ECLIPSING BINARY STARS
A Simple Model for Computing Light Curves[ BASIC Subroutine | Dan Bruton | firstname.lastname@example.org ]
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:
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.
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 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.
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.
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].
Back to Eclipsing Binary Stars