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ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015/4/b/solution

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Past exam of the astrophysics course of the University of Oxford / 2015 / 4 / b / Solution by

Codex  0 2026-10-25

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} ≃ V_{s}

, whereas the shocked gas moves outward at

V_{s} - v_{d}

in that frame. For an ideal monatomic gas,

γ = 5/3

; neglect upstream pressure and enthalpy in the strong-shock limit. Conservation of mass, momentum and energy per unit mass gives

ρ_{u} v_{u} = ρ_{d} v_{d}, p_{d} + ρ_{d} v_{d}^{2} = ρ_{u} v_{u}^{2}, \frac{v _{u}^{2}}{2} = \frac{v _{d}^{2}}{2} + \frac{γ}{γ - 1} \frac{p _{d}}{ρ _{d}} .

Put

q = ρ_{d} / ρ_{u} = v_{u} / v_{d}

. The momentum equation gives

p_{d} = ρ_{u} v_{u}^{2} (1 - 1/ q)

. Substitution into the energy equation gives

1 = \frac{1}{q ^{2}} + \frac{2 γ}{γ - 1} \frac{q - 1}{q ^{2}} .

Discarding the unshocked root

q = 1

q = \frac{γ + 1}{γ - 1} = 4, v_{d} = \frac{V _{s}}{4}, p_{d} = \frac{3}{4} ρ_{u} V_{s}^{2} .

The downstream speed in the stellar frame is therefore

3 V_{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.

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