Let the mean particle mass be $\mu m_p$. For a monatomic <ideal gas> with $\gamma=5/3$,
$$
P=\frac{\rho_0k_BT}{\mu m_p},
\qquad
c_s^2=\frac{\gamma k_BT}{\mu m_p}.
$$
A uniform sphere of mass $M=(4\pi/3)\rho_0r^3$ has Newtonian gravitational energy
$$
W=-\frac{3GM^2}{5r}.
$$
At the virial threshold, the pressure term $3\int P\,dV=3Mk_BT/(\mu m_p)$ balances $|W|$. Therefore
$$
M=\frac{5k_BT}{G\mu m_p}r.
$$
Eliminating $r$ gives the <Jeans mass>
$$
\boxed{
M_J=\left(\frac{5k_BT}{G\mu m_p}\right)^{3/2}
\left(\frac{3}{4\pi\rho_0}\right)^{1/2}
}.
$$
The numerical coefficient depends on the convention used to identify a finite cloud with a Jeans mode. For example, assigning the mass inside a sphere of radius half the standard <Jeans instability> wavelength gives
$$
M_J=\frac{\pi^{5/2}}6
\frac{c_s^3}{G^{3/2}\rho_0^{1/2}}.
$$
Both conventions have the physically invariant scaling
$$
M_J\propto T^{3/2}\rho_0^{-1/2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 347 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-347/1/a/solution