Past exam of the radiative processes and high energy astrophysics course of the University of Oxford / 2024 / 1 / d

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- Solution d

Solution

Work in the locally planar shock frame, taking

v_{u}, v_{d} > 0

towards the surface. For a stationary normal shock wave with negligible radiative loss inside the discontinuity, the Rankine-Hugoniot conditions for a perfect gas express conservation of mass, conservation of momentum, and conservation of energy per unit area as

ρ_{u} v_{u} = ρ_{d} v_{d} \equiv J,

P_{u} + ρ_{u} v_{u}^{2} = P_{d} + ρ_{d} v_{d}^{2},

J (\frac{v _{u}^{2}}{2} + h_{u}) = J (\frac{v _{d}^{2}}{2} + h_{d}), h = \frac{γ}{γ - 1} \frac{P}{ρ} .

Here

h

is specific enthalpy, including the pressure work; using only internal energy would omit that work. The gravitational potential is continuous across an infinitesimally thin front. For cold upstream gas,

P_{u}

and

h_{u}

are negligible, while a monatomic gas has

γ = 5/3

and

h = 5 P / (2 ρ)

.

The hydrodynamic equations apply to flow parallel to the guiding magnetic field when its stress contributions cancel across the front. An oblique magnetohydrodynamic shock requires magnetic stresses and Poynting flux; a radiation-mediated front or appreciable escaping radiation requires the corresponding radiative momentum and energy flux terms. These are additional physics, not terms hidden in

h

here.

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