PY 228: STELLAR ASTROPHYSICS

HOMEWORK #2: SOLUTIONS

1. Verify Kepler's three laws.
   • Increase the eccentricity to view an elliptical orbit.
     
     
   • For a highly elliptical orbit, in what part of the orbit is the star moving the fastest?
     When the stars are closest together (periastron).

   • Keeping the masses of the two stars constant, vary the binary separation and record the period. Does the orbit satisfy Kepler's Third Law?
     Plotting the log of the period versus the log of the orbital separation produces a straight line with a slope of 1.5, in agreement with Kepler's Third Law.
     

2. Set up the program to simulate the binary star Algol using the following parameters.
   • Primary Mass = 3.7 solar masses
   • Primary Radius = 2.9 solar radii
   • Primary Temperature = 13,000 K
   • Secondary Mass = 0.8 solar masses
   • Secondary Radius = 3.5 solar radii
   • Secondary Temperature = 4,500 K
   • Orbital Separation = 13.96 solar radii
   • Orbital Eccentricity = 0.015
   • Orbital Inclination = 81.4 degrees
   What is the orbital period?
   Orbital Period = 64 hours

   Watch the light curve and the "View from Earth". Is there a bug in this program? (Which star is more luminous?)
   The light curve shows a steep drop in luminosity when the red star is eclipsed, but almost no change in luminosity when the blue star is eclipsed (see figure below). This is WRONG! The blue star, being much hotter and of the same size as the red star, should be much more luminous than the red star. So the light observed from the system should drop when the blue star is eclipsed, not when the red star is eclipsed.
   

   Forgetting that you know the answers, begin from the observations (radial velocity curves and orbital period) and derive the masses of the two stars.
   The orbital velocities can be extracted from the radial velocity curves. Since the orbit is nearly circular, the maximum velocity on the radial velocity curve is the orbital velocity: v_red = 200 km/s, v_blue = 50 km/s (approximately!).
   

   First, get the distances from the center of mass given the velocities and period...
   
   Next, get the sum of the masses using Kepler's Third law...
   
   Finally, use the ratio of the masses (m1/m2=r2/r1) to get individual masses...
   

3. Adjust the orbital inclination to 0 degrees. Why is such a binary system much less useful in obtaining stellar parameters?
   The radial velocity is always zero, so the orbital velocities, and hence stellar masses, are unknown (unless the orbital separation can be resolved).

4. Set up a close binary system with a primary star of spectral type B0V, a secondary star of spectral type G0V, and a separation of 60 solar radii. What is the orbital period? Notice the slight wobble of the primary star. Now change the luminosity class of the primary to a supergiant, choosing a spectral type that gives roughly the same mass as the star had when it was on the main sequence. What will happen to this system when the primary star evolves into a supergiant?
   Initial system with orbital period = 313 hours.

   K5 Supergiant becomes larger than entire binary system! The companion star would be swallowed up by the supergiant.