Stability of radiation-pressure dominated disks

I. The dispersion relation for a delayed heating α-viscosity prescription

¹ Astronomical Observatory of the Jagiellonian University, ul. Orla 171, 30-244 Kraków, Poland
e-mail: adam.ciesielski@uj.edu.pl
² M. Smoluchowski Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland
³ Institute of Micromechanics and Photonics, Warsaw University of Technology, ul. Św. Andrzeja Boboli 8, 02-525 Warszawa, Poland
e-mail: maciek.wielgus@gmail.com
⁴ Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warszawa, Poland
e-mail: wlodek@camk.edu.pl; asadowski@cfa.harvard.edu
⁵ Department of Physics, Göteborg University, 412-96 Göteborg, Sweden
e-mail: marek.abramowicz@physics.gu.se
⁶ Institut d’Astrophysique de Paris, UMR 7095 CNRS, UPMC Univ Paris 06, 98bis Boulevard Arago, 75014 Paris, France
e-mail: lasota@iap.fr
⁷ Kavli Institute for Astrophysics and Space Research, MIT, Cambridge, MA 02139, USA
e-mail: pao@space.mit.edu

Received: 15 June 2011
Accepted: 20 December 2011

Abstract

We derive and investigate the dispersion relation for accretion disks with retarded or advanced heating. We follow the α-prescription but allow for a time offset τ between heating and pressure perturbations, as well as for a diminished response of heating to pressure variations. We study in detail solutions of the dispersion relation for disks with radiation-pressure fraction, 1 − β, and ξ, the ratio of viscous stress response to pressure perturbations. For τ < 0 (advanced heating) the number and sign of real solutions for the growth rate depend on the values of τ, and ξ: if the magnitude of τ is larger than a critical value (e.g., more than twice the thermal time, −τ > 2   τ_th, for β = 0 and ξ = 1) two real solutions exist, which are both negative. These results imply that radiation-pressure dominated accretion disks may be stabilized when there is a time delay between stress fluctuations and fluctuations in heating.