Parameter Space of Shock Formation in Adiabatic Flows

Extract

We study shock formation in a stationary, axisymmetric, adiabatic flow of a perfect fluid in the equatorial plane of a Kerr geometry. For such a flow, there exist two intrinsic constants of motion along a fluid world line, namely the specific total energy, E = −hu_t, and the specific angular momentum, l = −u_φ/u_t, where the u_μ’s are the four velocity components, h is the specific enthalpy, i.e., h = (P + ε)/ρ, with P, ε, and ρ being the pressure, the mass-energy density, and the rest-mass density, respectively.

As shown in Fig. 1 (Fig. la is for a Schwarzschild black hole, i.e. the hole’s specific angular momentum a = 0; Fig. lb is for a rapid Kerr hole, i.e. a = 0.99M, where M is the black-hole mass, and prograde flows: and Fig. 1c is for a = 0.99M and retrograde flows), in the parameter space spanned by E and l there is a strictly defined region bounded by four lines: three characteristic functional curves l_k(E), l_max(E), and l_min(E), and the vertical line E = 1. Only such a flow with parameters located within this region can have two physically realizable sonic points, the inner one r_in, and the outer one r_out. In between there is still one more, but unrealizable, sonic point, r_mid. The region is divided by another characteristic functional curve l_c(E) into two parts: in region I (= Ia + Ib) only τ_out is realized in a shock-free global solution (i.e., that joining the black-hole horizon to large distances), while in region II (= IIa + IIb) only r_in is realized.