Radiatively driven general relativistic jets
aa r X i v : . [ a s t r o - ph . H E ] J a n J. Astrophys. Astr. (2016) : / s78910-011-012-3 Radiatively driven general relativistic jets
Mukesh K. Vyas & Indranil Chattopadhyay Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital-263002, India * Corresponding author. E-mail: [email protected]
MS received ; revised ; accepted
Abstract.
We use moment formalism of relativistic radiation hydrodynamics to obtain equations of motion ofradial jets and solve them using polytropic equation of state of the relativistic gas. We consider curved space-timearound black holes and obtain jets with moderately relativistic terminal speeds. In addition, the radiation fieldfrom the accretion disc, is able to induce internal shocks in the jet close to the horizon. Under combined e ff ect ofthermal as well as radiative driving, terminal speeds up to 0.75 (units of light speed) are obtained. Key words.
Radiation hydrodynamics, Hydrodynamics, Shocks, Black holes, Jets and outflows
1. Introduction
Jets are ubiquitous in astrophysical objects like activegalactic nuclei (AGN e.g., M87), young stellar objects(YSO e.g., HH 30, HH 34), X-ray binaries ( e.g.,
SS433,Cyg X-3, GRS 1915 + et al. etal. et al. r s ) around the central object(Junor et al. . 1999; Doeleman et al. . 2012). Thisimplies that entire accretion disc doesn’t take part injet generation. Following this, we assume that jets arelaunched within the accretion funnel close to the BH.Further, numerical simulations (Molteni et al. et al. et al. et al. . 2013;Kumar & Chattopadhyay 2017) showed that additionalthermal gradient term in the accretion corona is able togive rise to bipolar outflows close to the BH.After the very first theoretical model of accretiondiscs (Shakura & Sunyaev 1973) being Keplerian in na-ture, there have been numerous attempts to understandthe interaction of radiation with jets (Icke 1980; Sikora& Wilson 1981; Paczy´nski & Wiita 1980; Fukue 1996;Chattopadhyay & Chakrabarti 2000a, Chattopadhyay & Chakrabarti 2000b, Chattopadhyay & Chakrabarti2002, Chattopadhyay et al. et al. (1985) studied isothermal and non-radial fluid jets under Newtonian gravity having arbi-trary radiation field in special relativistic regime. Theyobtained mildly relativistic jets and shocks induced bynon radial nature of the cross section. Isothermal as-sumption does not contain the e ff ect of the thermal gra-dient term which is a significant accelerating agent andis very e ff ective close to the BH. It is also the same re-gion where one needs to consider the e ff ects of generalrelativity as well. In this series Vyas et al. (2015) stud-ied special relativistic jets under radiation field consid-ering pseudo-Newtonian gravitational potential. Theyobtained relativistic jets but no multiple sonic pointsor shock transition was obtained. This paper extendsthe work of Ferrari et al. (1985) and Vyas et al. (2015)by considering fluid jets in curved space-time. Further,as Vyas & Chattopadhyay (2017) showed that non ra-dial jets, even without radiation field do create internalshocks while radial jets do not. Here we explore thepossibility that radial jets can form shocks under theimpact of su ffi ciently intense radiation field.The equations of motion of radiation hydrodynam-ics were developed by many authors (Hsieh & Spiegel1976; Mihalas & Mihalas 1984) and later their generalrelativistic version was obtained in further studies (Park et al. et al. c (cid:13) Indian Academy of Sciences 23
J. Astrophys. Astr. (2016) :
In next section 2., we present assumptions, equa-tions of motion and brief account of the procedure tocompute the radiation field. The methodology to obtainsolutions is narrated in 3.. Finally, we present resultsand draw conclusions in section 4.
Figure 1 . Cartoon diagram of cross-sections of axis-symmetric accretion disc and the associated jet in ( r , θ, φ coordinates). The outer limit of corona x sh , the intercept ofouter disc on the jet axis ( d ), height of the corona H sh , theouter edge of the disc x are marked.
2. Assumptions and governing equations
Assumptions
We invoke general relativity to take care of space-timecurvature, which around a non-rotating BH is describedby Schwarzschild metric: ds = − g tt c dt + g rr dr + g θθ d θ + g φφ d φ = − − GM B c r ! c dt + − GM B c r ! − dr + r d θ + r sin θ d φ (1)Here r , θ and φ are usual spherical coordinates, t is time, g µµ are diagonal metric components, M B is themass of the central black hole and G is the universalconstant of gravitation. Hereafter, we have used ge-ometric units (unless specified otherwise) with G = M B = c = M B , r g = GM B / c and t g = GM B / c respectivelyfor which, the event horizon is at r S =
2. The jet isassumed to be in steady state ( i.e., ∂/∂ t =
0) and as therelativistic jets are collimated, we consider on-axis ( i.e.,u r = u φ = ∂/∂ r =
0) and axis-symmetric ( ∂/∂φ = r , the physical variables of the jet remain same along its breadth. The jet is as-sumed to expand radially, perpendicular to the accre-tion plane. Further, a jet should have low angular mo-mentum else it cannot remain collimated and followingthe e ff ective angular momentum removal by radiationand magnetic fields, we assume jets to be non-rotating.The cartoon diagram of disc jet system is shown in fig-ure (1). The accretion disc has an outer disc and theinner torus like corona. The outer edge of corona andinner edge of outer disc is presented by x sh . The heightof the corona is assumed to be H sh = . x sh . Accre-tion disc works as a source of radiation emitting viasynchrotron, bremsstrahlung and inverse Compton pro-cesses along with assumption that magnetic pressure inthe disc is a fraction β of the gas pressure. We take β = . ff ects on radiation field observed are alsoincorporated. The relativistic e ff ects in the radiativetransfer explicitly appears in the equations of motionwhile the e ff ects of photon bending in radiation field areapproximated taking the help of Beloborodov (2002);Bini et al. I j f ) into curved space-time are given as I j = I j f − r a ! (2)Here r a is the radial coordinate of the source point onthe accretion disc. The su ffi x j → OD , C signifies thecontribution from the outer disc and the corona, respec-tively. The square of redshift factor (1 − / r a ) shows thatcurved space-time reduces the observed intensity.Further, as photon moves in curved path, the trans-formed expressions of the direction cosine and solid an-gle are given in terms of their flat space counterparts as(Beloborodov 2002), . Astrophys. Astr. (2016) : l j = l j f − r a ! + r a d Ω j = − r a ! d Ω j f (3)Using these transformation laws, the radiative momentsnamely radiation energy density ( R ), radiation flux ( R ),radiation pressure ( R ) and disc luminosities are calcu-lated using similar procedure as in Vyas et al. (2015).Here we have excluded radiation contribution from Ke-plerian disc as its contribution in the various compo-nents of total radiative moments was found to be negli-gible. The luminosity of the OD is obtained by integrat-ing specific intensities over the disc surface and thenusing the luminosity ratio relation between OD and C (Vyas et al. M B = M ⊙ we obtain luminosi-ties of C or corona. The total luminosity ( ℓ ) of the discthen is addition of both luminosities and is shown inunits of Eddington luminosity in this paper. We treat ℓ as an input parameter.2.2 Governing equations
Equation of state
Equation of state (EoS) is aclosure relation between internal energy density ( e ), pres-sure ( p ) and mass density ( ρ ) of the fluid. In this study,we consider the jet fluid obeying polytropic EoS havingfixed adiabatic index ( Γ = .
5) given as, e = ρ + p Γ − a in relativis-tic regime and enthalpy h are given by a = Γ pe + p = ΓΘ + N ΓΘ ; h = e + p ρ = + Γ N Θ (5)Here N ( = Γ − =
2) is polytropic index of the flow andnon-dimensional temperature is defined as
Θ = p /ρ .2.2.2 Dynamical equations of motion
In relativisticnotation the equations of motion of the any system areobtained when the four divergence of the energy-momentumtensor T αβ = T αβ R + T αβ M is set to zero. i.e.,T αβ ; β = ( T αβ R + T αβ M ) ; β = T αβ R and T αβ M stand for jet matter and radiationfield respectively and are given by (Mihalas & Mihalas1984) T αβ M = ( e + p ) u α u β + pg αβ ; T αβ R = Z I ν l α l β d ν d Ω , (7) The metric tensor components are given by g αβ , u α arethe components of four velocity, e and p the fluid en-ergy density, pressure in local co-moving frame. Fur-thermore, l α s are the directional derivatives, I ν is thespecific intensity of the radiation field with ν being thefrequency of the radiation. Ω is the solid angle sub-tended by a source point at the accretion disc surfaceon to the field point at the jet axis.In absence of particle creation / destruction, conser-vation of four mass-flux is given by,( ρ u β ) ; β = , (8)where, ρ is the mass density of the fluid. From above setof equations (eq. 6), the momentum balance equationin the i th direction is obtained using projection tensor,( g i α + u i u α ). i.e., ( g i α + u i u α ) T αβ M ; β = − ( g i α + u i u α ) T αβ R ; β (9)For an on axis jet in steady state it becomes (Park etal. u r du r dr + r = − − r + u r u r ! e + p d pdr + ρ e σ T m p ( e + p ) ℑ r , (10)Here ρ e is lepton mass density, m p is the mass of theproton and ℑ r is the net radiative contribution and isgiven by; ℑ r = p g rr γ " (1 + v ) R − v g rr R + R g rr ! (11)Here we define three velocity v of the jet as v = − u i u i / u t u t = − u r u r / u t u t = ⇒ u r = γ v p g rr and γ = − u t u t is the Lorentz factor. R , R and R arezeroth, first and second moments of specific intensity.Similarly, the energy conservation equation is obtainedby taking u α T αβ M ; β = − u α T αβ R ; β (12)In the scattering regime, it becomes dedr − e + p ρ d ρ dr = , (13)Absence of emission / absorption makes the right side ofequation (13) zero. It is a consequence of scattering J. Astrophys. Astr. (2016) : regime assumption and shows that the system is isen-tropic. From continuity equation (eq. 14) the mass out-flow rate is given as˙ M out = Ω ρ u r r ; Ω= geometric constant (14)The di ff erential form of the outflow rate equation is,1 ρ d ρ dr = − r − u r du r dr . (15)Equation (13) can be integrated with help of equa-tion (4) to obtain isentropic relation between p and ρ , p = k ρ Γ where, k is entropy constant of the flow. This equationsenables us to replace ρ from equation (14), and we ob-tain the expression for entropy-outflow rate as,˙ M = Θ N u r r (16)˙ M remains constant along the streamline of the jet, ex-cept at the shock.Integrating equation (10), we obtain generalized,relativistic Bernoulli parameter for the radiatively drivenjet, E = − hu t exp Z dr σ T (1 − Na ) ℑ r m p γ (1 − / r ) ! (17)Momentum balance equation (eq. 10), with the help ofequation (15), is simplified to γ vg rr r − a v ! dvdr = a (2 r − − + ℑ r r (1 − Na ) m p γ (18)Using energy conservation equation (13) along with theEoS (eq. 4), the expression of temperature gradientalong r is obtained to be d Θ dr = − Θ N " γ v dvdr ! + r − r ( r − (19)Equations (6) and (8), the two equations of mo-tion reduces to two di ff erential equations (18) and (19),which describes the distribution of two flow variables v and Θ .
3. Methods of analysis
The solution for radiatively driven jet can be obtainedif equations (18) and (19) are solved. Since jets arelaunched from accretion disc, close to the central ob-ject, so the injection speed will be small, while the tem-perature will be high. So at the base, jets should besubsonic. Far away from the base, jets are observed tobe moving with relativistic speed and therefore super-sonic. Hence such flows are transonic in nature. Thedistance ( r = r c ) at which the bulk speed ( v = v c )crosses the local sound speed ( a = a c ), is called thesonic point. Equation (18) shows that the sonic point isalso critical point since at r = r c , dv / dr → /
0. Thisproperty gives the sonic or critical point conditions, v c = a c ; (20)and a c (2 r c − − + ℑ rc r c (1 − N c a c ) m p γ c = ffi x c denotes that the values are to be obtained at thesonic point ( r = r c ). For a given r c , we solve equa-tion (21) to find a c and then Θ c (equation 5). We canalso compute ˙ M c , E c at r c (using equations 16 and 17).Since E c = E for a particular solution, therefore, fora given E , r c is determined and vice versa. In otherwords, sonic point is a mathematical boundary. So wefirst obtain all the variables at r c and then calculate | dv / dr | c by using the L’Hospital’s rule in equation (18)at r = r c . This leads to a quadratic equation for | dv / dr | c ,which can admit two complex roots having ‘spiral’ typesonic points, or two real roots but with opposite signs(called X or ‘saddle’ type sonic points), or real rootswith same sign (known as nodal type sonic point). For agiven boundary values at the base of the jet ( r = r b = E and ˙ M of the flow giving the values atthe outer boundary r ∞ (defined by r = r ∞ = ). Weintegrate equations (18 and 19) simultaneously inwardand outward from the r c using 4 th order RungeKuttamethod.3.1 Shock conditions
The existence of multiple sonic points in the flow opensup the possibility of formation of shocks in the flow. Atthe shock, the flow is discontinuous in density, pressureand velocity. The relativistic Rankine-Hugoniot condi-tions relate the flow quantities across the shock jump(Chattopadhyay and Chakrabarti 2011)[ ρ u r ] = , (22) . Astrophys. Astr. (2016) : [ ˙ E ] = T rr ] = [( e + p ) u r u r + pg rr ] = ff erence of quantitiesacross the shock, i.e. [ Q ] = Q − Q with Q and Q being the quantities after and before the shock respec-tively.Equation (23) states that the energy flux remainsconserved across the shock. Dividing (24) by (22) anda little algebra leads to(1 + Γ N Θ ) u r + Θ g rr =
4. Results and discussion
Nature of radiation field
Figure 2 . Radiative moments R , R and R from accretiondisc corresponding to accretion rate ( ˙ m ) to be 9 .
145 or discluminosity ( ℓ ) to be 0 .
5. Both ˙ m and ℓ are in Eddington units. In fig. (2) we show radiative moments R (solid,black), R (long-dashed, blue) and R (red, dashed) asfunctions of r calculated at the jet axis for ˙ m = . x sh = .
67 and ℓ = .
5. Thefirst peak ( < ∼
10) in the moments, is due to the radiationfrom the corona, and the second peak (55) is due tothat from the outer disc. Due to the shadow e ff ect from the post shock disc, all moments from the outer discare zero for r <
30. Since the corona is geometricallythick, the radiative flux R is negative in the funnel likeregion. The magnitude of the moments rise as the jetsees more of the disc as it propagates upward and theydecay after reaching a peak value. The moments followan inverse square law at large distances. The negativeflux in the funnel pushes the jet material downward andworks against the motion, to the extent that, it may driveshock in jets.4.2 Nature of sonic points and behaviour of flow vari-ables
10 1000.010.10.010.11
Figure 3 . Variation of Θ c (a), ˙ M c (b) and a c (c) with r c forvarious values of ℓ . Using the procedure explained in section (3.), weprovide sonic point r c and calculate physical variablesthere. In Fig. (3a, b,and c) we show variation of Θ c ,˙ M c and a c with r c , respectively. Various curves areplotted for di ff erent luminosities as ℓ = .
85 (dotted,black), 1 .
76 (dashed, blue), 0 .
80 (long-dashed, red),0 .
035 (dashed-dotted, magenta) and these all are com-pared with the thermal flow ℓ = .
00 (solid, black).Physically di ff erent sonic points mean di ff erent choicesof boundary conditions that give di ff erent transonic so-lutions, similarly, choice of an r c implies a solution witha unique choice of E and ˙ M . For all possible values of r c , thermal jets harbour real roots of a c (or, correspond-ing Θ c ). While radiation field limits the region where J. Astrophys. Astr. (2016) : a c can have real values. For r c > r c max , one obtains com-plex values of a c (equation 21). It is also found that r c max always lies inside the corona funnel ( i.e.,r c max < H sh ).Physically, the critical points where a c is found imag-inary, correspond to solutions where fluid approxima-tion breaks down or physical temperatures are not de-fined.
10 1000.1
Figure 4 . (a) Variation of three velocity v (solid black) and a (long-dashed blue), (b) ˙ M , and (c) Θ with r for ℓ = .
80. (d)Comparison of v for ℓ = . v for ℓ = E = . In Fig. (4) we show a typical nature of flow vari-ables along r for ℓ = . E = .
43. In Fig. (4a),we show variation of three velocity v (solid, black) and a (long-dashed, blue). The e ff ect of negative flux isclearly seen as v decreases inside the funnel and then itaccelerates above it. In Fig. (4b), we plot the entropyoutflow rate ˙ M which remains constant since scatter-ing is an isentropic process. Figure (4c), shows sharpdecline of temperature due to adiabatic expansion. InFig. (4d), we compare v of radiatively driven jet (solid,black), and thermally driven solution (dashed, red), bothhaving the same E . We see that radiative accelerationdominates over radiative drag and terminal speed of thejet is higher in presence of radiation field.Now in figure (5a-d) we investigate the behaviourof jet speed with di ff erent boundary conditions, i.e., di ff erentchoices of E . We choose ℓ = .
76 and plot E c with r c in Fig. (5a). For very high value of E = E c = . ff ective (Fig. 5b).The jet accelerates due to the thermal gradient term andbecomes transonic at r c = .
2. As the jet expands,the temperature decreases and radiation becomes ef-fective. The combined e ff ect of negative flux in thefunnel and radiation drag term decelerates the speed.Above the funnel radiation flux become positive andstarts to accelerate the jet and it achieves terminal speedof v T = .
76. If one chooses lower values of E = . Figure 5 . (a) Variation of E c with r c ; Variation of v with r for (b) E = .
83, (c) E = .
33 and (d) E = .
01. For all thecurves, ℓ = . r = .
4. Because the energy is low, radiation is moree ff ective. Radiation flux opposes the outflowing jet in-side the funnel more vigorously and causes a shocktransition — a discontinuous transition from supersonicbranch to subsonic branch at r = .
78 and then aftercoming out of the accretion funnel, it again acceleratesunder radiation push and becoming transonic formingan outer sonic point at r = .
77. The terminal speedachieved for this case is ∼ .
7. Here the jet crosses twosonic points with ˙ M to be higher for outer sonic point( ˙ M = . M = . E = .
01 (Fig. 5d), the ra-diation is even more e ff ective, and the jet speed is dras-tically reduced within the funnel. However, above thefunnel it is accelerated very e ffi ciently, becomes tran-sonic through a single sonic point and achieves terminalspeed of about v T ∼ . v with r for ℓ = .
85 (solid, black), ℓ = .
76 (dotted, black), ℓ = . ℓ = .
21 (long-dashed, magenta). Weobserve that higher radiation accelerates the jets up togreater speeds. In Fig. (6b), we plot the terminal speeds( v T ) as a function of ℓ . The base speed of these jetsare very low. For super Eddington luminosities, like ℓ = .
85 the jet achieves terminal speeds to be around0 . . Astrophys. Astr. (2016) : Figure 6 . (a) Variation of v with r for various luminositiesranging from ℓ = .
21 to 2.85 . (b) v T as a function of ℓ . E ff ect of corona geometry, magnetic pressure indisc ( β ) and Γ In this paper we have considered thickdiscs with corona height being 2 . ff ects of di ff erent geome-tries, we choose H sh = . x sh as in Vyas et al. (2015)and generate velocity profiles of the jet for ℓ = .
76 and E = .
33 (same parameters as in Fig. 5c). The profilesare plotted in Fig (7a) for thicker ( H sh = . x sh , solid,black) and thinner corona ( H sh = . x sh , dashed, red).It is clear that radiation from geometrically thick coronais more capable to produce shock, as the jet faces neg-ative flux after being launched. For thinner corona, theradiation resistance is relatively less and the jet is un-able to form shock inside the funnel. Further, lesserresistance inside the funnel of thinner disc, makes ra-diative acceleration more e ff ective and as a result theterminal speed is greater.Choice of the value of Γ is a tricky issue. This isbecause the base of the jet is hot and Γ should be closerto, but not exactly 4 / /
3, therefore, we took themedian value of 1 . ff erent values of Γ , then the behaviour ofthe jet changes because di ff erent choices of Gamma al-ter the net heat content of the flow.. In Figure (7b) weplot v T as a function of E for ℓ = .
80 and Γ ( = . Γ ( = .
5, red dashed) and Γ ( = .
6, longdashed magenta). Smaller value of Γ results in higherthermal driving and produces faster jets.In this study β parameter is introduced to computethe synchrotron cooling from stochastic magnetic field.Therefore in steady state it is most likely that β < β = . β would increase synchrotronradiation, but would not increase bremsstrahlung be-cause ˙ m is not being changed. Moreover, the numberof hot electrons which inverse-Comptonize soft pho- tons also do not change much, so although increasing β amounts to increasing ℓ , but the distribution of ℓ isdi ff erent and therefore, the response of v T to β is di ff er-ent than ˙ m or ℓ , as was shown in Vyas et. al. (2015). InFig. 8 shows the variation of v T as a function of β for agiven ˙ m and H sh = . x sh . Figure 7 . (a) Variation of v with r for various disc heightratios, H sh = . x sh (solid black) and H sh = . x sh (reddashed) for ℓ = .
76 (b) v T as a function of E for varying Γ keeping ℓ = . Figure 8 . v T is plotted as a function of β , for ˙ m =
10 and H sh = . x sh . Final Remarks