Chapter 16, Problem 16P

Collapse of a dust ball under Newtonian gravity. Suppose that a spherical ball of dust with no internal pressure and with an initial radius R₀ and total mass M gravitationally collapses from rest.

1. (a) Show that the outer shell of the collapsing dust ball obeys the equation of motion

   $\ddot{R}=-\frac{GM}{R^2}$

2. (b) Solve this equation of motion and show that the time to collapse is given by

   $t_{\text{c}}=\frac{\pi }{2}\sqrt{\frac{R_0^3}{2GM}}$

   Hint: Write Newton’s law as $\ddot{R}=f(R)$. Let f(R) = dF(R)/dR and multiply both sides of by $\ddot{R}=dF/dR$ by $\stackrel{\dot{}}{R}$. The resulting equation may be immediately integrated with respect to time by recognizing that $\ddot{R}\stackrel{\dot{}}{R}=\frac{d}{dt}\left(\frac{{\stackrel{\dot{}}{R}}^2}{2}\right)$. Integrating again with respect to time, this general technique yields t as a function of R where $t=-{\displaystyle \int \frac{dR}{\sqrt{2(F(R)+C)}}+C_1}$ and C and C₁ are constants determined by the initial conditions.

3. (c) Find the time of collapse of a dust ball with a mass and radius equal to that of the Sun.

   (R_S = 6.96 × 10¹⁰ cm, M_S = 1.99 × 10³³ g, G = 6.67 × 10⁻⁸ cgs units)

4. (d) What prevents the Sun from collapsing in this way?