Transonicity in black hole accretion -- A mathematical study using the generalized Sturm chains

TL;DR

This paper introduces a purely analytical method using generalized Sturm chains to determine the number of sonic points in black hole accretion flows, advancing understanding of transonic properties in astrophysics.

Contribution

It presents the first analytical approach applying algebraic polynomial theory to assess the number of critical points in black hole accretion flows.

Findings

Demonstrates how many critical points an accretion flow can have.

Provides a method to check for multiple sonic transitions.

Generalizes to analyze equilibrium points in dynamical systems.

Abstract

By applying the theory of algebraic polynomials and the theory of dynamical systems, we construct the generalized Sturm sequences/chains to investigate the transonic properties of hydrodynamic accretion onto non-rotating astrophysical black holes, to demonstrate, completely analytically, how many critical point such an accretion flow can have. Our work is significantly important, because for the first time in the literature, we provide a purely analytical method, by applying certain powerful theorem of algebraic polynomials in pure mathematics, to check whether certain astrophysical hydrodynamic accretion may undergo more than one sonic transitions. Our work can be generalized to analytically calculate the maximal number of equilibrium points certain autonomous dynamical systems can have in general (Abridged).