Consider a spherically symmetric, nonrotating star of mass m and radius r. If it gets too close to the supermassive black hole of mass M≫m at the center of a galaxy, it will be ripped apart by tidal forces. Throughout this problem, neglect relativistic effects and give answers in terms of G, M, m, and r. When asked for numbers, assume the star is Sun-like, so that m=2×1030 kg, r=7×108 m, and M=106m, and give all numeric answers to one significant figure.
a. Suppose the star orbits the black hole in a circle of radius R. What is the radius of curvature of the trajectory of each point on the star's surface? Note that the star does not rotate as it orbits.
b. Gas on the star's surface facing the black hole is attracted to the star by the star's gravity, and pulled away from the star by the black hole's tidal force. Write an expression for the tidal acceleration at the point closest to the black hole.
c. At a radius R where the gravitational and tidal forces are in equilibrium, the star will start being tidally ripped apart. This effect can only be observed when R is outside the black hole's Schwarzschild radius Rs=2GM/c2. Find an expression for R and numerically evaluate the ratio R/Rs.
For the rest of the problem, we consider a star on a parabolic orbit whose periapsis (distance of closest approach) is equal to the radius R found in part c. Near periapsis, the star will be torn apart in a "tidal disruption event". As a very rough model of this phenomenon, assume the star is initially a single rigid body. The moment it reaches periapsis, it fragments into many rigid pieces which do not interact with other, and feel only the black hole's gravity.
d. Afterward, some of the fragments will escape from the black hole, while the rest remain bound. Numerically evaluate the fraction of the mass which escapes.
e. An escaping fragment has speed vf when it is far from the black hole. Find the maximum possible value of vf, among all fragments, and evaluate it numerically.
f. A bound fragment orbits with period Tf. Find the minimum possible value of Tf, among all fragments, and evaluate it numerically.