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Radiatively and thermally driven self-consistent bipolar outflows from accretion discs around compact objects

We investigate the role of radiative driving of shock ejected bipolar outflows from advective accretion discs in a self consistent manner. Radiations from the inner disc affects the subsonic part of the jet while those from the pre-shock disc affects the supersonic part, and there by constitutes a multi stage acceleration process. We show that the radiation from the inner disc not only accelerate but also increase the mass outflow rate, while the radiation from the pre-shock disc only increases the kinetic energy of the flow. With proper proportions of these two radiations, very high terminal speed is possible. We also estimated the post-shock luminosity from the pre-shock radiations, and showed that with the increase of viscosity parameter the disc becomes more luminous, and the resulting jet simultaneously becomes faster. This mimics the production of steady mildly relativistic but stronger jets as micro-quasars moves from low hard to intermediate hard spectral states.

inating close to the black hole, the connection between growth of QPO frequency, Comptonized component and the radio flare, to the disappearance of jet activity in the soft state (no or very weak shock). Das et al. (2001) showed (Fig. 5 of their paper) that the correlation between thermally driven jets from shocked accretion discs and the spectral states from the same discs seems to follow the conclusions of observations of 10 microquasars (Gallo et. al. 2003). Since post-shock disc produces the jets therefore when shock is absent or very weak, jets disappear mimicking the soft state. But thermally driven jets can achieve terminal speed up to v∞ < ∼ [2{a 2 b /(γ − 1) − Φ b }] 1/2 , where a b is the sound speed at the base, which for a shocked disc is the sound speed of the post-shock disc, and Φ b is the gravitational potential at the jet base. Now the maximum sound speed physically possible is amax = c/ √ 3, and that the value of Φ b is quite significant because the jet base is rather close to the horizon (Junor et al. 1999). Given these facts, the expression of terminal speed given above indicates that, truly relativistic jet terminal speed is not possible by only thermal driving. Furthermore, jet states are correlated with the hard spectral states of the accretion disc, therefore, can the disc radiation accelerate the outflowing jet material? In this respect, one may raise the issue that jet activities are not always observed during the state transition, which might be due to the lack of availability of simultaneous X-ray and radio/infra-red measurements. However, Gallo et. al. (2003) studied about 10 sources to draw the correlation between jet activity with the hard states, and the conclusion that jets are not seen during the canonical soft states. We would therefore like to study the interaction of radiations from the hard/hard intermediate states with the emanating jet, or in our parlance, the acceleration of jets with the radiation from a shocked disc, and how this radiative acceleration of jets is influenced with the change in viscosity parameter of the disc. It is to be remembered that, since we are investigating the disc in steady state, we are actually studying the generation and acceleration of steady, mildly relativistic jets generally associated with hard to hard intermediate states. The strong radio flares associated with relativistic blob ejections during hard intermediate to soft intermediate state transition (Miller-Jones et al. 2012) is intrinsically a time dependent phenomenon and has not been addressed in this paper. Since we have not incorporated all the physical aspects to mimic the transient phenomena, in this limited sense the results depend on the model assumptions.
Interaction of the disc radiation with the outflowing matter has been followed by many authors. Icke (1980) studied radiative acceleration of jets above a Keplerian disc, but in absence of radiation drag. Sikora & Wilson (1981) showed that radiation drag is important for jets powered by radiation from a thick accretion disc (Lynden-Bell 1978). Including radiation momentum deposition on an axial particle jet illuminated by an infinite thin disc in presence of drag term, Icke (1989) showed that the upper limit of the terminal speed is around 45% of c. Investigations on radiatively driven jets were extended by Fukue (1996); Fukue et al. (2001) to generate relativistic terminal speeds for particle jets. Moreover, the interaction of radiations from shocked accretion disc and jets were studied too (Chattopadhyay & Chakrabarti 2000Chattopadhyay et al. 2004;Chattopadhyay 2005). Since shocked discs have two radiation sources, namely, the post-shock and pre-shock disc, therefore the redistribution of radiative power amongst these sources, could efficiently accelerate the jet around its axis. However, jets studied in the above mentioned papers were decoupled from the disc. In this paper we would like to study radiatively driven jet which is generated self consistently from the underlying accretion disc. In other words, we combine the accretion-ejection physics described by Kumar & Chattopadhyay (2013) with the effect of radiative momentum deposition on the jets as shown by .
In the next section, we present the simplifying assumptions and equations of motion. In section 3, we present the methodology of solution. And in the last section we present the solutions and discussion.

EQUATIONS OF MOTION AND ASSUMPTIONS
In this paper, we have assumed the axis-symmetric disc-jet system to be in steady state. The black hole is assumed to be nonrotating described by the pseudo-Newtonian potential introduced by Paczyński & Wiita (1980). The viscosity prescription in the accretion disc is described by the Shakura-Sunyaev α viscosity. The jets are tenuous and should have less differential rotation than the accretion disc, as a result the viscosity in jets can be ignored. To negate any resulting torque, the angular momentum at the jet base is assumed same as that of the local value of angular momentum of the disc. We estimate the radiative moments, namely, the radiative energy density, flux, and radiative pressure from the disc. The jets being optically thin are subjected to radiation force including the drag terms. Figure 1, shows the schematic diagram of the disc-jet system, where the pre-shock, post-shock disc and the jet are clearly marked. The black hole is at BH. The accretion disc occupies the space on or about the equatorial plane, and the jet flow geometry is about the axis of symmetry. We have used the geometric unit system where, 2G = M = c = 1 (M is the mass of the black hole, G is the Gravitational constant). Therefore, in this representation the unit of length is, the Schwarzschild radius or rs = 2GM/c 2 , the unit of mass is that of the central black hole M , and that of time is rs/c = 2GM/c 3 , respectively, consequently the unit of speed is c. Figure 1. Cartoon for disc-jet system. BH is the acronym for black hole. Both pre-shock and post-shock disc shines radiation on the bipolar jet.The shock location xs, the axis (vertical dotted) and equatorial plane (horizontal dotted) are shown. The half height h(x), and the opening half angle θs of the funnel like region between the inner torus is shown.

Equations of motion for accretion and ejection system
Since we are investigating self-consistent accretion-ejection system, we should present the governing equations together. However, since the accretion and jets follow different flow geometries, we present them separately. In section 2.1.1, we present equations of motions, which when solved gives us the accretion solution. In section 2.1.2, the equations governing the jet solution are presented. And finally in section 2.2, we present the solution procedure, i.e., how a jet solution is simultaneously obtained from accretion disc through shocks.

Equations governing the accretion disc
The equations of motion of the accreting matter around the equatorial plane, in cylindrical coordinates (x, φ, z), have been extensively presented in the literature (Chakrabarti 1996;Kumar & Chattopadhyay 2013). Here we give a brief account of them.
These equations are the radial momentum equation, the accretion rate equation for a disc in vertical equilibrium, the angular momentum conservation equation and the entropy generation equation, and they are The local variables u, p, ρ and λ in the above equations are the radial velocity, isotropic pressure, density and specific angular momentum of the flow, respectively. Here, Σ = 2ρh is vertically integrated density with h being the local disc height from equatorial plane and W xφ is the viscous stress tensor. Moreover, s, T are the entropy density and the local temperature respectively. The local heat gained (due to viscosity) and lost by the flow are given by Q + = W 2 xφ /η and Q − . The viscous stress, the local half height and the sound speed are given by where, η = ρνh is the dynamic viscosity coefficient, ν = αa 2 /(γΩ k ) is the kinematic viscosity, α is the Shakura-Sunyaev viscosity parameter, Ω, Ω k and γ are the local angular velocity, local Keplerian angular velocity and adiabatic index, respectively. Here, all the variables have been made dimensionless by employing the unit system mentioned above section 2.1, e.g., ifr,ũ etc are dimensional radial coordinate and velocity, then x =r/rs, u =ũ/c etc. Integrating equation (1) with the help of Eq. (2-4), we find another constant of motion called the specific grand energy (Gu & Lu 2004) of the flow and is given by where, a, λ0 are local sound speed and the specific angular momentum of the flow on the horizon which is one of the constants of integration.
Using the adiabatic law, the definition of sound speed, and the mass-accretion equation (Eq. 2), entropy-accretion rate is definedṀ where n = 1/(γ − 1).Ṁ is the measure of local entropy and is constant for an inviscid and adiabatic flow. We rearrange and simplify Eqs.
(1-4) to the following forms (see, Kumar & Chattopadhyay 2013 for details), Since accreting matter at x → ∞ is subsonic, and close to the horizon it is supersonic, at some point in between, the flow would become transonic, where du/dx → 0/0. This gives the so-called sonic point conditions or critical point conditions. And at x = xc or sonic point, the gradient is calculated by employing L'Hospital rule. The accretion solutions are obtained by solving Eqs. (8) with the help of sonic point conditions.

Jet equations of motion
The jet streamline is approximated from the numerical simulations (Molteni et al. 1996a), where the jet flows through the funnel wall (FW) and the centrifugal barrier (CB). A detail jet geometry has been described in Kumar & Chattopadhyay (2013). All the jet variables in the equations below have been made dimensionless by dividing the distances with rs, velocities with c, mass with M etc.
The disc photons accelerate the jet by depositing the radiation momentum onto the jet. Since the jet is optically thin so the radiation actually penetrates the jet. So every fluid parcel of the jet is accelerated by the radiation from behind, but the radiation field ahead also drags and decelerate the parcel. The momentum balance equation of the jet, correct up to first order in velocity, is given by (Mihalas & Mihalas 1984; vj dvj dr Here, Fr j = σT Fr j /mec, Er j = σT Er j /me, and Pr j = σT Pr j /me, where, σT is the Thompson scattering cross section, me is the electron mass, and Fr j , Er j , Pr j are radiative flux, radiative energy density and radiative pressure, respectively. The streamline is computed as rj = x 2 j + y 2 j . The first term in r. h. s of Eq. (9) is the acceleration term, while negative terms are the radiative drag term. All the terms with the suffix 'j' represents jet quantities. If Fr j = Er j = Pr j = 0, then Eq. (9) can be integrated to give the Bernoulli equation for jet, For purely thermally driven flow Ej is a constant of motion, however, for a radiatively driven flow Ej is a variable, as we will show later (Figs. 4e, 5e). The jet outflow rate is given bẏ where A is the cross-sectional area of the jet. Since absorption and emission terms in the jet is zero, therefore, it is described by the polytropic equation of state (pj = Kj ρ γ j ). The entropy accretion rate (from Eq. 11) for the jet is given bẏ Mj is a measure of entropy of the jet, and is constant along the jet since heating and cooling is ignored. Equations (9,12 ) are simplified, to obtain, where, and, The sonic point (rjc) condition for jet is obtained for the condition rj → rjc, dvj /dr → 0/0, where, The gradient of jet velocity on the sonic point is obtained by L'Hospital rule To obtain the jet solutions one has to integrate Eq. 13, with the help of Eqs. (16)(17). In this paper we have chosen the adiabatic index to be γ = 1.4, and has been shown to be the typical value close to the black hole (Chattopadhyay & Chakrabarti 2011).

The method to calculate self-consistent accretion-ejection solution
The jet solution is dictated by the accretion solution, and now we discuss how we obtain the self-consistent accretion-ejection solution. The accretion solution depends on the following constants of motion E, λ0, and the viscosity parameter α. Therefore, our first step is to obtain the accretion solution.
Step 1: Initially we assume no jet i.e.,Ṁout = 0 and integrate Eqs. (1-4) i.e., Eqs. (8) outwards and find the inner sonic point (xci) iteratively by choosing appropriateṀ and simultaneously checking for the sonic point conditions (Eqs. 16-19, of Kumar & Chattopadhyay 2013). The first hurdle to obtain the accretion solution is that although flow parameters are regular but there is a coordinate singularity on the horizon. However, the asymptotic values of u(x), and λ(x) very close to the horizon at xin, are obtained for appropriate values ofṀ (Becker et al. 2008), and are given by, and γ+1 and the free fall velocity in the pseudo-Newtonian potential is given by u f f (x) = 1/ (x − 1). Without any loss of generality we choose xin = 1.01, and with these asymptotic values at xin we start the integration of Eqs. 8 and find the sonic points of the accretion solution. All kinds of solutions can be obtained, ranging from solutions passing through one sonic point (ADAF type and for low angular momentum flow BONDI type, see Kumar & Chattopadhyay 2013), to flow solutions harbouring multiple sonic points. Only flows harbouring multiple sonic points can have shocks. Once a flow passes through inner type sonic points xci, we integrate the flow equations while checking for the shock conditions (Eq. 20-22) and sonic point conditions to calculate shock location (xs) and outer sonic point (xco) simultaneously. The shock conditions are, and the mass fluxṀ where + and − represents the post-shock and pre-shock quantities, respectively and W is vertically integrated pressure. Here, R = u−/u+ = Σ+/Σ− is the compression ratio at the shock. If there is no shock then R = 1, and otherwise R > 1. As has been mentioned above, the initial accretion solution is obtained by putting Rṁ = 0.
Step 2: After we obtain the accretion temperature and number density distribution from the accretion solution, we compute the radiative moments Erj, Fr j , and Pr j as is discussed in section 2.2.1. These variables are used in jet equation.
Step 3: The extra compression in the post-shock flow drives bipolar outflows, and the entire post-shock disc participates in jet generation (Molteni et al. 1994(Molteni et al. , 1996aNagakura & Yamada 2009;Nishikawa et al. 2005). The jet base is the post-shock disc, so the jet is launched with the post-shock disc quantities. If x = x b is the jet launch site in the post-shock disc, then the specific energy of the disc at x b i.e., E b = E (x b ), and the post-shock specific angular momentum at x b i.e., λ b = λ(x b ) and the , are plotted with log(r j ), and (c) Er j ps/J0 (solid, online black), Fr j ps/J0 (dotted, online blue) and Pr j ps/J0 (dashed, online red), the moments of the radiation field due to the pre-shock disc are plotted. (d) Total moments of radiation field due to post-shock and pre-shock disc, are generated by an accretion disc with parameters E = 3.5 × 10 −3 , λ 0 = 1.5435, α = 0.02, γ = 1.4 (forming shock at xs = 15.18). The radiation field is for equal post-shock and pre-shock luminosities ℓs = ℓps = 0.4. density at the top of the disc at x b i.e., ρ b , are the flow variables of the jet at its base. Here we have taken x b = (xci + xs)/3, and since the disc is assumed to be in hydrostatic equilibrium along the z direction, ρ b = ρ(x b )exp(−2y b /hs), where, y b is the height of the disc at x b , and hs is disc height at shock location. The jet velocity v jb and the sound speed a jb at the jet base are not arbitrarily assigned, but are computed self consistently. We eliminate aj from Eq. (12) and Eq. (10), and express E b in terms ofṀj, v jb , λ b etc. Since E b and λ b are obtained from the post-shock disc and therefore known for the jet, we iterate with various values of entropy i.e.,Ṁj to obtain the correct value of v jb and a jb , with which the unique transonic solution is determined by integrating Eq. (13) and checking for the sonic point conditions Eqs. (16,17). Once the transonic jet solution is obtained, then Rṁ is easily calculated from Eq. (22).
Step 4: Now the information of Rṁ is used in the shock condition (Eqs. 20 -22) and then go back to Step 1 to recalculate the new accretion solution, the new radiative moments (Step 2), and the new jet solution (Step 3). This process is repeated till the shock location converges to a value (xs), and we find out the self-consistent, converged accretion-ejection solution.
Therefore, there are three iteration processes, namely, (1) to find out inner sonic point of accretion disc, (2) accretion shock and the outer sonic point, and (3) the jet sonic point when launched with the post-shock disc variables.

Radiative moments
Now as the jet is generated due to compression as well as shock heating from the post-shock disc (see Fig. 1), the jet will be moving through the radiation field of the accretion disc. We now outline the method to compute radiative moments and the net radiative acceleration of the jet. The radiative acceleration term is proportional to the radiative flux Fr j , but the radiation drag term (negative terms in r. h. s of Eq. 9) depends on jet flow velocity vj , the radiation energy density Er j and the radiation pressure Pr j . Figure 1 shows the schematic diagram of the radiations coming from various parts of the disc, and how it can interact with the outflowing jet, generated due to shock heating of the inner disc. The radiative moments along the jet streamlines (rj) are calculated from the post-shock (s) and pre-shock (ps) disc respectively. The radiative terms in Eq. 9 are given by, Pr j = σT mc IsrjrjdΩs + IpsrjrjdΩps = σT m Pr j s + Pr j ps = Pr j s + Pr j ps (25) Since the jet streamline is close to the axis of symmetry, we calculate the radiative moments on the axis and approximate these to hold at the same radial distance on the jet streamline. In this paper, we are not dealing with the detailed features of the radiation spectrum from the accretion disc. Rather, we are interested to see the effect of the total pre-shock and postshock radiation on the acceleration of the jet. Hence, we do not include radiative transfer dynamically into the hydrodynamic solution. To calculate the moments of radiation due to the pre-shock disc, as an example, we consider only synchrotron processes from the pre-shock disc. The synchrotron emissivity is due to the presence of stochastic magnetic field, where the magnetic pressure (pm) is in partial equipartition with the gas pressure (pg), i.e., here, for β = 0 implies no magnetic field and therefore the total pressure in Eq.
(1) p = pg, while β = 1 implies strict equipartition between gas and magnetic pressure and therefore p = pg + pm. The analytical expression for synchrotron emissivity is given by (Shapiro & Teukolsky 1983), and the resulting intensity is, where, Θ, ne, h and θps are the pre-shock local dimensionless temperature k b T /(mec 2 ), electron number density, disc half height and angle from the axis of symmetry to the pre-shock disc surface, respectively. The dependence of Ips on disc radius is through flow variable like T & ne. Integrating Ips over the pre-shock disc gives us the pre-shock luminosity ℓps. The radiative intensity from the post shock disc is Is = ℓs/As, where ℓs is the post-shock luminosity in units of Eddington luminosity and As is the total surface area of the post-shock disc. The total luminosity is ℓ = ℓs + ℓps. The moments of the radiation field above the accretion disc was calculated before (Chattopadhyay & Chakrabarti 2000Chattopadhyay et al. 2004;Chattopadhyay 2005), and for the above mentioned approximations, they are given by, where, θs = tan −1 (xs/hs), hs = (2/γ)as+x 1/2 s (xs − 1) and θps = tan −1 (x/h) for pre-shock disc, i.e., for x > xs. Moreover, and, J0 = 2.93 × 10 34 e 4 µ 2 βσTṁ 2 3πm 4 e c 2 secθpsγ 5/2 G 2 M 2 ⊙ where,ṁ is the accretion rate in units of Eddington accretion rate, σT is Thomson scattering cross section, µ is mean molecular weight of the plasma, mp is the proton mass, me is the electron mass and M⊙ is the solar mass. It is to be noted that the pre-shock radiation would depend on the product βṁ 2 . It is interesting to know that, the post-shock region by virtue of its geometry will block some of the pre-shock photons to the base of the jet, an effect coined as the shadow effect of the post-shock disc (Chattopadhyay et al. 2004;Chattopadhyay 2005). That is to say, if the height of the jet is yj < y jl , then Er j ps = Fr j ps = Pr j ps = 0, where here, hinj & xinj are height and radius at outer edge of the disc. Additionally due to the shadow effect, the inner edge of the pre-shock disc as seen by an observer at some height yj will be, In this paper, we use βṁ 2 as a supplied parameter to calculate ℓps. Also we use ℓs initially as a parameter (except Fig. 8 and Fig. 9) in order to understand the effect of the relative proportions of radiation coming from different parts of disc on the ejected jet, but finally we will use the relation ℓs/ℓps as a function of xs (see Appendix), to compute ℓs from the pre-shock radiation, and solve the accretion-ejection solution.
In Figs. 2a-d, all the plots are generated for a disc characterized by (E, λ0, α, γ) = (3.5 × 10 −3 , 1.5435, 0.02, 1.4), which generates a shock at xs = 15.18. In Fig. 2a, Ips is plotted as a function of x, starting from xs outwards. In Fig. 2b, we plot Er j s/K0 (solid, online black), Fr j s/K0 (dotted, online blue) and Pr j s/K0 (dashed, online red), the radiation moments due to the post-shock disc. In Fig. 2c, we plot Er j ps/J0 (solid, online black), Fr j ps/J0 (dotted, online blue) and Pr j ps/J0 (dashed, online red), the radiation moments due to the pre-shock disc. The shadow effect of the post-shock disc is clearly shown in the figure. In Fig. 2d, we plot Er j (solid, online black), Fr j (dotted, online blue) and Pr j (dashed, online red) taking equal post-shock and pre-shock luminosities, ℓs = ℓps = 0.4, respectively. From the above figures it is clear that radiative moments from the post shock region peaks at a height closer to the black hole and the moments from the pre-shock disc peaks typically at a distance few times larger. However, the space dependent parts of the moments (Figs. 2b-c) shows that the moments of radiation from the pre-shock disc is typically an order of magnitude weaker than those due to the post-shock disc. However, the second distinct bulge in Fig. 2d shows, if the pre-shock luminosity is comparable to post-shock luminosity then the radiation moments peaks at two places, and hence presents a prospect for multi-stage acceleration scheme for the jets.

RESULTS
In Kumar & Chattopadhyay (2013) we have shown in details various cases of accretion solutions, and also have shown that post shock disc naturally produces bipolar outflows. In this paper, therefore we discuss only the shocked accretion solution.
The accretion solutions are characteristic by flow parameters like the grand energy E, the specific angular momentum at the horizon λ0 (conversely, λinj at the outer edge of the disc) and the viscosity parameter α. The jet is launched from the disc with the specific energy (Ej and angular momentum (λj ) of the post-shock disc at the jet launch site (r b ). We also compute the moments of radiation fields (Er j , Fr j , Pr j ) above the disc, and are used to accelerate the jets. The pre-shock disc radiation depends on βṁ 2 , where, β is the ratio of the magnetic and the gas pressure andṁ is the accretion rate in units of Eddington rate. In order to find the effect of post-shock radiation and pre-shock radiation on jet acceleration, the post-shock luminosity ℓs and βṁ 2 (conversely, ℓps the pre-shock luminosity) are supplied as independent parameters. However, since post-shock radiation is produced by inverse-Comptonization of self generated and intercepted photons from the pre-shock disc, we compute the post-shock radiation self-consistently by employing the techniques of . And in Figs. (8, 9, 10) the jets are accelerated by self consistent estimation of radiation field, both from pre and post-shock discs.

Effect of post-shock radiation on jet acceleration
The radiations produced from accretion is governed by the solution. However, we will now treat the post-shock luminosity as a parameter just to see how it affects the jet acceleration. The accretion Mach number M = u/a with log(x) is plotted for the parameters E = 3.3 × 10 −3 , λ0 = 1.353, α = 0.1, in Fig. 3. We first consider ℓs = 0 and βṁ 2 = 0.01, the shock is at xs = 21.546 (vertical dashed, online black), and drives bipolar outflow, where the jet Mach number Mj (dashed, online black) is plotted with log(rj), and the mass outflow rate is Rṁ = 0.032 and the sonic point of the jet is at rjc = 24.160. Radiation field from the accretion disc deposit its momentum and accelerate the shock generated jets. Assuming ℓs = 0.1 for same set of accretion boundary condition, accretion-ejection solution is computed. The shock is found to be at xs = 17.92 (solid, online blue) and the jet Mj distribution (solid, online blue) has a sonic point at rjc = 9.59 and Rṁ = 0.046. The total luminosity for this case is ℓ = (ℓs + ℓps) = 0.151 (ℓs = 0.1). Keeping βṁ 2 same, we increase ℓs = 0.2, so the total luminosity increases to ℓ = 0.274, The accretion-ejection solution shows that the mass outflow increases to Rṁ = 0.058 and the shock in accretion decreases to xs = 14.4 (dotted, online red). In this case too, due to the increase of radiation, the sonic point decreases to rjc = 6.36, indicating stronger jet. It is interesting to note that, the pre-shock luminosity increases even though βṁ 2 is kept constant, because, with the decrease in xs, the size of the pre-shock disc increases. The jet solutions are explored in more details in the following figure. Jet variables aj (Fig. 4a),Ṁj (Fig. 4b), vj (Fig.  4c), ρj (Fig. 4d), Ej (Fig. 4e) and Tj (Fig. 4f) are plotted with log(rj). Each curve corresponds to ℓs = 0 (dashed, online black), ℓs = 0.1 (solid, online blue), and ℓs = 0.2 (dotted, online red), and which are exactly the same cases of jet solutions as in Fig. 3. The increase in ℓs, accelerates the jets further, and therefore increases Rṁ, this in turn decreases the post-shock pressure and the location of the shock front moves close to the horizon. As a result the jet base moves closer. At a given rj, we find vj is higher for higher ℓs, but difference in aj or Tj are imperceptible. This shows, that enhanced jet acceleration is due to the radiative momentum deposition onto the jets and not due to conversion of thermal energy to the kinetic one. Since higher ℓs accelerate the jet, the sonic point is formed closer to the horizon. Higher ℓs not only means faster jet, but also a jet with higher entropy (Fig. 4b).
The jet terminal speed is defined as v∞ = v (at rj → large) where dvj /drj → 0. As shown in Figs. 4, v∞ increases appreciably with the increase of ℓs and ℓps, for a given value of disc viscosity parameter α. We would like to see whether this behaviour of v∞ holds true for a range of α. In Fig. 5a, v∞ is plotted with ℓs, in Figs. 5b & 5f, v∞ is plotted with xs and the jet base specific energy E b . And Rṁ is plotted with ℓs (Fig. 5c) and xs (Fig. 5d). Each curve correspond for α = 0.04944 (solid, online red), 0.049887 (dotted, online blue), and 0.050113 (dashed, online black). All these figures are generated with outer boundary condition E = 0.001 and λinj = 70.6 at xinj = 10 4 . It is clear that v∞ increases with ℓs at a given value of α, as well as, increases with α at a given ℓs. Since xs moves closer with the increase of both ℓs and α, the jets are launched with higher E b , which in turn increases v jb . This is also the reason that the relative mass outflow rate Rṁ increases with the increase of both ℓs and α. Since v jb increases with ℓs and α, therefore the jets become supersonic at a distance nearer to the jet base, i.e., rjc decreases with the increase of ℓs (Fig. 5e). Rṁ is plotted with ℓs (c) and xs (d). Moreover, r jc is plotted with ℓs (e) too. The accretion solution corresponds to disc parameters E = 0.001 and λ inj = 70.6 at disc outer boundary x inj = 10 4 and keeping pre-shock luminosity is ℓps = 0.0. Each curve correspond for α = 0.04944 (solid, online red), 0.049887 (dotted, online blue), and 0.050113 (dashed, online black).

Effect of pre-shock radiation on jets
The effect of radiation from pre-shock disc as it impinges on the jet is illustrated through Figs. 6b-f. In Fig. 6a, the Mach number M of the accretion solution is plotted with log(x) for disc parameters E = 3.3 × 10 −3 , λ0 = 1.353, and α = 0.1. The shock is at xs = 17.92. We choose ℓs = 0.1, but the pre-shock radiation is changed by varying βṁ 2 . The solutions correspond to βṁ 2 = 0.01 (ℓ = 0.151) (solid, online red), βṁ 2 = 0.02 (ℓ = 0.202) (dotted, online blue), and βṁ 2 = 0.04 (ℓ = 0.303) (dashed, online black). The post-shock disc actually hides the base of the jet from most of the radiation from the pre-shock disc, while shines its own light onto the jets (see Fig. 1). As a result, if the post-shock radiation remains unaltered and the jet sonic point is formed in the portion of the funnel like region where pre-shock radiation is negligible, then the jet base velocity v jb , the jet base r jb or the jet base properties are likely to remain roughly same, keeping the massloss rate unaltered. Consequently, the change in the accretion shock is imperceptible (Fig. 6a). All the jet variables closer to the base, e.g., Mj (Fig. 6b), vj (Fig. 6c), Ej (Fig. 6e), and Tj are indistinguishable, while they differ from each other in the supersonic region, where the interaction of pre-shock radiation with the jet is significant too. It is to be remembered, that the temperature plotted here is the single temperature of the outflow. The corresponding electron temperature should be about 2 orders of magnitude less. However, the entropy accretion ratesṀ (Fig. 6d) are distinguishable even at the base. Once again it is clear from the temperature plot, that radiative driving is significant. It is also interesting to note from Figs. 3-4, that increasing ℓs, would result in faster jets, with higher Rṁ and lower jet sonic point (rjc). While increasing the pre-shock radiation also results in faster jets, but with almost no change in Rṁ and rjc. Since relative mass outflow rates affects the accretion solutions (Eq. 22), so the feed-back effect of the jet on the disc due to ℓs might be more significant than that due to ℓps.
In Figs. 7a-d, we investigate how ℓps affects the jet solutions for a variety of α, but for the same outer boundary condition as in Fig. 6 with constant ℓs = 0.2. In Fig. 7a, we plot v∞ as a function of ℓ, where each curve represent disc solutions with α = 0.04944 (solid, online red), 0.049887 (dotted, online blue), and 0.050113 (dashed, online black). In all these plots ℓps varies from 0 → 0.6. Due to radiative driving, v∞ increases with ℓ, and at a given ℓ, it increases with α. As has been explained in connection to the previous figure, increasing α for a fixed outer boundary decreases xs, which means the jet base energy E b increases, resulting in faster jet. However, since the pre-shock disc primarily shines radiation on the supersonic part of the jet, therefore ℓps has marginal effect on rjc, and v jb . Therefore, Rṁ is almost constant with the change of ℓps (Fig. 7d), which in turn keeps xs almost unchanged (Fig. 7b), and the jet sonic point rjc also remains unchanged (Fig. 7c). In other words, we may conclude, that the radiation from the inner torus of the accretion disc accelerate the jet, but also increases net mass-loss. On the other hand, radiation from pre-shock disc or the outer disc, accelerates the jet appreciably, although, has almost no effect on Rṁ. Therefore, in the second case we may obtain jets with higher kinetic luminosity. This conclusion is valid for any value of α which admits accretion shock.

Radiative driving of jets with computed post-shock and pre-shock radiations
Having investigated the role the pre-shock and post-shock radiation may have on jets, we actually calculate the ℓs from ℓps. In appendix A, we have discussed the different radiation processes in a general radiative transfer model . From the accretion disc spectrum, we calculate the ratio of post-shock to pre-shock luminosity (ℓs/ℓps) using the viscous transonic solution. We then use a fitting formula of ℓs/ℓps (Eq. A1) to calculate the ratio at any given shock location (xs). In (Fig. A1b), a typical ℓs/ℓps dependence on xs is obtained where the dots are the data points from model  and solid line represents the fitting function. We assume that the behaviour of this ratio with shock location is generic. The procedure for calculating ℓs is as follows -for a given set of values of E, λ0, α, the accretion solution shows a shock at xs and a bipolar outflow with some Rṁ. We then calculate ℓps (Eq. 26) by supplying βṁ 2 and the density and temperature distribution between xinj and xs. We use the fitting formula of ℓs/ℓps (Eq. A1) to calculate ℓs. Using these the jet solution is obtained. All the solutions presented in Figs. (3-7), we have solved the accretion ejection solution with the following fluid parameters E, λ0 (at the horizon, or equivalently, λinj at the outer boundary), α, and in addition the radiation parameters ℓs and βṁ 2 (equivalently ℓps). Now, following the procedures described in appendix A, we reduce one parameter, namely, ℓs. In Fig. 8a, we plot xs with λ0, in fig. 8b, we plot the compression ratio R with xs, and in fig. 8c, mass outflow rate Rṁ with the compression ratio R. All the plots are for accretion disc parameters E = 0.004, α = 0.01, and various results has been obtained by varying λ0. The curves are for thermally driven jet (i.e., βṁ 2 = 0; dashed, online red), and thermal plus radiatively driven jets (i.e., βṁ 2 = 0.005; dotted online blue, and βṁ 2 = 0.01; solid, online black). This shows that as the compression at the shock increases, it forces more matter into the jet channel. Although R increases as xs decreases, but smaller post-shock region means less matter can be driven as jets, so Rṁ maximizes at some intermediate R.
It has also been shown earlier (Chakrabarti 1999;Das et al. 2001), that for R = 1 i.e., no shock, Rṁ ∼ 0 i.e., for no shock there is no outflow. In Figs. 9a-l, we have plotted accretion and jet solutions for various α and ℓps. All the accretion solutions (M with log(x)) are for outer boundary parameters E, λinj = 0.001, 18.592 at the outer boundary xinj = 10 4 . The viscosity parameters are α = 9.115 × 10 −3 (9a), 9.315 × 10 −3 (9d), 9.626 × 10 −3 (9g), and 9.875 × 10 −3 (9j). The vertical jumps show the location of accretion shocks, and they are at xs = 125.896 (9a), 45.9986 (9d), 18.2871 (9g), and 8.5741 9(j). The jet    solutions corresponding to these accretion solutions, are presented by Mj (9b, 9e, 9h, 9k) and vj (9c, 9f, 9i, 9l). As α increases xs decreases, therefore increasing the pre-shock disc. Moreover, with decreasing xs, the post-shock disc becomes smaller and hotter. So as xs decreases, initially both ℓs & ℓps will increase, but at around xs ∼ 100, further decrease of xs will reduce ℓs/ℓps and significantly increase ℓps (Fig. A1b). In Figs. 9a, d, g, j, increase of α, causes a shift of xs = 125.896 → 8.5741. Consequently, ℓps increases from 0.0054 → 0.2337. The resulting jets are accelerated and the terminal velocity increases from v∞ = 0.0534 → 0.3519 as shock shifts from xs = 125.896 → 8.5741, with the corresponding change in luminosity. By considering the relative proportions of post-shock and pre-shock radiations, decrease of xs with increasing α resembles the disc to move from hard state to hard intermediate state, and simultaneously the jet becomes stronger and faster (both v∞ and Rṁ increases). In Fig. 10a-c, we show the comparison of shock parameter space (E − λ0) of the accretion disc without massloss (dotted, online red) and with massloss but disc parameter βṁ 2 = 0.001 (long dashed, online black) and βṁ 2 = 0.01 (dashed, online blue), and for various viscosity parameter α = 0 (Fig. 10a), α = 0.1 (Fig. 10b) and α = 0.2 (Fig. 10c). It is to be noted, that the bounded regions in E − λ0 parameter space, show the parameters for steady state shocks to occur, but non-steady shocks still exist outside the bounded region. The parameter space shrinks when massloss is considered, because with massloss, the post-shock pressure decreases, and the entire range for which steady shock may have existed in absence of jets, will not be able to satisfy the momentum balance across the shock front. Moreover, shocks seem to exist for fairly high viscosity and in presence of massloss.

DISCUSSION AND CONCLUSION
Our main focus in this paper has been to study radiative and thermal driving of bipolar jets from dissipative accretion disc. The thermal driving is determined by the thermal energy of the jet base or the post-shock region. In other words, hotter the post-shock region, faster will be the jet. However, the thermal energy in the inner disc though hot, but cannot produce jets with v∞ > 0.1 (Chakrabarti 1999;Chattopadhyay & Das 2007;Kumar & Chattopadhyay 2013), for any realistic temperature ranges of the accretion disc. As has been explained in section 1, only thermally driven outflows can achieve terminal speeds of the order of the sound speed at the base. Radiative driving of jet is due to the momentum deposition of the accretion disc photons on to the shock driven jet. The issue of radiative acceleration of jets have been studied earlier  Figure 10. Shock parameter space or E − λ 0 space for accretion, considered without massloss (dotted), and with massloss (dashed), and for α = 0 (a), α = 0.1 (b), and α = 0.2 (c). (Chattopadhyay & Chakrabarti 2000, however, the jet base conditions in those works were assumed and were not self consistently obtained. In this paper, we do study the radiative acceleration of jets, but also by self-consistently solving the jet solution from the accretion disc solution itself. This is an exploratory study to investigate whether or not radiative driving is meaningful acceleration mechanism for jets. Radiative moments were calculated a posteriori from viscous accretion solutions. We initially considered the post-shock luminosity (ℓs) as a parameter, and calculated the pre-shock radiation from the disc solution (a, ρ) and by supplying βṁ 2 . Similar to Kumar & Chattopadhyay (2013), in this paper too, we find that with the increase of α for disc solutions starting with same outer boundary condition, the shock location moves towards the horizon. As xs decreases for fixed values of ℓs, the jets are stronger and faster since the post-shock intensity increases (Figs. 6). For fixed values of α, if ℓs is increased, the jet is accelerated, as well as, more matter flows into the jet channel. The depleted post-shock flow causes xs to decrease, in order to regain pressure balance across the shock. Since enhanced ℓs increases the base velocity or v jb , the jet sonic point decreases too.
The effect of pre-shock luminosity is quite different. Although pre-shock luminosity can accelerate as efficiently, as the post-shock radiation, however, since pre-shock radiation can 'see' mostly the supersonic branch of the jet, so neither Rṁ nor rjc is affected appreciably. Therefore, one may conclude that post-shock radiation both accelerates and controls the mass outflow rate, pre-shock radiation increases the kinetic energy of the jet. And since for both pre-shock and post-shock radiations 2Fr j ∼ (Er j + Pr j ) at rj ∼ few×100rg , radiation drag effect is nullified at around that distance (Chattopadhyay et al. 2004;Chattopadhyay 2005). Which means faster jets can be obtained if enough radiation power can be supplied. And indeed we obtained terminal speeds of v∞ > ∼ 0.4 for total disc luminosity ℓ ∼ 1. Although we have used ℓs 0.5 (which is acceptable for luminous sources), but assigning arbitrary values of ℓs may give wrong results, because ℓs crucially depends on pre-shock radiation. So in Appendix 1, we have estimated ℓs/ℓps(xs) from correct radiative losses from a set of solutions of advective disc. We used this relation to estimate ℓs, for given values of ℓps and xs. We then increased α for disc solutions starting with same E & λinj , and calculated simultaneous disc-jet solution. We showed that as α is increased, xs decreases and for reasons explained above, not only the disc moved from low intensity disc to brighter disc, but the jet became stronger and faster. Although, we have not considered the Keplerian disc component in our solution, but it resembles qualitatively, the transition of the disc from hard to intermediate hard states with associated strengthening of the jet from slow jets to faster and stronger jet, as has been reported in observations. We are still to include a few more physical processes in our disc-jet model, but our results qualitatively shows the correlation of disc spectral states and the jet states, as are observed in micro-quasars. Moreover, our result also indicates that, with multi-stage acceleration mechanism, truly relativistic jets from accretion disc are a distinct possibility. Figure A1. Variation of (a) shock location(xs) with disc viscosity parameter(α), and (b) ratio of ℓs/ℓps with shock locations, and (c) the photon index Γ for the same injection values E = 0.001, λ inj = 18.592 at disc outer boundary, x inj = 10 4 but varying viscosity parameter(α) from 0.0091 to 0.01. In plot (b) dot points are actual data points and solid line is the fitted polynomial.