Subfield: Modern Physics

Problem T3. Protostar formation

Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\mu$ and its adiabatic index is $\gamma>\frac{4}{3}$. Assume that $G \frac{m \mu}{r_{0}} \gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.
Context question:
i. During much of the collapse, the gas is so transparent that any heat generated is immediately radiated away, i.e. the ball stays in thermodynamic equilibrium with its surroundings. What is the number of times, $n$, by which the pressure increases when the radius is halved to $r_{1}=0.5 r_{0}$ ? Assume that the gas density remains uniform.
Context answer:
\boxed{$n=8$}


Context question:
ii. Estimate the time $t_{2}$ needed for the radius to shrink from $r_{0}$ to $r_{2}=0.95 r_{0}$. Neglect the change of the gravity field at the position of a falling gas particle.
Context answer:
\boxed{$\sqrt{\frac{0.1 r_{0}^{3}}{G m}}$}


Question: iii. Assuming that the pressure remains negligible, find the time $t_{r \rightarrow 0}$ needed for the ball to collapse from $r_{0}$ down to a much smaller radius, using Kepler's Laws.

Solution:
Gravitationally the outer layer of the ball is influenced by the rest just as the rest were compressed into a point mass. Therefore we have Keplerian motion: the fall of any part of the outer layer consists in a halfperiod of an ultraelliptical orbit. The ellipse is degenerate into a line; its foci are at the ends of the line; one focus is at the center of the ball (by Kepler's $1^{\text {st }}$ law) and the other one is at $r_{0}$, see figure (instead of a degenerate ellipse, a strongly elliptical ellipse is depicted). The period of the orbit is determined by the longer semiaxis of the ellipse (by Kepler's $3^{\text {rd }}$ law). The longer semiaxis is $r_{0} / 2$ and we are interested in half a period. Thus, the answer is equal to the halfperiod of a circular orbit of radius $r_{0} / 2$ :

$$
\left(\frac{2 \pi}{2 t_{r \rightarrow 0}}\right)^{2} \frac{r_{0}}{2}=\frac{G m}{\left(r_{0} / 2\right)^{2}} \Longrightarrow t_{r \rightarrow 0}=\pi \sqrt{\frac{r_{0}^{3}}{8 G m}}
$$

centre of the cloud initial position of a certain parcel

strongly elliptical orbit of the gas parcel of gas area covered by the radius vector

Alternatively, one may write the energy conservation law $\frac{\dot{r}^{2}}{2}-\frac{G m}{r}=E$ (that in turn is obtainable from Newton's II law $\left.\ddot{r}=-\frac{G m}{r^{2}}\right)$ with $E=-\frac{G m}{r_{0}}$, separate the variables $\left(\frac{d r}{d t}=-\sqrt{2 E+\frac{2 G m}{r}}\right)$ and write the integral $t=-\int \frac{d r}{\sqrt{2 E+\frac{2 G m}{r}}}$ This integral is probably not calculable during the limitted time given during the Olympiad, but a possible approach can be sketched as follows. Substituting $\sqrt{2 E+\frac{2 G m}{r}}=\xi$ and $\sqrt{2 E}=v$, one gets

$$
\begin{aligned}
\frac{t_{\infty}}{4 G m} & =\int_{0}^{\infty} \frac{d \xi}{\left(v^{2}-\xi^{2}\right)^{2}} \\
& =\frac{1}{4 v^{3}} \int_{0}^{\infty}\left[\frac{v}{(v-\xi)^{2}}+\frac{v}{(v+\xi)^{2}}+\frac{1}{v-\xi}+\frac{1}{v+\xi}\right] d \xi
\end{aligned}
$$

Here (after shifting the variable) one can use $\int \frac{d \xi}{\xi}=\ln \xi$ and $\int \frac{d \xi}{\xi^{2}}=-\frac{1}{\xi}$, finally getting the same answer as by Kepler's laws.

Final answer: $t_{r \rightarrow 0}=\pi \sqrt{\frac{r_{0}^{3}}{8 G m}}$