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University Microfilms international 300 N. ZEEB RD.. ANN ARBOR. Ml 48106 8129048
K u z m a , T h o m a s Jo h n
O N SOME ASPECTS OF THE RADIATIVE INTERACTION IN A CLOSE BINARY SYSTEM
The O hio S tale University PH.D. 1981
University Microfilms International 300 N. Zeeb Road, Ann Arbor, MI 48106 ON SOME ASPECTS OF THE RADIATIVE INTERACTION
IN A CLOSE BINARY SYSTEM
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy In the Graduate
School of The Ohio State University
By
Thomas John Kuzma, B.S.
*****
The Ohio State University
1981
Approved By Reading Committee:
Geoffrey Keller
William M. Protheroe
George W. Collins II
Department of Astronomy It gives me great pleasure to dedicate this dissertation to Michael
Cole and Alonda Crowell for putting up with me while I wrote it, Steven
McGuire, Laura Greene, and Patricia Bobal for many years of friendship and support, Snezna Roglej for pushing me into this in the first place, and my parents for giving me my first telescope when I was ten.
ii ACKNOWLEDGMENTS
First and foremost I would like to express my gratitude to m y adviser, Dr. George W. Collins II, for introducing me to the wonderful world of stellar atmospheres and proving to me that theory can be fun.
I would also like to thank Dr. George Sonneborn and Mr. Kenneth
Carpenter for setting up the computer storage space necessary to do this problem and for many long helpful discussions concerning metallic line blanketing.
Thanks are also due Drs. William M. Protheroe and Paul F. Buerger for providing their model of V Puppis, Mr. Kenneth Rumstay for doing some of the figures for this dissertation, and Ms. Frances Crowell for dilligently typing my illegible manuscript.
Finally I would like to thank The Ohio State University for supplying the computer time necessary to complete this project.
iii VITA
November 17, 1949 ...... Born - Elizabeth, New Jersey
1970-1972 ...... Teaching Assistant, Department of Astronomy, Villanova University, Villanova, Pennsylvania
1972 ...... B.S., Villanova University, Villanova, Pennsylvania
1973-1974 ...... Graduate Teaching Associate, Department of Astronomy, The Ohio State University, Columbus, Ohio
1974-197 5 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio
1975-1977 ...... Graduate Teaching Associate, Department of Physics, The Ohio State University, Columbus, Ohio
1977-1979 ...... Graduate Teaching Associate, Department of Astronomy, The Ohio State University, Columbus, Ohio
1979-1981 ...... Graduate Teaching Associate, Department of Physics, The Ohio State University, Columbus, Ohio
PUBLICATIONS
"On Some Aspects of Balmer-Line Formation in Rotating White Dwarfs." Ap. J., 227, 548, 1979.
"Effects of Stellar Rotation on Spectral Classification." (with A. Slettebak) I.A.U. Colloq. 47: Spectral Classification of the Future, ed. M.F. McCarthy, A.G.D. Philip, and G.V. Coyne, (Vatican Observatory), p. 87, 1979.
iv VITA (continued)
"The Effects of Enhanced Helium Abundance on the Visible Spectra of Luminoius F and G Stars." (with G. Sonneborn and G. W. Collins II), Ap. J., 232, 807, 1979.
"Effects of Stellar Rotation on Spectral Classification." (with A. Slettebak and G.W. Collins II), Ap. J., 242, 171, 1980.
FIELDS OF STUDY
Major Field: Theoretical Astrophysics
Studies in Stellar Atmospheres and Radiative Transfer. Professor George W. Collins II
Studies in Stellar Rotation. Professors Arne Slettebak and George W. Collins II
v TABLE OF CONTENTS
Page
DEDICATION...... ii
ACKNOWLDEGMENTS ...... ill
V IT A ...... iv
LIST OF TABLES ...... vii
LIST OF FIGURES ...... viii
Chapter
I . INTRODUCTION ...... I
II. GEOMETRICAL CONSIDERATIONS ...... 12
III. TRANSPORT OF POLARIZED RADIATION ...... 35
IV. MODEL CALCULATIONS ...... 48
V. RESULTS AND D I S C U S S I O N ...... 63
LIST OF REFERENCES...... 136
vi LIST OF TABLES
Table Page
1. Plane Parallel and Uniform Illumination Incident Terms for the Moment Equations...... AO
2. Model Parameters for V P u p ...... 55
3. Effective Temperatures and Gravities for the Grid of Atmospheres on the Illuminating S t a r ...... 57
A. Effective Temperatures and Gravities for the Grid of Atmospheres on the Illuminated S t a r ...... 59
5. Temperatures and Horizontal Fluxes for the Inner Hemisphere of the Illuminated S t a r ...... 6A
6. Theoretical Spectral Types ...... 129
vii LIST OF FIGURES
Figure Page
1. Star centered reference frames for the binary system......
2. Cross section in the orbital plane of several equipotentials for the Roche model with a mass ratio of % ...... 17
3. The rotated coordinate frame...... 23
4. Star centered, and observed coordinates on the illuminating star...... 26
5. The projection of £ onto the observed plane of the sky 27
6. Relation between the observed coordinates p,f5 and the incident angles 6,4>...... 31
7. Coordinate system to define the emergent angles toward a distant observer...... 33
8. Local coordinate frame to define the incident flux...... 44
9. The binary system V Puppis...... 56
10. Flux distributions for the four models at the sub-stellar point...... 67
11. Flux distributions for the four models at point 2...... 68
12. Flux distributions for the two models at point 3...... 69
13. The incident and intrinsic (non-illuminated) flux distributions at the sub-stellar point...... 70
14. He I 4471 and Mg II 4481 residual intensities for the four models at the sub-stellar point...... 72
15. He I 4471 and Mg II 4481 residual intensities for the four models at point 2...... 73
viii 16. He I 4471 and Mg II 4481 residual intensities for the two models at point 3 ...... 74
17. Temperature structure as a function of for the four models at the sub-stellar point...... 75
18. Temperature structure as a function of xD for the _ , , „ Ross -,r four models at point 2 ...... /o
19. Temperature structure as a function of t^oss for the two models at point 3...... 77
20. Temperature structure as a function of T^Qg for the four models at point 2, indicating the Rossland depth at which the optical depth in the line cores reaches unity for Mg II 4481 (MG), and the permitted (HE) and forbidden component (HE) of He I 4471...... 79
21. The number of electrons contributed per He atom as a function of t„ for three models at point 2. LB is plane parallel Illumination with line blanketing, NLB is plane parallel illumination without line blanketing, and EN is the enhanced flux model...... 80
22. Spectral types for the models at the three points, using monochromatic magnitude differences. LB, NLB, and EN are as in figure 21 while FSA indicates the model with finite solid angle illumination...... 82
23. Spectral types for the models at the three points estimated using the ratio of He I 4471/Mg II 4481 equivalent widths.. 84
24. The departure coefficients for the first three levels of hydrogen for plane parallel illuminated models with (upper) and without (lower) line blanketing at point 2...... 86
25. Flux distributions for model atmospheres at point 2 with and without metallic line blanketing and non-LTE 87
26. He I 4471 and Mg II 4481 residual intensities for the model atmospheres at point 2 with and without metallic line blanketing and non-LTE...... 88
ix 27. Percent polarization as a function of
28. Percent polarization as a function of 29. Percent polarization as a function of for a preliminary finite solid angle illuminated model with the center of the disk at p = 0.5. Also shown are results for plane parallel illuminated models with angles of incidence of p = 0.3, 0.5, and 0.707...... 93 30. Percent polarization as a function of 31. The angle x as seen at 32. Percent polarization as a function of wavelength for three views of the illuminating star...... 98 33. Percent polarization as a function .of wavelength for three views of the illuminated star...... 99 34. Local normals on the illuminating star projected onto the plane of the sky...... 100 35. Local polarization for the illuminating star at 930 X. The values along the limb are shown reduced by a factor of ten. SS is the sub-stellar point...... 102 36. Local polarization for the illuminating star at 5000 X. The values along the limb are shown reduced by a factor of 100...... 103 37. Local polarization for the illuminated star at 930 X. The values along the limb are shown reduced by a factor of ten...... 104 38. Local polarization for the illuminated star at 5000 X. The values along the limb and those with arrowheads are shown reduced by a factor of 100...... 105 x 39. The polarization shown in Figure 35 rotated into the observer's frame...... 106 40. The polarization shown in Figure 36 rotated into the observer's frame...... 107 41. The polarization shown in Figure 37 rotated into the observer's frame...... 108 42. The polarization shown in Figure 38 rotated into the observer's frame...... 109 43. Limb darkening curves for the inner hemisphere of the illuminating star...... Ill 44. Limb darkening curves for the inner hemisphere of the illuminated star...... 113 45. Limb darkening curves for the terminator of the illuminating star...... 114 46. Limb darkening curves for the terminator of the illuminated star...... 113 47. Limb darkening curves in the equatorial plane of the illuminating star...... 116 48. Limb darkening curves in the equatorial plane of the illuminated star...... 117 49. Isophotes on the illuminating star for the inner hemisphere (top), side view, and outerhemisphere (bottom) at 5000 A. Contour intervals are 0.15 magnitudes...... 118 50. Isophotes on the illuminating star for the inner hemisphere (top), side view, and outer hemisphere (bottom) at 930 A. Contour intervals are 0.3 magnitudes...... 119 51. Isophotes on the illuminated star for the inner hemisphere (top), side view, and outer hemisphere (bottom) at 5000 A. Contour intervals are 0.15 magnitudes...... 120 52. Isophotes on the illuminated star for the inner hemisphere (top), side view, and outer hemisphere (bottom) at 930 A. Contour intervals are 0.3 magnitudes...... 121 xi 53. Flux distributions for three views of the illuminating star...... 123 54. Flux distributions for three views of the illuminated star...... 124 55. He I 4471 and Mg II 4481 residual intensities for the inner hemisphere, side view, and outer hemisphere of the illuminating star. The small variations present are masked by the thickness of the line...... 126 56. He I 4471 and Mg II 4481 residual intensities for three views of the illuminated star...... 127 57. The flux distribution from the system seen at phase =0.25 compared with the flux distribution obtained by Protheroe and Buerger...... 130 58. The ratio horizontal flux/normal flux on the surface of the inner hemisphere of the illuminated star seen projected onto the plane of the sky...... 132 xii I INTRODUCTION Binary stars of of paramount importance in astronomy, for it is only through the study of these stars that stellar masses and absolute stellar dimensions may be obtained. Widely separated stars forming visual systems are the basis for mass determinations. However, this sample is severely limited to the nearby stars, as most systems are suf ficiently distant as to make them unresolvable. Also the majority of stars more massive than about two solar masses do not occur in widely separated systems. Thus, eclipsing and spectroscopic binaries offer the largest data base for obtaining masses as well as absolute dimen sions of stars. Since one must deal with the total light of the system, rather than two resolved individual stars, the analysis of these objects is very complex. Not only must the properties of the individual stars be computed, but the effect of each star upon the other must be taken into account. Ignoring such manifestations as gas streams and mass transfer, the most commonly observed results of binary interaction are variations in light, color, polarization and spectral line intensities as a func tion of orbital phase. These variations are primarily due to two effects. First, due to rotational and gravitational distortion, the components will not be spherical and thus the apparent stellar surface area, and hence flux, seen by an observer will vary due to the orbital motion of the system. 1 This so called ellipticity effect manifests itself as outside of e- clipse variations in stars such as the W UMa systems, as well as in the ellipsoidal variables, whose only light variations are due to the changing viewing aspect. The shape distortion in turn will cause a spatial redistribution of the emergent flux. For stars in radiative equilibrium, the net flux is proportional to the potential gradient and is everywhere paral lel to it. The shape distortion will result in non-spherical equipoten- tial surfaces and therefore there will be a variation in flux and hence effective temperature as a function of position on the stellar surface (von Zeipel 1924). For stars in convective equilibrium a similar pro cess is thought to occur. However, the situation is less precisely understood, with some uncertainty in the exact dependence of the flux on the local gravity (Lucy 1967, Anderson and Shu 1979). The second complication introduced by the interaction of the two stars is the misnamed reflection effect. Here, the light from one star will be scattered, and or absorbed and reemitted by the atmosphere of the companion. This leads to non-uniform heating of the companion which could greatly alter the locally produced continuous and line spec trum of the star. Of course, the manner in which the spectrum will vary will be determined by the temperature dependence of the opacity sources in question. Also, the incident flux will alter the tempera ture gradient in the atmosphere, leading to a more isothermal structure, which would cause a weakening of the spectral lines. Since we are dealing with a flux, it is the sum of these local effects that deter mine the observed photoelectric and spectroscopic variations. On a global scale, we have the situation where limb darkening en hances the region of the stellar surface nearest the observer, gravity darkening enhances the region of the surface closest to the center of the star, while the radiative interaction will enhance the substellar point. The net effect will be to yield non-concentric isophotal lines on the stellar surface, and could in fact displace the observed center of the stellar surface from the geometrical center. The changing interplay of these phenomena due to the orbital mo tion of the system, greatly complicates the solution for eclipsing sys tems, and may yield erroneous data for radial velocity studies. Thus, we must consider all the effects if reliable data are to be obtained for these systems. The problem of binary star systems has been the subject of exten sive study during the last 70 years. The backbone of the study into the problem was laid out in the fundamental work of Russell (1912) and Russell and Shapley (1912), where the basis for determining the param eters for a system of spherical stars in circular orbits was developed. Later, refinements were added to remove outside of eclipse variations (rectification) and to account for different amounts of limb darkening (Merrill 1950, 1951, and Russell and Merrill 1952). Nonetheless, these approximations to the problem did not adequately represent the physical variations present in many binary systems. Subsequent research in the binary star problem took on a two step approach. First, theoretical work was done to better understand the physical processes involved in binary star interaction. The result of this work was then utilized to construct models in attempts to repro duce observed light curves. The problem of shape distortion was first investigated by von Zeipel (1924), who looked at tidal distortion of a star by a point source. This was extended by Chandrasekhar to include the effects of rotation and tidal distortion on polytropic configurations (1933) (see also Jackson 1970). This distortion led to the concept of gravity darkening as induced by rotation and tidal effects, and was included as an additional parameter in the Russell method (Russell 1939). The results of this work were utilized by Sterne (1941) who used a combina tion of rotational and tidal distortion to predict outside of eclipse variations in close binaries. Simultaneously, investigations into the radiative interaction of two stars were under way. The problem was first studied by Eddington in 1926, and he showed that the incident radiation would not affect the deep structure of the atmosphere. Furthermore, for a star in radiative equilibrium, the illuminated star would have a bolometric albedo of unity, that is, all of the incident radiation will be reemitted, so the total flux of the star is the sum of the internally generated flux plus any incident radiation. However, for stars in convective equilibrium, the situation is more complex. Rucidski (1969) has shown that for fully convective stars, part of the incident radiation goes into changing the structure of the envelope rather than being totally reemitted, thus giving albedos less than one. Although his results are model dependent, the low observed albedos of several Algol systems (Hosokawa 1957, 1959, Sobieski 1965) can be explained by this effect. Milne (1926, 1930), using a two stream Schwarzchild atmosphere with parallel incident radiation, has shown that it is important to con sider not only the amount of incident radiation, but the angle of inci dence as well. In investigating several systems, he determined that the mean surface temperature of the illuminated atmosphere can be greatly increased, with values obtained for the temperature enhancement ranging from 6% up to 100%. The work of Milne was then utilized by several investigators to predict theoretical changes in binary bolometric light curves. Takeda (1934) used two equal ellipsoids to determine the effect of gravity darkening, but assumed spherical stars to investigate the effects of illumination in close binaries. This work was extended by Sobieski (1965a, 1965b) to include small departures form grayness in the atmo sphere, and was used to calculate monochromatic reflection coefficients for several systems. More recent investigations have calculated the gravity and temper ature variations over the surface of a tidally distorted, limb darkened Roche model (Chen and Rhein 1969, Peraiah 1970, RuciAski 1971). Also several authors have made model systems including the effects of plane parallel illumination, limb darkening and gravity darkening to varying degrees of sophistication (Hill and Hucthings 1970, Wood 1971, Wilson and Devinney 1971), and were able to reproduce light curves for several systems. However, with the advent of good model stellar atmosphere tech niques, the binary problem can be attacked much more rigorously. RuciAski (1970) has modeled the system Sco, which is composed of an early B primary and a late B companion. Using plane parallel illumina tion and LTE model atmospheres, he computed the emergent flux of the secondary at two points on the illuminated side, and one point on the non-illuminated side. For the substellar point, he found that the flux intensity corresponded to a spectral type of B4 but there was a redis tribution of the flux in the Balmer continuum, in that there was an UV excess just shortward of the Balmer discontinuity, while the flux in the far UV was below that for a normal B4 star. This could account for the UV excess observed in many systems (Devinney et al. 1970) without re sorting to composition anomalies. Further work by Muthsam (1973) has shown that the temperature structure of the outer layers of an illuminated star is strongly af fected by the spectral distribution of the incident radiation, rather than just the total amount. However, the general trend in each case investigated was to create a temperature rise at the surface. Further more, in hotter stars, that is stars in which H“ is not the dominate opacity source, non-LTE effects can similarly alter the temperature structure. This temperature rise can be further enhanced by an in crease in metal abundance in both LTE and non-LTE models. Thus is it plausible that the inclusion of UV metal line blanketing in the calcu lations will add further to the temperature rise at the surface, which could greatly alter the shapes of spectral line profiles. However, not as much work has been done on the problem of spectral line formation in binary systems as has been done on the shape distor tion problem. The formation of spectral lines in rotationally dis torted stars has been investigated for both weak lines and the Balmer absorption lines by Slettebak, Kuzma, and Collins (1980), Collins and Sonneborn (1977), Collins (1969) and Collins and Harrington (1966). This work has shown that the lowering of the surface gravity and ef fective temperature due to rotation will cause a shift in the observed spectrum to a somewhat later type. Although there will no longer be azmuthal symmetry, a similar effect can be expected from gravitational distortion. The effects on line formation on a global scale due to radiative interaction has been investigated by Ovenden (1963). He has shown that illumination could give rise to lines that have central wavelengths that are shifted with respect to lines that would be formed if the region of highest light intensity corresponded to the geometrical center of the stellar surface. This result has important ramifications for radial velocity studies. The most ambitious work to date is that of Buerger (1969). Using a Roche configuration, he computed light and spectral line variations for two binary systems. The model atmosphere code ATLAS was used to compute the models which were affected by limb darkening, gravity dark ening, as well as shape distortion. The radiative interaction was treated with parallel incident radiation illuminating the secondary, with the primary being unaffected by flux received from its companion. 8 These atmospheres were then used to calculate color and spectral line variations as a function of orbital phase. In the field of binary star studies, the investigation of polari zation induced by distortion and radiative interaction is a rather re cent innovation. The transfer of polarized radiation in a pure scatter ing gray atmosphere was first investigated by Chandrasekhar (1946a, 1946b, 1947). This work was extended by Code (1950) to include a pure absorption component. The techniques developed here were then used to investigate polarization due to rotational distortion (Harrington and Collins 1968) and tidal distortion (Buerger and Collins 1970). This field of investigation has culminated in the work of Collins (1972) and Collins and Buerger (1974) who have constructed self consistent non gray stellar atmospheres illuminated by an arbitrarily polarized plane parallel incident radiation field. A slightly different line of investigation has been the study of phase locked polarization variations in a binary system by Brown et al. (1978) and Rudy and Kemp (1978). By a series of approximations, they have calculated the polarization due to a single scattering in an op tically thin electron cloud that shows mirror symmetry about the or bital plane. Their results show that the Stokes parameters U and Q will sweep out an ellipse in one-half an orbital period, whose eccen tricity is a direct measure of the inclination of the orbit. Several systems have been analyzed by this technique to yield inclinations even when no eclipses are present. In all binary systems modeled so far, the incident radiation has been assumed to be plane parallel; the origin of this radiation being located at the center of the illuminating star. In reality however, the illuminating star may cover a large portion of the sky. This il lumination from a finite solid angle will affect not only the local atmospheric structure, but the total area that is illuminated on the companion as well (see Kopal 1959, 1978). The purpose of this dissertation is to set up the formalism nec essary to calculate the effects due to illumination of a stellar atmo sphere from a finite solid angle. This will be coupled with previous work on shape distortion, illumination and polarization to form a first step in constructing a realistic model for a binary star system. Since the surface of a distorted star can be thought of as being composed of many plane parallel atmospheres, each having its own dis tinct effective temperature and gravity, by choosing a representative atmosphere the relative effects of line blanketing and non-LTE as well as the spatial distribution of the incident flux may be determined. In turn, by summing over these local atmospheres, the global properties of the star may be represented, giving the brightness distribution over the stellar surface, as well as the total emergent flux and polariza tion as a function of viewing angle. In modeling a binary system, the use of two similar stars would greatly reduce the work involved since both would have the same atom- spheric structure. However, both stars would show similar variations, which would limit the information obtained. Since one wishes to study the effects due to the interaction of the components, two dissimilar stars will produce the greatest variations. 10 In choosing two different stars, we are limited both theoretically and observationally. The model atmosphere code ATLAS 6 is capable of calculating models over a wide range of spectral and luminousity types: roughly B0-G2 and luminousity classes V-I. However, due to the compli cations introduced by convection and uncertainty in the gravity darken ing law for convective equilibrium, it would be advantageous to avoid stars later than about A5. The restrictions due to observational con siderations are more stringent. The two stars must have similar abso lute magnitudes, otherwise the light from the primary will mask the com panion, making its spectral features unobservable to a distant viewer. This would imply utilizing a hot dwarf with a somewhat cooler giant or subgiant companion: the standard early type semi-detached sys tem. There are many examples of such systems including u Her (B2+ B9), y1 Sco (B2 + B6), and V Pup (B1 + B3) . A theoretical light curve for V Pup has been calculated by Protheroe and Buerger (1981) assuming plane parallel illumination with no metallic line blanketing present. For this present work, it was de cided to use Buerger's model parameters, thus allowing for a direct com parison of the emergent flux and polarization for a model with and with out metallic line blanketing and illumination from a finite solid angle. As in the previous work, the more massive star will be assumed to be the illuminating star, and the reradiation back onto this component will be ignored. In addition, to determine the effects on spectral line formation, theoretical profiles for He I 4471 and Mgll 4481 will be calculated. In chapter 2 we will discuss the assumptions going into the model construction, as well as dealing with geometrical considerations. Spe cifically we will set up the coordinates for the binary star system and the defining shape equations. Also the methods for determining posi tions on the stellar surface as measured on the star itself and as viewed from its companion will be outlined. The third chapter will deal with the theoretical aspects of radiative transfer of polarized radiation, including the modifications necessary to take into account illumination from a finite solid angle. In chapter 4 the integration and interpolations schemes necessary to take into account illumination from a finite solid angle will be outlined, and the construction of the atmospheres is discussed. Final ly, the theoretical light, spectral line and polarization calculations are presented in chapter 5. II GEOMETRICAL CONSIDERATIONS In performing any model calculations, simplifying assumptions are usually made; however, these assumptions must be based on observational and theoretical considerations. For the model presented here, the two stars were assumed to be rigidly rotating while revolving in a circular orbit, with their rotational axes perpendicular to the orbital plane. In addition, the stars were assumed to be tidally locked in a synchro nous orbit. Work done by Zahn (1966a, 1966b, 1966c, 1975, 1977) and Kopal (1978) has shown that tidal breaking in stars having either convective or radiative envelopes can occur on relatively short time scales. While this does depend on the ratio of the stellar radius to binary separa tion, observations (Struve 1950, Olson, 1968, Mallama, 1978, Rajamohan and Venkatakrishnan 1980) have indicated that many systems are indeed synchronous. This would imply that the orbits are sensibly circular and that the rotational axes are indeed more or less perpendicular to the orbital plane. Observational evidence for differential rotation in stars other than the sun is marginal at best (see for example Gray 1976) and there fore could be considered as a minor pertubation that can be neglected. Furthermore, as pointed out by Swihart (1964), for gray atmosphere at least, the time scale for an illuminated atmosphere to reach thermal equilibrium is on the order of minutes. Therefore, if the system were 12 13 not tidally locked, the rapid readjustment of the atmosphere would elim inate the possibility of any observational effects being produced, thus giving further, justification to the assumptions of rigid rotation and synchronous orbits. Since it has been shown (Plavec 1958, Limber 1963) that shape dis tortion due to non-synchronous orbits differs considerably from that obtained from synchronous rotation, allowing the system to be tidally locked simplifies the situation, in that it is then possible to express the potential of the system in closed form. As shown by Orlov (1961) the shapes of polytropes with finite cen tral concentration are the same as those that are infinitely centrally condensed, to better than 0.5%. Thus each star can be represented as a Roche configuration, and one can easily obtain the value for the total potential at any point in the system. Surfaces of constant potential are then taken to represent the stellar surfaces. Following Kopal (1959) we pick two masses, m^ and m^ and let their centers be separated by R as shown in Figure 1. We set up a right hand rectangular coordinate system centered at each mass, with the x axes pointed along the line of centers. With respect to the m^ refer ence frame the coordinates, x, y, z, of the center of the mass of the system are then given by m.^ m2 , 0 , (2.1) The total potential at any point P (x, y, z) in the system is determined by three effects. First we have the gravitational potential, G m^/r^, at P due to m.^. G is the gravitational constant, and r^ m m Figure 1. Star centered reference frames for the binary system 15 2 2 2 h (x + y + z ) is the distance to point P from the center of m^. Simi larly, there is the contribution of the gravitational potential due to 2 2 2 h n^, G m^/r^, where r^ = ((x-R) + y + z ) . Finally, an additional term |2 ^((x-m2 R )2 + y2 \ (2 .2) is needed to account for the orbital motion of the system about the cen ter of mass. The angular velocity, w, is assumed to be Keplerian 2 G (m + m ) w = • (2.3) R By using the separation of the centers, R, as the unit of length and the sum of the masses as the unit of mass, we can obtain an expres sion for the normalized total potential of the system. The problem can be further simplied by transferring into a spherical coordinate system by defining ^ cos 0 = aX (2.4) ^ ^ sin 6 cos 4) = ay (2.5) Z IT — = — sin 0 sin <|> = a v » (2.6) R R where 0 is measured from the x axis and tp is measured from the y axis in the y-z plane (see Figure 1)• Thus the expression for the total potential _ Gm Gm 2 m R , „ Ifc_ ^ _ > 2 + Jr2,. (2.7) becomes upon normalization 16 VR n = (2.3) G(m^+ m^) a “— 2~T£ + j[a2 (l-v2)-2qoA + q2], (l-2aX+a ) where q equals the reduced mass, + m2' The surfaces generated by setting ft equal to a constant are called the Roche equipotentials. The special case of the equipotential surface that contains the center of mass of the system is referred to as the Roche limit, for at the center of mass the acceleration of gravity van ishes forming the first Lagrangian point, L^, through which mass trans fer is thought to occur. For values of ft larger than this limiting value, the stars do not fill their Roche lobes, and therefore can be treated as discrete bodies (see Figure 2). tlany of the computations necessary are most easily performed in spherical coordinates, while other forms lend themselves to calculations performed in rectangular coordinates. The transformation between co ordinate systems is performed using the matrix * ■ ’ ^ ■ r COS0 sin0cos<|> sin0sin <8 ► = i-sin0 COS0COS A ♦ 0 -sin (l-2aX + cx2) ~ 2 = ZPn (A)an = 1 + Act + E Pn Q ) a n . n=0 n=2 (2 .10) 2 M Figure 2. Cross section in the orbital plane of several equipotentials for the Roche model with a mass ratio of \ 18 Substituting this into equation (2.8) and rearranging terms we have (fl-q Jsq2)a = 1-qfq E Pn (X)an+1 + JsaV-v2) . (2.11) n=0 Since a is less than 1, a first apprpxiamtion can be made by neglecting the last two terms in equation (2.11) to give 1-q “0 = 2 ~ (n-q->5q*) . (2.12) For the next approximation we let = a0 (1+A'«), which upon sub stitution into equation (2.11) yields 1 ^ M q l P U ) a 0n+1+ ?5a03(l-v2))/(l-q) . (2.13) “° n=2 n Similarly, successive approxmations can be made by letting , A ' a , A A 11 a* a2 = a0(l + ---+ ---- ) , ot0 aD " ' T a. = a0 (1 + 7— ... -— ). (2 .1‘t) J Cio As might be expected, this approximation becomes less reliable in the vicinity of when the star fills its Roche lobe, since the stellar surface comes to a point. To obtain the normalized radius in this re gion, we rewrite equation (2.8) as a polynomial in a by multiplying both 2 sides by a(l-2aX+a ) and squaring. Rearranging terms gives the eighth order equation 19 a8(Jj(l-v2)^ + a7(—1 iX (1-v2) ) + a6 (%(l-v2)^ + 2X2q(l-v2) + (1-v2) (hq2-fi) + q2X2) + a5 (-qX (1-v2) - 2X (1-V2) (igq2—S2) - 2X3q2 + (1-q) (1-V2) - 2qX (^q2-^) ) + a4((1-v2) (^q2-^) + q2X2 - 2X(l-q) (1-V2) + 4X2q0sq2-J2) - 2qX(l-q) (%q2-fi)^) + a3 ((1-V2) (1-q) -2qX(J5q2-n) +4 X 2q(l-q) - 2X (!j-fl) 2 + 2 (*sq2- Jl ) (1-q) ) + a2(-2qX(l-q) + (hq-ti)2 - 4X(1-q)(%q2-fi)+ (1-q)2 - q2) + a(2(1-q)(^q2-^) - 2X(l-q)2) + (1-q)2 = 0 . (2.15) When an approximate value of a real root of this equation has been obtained, its accuracy may be tested by substituting back into equation (2.8) to obtain a calculated value to the potential, If - 3 < 1 0 , n the value of the root is considered to be sufficiently accurate. Be cause the equipotential surface consists of two lobes, more than one root can satisfy this criterion, in which case the minimum root is used. Letting j = 7 in equation 2.14 gives the correct normalized radius to better than 0.5% for 0 greater than 20°. For values of 0 less than this amount, the roots of equation 2.15 were found. Once the normalized radius is obtained, it is very straight for ward to calculate the components of the local gravity at this point on the stellar surface, since they are merely the gradients of the total potential. Thus, ■ '“ f * S S s w * -q sin 0 av2 X g 0 = -3------j , + q sin 0 , (2.17) (l-2aX+a2) sin 0 -gyiv (2 .18) sin 0 ’ with the magnitude of the gravity given by (2.19) As mentioned previously, by von Zeipel's theorem, the net flux is proportional to the local gravity: Fnet = K(q’n)ig| • (2.20) The proportionality constant, von Zeipel's constant, can be obtained from the total luminousity of the star, which is just the flux integrat ed over the stellar surface. Therefore (2.21) L = U Fnet d0 = K(q’a) /A lilda » or K(q,n) = ------(2.22) /A lg|da By specifying the masses, separation of the centers, and the total potential, we may then calculate the radius and gravity which are needed to specify the local atmospheric structure at any point P (6,$). How ever, the local atmospheric structure will be modified by the incident radiation field. The intrinsic flux of the illuminating star (star 2) is determined by the physical parameters of that component. However as one moves over the surface of the illuminated star (star 1), the char acteristics of the observed incident radiation field will change. Due to the shape distortion of star 2, as an observer moves over the surface of star 1 he will be presented with a change in the shape of the apparent surface area of star 2 seen projected onto the plane of the sky. This of course will be accompanied by a change in the 21 apparent surface brightness distribution as different areas of the il luminating star come into view. Furthermore, the position in the sky of the star will change which will also modify the incident flux. This is most obvious in the extreme case where at certain positions on star 1 , portions (or all) of star 2 will be below the horizon. The intrinsic radiation from some point ( ) on star 2 will be determined by the temperature and gravity at that point as well as the angle 0O2 at which the radiation leaves the surface measured with re spect to the local normal (actually it is the cosine of the angle which is needed). For a non-illuminated star the radiation field exhibits axial symmetry about the local normal so it is only the quantity cos 0o2 that need be found. However, since it is the apparent surface and its position in the sky that will determine the incident radiation field, what is needed is the angle 0, the angle the incident radiation from any point on star 2 makes with the local normal on star 1. Moreover, due to non-uniform surface brightness of the illuminating star, there will also be an az- muthal dependence on the incident radiation. This can be characterized by an angle Unlike a non-illuminated atmosphere, the radiation emergent from star 1 will not exhibit azmuthal symmetry. Therefore to obtain the flux as seen by a distant observer we need not only 0iq, the angle the emer gent radiation makes with the local normal, but the azmuthal angle <|>iq as well. 22 Thus we have a two step process. First to calculate the structure of the illuminated atmosphere we need to go from the star centered co ordinates of the illuminating star 02, Of course the problem is symmetric, that is, in the derivations that follow, the component chosen as the illuminating star and the one acting as the illuminated star may be interchanged. For any chosen point 0i,4>i on the illuminated star, we can calcu late the radius and the gravity using equation 2.14-2.18. To obtain the incident angles 0, These coordinates are given by first rotating the star centered co ordinate system of star 2 about its x axis by an angle £, where £ = ■j <^>1. A rotation about the new y axis, y^ » by an angle £ = tan ^(r^sin 0i) where d^ = 1 - r^cos 0^, (see Figure 3) points the x^ axis at the chosen point on star 1 , and the apparent disk of star 2 then lies in the y^ -z£ plane which also defines the plane of the sky as seen from star 1. 02 is then measured from the axis an The transformation between the primed and unprimed coordinate sys tems is obtained through the following rotation matricies. The rota tion about the x axis gives [ 1 0 0 0 cos £ -sin £ V , (2.23) ^0 sin £ cos £j while rotating about the y' axis gives cos £ 0 sin £"\ 0 1 0 f • (2-24) V;-sin £ 0 cos £^J The total rotation matrix is therefore given by ► 1 A. f \ M i' COS £ sin £ sin £ sin £ cos £ i a A 0 cos £ -sin £ > < j > (2.25) r A k' -sin £ cos £ sin £ cos £ COS £ k 1 4 4 Since this matrix is orthogonal, we use the transpose to go from the primed to the unprimed coordinates. With respect to the coordinate system of star 1, the x„' axis de- A fines a direction 0 such that a A a ^ 0 = cos £ i + sin £ sin £ j - sin £ cos £ k . (2.26) This axis makes an angle 0O with respect to the local gravity which may be obtained by taking the dot product 0 • g to give COS 0O = . (2.27) l i l Since this direction points to the center of the apparent disk of star 2 , we may use this line as a reference from which to measure co ordinates on the apparent disk. In the reference frame of the 25 illuminated star, each dz , Figure '4) is measured from so that equals 3 plus the projection of 5 onto the y'-x'plane (angle rj). As can be seen in Figure 5, we have the relations tan 5 = j , (2.28) tan n = > (2.29) e cos £ = - C ’ (2.30) then C tan ^ /o oi\ tan n = ------= » (2.31) A cos £ cos c so that 3 = Of primary importance is the determination of Pmax» the maximum angular extent of the apparent disk in the sky for any given 8. Since the stars are not round p will vary as a function of position along max the limb, and will not generally occur at §z = -|- As can be seen in Figure 4, any angular measure, p, along a radius on the apparent disk specified by a given 8 can be obtained from the relation tan p = _tr2______sin 92* , n U.3-3) d* - r2cos 02 where d'= d^/cos £. Letting 62' increase from zero, causes p to become larger than smaller, reaching a maximum at p max z2 z '2 02. Figure A. Star centered, and observed coordinates on the i1luminating star to O' 27 / A Figure 5. The projection of £ onto the observed plane of the sky 28 Therefore at p max = o . (2.34) d02 Differentiating both sides of equation 2.33 and setting the result to zero, we have dx d tan p _ cos 69' - sin 0? d0?' -1 — d 0 7 “ " ------0 , (2.35) (x - COS 02' )2 or cos 02 = (1+sin 02 )/x . (2.36) p doo max z Since x = — r 2 d02 dx _ x dr (2.37) d02' ” r d02 d02' Also, the total potential SI is a constant, so that dfi = 0. We can therefore write = 0 = f dr+ I f d6+ If d We then have | | d r = ~ffd0, (2.39) or ±r.= - 3fl/30/3fi/3r , (2.40) d0 which is merely Therefore combining equations 2.36, 2.37 and 2.39 we have COS 02* = 1 , • at 80 d02 zp — + sin 0 — — - • /, / n max x g d0' (2.41) &r 29 Using the transpose of the rotation matrix, equation 2.25, we can write cos 02 = cos cos ? - sin 02* sin Differentiating both sides with respect to 6'we obtain sin 02 = sin ©2* cos 5 + cos ©2' sin 02* sin C . (2.43) d02' d02 Solving for -rr-7 and substituting into equation 2.41 yields . . 1 sin 0? cos C + cos 02' sin ? sin qio 8 . COS *<>,.„max ‘ * + ------rrr-g— sin 02 ------fcfVn & 92’ . (2.44) * r This new 02' can then be used to calculate new values of gg and g^ which in turn will generate a new Qz' This process can be iter ated to consistency to obtain Pmax* Thus any point 02, It is this set of coordinates p,8 which are then used to determine 0, A unit vector along the local gravity has components in the star 1 coordinate frame of g , g ,g . The angle given by the tangent of X1 yi Z1 g /g then gives the orientation of the local gravity with respect to Z1 yi 30 the orbital plane which is defined by x, , y, . Since the horizon is perpendicular to the gravity we have y = /tt -1 . .\ \2 - Defining 6 = S+Y we have from Figure 5 sin 0 = sin 6 sin p , (2 .45) where 0 is the angle the line to any point p,6 on the apparent disk makes with 0O. Therefore 0 = 0O - 0 and if 0 is greater than 90°, the point lies below the local horizon, and will not contribute to the in cident flux. Similarly, we can define sin These angles 0, We can define a unit vector 0j, from the illuminating atmosphere on star 2 to the illuminated atmosphere on star 1. This unit vector (see Figure 4) has the components 0 = cos p x 0^ = sin p cos $2 0z = sin ip2 sin p Rotating these into the unprimed coordinate system, we can calculate 0i • -g cos 0o2 “ * 2 , (2.49) I s2 1 where 0O2 is angle at which the emergent radiation leaves star 2 with respect to the local normal. Unlike a non-illuminated star, the emergent radiation from the il luminated star in the direction of an external observer is a function p r o j e c t io n o f LOCAL HORIZON ORBITAL PLANE Figure 6. Relation between the observed coordinates p,B and the incident angles 0, Following Buerger and Collins (1970), if we define a unit vector i0 in the direction of the observer, the theta dependence is merely cos 0io = io’ ijj, where is a unit vector parallel to the local nor mal. Since the angle Taking the cross product i^ x 0 gives a unit vector i^ which lies A A in the plane of the surface (see Figure 7). The cross product i^ x i^ A gives another unit vector i which also lies in the surface plane but K is parallel to the projection of 0 on the surface. The angle iQ makes _l/> . \ with i„ in the plane is just tan / i • i„ \. Since (bin is measured K f o M 1 1 ° from the projection of 0 on the surface, we have (PlO = Tr + tan (2.50) Of course from the non-illuminated star, all that is needed is the cosine of the angle the emergent radiation makes with the local nor- A A mal which is easily obtained from the dot product i • 1^. By using the formalism developed in this chapter, we may calculate the radius and gravity at any point on the stellar surface, by speci fying the total potential, the masses, and the separation of the two stars in the binary system. In turn, through the use of von Zeipel's theorem, we may obtain the local effective temperature, which coupled IN C ID E N T m > BEAM \ i« EMERGENT BEAM J SURFACE PLANE Figure 7. Coordinate system to define the emergent angles toward a distant observer 34 with g will define the local atmospheric structure of the star. Fur thermore, any point on the observed disk of the companion star can be assigned to a coordinate &2 >$2 » which, using the transpose of matrix 2.25, can be converted to a star centered coordinates $2*^2* thus al lowing the atmospheric parameters to be specified at that point. The angle at which the specific intensity leaves the stellar surface can be calculated, as well as the angle of incidence that specific inten sity makes with the local normal on the illuminated star. Using these quantities we can then construct an illuminated atmosphere and calcu late the emergent radiation in the direction of a distant observer. In chapter 3, the geometrical relations obtained here are used to develop the transfer equations for a stellar atmosphere illuminated from a finite solid angle. Ill TRANSPORT OF POLARIZED RADIATION It has been shown (see for example Chandrasekhar 1960 and Pomraning 1973) that the most general representation of a beam of light Is a com bination of an unpolarized beam and an independant beam of elliptically polarized radiation. To completely describe the elliptically polarized component, four parameters are needed: the intensity of the beam, I, the degree of polarization, the orientation of the ellipse with respect to some fixed set of axes (angle x ) » and the ellipticity or ratio of the axes of the ellipse, 3. These quantities are given by the four Stokes parameters 1^, 1^, U, and V: where I, and I are the intensities of the beam in two planes 1 r perpendicular to each other, and the parameters U and V describe the ellipticity through the relations U = I cos 23 sin 2\ V = I sin 23 (3.1) with I = I, + I . Alternately we may use the forms 1 r \ I = I. + I 1 r Q = I. - I ^ 1 r > U = (I1 - Ir)tan 2x V = I sin 23 . (3.2) It was in this latter form that these equations were first presented by Stokes in 1852. 35 36 When a photon undergoes a scattering event, the state of polariza tion, and hence the Stokes parameters will change. For polarization in early type stellar atmospheres, the most applicable mode of scattering is Thompson scattering by free electrons, which can be represented by the scattering matrix given by Chandrasekhar (1960); (1,1)' (r,1)‘ (1,1) (r,l) 0 (l,r)‘ (r,r): (l,r)(r,r) 0 2(1,1)(l,r) 2, (r,r)(r ,1) (1,1)(r,r)+(r,l)(l,r) 0 ►(3.3) 0 0 (1,1)(r,r) (l,r)(r,l) The matrix elements are defined by (r,r) = cos( (l,r) = -cos 0f sin(cj>r —<(>) A3.4) (r,l) = cos 0 sin (<^’ ~4>) (1 ,1) = sin 0 sin 0'+ cos 0 cos 0' cos( Since four parameters are needed to specify the state of polariza tion, it follows that four equations of transfer are necessary to de scribe the transport of polarized radiation. A method for calculating the transfer of polarized radiation in an illuminated stellar atmosphere has been developed by Collins (1972, see also Collins and Buerger 1974) in which the depth dependence has been separated from the angle depen dence, thus allowing solutions to be obtained in situations where little or no symmetry is present. 37 The equation of transfer for the diffuse field is jidid(*,e,T ) d d — ------Ia (d>.e,Tv) - SQ («J),0,t ) , (3.5) V where the source function is given by sd(*,e,Tv ) = s b v(,v) + ( i ^ ) ,e- )[id (*- ,e' ,t ) + 4ir v Ii( In this representation e = < /(< +a ), B (t ) has the components V V V V V [%B ,%B ,0 ,0 ] and ,6' ,0 ) is the incident radiation field with the v v components £l£, I®, U°, V°]. 6 is the scattering matrix (equation 3*3)* Thus we have the four equations necessary to describe the transfer of arbitrarily polarized radiation. Following Collins (1970), we can express the source function in terms of moments of the radiation field of the form V n ( V - h <»'> ,e' . O ,e’ »0>e_Tv^cos6' ] sin 0’ d0' d In this notation, the subscript m represents the appropriate Stokes parameter, and thus takes the values 1, r, u, or v. The values for the moment itself, and the associated functions f(0' ) are given by M = J f(0r ) = h F 2 cos 0* A K h cos 0' > (3.8) G 2 cos 0' L h cos 0' ), 38 Finally, the second subscript, n, will be denoted j, f, k, g, or 1 to 2 signify g ( F i ("O = '~ cos ' sin -t,,/cos 0' ] sin 0' d0* d Using this moment notation, the four source functions (see Collins 1972) corresponding to the four Stokes parameters can be expressed as S1 (*,0,tv) = h evB(tv ) + 3/4(2 X (tv)-cos2 0[Y(Ty) + Z(t^) c o s 2 2T(tv) sin 2tJ>] + cos 0 sin0(M(x^) cos S_($,0,t ,) * 1je4B(Tti) + 3/4{2X(tm) - Y(xt|) + Z(xvj) cos 2 4> -2T(xv)sinS \(3.10) S ( (-M(xy ) cos (J> +N(x^) sin<()) sin 0 } Sv ( Y(tu ) = I1-=VCTV> ] (Tv> - Z(TV) - 1 [K1:j(Tv) - 2Ky(Tv) + 2J(tu) - Jrj (rj - hFul(,J] *<■■’*> * I1 ' W » KU < V - W u k < V + l/8V \ > 1 X (3 .U, S(tv) - [l-E/T^l^L^O W(tv) - H - % ( T a)][Gvf(t3)l P(x ) = [1-e (x )][F ,(x )] V v V 1 vf V 39 As can be seen in equation 3.10, the moment equations are separ able into two parts representing the diffuse field and the incident field. By treating the moments in two parts we can evaluate the second part of the integral directly in terms of the incident radiation field. The exact form of this integral will then be determined by the geomet rical form of the incident field. The most general form for the incident term is illumination by a finite solid angle. This is bounded by the two extremes of plane parallel illumination and uniform illumination from a hemisphere. For illumination from a finite solid angle, by setting y = cos 0, we can write the incident part as X1 ^ ) = ^ / J V ( l - u 2)lJ (e, U i (0,«j),O)]e_i:v/'Jdyd(() I M 1 (t ) = f - /27r/1[u(l-y2)lscos 1 ,1 = h ; /J 7T/ 1otijsin(t,4 (e ’ <,,’0) ■ 2(1_y2)J5cos Wi(t ) = — /2u/\l-y2)^cos4>V1(0,$,O)e~Tv/udyd In the cases of plane parallel and uniform illumination, these terms simplify considerably, and are presented in Table 1, where the terms corresponding to plane parallel illumination are those given by 40 Table 1 Incident Terms for the Moments ane Parallel Uniform Illumination 1 - T /jj0 2 v T o 1/2 1^ IE2(tv)-E4 (tv)] X * 4^ e v “ ( l - V o )IX0 Y = ^ [(2-3po 2 ) ^ 0 - Ir° ]e"Tv/l10 1/2 IIj0 (2E2(tv)-3E4 (tv)] - t V W 1 Z 1 1 1 2 \^ST o ” T /Po M = - p0(l~Po ) I, e V 0 (t V ) TT 1 N ( t = ffd -P o 2)55 U°e"Tv/i)o 0 V) 0 W ( t ) = -(l-P c 2)*5 V°e"Tv/lJo V 7T 1 „o -T /po P ( x ) ■ — P o V e v 0 V ir 41 Collins (1972). As can be expected, for uniform illumination, all but the X and Y terms integrate to zero due to azmuthal symmetry. Generally we have equations of the form Jn I g(e, Jn I g(6,4')p: , (3.14) where D is the distance separating the two atmospheres. The term dA is the differential area on the surface of the illuminating star, dS, fore shortened along the line of sight by an amount cos 0O2» where once again 0O2 is the angle at which the specific intensity emerges with re- cos 0O2 2 spect to the local normal. Thus dA = cos 0O? dS = — — -— r„ sin ©2' cos o z d02' d (§2 r2 TT _ .. Q 2 sin ©2' d02' d^' J 2 max J I g(0,40cos 602^ z 0 0 2 * U.xoy C O S G D with g (0,4>) representing the values sin 0, cos 0, sin Using equations 2.32, 2.45, 2.46 and 2.47, and letting sin p = rr sin 02’ /D cos p = (d* — r2 cos 02' )/D }. I , (3.16) we have - —1 —1 tan E r_sin02 sin 0 = sin(0o - sin (sin( D cos 0 with D = (d'2 - 2r2d’ cos 02f + r2)^ . Our equations for the incident terms then become i a' , - i 2. /B „ r2sin02 d02' dd>2* xi(T f-2max/2MlJ(92',^,0)(lV)e ./ cos6o2J \ cos o D - T /B Y X (t ) - j - / 02'™ax /27rI (2-3B2)lJ(e2T A z ,0)]cos 0o2 e"Tv /B x V HIT o o 1 r2sin 02' d02'd cos a D z l (Tv ) = ^ J^'max /27r[l-2E2) (I^(02' A z ,O)B2 - l J ( 02' A z ,0)) -2BCEJ U*(02' t^z »0)1 e Ty/,Bcos 0„2 r^sin 02' d02' d cos a D t1(tv) = h ^02'maX^ r [{B 2lJ(e2 .*2 .O)-lJ(02' .*2 »0)CE+ BU 1 >(3.18) t . 1 ■ t.2m o r;^sin02,d 02,d<^>2, (02 ,$2 .0) es-E ) ]e v cos9o9 2 cos a D M 1 (x ) = j - / 92'max /2ir[4BAEIX (e2' A z ,0) + 2ACU1 (02' , N'L(tv) = ^ /9 2 max /^[4BAClJ(02' ,4>2’ .0) - 2AEUi (02' , „i/ \ 1 f02 r2ir,T7Tri/o * * • f>\ _T /B Q rjsln 02' d02’ d _,i, . 1 (02 i x i n\ "T /B q r2sin 02 d02 d These incident terms, coupled with the moments for the diffuse field, will specify the source function (equation 3.10) , thus allowing the equation of transfer to be solved for the transport of arbitrarily polarized radiation incident from a finite solid angle. Since the V Stokes parameter and the corresponding W(t^) and ^(t^) moments describe the transport of circularly polarized radiation, they will not be used in this work. There will be a net local horizontal flux produced by the incident radiation. If we set up orthogonal axes I, J, K (see Figure 8) with I being parallel to the local normal, and K parallel to the projection onto the stellar surface of the axis, we can express the total flux in thethree coordinates as t? 1 f 2 it fTr. rTd, Td*i r Ti , Td . /co s 0. I ” 4ir ■'o l r ~ 1 r ^cos 9 sin 9 d9 d FJ = h C ^ o i[1l + 1r ] + [1l + I i ] e"Tv/cOS9>sin2 6 sin (3.19) Pk = T ~ J2T U I ? + Id]- [ i S i e_Tv/c°se}cos It is clear that the diffuse field exhibits azmuthal symmetry in the plane, and thus the leading terms in the expressions for Fj and F^ will integrate to zero. Thus equations for the flux become INCIDENT BEAM Figure 8. Local coordinate frame to define the incident flux -p- 45 Ft = 7— J'27rJ'TrtX^ + Id ] cos 0 sin 0 d 0 d <}>— J"^*max/0max[I?‘+I*] ^ x 4tt o o X r n 7 o x r max -T /COS0 e V cos 0 sin 0 d0 d«f> >(3.20) 1 (0 r_i, _i. -t /cos© . 2- 7 - - J = 4ir m a x i max[I^+Ir] e v sin 0 sin F t = 7— /27r/ir[I ? + 1^] cos 0 sin 0 d 0 d <|>- 7— /02 max fZlr[I* + I"*"] I4ir-'oo l r t 4it''o '0lr e_Tv/B B r2sin 02 d02' d F = X - /02 max /27T{I1 + I1} e"Tv /B AC r^ sin 02' de2' d Due to the fact that the amount of polarization is usually quite small, the second form of the Stokes parameters (equations 3.2 ) are usually used to minimize the effects of round off errors. Thus rather than using the source functions as given by equation 3.10, what is calculated is (S^ + Sr ), (S^ - Sr ), and S Since we are dealing with polarized radiation, the orientation of this radiation must be taken into account. The two beams of radiation 1^ and I may be thought of as vector-like quantities, in that while they do not have a direction associated with them, they do have an 46 orientation. For our purposes, 1^ is defined to lie in the plane de fined by the normal to the surface at some point and the line of sight to that point, while I is perpendicular to it. These planes of orientation apply to both the incident and emer gent radiation. Thus the emergent radiation from each point on star 2 must be rotated to line up with the local gravity at each point on star 1. This rotation is most easily accomplished by projecting the local garvity at each point of the illuminated star onto the plane of the sky. Thus each gxl> gyl> g^, can be expresed as g ^,, gy2,, gz2,, by letting ?x2 * -g x l ’ 8y2 gy2 “ “8v;p gz2 " gzl and uslng the rotation matrlx 2 -25- The “-1* R angle £ obtained from £ = tan . ^ 2' gives the orientation of the local gz2' gravity on the plane of the sky measured with respect to the z2, axis. Now each point on the illuminating star has a local gravity Gx2, Gy2 G , which can also be expressed in terms of the primed coordinate frame 2 2m aS Gx2' ’ Gy2' * Gz2' ‘ T^e an^ e 1 = tan”^(G £ ^Gz2' ^ Sives the orienta tion of the emergent beam with respect to the z2, axis. The full rota tion angle ip is then simply ip = i - e Following Chandrasekhar (1960) we can write (I1 + v 1 - ri + \ (3^ - *r)’ = cos + Usin 2ip > (3-22) U* = -(^ - Ir) sin 2rp + Ucos 2ip A similar rotation must be performed to obtain the flux seen by a distant observer. For this present work, we will be concerned with the 47 flux obtained from the inner "hemisphere", the outer "hemisphere", as well as the flux seen when the star is viewed along the y axis. We will take the reference line for the rotation to be the z axis, then the rotation angle for each point on the star as simply tj> = tan ^ (g /g ) for the first two cases, and = tan ^(g /g ) for the latter y 2 X 2 case. For a more general method of obtaining the rotation angle see Buerger and Collins (1970). In this chapter we have used the geometry of chapter 2 to derive expressions for the source function and incident flux for a stellar at mosphere illuminated from a finite solid angle. In order to calculate the source functions, it is necessary to numerically integrate the variables defined by equations 3.54 through 3.61. However, as one moves over the surface of the star, the apparent surface area of the compan ion will change, that is the observed disk will be defined by a differ ent set of coordinates, ©2r *d>2f» f°r each point on the illuminated star. Since it is not feasible to calculate the atmospheric structure at every different 62' ,$2 ^ becomes necessary to establish a fixed grid of atmospheres which may be interpolated to give the structure at the desired points. The construction of the grid of atmospheres and the necessary numerical integration and interpolation are the subjects of chapter 4. IV MODEL CALCULATIONS The model stellar atmosphere code ATLAS has been modified by Buerger and Collins to account for plane parallel illumination. The subroutines that calculate the radiation pressure, statistical equili brium for non-LTE and the temperature correction (see Buerger 1972) each have a term that corresponds to the normal component of the inci dent flux. For plane parallel illumination this term has the form pF, where p is the cosine of the angle of incidence measured with respect to the normal. For illumination from a finite solid angle, this term is replaced by the incident part of equation 3.18 where B once again is the cosine of the angle of incidence for each point on the illuminating star. As pointed out by Collins (1972), it is only the moments X(ty) and Y(t ^) that are coupled directly to the atmospheric structure. This coupling is through the Planck function, therefore to determine the temperature distribution in the atmosphere, only these moments are used. The remaining four moments are coupled to the atmospheric struc ture through the scattering term which is fully conservative. There for they are used to determine the characteristics of the emergent ra diation field, but do not play a role in determining the atmospheric structure. Since the effect of the incident radiation on the structure of the atmosphere is only determined by the incident terms of X(t ^) and Y(tv) moments, the exact form of which is dictated by the spatial 48 49 distribution of the incident flux, the only modification necessary to perform the radiative transfer calculations, is to substitute the fi nite solid angle forms of the incident terms (equations 3.54, 3.55) for the plane parallel forms. Once the atmospheric structure is determined, the remaining four moments may be calculated. For these calculations, a similar modifi cation must be made, that is replacing the plane parallel form of the incident terms by the finite solid angle forms. Also (see Collins 1972), for plane parallel illumination, the T*(xv) and N*(t v) moments are obtained from simple transformations of the Z*(t v) and M*(tv) mo ments. In the case of illumination from a finite solid angle, no sim ple transformations were found and the T*(t v) and N^x^) moments were calculated separately. The incident terms for the moments are obtained from integrals over 021 and on the apparent surface of the star in the sky. Since the integrals are not of closed form, it is necessary to use a numeri cal quadrature scheme. The Gaussian schemes have the advantage of being exact for polynomials of order 2k+1 where k is the order of the quadrature (see for example Hildebrand 1958). Unfortunately, in order to estimate the accuracy of an integration scheme, as well as to maximize the efficiency of the procedure, it is necessary to perform the computations several times using quadratures of increasing order. Due to a shortage of computer funds, it was not feasible to run exten sive tests on the moment integration. However, to get some idea of the accuracy involved and the order necessary to accurately calculate the moments, the area of a Roche 50 surface was computed using a 90,000 point Simpson's rule as well as Gauss Legendre and Gauss Chebychev integration schemes of various or ders. Since the flux and hence the moments are smoothly varying func- _2 tions of r , which in turn is a smoothly varying function of 0 and <£> an estimate of the accuracy of the quadrature obtained by calculating a surface area should be fairly representative. A 15 point Gauss Legendre quadrature in both 0 and Chebychev scheme in 4> was accurate to about 0.5%. Additionally, the flux from a spherical star of known effective temperature was calcu lated. Again both Gaussian schemes were highly accurate, giving re sults to better than 0.1%. While both Gaussian schemes are undoubted ly over estimates of the order needed, to be on the safe side the 15 by 15 point Gauss Legendre quadrature was used. In performing the radiative transfer for the emergent flux and the polarization a fixed scale of 15 points was used. For a non-illu- minated atmosphere, the emergent radiation at each frequency is deter mined by the diffuse part of the X(t ^) and Y(x^) moments, the fraction of the source function due to scattering, and the Planck function. For each atmosphere composing the grid of the illuminating star, these four quantities were calculated and stored on disk for each frequency point at the 15 standard depths. As one moves over the surface of the illuminated star, the appar ent surface area will change and in general the observed coordinates 51 will not correspond to any of the star centered 62^2 Points» thus necessitating an interpolation on' the fixed grid. For this interpolation, the ATLAS subroutine MAPI was used. This is a parabolic interpolation scheme which constructs oneparabola us ing the two previous fixed grid points and the one following point. A second parabola using the one previous point and the two following grid points is also calculated. The interpolation is performed on the mean parabola which is obtained by weighting the two parabolas by the inverse of their second derivatives, thus emphasizing the parabola with the smallest curvature. The apparent disk was divided into 21 lines of constant $2 at 18° intervals, the twenty-first point being of course 360°. For each of these lines of constant d>V , the maximum theta point B?’ was obtained ^ ‘max using the process outlined in equations 2.32 through 2.43. The 15 quadrature points 62' were then calculated for the interval 0 to ^ ^ a x ’ For each of these 315 points, the corresponding star centered coordin ates &2$2 were computed, as well as the angle of the emergent radiation 0o2> the rotation angle the incident angles 0 and i, and the "con- 2 2 stant" term in the moment equations cos0O2r2 sin02' /cos a D . Due to the symmetry of the star, the atmosphere calculated at 02 = 25°, 25° for $2 ~ 162°, -18° and -162°. Thus any 02 corresponding to For each point (02 and Y(x^) moments, the scattering term and the Planck function were first mapped to the desired 02 along the two previous and two following lines of constant <|>2' These four points were then used to interpolate to the correct 315 atmosphere at the 15 points and at each frequency. Utilizing these values, the quantities (I^+Ir), (1^ - Ir) and U were then com puted for the appropriate emergent angle 0O2* These in turn were ro tated through ip to align them with the local gravity on the illuminated star. The integration was first carried out in 02f along the 21 lines of constant contribution to the integral was set to zero. These 21 values were then mapped onto the 15 Gauss Legendre quadrature points in the range 0-2ir. Once again to minimize roundoff error in the polarization, the incident part of the moments were calculated using (I^+Ir) and (1^ - 1^) rather than 1^ and I separately. By adding or subtracting these values and dividing by 2, the moment with the correct 1^ and 1^ de pendence can be found. Thus for example, in calculating the incident term of the Y( t ^) moment, what is actually computed is ) = ± - { ( [ E Z (2-3B2)(Ii± + l J S + E E (2-3B2) (I 1 - IJl) ]/2 - 02*+2’ 02* *2’ 53 [E E ( i / t l S - Z E (I /-I JS/2] cos0o2 e"Tv /B r A i n e z ' *1 * 2 *l' *2’ — ----- 2">- (4-D cos a 0 In addition to the six moments, the three components of the incident flux (equation 3.21) were calculated at every frequency and point for each of the 29 atmospheres on the illuminated hemisphere of star 1. Since Atlas uses an interative procedure to arrive at the correct atmospheric structure, it would be advantageous to include the effect of the incident flux on the initial guess for the temperature distribution. Following Milne (1930), we can assume we have two separate temperature distributions: T1 (t ) corresponding to the diffuse field and T2(x) which corresponds to the incident field. By demanding radiative equilibrium, we can attribute a temperature distribution to the incident flux of the form t2(t) = j(—~ ) (v + h) e ~ T / p ]k , (4.2) where F. is the total incident flux, a is the Stefan Boltzmann con- in stant, and p is the cosine of the angle of incidence. In the case of illumination from a finite solid angle of course, the radiation is inci dent over a range of angles, but since this is only a first approxima tion to the temperature distribution, p may be taken to be cos0o» and in cases where 0o>9O°, an average value of p for the portion of the disk above the horizon may be used. This temperature distribution T2(x) is then added to the initial gray temperature distribution T1 (t ) of the diffuse field, to form a first guess for the total initial tem perature distribution. 54 For this dissertation, it was decided to utilize the model param eters given by Protheroe and Buerger (1981) for the early type semi detached binary system V Pup (see Table 2). Since they assumed plane parallel illumination and did not include metallic line blanketing in their model, it will be possible to make a direct determination of the effects of line blanketing and illumination from a finite solid angle on the emergent flux and polarization of the system. As can be seen in Table 2 and Figure 9, the less massive cooler com ponent nearly fills its Roche lobe, while its more massive hotter com panion is much less distorted. For these parameters, we obtain a polar effective temperature for the more massive star of about 23,890°K which corresponds to a spectral type of slightly cooler than B1 V, while the polar effective temperature of the less massive component, 18,220°K is slightly hotter than B3 V (see Slettebak et al. 1980). Since each star exhibits symmetry about the orbital plane, as well as about a plane through the rotation axis and the center of mass of the system, it is only necessary to compute atmospheres for the points which lie in the range 0 <_ d>2 .1 90°, 0 62 1. 180°. A fixed grid of 80 atmo spheres was used to cover this regime for the more massive star, while a 51 point grid was used for the less massive component. The coordin ates and corresponding effective temperatures and gravities for these two grids are shown in Table 3 and Table 4. To calculate the structure at each point, the model stellar atmo sphere code ATLAS 6 (Kurucz 1970, Kurucz et al. 1974) was used. In cluded in the opacity calculations were contributions due to bound-free and free-free transitions of HI, H", H+ , Hel, Hell, Cl, Mgl, Sil, All, 55 Table 2 Parameters for the System V Pup Period if 454 Separation of centers 1.15 x 10^ Km Star 1 Star 2 Masses 11.03 Me 18.71 Mo Luminousity 2265.1 Lo 9978.9 Lo Roche Constant 2.0379 2.1520 Polar Radius 5.010 R® 6.060 Ro Equatorial Radius 5.206 Re 6.314 Ro Inner Radius 6.145 Re 6.927 Ro Outer Radius 5,632 Re 6.600 Ro Figure 9. The binary system Ln ON Table 3 Effective Temperatures and Gravities for the Grid of Atmospheres on the Illuminating Star 9 (degrees)______it (degrees)______Teff °K______log g_ 0 0 20266 3.870 0 2 20280 3.871 0 7 20431 3.884 0 15 20899 3.923 0 25 21571 3.978 0 35 22130 4.023 0 45 22529 4.054 0 55 22785 4.073 0 70 22958 4.086 0 90 22916 4.083 0 110 22685 4.066 0 130 22378 4.042 0 150 22095 4.020 0 165 21953 4.009 0 180 21902 4.005 18 2 20280 3.871 18 7 20434 3.884 18 15 20914 3.924 18 25 21601 3.981 18 35 22175 4.026 18 45 22588 4.058 18 55 22858 4.079 18 70 23051 4.093 18 90 23021 4.091 18 110 22785 4.073 18 130 22449 4.048 18 150 22128 4.022 18 165 21962 4.009 36 2 20281 3.871 36 7 20444 3.885 36 15 20951 3.927 36 25 21677 3.987 36 35 22289 4.035 36 45 22740 4.070 36 55 23045 4.093 36 70 23281 4.111 36 90 23281 4.111 36 110 23032 4.092 36 130 22631 4.062 36 150 22214 4.029 36 165 21986 4.011 58 Table 3 continued 9 (degrees) j) (degrees.) Teff K log 8 54 2 20283 3.871 54 7 20457 3.886 54 15 20996 3.931 54 25 2i769 3.994 54 35 22425 4.046 54 45 22918 4.083 54 55 23262 4.109 54 70 23547 4.130 54 90 23583 4.133 54 110 23315 4.113 54 130 22843 4.078 54 150 22317 4.037 54 165 22016 4.014 72 2 20283 3.871 72 7 20467 3.987 72 15 21032 3.934 72 25 21841 3.999 72 35 22531 4.054 72 45 23056 4.094 72 55 23429 4.122 72 70 23748 4.145 72 90 23808 4.150 72 110 23528 4.129 72 130 23006 4.090 72 150 22398 4.044 72 165 .22040 4.016 90 2 20284 3.871 90 7 20470 3.887 90 15 21046 3.935 90 25 21868 4.002 90 35 22571 4.057 90 45 23107 4.097 90 55 23492 4.126 90 70 23823 4.151 90 90 23891 4.156 90 110 23607 4.135 90 130 23065 4.095 90 150 22428 4.046 90 165 22049 4.016 Table 4 Effective Temperatures and Gravities for the Grid of Atmospheres on the Illuminated Star 6 (degrees)______6 (degrees)______Teff °K______logg 0 0 13844 3.622 0 5 14006 3.642 0 15 14897 3.749 0 30 16184 3.893 0 50 17169 3.996 0 70 17548 4.034 0 90 17533 4.032 0 110 17264 4.005 0 130 16859 3.964 0 150 16450 3.921 0 175 16231 3.898 0 180 16150 3.8895 30 5 14014 3.643 30 15 14936 3.754 30 30 16267 3.902 30 50 17298 4.008 30 70 17714 4.050 30 90 17920 4.051 30 110 17487 4.024 30 130 17001 3.978 30 150 16521 3.929 30 175 16252 3.900 45 5 14022 3.644 45 15 14978 3.759 45 30 16348 3.911 45 50 17420 4.021 45 70 17370 4.065 45 90 17896 4.068 45 110 17618 4.041 45 130 17136 3.992 45 150 16591 3.936 45 175 16273 3.903 60 Table 4 continued 8 (degrees)______ (degrees!______Teff °K______log 6 60 5 14030 3.645 60 15 15019 3.763 60 30 16427 3.919 60 50 17537 4.033 60 70 18018 4.080 60 90 18062 4.084 60 110 17779 4.056 60 130 17264 4.005 60 150 16658 3.905 60 175 16294 3.891 90 5 14037 3.646 90 15 15059 3.768 90 30 16503 3.927 90 50 17649 4.043 90 70 18158 4.093 90 90 18219 4.098 90 110 17931 4.071 90 130 17386 4.018 90 150 17624 3.950 90 175 16314 3.907 61 Sill, Mgll, Call, NI, and 01. The effects of electron scattering and HI Rayleigh scattering were also taken into account. Balmer and Lyman hydrogen line blanketing and metallic line blanketing were also included. The standard ATLAS abundances were used. A total of 214 frequency points were used, covering the range 472A to 200,000A. Of these 214 frequency points, 99 were used to determine the structure of the atmosphere, while an additional 115 points were used to compute spectral line profiles for He I 4471 and the Mg II doublet 4481. The He I 4471 line profiles were calculated using the tables and asymptotic formulae of Griem (1968). The Mg II doublet was taken to be the sum of two Voigt profiles centered at 4481.13 and 4481.33A. The collisional brodening constants used are those of Griem (1974) while the f-values of Wiese et al. (1966) were used. For the illuminating star, each atmosphere converged after ten iterations, with the depth weighted flux error being less than 0 .2 %, and the depth weighted flux derivative being on the order of 2% or less. Using the interpolation and integrations schemes outlined in this chapter, these atmospheres were then used to calculate the incident part of the moments for the illuminated star. The convergence rate for star 2 was much slower, taking on the order of 17 iterations. The total moments calculated for each star are then used to calculate the source function and the emergent radiation field, which is just the integral of the source function over t v , for any value of the viewing angle desired. 62 In chapter 5 we present the- effects of illumination from a finite solid angle, non-LTE and metallic line blanketing on the structure, polarization and emergent continuous and line spectra of a stellar atmosphere. Using the integration scheme described in this chapter, the emergent flux and polarization distribution of each star will be presented* In addition, the surface brightness distribution for each star will be shown for various viewing angles. V RESULTS AND DISCUSSION When a binary star is observed, what is obtained is the total in tegrated flux from the system. Of course due to the relative motion of the two components, it is possible to separate the two spectra to some extent. However, it is only during a total eclipse that an uncon taminated spectrum may be obtained, and as can be seen in Tables 3, 4, and 5, this "back" portion of the star is not representative of the physical parameters of the star as a whole. Based on the observed flux distribution and line profiles it is possible to assign a spectral type to each component. While it is obvious that these quantities represent some average of the physical properties of each star, what is not ob vious is how the relative contributions of shape distortion and illu mination enter into the picture to effect the observed flux from each star. In order to understand the observed flux, it is necessary to look at the local properties of the star. By investigating the structure and emergent radiation as a function of position on the stellar sur face, it will be possible to obtain a more physical interpretation of the observed flux. In this work, we will calculate the effects of illumination on the smaller star, while the re-radiation back onto the primary will be ignored. While this is not physically correct, it will allow us to separate the relative effects of shape distortion and illumination. 63 64 Table 5 Temperatures and Horizontal Fluxes for the Inner Hemisphere of the Illuminated Star Teff ° K, Horizontal Flux 9 (degrees) $ (degrees) Teff ° K Illuminated_____Normal Flux 0 0 13845 21584 .019 0 5 14006 21467 .189 0 15 14844 20806 .478 0 30 16183 19900 .721 0 50 17170 18762 .784 0 70 17548 17927 .416 0 90 17533 17540 .045 30 15 14936 20852 .485 30 30 16267 19923 .723 30 50 17298 18846 .775 30 70 17714 18084 .448 30 90 17720 17727 .043 45 5 14022 21465 .197 55 15 14978 20849 .606 45 30 16349 19948 .724 45 50 17420 18929 .765 45 70 17870 18228 .436 45 90 17897 17904 .046 60 15 15019 20860 .512 60 30 16426 19971 .724 60 50 17537 19009 .757 60 70 18018 18343 .398 60 90 18062 18068 .041 90 5 14037 21462 .020 90 15 15059 20843 .503 90 30 16503 19996 .725 90 50 17649 19089 .749 90 70 18158 18493 .379 90 90 18219 18225 .041 65 As a representative sample we will look at three atmospheres on the illuminated star that lie in the orbital plane, $ - 0°, with values of 0=0° (the substellar point), 30° (point 2) and 70° (point 3). For this last atmosphere the apparent center of the illuminating star lies on the local horizon. Thus for plane parallel illumination, this latitude represents the borderline between the illuminated portion of the star and the non-illuminated portion, when in reality, however, half of the illuminating star is still above the horizon. In order to determine the effects of metallic line blanketing and illumination from a finite solid angle, four model atmospheres were constructed for both the substellar point and point 2. In addition to the finite solid angle illuminated atmosphere with metallic line blanketing, the total incident radiation at each point was used to construct plane parallel illuminated atmospheres both with and without metallic line blanketing. Also a comparison atmosphere, an enhanced flux model was constructed, which was not illuminated, but had an emergent flux equal to the intrinsic flux of the atmosphere plus the incident flux. As shown by Muthsam (1978) the inclusion of the effects of non-LTE can influence the structure of the illuminated atmosphere. Because all of the models calculated for this study were assumed to be in non-LTE, to get some idea of the relative effects of non-LTE and metallic line blanketing, LTE plane parallel illuminated atmospheres with and without line blanketing were also constructed for point 2. Since the third point would not be illuminated by plane parallel incident radiation, 66 only two atmospheres were constructed: the finite solid angle illuminated model and a reference non-illuminated atmosphere. The flux distributions at the three points are shown in Figures 10, 11, and 12. As can be seen the flux shortward of the Lyman jump is greatly enhanced due to the illumination. This is primarily due to the spectral energy distribution of the incident radiation field. Figure 13 shows the energy distribution for the incident radiation and for the non-illuminated atmosphere at the sub-stellar point. As can be seen the incident radiation reaches a maximum at very short wavelengths, and is several orders of magnitude higher in the Lyman continuum than the corresponding intrinsic radiation field. This increase in the short wavelength flux is further enhanced by the inclusion of metallic line blanketing, which greatly increases the opacity and hence the absorption in this wavelength region. These figures also show that the emergent flux for the finite solid angle illumination is slightly smaller than for the plane parallel illumination. Although the total amount of incident radiation is the same for both models, the spatial distribution is different. What is important is the incident flux, that is the normal component of the radiation field, for it is this component that determines the temperature structure of the atmosphere. For the plane parallel case at the sub-stellar point, all of the incident radiation strikes the surface parallel to the local normal. For the finite solid angle, however, much of the incoming radiation is incident at some angle to the normal. For the model presented here, o o o => _1 U- O o 5—1 ° (O . ENMNCEO FLUX MODEL o PLANE PAHALLEL ILLUHINAT1ON-N0 LINE BLANKETING PLANE PAAALLEL ILLUMINATION-LINE BLANKETING FINITE SOLID ANGLE ILLUNINATION-LINE BLANKETING o o • CO 0.00 1.00 2.00 3.00 FREQUENCY X 10-15 Figure 10. Flux distributions for the four models at the sub-stellar point o\ ^4 iue 1 Fu itiuin fr h fu oesa pit 2 point at models four the for distributions Flux 11. Figure LOG FLUX I 0.00 8.00 -7.00 -6.00 -S.00 -1.00 -3.0 1.00 ENHANCED FLUX HOOEL FLUX ENHANCED PLANE PARALLEL ILLUHINAT1ON-N0 LINE BLANKETING LINE ILLUHINAT1ON-N0 PARALLEL PLANE FINITE SOLID ANGLE ILLUHINATION-LINE BLANKETING ILLUHINATION-LINE ANGLE SOLID FINITE BLANKETING ILLUMINATION-LINE PARALLEL PLANE RQEC X 10-15 X FREQUENCY 2.00 3.00 4.00 oo 1 e o X ID O o —I® . I NON-ILLUMINATED ATMOSPHERE FINITE SOLID ANGLE ILLUMINATION o s 1 1------0.00 1.00 2.00 3.00 0.00 FREQUENCY X 10-15 Figure 12. Flux distributions for the two models at point 3 O' VO o o in U. O o 3—I °ID NON 1LLUNINRTE0 INCIDENT FLUX o «o 0.00 1.00 2.00 3.00 4.00 FREQUENCY X 10-15 Figure 13. The incident and intrinsic (non-illuminated) flux distributions at the sub-stellar '-j point o 71 the Illuminating star subtends approximately 60° in the sky. Thus the angle of incidence for the incoming specific intensity from each point on the illuminating star covers the range 1_>VL> 0.866. Therefore the total normal component of the incident radiation field is less for the finite solid angle model. This becomes less important as the illuminating star gets lower in the sky, for we now have the situation where half of the illuminating star is above the line defining the direction of the plane parallel incident radiation and half is below. Thus radiation from the upper half, striking the surface closer to the normal, compensates for the larger angle of incidence for the. radiation from the bottom half of the star. For the sub-stellar point, the plane parallel flux is approximately 5% greater than the finite solid angle illumination, while for point 2 the increase is only about *s%. Along with the modification of the total emergent energy distributions, we also have changes produced in the spectral lines. Figures 14, 15, and 16 show the He I 4471 and Mg II 4481 spectral line profiles. In all cases illumination causes a weakening of the lines, and is most dramatic for the illuminated models with metallic line blanketing. In order to fully understand these changes, it is necessary to look at the structure of the atmosphere, more specifically the structure of the line forming region. Figures 17, 18, and 19 show the temperature structure as a function of tRoss f°r atmospheres at the three points under consideration. As might be expected, the inclusion of radiation incident on the stellar surface causes a temperature rise. Once again this effect is enhanced iue 4 H 47 adM I 41 eiul neste fr h fu oes t h sub-stellar the at models four the for intensities residual 4481 II Mg and 4471 I He 14. Figure o> o o o CO o o o m o “ o CO o o e o o o> (M RESIDUAL INTENSITY • • _ • • _ point FINITE SOLID ANGLE ILLUMINATION-LINE BLANKETING ILLUMINATION-LINE ANGLE SOLID FINITE BLANKETING ILLUMINATION-LINE PARALLEL PLANE BLANKETING LINE ILLUHINATION-NO PARALLEL PLANE MODEL FLUX ENHRNCEO AEEGH (ANGSTROMS) WAVELENGTH E 47-G I 4481 II 4471-MG 1 HE iue 5 H 47 ad gI 41 eiul neste fr h fu oesa on 2 point at models four the for intensities residual 4481 II Mg and 4471 I He 15. Figure o" o> oo o o o o o o o m RESIDUAL INTENSITYin o to o • 880 86.0 880 87.0 17.0 880 87.0 800 88.0 8882, 8881.00 8880.00 8876.00 8878.00 81172.00 8870.00 8868.00 8866.00 8868.00 FINITE SOLID ANGLE ILLUHINATION-LINE BLANKETING ILLUHINATION-LINE ANGLE SOLID FINITE BLANKETING ILLUMINATION-LINE PARALLEL PLANE BLANKETING LINE 1LLUH1NAT10N-N0 PARALLEL PLANE HODEL FLUX ENHANCED AEEGH (ANGSTROMS) WAVELENGTH E 47-G I 4481 II 4471-MG I HE o o HE I 4471-MG II 4481 o o> CD ■ _ O' NON ILLUMINATED FINITE SOLID ANGLE ILLUMINATION • _ o o m __ o 0060.00 HUM. 00 0060.00 0070.00 0072.00 0070.00 0076.00 0080.00 0061.00 0082, WAVELENGTH (ANGSTROMS) Figure 16. He I 4471 and Mg II 4481 residual intensities for the two models at point 3 iue 7 Tmeaue tutr a ucino x fr the for xw of function a as structure Temperature 17. Figure TEMPERATURE X 10-3 K 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 45 -.0 25 -.0 05 05 15 2.50 1.50 0.50 -0.50 -1.50 -2.50 -3.50 -4.50 ormdl t h sbselrpit OSS point sub-stellar the at models four PLANEPARALLEL NOLB ENHANCEDFLUX MODEL FINITE SOLIDANCLE PLANEPARALLELLB O (TAU-ROSS) LOG 75 Figure 18. Temperature structure as a function of of function a as structure Temperature 18. Figure TEMPERATURE X 10-3 K 45 -3.50 -4.50 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 ormdl tpit 2 point at models four FINITESOLID ANGLE PLANEPARALLEL LB PLANEPARALLEL NOLB ENHANCEDFLUX H00EL O (TflU-ROSS) LOG -1.50 05 05 1.50-2.50 0.50 -0.50 t R s s o o the for 2.50 76 Figure 19. Temperature structure as as structure Temperature 19. Figure TEMPERATURE X 10-3 K 10.00 15.00 20.00 25.00 30.00 35.00 10.00 15.00 15 35 -2.50 -3.50 -1.50 w oesa on 3 point at models two FINITESOLID RNCLE NON ILLUMINATEDNON RTM0SPHERE O (TRU-R0SS) LOG 15 -0.50 -1.50 a function of of function 0.50 t R s s o 1.50 o the for 77 78 by the inclusion of metallic line blanketing due to the increased opacity. As mentioned previously, since the normal component of the incident radiation will determine the temperature structure, the slightly smaller amount from the finite solid angle is reflected as a lower temperature in the atmosphere. Figure 20 shows the temperature distributions for the atmospheres at point 2 (see Figure 18), where the places marked indicate the points in the atmosphere where the optical depth reaches unity in the line cores for the helium permitted and forbidden components, and the magnesium doublet. As can be seen, the forbidden component of the helium, and the magnesium doublet are formed deepest in the atmosphere. The initial flattening of the temperature gradient due to illumination causes a drastic weakening of these lines; however, the additional flattening due to line blanketing is very small at these depths in the atmosphere and is reflected in the smaller changes in the line profiles. ■+* The permitted component of the helium line, however, is formed higher in the atmosphere where it is much more influenced by the illumination and the enhanced flattening of the temperature gradient due to line blanketing. This effect is clearly seen in the line profiles. Also, for the helium line a second effect comes into play. The neutral helium lines reach maximum strength at spectral type B2 which has a temperature distribution much like the enhanced flux model shown here. In these stars the line core is formed high in the atmosphere in a relatively cool region. As can be seen in Figure 21 the temperature rise caused by illumination is sufficient to ionize the helium, Figure 20. Temperature structure as a function offunction structure asa Temperature 20.Figure TEMPERATURE X 10-3 K • . 5 -.0 25 -.0-.00.50 -0.50 -1.50 -2.50 -3.50 -•I. 10.0050 15.00 20.00 25.00 30.00 35.00 HO.00 M5.00 models at point 2, indicating the Rossland depth at which atwhich depth 2, indicatingthe pointRossland at models 4481 (MG), and the permitted (HE) and forbidden componentforbidden (MG), (HE) the and 4481 andpermitted the optical depth in the line cores reaches unity for Mg II for Mg reachesunity incoresthe linethedepth optical (HE')I4471 of He IIE OI ANCLE SOLID FINITE LB PARALLEL PLANE PLANE PARALLEL NO LB NO PARALLEL PLANE MODEL FLUX ENHANCED O (TflU-ROSS) LOG tr o h four forthe 1.50 2.50 79 80 1.0 0.9 0.8 0.7 CO 0.6 § LU 0.5 OU_ C£L 0.3 •2 ■1 0 ^ ^ ROSS^ Figure 21. The number of electrons contributed per He atom as a function of gg for three models at point 2. LB is plane parallel illumination with line blanketing, NLB is plane parallel illumination without line blanketing, and EN is the enhanced flux model 81 drastically reducing the number of absorbers. For the wings which are formed deeper in the atmosphere, both effects are greatly reduced, resulting in much smaller changes. The masses obtained from the solution of the light and radial velocity curves are assigned to stars of a given spectral type. The spectral type is determined by the observed flux distribution, and spectral line profiles. To see how illumination would effect the spectral type we can look at the difference in monochromatic magnitudes at different wavelengths, as well as the ratio of the equivalent widths of He I 4471 to Mg II 4481. The spectral types were obtained from the models of Slettebak et al. (1980). Figure 22 shows the difference in magnitudes at wavelengths of 1000 A-2500 A (m^o-m25)» 2500 A - 3500 A (m25~m35), 3500 A - 4500 A (13135- 1 1145), and 3712 A - 5800 A (11137-1 11 5 8 ) for the sets of atmospheres at the three points. For the illuminated models the flux longward of the Lyman jump is lower than for the non-illuminated model, resulting in a flatter gradient in the Balmer continuum which mimics a slightly cooler spectral type. At longer wavelengths, the effect of illumination is much smaller causing only minor changes in the Balmer decrement in the near ultraviolet. In the visual the effect is reversed. The inclusion of incident radiation enhances the region just longward of the Balmer jump, steepening the Paschen gradient, and causing the star to appear to be of a slightly hotter spectral type. 82 BO - m37~ "ss B2 - B3 - B1 - B 5 - B6 - POINT 2 B3- EN NLB LB FSA Figure 22. Spectral types for the models at the three points, using monochromatic magnitude differences. LB, NLB, and EN are as in Figure 21 while FSA indicates the model with finite solid angle illumination The large variations in the raio-11125 and m37-m5g color curves for the non-line blanketed models are due to the fact that the hydrogen line profiles are calculated explicitly and the points at 1000 & and 3712 %. lie in the wings of hydrogen lines, while for the line blanketed models, the opacity due to hydrogen is included, but the line profiles are not computed. Figure 23 shows the spectral type obtained from the ratio of He I 4471/Mg II 4481 equivalent widths. While the calibration of Slettebak et al. (1980) was for main sequence stars, the low latitude points for the illuminated star would be classified as subgiants according to their gravity. Although the absolute classification may be slightly in error the general trends should still be valid. As was seen, the absolute strengths of the spectral lines change drastically due to illumination. However, since it is the ratio of the line strengths that are used for classification the effects are less dramatic. Specifically, at the substellar point, the ratio stays approximately the same, indicating no change in spectral type. In general, the trend due to the weakening of the lines is to make the star appear too early by one or two subclasses. While this may appear to be a small effect, it must be remembered that the spectral types are a very non-linear function of temperature. In the early type stars for example, a change in spectral type of B1 to B3 corresponds to a change in effective temperature of approximately 7500°K while the entire class of A stars covers a range of only 2000°K. Figure 23. Spectral types for the models at the three points estimated three thepoints at the for typesmodels Spectral 23.Figure B3 B1 TYPE BO sn h ai fH 47/gI 41 equivalentIIwidths 4481 14471/Mg Heofratio the using N U L FSA LB NUB EN SUBSTELLAR POM POM 2 84 85 As seen in Figure 24, the departure coefficients for the low lying levels of hydrogen differ considerably from unity. The inclusion of metallic line blanketing causes an overpopulation of the levels, while illumination alone causes a depletion of these populations. To investigate the effects of non-LTE on the emergent spectra, two additional plane parallel models were calculated for point 2, but were assumed to be in LTE. As can be seen in Figures 25 and 26, the inclusion of the effects of non-LTE produces no change in the observed spectrum. In fact, the Balmer and Lyman jumps were changed by less than 0.001 magnitudes. In the temperature regime considered here, the major opacity source is electron scattering, with hydrogen playing a very subordinate role. Hence, changes in the level populations produce no change in the emergent spectrum. What should be most effected are the hydrogen line profiles, and further work is needed to examine this effect. The emergent continuous and line spectrum of a stellar atmosphere can be greatly altered by incident radiation. In order to accurately model a close binary system, it is not sufficient to use non-illuminated enhanced flux model atmospheres, and the incident radiation must explicitly be taken into account. The effects on polarization due to line blanketing and illumination from a finite solid angle were also investigated. In these early type stars the source of the polarization is Thompson scattering off of free electrons. When a photon undergoes a scattering event, the energy of the photon causes the electron to oscillate. Since an accelerated Figure 24. The departure coefficients for the first three levelsofthree the forfirst coefficients The departure 24.Figure n ihu (oe) iebaktn tpit 2 (lower)pointblanketing line at and without (upper)with illuminated forparallelplanemodels hydrogen DEPARTURE COEFFICIENTS 1.0 1.5 0.8 1.2 0.7 0.9 1.3 0.6 1.1 -4 ■3 LOG N O N L I N E B L A N K E T E D 0SS) r 7 ( ■2 86 m, X u. O o g°. — i PARALLEL ILLUHINATION-NO LINE BLANKETING o PLRNE PARALLELI. ILLUH1NRT10N-N0 LINE BLANKETING NON-LTE PRRRLLEL ILLUHINRTION-LINE BLANKETING PLANE PARALLEL ILLUMINATION-LINE BLANKETING NON-LTE (B 0.00 1.00 2.00 3 . 0 0 11.00 FREQUENCY X 10-15 Figure 25. Flux distributions for model atmospheres at point 2 with and without metallic line blanketing and non-LTE Figure 26. He I IIand4471 He Mg Figure26. o CO o 0 o o' oa o <0 o' o” o RESIDUALm o' INTENSITY > VV0 U6.0 U80 <400 47.0 440 47.0 400 48.0 4482.00 4481.00 4480.00 4476.00 4474.00 4472.00 <<470.00 AU68.00 UU66.00 VV6V.00 and without metallic lineand non-LTE blanketing andmetallic without NON-LTE PLANE PARALLEL ILLUNINATION-LINE BLANKETING ILLUNINATION-LINE PARALLEL PLANE ^ P A R A L L E L ILLUHINATION-NO ILLUHINATION-NO L E L L A R A P ^ I j ^NE ^NE 1 8 4 4 PARALLEL ILLUHINATION-NO LINE BLANKETING LINE ILLUHINATION-NO PARALLEL l e l l a r a p AEEGH (ANGSTRGHS) WAVELENGTH residual intensities for the model atmospheres at point 2at point for with atmospheresintensitiesthe model residual E 47 MG I 4481 II G -M 4471 I HE n o i t a n i n u l l i - e n i l e n i l g n i t e k n a l b g n i t e k n a l b / / 89 charge will radiate, the photon will be re-emitted in a different direction. In order to see the radiation emitted by an accelerated charge, the motion must be across the line of sight. If one views a scattering event from different angles, there will always be some component of the motion across the line of sight regardless of the orientation of the plane of oscillation. With an ensemble of photons, some polarization will result from each event, and the net result approaches zero. However, if the scattering event is seen at right angles to the incident beam, the only component of motion across the line of sight is also at right angles to the incident beam so that we could expect the maximum amount of polarization to occur at right angles to the photon path. In a non-illuminated atmosphere, the flux is normal to the surface, so we obtain the maximum polarization, (Ij-Ir)/ (Ij+Ir), at the limb. However, the sign will be negative since 1^ is defined to lie in the plane containing the normal to the surface and the line of sight, while Ir is perpendicular to it. For an illuminated atmosphere, we have an additional component of the polarization due to the scattering of the incident radiation. As can be seen in Figures 27 and 28, which show the polarization for the atmospheres at point 2, the inclusion of metallic line blanketing will decrease the amount of local polarization in the ultraviolet due to the increase in the number of absorbers as compared to the number of scatterers in the total opacity. The magnitude of the polarization remains positive due to scattering perpendicular to the incident beam. However, in the visual where the effects of line blanketing are negligible, there is no change in the polarization. Figure 27. Percent polarization as a function of - PERCENT POLARIZATION 1 1 - - 0 for the plane parallel illuminated atmospheres at point 2 pointat illuminatedatmospheres for the parallelplane 20 40 OBSERVED 0.05 AT N Q N L I E B L A N K E T E D 080 60 5000 A 0 10 4 1 180 0 16 140 120 100 91 92 To give some indication of the effects of illumination from a finite solid angle, Figure 29 shows results from a preliminary model where the center of the illuminating star makes an angle p = 0.5 with respect to the.local normal. This is bracketed by plane parallel illuminated atmospheres with u = 0.3, 0.5, and 0.707. The polarization from the finite solid angle lies betwee the values obtained for the plane parallel illuminated models. As the incident beam becomes more normal to the- surface, the amount of positive polarization is decreased as the plane of scattering approaches the surface. In the same manner, the more normal component of the incident radiation from the finite solid angle reduces the positive polarization. This is also shown in Figure 30 which depicts the polarization from the finite solid angle illuminated atmosphere at point 2. In addition, the amount of positive polarization is reduced compared to the results shown in Figures 27 and 28. The relative effects on the observed polarization due to metallic line blanketing and illumination from a finite solid angle can also be seen in Figure 31, where the angle x is plotted as a function of 0O for the different models representing point 2. This angle, which indicates the plane of polarization, is measured from the local normal, thus for values of X less than 45° the polarization is positive, while for X > 45° the polarization is negative. At point 2 the angle of incidence of the plane parallel illumination is 54°.14. At 5000 A for the models with and without metallic line blanketing, X reaches a value of 35°.86, the compliment of iue2. ecn oaiaina ucino forapreliminary function of a as polarization Percent 29.Figure PERCENT POLARIZATION -6 ■5 ■7 •8 0 n 0.707. and disk at y= 0.5. Also shown are results for plane parallelforplane results are shown Also 0.5.y=at disk illuminated models with angles of incidence of angles with illuminatedmodels finite solid angle illuminated model with the center of thetheof center with illuminatedmodel solid finiteangle O B S E R V E D A T 20 912 > A U= 6040 0.3 0 0 10 4 10 180 160 140 120 100 80 U= .707 ofy= FSA Mr 03 0.5.0.3, .3 93 Figure 30. Percent polarization as a function of 930 A 5000A 94 Figure 31. The angle x as seen at at seen as x angle The 31. Figure X DEGREES 10 - 0 5 - 0 4 - 0 3 - 0 3 10 - o - o - - 10 FINITE SOLID SOLID FINITE ANGLE LINE BLANKETED LINE 20 30 NO LINE BLANKETING NO ij> 050 40 NO LINE BLANKETING LINE NO = 0 o temdl t on 2 point at models the for 90° = ^ ^ = LN BIANKETED ■LINE FINITE SOLID ANGLE SOLID FINITE 90 5000 A 60 930 A 70 95 96 of the angle of Incidence, indicating that the light received is due primarily to the scattered incident radiation. For the models at 930 &, however, X has a slightly larger value, thus the plane of polarization is more parallel to the surface. The higher opacity at this wavelength causes more of the incident radiation to be absorbed and re-emitted adding a horizontal component to the net polarization. Of course this effect is enhanced by the inclusion of metallic line blanketing. At both wavelengths, the value of X for the finite solid angle illumination has a value larger than 35°.86. At 5000 &, where the absorption is quite small, the observed light is mostly due to scattered incident radiation and this value of X reflects the more normal component of the incident radiation. This increase in the value of X is enhanced in the ultraviolet where the increased opacity adds its additional horizontal component, yielding a net negative polarization. Of all the quantities investigated in this study, the polarization is the most complex. On a local scale, by including metallic line blanketing the amount of polarization produced from the scattering of the incident radiation can be reduced due to the increased absorption. Illumination on the other hand may increase or decrease the amount of polarization, the plane of which is determined by the spatial distribution of the incident radiation. To produce a net polarization on a global scale, the source must be asymmetrical either in the sense of a physical distortion or an apparent asymmetry due to brightness variations. Thus, to understand the total polarization from the system the global properties of the stars must be studied. 97 Figures 32 and 33 show the total polarization obtained from each of the two stars as seen from three different angles. Generally, for the illuminating star, we have a net negative polarization in the ultraviolet, reflecting the steep source function gradient in this region, while the more shallow gradient in the visual produces the smaller positive polarization. The magnitude of the polarization indicates the amount of asymmetry present. The small amount of distortion and accompanying gravity darkening for the outer hemisphere yields a small amount of net polarization, while the larger distortion of the inner hemisphere and the side view gives a corresponding larger amount. If the incident radiation were ignored, we would still have slightly more polarization from the illuminated star due to the larger amount of shape distortion. The high degree of polarization in the side view of the illuminated star is due to the scattered incident radiation, for at this viewing angle, the scattering angle is close to 90°. Furthermore, the decrease in the amount of polarization due to an increase in absorption can be easily seen just shortward of the Balmer and Lyman jumps and in La. To understand the polarization obtained in more detail, it is necessary to look at the contriubtion of the local polarization to the global picture. Figure 34 shows the local normals on the illuminating star projected onto the plane of the sky. Locally a positive degree of polarization is defined to lie along these projected normals, while a negative polarization is oriented perpendicularly to them. As shown in 0.0 - H INNER HEMISPHERE OUTER 1 [EMISPHERE - 0.1 - SIDE ILUMINATING STAR 1000 2000 3000 4000 5000 WAVELENGTH (A) Figure 32. Percent polarization as a function of wavelength for three views of the illumlnaLlug star vo oo 0.0 - H - 0.1 INNER HEMISPHERE 'OTTER HEMISPHERE 8 SIDE - 0.2 ILLUMINATED STAR 1000 2000 4000 50003000 WAVELENGTH (A) Figure 33. Percent polarization as a function of wavelength for three views of the illuminated star VO vO Figure 34. Local normals on the illuminating star projected onto the plane of the sky 100 101 Figures 35 and 36 the local polarization in the ultraviolet is negative over the surface, while in the visual a positive degree of polarization is obtained. Since virtually all of the light obtained from the limb is due to scattering, we have a very large degree of negative polarization produced in both cases. For the illuminated star the situation is more complex. In Figure 37 we see a general negative local polarization in the ultraviolet; however, some positive polarization is produced by the scattering of the incident radiation. Once again with the reduced absorption in the visual* the polarization from the scattered incident radiation is much more prominent (see Figure 38). On a global scale, however, it is more convenient to discuss the polarization in the observer's reference frame. In this case, a positive polarization is now defined along the line parallel to the rotational axis and negative polarization lies in the equatorial plane. Firgures 39, 40, 41, and 42 show the local polarization from the previous four figures rotated into the observer's frame. While it is fairly straight forward to visualize how the local polarization would add up to give a net negative polarization in the ultraviolet and a net positive polarization in the visual for the illuminating star, the situation for the illuminated star is not at all obvious. For the observer, the polarization distributions are modified by the brightness distribution of the surface. Some indication of the brightness distribution can be obtained from looking at the limb darkening curves for the two stars. Figure 43 shows / \ Figure 35. Local polarization for the illuminating star at 930 A. The values along the limb are shown reduced by a factor of ten. SS is the sub-stellar point Figure 36. Local polarization for the illuminating star at 5000 A. The values along the limb are shown reduced by a factor of 100 103 930 A SS Figure 37. Local polarization for the illuminated star at 930 A. The values along the limb are shown reduced by a factor of ten 104 0.1% 5000 SS Figure 38. Local polarization for the illuminated star at 5000 A. The values along the limb and those with arrowheads are shown reduced by a factor of 100 '— I— I Figure 39. The polarization shown in Figure 35 rotated into the observer's frame 5000 A — ss Figure 40. The polarization shown in Figure 36 rotated into the observer s frame 107 930 A SS b r Figure 41. The polarization shown in Figure 37 rotated into the observer's frame 108 5000 A 0.1% Figure 42. The polarization shown in Figure 38 rotated into the observer's frame Figure 43. Limb darkening curves for the inner hemisphere of the illuminating star. 110 Ill MERIDIAN EQUATOR 5000 A INNER HEMISPHERE ILLUMINATING STAR —]---- 1---- 1---- 1---- 1---- 1---- 1---- i---- r 10 20 . 30*105060e 70 8090 Figure 43. 112 the limb darkening for the inner hemisphere of the illuminating star. The shape distortion causes the substellar point to be the coolest region on the stellar surface, resulting in an initial brightening as one moves toward the limb. The different degree of shape distortion and corresponding gravity darkening in the equatorial and meridional planes results in the star appearing brighter as one moves to the pole as compared to the intensity along the equator. For the inner hemisphere of the illuminated star (Figure 44) the brightness distribution is dominated by the incident radiation with only a small indication of the effect of gravity darkening in the far ultraviolet. As viewed from the side, the limb darkening curves (Figures 45 and 46) along the "terminator" ( 6 = 90°) for the two stars are very similar, showing a much faster drop in brightness in the ultraviolet as one approaches the limb. In the equatorial plane of the illuminating star (Figure 47) the limb darkening curves are similar but with a slight asymmetry due to the larger shape distortion of the inner hemisphere, 0 < 90. In Figure 48, the limb darkening curves for the equatorial plane of the illuminated star are shown. The incident radiation totally dominates the brightness distribution, emphasizing the region of the inner hemisphere. This is more dramatically shown in Figures 49, 50, 51, and 52 where isophotes are presented for the two stars. As viewed from the side, the shape distortion of the larger star will shift the region of maximum brightness slightly toward the outer hemisphere. For the illuminted star, however, the incident radiation shifts the apparent center of the 113 1.0 0.9 0.8 930 A 5000 A 0.7 0.5 o ^ 0.4 0.3 0.2 INNER HEMISPHERE 0.1 ILLUMINATED STAR 0.0 EQUATOR 0 10 20 30 40 50 60 70 80 90 e Figure 44. Limb darkening curves for the inner hemisphere of the illuminated star 114 1,0 0.9 0.8 0.7 0.6 930 A 0.5 5000 A ILLUMINATING STAR 0.3 0.2 0 10 20 30 40 50 60 70 80 90 Figure 45. Limb darkening curves for the terminator of the illuminating star 115 1.0 i9 0.8 0.7 0.6 930 A 5000 A 5 0.3 ILLUMINATED STAR 0.2 0.1 0 Figure 46. Limb darkening curves for the terminator of the illuminated star 116 1.0- 0,8- 0.5- 0 > — 930 A 0.3- — 5000 A 0,2- 0.1- ILLUMIMATIN6 STAR 0 20 AO 60 80 e 100 120 MO 160 180 Figure 47. Limb darkening curves in the equatorial plane of the illuminated star 117 1.7 ILLUMINATED STAR 1.6 1.5 -930 A 5000 A 1.2 1.1 1.0 0.9 0 20 40 60 80 100 120 1A0 160 180 0 Figure 48. Limb darkening curves in the equatorial plane of the illuminated star 118 Figure 49. Isophotes on the illuminating star for the inner hemisphere (:top), side view, and outer hemisphere (bottom) at 5000 A. Contour intervals are 0.15 magnitudes. 119 Figure 50. Isophotes on the illuminating star for the inner hemisphere (top), side view, and outer hemisphere (bottom) at 930 A. Contour intervals are 0.3 magnitudes 120 Figure 51. Isophotes on the illuminated star for the inner hemisphere (top), side view, and outer hemisphere (bottom) at 5000 A. Contour intervals are 0.15 magnitudes. 121 Figure 52. Isophotes on the illuminated star for the inner hemisphere (top), side view, and outer hemisphere (bottom) at 930 A. Contour intervals are 0.3 magnitudes 122 observed disk away from the geometrical center and toward the center of mass of the system. This effect could have important ramifications for radial velocity studies. It is these brightness distributions coupled with the polarization distributions that are needed to understand the net polarization. For the large star, the limb darkening emphasizes the central portions of the observed disk, giving the net positive polarization in the visual and the net negative polarization in the ultraviolet. On the other hand, for the illuminated star, most of the light received is from the equatorial region of the inner hemisphere resulting in a net negative polarization over the entire wavelength range considered. Similarly, these brightness distributions will act as weighting factors in determining the total emergent radiation from each star as well as the system as a whole. Figures 53, 54, 55, and 56 present the theoretical flux distributions and spectral line profiles for each star as seen from three different viewing angles. The illuminating star appears hottest when seen edge on due to limb darkening emphasizing the center of the apparent disk which in this case represents the hottest portion of the equatorial region. It is also this configuration that yields the greatest amount of total flux due to the large surface area presented. As expected, the inner hemisphere of the illuminated star appears hottest, with lower temperatures seen as one moves to the outer hemisphere. This trend is born out by the spectral types obtained from the theoretical monochromatic color differences and line ratios (see m, o o o in, U. INNER HEMISPHERE SIDE VIEW OUTER HEMISPHERE o 0.00 1.00 2.00 3.00 U. 00 FREQUENCY X 10-15 Flguce 53. Flux distributions for three views of the illuminating star CO, o in, b. INNER HEMISPHERE o SIDE VIEW OUTER HEMISPHERE o 0.00 1.00 2.00 3.00 FREQUENCY X 10-15 124 Figure 54. Flux distributions for three views of the illuminated star Figure 55. He I 4471 and Mg II 4481 residual intensities for the inner hemisphere, side view, and outer hemisphere of the illuminating star. The small variations present are masked by the thickness of the line. 125 •c GO RESIDUAL INTENSITY e o c» ° n e 38 c 38 to 3D CD tO o r Figure 55 9ZT HE I 4471-MG II 4481 O o> • . o CD O' O' —I C E 2 INNER HEMISPHERE SIDE VIEW OUTER HEMISPHERE o O O' 4464.00 4466.00 *1488.00 4*170.00 4472.00 4474.00 4476.00 4480.00 4481.00 4482.00 HAVELENGTH (ANGSTROMS) 127 Figure 56. He I 4471 and Mg II 4481 res.idual intensities for three views of the illuminated stai 128 Table 6). Since these are results for the integrated flux, the large variations that were locally produced, tend to average out giving remarkably consistant spectral types from the different criteria utilized. This is most obvious for the system as a whole as seen at 0.25 phase. Of course the continuum colors could be slightly modified due to the relative doppler shifts of the two components, but this effect will be small. No line ratio was used since the lines from the two stars will be at maximum separation due to their radial velocities. The maximum monochromatic color variation due to orbital motion that is predicted from this model corresponds to approximately 1.5 subclasses in spectral type. This will be modified to some extent during primary eclipse where an annulus of the larger star will still be seen, thus diluting the light from the outer hemisphere of the illuminated star. The ratio of equivalent widths of He I 4471/ Mg II 4481 shows a slightly larger variation. However, in most instances the lines from the two components will be blended, thus complicating the spectral classification. We can also compare the flux distribution of the system to that obtained by Protheroe and Buerger for their non-line blanketed plane parallel illuminated model. VThile their model takes into account the inclination of the system (= 81°) the model presented here is seen equator on. However, the inclination efects will be small. As can be seen in Figure 57 the general trends presented earlier for the individual atmospheres are carried through to the integrated flux. Once again there is a flux enhancement shortward of the Lyman jump with a Table 6 Theoretical Spectral Types MODEL CRITERION m10-m25 m2 5-m3 5 m3 5-m4 5 m37-m58 He I/Mg II Illuminating Star inner hemisphere B2.3 B1.5 B1.5 B1.5 BO.9 side view B1.2 B1.3 B1.4 B1.4 BO.8 outer hemisphere B1.3 B1.4 B1.4 B1.5 BO.9 Illuminated Star inner hemisphere B1.5 B2.2 B1.4 B1.5 BO.7 side view B2.9 B2.4 B2.4 B2.3 B2.4 outer hemisphere B3.2 B2.7 B3.5 B2.7 B3.3 System seen at ■ phase = 0.25 B1.5 B1.7 B1.7 B1.7 --- 129 -! * * . . 8 - 9 - |1 0 - K i i - _ PROTHEROE and BUERGER * KUZMA 12 _ PHASE =0.25 1 3 - 1000 2000 3000 4000 5000 WAVEIENCTH (A) 130 '•’ip;urc 37. The flux distribution from the svstom seen ;it phase =0.25 eomparod witli the flux distribution obtained by Protheroe and linerger 131 decrease in the flux just to the redward side. In turn, there is a flattening of the Balmer and Paschen gradients due to the flux redistribution to longer wavelengths. Furthermore, there is a slight increase in total flux due to the larger surface area of the illuminated star that is effected by the incident radiation from a finite solid angle. Thus the flux distribution from the system will mimic that of a star of slightly cooler spectral type. Also the large negative polarization from the illuminated star seen at 0.25 phase totally dominates the system, giving a negative polarization at all wavelengths considered. One final product of the incident radiation that has not been mentioned is the horizontal flux. As can be seen in Table 5 and Figure 58, the horizontal flux reaches almost 0.8 of the total normal flux in the atmosphere. To get some idea of the effect of this flux on the atmosphere, we can make a crude estimate of the velocity induced. The radiative acceleration is given by the integral over wavelength of the product of the total opacity and the component of the flux in the direction of the acceleration. If we assume that the ratio of the horizontal flux to the normal flux equals the ratio of the horizontal and verticle acceleration, we derive a value of approximately 125 cm/sec^ for the acceleration at the surface of the atmosphere at point 2. Neglecting viscosity, we obtain the result that the surface material will be accelerated to sound speed (20 km/sec) over approximately 0.3 of a solar radius, which is a small fraction of the dimensions of the illuminated star. While this is 132 1.0 Figure 58. The ratio horizontal flux/normal flux on the surface of the inner hemisphere of the illuminated star seen projected onto the plane of the sky 133 just a crude estimate, it appears that the incident radiation could produce large scale mass motions over the stellar surface, and should be investigated more fully. For the Roche model, surfaces of constant potential correspond to surfaces of constant temperature. For this work the surface of the star was taken to be t r oss “ 0. A cursory glance at Table 5 shows this is definitely not a surface constant temperature, and that a scaling of the radial distance for each atmosphere is indicated. However, there is a problem as to which temperature to use as a reference frame. A first guess might be to use the effective temperature of the polar atmosphere, which in this case is 18250°K. Due to the illumination and subsequent heating, however, the substellar atmosphere does not become cooler than 20,000°K. On the other hand, in order to use one of the illuminated atmospheres as a reference point, one must first know the radial distance in order to calculate the effects of illumination. In turn, the illumination will alter the physical height of the atmosphere. If we use the illuminated substellar point as the reference, we must change the position of the polar atmosphere by about 4 X 10^ km which is approximately 1% of the stellar radius. Thus we have a coupled situation, where one should calculate the illuminated atmospheres, scale their positions, and then re-do the computations. While this is not currently practical, nor of major concern for this model, as the star fills more of its Roche lobe and hence becomes more extended, this effect can become of extreme importance. Furthermore, radiatively driven large scale mass motions could result in the equipotentials no 134 longer being surfaces of constant temperature, thus making the problem much more complex* In this dissertation, we have investigated the radiative interaction in a close binary system. By assuming a Roche model, we can calculate the atmospheric parameters of any point on the stars. In turn, the geometry has been set up to calculate the radiation incident on any point on the companion. Using the general form of the incident terms of the moment equations, we have calculated the structure for a grid of atmospheres on the illuminated star. In the model calculated, the major effect on the structure is due to the inclusion of metallic line blanketing. The large opacity in the ultraviolet causes heating in the atmosphere and a re-arrangement of the emergent flux. The spectral line profiles are greatly altered due to the flattening of the temperature gradient. This is enhanced for the neutral helium lines due to an increase in the amount of ionization caused by the higher temperatures. The colors and line profiles obtained from the illuminated atmosphere are very different from those calculated using a non-illuminated atmosphere with the same total emergent flux. Metallic line blanketing reduces the amount of polarization in certain wavelength regions due to the large increase in the relative number of absorbers to scatterers. While illumination from a finite solid angle only weakly effects the temperature structure and emergent radiation, the locally.produced polarization is altered by the spatial distribution of the incident radiation. On a global scale, the large 135 net negative polarization from the illuminated star dominates the system, giving a net negative polarization in the visual as well as the ultraviolet for the system seen at phase = 0.25. The shape distortion and especially illumination produce non-concentric isophotal lines which shift the apparent center of the observed surface of the illuminated star closer to the center of mass of the system. Finally, the horizontal flux can be a non-trivial fraction of the normal flux in the atmosphere, and may be resonsible for large scale mass motions on the stellar surface. Using the methods discussed here, the illuminated star can then be used to calculate the effect of the incident radiation on the primary. 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