The Orbital Inclination

There is another effect of the tidal forces of the compact object on the secondary star, namely, that the secondary star is not spherical. Rather, it has a characteristic teardrop shape. The non-spherical shape is essentially a giant tide on the star.

The consequence of this non-spherical shape is that one sees different cross-sections of the star as it orbits around its companion. Consider the diagram below, showing a binary system composed of a compact object (solid black circle) and its companion star:



When viewed side-on, as on the left of the diagram, one sees a larger cross-section than if one views the star face-on. Thus twice per orbit (when viewed side-on) the star appears brighter. So if one creates a light curve, that is to say a plot of brightness vs. time, one sees two maxima and two minima during each orbit, describing a characteristic double-humped lightcurve.

The importance of these double-humped lightcurves is that the difference between the maximum and minimum brightness is determined by the orbital inclination of the system. If the binary system is observed face-on, where i=0° (as on the right side of the above diagram), then one always looks down at the top of the star, and the cross-section doesn't change at all during the orbit. But if the system is viewed side-on (i=90°), one sees the star first from the side, then from the end, then from the side, and so on. This creates the maximum amount of difference in the brightness during the orbit. In general, the system is seen somewhere between the two extremes, and inclination can be determined by the amount the brightness changes.

Once we know the inclination of the binary system relative to the observer (i), as well as the mass function (ƒ) and mass ratio (q), we can calculate the mass of the black hole by solving the mass function for M1.

There are some possible problems with this method. In particular, there is often light from the accretion disk in addition to light from the secondary star. This extra light dilutes the changes due to the ellipsoidal variability, and makes the system appear more face-on than it actually is. Also, there are other sources of light variations in binary star systems, including among other effects hot spots in the disk, cool spots on the star, and eclipses. Fortunately, all of these kinds of changes in brightness are also associated with changes in color, so by observing the lightcurves at many different wavelengths of light it is often possible to sort out what is causing the changes. A successful example of this process can be seen here.

 

• A detailed discussion of the light curves of binary stars can be found here.