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Effect of the flow composition on outflow rates from accretion discs around black holes

We studied the outflow behaviour from accretion discs around black holes taking into account the vertical equilibrium accretion flow model. The outflow rate is found to depend crucially on flow composition. Our approach is to study the outflow behaviour as function of inflow around black holes with an equation of state which allows flow to be thermally relativistic close to black holes and non relativistic far away from black holes. We studied shock ejection model. A pure electron positron pair flow never undergoes shock transition while presence of some baryons (common in outflows and jets) makes it possible to have standing shock waves in the flow. It can be concluded that the presence of protons is necessary for the flow to show the outflow behaviour. The outflow rate is maximum when the flow contains the proton number density which is 27% of the electron number density. We conclude that a pure electron-positron jet is unlikely to form.


INTRODUCTION
One of the spectacular phenomena associated with many classes of astrophysical objects is the formation of outflows and jets. Systems which have been associated with winds and jets are active galactic nuclei (AGNs e.g., M87), young stellar objects (YSOs e.g., HH 30, HH 34), high mass X-ray binaries (HMXB e.g., SS433, Cyg X-3), black hole X-ray transients (e.g., GRS 1915+105, GRO 1655-40) and low mass X-ray binaries (LMXB e.g., Cir X-1). In matter, Chakrabarti & Titarchuk (1995) showed that the soft, hard and the intermediate spectral states can be satisfactorily explained. The post-shock disc inverse-Comptonizes the intercepted soft photons to produce the hard power-law tail and produces the so-called hard spectral state, while absence of or weakened post-shock disc produces essentially the soft spectral state. These assertions are vindicated by observations (Smith et al. 2001(Smith et al. , 2002(Smith et al. , 2007, and also by numerical simulations (Giri & Chakrabarti 2013). Moreover, oscillating and transient shocks explain the low-frequency quasi periodic oscillation (QPO) and its evolution during spectral state transitions (Nandi et al. 2012).
However, very few addressed the most important issue of the outflow rate as a function of the inflow parameters. Chakrabarti (1999) attempted to estimate the outflow rate by considering simple hydrodynamical equations such as those applicable to conical inflows and outflows.
Majority of these works mentioned above, however, used a constant polytropic index and did not consider the variation of it from a large distance to the black hole and from the disc phase to the jet phase. Realistically, at large distances away from the central object, matter should be cold, but as it dives into the central object its thermal and kinetic energy increases. A flow with an equation of state described by a fixed Γ (adiabatic index) need not describe the behaviour of accreting matter properly from infinity to the horizon, because for ultra-relativistic non-degenerate gas Γ = 4/3, and that for non-relativistic one, it is Γ = 5/3. Γ takes intermediate values for trans-relativistic thermal energy. The equation of state (hereafter, EoS) of a single species relativistic gas was computed by Chandrasekhar (1938); Synge (1957), which gave the correct description of the flow in all temperature range, and we call the EoS as RP (relativistically perfect). However, RP is a combination of modified Bessel's functions and which makes its usage non-trivial and fairly expensive from computational point of view. Hence many workers proposed approximate EoS for single species flow, which are close to RP and yet easy to use (Taub 1948;Mathews 1971;Ryu et al. 2006). For the very first time Blumenthal & Mathews (1976, hereafter BM76), used the EoS of Mathews (1971)  EoS for electron-proton flow was suggested earlier (Fukue 1987). Chattopadhyay (2008); Chattopadhyay & Ryu (2009) showed in a general relativistic investigation that, a purely electron-positron adiabatic flow is the slowest, thermally least relativistic flow, compared to flows containing protons and leptons, and consequently, electron-positron flows do not show accretion shocks (Chattopadhyay 2008;Chattopadhyay & Chakrabarti 2011). It was also shown that electron-proton (i.e.,equal number of electrons & protons) flow is not the most relativistic case. However, when the proton number density is ∼ 20% of the electron number density, the flow becomes most relativistic. Therefore, an estimation of the outflow rate from the inflow parameters using an equation of state of a fully ionized flow would be an important contribution. In this paper, we precisely do this. The plan of our paper is the following: in the next Section, we present the assumptions and equations governing inflow and outflow from accretion discs around black holes. In Section 3, we present a detailed analysis of critical point behaviour and solution of accreting flow. We then present our results exhibiting the dependence of outflow rates on flow composition, energy and angular momentum of the accretion disc. Finally, in Section 4, we carry out discussion and make concluding remarks.

ASSUMPTIONS AND EQUATIONS
We consider a fully ionized, adiabatic, rotating and accreting axisymmetric disc around a Schwarzschild black hole. For mathematical simplicity, space-time around the black hole is described by the Paczyński-Wiita (hereafter PW) pseudo-Newtonian potential (Paczyński & Wiita 1980). We have used PW potential in order to simplify our calculations while retaining all the essential qualitative features of strong gravity. Moreover, it is easier to incorporate more complicated physics in pseudo-Newtonian scheme. Earlier investigations of disc-jet system had been to employ general relativity and fixed Γ EoS, or general relativity with relativistic EoS, or PW approach with fixed Γ EoS. So far, the approach with PW potential which is coupled with a variable Γ EoS was not attempted and ours is the first to do so.
Effect of flow composition on outflow rates 5

Equation of state
Following Chattopadhyay (2008); Chattopadhyay & Ryu (2009), we consider flows which are composed of electrons (e − ), positrons (e + ) and protons (p + ) of varying proportions, but always maintaining the overall charge neutrality. The number density is given by, where, n e − , n e + and n p + are the electron, positron and proton number densities, respectively. Charge neutrality condition over a reasonable volume element demands that which implies, n = 2n e − , and n e + = n e − (1 − ξ), where ξ = n p + /n e − is the relative proportion of protons. The mass density is given by where, η = m e − /m p + , and m e − and m p + are the electron and proton masses, respectively.
For single temperature flows, the isotropic pressure is given by The EoS for multi-species flow is (Chattopadhyay 2008;Chattopadhyay & Ryu 2009) The non-dimensional temperature is defined with respect to the electron rest mass energy, Θ = kT /(m e − c 2 ). Using equations (1)-(5), the expression of the energy density in equation where, The enthalpy is given by, The expression of the polytropic index is given by, The adiabatic index is

Equations of motion
The length, time and velocity scales are measured in units of r g = 2GM BH /c 2 , r g /c = 2GM BH /c 3 and c, respectively, where, r g is the Schwarzschild radius, M BH is the mass of the black hole, G is the gravitational constant, and c is the velocity of light, this amounts to 2G = M BH = c = 1. Hereafter, all equations are written using the above unit system. It is to be noted though, we have retained the same symbols for the thermodynamic quantities like ρ, p etc, in the dimensionless form too.

Inflow
In our study we neglect the non-conservative processes and magnetic fields. We consider the steady and radial flow assumptions.
In this case, the radial momentum equation is given by, Here, λ, ϑ, and x are the specific angular momentum, the infall velocity, and the radial coordinate in the units described above. The integral form of the eq.(11) gives the Bernoulli parameter and is written as and is also called as specific energy of the flow.
The mass conservation equation is given by, where Σ = 2ρH is the vertically integrated surface density of the flow.
The disc matter is in hydrostatic equilibrium in the vertical direction, and the half height is given by (Chakrabarti 1989) The entropy generation equation or the first law of thermodynamics is given by, Effect of flow composition on outflow rates 7 here, e =ē/ρ = f /K, where c = 1. Integrating eq.(15), and using the definition of e, ρ (eq. 4), and p (eq. 5), we get where, k 1 = 3(2 − ξ)/4, k 2 = 3ξ/4, and k 3 = (f − K)/(2Θ). This is the adiabatic equation of state for multispecies flows and K is the constant of entropy. Using equations (13) and (16), we can define entropy accretion rate (Ṁ) aṡ here,Ṁ is also constant for inviscid multispecies relativistic flows.

Critical-point conditions
Gravity, by the very act of pulling matter towards the gravitating centre, primarily causes the increase of the infall speed of the matter, but as a secondary effect, also compresses the in flowing matter and thereby increasing its temperature, and consequently, the sound speed. This causes the flow velocity to cross the local sound speed at a finite distance, which is termed as the sonic point or the critical point of the flow. If the matter is rotating, then the centrifugal force opposes the effect of gravity. Since the centrifugal force is ∝ x −3 and is axisymmetric, it breaks the spherical symmetry of gravity, causing the formation of multiple sonic points in a significant range of E − λ parameter space (Liang & Thompson 1980;Chakrabarti 1989). At the sonic point dϑ/dx → 0/0, which gives us the critical or sonic point conditions, and The subscript c denotes the quantities at the critical point during accretion flow. At a critical point, dϑ/dx is obtained by using the l'Hospital rule, where, N and D are the numerator and denominator of equation (19). The equation becomes, where, Since eq.(23) has two roots, they may be either real or complex conjugates depending on Effect of flow composition on outflow rates 9

Shock conditions
In the domain of multiple sonic points, supersonic matter through outer sonic point, may be slowed down due to the centrifugal barrier at x < ∼ few×10 r g . If the barrier is strong enough it may produce a centrifugal barrier mediated shock transition. In presence of mass loss, the mass and the momentum flux condition across the shock is given by (Chattopadhyay & Das 2007;Kumar & Chattopadhyay 2013), and, Here, subscripts minus (-) and plus (+) denote the quantities of subsonic and supersonic branch of the accretion flows, respectively, and, if the mass outflow rate isṀ out , then the relative mass outflow rate is Rṁ =Ṁ out /Ṁ − . Moreover, W = 2pH is the vertically integrated thermal pressure. The third condition will define the type of shock. If there is no energy loss at the shock front, then we have an adiabatic shock.
If there is some energy exchange, it becomes a dissipative shock. In this case, here, f e is the dissipation parameter. If the entropy is conserved across the shock front, then it is called an isentropic shock, and is given by, Using the adiabatic shock conditions i.e.,(24-26), the supersonic branch temperature and bulk velocity can be obtained from post-shock quantities and vice versa, here, c 0 = (1 − Rṁ)[2Θ + /K + ϑ 2 + ]/ϑ + and c 1 = ϑ 2 + /2 + h + . Both the expressions (ϑ − , Θ − ) in eq. 29 are obtained simultaneously in terms of post-shock quantities which gives us the shock location x s .

Outflow
Simulations (Molteni et al. 1996;Giri & Chakrabarti 2013) have shown that extra thermal gradient force along the vertical direction, drives bipolar outflows, therefore the outflow and the accretion solution is connected by the shock. As has been suggested by Molteni et al. (1996) and is described in details by Kumar & Chattopadhyay (2013), the outflowing matter tend to move out through the region between the funnel wall (FW) and the centrifugal barrier (CB). The cylindrical radius of the CB is, and that of FW is, where, the spherical radius of CB is r CB = (x 2 CB + y 2 CB ) 1/2 , while y CB is the local height of CB. The streamline is computed as, r j = x 2 j + y 2 j , where x j = (x F W + x CB )/2, and y j = y F W = y CB . The cross-sectional area of the jet is defined as, The momentum balance equation along the jet stream line is given by here the derivative is with respect to r = r CB . Suffix 'j' denotes jet quantities. The integrated continuity equation along the jet stream line is given bẏ This integration constant is called as mass flux which is moving along the stream line of jet.
By using eqs. (16, 36), we can write entropy-outflow rate of the jet.
Effect of flow composition on outflow rates 11 Now, using eqs.(6,36) in eq.(15), we get The jet velocity gradient equation obtain by combining eqs.(8, 35-38), we get where, The critical point conditions are, The derivatives at r jc is obtained as, The subscript c denotes the quantities at critical point in the outflow.
So, the mass outflow rate in term of flow variables at the shock front is given by, where, ρ j = ρ(r b )exp(−x j /h(r b )), ϑ j (r cb ) and A(r cb ) are jet base density, base velocity and base cross section, respectively. Here, r b = (x ci + x s )/3 and r CB = r = r 2 b + h 2 b , h b being the height of the disc at r b . The angular momentum of the jet λ j = λ| (x=r b ) = λ, i.e.,the jets are launched with the angular momentum of the accretion disc at the base of the jet, and since presently we assume inviscid flow, the jet and the accretion disc has the same angular momentum.

Solution Procedure
The solution procedure is the following, given E, λ, ξ the sonic points of the accretion discs (29), the solution will jump to the supersonic branch through x co . Once the shock is found, we solve eqs. (39-38) to find the jet solutions and the corresponding Rṁ (eq. 42). This value of Rṁ is supplied to eq. (29) and the whole solution procedure is repeated to obtain another shock location. This is repeated till the shock location converges.
Effect of flow composition on outflow rates 13

Nature of critical points and accretion flow solutions
We discuss the properties of the sonic points of the flow around blackholes with an approximate relativistic equation of state and how they are affected by ξ and λ parameters. We plot E c (Fig. 1a),Ṁ c (Fig. 1b), Γ c (Fig. 1c) and log(T c ) ( Fig. 1d)  its thermal energy is comparable to its rest energy i.e.,Θ = kT /m e c 2 > ∼ 1, and the adiabatic index will be reflected by a value 4/3 Γ < 5/3. Although, e − − e + is the lightest flow, its T c is so low that for most values of x c , Γ c ∼ 5/3 (long dashed-dotted in Fig. 2c). Therefore, at a large distance away from the black hole, physical sonic points are not formed. Closer to the horizon, the flow is hot enough to produce one physical sonic point. As ξ increases, the rest energy increases but the temperature increases even more, which makes flow with protons much hotter and thermally more relativistic (Fig. 2c). Therefore, for a given E > 1, one or more sonic points may form.   shock shown by S 4 → S 2 or T 2 transition. An isentropic shock is parallel to the ordinate represented by eq. 31, and by the transition S 4 → S 3 or T 3 transition. The energy and entropy jumps for the respective transitions are marked in the Figure. We plot E c withṀ c for λ = 1.6 and ξ = 1.0 (Fig. 5a), ξ = 0.1 (Fig. 5b), ξ = 0.035 (Fig. 5c)  The temperature and the entropy is higher for higher ξ, and the shock location is also at a larger distance for flow with higher ξ. Figures 6c, g & k, show Γ is variable for flows with any ξ. At x → large, Γ ∼ 5/3, irrespective of the value of ξ. However, since ξ = 0.0 flow has very low thermal energy, Γ ∼ 5/3 up to x ∼ 100. But for flows with ξ = 0.0, we find Γ ξ=0.5 < Γ ξ=1.0 at x <few×100. At x ∼ 1, Γ ξ=0.5 ∼ 1.4, Γ ξ=1.0 ∼ 1.44. This means that the flow becomes thermally more relativistic with the reduction of ξ up to a certain value (i.e.,around ξ ∼ 0.27), and then becomes less relativistic with further reduction of ξ.

Accretion and ejection solutions
In this Section, we present self-consistent accretion-ejection solutions. The methodology to simultaneously and self-consistently compute the accretion-ejection solution has been presented in Section 2.2.5. In Figs Figs. 8a-c), and launches a thermally driven bipolar jet whose sonic point is at r jc = 209.139. The relative mass outflow rate is Rṁ = 0.038. The jet starts with the Γ value of the post-shock disc but since it is thermally driven and is powered by converting the thermal energy to kinetic energy, Γ approaches non-relativistic values far away from the central object. At the shock, the entropy of the jet jumps up from pre-shock to post-shock value (Fig. 8c). Interestingly, the entropy of the jet is also much higher than the pre-shock disc. This entropy condition ensures that although most of the matter flows through the inner sonic point into the black hole, a significant amount of matter also flows out as jet. Now we plot x s with λ ( Fig. 9a), Rṁ with x s (Fig. 9b), compression ratio R = Σ + /Σ − with x s (Fig. 9c), and Rṁ with R (Fig. 9d), the composition of the flow is given by ξ = 0.27.  (1999). For a given E, x s decreases with decreasing λ. This increases the compression, and drives more matter into the jet channel. But with decreasing x s , the post-shock area decreases too, and that also limits the total amount of matter leaving the disc. Hence Rṁ will depend on increasing R, as well as the decreasing total post-shock area, and hence Rṁ peaks at some intermediate value of R.
Keeping E = 1.0003, we now vary λ, of flow with following compositions ξ = 1.0 (solid), ξ = 0.635 (dotted), and ξ = 0.27 (dashed). We plot x s with λ (Fig. 10a), Rṁ with x s (Fig.   10b), R with x s (Fig.10c), and Rṁ with R (Fig. 10d). We reconfirm that indeed Rṁ increases  with the increasing R or decreasing x s , although decreasing post-shock surface area finally decreases Rṁ. Rṁ is highest for ξ = 0.27, compared to flow of other compositions. To ensure this we scan the entire E & λ parameter space for a given value of ξ, and find out the maximum mass outflow rate possible for that particular ξ. In Fig. 11, we plot the maximum mass-outflow rate i.e.,R ṁ m = max(Rṁ), as a function of ξ. It easily shows that maximum outflow is possible for ξ = 0.27.
In Figs. 12a-b, we show the parameter space where standing shocks form (the so-called 'shock-domain') for various ξ. Similar to the behaviour of MCP domain, the shock domain in E − λ parameter space shifts to the higher energy and angular momentum region as ξ is decreased from its e − − p + value up to ξ = 0.27. If ξ is decreased further, the flow becomes less energetic and the shock-domain moves towards the lower energy and lower angular momentum corner, with subsequent decrease in the bounded area of the shock space. Finally, the shock domain disappears for ξ = 0.0. In Fig. 12c, we compare the shock domain for ξ = 1.0, of the accretion flow which includes mass loss (solid) and which does not include mass loss (dotted). In Fig. 12d, we compare the shock domain for ξ = 0.25, of the accretion flow which includes mass loss (solid) and which does not mass loss (dotted). This shows that the reduction of post-shock pressure due to mass loss, reduces the steady shock  Figure 11. The plot of maximum mass outflow rate R ṁ m with the composition parameter ξ.  domain. Flow from that part of the parameter space which admits shock without massloss, actually do not show steady shock when massloss is allowed. So there is a possibility of massloss driven shock oscillation too.

DISCUSSION AND CONCLUDING REMARKS
In this paper, we studied the outflow behaviour taking into account flow composition and approximate relativistic equation of state of the flow onto black holes. We have re-established our earlier findings that e − − e + is the least relativistic flow, but e − − p + is not the most relativistic flow. Thermally, the most relativistic flow is when ξ = 0.27. As has been explained earlier, the flow starting with the same E implies a flow starting with the same sound speed at infinity. Since the sound speed is a measure of the thermal speed, if we compare two flows with the same thermal speed, the constituent particles will transfer less average kinetic energy (i.e.,thermal energy) in the flow where these particles are lighter. Therefore, the temperature of e − −e + flow is lesser than the flow containing both electrons (and/or positrons) and heavier protons. The temperature of e − − e + is so low that the lower inertia of the flow cannot compensate for its lack of thermal energy and therefore thermally it is the least relativistic, while the flow with some amount of protons are more relativistic. Although the conclusions are consistent with the previous papers (Chattopadhyay 2008;Chattopadhyay & Ryu 2009;Chattopadhyay & Chakrabarti 2011), in this paper, we derived the analytical form of the adiabatic equation of state for multispecies flow (e.g.,eq. 16). Using the adiabatic equation of state, we present the form of the entropy-mass flow rate for accretion (eq. 17), as well as for outflow (eq. 37). Since the e − − e + flow is not hot enough, they show only one physical sonic point, and has been shown in this paper. However, the flow with some amount of protons show formation of multiple sonic points and shocks.
Post-shock disc drives bipolar outflows. The mass outflow rate is calculated self-consistently and is found to be around a few percent. Interestingly, the mass outflow rate is also generally higher for flows with ξ ∼ 0.27. Since, the shock forms at a larger distance for flows with a higher angular momentum, the mass outflow rate is also generally lower for high angular momentum flow. The mass outflow rate depends on many factors, the shocks location, the angular momentum, energy or composition of the accretion flow. If the shock location is smaller (say for lower angular momentum), the compression ratio is larger, and the mass outflow increases. But we have shown that the mass outflow rate do not increase monoton-ically with the compression ratio, rather the mass outflow rate peaks at some intermediate value.
We also plotted the parameter space for the shock in presence of the mass loss. Due to the loss of matter, the post-shock pressure is reduced, and therefore the entire range of parameters for which steady shock condition is satisfied (the so-called shock-domain) for flow without mass loss, also shrinks. Hence there is a possibility of mass loss induced shock instability too. Since shocks do not form for ξ = 0.0 or e − − e + flow, purely e − − e + jet solutions are not found, although lepton dominated (0 < ξ < 1) jets are possible. It must be noted that a purely e − − e + flow is essentially a flow described by fully ionized single species EoS. At temperatures T > 10 5 the flow is likely to be fully ionized. A fully ionized flow comprised of similar particles can only be e − − e + flow. However, at the same time, an accretion disc from large distances to the horizon and made up of only the pair plasma is not likely to happen in nature. We compared e − − e + flow with other flows containing protons, only to show the stark contrast between a physically plausible flow and a flow which is not viable.
Moreover, it is to be noted that in principle, ξ should be computed from physical processes self-consistently and not supplied as a parameter. In a given disc, ξ must be a function of the radial distance. We are working on this question and the results will be communicated elsewhere.
ACKNOWLEDGMENT CBS acknowledges the Visiting Students Programme in ARIES, where this work was initiated.