Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2015 / Paper 1 / 4

For a uniform sphere, $M=4\pi \rho R^3/3$ and its gravitational potential energy is $\Omega =-3G{M}^2/(5R)$. The ideal-gas kinetic energy is $E_{\mathrm{kin}}=3M{k}_{B}T/(2\bar{m})$. Equating these magnitudes using the virial theorem gives $GM/R=5k_{B}T/\bar{m}$. Eliminating $R$ yields the uniform-sphere pressure-work collapse criterionIn particular, for fixed composition,The same Jeans mass scaling follows from $c_s^2{k}^2\sim 4\pi G\rho$ in the Jeans instability dispersion relation and a characteristic size $k^{-1}$. Pressure communicates support in a sound-crossing time; sufficiently strong self-gravity wins when the gravitational time is shorter.

The numerical coefficient depends on the equilibrium, boundary and wavelength-to-cloud conventions. A sphere with both exactly uniform density and uniform temperature is not an exact hydrostatic self-gravitating ideal-gas equilibrium without additional stresses. The uniform-sphere argument establishes the characteristic mass scaling; it does not prove an exact stability boundary for that nonexistent equilibrium.

Let $\mathcal{L}$ be net cooling per unit volume, positive when energy is lost. At thermal equilibrium $\mathcal{L}=0$. For a slow perturbation that maintains pressure balance, the linearized thermal equation isThe heat capacity factor is positive. A positive derivative restores equilibrium: hotter gas cools more strongly, while cooler gas is heated. This is isobaric thermal stability of a heated ideal gas.

For fully ionized hydrogen, both protons and electrons contribute to the ideal gas law, so $P=2n{k}_{B}T$. Treat the stated constant heat input $\Gamma$ as a fixed heating rate per unit volume. Substituting $n=P/(2k_{B}T)$ into the cooling givesAt equilibrium $T_0$, its derivative isWith a temperature-independent recombination coefficient, the derivative is $-\Gamma /T_0$: the equilibrium is thermally unstable to isobaric perturbations. More generally, any nonincreasing ${\alpha }_B(T)$ also gives instability; stability requires its logarithmic slope to exceed one. Heating at fixed volume and pressure-balanced perturbations are distinct assumptions: the density falls as a parcel is heated, reducing the quadratic-density cooling.

No temperature law for ${\alpha }_B$ is supplied, so the displayed slope criterion states the general result without inventing one. A constant incident ultraviolet flux need not imply constant local volumetric heating in a self-consistent ionization-balance calculation. If the heating instead varies with density, temperature or neutral fraction, its derivative must be included; the conclusion here uses the explicit constant-$\Gamma$ model.

In the stated gravity-dominated approximation, neglect the pressure force in the radial momentum equation. A steady outward stellar wind then obeysAt $R_e=R_{\mathrm{esc}}$, the escape velocity is $v_e=\sqrt{2G{M}_{\ast }/R_e}$, so $C=0$. The ballistic marginal-escape stellar wind hasThe outward flow decelerates and approaches zero speed at infinity; it is exactly at the escape threshold in this pressure-negligible model. An initial speed above that threshold gives a nonzero asymptotic speed.

Adiabaticity alone would instead conserve the full Bernoulli equation $v^2/2+h-G{M}_{\ast }/r=B$. If finite pressure is retained and $v_e$ equals the mechanical escape speed, then $B=h_e$, not zero. With $p=K{\rho }^{\gamma }$ and conservation of mass, the more general implicit solution isIt needs $h_e$ and $\gamma$ in addition to the escape-speed datum. The boxed explicit power law is the leading gravity-dominated result, not an exact solution for an arbitrary warm adiabatic wind.

For a steady spherical wind, conservation of mass givesThusHere $n_e$ denotes the given number density at the escape radius, rather than electron density; the composition is atomic hydrogen. The reference speed is $v_e\simeq 4.22\times 10^4\mathrm{m}{\mathrm{s}}^{-1}$. This boundary flux remains fixed even if pressure changes the subsequent radial velocity profile.

Insert the pressure-negligible velocity law into the steady spherical mass flux. With $x=r/R_e$,An adiabatic equation of state $p{\rho }^{-\gamma }=\text{constant}$ and the ideal gas law giveFor a monatomic gas with $\gamma =5/3$,The normalization $T_e$, and hence $p_e=n_e{k}_B{T}_e$, was not specified and remains a boundary parameter. If it is large enough for the pressure gradient to affect acceleration, replace the ballistic velocity by the full adiabatic Bernoulli equation solution. The general mass-flux and thermodynamic relations still hold, but these particular radius powers then need not hold.