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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Modelling Spectro­photometric Characteristics of
Eclipsing Binaries
Grzegorz Polubek
Astronomical Institute of the Wroclaw University PL­51­622 Wroclaw,
ul. Kopernika 11, Poland
Abstract. We present a computer program for modelling energy flux
distribution and light curves of eclipsing binaries from far ultraviolet to
infrared regions. The Roche model is assumed. Proximity e#ects of com­
ponents (reflection of radiation and surface gravity darkening) are taken
into account. The calculations are made in the absolute flux units and
the newest Kurucz's (1996) models of stellar atmospheres are used. As
an example, we consider the Algol (# Persei) system, where an eclipsing
pair (Algol A­B) is accompanied by the third component (Algol C).
1. Model Description
The photospheres of eclipsing components are represented by a grid of surface
elements, which are treated as plane­parallel and homogeneous ones. In order to
calculate the synthetic fluxes of radiation an integration over all surface elements
visible at a given orbital phase, #, should be done. For this purpose geometrical
parameters (i.e., area of elements and their positions) as well as physical ones
(temperatures and surface gravities) for all elements have to be known. To de­
scribe the system's geometry we assumed the Roche model presented by Kopal
(1959) and Limber (1963) for synchronous and nonsynchronous rotation, respec­
tively. Assuming synchronous rotation of the components, the total potential #
can be written as:
# = G MA
r 1
+G MB
r 2
+ # 2
2 # (x - aMB
MA +MB ) 2 + y 2 # , (1)
where MA and MB are stellar masses, a -- orbital separation of components, # --
angular velocity of the system about the z­axis, G -- the gravitational constant,
r 1 2 = x 2 +y 2 +z 2 , and r 2 2 = (a-x) 2 +y 2 +z 2 . Using spherical polar coordinates
(r, #, #) equation (1) can be rewritten as:
# = 1
r
+ q # 1
(1 - 2lr + r 2 ) 1
2
- lr # + 1
2 (1 + q)(1 - n 2 )r 2 , (2)
where q = MB /MA
,# is the dimensionless Roche potential
# = a#
GM -
1
2
q 2
1 + q
, (3)
19

20 Polubek
12850 K
13000 K
4000 K
6000 K
Algol A
Algol B
Figure 1. The surface temperature distribution for Algol A­B system
before (upper plot) and after (lower plot) correction for temperature
due to the irradiation e#ect. It is clearly seen that the irradiation
e#ect a#ects mainly the B component's temperature distribution. The
binary system is displayed at orbital phase # = 0.3.
and l, m, n are direction cosines of the radius vector, i.e. l = sin # cos #, m =
sin # sin #, and n = cos #. If we assume that the photospheres of primary and
secondary components are equipotential surfaces with
potentials# 1
and# 2 ,
respectively, then the equation (2) allows us to determine the total surfaces
of both components. We solve this equation by the iterative Newton­Raphson
method.
The local surface gravity follows from the equation
#g = -grad# = - # ##
#x
,
##
#y
,
##
#z
# . (4)
If we neglect the e#ect of irradiation by the companion, the temperature at
(i, j)­point of the surface grid depends only on the local value of gravity
T e# (i, j, 0) = T e# (pole, 0) # g(i, j)
g(pole)
# #
, (5)
where # is the gravity­darkening exponent. Index 0 refers to e#ective temper­
atures in case of neglecting of the irradiation e#ect (hereafter referred to as
intrinsic ones). For photospheres in radiative equilibrium the integral flux is
proportional to the local value of gravity (von Zeipel 1924) which implies # =
0.25. The gravity­darkening exponent is smaller for photospheres with convec­
tion and in case of thick convective envelope # = 0.08 (Lucy 1967). If we take
into account the irradiation e#ect, the local temperatures have to be modified.
For this purpose an iterative procedure based on the Chen & Rhein (1969) ap­
proach was used. In the first step, the intrinsic e#ective temperatures of both

Modelling Eclipsing Binaries 21
1100 1600 2100 2600 3100
0.0
1.0e­09
2.0e­09
3.0e­09
wavelength [A]
flux
Figure 2. Examples of IUE spectra of Algol (solid lines) in
erg cm -2 s -1 š A -1 , in comparison with the best­fit model calculations
(circles) for orbital phases # = 0.925, 0.979, 0.996.
components were set. Then, for each surface element (i, j) of the given com­
ponent the flux F (i, j) received from the other component was evaluated. The
corrected temperature T e# (i, j, c) follows from the equation
T 4
e# (i, j, c) = T 4
e# (i, j, 0) +A F (i, j)
#
, (6)
where A is bolometric albedo, # ­ Stefan­Boltzmann constant. A was assumed
to be constant over the stellar surfaces. In the case of radiative equilibrium
all absorbed energy must be re­emitted; this means that A equals 1.0. For the
convective models the bolometric albedo can be smaller than 1.0 and we assume
A = 0.5 (cf. Rucinski 1969). The influence of the irradiation e#ect for the Algol
eclipsing pair is shown in Figure 1. Next, we calculate the radiation flux received
from the system:
F # = # I # (µ)µ dS , (7)
where integration extends over visible parts of all components, µ = cos # (# is
the angle between the surface normal and the line of sight). In this paper we use
the second­order limb­darkening law for the evaluation of the specific intensity
I # (µ) = I # (1)[1 - u 1 (1 - µ) - u 2 (1 - µ) 2 ]. (8)
The coe#cients u 1 and u 2 , defined by Wade & Rucinski (1985), were calculated
for Kurucz's (1996) unpublished models of stellar atmospheres.
2. An Example
We analyse UV spectra of Algol collected by the IUE satellite. We use 15 pairs of
IUE spectra from short wavelength primary (SWP) and long wavelength primary

22 Polubek
0.3 0.5 0.7 0.9 1.1 1.3
4.8
4.6
4.4
4.2
4.0
3.8
0.3 0.5 0.7 0.9 1.1 1.3
3.3
3.1
2.9
2.7
2.5
2.3
2.1
phase phase
(a) (b)
magnitude
Figure 3. Comparison of the theoretical light curves of Algol (solid
lines) with observations (dots) in Johnson V band (a) and in the in­
frared at 1.6 µm (b).
(LWP) cameras. We found T pole
e# (Algol A) = 13000 K, T pole
e# (Algol B) = 5000
K, T pole
e# (Algol C) = 7000 K, i A-B = 82. o 2 and other parameters similar to
those given by Tomkin & Lambert (1978). The best­fit solution for the three
pairs of spectra is displayed in Figure 2. This model was further verified by
UBV observations of Algol taken from Wilson et al. (1972) and by the infrared
observations of Chen & Reuning (1966). The comparison of these observations
with the theoretical light curves are shown in Figure 3.
Acknowledgments. This work was supported by the research grant No. 2
P03D 001 08 from the Polish Scientific Research Committee (KBN).
References
Chen, K.­Y., & Reuning, E. G., 1966, AJ, 71, 283
Chen, K. ­Y., & Rhein, W. J., 1969, PASP, 81, 387
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Lucy, L.B., 1967, Z. Astrophys., 65, 89
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