In astrophysics (e.g., in supernova explosions and in jets emerging from the vicinities of black holes), one

Question:

In astrophysics (e.g., in supernova explosions and in jets emerging from the vicinities of black holes), one sometimes encounters shock fronts for which the flow speeds relative to the shock approach the speed of light, and the internal energy density is comparable to the fluid’s rest-mass density.

(a) Show that the relativistic Rankine-Hugoniot equations for such a shock take the following form:

Here,

(i) We use units in which the speed of light is 1.

(ii) The volume per unit rest mass is V ≡ 1/ρ_o, and ρ_o is the rest-mass density (equal to some standard rest mass per baryon times the number density of baryons; cf. Sec. 2.12.3).

(iii)We denote the total density of mass-energy including rest mass by ρ_R and the internal energy per unit rest mass by u, so ρ_R = ρ_o(1 + u). In terms of these the quantity η ≡ (ρ_R + P)/ρ_o = 1+ u + P/ρ_o = 1 + h is the relativistic enthalpy per unit rest mass (i.e., the enthalpy per unit rest mass, including the rest-mass contribution to the energy) as measured in the fluid rest frame.

(iv) The pressure as measured in the fluid rest frame is P.

(v) The flow velocity in the shock’s rest frame is v, and

(not the adiabatic index!), so vγ is the spatial part of the flow 4-velocity.

(vi) The rest-mass flux is j (rest mass per unit area per unit time) entering and leaving the shock.

(b) Use a pressure-volume diagram to discuss these relativistic Rankine-Hugoniot equations in a manner analogous to Fig. 17.11.

(c) Show that in the nonrelativistic limit, the relativistic Rankine-Hugoniot equations (17.34) reduce to the nonrelativistic ones (17.31).

(d) It can be shown (Thorne, 1973) that relativistically, just as for nonrelativistic shocks, in general P₂ > P₁, V₂ 1, and v₂ 1. Consider, as an example, a relativistic shock propagating through a fluid in which the mass density due to radiation greatly exceeds that due to matter (a radiation-dominated fluid), so P = _ρR/3 (Sec. 3.5.5). Show that v₁v₂ = 1/3, which implies v₁ > 1/√3 and v₂ 2/P₁ = (9v₁² − 1)/[3(1− v₁²)].