Limits on thickness and efficiency of Polish doughnuts in application to the ULX sources

¹ N. Copernicus Astronomical Center, ul. Bartycka 18, 00-716 Warszawa, Poland
e-mail: maciek.wielgus@gmail.com
² Institute of Micromechanics and Photonics, Warsaw University of Technology, ul. Św. A. Boboli 8, 02-525 Warszawa, Poland
³ Institut d’Astrophysique de Paris, CNRS et Sorbonne Universités, UPMC Univ. Paris 06, UMR 7095, 98bis Bd Arago, 75014 Paris, France
⁴ Physics Department, Gothenburg University, 412-96 Göteborg, Sweden
⁵ Institute of Physics, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic

Received: 2 December 2015
Accepted: 10 January 2016

Abstract

Polish doughnuts (PDs) are geometrically thick disks that rotate with super-Keplerian velocities in their innermost parts, and whose long and narrow funnels along rotation axes collimate the emerging radiation into beams. In this paper we construct an extremal family of PDs that maximize both geometrical thickness and radiative efficiency. We then derive upper limits for these quantities and subsequently for the related ability to collimate radiation. PDs with such extreme properties may explain the observed properties of ultraluminous X-ray sources without the need for the black hole masses to exceed ~10 M_⊙. However, we show that strong advective cooling, which is expected to be one of the dominant cooling mechanisms in accretion flows with super-Eddington accretion rates, tends to reduce the geometrical thickness and luminosity of PDs substantially. We also show that the beamed radiation emerging from the PD funnels corresponds to isotropic luminosities that obey L_col ≈ 0.1Ṁc² for Ṁ ≫ Ṁ_Edd, and not the familiar and well-known logarithmic relation, L ~ lnṀ.