An Analytical Study on the Multi-critical Behaviour and Related Bifurcation Phenomena for Relativistic Black Hole Accretion

TL;DR

This paper analytically investigates the multi-critical behavior and bifurcation phenomena in relativistic black hole accretion flows using algebraic polynomial theory and Sturm's chain algorithm.

Contribution

It introduces an analytical method to determine the maximum number of critical points in black hole accretion flows, advancing understanding of their multi-critical properties.

Findings

Analytically determines the maximum number of critical points in accretion flows.

Demonstrates the transition between mono-critical and multi-critical flows via bifurcation.

Provides a novel analytical approach to study accretion flow stability and bifurcations.

Abstract

We apply the theory of algebraic polynomials to analytically study the transonic properties of general relativistic hydrodynamic axisymmetric accretion onto non-rotating astrophysical black holes. For such accretion phenomena, the conserved specific energy of the flow, which turns out to be one of the two first integrals of motion in the system studied, can be expressed as a 8$^{th}$ degree polynomial of the critical point of the flow configuration. We then construct the corresponding Sturm's chain algorithm to calculate the number of real roots lying within the astrophysically relevant domain of $\mathbb{R}$. This allows, for the first time in literature, to {\it analytically} find out the maximum number of physically acceptable solution an accretion flow with certain geometric configuration, space-time metric, and equation of state can have, and thus to investigate its multi-critical properties {\it completely analytically}, for accretion flow in which the location of the critical points can not be computed without taking recourse to the numerical scheme. This work can further be generalized to analytically calculate the maximal number of equilibrium points certain autonomous dynamical system can have in general. We also demonstrate how the transition from a mono-critical to multi-critical (or vice versa) flow configuration can be realized through the saddle-centre bifurcation phenomena using certain techniques of the catastrophe theory.