Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2022 / Section A / 2

Interpret “reach” first as remaining an energetic, supersonic blast wave, rather than as the arrival of any acoustic wave. An atmospheric blast-wave range estimate must account for the thin atmosphere: the distance $D=8.5\times 10^6\mathrm{m}$ is much greater than its height $H=10^4\mathrm{m}$.

For an optimistic estimate, suppose the deposited energy $E$ remains in a horizontally expanding atmospheric layer. If its representative mass density is ${\rho }_a$, the swept mass is of order $\pi D^{2}H{\rho }_a$. Conservation of energy gives the self-similar blast wave scalingUsing ordinary atmospheric estimates ${\rho }_a\sim 1\mathrm{kg}{\mathrm{m}}^{-3}$ and speed of sound $c_a\sim 300\mathrm{m}{\mathrm{s}}^{-1}$, this givesThese are scaling estimates, not an exact shock wave solution. The strong shock wave approximation would already have failed atEquivalently, keeping the disturbance strongly supersonic out to $D$ would require substantially more than the sonic-transition energy scalewhich exceeds $10^{22}\mathrm{J}$ by about a factor of twenty. Reasonable atmospheric averages and order-unity geometrical coefficients do not reverse the comparison. Escape of energy through the top of the atmosphere makes the fully confined estimate optimistic; Earth's curvature changes its area estimate by an order-unity factor.

The strong atmospheric blast will not survive to Lansing in this model. A weakened acoustic wave can nevertheless travel there. The supplied energy and atmospheric height do not determine its detectable amplitude, detailed attenuation or damage: those require an atmospheric propagation model. Thus “no disturbance of any kind reaches Lansing” would be an unjustified stronger conclusion.