Model Dependence of Transonic Properties of Accretion Flows Around Black Holes

We analytically study how the behaviour of accretion flows changes when the flow model is varied. We study the transonic properties of the conical flow, a flow of constant height and a flow in vertical equilibrium and show that all these models are basically identical provided the polytropic constant is suitably changed from one model to another. We show that this behaviour is extendible even when standing shocks are produced in the flow. The parameter space where shocks are produced remain roughly identical in all these models when the same transformation among the polytropic indices is used. We present applications of these findings.


INTRODUCTION
Fully self-consistent study of any astrophysical system is generally prohibitive. Very often, for simplicity, it is necessary to construct models which have all the salient features of the original problem. However these models need not be unique. In the present paper we make a pedagogical review of three different models of rotating accretion flows and show that even though they are based on fundamentally different assumptions, they have identical physical properties. What is more, results of one model could be obtained from the other by changing c 2000 RAS a physical parameter, namely, the polytropic constant. In other words, all these models are identical.
Accretion disk physics has undergone major changes in the last fifty years. Bondi (1952) studied spherical accretion and found the existence of only one saddle type sonic point in an adiabatic flow. Later, the Keplerian disk model of Shakura and Sunyaev (1973) and thick disk model of Paczyński and Wiita (1980) the disk solutions became more realistic, though none of them was transonic, i.e., none was passing through any sonic point. Meanwhile, Liang and Thompson (1980) generalized this work for a flow which included angular momentum and discovered that there could be three sonic points. Matsumoto et al. (1984) tried to let the flow pass through the inner sonic point only and found that flow could pass through nodal type sonic points. Chakrabarti (1989Chakrabarti ( , 1990; hereafter referred to as C89 and C90 respectively) studied transonic properties of accretion flows which are conical in shape in the meridional plane ('Wedgeshaped Flow') and also flows which are in vertical equilibrium. Subsequently, Chakrabarti & Molteni (1993) studied flows of constant height and also verified by time dependent numerical simulations that the flow indeed allows standing shocks in it. In a Bondi (1952) flow, to specify a solution one requires exactly one parameter, namely the specific energy E of the flow. This is in turn determined by the temperature of the flow at a large distance.
In an inviscid, rotating axisymmetric accretion flow, one requires two parameters, namely, specific energy E and specific angular momentum λ. Once they are specified, all the crucial properties of the flow, namely the locations of the sonic points, the shocks, as well as the complete global solution are determined. C89 numerically studied the properties of the parameter space rather extensively, and divided the parameter space in terms of whether standing shocks can form or not. In the present paper, we compare these models completely analytically and show, very interestingly, that one could easily 'map' one model onto another by suitably changing the polytropic index of the flow. In other words, we show that these models are roughly identical to one another as far as the transonic properties go.
In the next Section, we present a set of equations which govern the steady state flow in all the three models. In §3, we present the sonic point analysis and provide the expressions for the energy of the flow in terms of the sonic points. We observe that these expressions are identical provided there is a unique relation among the polytropic indices of these model flows. In §4, we compare shock locations in all the three models. We also compare the parameter space which allows shock formation in these models with the regions obtained using purely numerical methods. In §5, we show that in fact if the relations between the polytropic indices are used, the shock locations in all these models are also roughly identical. Consequently, the apparently disjoint parameter spaces drawn with the same polytropic index overlap almost completely when the aforementioned relations among polytropic indices is used. This remarkable behaviour shows underlying unity in these apparently diverse models.
Finally, in §6, we draw our conclusions.

MODEL EQUATIONS
As discussed in the Introduction, we shall be concerned about three axisymmetric and inviscid models: (a) Model H: the flow has a constant height everywhere; (b) Model C: the flow cross-section in meridional plane is conical in shape and (c) Model V: the flow is in equilibrium in the transverse direction. Figure 1 shows a cartoon diagram of these three models. Filled circle at the centre correspond to the black hole. Region of the disk shaded in light corresponds to the pre-shock flow while the region shaded in dark corresponds to the post-shock flow. We also assume that the distances are measured in units of r S = 2GM BH /c 2 , where G is the gravitational constant, c is the velocity of light, M BH is the mass of the black hole. Velocities and angular momenta are measured in units of c and cr S = 2GM BH /c respectively. In all the three models, the dimensionless energy conservation law can be written as, where, g(x) is the pseudo-Newtonian potential introduced by Paczyński & Wiita (1980) and is given by, g(x) = − 1 2(x−1) . Here, ϑ e and a e = γp/ρ are the non-dimensional radial and the sound velocities respectively, x is the non-dimensional radial distance, the subscript e refers to the quantities measured on the equatorial plane. The flow has been chosen to be adiabatic with equation of state, P = Kρ γ , where K is a constant which measures the entropy of the flow, and γ is the polytropic exponent. The energy equation is the integral from of the radial momentum balance equation.
The mass flux conservation equation, which comes directly from the continuity equation depends on specific geometry of the models. Apart from a geometric constant, the conservation equation is given by, (2) where β, ζ and δ are constants. For Model V (Chakrabarti, 1989), β = 3/2, ζ = 1, δ = 1.
For Model C (Chakrabarti, 1990), β = 2, ζ = 0, δ = 0. For Model H (Chakrabarti, 1992;Chakrabarti & Molteni, 1993), β = 1, ζ = 0, δ = 0. Note that since the local disc height h(x) depends on sound speed h(x) ∼ a e x 1/2 (x − 1), so a factor of a ζ e is applicable for this Model. Though it is customary to deal with the conserved mass accretion rate of the flow, since we incorporate shock formation where entropy is increased, it is more convenient to re-write the mass flux conservation equation in terms of ϑ e and a e in the following way, where, α = 2n + ζ, α = 2n and α = 2n for Models V, C and H respectively and f (x) = x β (x − 1) δ . We shall use the phrase "entropy-accretion rate" for the quantity,Ṁ =Ṁ K n γ n .
In a flow without a shock, this quantity remains constant, but in presence of a shock it changes because of the generation of entropy.
Since the flow is expected to be sub-sonic at a long distance and supersonic on the horizon, the flow must pass through sonic points. At the sonic point a few conditions are to be satisfied. They can be derived in the following way: First, we differentiate the energy equation and the mass conservation equation and eliminate da/dx from them to obtain, is the effective potential and β ′ = β + δ. Since the flow is assumed to be smooth everywhere, if at any point of the flow denominator vanishes, the numerator must also vanish there. The vanishing of the denominator gives, The vanishing of the numerator gives, The subscript c denotes quantities at the critical points. Here, λ K is the Keplerian angular momentum defined as λ 2 . It is to be noted that since square of the sound speed (eq. 6) is always positive, angular momentum at the sonic point must be sub-Keplerian, i.e., λ < λ K . When the above expression for the velocity of sound is inserted in the expression for the specific energy, we get, for the Vertical Equilibrium (V) Model, for the Conical Flow (C) Model, and for Constant Height Flow (H) Model, Here, we have used the subscripts V, C and H under specific energy and polytropic index n to denote specific models. For a given angular momentum and at the same sonic point, the energy expression will be the same provided, where, we have used for Model V.
The relations in Eq. (8) are very important. If these relations are satisfied, then transonic properties of Model C with polytropic index n C would be identical to those of Model H with index n H and those of Model V with index n V respectively.

SHOCK INVARIANTS AND LOCATIONS IN DIFFERENT MODELS
In between two sonic points, the flow can undergo a standing shock transition. For an inviscid flow, at the shock, a set of conditions are to be satisfied. These are known as the Rankine-Hugoniot conditions (Landau & Lifshitz, 1959). These conditions are different for different models (C89, C90). For Model V, the shock conditions are as follows: the energy flux conservation equation, the pressure balance condition, and the baryon flux conservation equation, where subscripts "−" and "+" refer, respectively, to quantities before and after the shock.
Here, W and Σ denote the pressure and the density, integrated in the vertical direction (see, e.g., Matsumoto et al. 1984), i.e., and where, I n = (2 n n!) 2 (2n+1)! , n being the polytropic index as defined previously. For Models C and H, the shock conditions are as follows: the energy flux conservation equation, the pressure balance condition, 11b) and the baryon number conservation equation, The subscripts "−" and "+" have the same interpretation as before. Here, P and ρ denote the local pressure and the local density. In the subsequent analysis we drop the subscript e if no confusion arises in doing so.
The expressions for the conserved quantities could be combined to obtain the so-called Mach number relation which must be satisfied at the shock. For Model V, we obtain this relation as follows: We rewrite the energy conservation Eq. (9a), and the pressure balance Eq. (9b) in terms of the Mach number M = ϑ/a of the flow, where, ν ′ = 2γ γ−1 , and a ν where, ν = 3γ−1 γ−1 and x s is the location of the shock. f (x s ) = x 3/2 s (x s − 1) is the term in accretion rate which is explicitly a function of x and is the same both before and after the shock. From Eqs. 12(a-d) one obtains the following equation relating the pre-and post-shock Mach numbers of the flow of Model V at the shock (C89a), The constant C is invariant across the shock. The Mach number of the flow just before and after the shock can be written down in terms of C as, The product of the Mach number is given by, Similarly, one can obtain the Mach-number relations and the expression for Mach numbers for the other two models. The relation between the pre-and the post-shock Mach So far, in the literature, analytical shock studies have been carried out in models of vertical equilibrium (Das, Chattopadhyay and Chakrabarti, 2001) by using the Mach invariant relations (Eq. 13) when two parameters, namely, the specific energy and the specific angular momentum are given. Presently we carry out the same analysis using Eq. 16 for Models H and C respectively and obtained shock locations and parameter space boundaries for all the three models. Figure 2 compares these results where plots of specific energy (Y-axis) is given as function of specific angular momentum (X-axis). Solid boundaries mark regions for which standing shocks form in different models. Shaded regions are obtained from the analytical method (Das, Chattopadhyay & Chakrabarti, 2001) and results of these two methods roughly agree. We note that constant height flows occupy much larger region than that of the Conical or vertical equilibrium.
We have already noticed that one could use a relation (Eq. 8) which maps one model on to another, as far as the transonic properties go. If, for instance, we choose n H = 3, we find that n V = 31/4 and n C = 13/2 respectively. This means that for a given What about the shock locations? In Fig. 3, we compare the locations of the standing shocks around a black in these three models. Solid curves are for Model V, small circles are for Model H and crosses are for Model C respectively. The polytropic indices n V , n C and n H are as above. We note that though models are different, the shock locations are also remarkably close to one another. In Fig. 4, a comparison of the parameter space is shown once more (cf. Fig. 2). However, the polytropic indices are chosen as above. Unlike disjoint regions in Fig. 2, we find that the regions are almost completely over-lapping when Eq. (8) was used. This also shows that the Eq. (8) is valid even for the study of shock waves. We thus believe that generally speaking, the three models are identical when the Eq. (8) is taken into account.
What could be possible applications of the pedagogical exercise we carried out? One could imagine that certain models are easier to study (say, using numerical simulations) than the others. For instance, Chakrabarti & Molteni (1993) and Molteni, Gerardi and Chakrabarti (1994), Chakrabarti & Molteni (1995)

studied constant height disks using Smoothed Particle
Hydrodynamics. This was done because a flow in vertical equilibrium cannot be forced on a time dependent study. However, one could question whether one can draw any conclusion about the behaviour of flows in vertical equilibrium using a simulation of constant height.
Our present study shows that it does. Since three models are shown to be identical, running simulation for one model would give results for other models in a straight forward manner.
Similarly, study of stability analysis of a model of constant height may be simpler and stability of one model would imply stability of others.  . Nature of the boundary of the parameter space for the three models of the accretion flows. Solid, short-dashed, and long-dashed curves are drawn for Models C, H and V respectively with polytropic indices n V = 31/4, n C = 13/2 and n H = 3 which obey Eq. (8). The roughly similar parameter space shows that the mapping of the indices based on the transonic properties remain roughly the same even when standing shocks are considered. The asterisk mark on 'Vertical Equilibrium Flow' indicates that condition 8a has been utilized.

CONCLUDING REMARKS
In this paper, we discovered a unique relation among the polytropic indices of three different models of the axisymmetric accretion flows which ensures identical transonic properties in the sense that if all these models have the same conserved energies and angular momenta, then the sonic points also form exactly at the same place. When we proceeded further to compute the shock locations, we found that even the shocks form roughly at the same places.
Apparently, disjoint parameter spaces for shock formation with the same value of polytropic index in three different models exhibits considerable overlap when the same unique relation (Eq. 8) was used. This shows that the models are virtually identical in properties and various disk models belong to one parameter family. Our finding has given some insight into the relation between the nature of a flow with its equation of state. It seems that the relativistic equation of state in a flow in vertical equilibrium behaves similar to a roughly isothermal flow in a disk of constant height or in a conical flow. It is possible that in the latter models (Model C and H) the geometric compression is smaller and hence it is easier to keep them roughly isothermal while conserving energy as well.
Though our work has been mainly pedagogical, we believe that it could have several applications. For instance, linear and non-linear stability analysis and time dependent calculations (numerical simulations) are easier to perform when the disk is of constant thickness.
Our work indicates that once certain properties regarding stability are established in one flow model, they would remain valid in other models as well provided the relation among the polytropic indices is incorporated.