## ECLIPSING BINARY STARS

### A Simple Model for Computing Light Curves

[ BASIC Subroutine | Dan Bruton | astro@sfasu.edu ]

### INTRODUCTION

Eclipsing binary stars are just one several types of variable stars. These stars appear as a single point of light to an observer, but based on its brightness variation and spectroscopic observations we can say for certain that the single point of light is actually two stars in close orbit around one another. The variations in light intensity from eclipsing binary stars is caused by one star passing in front of the other relative to an observer. If we assume that the stars are spherical and that they have circular orbits, then we can easily approximate how the light varies as a function of time for eclipsing binary stars. These calculations can be performed in a relatively short computer program.

### THE ORBIT AND STAR PARAMETERS

A brightness versus time plot for a variable star is know as light curve. For close binary systems, time is usually expressed as phase, for which one unit of time is the orbital period. Measured light curves for periodic variable stars are usually "folded", which means that successive cycles are plotted atop one another. An example of an experimentally measured light curve is given in Figure 1. Note that the vertical axis of the graph is labeled "magnitude" which is a logarithmic scale of brightness.

Figure 1: Photometry of Beta Lyrae in 1992-1993.

The shape of the light curve for an eclipsing binary star system depends mostly on the relative brightness and size of the two stars as well as their orbital inclination as seen from Earth. We will now see how you can use simple geometry to generate a light curve. For this simple spherical star model, the parameters that are needed to generate a light curve are:

i = Orbital Inclination
M1, M2 = Masses of the Stars
L1, L2 = Luminosities of the Stars
R1, R2 = Radii of the Stars

The orbital inclination is measured relative to the observer in degrees and is 90 degrees for edge-on orbits. The star masses and luminosities can be expressed in arbitrary units and the star radii will be expressed in terms of a fraction of the orbit radius of the binary star orbit.

### THE ECLIPSE EQUATIONS

If the orbits of the binary stars are assumed to be circular orbits then the coordinates of star 1
(x1,y1,z1) and star 2 (x2,y2,z2) can be found using the following equations:

The x and y positions of the two stars can be used in a computer program to draw the stars on the screen and the z coordinate can be used to determine which star is in front during an eclipse. The equations above are discussed in books like Herbert Goldstein's Classical Mechanics [1] or Jerry B. Marion's Classical Dynamics [2].

Luminosity is defined to be the amount of energy escaping from a star's surface per unit time. The energy flux, F, at the surface a star is the amount of energy radiated per unit area per unit time. For spherical, uniformly lit stars we can write

If we ignore limb darkening, then we can approximate the measured brightness of the binary star system as seen by an observer as

where A1 and A2 are the areas of the star disks seen by the observer and K is a constant that can be determined from the area of the observer's detector and the distance between the Earth and the binary star system. A1 and A2 can be found by considering the geometry of the eclipses. To find these areas we will need to know the apparent distance between the two stars as seen by an observer. Using Figure 2, we see that this distance is

The values for A1 and A2 for the different stages of the eclipses are given in the tables below. These tables are only valid when R1 > R2.

The area of a portion of a circle cut by a line segment can be found using the following equation:
where r is the radius of the circle and theta (in radians) is the angle subtended by the line segment. Therefore,

The theta's can be found using the law of cosines as follows:

Note: The theta's here are assumed to be between 0 and 180 degrees. If your computer computes theta's that are greater than 180 degrees, then you will need to subtract them from 360 degrees to get the correct values.

Another Simpler Approach: If your computer computes theta's that are greater than 180 degrees, then you may also replace (not swap) the equations for the Deep Eclipse with those from the Shallow Eclipse in Table 2 above.

### SUMMARY AND RESULTS

Given i, M1, M2, L1, L2, R1, R2, and the phase of the orbit one can use the equations above to determine the relative positions of the stars, (x1,y1,z1) and (x2,y2,z2), and the brightness of the system, I. Figure 3 shows an example light curve generated using the equations above.

Figure 3: An example light curve with masses and luminosities expressed in arbitrary units and the star radii expressed in terms of a fraction of the orbit radius of the binary star orbit

You will notice that the masses have no effect on the light curve for this simple model. However, one will see the effects of the masses by observing the stars in orbit about the center of mass in an animation. For a large mass ratios q=M2/M1 (M2 >> M1), star 2 will not move much. For small mass ratios, star 1 will not move much. The orbital inclination, i, will effect the depth of the eclipses with the minimum possible intensities occurring when the orbital inclination is 90 degrees. The radii and luminosities of the stars also effect the depth of the eclipses as well as the general shape of the light curves. There is software and code available that use the simple model discussed here. To improve this model, the effects of limb darkening, tidal distortion, elliptical orbits and other phenomena can be considered. There are several books and articles that discuss these effects [3,4,5,6,7].

### REFERENCES

1. Herbert Goldstein, Classical Mechanics, Second Edition, Addison-Wesley, 1981,
Equations 3-64, 3-8 and 3-2.

2. Jerry B. Marion, Classical Dynamics, Equations 7.52, 7.10, and 7.3.

3. David Gossman, "Light Curves and Their Secrets", Sky & Telescope, October 1989, p.410.

4. Robert E. Wilson and Edward J. Devinney, "Realization of Accurate Close-Binary Light Curves: Application to MR Cygni", Astrophysical Journal, vol. 166, 1971, p. 605-619.

5. Robert E. Wilson, "Eccentric Orbit Generalization and Simultaneous Solution of binary Star Light and Velocity Curves", Astrophysical Journal, vol. 234, 1979, p. 1054-1066.

6. Zdenek Kopal, Language of the Stars, D. Reidel, 1979.

7. Zdenek Kopal, The Roche Problem, Kluwer Academic, 1989.

Notes: The equations in this document were created using MathType 3.0. A Microsoft Word document was created with the equations and figures. The document was then saved as an RTF (rich text format) file from Microsoft Word and then RTFtoHTML was used to create the HTML file. This document also uses the HTML subscript feature <sub> </sub>.

Dan Bruton
astro@sfasu.edu