Dependence of acoustic surface gravity on disc thickness for accreting astrophysical black holes

For axially symmetric accretion maintained in hydrostatic equilibrium along the vertical direction, we investigate how the characteristic features of the embedded acoustic geometry depends on the background Kerr metric, and how such dependence is governed by three different expressions of the thickness of the matter flow. We first obtain the location of the sonic points and stationary shock between the sonic points. We then linearly perturb the flow to obtain the corresponding metric elements of the acoustic space-time. We thus construct the causal structure to establish that the sonic points and the shocks are actually the analogue black hole type and white hole type horizons, respectively. We finally compute the value of the acoustic surface gravity as a function of the spin angular momentum of the rotating black hole for three different flow thicknesses considered in the present work. We find that for some flow models, the intrinsic acoustic geometry, although in principle may be extended up to the outer gravitational horizon of the astrophysical black hole, cannot be constructed beyond a certain truncation radius as imposed by the expressions of the thickness function of the corresponding flow.