Problem Set 5

1 The Shock Jump Conditions

Some models for high-energy radiation from compact objects involve gas free-falling onto the surface of the object at a velocity near the object’s escape speed. This speed is very supersonic relative to the star, so the gas must pass through a shock (called either an “accretion” or “standoff” shock) before it settles on the surface. Assuming that this is a strong shock, estimate the post-shock temperatures for a white dwarf (with M? = 1 M and R? = 109 cm) and a neutron star (with M? = 1 M and R? = 106 cm). You can assume γ = 5/3, μ = mp, and that the standoff shock is just above the surface of the star. (For the pedantic among you: you can use the expressions in this problem set, even though this shock is actually spherical rather than planar.)

using DrWatson
using Unitful: Mass, Length
include(srcdir("astr145/main.jl"))

function postshock_temperature(u1; μ=mp, γ=5 / 3)
    2 * (γ - 1) / (γ + 1)^2 * μ * u1^2 |> upreferred
end

function postshock_temperature(M::Mass, R::Length)
    v_esc = escape_velocity(M, R)
    kT = postshock_temperature(v_esc)
    T = uconvert(u"K", kT / Unitful.k)
    @show v_esc kT T
end

postshock_temperature(1u"Msun", 1e9u"cm")
postshock_temperature(1u"Msun", 1e6u"cm")