[Paper Review] 1/2, 1, and 3/2--Law Non-Radiative Accretion Flows

Andrei Gruzinov|arXiv (Cornell University)|Nov 22, 2013

Solar and Space Plasma Dynamics3 citations

TL;DR

This paper proposes that non-radiative accretion flows around black holes can exhibit three self-similar solutions with density profiles scaling as $\rho \propto r^{-p}$, where $p = 1/2$, $1$, or $3/2$. The $p=1$ solution, interpreted as a collection of turbulent Prandtl jets, may explain numerical simulations showing $p \approx 1$, suggesting such flows could persist even in collisionless plasma due to momentum flux conservation.

ABSTRACT

Assuming self-similarity of the first kind, we get three possible values p=1/2, 1, 3/2 for the exponent describing the density profile, rho r^{-p}, of a non-radiative (and hence quasi-spherical) accretion flow. The high and low p cases are known as Bondi and Convection-Dominated accretion flows. The 1-law flow we tentatively identify with the so-called Magnetically-Frustrated accretion flow. If our interpretation is correct, the accretion flow must be, roughly speaking, a collection of Prandtl's turbulent jets. The 1-law flow, being a first-kind self-similar solution, may actually occur in nature (in collisionless plasma).

Motivation & Objective

• To identify physically plausible self-similar solutions for non-radiative accretion flows around black holes.
• To resolve the discrepancy between theoretical Bondi $p=3/2$ predictions and numerical simulations showing $p \approx 1$.
• To explore whether the $p=1$ solution, interpreted as magnetically frustrated or turbulent jet flows, can remain valid in collisionless plasma.
• To assess the role of self-similarity of the first kind in constraining possible accretion flow profiles.
• To evaluate whether the $p=1$ solution can explain the observed $p \approx 1$ in simulations like Pang2010, even under non-ideal plasma conditions.

Proposed method

• Assumes self-similarity of the first kind, where all physical quantities scale with radius $r$ as $r^{-1/2}$, leading to power-law solutions for density, velocity, and fluxes.
• Derives scaling laws for mass flux $\Phi \propto r^{3/2-p}$, momentum flux $F \propto r^{1-p}$, and energy flux $L \propto r^{1/2-p}$, which constrain possible values of $p$.
• Identifies three possible $p$ values—$1/2$, $1$, and $3/2$—from the requirement that fluxes remain consistent under self-similar scaling.
• Interprets the $p=1$ solution as a system of turbulent jets that maintain constant momentum flux via entrainment and energy transfer to the ambient plasma.
• Uses flux conservation and entropy arguments to rule out the $p=3/2$ (Bondi) solution in non-radiative, collisionless plasmas due to magnetic energy buildup.
• Compares the $p=1$ solution to observed simulations (e.g., Pang2010) and suggests it may represent a physically viable, self-consistent accretion mode.

Experimental results

Research questions

• RQ1 What are the physically allowed values of the density profile exponent $p$ in self-similar, non-radiative accretion flows?
• RQ2 Why do numerical simulations (e.g., Pang2010) find $p \approx 1$ when standard theory predicts $p=3/2$?
• RQ3 Can the $p=1$ solution, interpreted as a turbulent jet system, remain stable and self-consistent in collisionless plasma?
• RQ4 How does magnetic field amplification via spaghettization rule out the standard Bondi $p=3/2$ solution in non-radiative flows?
• RQ5 Is the $p=1$ solution a first-kind self-similar solution or a second-kind one, and what does this imply for its robustness in real astrophysical systems?

Key findings

• The $p=1$ solution is a valid first-kind self-similar solution where momentum flux is constant ($F=\text{const}$), mass flux vanishes ($\Phi=0$), and energy flux vanishes ($L=0$).
• The $p=1$ flow is interpreted as a collection of Prandtl-type turbulent jets that entrain ambient plasma and transfer momentum, maintaining constant momentum flux.
• The $p=1$ solution may explain the $p \approx 1$ result observed in numerical simulations by Pang2010, suggesting it is physically plausible.
• The $p=3/2$ (Bondi) solution is ruled out in non-radiative flows due to magnetic energy buildup from flux amplification, which violates entropy and energy constraints.
• The $p=1/2$ solution corresponds to a convectively unstable, nearly non-accreting atmosphere, but is not supported by direct numerical simulations.
• If $p=1$ is confirmed as a first-kind solution, it may remain valid even in collisionless plasma, unlike second-kind solutions which may not survive such conditions.