriving equations (C—3) and (C-4) it has been assumed that all of the incident external radiation is absorbed and that the "heating" is uniform over the inner faces. Thus for the inner hemisphere of the hotter star we have 138 139 1 _ 1 c _ 4 2 E-Lh - 51h,b + ALh - 0 Th znrh , (C—S) where Th is the effective temperature of the hotter face Of the hotter component. Similarly, for the cooler component l-L = l-L + AL = o T 2nr (C-6) 2 2 c - Define the luminous efficiency, with o the Stefan-Boltzmann constant, as JT(A) 0T4 ’ E(T) = (C-7) where JT(A) is the wavelength distribution of the emitted energy. Then 1 4 2 Eh( -2--Lh,b + AL; ) = Eh oTh 2nrh (C-8) (C-9) 2 JTh(X) Zurh Define the increase in radiation at effective wavelength A caused by the incident external radiation as 28 . Then h 25 = E (T )( l-L + ALc ) — 1-5 (T )L (c-10) h h h 2 h,b h 2 h h,b h,b = E (T )ALC + l-[E (T ) - E (T ))L (c-11) h h h 2 h h h h,b h,b ° If, as is customary, it is assumed that the change in effective tempera- ture is small, then c zsh — Eh(Th)ALh (c-12) Similarly for Sc 25 = E (T )ALh (c-13) c c c c ' 140 Thus we have for the ratio of reflected lights _ = _S___. = —-———— ((3-14) = —E_.———— (c-15) h Substituting for T4 (see equation (C-7)), we have = (C-16) If it is assumed that the stars radiate like black bodies, Planck's law gives the emergent surface flux distribution as a function of wave- length, c1 A's JT(A) = ‘E;7XT“" (c-17) e — 1 Here c1 and c2 are Planck's first and second radiation constants, A is the effective wavelength, and T is the absolute temperature. The total emergent surface flux is given by the Stefan-Boltzmann relation Jb = 0T4 , (C-18) where o is the Stephen-Boltzmann constant. Let x = __. (c-19) 141 and define J;(A) as The luminous efficiency is E(X) = JTCA)/Jb = 3;;—;' e -1 Define E’(x) as E(x) 4 E’(x) _ _ x cl/(oc24A) ex-l The maximum value of this function is U1 ll 4.7798404 , max which occurs at x , where max x 3.9206904 max / (Jurkevich 1964, p. 143). The values of T and J max T,max as ) T = c2/(Axma max X and S / = (Tmax xmax) T,max (exmax _ 1) (C-20) (C-Zl) (C-22) (C-23) (C-24) then follow (c-zs) (C-26) Note that A is fixed for a given set of Observations and this discussion relates the maximum value of E to an effective temperature. For the known spectral type a value of Th is Obtained from a plot of color temperature versus spectral type, as in Figure 15. The data for 142 this plot is from Harris (1963). We then have xh = cz/(ATh) (C-27) and 5 JTh(A) = -3q;-- (c-zs) (e -1) The values of J; (A) and E’ normalized to the values at x are h h max J “1 --JI A J’ A (:29 Th( 1 - Th( 1/ max( 1 ( - 1 and E“ - E’ /E’ (c-so) h - h max ’ where x 4 I h Eh = T— (CI-31) (6 -1) With this normalization the range Of En is [0,1]. h The spectral type of the cooler star cannot always be determined. An alternate method of determining TC is thus necessary. The ratio of the mean surface brightnesses Of the smaller and larger component is given by L. 1-£(p=p ) . = Y(k,po) O occultation (C-32a) _§. Jg 1—l(p=po) transit where pO is the geometrical depth of maximum eclipse and is defined as as -l for total eclipses (KOpal 1959, p. 338 and p. 348). Approximating Y(k,po) as unity (Kopal 1959, p. 343) 143 J 1-£(po) occultation __5_ J 1- £(po) g transit (C-32b) Associating the deeper eclipse with the eclipse of the hotter star and defining a_as the ratio of surface luminosities Of the hotter and cooler components : JThCA) 2 1-IL(pO)primary minimum___ a JTc(A) 1-'9'(po)secondary minimum Thus .1’ (1) 1 (T x )S ’ T h h J (A) =._JL.__= _. T° a a (exh-U D S 5 (Tcxc) = 1_(Thxh) (eXC-11 3 (e‘h-11 Solving equation (C-35) for xc, we have xC = 1n[ E(eXh-l) + 1 1 and thus TC = cz/(Axc) Continuing with the calculation for Sc/Sh’ we have 5 I (T x ) C (8 C_1) : with its normalized value JT:(A) = J{c(11/Jéax(1) , and (c—ss) I“ (c-34) ’- (C-35) (c-se) (c—37) (C-38) (C-39) 144 4 , x E = ——°—— (c-4o) C xc (e '1) a with its normalized value E" - EI/E' ((1-41) c - c max ' Finally, combining equation (C-29) and (C-39) with equation (C-16) we have n n 2 S JThcx1/(Bh1 C sh JT2(A1/(E212 In summary, we have assumed: (1) Stars radiate like black bodies. (2) A11 incident radiation is absorbed and re-emitted at the effective temperature of the absorbing star. (3) The luminous efficiency is not significantly changed by the external incident radiation. (4) The heating is uniform over the inner face Of the star. 145 Effective temperatures and color temperatures from Harris (1963). 40 - 4250A T (103 0K) 5000A 30 _ Teff 3500A 20 _ 10 H. F — h- _ B A F G Spectral Type Figure 15. Effective Temperature and Color Temperature. APPENDIX D. THE FUNCTIONS xf°c AND xftr As was mentioned in Section A3 of Chapter II, it is convenient to express the geometrical dependence of the theoretical value of the light of an eclipsing binary as a function of the ratio of radii k and the geometrical depth of eclipse p. Thus x 2 = u - £(k.p1L . (0-11 where 2 is the theoretical value of the light of the eclipsing binary, U is the unit of light, L is the total light Of the eclipsed star (with limb-darkening coefficient x), and xf(k,p) is the fraction of the light L lost by eclipse at geometrical depth p. The fractional loss of light for occultation is expressed as x oc x oc f (k.p) = or 091)) . (D-Z) while the fractional loss Of light for transit is xf”(k,p1 = chk1 Xa”(k.p1 . (0-31 where XT(k) - 3(; i) k2 + 3 {xx 10100 (D-4) with 10T(k) = %-( sin‘l/E'+ %-(4k — 3)(2k + 1)¢ETT":‘E7‘) . (D-S) The a function is the fractional amount of light lost at geometrical depth p normalized to the fractional amount of light lost at internal tangency (p = -1) for the respective eclipse. In turn, the a functions 146 147 may be expressed in terms of the a functions for uniform and completely limb-darkened stars, 0a and 10a, respectively. xaoc = 3(1 - x) 0o + 2x 10aoc 3 - x 3 - x xatr = (1 _ utr)a0 + where tr _ x o - 1 - x + x¢ and 2 4» = ——7 ”mo 3k tr tr u 0 Thus 3 ’ (D-6) (D-7) (0-8) (D-9) These relations are discussed by Irwin (1947) and KOpal (1950, p. 34ff). Merrill (1950) gives the generating equations for the various a functions. The special forms used in the computer routines are given by Linnell (1965a,b; 1960a,b,c). APPENDIX E. PARTIAL DERIVATIVES OF xfoc AND xftr axf axf axf axf x The generating expressions for arg, ars, 86 , and 3x , where f is xf0c or xftr, have been given by Kopal (1946, p. 78ff) and Irwin (1947, p. 386). The function xf is a homogeneous function of 6/rg and 6/rS of order zero (Kopal 1950, p. 88). The dependence Of xf on coszi, e, w, and to is through 6. Previous methods for including the effects of orbital eccentricity on eclipsing binary light curves employed "fictitious" elements and were correct to second or third order in the orbital eccen- tricity (Kopal 1950, p. 106). A more direct approach is possible. We have axf = axf as ecoszi 36 ecoszi , (E-l) if: . 8311i (5-2) Be 36 3e , axf _ axf as aH ' as aw , (E-3) axf _ axf as at ‘ as at , (E-4) o o where 1’ 6 = R(sinze sinzi + coszi)2 . (E-5) Here 2 R = 3(1 ' e ) (5'6) 1 + e cos 0 where a is the semi-major axis of the orbit (taken as unity), u is the 148 149 true anomaly measured from periastron, and 6 is the phase angle measured from minimum 6 (primary minimum). The phase angle 0 is defined by e = u + H - 90° , (E-7) with the true anomaly 0 given by 0 [1+6 E tani- = 1:;- tan??- (5.8) (Binnendijk (1960, p. 101)). The eccentric anomaly E needed for equation (E-8) results from the solution Of Kepler's equation M = E - e sin E , (E-9) where M is the mean anomaly with respect to periastron _ 2“ _ - M — T (t tpp) . (E 10) (Binnendijk (1960, p. 102) discusses Kepler's equation in greater detail.) Here t is the time Of periastron passage. We wish to express M is terms Of to, the time Of minimum 6 at primary eclipse, since to is more easily estimated from the light curve. Let the sub- script "o" refer to a quantity evaluated at minimum 6, 60. Then the mean anomaly at minimum 6 is M = -—— (t - t ) . (E-ll) Substituting tpp from equation (E-ll) into equation (E—lO) we have 2n M - -F-(t - to) + MO . (E-12) We can Obtain the value of MD from the geometrical and orbital parameters 150 by integration of Kepler's second law. From KOpal (1946, p. 94) for the time interval (t2 - t1), we have U 2n 1 2 2 —§-(t2 — t1) — —§-———-—E- 8 R dU (E-l3a) a l - e U1 3 U = (1 - e2)"2 2 1 dU (E-13b) U1 (1 + e cos 0) ’ where the indefinite integral is evaluated as S do = 1 [ 2 tan-l[ 1-e tan a} e sin U_] (l + 6 cos 0)2 l-e2 Jl-e2 1+6 2 1 + e cos 0 (E-14) and u and u are the true anomalies at t and t respectively. Thus 1 2 l 2 applying (E-13) and (E-14), we have for MD M = -—-(t - t ) (E-IS) -e UO ]_ eJl-e2 sin U0 1 + e cos uo (E—16) II N H 93 :3 I r——-\ 9 where u = a - H + 90° . (E-17) O O The required value Of 00 is derived from the evaluation of the minimum geometrical depth of eclipse 60. The expression for 6 is 2 . 2 . 2. 2. 1/2 = a(l - e )(51n 0 Sin 1 + cos 1) (E-18) 1 + e cos(6-w+90°) The requirement for minimum 6 is given by-%% = 0. From Kopal (1950, p. 106) this expression is 151 (l-e sin(e -w))sin2i sin 26 + 2e cos(6 -w) (l-cosze sinzi) = 0 O O o O (E-19) This can be solved numerically for 60. Thus we have 2. 60 - 60(cos 1,e,w) . (E-20) With the evaluation of 00 the equations necessary for the evaluation of MD are complete. The evaluation of the derivatives of 6 with respect to coszi, e, w, and t0 is required. This proceeds as follows: We have 1/ 6 = R (sinze sinzi + coszi) 2 (E-21) and 2 _ a(1-8) R - l + e cos 0 (E-ZZ) Thus 35 BR 0 R2 2 89 37c....= _E—.§-+ -E-51n 1 $1n 6 cos 6 52—- ’ (5‘23) 0 O O 36 BR 6 R2 2 30 c0526 ____§_.= __——7f—'_'+'_- [sin i sin 0 cos 6 ————§—-+ (E-24) 3cos i 8cos i R 5 Bees 1 2 ’ 85 BR 6 R2 2 BO 33': SE'R'+ —E-51n 1 Sin 9 cos 0 53- , (5‘25) as aR s 2 2 as 56-: 55-§-+ -E-51n 1 Sin 0 cos a 35- , (E-26) where from (E-7) 30 _ 30 at ' at , o O 36 = Bu Ecoszi Bcoszi , 36 80 32.- 1 22. am ‘ aH , and GR = R e sin 0 30 3t 1 + e cos 0 at , O 0 8R = R e sin u 30 3coszi l + e cos 0 Ecoszi , 8R _ . 30 53-- ( R 3 Sin 0 53-- R cos 0 - 2 a e ) / (l + e cos u] , 8R _ R e sin 0 30 55" l + e cos 6 8w Solving equation (E-8) for u we have l-e tan f U = 2 tan-1[ 1+e E ] The partial derivatives of u with respect to to, coszi, e, and thus a“ “6 coszaa [1 +1'e Hm] LE at l-e 1+e 2 at , o o -§2——-= 1+e cosz(%) [1 + 1;: tan2[%]] -§jij?— acos i '6 acos i : (E-27) (E-28) (E-29) (E-30) (E-31) (E-32) (E-33) (E-34) (E-35) are (E—36) (E-37) From equation (E—9 Implicit different where we have used and ) and equation (E-12) we have 2n _ . —§-(t - to) + Mo — E - e $1n E iation of this equation gives a . —' 2. OCOSZI R 8cos 1 , 25-1.3149. Sui-R861 ) 21140 at ’0 O R = a(l - e cos E) (E-38) (E-39) (E-40) (E-41) (E-42) (E-43) (E-44) (E-4S) (E-46) (See Bennendijk (1960, p. 101) for a proof of the latter equation.) 154 Repeating equation (E-l6) we write the expression for M0 in terms Of e and O O - - u 1 - 2 ' M = 2 tan 1 ’l—E—tan -9- - e 1 e $1n U0 (E-47) o 1+e 2 l + e cos U0 . Thus the derivatives of MD are 3cos i 300 3cos i , EN = BMQ 300 - :an 30 2 j1+e 2 2 Be -e u _ OUO 38 1 + 1+e tan (20) 1 e (1+e) _ sin U0 l—e2 _ e2 (E-49) 1 + e cos 0 l + e cos 6 l-e o O ' a a Go 300 So , where 3Mo = 2 'l-e 1 3 l-e 2 go 1+e 1 + e cos 0 U0 1 + m tan (2 ) O eJl-e2 e sianO - cos 0 + (E-Sl) 1 + e cos 0 O 1 + e cos 0 . O 0 With 0 00 = 60 - w + 90 , (E-SZ) we have 300 300 = ——-———— (E-53) 2. 2. 8cos 1 Boos 1 , 155 3U 36 O o -— = -— E-54 3e 8e , ( ) 30 30 ——O = -1 + O (E-SS) 8m 8w , Since 60 is the root Of equation (E-19), we may differentiate equation (E-19) implicitly to Obtain the derivatives of 60 with respect to coszi, e, and w. With D = 2(1 - e sin(60-w))sinzi cos 260 + e cos(60-w)(sinzi sin 260) - 2e sin(eo-w)(1 - c05200 sinzi) , (E-56) we have 360 = ((1 - e sin(6 -o)) sin 20 - 2e cos(6 -w) c0526 ) / D . O O O O ’ BCOS 1 (5-57) 36 2 2 2 0 _ _ _ _ . . . _ . . . SE___ ( 2 cos(eo w)(l cos 60 $1n 1) + $1n(eo w) Sin 1 Sin 200) / D, (E-58) 392.: (- e cos(6 -H) sinzi sinze - 2e sin(e -H)(1 - c0526 sinizi)/ D 3w 0 O 0 . (E-59) The equations necessary for the evaluation Of the partial derivatives x cc of f and xftr are now complete. If it is desired to calculate the corrections to e and w in the form A(e sin w) and A(e cos w) the following transformations may be applied 8f . 8f cos w 8f 8(e sin w) 3 Sin w BED+ e 55' , (E'6O) af _ 3f sin H at 3(e cos w) — COS w 8e - e 55' . (5'61) APPENDIX F. ERRORS IN THE PUBLISHED DATA This appendix contains a list of points that were found to contain errors in their published phase values. These points are listed with corrected phase values. Also included in the table are points that were omitted from subsequent iterative solutions because their residuals were greater than 3 standard deviations from the calculated light curve. These points are listed without corresponding phase values. It is felt that most of these errors are typographical in nature. Table 22. Errors in published data. Corrected System Color J'D'Hel. Phase Values CO Lac B 2439033.5594 .5141 2439034.3798 .0461 2439060.4350 .9409 V 2438990.4732 .5760 2439029.6320 .9675 CM Lac U 2434595.694 2434643.821 2437201.7242 2437201.7465 B 2434595.694 2434606.872 2434643.753 2434643.813 V 2434595.694 RX Ari B 2437984.6910 .0301 V 2437637.7173 .0400 2437639.6936 2438315.7749 2438398.6477 2438398.6513 V338 Her V 2439648.4767 .9683 Y Leo IR 243663l.7214 .0104 RW Mon IR 2439454.8463 156 157 Table 22 (cont'd) Corrected System Color J'D'Hel. Phase Values BV 412 B 2439036.86266 .86950 V 2439094.72037 .87819 SW Lyn V 2439598.3325 14 observations begining with 2439615.3140 B 13 observations begining with 2439598.3201 BIBLIOGRAPHY BIBLIOGRAPHY Aitken, R. G. 1964, The Binary Stars (New York: Dover Publications, Inc.). Alexander, R. S. 1958, A. J. 63, 108. Baize, P. 1950, J. d. Obs. 333 1. Barnes, R. C. , Hall, D. S., and Hardie, R. H. 1968, P. A. S. P. 80, 69. 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