Dynamics of Hot Accretion Flow with Thermal Conduction

The purpose of this paper is to explore the dynamical behaviour of hot accretion flow with thermal conduction. The importance of thermal conduction on hot accretion flow is confirmed by observations of the hot gas that surrounds Sgr A$^*$ and a few other nearby galactic nuclei. In this research, the effect of thermal conduction is studied by a saturated form of it, as is appropriate for weakly collisional systems. The angular momentum transport is assumed to be a result of viscous turbulence and the $\alpha$-prescription is used for the kinematic coefficient of viscosity. The equations of accretion flow are solved in a simplified one-dimensional model that neglects the latitudinal dependence of the flow. To solve the integrated equations that govern the dynamical behaviour of the accretion flow, we have used an unsteady self-similar solution. The solution provides some insights into the dynamics of quasi-spherical accretion flow and avoids from limits of the steady self-similar solution. In comparison to accretion flows without thermal conduction, the disc generally becomes cooler and denser. These properties are qualitatively consistent with performed simulations in hot accretion flows. Moreover, the angular velocity increases with the magnitude of conduction, while the radial infall velocity decreases. The mass accretion rate onto the central object is reduced in the presence of thermal conduction. We found that the viscosity and thermal conduction have the opposite effects on the physical variables. Furthermore, the flow represents a transonic point that moves inward with the magnitude of conduction or viscosity.

💡 Research Summary

The paper investigates how saturated thermal conduction influences the dynamics of hot, quasi‑spherical accretion flows that are also subject to viscous stresses described by the standard α‑prescription. Recognizing that the inner regions of low‑luminosity accretion flows (e.g., around Sgr A*) are weakly collisional, the authors adopt the Cowie & McKee (1977) saturated heat‑flux formulation, introducing a dimensionless saturation constant φ_s (0 < φ_s ≤ 1). The governing equations—mass continuity, radial momentum, azimuthal momentum, and energy—are written in spherical coordinates, neglecting latitudinal and azimuthal variations, and using a Newtonian gravitational potential. Viscous heating (Q_vis), radiative cooling (Q_rad), and conductive transport (Q_cond) appear explicitly in the energy equation; the latter is expressed as the divergence of the saturated flux.

To explore time‑dependent behavior, the authors employ the unsteady self‑similar method originally used by Ogilvie (1999). They define a similarity variable ξ = r (GM_* t²)^{‑1/3} and assume that density, pressure, radial velocity, angular velocity, and mass‑accretion rate separate into ξ‑dependent dimensionless functions multiplied by power‑law time factors (ρ ∝ t^{‑1}, p ∝ t^{‑5/3}, v_r ∝ t^{‑1/3}, Ω ∝ t^{‑1}, Ṁ = Ṁ₀ ṁ(ξ)). Substituting these forms reduces the original partial differential equations to a set of four coupled, nonlinear ordinary differential equations (ODEs) for the dimensionless variables R(ξ), Π(ξ), V(ξ), and ω(ξ).

An asymptotic analysis near the origin (ξ → 0) yields power‑law behaviors (ρ ∝ ξ^{‑3/2}, p ∝ ξ^{‑5/2}, v_r ∝ ξ^{‑1/2}, Ω ∝ ξ^{‑3/2}) with coefficients R₀, Π₀, V₀, ω₀ linked by algebraic relations derived from the ODEs. The mass‑accretion rate at the inner boundary is ṁ_in = ‑4π R₀ V₀. Crucially, R₀ satisfies a fifth‑order algebraic equation (Eq. 35) that contains the saturation constant φ_s, the viscosity parameter α, and the advection factor f. When φ_s = 0 (no conduction) the equation simplifies and can be solved analytically; for realistic φ_s ≠ 0 the authors solve it numerically.

The ODE system is integrated outward from a small ξ_in using a Runge‑Kutta‑Fehlberg 4‑5 scheme, with the inner asymptotic expansions providing the initial conditions. The numerical solutions reveal several systematic trends:

1. Temperature and Density: Increasing φ_s enhances conductive heat loss, lowering the dimensionless temperature Π/R and raising the density R. This reproduces the cooling and densification seen in 3‑D MHD simulations (e.g., Sharma et al. 2008; Wu et al. 2010). Conversely, larger α boosts viscous heating, raising temperature and reducing density.

2. Angular and Radial Velocities: Conductive cooling weakens the viscous torque, reducing angular‑momentum transport efficiency. Consequently, the flow spins faster (higher Ω) while the inward radial speed v_r declines. Larger α has the opposite effect: stronger torque slows rotation and accelerates infall.

3. Mass‑Accretion Rate: In the unsteady self‑similar framework the accretion rate varies with ξ. Higher φ_s lowers Ṁ(ξ) because conductive energy transport reduces the amount of energy retained to drive inflow. Higher α modestly raises Ṁ by increasing pressure support. This ξ‑dependence contrasts with steady‑state models where Ṁ is constant, aligning instead with recent studies that allow radially varying accretion rates.

4. Transonic Point: The flow exhibits a sonic transition (Mach = 1) whose radial location moves inward as either φ_s or α increase. Conductive cooling flattens the pressure gradient, moving the sonic point closer to the black hole; enhanced viscosity steepens the velocity gradient, producing a similar shift.

Overall, the paper demonstrates that saturated thermal conduction and viscous stresses exert opposite influences on the structure of hot accretion flows. Conductivity cools and densifies the plasma, slows radial inflow, and speeds up rotation, while viscosity heats, thins, and accelerates infall. Their combined effect determines the location of the transonic point and the radial profile of the mass‑accretion rate. By integrating a realistic saturated conduction model into an unsteady self‑similar solution, the authors bridge the gap between analytic theory, numerical simulations, and observations of low‑luminosity galactic nuclei such as Sgr A*. The work provides a useful analytic framework for interpreting future high‑resolution observations and for benchmarking more sophisticated multidimensional simulations.