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A model on transition between steady states of sub-Keplerian accretion discs: implication for spectral states and hot corona above the disc

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Abstract

We present here a simple hydrodynamic model based on a sequence of steady states of the inner sub-Keplerian accretion disc to understand its different spectral states. Correlations between different hydrodynamic steady states are studied with a goal to understand the origin of, e.g., the aperiodic variabilities. The plausible source of corona/outflow close to the central compact object is shown to be a consequence of steady state transition in the underlying accretion flow. We envisage that this phenomenological model can give insight on the influence of viscosity, efficiency of energy advection, nature of the background flow and environment on the evolution of the inner sub-Keplerian accretion disc.

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Acknowledgements

We thank Harsha Raichur for critical comments on an earlier version of the manuscript. This work is partly supported by facilities provided by the Department of Science and Technology, Government of India under the FIST programme. AA thanks the Council for Scientific and Industrial Research, Government of India, for the research fellowship. SRR thanks IUCAA, Pune for the Visiting Associateship Programme.

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Authors and Affiliations

  1. Department of Physics and Research Centre, Sanatana Dharma College, University of Kerala, Alappuzha, 688003, Kerala, India

    Arunima Ajay & S. R. Rajesh

  2. Research Centre, University of Kerala, Thiruvananthapuram, 695034, Kerala, India

    Arunima Ajay

  3. Inter-University Centre for Astronomy & Astrophysics, Ganeshkind, Pune, 411007, Maharashtra, India

    Nishant K. Singh

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  1. Arunima Ajay
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  2. S. R. Rajesh
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  3. Nishant K. Singh
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All authors contributed equally to the manuscript.

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Correspondence to S. R. Rajesh or Nishant K. Singh.

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Appendices

Appendix A: Some further details of our model

The hydrodynamic variables of the intermediate steady state as well as the total mass accretion rate are expressed as polynomial series in the parameter \(\varepsilon \) (Eqn. (8), (9) and (14)). Therefore we may expect to express the correlation between the intermediate steady state viscosity parameter \(\alpha \) or the intermediate steady state advective factor \(f\) to the corresponding initial steady state parameters \(\alpha _{0}\) or \(f_{0}\) as polynomial series in \(\varepsilon \). If we express \(\alpha \) or \(f\) as series in increasing power of \(\varepsilon \) then we will arrive at the following unphysical conclusions: (1) The steady state hydrodynamic equations of the zeroth order flow variables of the intermediate state will be completely independent of the first order and other higher order flow variables. But the steady state hydrodynamic equations of the first order flow variables will depend on the zeroth order flow variables, and independent of the second order and other higher order flow variables and so on. Thus the fundamental principal of action and reaction is not captured. (2) By definition \(\alpha _{0}\) and \(\alpha \) have values between zero and one. Therefore if we express \(\alpha \) as series in increasing power of \(\varepsilon \), then for the term \(\varepsilon \alpha _{1}\) to have any significant value, the magnitude of \(\alpha _{1}\) should be close to one (since \(\varepsilon \ll 1\)). If we continue like this, magnitudes of the higher order parameters \(\alpha _{2}\), \(\alpha _{3}\,\ldots \), will become greater than unity and the values of these parameters will not converge as order increases. Similar argument is valid for \(f_{0}\), \(f\) and higher order advective factors \(f_{1}\), \(f_{2},\,\ldots \).

Series expansions as given in Eqn. (16) are the appropriate representation of \(\alpha \) and \(f\) because of the following reasons: (1) The hydrodynamic equations of flow variables of different orders are coupled. Therefore the action reaction principle is maintained. (2) The values of \(\alpha _{0}\) and \(\alpha \) are between zero and one. Since \(\varepsilon \ll 1\), \(\varepsilon ^{-n}\) increases rapidly as \(n\) increases. Therefore the magnitudes of the set of numbers, the higher order viscosity parameters \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\)...are less than one and decrease rapidly as the order \(n\) increases. Similar explanation is valid for higher order advective factors. (3) For given values of \(\alpha _{0}\), \(\alpha \) and \(\varepsilon \), we can choose the set of numbers \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\,\ldots \). Similarly for given values of \(f_{0}\), \(f\) and \(\varepsilon \), we can choose the set of numbers \(f_{1}\), \(f_{2}\), \(f_{3}\,\ldots \). By definition \(\varepsilon \) characterises the amount of matter lost by the disc due to state transition. Therefore if there is no state transition that is \(\alpha \rightarrow \alpha _{0}\) and \(f \rightarrow f_{0}\) then \(\varepsilon \rightarrow 0\). For this particular combination of \(\alpha _{0}\), \(\alpha \) and \(\varepsilon \), the only possible set of physically acceptable higher order viscosity parameters is \(\alpha _{1} \, , \, \alpha _{2}\, , \, \alpha _{3}.... = 0\). Similarly for this particular combination of \(f_{0}\), \(f\) and \(\varepsilon \), the only possible set of physically acceptable higher order advective factors is \(f_{1} \, , \, f_{2}\, , \, f_{3}.... = 0\). In other words, \(\alpha _{n}\) and \(f_{n}\) decrease faster than the increase in \(\varepsilon ^{-n}\) for \(n>0\) in the series expansion of Eqn. (16).

Appendix B: Expressions for the first order conservation equations discussed in §2.5

Substituting expressions of the flow variables (Eqn. (8)-(9)), vertically integrated stress tensor (Eqn. (17)), vertically integrated energy advection (Eqn. (18)) and total mass accretion rate (Eqn. (14)) in Eqn. (1) to (4), and equating the first power of \(\varepsilon \) we obtain the first order conservation equations:

$$ 4 \pi r \rho _{0} h_{0} v_{1} + 4 \pi r\rho _{1} h_{1} v_{0} = \dot{M_{1}} $$
(B.1)
$$\begin{aligned} &\rho _{0} v_{0} \frac{\partial v_{1} }{\partial r} + \rho _{1} \phi ^{ \frac{1}{2}} v_{0} \frac{\partial v_{0} }{\partial r} +\rho _{0} v_{1} \frac{\partial v_{0} }{\partial r} \\ &\qquad{}- \rho _{1}\phi ^{\frac{1}{2}} \Omega _{0}^{2}r - 2\rho _{0}\Omega _{0}\Omega _{1} r \\ &\quad = - \frac{\partial (\rho _{0} c_{s1}^{2}) }{\partial r}- \phi ^{ \frac{1}{2}} \frac{\partial (\rho _{1} c_{s0}^{2}) }{\partial r} - \frac {\rho _{1}\phi ^{\frac{1}{2}}}{r^{2}} \end{aligned}$$
(B.2)
$$\begin{aligned} &\rho _{0}h_{0} v_{1} \frac{\partial (\Omega _{0} r^{2}) }{\partial r} + \rho _{1}h_{1} v_{0} \frac{\partial (\Omega _{0} r^{2}) }{\partial r}+ \rho _{0} h_{0}v_{0} \frac{\partial (\Omega _{1} r^{2}) }{\partial r} \\ &\quad = \frac {1}{r} \frac{\partial \left (r^{2} \alpha _{0} D_{1} \, + \, \alpha _{1} D_{2} \right )}{\partial r} \end{aligned}$$
(B.3)
$$\begin{aligned} &\frac{v_{0}}{(\gamma -1)} \biggl[ h_{0} \frac{\partial (\rho _{0}c_{s1}^{2}) }{\partial r} +h_{1} \frac{\partial (\rho _{1} c_{s0}^{2}) }{\partial r} \\ &\qquad{} - \gamma c_{s1}^{2} h_{0} \frac{\partial \rho _{0} }{\partial r} - \gamma c_{s0}^{2} h_{1} \frac{\partial \rho _{1} }{\partial r} \biggr] \\ &\qquad{} + \frac{v_{1} h_{0}}{(\gamma -1)} \biggl[ \frac{\partial (\rho _{0} c_{s0}^{2}) }{\partial r} - \gamma c_{s0}^{2} \frac{\partial \rho _{0} }{\partial r} \biggr] \\ &\quad = f_{0} F_{1} + f_{1} F_{2} \end{aligned}$$
(B.4)

The left hand side of the first order energy equation (Eqn. (B.4)) has two parts. The first part is the energy due to the interaction between the zeroth order flow variables and the first order flow variables, which is advected by the zeroth order radial velocity. The second part is the energy due to the zeroth order flow variables, which is advected by the first order radial velocity.

Appendix C: Expressions for \(L\)-coefficients that appear in §4

The terms in the fully nonlinear first order equations with respect to a self-similar background disc (Eqn. (38)-(40)) are:

$$\begin{aligned} \begin{aligned} & L_{1} = ACr^{-2} + Br^{-1}\phi ^{\frac{1}{2}}k_{6} \;;\\ & L_{2} = Ar^{\frac{-3}{2}} + Br^{-1}\phi ^{\frac{1}{2}}k_{5} \end{aligned} \end{aligned}$$
(C.1)
$$\begin{aligned} & L_{3} = \biggl[ \frac{1}{2} ACr^{-3} v_{1} + 2AD r^{-2}\Omega _{1}+ \frac{3}{2} Ar^{\frac{-5}{2}} cs_{1}^{2} \biggr] \\ &\phantom{L_{3} =}\quad{}- \rho _{1}\phi ^{ \frac{1}{2 }} \biggl[ 1- \frac{C^{2}}{2} -D^{2} - B_{f} \biggr] r^{-2} \\ &\phantom{L_{3} =}\quad{}- Br^{-1}\phi ^{\frac{1}{2}} k_{4} \end{aligned}$$
(C.2)
$$\begin{aligned} & L_{4} = 3ABrS - \frac{9}{4}ADr^{\frac{-1}{2}} cs_{1}^{2} - 3BD \rho _{1} \phi ^{\frac{1}{2}} \\ &\phantom{L_{4} =}{}- \frac{3}{2}BD r \left [ \frac{\phi +1}{2\phi ^{\frac{1}{2}}} \right ]k_{4} \\ &\phantom{L_{4} =}{} - \frac{3}{8}BD \rho _{1} \left [\frac{\phi -1}{\phi ^{\frac{1}{2}}} \right ] \end{aligned}$$
(C.3)
$$\begin{aligned} & L_{5} = ABr^{2} ; \\ & L_{6} = - \biggl(\frac{3}{2}ADr^{ \frac{1}{2}} + \frac{3}{2}BD r \biggl[ \frac{\phi +1}{2\phi ^{\frac{1}{2}}} \biggr] k_{5} \\ &\phantom{L_{6} =}{} + \frac{3}{4}BD r \biggl[ \frac{\phi -1}{\phi ^{\frac{1}{2}}} \biggr] \frac{\rho _{1}}{cs_{1}^{2}} \biggr) \;; \\ \begin{aligned} & L_{7}= -\frac{3}{2}BD r \left [ \frac{\phi +1}{2\phi ^{\frac{1}{2}}}\right ] k_{6} ;\\ & L_{8} = 2AC \Omega _{1} + ACrS + \frac{1}{2} AD r^{-1} v_{1} + \frac{1}{2} CD \rho _{1} \phi ^{\frac{1}{2}} \end{aligned} \end{aligned}$$
(C.4)
$$\begin{aligned} \begin{aligned} & L_{9} = \frac{1}{\gamma -1} AC r^{-1} - BC \phi ^{\frac{1}{2}} r^{ \frac{-1}{2}} k_{5} ;\\ & L_{10} = - BC \phi ^{\frac{1}{2}} r^{ \frac{-1}{2}} k_{6} \end{aligned} \end{aligned}$$
(C.5)
$$\begin{aligned} \begin{aligned}&L_{11} = L_{111} + L_{112} + L_{113} + L_{114} + L_{115} ;\\ & L_{111} = \frac{3}{2} f_{0} ABD r^{\frac{-1}{2}} ( \alpha + \alpha _{0} ) S \end{aligned} \end{aligned}$$
(C.6)
$$\begin{aligned} \begin{aligned} & L_{112} = \left (\frac{3\gamma -5}{4(\gamma -1)}\right ) AB r^{ \frac{-5}{2}} v_{1} ;\\ & L_{113} = - \left ( \frac{9}{4} AD^{2}f_{0} \alpha _{0} - \frac{3}{2}AC \right )r^{-2 }cs_{1}^{2} \end{aligned} \end{aligned}$$
(C.7)
$$\begin{aligned} \begin{aligned} &k L_{114} = - \phi ^{\frac{1}{2}} \rho _{1} r^{\frac{-3}{2}} \left ( \frac{1}{\gamma -1} BC + \frac{9}{4} BD^{2} f_{0} \alpha _{0} \right ) ;\\ & L_{115} = - BC \phi ^{\frac{1}{2}} r^{\frac{-1}{2}} k_{4} \end{aligned} \end{aligned}$$
(C.8)
$$\begin{aligned} & \theta = \frac{A^{2}}{B^{2}} r^{-1} (cs_{1}^{2})^{2} +4 [ A\delta r^{\frac{-3}{2}} - \frac{A}{C} r^{-1} v_{1}]^{2} ; \\ &k = 4 [ A\delta r^{\frac{-3}{2}} - \frac{A}{C} r^{-1} v_{1}] \\ & k_{1} = 2k \left [ \frac{A}{C} v_{1}r^{-2} - \frac{3}{2} A\delta r^{ \frac{-5}{2}}\right ] - \frac{A^{2}}{B^{2}}\left ( \frac{cs_{1}^{2}}{r}\right )^{2} \end{aligned}$$
(C.9)
$$\begin{aligned} & k_{2} = 2 \frac{A^{2}}{B^{2}} r^{-1} cs_{1}^{2} \;;\quad k_{3} = - \frac{2Ak}{C} r^{-1} ; \\ &k_{4} = \frac{A}{4B} r ^{\frac{-3}{2}} cs_{1}^{2} \pm \frac{1}{4} \theta ^{\frac{-1}{2}} k_{1} \\ \begin{aligned} & k_{5} = -\frac{A}{2B} r ^{\frac{-1}{2}} \pm \frac{1}{4} \theta ^{ \frac{-1}{2}} k_{2} ;\\ &k_{6} = \pm \frac{1}{4} \theta ^{\frac{-1}{2}} k_{3} \end{aligned} \end{aligned}$$
(C.10)

where

$$\begin{aligned} \frac{\partial h_{1}}{\partial r} ={}& \frac{h_{0}\phi ^{\frac{1}{2}} }{r}+ \frac{h_{0}}{2} \left ( \frac{{\phi}-1}{\sqrt{\phi}} \right ) \\ &{}\times\left [ \frac{1}{c_{s1}^{2}} \frac{\partial c_{s1}^{2}}{\partial r} - \frac{1}{\rho _{1}} \frac{\partial \rho _{1}}{\partial r} - \frac{1}{2r} \right ] \, \, \text{and} \\ & S=\frac{\partial \Omega _{1}}{\partial r} \end{aligned}$$
(C.11)

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Ajay, A., Rajesh, S.R. & Singh, N.K. A model on transition between steady states of sub-Keplerian accretion discs: implication for spectral states and hot corona above the disc. Astrophys Space Sci 369, 55 (2024). https://doi.org/10.1007/s10509-024-04318-2

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