Fourier analysis of advection-dominated accretion flows

TL;DR

This paper introduces a semi-analytical Fourier-based method to solve the complex differential equations of advection-dominated accretion flows, enabling analysis over a broad parameter space without typical numerical restrictions.

Contribution

The authors develop a novel Fourier expansion approach to solve ADAF equations, including a new varying alpha viscosity model and outflow/inflow solutions, improving computational robustness.

Findings

Method converges reliably for various parameters.

New outflow and inflow solutions are identified.

The approach overcomes numerical issues near the polar axis.

Abstract

We implement a new semi-analytical approach to investigate radially self-similar solutions for the steady-state advection-dominated accretion flows (ADAFs). We employ the usual $\alpha$-prescription for the viscosity and all the components of the energy-momentum tensor are considered. In this case, in the spherical coordinate, the problem reduces to a set of eighth-order, nonlinear differential equations with respect to the latitudinal angle $\theta$. Using the Fourier expansions for all the flow quantities, we convert the governing differential equations to a large set of nonlinear algebraic equations for the Fourier coefficients. Using the Newton-Raphson method we solve the algebraic equations and ADAF properties are investigated over a wide range of the model parameters. We also show that the implemented series are truly convergent. The main advantage of our numerical method is that it does not suffer from the usual technical restrictions which may arise for solving ADAF differential equations near to the polar axis. In order to check the reliability of the approach, we recover some widely studied solutions. We also introduce a new varying $\alpha$ viscosity model and new outflow and inflow solutions for ADAFs are presented using Fourier expansion series.