[Paper Review] A simple accretion model of a rotating gas sphere onto a Schwarzschild black hole

E. A. Huerta, S. Mendoza|ArXiv.org|Mar 12, 2006

Astrophysical Phenomena and Observations14 references3 citations

TL;DR

This paper presents a fully general relativistic model of a rotating gas sphere accreting onto a Schwarzschild black hole, using exact analytic solutions in terms of Jacobi elliptic functions. It demonstrates that the equatorial accretion disc radius increases without bound as the specific angular momentum approaches the critical value $ h = 2r_g $, softening the particle density divergence seen in the Newtonian Ulrich model and revealing significant deviations from pseudo-Newtonian approximations near the horizon.

ABSTRACT

We construct a simple accretion model of a rotating pressureless gas sphere onto a Schwarzschild black hole. We show how to build analytic solutions in terms of Jacobi elliptic functions. This construction represents a general relativistic generalisation of the Newtonian accretion model first proposed by Ulrich (1976). In exactly the same form as it occurs for the Newtonian case, the flow naturally predicts the existence of an equatorial rotating accretion disc about the hole. However, the radius of the disc increases monotonically without limit as the flow reaches its maximum allowed angular momentum.

Motivation & Objective

• To develop a fully general relativistic extension of Ulrich's (1976) Newtonian accretion model for a rotating gas sphere.
• To resolve inconsistencies in prior pseudo-Newtonian approximations, particularly the Paczynski & Wiita (1980) model, which misrepresents particle trajectories near the event horizon.
• To analytically derive velocity, density, and streamline fields for ideal, pressureless, ballistic flow in Schwarzschild spacetime.
• To investigate the behavior of the accretion disc radius and particle density profile as the specific angular momentum approaches the minimum allowed value $ h = 2r_g $.
• To compare the exact relativistic solution with the widely used pseudo-Newtonian approximation, highlighting key physical discrepancies.

Proposed method

• The model assumes ideal, pressureless, ballistic flow with negligible viscosity and radiative heating, valid for supersonic, transonic flows.
• The equations of motion are derived from geodesic trajectories in Schwarzschild spacetime, using conserved energy and angular momentum per unit mass.
• Analytic solutions for particle trajectories and velocity fields are constructed using Jacobi elliptic functions, enabling exact parametric expressions for streamlines.
• The particle number density is computed from the continuity equation, assuming a polytropic equation of state for thermodynamic consistency.
• The solution is validated by showing convergence to Michel (1972) and Ulrich (1976) models in the appropriate limits: $ h \to 0 $ and $ \alpha \to 0 $.
• The extreme case $ \alpha = 1/8 $, corresponding to $ h = 2r_g $, is solved explicitly using hyperbolic functions, enabling direct comparison with pseudo-Newtonian models.

Experimental results

Research questions

• RQ1 How does the general relativistic treatment of a rotating gas sphere accreting onto a Schwarzschild black hole differ from the Newtonian model of Ulrich (1976)?
• RQ2 What is the behavior of the equatorial accretion disc radius as the specific angular momentum approaches the critical value $ h = 2r_g $?
• RQ3 How do the exact relativistic streamlines and density profiles compare with the pseudo-Newtonian Paczynski & Wiita (1980) approximation?
• RQ4 Why is the pseudo-Newtonian model inadequate for describing particle dynamics near the event horizon in this context?
• RQ5 What is the physical significance of the divergence in particle number density at the disc edge in the Newtonian limit, and how is it resolved in the relativistic case?

Key findings

• The equatorial accretion disc radius increases monotonically without bound as the specific angular momentum approaches $ h = 2r_g $, unlike in the Newtonian case.
• The particle number density diverges at the disc edge only in the Newtonian limit ($ \alpha \to 0 $); this singularity softens and disappears as $ \alpha \to 1/8 $, indicating a physically smoother profile in full relativity.
• The exact relativistic solution shows significant deviations from the pseudo-Newtonian Paczynski & Wiita approximation, especially near the Schwarzschild radius, where some particles are misclassified as swallowed in the pseudo-Newtonian model but are instead injected into the disc in the full solution.
• For the extreme case $ \alpha = 1/8 $, the solution simplifies to hyperbolic functions, enabling analytical verification and direct comparison with numerical approximations.
• The model converges to Michel’s (1972) solution in the limit of vanishing angular momentum ($ h \to 0 $), confirming consistency with known general relativistic results.
• Streamline and density contour plots show that as $ \alpha $ increases toward $ 1/8 $, streamlines spread out and density peaks at the origin, with no edge divergence, confirming the physical regularization of the disc structure.