Modeling an Eclipsing Binary Star

posted Jan 24, 2010, 11:00 AM by James Roe
While observing a star known as LPH128 (a high mass xray binary), I learned that a star nearby had been noticed to be a variable but had not been characterized very well at all, so I decided to do double duty by moving LPH128 over a bit in my field of view to bring the suspected variable into the same field.  The star, known as GSC 3973-1124 is located at 22h09m18s and +54 d37m20s in the northern constellation of Cephus.  It is about 10th magnitude in a V filter.  During my first night of observing this star I detected a minimum of about 0.3 mag depth - nice and encouraging.  The next night I detected another minimum almost exactly 24 hours later.  OH NO - a period near one day would mean it might take a long time to get the complete light curve.  Fortunately, a friend in Cyprus offered to look at the star and he also got a minimum just a few hours earlier than mine.  Wow.  To make the story shorter, it turns out the star has a period of 0.4904225 days (11.77 hours) with a minimum (primary or secondary) every 5.39 hours.

The period of such an object is usually defined and refined by oberving successive minima but the overall shape of the light curve is studied with a phase plot.  In effect, data widely scattered in time is "folded" to represent just one cycle of the light curve.  Below is the phase curve I obtained for this object using the period mentioned above.

The small red crosses are the actual measured points while the blue dots are from a model of what's going on (discussed below).  Much can be noted by study of this curve.  In particular, the continually varying light from the star  suggests the stars are very close to each other and not undergoing total eclipses.  That the depth of the first minimum (at Phase = 0.0) is somewhat deeper than the secondary minimum at Phase 0.50 suggests one star is probably hotter than the other (hotter stars are brighter).

There is a wonderful tool called Binary Maker that can be used to estimate the properties of these two stars.  While there are some guides from the light curve (ie, one star appears to be hotter than the other, both stars are probably not spherical (filled their Roche lobes), etc), the process is essentially trial and error by comparing predicted results to the actual data.  The predicted results shown above as the blue dots came from the following assumptions.  The stars have a mass ratio of 0.4 (ie, the smaller star is 0.4x as heavy as the bigger star), the stars atmospheres are in contact, the bigger star is about 270 deg C hotter than the smaller star and we are viewing it from about 55 deg above its equatorial plane.  There are other parameters such as limb darkening and reflection coefficients that are taken from detailed studies of such systems.  The model of the system is shown below.

The two outer crosses mark the centers of mass of the two stars while the center cross mark shows the barycenter for the system about which both stars rotate.  The smaller red circle shows the path of the center of the larger star and the larger dotted circle shows the path of the center of the smaller star.  We can also calculate the relative radial motions of the two stars as shown below.

If we can get someone to measure the actual radial velocities (using a spectroscope), we can determing the actual orbital dimensions of the pair and their masses.  All in all, a fun exercise.