Slim disks

Models of thin disks (Shakura & Sunyeav 1973, Novikov & Thorne 1973) assume that accretion is radiatively efficient. This assumption means that all the heat generated at a given radius by viscosity is immediately radiated away. In other terms, the viscous heating is balanced by the radiative cooling and no other cooling mechanism is allowed. All of the assumptions of the thin disk models are satisfied as long as the accretion rate is small. At some luminosity (L \approx 0.3 L_{Edd}), however, the radial velocity is large, and the disk is thick enough, to trigger another mechanism of cooling: advection. It results from the fact that the viscosity-generated heat has not enough time to transform into photons and to leave the disk before being pushed inwards by the gas inward motion. The higher the luminosity, the more significant advective cooling is. It becomes comparable with the radiative cooling at high luminosities (Fig. 1), and the standard, thin disk approach can no longer be applied.


Fig. 1: The advection factor profiles for mdot=0.01, 1.0 and 10.0. Profiles for alpha=0.01 and 0.1 are presented with solid black and dashed red lines, respectively. The fraction f_{adv}/(1+f_{adv}) of heat generated by viscosity is accumulated in the flow. In regions with f_{adv}<0 the advected heat is released.

Such a problem of accreting disk with additional cooling mechanism has to be treated in a different way than the radiatively efficient flows. It is not possible to solve the equations describing the flow in analytical way. To find a solution, one has to solve a two-dimensional system of ordinary differential equations with a critical point --- the radius where the gas exceeds the local speed of sound (the sonic radius). It was done for the first time by \cite{slim} who forged the term ``slim disks``.  

Such slim accretion disks are both sub- and supersonic and extend down to BH horizon, as opposed to thin disks that formally terminate at the marginally stable orbits. Slim disks may rotate with the angular momentum profile significantly different than the Keplerian one --- the higher the accretion rate, the more significant departure (Fig. 2). The disk thickness also increases with the accretion rate. For rates close the Eddington limit, the maximal H/R ratio reaches 0.3. Finally, the flux emerging from the slim disk surface is modified by the advection. At high luminosities large fraction of viscosity-generated heat is advected inward and released closer to the BH or even not released at all. As a result, the slope of the flux radial profile changes, and radiation is emitted also from within the marginaly stable orbit (Fig. 3). Due to the increasing rate of advection, the efficiency of transforming gravitational energy into radiative flux decreases with increasing accretion rate. Despite highly super-Eddington accretion rates, the disk may remain only moderately luminous (Fig. 4).


Fig. 2 Profiles of the disk angular momentum u_\phi for \alpha=0.01 (left) and \alpha=0.1 (right panel) for different accretion rates. The spin of the BH a_*=0.




Fig. 3 Flux profiles for different mass accretion rates in case of a non-rotating BH and two values of alpha: 0.01 (black solid), 0.1 (red dashed lines). For each value five lines for the following mass accretion rates:  0.01, 0.1, 1.0, 2.0 and 10.0 \dot M_{Edd} are presented. BH mass is 10\Msun.



Fig. 4 Top panel: Luminosity vs accretion rate for three values of BH spin (a_*=0.0, 0.9, 0.999) and two values of \alpha=0.01 (black) and 0.1 (red line). Bottom panel: efficiency of accretion \eta=(L/L_{Edd}) / (\dot M / \dot M_{Edd}).

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