Past exam of the astrophysics course of the University of Oxford / 2015 / 4

Write $R=C{E}^a{\rho }_0^b{t}^d$ for a uniform ambient mass density. Equating mass, length and time dimensions gives $a+b=0$, $2a-3b=1$ and $-2a+d=0$. Thus the Sedov-Taylor blast wave hasThe dimensionless coefficient depends on the gas's adiabatic index and on whether $R$ denotes the shock radius. Solving the similarity profile for a monatomic ideal gas gives $\xi \simeq 1.15$, so taking it as unity is an order-of-magnitude approximation rather than an extra dimensional result.

Timed photographs of the 1945 atomic explosion provide a direct scale test: the measured fireball radius follows $R\propto t^{2/5}$ over the relevant blast-wave interval, and inserting the independently established explosion energy and atmospheric density gives a coefficient of order unity. The published photographic analysis tested the radius-time law and used its normalization to infer an explosion energy of the correct nuclear-test scale. This illustrates both the success and the limits of dimensional reasoning: the power law is fixed dimensionally, whereas the numerical coefficient requires a profile calculation or measured normalization.^[ref]

Neglecting the exterior temperature requires a strong shock, ${\mathcal{M}}_s=V_s/c_{s,u}\gg 1$, or ${\rho }_0{V}_s^2\gg p_u$. The swept-up mass must dominate the initial ejecta for this energy-conserving similarity stage, while radiative losses, gravity, ambient gradients and continuing energy input must be unimportant. A very hot exterior or a late weak disturbance introduces a sound-speed scale and invalidates the cold-upstream similarity assumptions.

Use the local shock rest frame. Let $u$ and $d$ label upstream and downstream gas, with positive normal speeds $v_u,v_d$ through the shock. The unshocked ISM is almost stationary in the stellar frame, so $v_u\simeq V_s$, whereas the shocked gas moves outward at $V_s-v_d$ in that frame. For an ideal monatomic gas, $\gamma =5/3$; neglect upstream pressure and enthalpy in the strong-shock limit. Conservation of mass, momentum and energy per unit mass givesPut $q={\rho }_d/{\rho }_u=v_u/v_d$. The momentum equation gives $p_d={\rho }_u{v}_u^2(1-1/q)$. Substitution into the energy equation givesDiscarding the unshocked root $q=1$ leavesThe downstream speed in the stellar frame is therefore $3V_s/4$, not $V_s/4$. These conditions describe immediate shock heating; subsequent cooling and evolution behind the front require the downstream flow profile. Cosmic-ray pressure, radiative energy loss inside the shock layer or a different equation of state would alter the compression ratio.

Let the mean particle mass be $\bar{m}=\mu m_H$, with $n_u$ counting total particles so that ${\rho }_u=n_u\bar{m}$. The ideal-gas law and the strong-shock pressure give the immediate post-shock temperatureThe robust prediction is $T_d\propto t^{-6/5}$. For a transparent numerical convention take $\mu =1$, $n_u=10^6{\mathrm{m}}^{-3}$ and $\xi =1$. ThenFor an ionized mixture with $\mu =0.6$ under the same total-particle-density convention, these normalizations become $5.13$ pc and $5.49\times 10^7$ K. The paper does not specify composition or whether its quoted density instead counts hydrogen nuclei; that convention must be stated, not hidden in a precise age.

The ambient-temperature limit of an adiabatic blast wave gives a formal matching estimate, $T_d\sim 10^4$ K, givesWith $\mu =0.6$ the age is about $1.3\times 10^6$ yr and the radius stays about 90 pc. This matching criterion already lies outside a very strong shock: $T_d=T_u$ in the strong-shock formula corresponds to ${\mathcal{M}}_s=\sqrt{16/(3\gamma )}\simeq 1.79$. A separate sonic estimate $V_s\sim c_{s,u}$ gives about $4.4\times 10^6$ yr and 133 pc for $\mu =1$; neither extrapolation is a precise transition solution. Using $\xi =1.15$ increases these inferred matching ages and radii by factors ${\xi }^{5/3}$.

In a real supernova remnant, substantial radiation losses commonly end the energy-conserving phase before these formal pressure-merging limits. The relevant comparison is $t_{\mathrm{cool}}\sim (3/2)n{k}_{B}T/(n_e{n}_i\Lambda (T))$ against the expansion time. A cooling function, composition and ionization state are needed to calculate that earlier age and radius; they are not supplied. The numerical values above answer the ambient-temperature breakdown estimate under explicit assumptions and must not be presented as a uniquely determined radiative shell-formation age.

After cooling, an interior-pressure-driven shell can approximately follow $R\propto t^{2/7}$ while pressure remains important. In the limiting momentum-conserving stage, $M_s\dot{R}\simeq \mathrm{const}$ and $M_s\propto {\rho }_u{R}^3$ give $\boxed{R\propto t^{1/4}}$. Ultimately ambient pressure and turbulent motions make the disturbance merge with the ISM. These slower stages respect the reduced energy budget rather than continuing an adiabatic strong-shock law indefinitely.