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

Let the mean particle mass be $\mu m_p$. For a monatomic ideal gas with $\gamma =5/3$,A uniform sphere of mass $M=(4\pi /3){\rho }_0{r}^3$ has Newtonian gravitational energyAt the virial threshold, the pressure term $3\int PdV=3M{k}_{B}T/(\mu m_p)$ balances $|W|$. ThereforeEliminating $r$ gives the Jeans massThe 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 givesBoth conventions have the physically invariant scaling