## ECLIPSING BINARY STARS## A Simple Model for Computing Light Curves[ BASIC Subroutine | Dan Bruton | astro@sfasu.edu ] |

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:

M

L

R

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.

(x

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 A_{1} and
A_{2} 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.
A_{1} and
A_{2} 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 A_{1} and
A_{2} for the different
stages of the eclipses are given in the tables below.
These tables are only valid when R_{1} >
R_{2}.

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.
*

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=M_{2}/M_{1}
(M_{2} >> M_{1}), 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].

- Herbert Goldstein,
*Classical Mechanics*, Second Edition, Addison-Wesley, 1981,

Equations 3-64, 3-8 and 3-2. - Jerry B. Marion,
*Classical Dynamics*, Equations 7.52, 7.10, and 7.3. - David Gossman, "Light Curves and Their Secrets",
*Sky & Telescope*, October 1989, p.410. - 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. - Robert E. Wilson, "Eccentric Orbit Generalization and Simultaneous
Solution of binary Star Light and Velocity Curves",
*Astrophysical Journal*, vol. 234, 1979, p. 1054-1066. - Zdenek Kopal,
*Language of the Stars*, D. Reidel, 1979. - Zdenek Kopal,
*The Roche Problem*, Kluwer Academic, 1989.

*
Back to Eclipsing
Binary Stars
*

*
*

Dan Bruton

astro@sfasu.edu