Variation of the gas and radiation content in the sub-Keplerian accretion disk around black holes and its impact to the solutions
Highlights
► The polytropic index (γ) and ratio of gas to total pressures (β) are shown to vary significantly in accretion flows. ► The values of γ and β could vary respectively, up to 20% and 300%. ► Solutions established based on constant γ throughout would misinterpret the physics of flows in certain regimes. ► This has a significant impact in explaining observed data.
Introduction
Gas flows in the vicinity of a compact object, particularly black hole and neutron star, are expected to be highly relativistic. However, far away of it, when infalling matter comes off, e.g., a companion star, flows are rather non-relativistic. Therefore, during accretion of matter from a large distance, flows should transit from the non-relativistic to relativistic regime while reaching the vicinity of the black hole horizon or the neutron star surface, when flow is expected to be sub-Keplerian in nature. Indeed, it is generally believed that the velocity and temperature at the inner edge of accretion disk are very high.
Not only the cases in accretion, highly relativistic flows are involved in astrophysical jets from galactic, extragalactic black holes, gamma-ray bursts etc. (e.g. Zensus, 1997, Mirabel and Rodríguez, 1999, Mészáros, 2002), with the velocity 0.9–0.98 times the speed of light. The collimated dipolar outflows emerging from deep inside collapsars, according to the collapsar model of gamma-ray burst (Woosley, 1993), are expected to achieve a Lorentz factor more than 100. As the accretion and outflow/jet are expected to be coupled (influencing each other; Bhattacharya et al., 2010), any change/transition (e.g. as mentioned above) in the accretion flow would influence the jet and hence the underlying inferences.
When the velocity varies from the non-relativistic to relativistic (ultra-relativistic) regime, the corresponding polytropic index (γ) in the equation of state (EOS), i.e. the ratio of gas to total pressure (β) in the present context, should not remain constant. Note that γ and β can be approximated as constant only if the flow remains non-relativistic or ultra-relativistic throughout. Therefore, the existing solutions for such flows with a fixed γ need to be verified and corrected if necessary with a relativistically correct EOS (see e.g. Chandrasekhar, 1938).
Most of the studies of accretion disks around compact objects have approximated γ to be constant throughout the flow, which cannot be the correct description for the reasons described above. Blumenthal and Mathews (1976) introduced, for the first time to best of our knowledge, variable γ and proposed an appropriate EOS in obtaining solutions for spherical accretion and wind around Schwarzschild black holes. Recently, Meliani et al. (2004) used their EOS and obtained the general solutions for the relativistic Parker winds incorporating varying γ according to the nature of the flow. Ryu et al. (2006) proposed a new EOS suitable for non-relativistic and relativistic regimes with varying γ, to implement in the simulations of relativistic hydrodynamics. Very recently, Chattopadhyay and Ryu (2009) showed that composition of the fluid plays an important role to determine the value of γ and then solutions of inviscid spherical flows around black holes. This would have impact on a hot accretion disk as well which was shown to exhibit significant nuclear burning, producing the variation of compositions in the disk (Chakrabarti and Mukhopadhyay, 1999, Mukhopadhyay and Chakrabarti, 2000). Earlier, while Chen and Taam (1993) discussed structure and stability of the transonic optically thick accretion disks without restricting the gas and radiation content therein, did not show the actual variation of the gas/radiation content as a function of radial coordinate and then its impact to the solutions.
The present work introduces a self-consistent variation of γ in solving viscous flows in the accretion disk around black holes. We essentially concentrate on the region of the flows which is expected to be sub-Keplerian in nature. Therefore, following previous work (Muchotrzeb and Paczynski, 1982, Abramowicz et al., 1988, Chakrabarti, 1996, Mukhopadhyay and Ghosh, 2003, Rajesh and Mukhopadhyay, 2010a) we model the flow from the Keplerian to sub-Keplerian transition radius (xt) to the black hole event horizon (x+). We plan to concentrate, in particular, the class of flows which is not geometrically very thick, yet sub-Keplerian in nature including a nonzero component of advection and gradient of pressure in general. Hence, the underlying equations can be integrated/averaged vertically without losing important physics, in obtaining the solutions. Therefore, following previous work (Muchotrzeb and Paczynski, 1982, Abramowicz et al., 1988, Chakrabarti, 1996, Mukhopadhyay and Ghosh, 2003, Rajesh and Mukhopadhyay, 2010a, Rajesh and Mukhopadhyay, 2010b), we consider vertically integrated infall consisting of gas and radiation. While the flow around xt might be non-relativistic with a high γ (depending on the size of the disk), as it advances towards x+, γ must be varying. We plan to investigate, how the self-consistent variation of γ over the disk radii affects the established solutions with a fixed γ.
In the next section we establish the model equations describing flows. Subsequently, we discuss solutions in Section 3 and finally summarize the results in Section 4.
Section snippets
Model equations
The basic equations describing conservations of mass and momentum in the vertically integrated disk remain same as of earlier works which assumed γ and then β to be constant throughout (e.g. Abramowicz et al., 1988, Chakrabarti, 1996, Rajesh and Mukhopadhyay, 2010b). Abramowicz et al. (1988) also showed that the geometrically slimmer flow is stable for β > 0.4. For simplicity, following e.g. Narayan and Yi, 1994, Chakrabarti, 1996, we assume that the heat radiated out to be proportional to the
Solution
First we discuss the solution for the optically thick flows which are geometrically slimmer. Subsequently, we address the solution for the optically thin and geometrically thicker flows.
Summary
We have investigated the sub-Keplerian general advective accretion flows (GAAF; Rajesh and Mukhopadhyay, 2010a) around black holes allowing evolution of the gas and radiation content into flows self-consistently. We have, in one hand, considered radiation to be optically thick blackbody in the geometrically thin/slim vertically integrated disks. On the other hand, we have also considered optically thin geometrically thicker flows in the assumption of bremsstrahlung radiation. Hence, as matter
Acknowledgments
This work is partly supported by a project, Grant No. SR/S2HEP12/2007, funded by DST, India. One of the authors (P.D.) thanks the KVPY, DST, India, for providing a fellowship.
References (28)
- et al.
New Astron.
(2010) - et al.
ApJ
(1995) - et al.
ApJ
(1988) - et al.
ApJ
(2010) - et al.
ApJ
(1976) ApJ
(1996)- et al.
A&A
(1999) An Introduction to the Study of Stellar Structure
(1938)- et al.
ApJ
(2009) - et al.
ApJ
(1997)
ApJ
RAA
A&A
A&A
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