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Iterative Relaxation Method to Obtain Global Transonic Flows around Compact Objects

Shilpa Sarkar, I. M. Kulikov

TL;DR

This work addresses the challenge of obtaining global transonic flows around compact objects, where gravity, rotation, heating, and cooling produce multiple sonic points and possible shocks. It introduces two complementary algorithms, IRM-SP and IRM-SHOCK, to locate sonic points and construct global accretion and wind solutions within a viscous, radiative framework using the PW potential and a variable-Γ EoS, with inner-boundary conditions driving the solutions. The approach unifies accretion and wind regimes, demonstrates MCPs and shock formation, and maps solution topologies in the $E$–$\lambda$ parameter space, highlighting how shocks arise via RH conditions and how global solutions connect to the central object or infinity. The results have implications for interpreting accretion physics in BH X-ray binaries and AGNs, and the methodology provides a rigorous computational foundation, with future work aimed at more realistic viscosity prescriptions and observational linkages.

Abstract

Flows around compact objects are necessarily transonic. Due to their dissipative nature, finding of sonic points is not trivial. Becker and Le in 2003 (BL03) proposed a novel methodology to obtain global transonic solutions, using iterative relaxation technique and exploiting the inner boundary conditions of the central object. In the current work, we propose a generic methodology -- IRM-SP and IRM-SHOCK to obtain any class of global accretion and wind solutions, given a set of constants of motion. We have considered viscosity in the system, which transports angular momentum outwards. In addition, it heats the system. Radiative processes like bremsstrahlung which cools the system is also incorporated. An interplay between heating and cooling process, along with gravity and centrifugal forces gives rise to multiple sonic points and hence shocks. The proposed methodology successfully generates any class of accretion as well as wind solutions, allowing us to unify them. Additionally, we report here rigorously the mathematical as well as the computational algorithm needed, to find sonic point(s) and thus obtain global transonic flows around compact objects.

Iterative Relaxation Method to Obtain Global Transonic Flows around Compact Objects

TL;DR

This work addresses the challenge of obtaining global transonic flows around compact objects, where gravity, rotation, heating, and cooling produce multiple sonic points and possible shocks. It introduces two complementary algorithms, IRM-SP and IRM-SHOCK, to locate sonic points and construct global accretion and wind solutions within a viscous, radiative framework using the PW potential and a variable-Γ EoS, with inner-boundary conditions driving the solutions. The approach unifies accretion and wind regimes, demonstrates MCPs and shock formation, and maps solution topologies in the – parameter space, highlighting how shocks arise via RH conditions and how global solutions connect to the central object or infinity. The results have implications for interpreting accretion physics in BH X-ray binaries and AGNs, and the methodology provides a rigorous computational foundation, with future work aimed at more realistic viscosity prescriptions and observational linkages.

Abstract

Flows around compact objects are necessarily transonic. Due to their dissipative nature, finding of sonic points is not trivial. Becker and Le in 2003 (BL03) proposed a novel methodology to obtain global transonic solutions, using iterative relaxation technique and exploiting the inner boundary conditions of the central object. In the current work, we propose a generic methodology -- IRM-SP and IRM-SHOCK to obtain any class of global accretion and wind solutions, given a set of constants of motion. We have considered viscosity in the system, which transports angular momentum outwards. In addition, it heats the system. Radiative processes like bremsstrahlung which cools the system is also incorporated. An interplay between heating and cooling process, along with gravity and centrifugal forces gives rise to multiple sonic points and hence shocks. The proposed methodology successfully generates any class of accretion as well as wind solutions, allowing us to unify them. Additionally, we report here rigorously the mathematical as well as the computational algorithm needed, to find sonic point(s) and thus obtain global transonic flows around compact objects.
Paper Structure (12 sections, 2 equations, 3 figures)

This paper contains 12 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Modelling and Equations used
  3. Sonic points and their properties
  4. Shocks
  5. Algorithm and Computational Scheme
  6. Transonic solutions connected with the central object (Method name: IRM-SP)
  7. Transonic solutions disconnected with central object (Method name: IRM-SHOCK)
  8. Analysis of the solutions obtained
  9. Energy-angular momentum ($E-\lambda$) parameter space
  10. Accretion and wind flow properties
  11. Conclusions & Future Work
  12. Appendix: Stability of Cash-Karp method - an explicit adaptive Runge-Kutta method

Figures (3)

  • Figure 3: (a) IRM-SP used to obtain TS passing through $r_{\rm ci}$, (b) IRM-SHOCK technique utilised to obtain $r_{\rm co}$ and therefore $r_{\rm sh}$, (c) global accretion (green curve) and wind solution (black curve)
  • Figure 4: Variation of solutions with change in $E$ and $\lambda$ (values written inset). $r_{\rm ci}$ and $r_{\rm co}$ are represented using grey star and grey circle respectively. Solid curves are global accretion solutions while dashed curves are global wind solutions. Grey curves represent non-global or truncated solutions because of sudden shock jump. Panels (a3) and (b2) harbour accretion shocks while (a4), (b3) and (b4) harbour wind shocks.
  • Figure 5: Flow variables for solutions represented in panel (b2) and (b4) of Fig. \ref{['fig:elam']}. Coloured curves are the global transonic solutions. Arrows represent the direction of the flow.