Consider a hypothetical elliptical galaxy that is spherical in shape and consists of $N$ identical stars of mass $m$. The galaxy is in virial equilibrium, has a radius of $R_0$, a mass of $M_0$, and total energy of $E_0 = T_0 + U_0$, where $T_0$ is the kinetic energy and $U_0$ is the potential energy. The kinetic energy is $T_0 = \frac{1}{2}M_0(v_0)^2$ where $(v_0)^2$ is the mean square velocity of the stars. The potential energy is related to the mass and radius by $U_0 \propto \frac{GM_0^2}{R_0}$.
Suppose that suddenly a fraction $f = \frac{1}{4}$ of the stars is removed randomly throughout the galaxy, with the result that the number density of stars at any given radius instantly drops by a factor of 4. This throws the galaxy out of virial equilibrium and it will need to reorganize itself.
At the instant the stars are removed, the mass changes from $M_0$ to $M' = (1 - f)M_0$. The kinetic energy becomes $T'$, the potential energy becomes $U'$ and the total energy becomes $E' = T' + U'$. At the instant the stars are removed, the galaxy has not yet had time to react, so its radius is still $R_0$ and the mean square velocity of the stars is still $(v_0)^2$.
a) At the instant the stars are removed (i.e., before the galaxy has time to react and start reorganizing itself), what will be values of $T'/T_0$ and $U'/U_0$?
b) Use your results from part a) to compute $E'/E_0$ where $E'$ is the total energy after the stars have been removed.
c) After a sufficient amount of time has passed, the galaxy will return to virial equilibrium. Once the galaxy has returned to equilibrium, the kinetic, potential and total energies will be $T_f$, $U_f$, and $E_f = T_f + U_f$. Assuming no loss of stars or energy in the process of returning to equilibrium (i.e., $M_f = M'$ is the mass of the galaxy and $E_f = E'$) what will be the ratio of the mean square velocity of the stars in the new equilibrium configuration, $(v_f)^2$, to the mean square velocity of the stars in the original configuration, $(v_0)^2$? Note: recall that the galaxy will be centrally-concentrated when virialized; it will not have uniform density and you should not assume that it does.
d) For $f = \frac{1}{4}$ it is certainly possible for the galaxy to reorganize itself and once again be in equilibrium. How large would $f$ need to be in order for it to be impossible for the galaxy to ever again achieve equilibrium after the stars have been removed from the original galaxy? Be sure to justify your answer both mathematically and in a few words. Hint: think about what the value of $E_f$ must be such that the system is bound together by gravity.