Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2014 / Paper 1 / 1

For an ideal gas of fixed composition, the central temperature satisfies $T_c\propto p_c/{\rho }_c=K{\rho }_c^{1/n}$. Keeping $T_c$ fixed therefore requires $K\propto {\rho }_c^{-1/n}$, rather than keeping the polytropic constant fixed. Substitution in the Lane-Emden equation scalings givesConsequently the homologous polytropes at fixed central temperature obeyThe polytropic index and mean particle mass must be common to this homologous sequence: varying either changes the dimensionless structural coefficient. The fixed-central-temperature assumption is the idealization being used here, rather than a universal property of real stars.

In an adiabatic outflow without additional heating, the reservoir's specific enthalpy supplies both escaping kinetic energy and the work against gravity. For a monatomic ideal gas, $\gamma =5/3$ and $h=c_s^2/(\gamma -1)=3c_s^2/2$, where $c_s$ is the reservoir's adiabatic sound speed. The Bernoulli equation, measured from a reservoir initially at rest at radius $r$, givesThe maximum terminal speed uses all available specific enthalpy, so $h_{\infty }\to 0$ andA freely escaping adiabatic flow needs a nonnegative radicand; equality is marginal escape. If the radicand is negative, this reservoir cannot supply an unheated escaping outflow. The initial rest condition is a stagnation-reservoir approximation: an exactly stationary finite-density surface cannot simultaneously carry a nonzero steady mass flow rate. Further heating, radiation forces or an imposed confining pressure would change the terminal-energy budget.

Once swept-up matter dominates the ejecta and radiative cooling is negligible, the Sedov-Taylor blast wave has its shock radius fixed by dimensional analysis:Here $\xi$ is a dimensionless constant of order unity fixed by the adiabatic index. Treating the ambient matter as hydrogen gives ${\rho }_0\simeq 10^6{m}_H\simeq 1.67\times 10^{-21}\mathrm{kg}{\mathrm{m}}^{-3}$, while $t\simeq 3.16\times 10^{10}\mathrm{s}$. ThusFor a monatomic gas, the usual similarity normalization $\xi \simeq 1.15$ instead gives about $5.3\mathrm{pc}$. An estimate of a few parsecs is therefore appropriate. Negligible ejecta mass does not by itself guarantee this energy-conserving stage: substantial radiation losses or a strongly nonuniform ambient mass density would invalidate the assumed similarity law.