Analytical model for the local vertical structure of thin accretion discs

Abstract

ABSTRACT We derive several analytical relations and approximate solutions for the local vertical structure of viscous thin accretion discs. Under the alpha prescription, when the viscous heating is proportional to the gas pressure p, we derive the analytical relation between the radiative flux F and the radiation pressure P: (F/F0)2 = [1 − (P/Pc)5/4]/[1 − (P0/Pc)5/4], where the subscript 0 means the value at the surface and the subscript c is the value at the disc centre. Both F and P are approximately integrated to yield the well-known uniform heating model. In this case, furthermore, the height z and density ρ are approximately fitted as a function of the optical depth τ. When the viscous heating is proportional to the radiation pressure P and the disc is almost isothermal, the flux F is proportional to z as F = (3/2)αΩPz, where α and Ω are the alpha parameter and angular speed, respectively. In this case, moreover, the height and density are analytically solved and expressed as $z=(\sqrt{2}c_{\rm T}/\Omega) {\rm erf}^{-1}(1-\tau /\tau _{\rm c})$ and ρ = ρcexp { − [erf−1(1 − τ/τc)]2}, cT being the isothermal sound speed, and erf−1 the inverse of the error function.