An analytical study on the multi-critical behaviour and related bifurcation phenomena for relativistic black hole accretion

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 R. This allows, for the first time in literature, to 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 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 monocritical to multi-critical (or vice versa) flow configuration can be realized through the saddle-centre bifurcation phenomena using certain techniques of the catastrophe theory.

Multi-criticality and related bifurcation in accretion discs around non-rotating black holes -- an analytical study

“…Now, these polynomials as will be shown in the subsequent sections are usually (except in a few special cases) of order n > 4 and hence, not analytically solvable. So, in order to obtain the number of real roots of the polynomial equations for a specified range of parameters within relevant astrophysical domain 3 , we need to use the Sturm method that we discussed earlier ( [14]).…”

“…Aforementioned papers have already solved them numerically and described the whole process from a semi-analytical approach. Very recently, a technique is developed [14] using Sturm analysis which gives us exact number of physically acceptable solutions (for which the critical points form outside the horizon) of a n-th order algebraic polynomial(of critical points) for a set of initial conditions. So, this method will help us to investigate the multi-transonicity of a general relativistic flow onto a Schwarzschild black hole completely analytically, and to get a real flavour of multi-transonic 1 except when the accreting matter starts supersonically from a reasonably large distance.…”

Dependence of acoustic surface gravity on geometric configuration of matter for axially symmetric background flows in the Schwarzschild metric ~

“…However, the number of roots of such equations lying between infinity and the event horizons can be estimated analytically using the generalized Sturm sequence algorithm. 87 The expressions for the critical values of (du/dr) and (dc s /dr) have been provided in the appendix.…”

Effective sound speed in relativistic accretion discs around Schwarzschild black holes

“…The critical points are obtained using the critical point analysis method -a technique borrowed from the dynamical systems theory. For many of the accretion scenarios, it may be possible to locate the critical points analytically (see (Agarwal et al, 2012) and references therein). Using certain eigenvalue techniques, one becomes able to gain, completely analytically, qualitative ideas about the phase portrait of the transonic flow structure close to the critical point (Ray, 2003;Chaudhury et al, 2006;Goswami et al, 2007;Chaverra et al, 2016;Mandal et al, 2007).…”