Relativistic spine jets from Schwarzschild black holes: "Application to AGN radioloud sources"
AAstronomy & Astrophysics manuscript no. 12920tex c (cid:13)
ESO 2018October 26, 2018
Relativistic spine jets from Schwarzschild black holes:
Application to AGN radioloud sources
Z. Meliani, , , C. Sauty , K. Tsinganos , E. Trussoni , and V. Cayatte Centrum voor Plasma Astrofysica, Celestijnenlaan 200B bus 2400, 3001 Leuven, Belgium Observatoire de Paris, LUTh., F-92190 Meudon, France IASA and Section of Astrophysics, Astronomy & Mechanics Department of Physics, University of Athens,Panepistimiopolis GR-157 84, Zografos, Greece INAF - Osservatorio Astronomico di Torino, Via Osservatorio 20, I-10025 Pino Torinese (TO), ItalyReceived ... / accepted ...
ABSTRACT
Context.
The two types of Fanaroff-Riley radio loud galaxies, FRI and FRII, exhibit strong jets but with differentproperties. These differences may be associated to the central engine and/or the external medium.
Aims.
The AGN classification FRI and FRII can be linked to the rate of electromagnetic Poynting flux extractionfrom the inner corona of the central engine by the jet. The collimation results from the distribution of the totalelectromagnetic energy across the jet, as compared to the corresponding distribution of the thermal and gravitationalenergies.
Methods.
We use exact solutions of the fully relativistic magnetohydrodynamical (GRMHD) equations obtained by anonlinear separation of the variables to study outflows from a Schwarzschild black hole corona.
Results.
A strong correlation is found between the jet features and the energetic distribution of the plasma of the innercorona which may be related to the efficiency of the magnetic rotator.
Conclusions.
It is shown that observations of FRI and FRII jets may be partially constrained by our model for spinejets. The deceleration observed in FRI jets may be associated with a low magnetic efficiency of the central magneticrotator and an important thermal confinement by the hot surrounding medium. Conversely, the strongly collimatedand accelerated FRII outflows may be self collimated by their own magnetic field because of the high efficiency of thecentral magnetic rotator.
Key words.
MHD, Relativity, Galaxies: active, Galaxies: jets, Acceleration of particles, Black hole physics
1. INTRODUCTION
According to the standard Active Galactic Nuclei (AGN)paradigm, their radio luminosity is related to the presenceof powerful relativistic jets (radio loud AGN), or, to mildlysub-relativistic outflows (radio quiet AGN). And, by as-suming a supermassive BH surrounded by an accretiondisk/torus, the different AGN phenomenologies observedin both classes are related to the orientation of the axis ofthe BH/disk system with respect to the line of sight, andthe thickness of the torus which is responsible for the ob-scuration effects (Urry & Padovani 1995). Typical examplesfor radio quiet AGN are the various types of Seyfert I - IIgalaxies, where uncollimated (or loosely collimated) windsare outflowing from the BH/disk system at a speed of a fewthousands km s − . However, besides their inclination to theline of sight the classification cannot be complete withoutinvoking another key parameter to explain the outflow dif-ferences between the various AGN. These differences maybe related to galaxy environment effects and/or intrinsicproperties of the AGN, as shown in Fig. 1 (see e.g. Kaiser& Alexander 1997; Celotti 2003; Kaiser & Best 2007).For radio loud objects a fundamental role is played byDoppler boosting, strongly affecting the luminosity and Send offprint requests to : [email protected] spectral properties of these AGN. In fact, these radiosources are associated with powerful relativistic jets whichreach at the parsec scale, high Lorentz factors γ ∼ − ∼ ◦ (Pushkarev et al. 2009). Innearby AGN recollimation is inferred from the inner ra-dio jet structure (Horiuchi et al. 2006 for Cen A; Kovalevet al. 2007 for M87). We recall that the main classes ofradio loud AGN are Radio Quasars, Flat Spectrum andBroad Line Radio Galaxies, BL Lacs, Fanaroff Riley I (FRI)and Fanaroff Riley II (FRII) objects. According to the uni-fied model, FRI objects are misaligned BL Lacs, while theparent population of FRII are Radio Quasars, Broad LineRadio Galaxies and the brightest BL Lacs, as sketched inFig. 1. Regarding in particular the FR I and FR II di-chotomy, we briefly outline in the following their main prop-erties (Fanaroff & Riley 1974):- In FR II sources the extended radio morphology shows aclear, generally one sided collimated (within a few degrees)thin jet, terminating into a hot spot and surrounded bydiffuse blobs. Conversely, in FR I sources the collimatedsymmetric jets smoothly merge into the extended emittingregions. a r X i v : . [ a s t r o - ph . C O ] J un Z. Meliani, et al.: Relativistic spine jets - FR II jets look highly relativistic and narrow along theirwhole length (tens of kpc). FRI jets are conversely relativis-tic only on pc scales (Bridle 1982), becoming subrelativisticand diffuse on kpc scales (Giovannini et al. 2005). However,in some FR I sources the structure of the jet in the kilo-parsec scale appears more complicated, with an inner spinethat remains relativistic and an outer shell that deceleratesand becomes sub-relativistic (Canvin et al. 2005).- FR II sources are more powerful than the FR I ones, withthreshold power ∼ W Hz − sr − increasing with theradio galaxy luminosity (Ledlow & Owen 1996).- FR II are usually found in poor gas environments, withjets probably collimated by their helical magnetic fields(Hardcastle & Worrall 2000; Asada et al. 2002; Zavala &Taylor 2005) and slightly interacting with the external gas.Rich environments harbor mostly FR I sources and theirjets, thermally confined (at least partially) and appearingto strongly interact with the intracluster medium (Kaiser& Alexander 1997; Laing et al. 1999; Gabuzda 2003). Themeasured transverse magnetic field suggests the presenceof internal shocks where the tangled magnetic field is com-pressed (Gabuzda et al. 1994; G´omez et al. 2008). Thoseshocks could be the result of the thermal collimation of thejet. In the following, we briefly discuss numerical versusanalytical modeling of multicomponent jets.There are two main theories to interpret the above ob-servational characteristics. The first explains the morpho-logical differences as mainly due to the different physicalproperties of the environment in which the relativistic jetpropagates (De Young 1993; Bicknell 1995; Laing et al.1999; Gopal-Krishna & Wiita 2000; Meliani et al. 2008).The second explains the dichotomy by involving a differ-ence in the nature of the central engine, the spin of thecentral black hole, the accretion rate and the jet composi-tion (Baum et al. 1995; Reynolds et al. 1996; Meier 1999;Meliani & Keppens 2009). Finally, there may be a combina-tion of external and engine factors to explain the FR I/FRII dichotomy, as we have suggested in Meliani et al. (2006a)and studied in Wold et al. (2007).Similarly to jets from Young Stellar Objects (Ferreiraet al. 2006), AGN jets probably have at least two compo-nents (Sol et al. 1989; Tsinganos & Bogovalov 2002) oneoriginating from the surrounding Keplerian disk (Baum etal. 1995; Meier 2002; Begelman & Celotti 2004) and theother from the inner corona surrounding the central blackhole. This corona can be created by the ”CEntrifugal pres-sure supported Boundary Layer” model (CENBOL) (Das &Chakrabarti 2002). The corona can also be created by themechanism presented in Kazanas & Elison (1986). A thirdalternative to produce such a corona with pair plasma is theBlandford & Znajek (1977) model where the jet is poweredby the spinning black hole. However observations indicatethat the jet should have both hadronic and leptonic com-ponents as explained in two-component models (Henry &Pelletier 1991; Fabian & Rees 1995).The MHD equations can be solved through numericalsimulations, which describe the evolution of the jet config-uration. The availability of more and more powerful com-puting facilities and sophisticated numerical codes allowsa quite complete description and understanding of the jetacceleration/collimation (Komissarov et al. 2007; Porth &Fendt 2010) and the accretion/ejection process (Koide etal. 1998, 1999; McKinney 2006; McKinney & Blandford2009; Gracia et al. 2006, 2009). Recent numerical simula- Fig. 1.
Standard classification of AGN sources followingUrry & Padovani (1995). The horizontal axis represents theinclination of the source axis with the line of sight. The ver-tical axis we suggest that it may be linked to the efficiencyof the underlying magnetic rotator to collimate the flow.tions have progressed to general relativistic magnetohydro-dynamic (GRMHD) jet launching, as in McKinney (2006)and Hardee et al. (2007), suggesting also the formation ofjets with two components. However computational limitsdo not allow yet to follow simulations for very long timesand reach exact stationary configurations. It also fails atanalyzing structures with very different scale lengths.Nevertheless, tremendous progress on understandingthe physics of relativistic jet acceleration/deceleration –and therefore the FRI/FRII dichotomy – has been donethanks to numerical simulations of jet propagation inthe asymptotic regions. Some investigated the relativis-tic hydrodynamic jet propagation through the interstel-lar medium (Duncan & Hughes 1994; Mart´ı et al. 1997;Komissarov & Falle 1998; Aloy et al. 1999; Rossi et al.2008). They show that the different dynamics of FR I andFR II jets may be a consequence of the power of the jet.Many groups had also investigated the two-dimensional rel-ativistic magnetized jet propagation in an external medium(van Putten 1996; Komissarov 1999; Leismann et al. 2005;Keppens et al. 2008) and 3D (e.g. Mizuno et al. 2007,Mignone et al. 2009). They showed that the interactionbetween the jet and the external medium depend on themagnetization of the jet and the density ratio betweenthe jet and the external medium. The role of the envi-ronment may be crucial in HYbrid MOrphology RadioSources (HYMORS), as shown by Gopal-Krishna & Wiita(2000, 2002). These radio sources appear to have a FR IItype on one side and a FR I type diffuse radio lobe onthe other side of the active nucleus. This last model forHYMORS has been recently confirmed by numerical simu-lations of two component jets (Meliani et al. 2008). Anotheralternative to the FRI/FRII dichotomy could come fromthe nature of the instabilities that develop in the jet, or,at the jet interface with the external medium, or, with . Meliani, et al.: Relativistic spine jets 3 another surrounding outflow component (Keppens et al.2008). This is consistent with the fact that hydrodynam-ical jets with high Lorentz factors are more stable (Mart´ıet al. 1997). Poloidal magnetic fields also help to stabilizethe jet (Keppens et al. 2008). On the last alternative notethat simulations by Meliani & Keppens (2009) confirm thatthe deceleration in FRI jets could be attributed to a strongRayleigh Taylor instability between the spine jet and thesurrounding component, supporting a two component jetstructure.In parallel to the development of time dependent simu-lations, steady jet solutions in GRMHD were first obtainednumerically by solving the transfield equation in the force-free limit (Camenzind 1986). This study has been furtherdeveloped in GRMHD by using first a Schwarzschild met-ric and then extending it to a Kerr metric (Fendt 1997).However this method cannot incorporate consistently themass loading of the jet.Exact models for disk winds can be constructed via anonlinear separation of the governing MHD equations. Thistechnique of radial self-similarity has been largely exploredin the context of stellar and relativistic jets by various au-thors (Bardeen & Berger 1978; Blandford & Payne 1982;Li et al 1992; Contopoulos 1994; Ferreira 1997; Vlahakis &K¨onigl 2004).Meridional self similarity is another way to variable sep-aration which is used to produce models of pressure drivenwinds (Meliani et al. 2006a). It is also complementary tomagnetically driven disk winds. Such models may describethe inner spine jet from the central object necessary to sus-tain the outer disk wind but where the radial self-similarmodels fail by construction. In this paper, using such solu-tions, we show that the collimation criterion developed inthe frame of this model can help understanding how theFRI/FRII dichotomy may influence the morphology of theinner spine jets. As a back reaction, the spine jet dynamicsinfluences the outer jet as it was demonstrated in numeri-cal simulations (Meliani et al. 2006b; Fendt 2009), even ifthe central jet is energetically very weak (Matsakos et al.2009).In Sect. 2 we recall briefly the main assumptions of themodel. In Sect. 3 we summarize the details of the standardAGN jet classification and how it helps to constrain theparameters. In Sect. 4 we present an interesting solutionfor FRI type spine jet and in Sect. 5 another one for jetsassociated with FRII objects. In Sect. 6 we discuss andsummarize the main implications of our model.
2. Model assumptions
We use the ideal GRMHD equations in the backgroundspacetime of a Schwarzschild black hole, and neglect theeffects of self gravity of matter outside the black hole. Thespacetime curvature at a distance r from the black hole isgiven by the lapse function, h = (cid:114) − r G r , (1)where r G is the Schwarzschild radius.Following Meliani et al. (2006a), all physical quantitiesare normalized at the Alfv´en radius r (cid:63) along the polar axiswhere the meridional angle is zero ( θ = 0). We define a dimensionless spherical radius R = r/r (cid:63) , cylindrical radius G , and magnetic flux function α , α = R G ( R ) sin θ , G = r sin( θ ) r (cid:63) sin( θ (cid:63) ) . (2)To describe the GRMHD outflow of the coronal plasma,we use the relativistic meridionally self-similar solutionspresented in Meliani et al. (2006a). The specific enthalpy,and density in the lab frame, together with the pressure,velocity and magnetic field are given in terms of functionsof the radial distance R , hγw = h (cid:63) γ (cid:63) w (cid:63) (cid:18) − µλ ν N B D α (cid:19) , (3) hγn = h (cid:63) γ (cid:63) n (cid:63) h (cid:63) M (cid:18) δα − µλ ν N B D α (cid:19) , (4) P = P + 12 γ (cid:63) n (cid:63) w (cid:63) c V (cid:63) Π( R )(1 + κα ) , (5) V r = V (cid:63) M h (cid:63) G √ δα (cid:18) cos θ + µλ ν N B D α (cid:19) , (6) V θ = − V (cid:63) M h (cid:63) G hF √ δα sin θ , (7) V ϕ = − hh (cid:63) λV (cid:63) G N V D R sin θ √ δα , (8) B r = B (cid:63) G cos θ , (9) B θ = − B (cid:63) G hF θ , (10) B ϕ = − λB (cid:63) G h (cid:63) h N B D R sin θ . (11)where N B = h h (cid:63) − G , N V = M h (cid:63) − G , D = h h (cid:63) − M h (cid:63) . (12)The free parameters δ and κ describe the deviation fromspherical symmetry of the ratio of number density/enthalpyand pressure, respectively, while λ is a constant controllingthe angular momentum extracted by the jet. The constants ν and µ measure the escape speed in units of the light speedand the escape speed in units of Alfv´en speed, at the Alfv´enpoint along the polar axis, respectively, µ = V ,(cid:63) c , ν = V esc ,(cid:63) V (cid:63) . (13)Thus, all physical quantities are determined in terms ofconstant parameters, δ , κ , λ , µ , ν and the three unknownfunctions, Π( R ), F ( R ) and M ( R ). Note that Π( R ) is thedimensionless pressure function, defined modulus a con-stant P , F is the expansion factor, and M is the poloidalAlfv´enic number, F = 2 − d ln G d ln R , M = 4 πh nwγ V B c . (14)These three unknown functions Π( R ), F ( R ) and M ( R ) aredetermined by three nonlinear equations. We start integrat-ing these equations from the Alfv´en critical surface, tak-ing into account there the corresponding regularity condi-tion and then integrate downwind and upwind, crossing theother critical points, for details see Meliani et al. (2006a). Z. Meliani, et al.: Relativistic spine jets
The light cylinder is defined by the function x = Ω L/E becoming unity, where Ω is the angular speed, L the totalangular momentum per unit mass and E the generalizedBernoulli integral. It is a measure of the energy flux of themagnetic rotator in units of the total energy flux. The s-sdescription is possible only if the jet is rotating at subrela-tivistic speeds. In such conditions x must remain small andtherefore the light cylinder effect have to be negligible forour solution to be valid.In this setup of the relativistic MHD problem an extraparameters exists (cid:15) which is constant everywhere (Melianiet al. 2006a): (cid:15) = M h (cid:63) R G (cid:18) F − h − κ R h G (cid:19) − ( δ − κ ) ν h R + λ G h (cid:63) (cid:18) N V D (cid:19) + 2 λ h N B D . (15)This is the relativistic generalization for a Schwarzschildblack hole of the classical constant found in Sauty et al.(2004) that measures the magnetic energy excess or deficiton a nonpolar streamline, compared to the polar one. Tofirst order, (cid:15) determines the fraction of Poynting flux car-ried by the jet in the asymptotic region, such that 1 − (cid:15) measures the fraction of the Poynting flux which is usedin the jet acceleration. Thus, if (cid:15) > (cid:15) <
3. New interpretation of the Fanaroff-Rileyclassification
We propose here to examine the vertical classification inFig. 1 by means of our parameter (cid:15) , or less ambitiously, tointerpret how the various observations we have on FRI andFRII jets may influence the formation of the inner spinejet component. As in the non relativistic case (Sauty et al.1999), this parameter (cid:15) allows to classify jets according tothe efficiency of the central magnetic rotator. Jets emerg-ing from efficient central magnetic rotators, (cid:15) >
0, colli-mate cylindrically without oscillations in the asymptoticregion. In this type of jets, the velocity increases monotoni-cally to reach its asymptotical maximum value. Conversely,jets associated with inefficient central magnetic rotators, (cid:15) <
0, are collimated mainly by the pressure of the exter-nal medium. Thus, this type of jets strongly interacts withthe ambient medium and this induces oscillations in theirshape at the asymptotic region. The speed of these jets alsodoes not increase monotonically but it oscillates too. Theplasma in those jets is accelerated until an intermediate re-gion where the speed reaches its maximum value. Then, theacceleration of the jet stops in this region. Further away inthe recollimation region the outflow slows down. Note thatsuch jet oscillations could lead finally to a more turbulentoutflow consistently with the numerical simulations of twocomponent jets we mentioned in introduction. After the rec-ollimating region we cannot exclude the presence of shocksor instabilities. Thus it is difficult to know if the oscillationsthat appeared in the solutions would be observable or not.In summary, we propose that a difference between thesetwo types of spine jets associated with FR I and FR II radio galaxies may result from the competition between the mag-netic and thermal confining mechanisms. Magnetic pinch-ing and pressure gradient tend to compensate the trans-verse expansion of the jet because of the centrifugal forceand the charge separation which also induces an outwardselectric force. These two expanding forces are characterizedby the free parameter λ , that measures the quantity of an-gular momentum carried along the streamlines. Magneticcollimation is controlled by the parameter (cid:15)/λ (see Sautyet al. 1999). On the other hand, thermal collimation is con-trolled by the parameter κ that defines the transverse vari-ation of the pressure.Hence, an appropriate manner to classify the differentjet solutions according to the nature of their collimation isbased on the two free parameters κ/ λ and (cid:15)/ λ (Sautyet al. 1999, 2002, 2004). The higher is the value of thoseparameters, the stronger is the collimation and the lower istheir terminal speed. This can be seen in Tab. 3 where weplot for various solutions the asymptotic jet speed in unitsof the escape speed from the base of the corona: as the effi-ciency of collimation increases, the efficiency of accelerationdecreases. In other words, tightly collimated jets (larger val-ues of (cid:15)/ λ , or, κ/ λ ) have lower terminal speeds. Thusthe strong interaction with the external medium would nat-urally result into a decelerated flow even if it remains stable.Parallely, the jet being denser would radiate more on largescale before the terminal shock with the ambient medium. Fig. 2.
In (a) we plot the Alfv´en number M and, in(b), the cylindrical jet cross section G as functions ofthe distance for κ/ λ = 0 .
05 and four values of (cid:15)/ λ ( − . , − . × − , . × − , . . Meliani, et al.: Relativistic spine jets 5 (cid:15) λ / κ λ − . − . − . − . − .
05 0.50 . − . −
30 13 7.0 5.77 1.069 1.660 . −
90 75 65 60 10 1.02
Table 1.
The asymptotic jet speed in units of the escape speed from the base of the corona, for various values of theparameters κ/ λ and (cid:15)/ λ Tab. 3 also shows that jets from inefficient magnetic ro-tators are more powerful in transforming thermal energyinto kinetic energy, than those from efficient magnetic ro-tators. There are two reasons for this.
First , in a EMR flow, the centrifugal force at the baseof the jet is important. Then, the last stable orbit of theplasma gets closer to the central black hole. Therefore, thecorona extends closer to the black hole horizon. Thus, ifthe available total amount of thermal energy is the samein the corona, the plasma gravitational potential increases.Consequently, in EMR as (cid:15)/ (2 λ ) increases, more thermalenergy is tapped in order to allow for the plasma to escapeand less is left for accelerating it. Second (see Fig. 2), an increase of the magnetic rota-tor efficiency limits the initial expansion of the outflow.The pinching magnetic force gets stronger after the Alfv´ensurface and the conversion of thermal energy into kineticenergy stops when the jet reaches its asymptotic cylindricalshape. This decrease of the jet asymptotic speed with theincrease of the magnetic rotator efficiency may seem contra-dictory with the usual picture. However, in axial outflowsthe contribution of the Poynting flux to the total accelera-tion remains weak. This is of course different from relativis-tic disk wind models where the acceleration is dominatedby the conversion of Poynting flux to kinetic energy flux(Li et al 1992; Contopoulos 1994; Vlahakis & K¨onigl 2004).Those jets are characterized by a strong inclination of themagnetic field lines and the rotation at the base of the out-flow is almost Keplerian. Note also that, as expected, theefficiency of the acceleration also increases with the degreeof expansion, i.e., as κ/ λ decreases. In the following sections, we mainly explore two examples oftypical solutions for relativistic jets. One is associated withan IMR such that the contribution of the external pres-sure is comparable to the magnetic one, while the secondis associated to an EMR such that the jet is self collimatedmagnetically. In order to find these solutions we first makean estimation of the free parameters of the model using ourknowledge of the properties of FRI and FRII jets, in par-ticular in the launching region on one hand and in the farasymptotic region on the other hand.At the base of the outflow, we assume that the coronastarts at the radial distance of the last stable orbit which isaround the radius of the Schwarzschild black hole, as dis-cussed below. We consider that the last open streamlines,which emerge from the corona, should be in sub-Keplerianrotation if they are anchored in the thick disk surroundingthe central black hole. The last open streamline in the coro-nal jet is the one that crosses the equatorial plane at theedge of the magnetic dead zone, which we assume to havea dipolar configuration. We use also some observational constraints in theasymptotic region of the jet. The opening angle of jet hasto be a few degrees. Six degrees is the value inferred for thewell measured jet of M87 (Biretta et al. 2002; Kovalev etal. 2007). We guess that the spine jet opening angle is evensmaller and we took a value of (cid:39) ◦ at 1 pc. Then, from theexpression of the dimensionless magnetic flux α , by know-ing the last streamline, we can deduce the asymptotic valueof G ∞ : G ∞ = r ∞ r (cid:63) √ α ext sin θ , (16)where α ext is the last open streamline in the jet. As areasonable estimate, we have taken α ext = 4. In fact, forsmaller values of α ext , the value of G ∞ becomes too largeand therefore unrealistic for our model. We further assumethat in the asymptotic region the Alfv´en number is of theorder of M ∞ (cid:39)
5: this is the order of magnitude found inthe literature for relativistic magnetohydrodynamical jetpropagation (Leismann et al. 2005, Keppens et al. 2008).We know that the asymptotic Lorentz factors shouldbe between ∼ ν , ν = µc V ∞ (cid:18) M ∞ h (cid:63) G ∞ (cid:19) . (17)We used on purpose a rather lower limit for the Lorentzfactor in order to avoid large effective temperatures in ourmodel. In fact, we can use the same solutions and scale themup to obtain higher Lorentz factors but then the effectivetemperature would attain extremely large values above themass temperature. However the corresponding high pres-sure could have a large contribution from a turbulent mag-netic or ram pressure component in the jet (see Aibo et al.2007, for the solar wind). In such a case the kinetic temper-ature would be lower. Nevertheless, as far as the collimationis concerned this does not affect qualitatively our discussionon the dichotomy between FRI and FRII and we kept thisrelatively low Lorentz factor.We used also the observed mass loss rate in the outflowto constrain the free parameter δ . However, the spine jetprobably carries only a small fraction of the observed en-ergy flux in AGN jets, L jet , Kin ∼ ergs / s (Allen et al.2006) The energy flux of the coronal wind remains weakcompared to the total mass carried by the disk-wind whichis supposedly denser (Vlahakis & Konigl 2004), a situationsimilar to stellar jets associated with Young Stellar Objects(Meliani et al. 2006b). Then δ is deduced from the assumedvalue for the mass loss rate,˙ M = 2 (cid:90) section m part h γ n V p d S ,
Z. Meliani, et al.: Relativistic spine jets = 4 π m part r µ / ν (cid:112) γ (cid:63) n (cid:63) w (cid:63) /c (cid:90) α ext (cid:115) M nw/c d α . (18)We get an equation for δ ,(1 + α ext δ ) × C − (2 C + 1) × (1 + α ext δ ) +(1 + α ext δ ) × C × (2 + C ) − × C × (1 + α ext δ ) + 1 = 0 , (19)where C is another constant given by, C = 3 µ / ν (1 − µ ) ˙ M πr G m part (cid:112) − r G /r ∞ γ ∞ n ∞ M ∞ α . (20)The variable n ∞ is the asymptotic density and m par theaverage mass of the particles. We consider a proton-electronfluid, m par = m proton .For the mass loss rate, we choose ˙ M = n − ˙ M Edd ,where ˙ M Edd is the Eddington mass loss rate. It correspondsto value found for the relativistic Parker wind (Melianiet al. 2004). The asymptotic density is taken equal to n ∞ = 10 − × n cm − (Meliani et al. 2004), with n adimensionless free parameter.We suppose that the rotation is sub-Keplerian on thelast open streamline of the jet at the equator. The param-eter η measures the deviation of the rotation function Ωfrom its Keplerian value,Ω = (1 − η ) (cid:115) r G r µ c , (21)From the definition of Ω (Eq. 21) we can deduce the valueof the free parameter λ which is the constant controllingthe angular momentum extracted by the jet, λ = (1 − η ) (cid:115) r G r c µ r (cid:63) √ δα h (cid:63) v (cid:63) c = (1 − η ) (cid:115) r G r c µ r (cid:63) √ δα h (cid:63) (cid:114) νµ . (22)As we mention earlier we assume that the corona formsabove the last stable orbit at r = 3 r G . We choose a typi-cal magnetic lever arm (i.e. Alfv´en radius) of 10 times theSchwarzschild radius, r (cid:63) = 10 r G . It gives, R = r r (cid:63) = 0 . . (23)The parameter κ , which is the relative variation of the pres-sure with latitude, is calculated from an approximate ex-pression of the coronal base where M → dM /dR isfinite in Eq. A.2 - A.5 (Meliani et al. 2006a) using, κ = δ − R λ ν (1 − µ ) . (24)To summarize, our free parameter estimates are:1. The Alfv´en radius r (cid:63) = 10 × r G → µ = 0 . . (25) 2. The jet opening angle at r ∞ (cid:39) (cid:39) ◦ whichgives the asymptotic value of G ∞ for the last streamline α ext = 4.3. For the asymptotic Alfv´en number we chose M ∞ ∼ γ ∞ ∼ n ∞ = n × − cm − .6. The new parameter η which measures how sub-Keplerian is the velocity (the values differ from solutionto solution).7. The corona is supposed to be formed above the laststable orbit, r c = 3 r G which gives κ .The parameter η is not independent from the rest ofthe model. However it measures precisely the rotation ofthe footpoints and controls the efficiency of the magneticrotator to collimate the jet. Thus we have an indirect wayto determinate the properties of the disk from the asymp-totic characteristics of the jet. If η → η → In polytropic relativistic winds (Meliani et al. 2004), thetemperature is usually defined by the ideal gas equationof state
P/n = k B T . Therefore, the knowledge of specificpressure P/n , the specific thermal energy e th and the den-sity describe completely the thermodynamics of the fluid(temperature T , enthalpy w and pressure P ). However, asdiscussed in Meliani et al. (2004), flows cannot be adia-batic. The polytropic approximation is just a convenientway to mimic heating going on in the flow. Therefore pres-sure, temperature and enthalpy are not the real ones buteffective quantities that hide the extra necessary heating.The temperature definition in meridional self-similarmodels is similarly delicate. In fact, as indicated in Sauty& Tsinganos (1999) and Meliani et al. (2006a) the total gaspressure is not necessarily limited to be the kinetic pressure.Moreover, the generalized specific thermal energy of themodel, e th = (cid:0) w − mc (cid:1) − Pn , is not restricted to the ther-mal energy. In self-similar model these two quantities arealso effective quantities. They account for different physicalprocesses of energy and momentum transport and dissipa-tion. They can include the contribution of magnetohydro-dynamic waves and viscous and/or radiative mechanisms.As a matter of fact, the complexity and variety of MHDprocesses that can contribute to the internal energy of amagnetized fluid, makes the definition of the real thermalenergy impossible.Therefore the quantities T eff = P/n and e th , eff = e th aresimply the specific effective temperature and thermal en-ergy imposed by the dynamics of the outflow. They do notnecessarily represent neither the kinetic temperature northe thermal energy. However they are simple tools to ana-lyze the energetics of the flow. In the following, we discussthe thermodynamical properties of the fluid with this ef- . Meliani, et al.: Relativistic spine jets 7 Fig. 3.
Plot of the morphology of the solution correspond-ing to a FRI-type spine jet. In (a) is shown the projec-tion of the streamlines on the poloidal plane. The solidlines in the center correspond to lines where the conditions x A G < − and (2 + δα ) / (1 + δα ) − < − are sat-isfied. The dashed lines correspond to x A G < − and(2 + δα ) / (1 + δα ) − < − . The dashed-dotted lines cor-respond to x A G > − and (2 + δα ) / (1 + δα ) − > − (see Meliani et al. 2006a for details).fective temperature and not the kinetic temperature whichwe cannot calculate. T eff = Pk B n = 12 k B γ (cid:63) w (cid:63) M Π 1 + κα δα , (26) e th , eff = w − mc − Pn = w − mc − k B γ (cid:63) w (cid:63) c M Π 1 + κα δα , (27)Thus along the polar axis, the specific thermal energyis deduced from the Bernoulli equation as follows, e th , eff = w (cid:63) h ,(cid:63) h γγ (cid:63) − mc − γ (cid:63) w (cid:63) M Π , (28)We also define Q the heat content added to the fluid, whichis the difference between the total effective internal energyof the fluid e th , eff and the internal energy of the fluid ob-tained if it were adiabatic: Q = e th , eff − (cid:113) mc + κ pol n − , (29)where κ pol is a constant. It is determined from pressureand density at the flow boundary either in the asymptoticregion or in the launching region (in adiabatic flows, wehave κ pol = P/n ).
4. Application I - Model of FRI spine jets
The first solution we show here is adapted to model thespine jet of radio-loud galaxies of FRI type. The environ-ment of such jets, i.e. the host galaxy, is known to be richand containing dense gas. Moreover, the properties of FRIjets on the pc scale are quite different from that on kpcscales. In fact, in the region close to the nucleus (on thescale of a pc), FRI jets are accelerated to highly relativisticspeeds.Beyond this region, the jets interact with the externalmedium which is denser. This interaction induces an ob-served deceleration of the jet. Thus, we assume that theoutflow is likely confined by the pressure of the ambientmedium, at least partially. In our model, these types of jetscorrespond to solutions associated with inefficient centralmagnetic rotators. In this solution η = 0 .
90 which gives aninefficient magnetic rotator.The solution corresponds to the following parameters, µ = 0 . ,ν = 0 . ,λ = 0 . ,δ = 1 . , (30) κ = 0 . ,(cid:15) = − . . As seen in Fig. 3, the jet solution shows an initial expan-sion up to a distance of 100 Schwarzschild radii which thenstops and the jet recollimates. The expansion of the jet isdue to the strong initial inertia of the plasma carried alongthe external streamlines. The jet becomes collimated onceit interacts with the ”external” ambient medium that com-presses it. What we call ”external” medium refers to thegas surrounding the last valid streamline of the solution;this can be the actual external medium of the host galaxy,but considering the transverse expansion of the solution,
Z. Meliani, et al.: Relativistic spine jets it more likely corresponds to the over-pressured gas of thesurrounding disk wind.The jet compression for inefficient magnetic rotatorsgenerates strong oscillations in the asymptotic region ofthe jet even in the relativistic case (Meliani et al. 2006a).The light cylinder of this jet solution is at infinity. Theseoscillations results from a transfer of energy between theenthalpy and the Lorentz factor as hγw remains constantto first order with respect to α . Fig. 4.
In (a) we plot the energetic fluxes normalized tothe mass energy of the first solution for a ”FRI-type” spinejet. In (b) we plot for the same solution, the Lorentz factor γ along four different streamlines. The solid line correspondto the polar axis and the dotted line to the last streamlineconnected to the central corona. Other lines are intermedi-ate ones. The jet acceleration occurs mainly in the intermediate re-gion (Fig. 4b) where gravity becomes weak. Thus, all theenthalpy still remaining in the outflow is converted to ki-netic energy.The maximum Lorentz factor in this solution ( γ = 2 . ≈ K). To explain such a high effective tempera-ture, part of the pressure must be of non kinetic origin, withcontribution from turbulent ram and magnetic pressure, aswe discussed in Sec. 3.1 Thus this value of γ is close to thelower observed ones, but already corresponds to a highlyturbulent medium. The model can produce higher Lorentzfactors provided the turbulent pressure level is sufficient.This is not the main topic of the qualitative discussion weaddress on the collimation of the jet itself. In the region of re-collimation, the increase of the pres-sure induces a deceleration of the jet. The Lorentz factordecreases from γ = 2 .
8, its maximum value before the re-collimation, to γ = 2 in the asymptotic region. As a matterof fact such a deceleration is indeed a characteristic of FRIjets as we mentioned. It is remarkable that this solutionshows clearly that the main effect of the re-collimation bysome external pressure is a global deceleration of the out-flow, as observed in FRI jets. The distance of recollimationin this solution is however smaller than the usual parsecscale. This may be due to the fact that here we are dealingonly with the inner part of the jet, while observations maycorrespond to the surrounding disk-wind, or, it can be dueto the fact that our Lorentz factor is too low. Moreover, weassume here that the external pressure of the host galaxysomehow is transmitted to the disk-wind, which in turnconfines the spine jet. Keeping in mind these necessary pre-cautions, this result by itself seems interesting, if we takeinto account the simplicity of our model. The temperature profile is characterized by four differentregimes. The three first are common with the ”FRII” jet so-lution and we shall discuss it later on. The fourth one corre-sponds to the asymptotic recollimated region of the ”FRI”jet. There, the effective temperature reaches the high valueof T eff ∼ K, because of the strong compression of theoutflow by the external medium (Fig. 5b). This effectivetemperature is high compared to the observed temperaturein AGN jets that are usually of the order of T ∼ K. Thislarge difference can be explained however by some increaseof the contribution of non thermal mechanisms to the effec-tive specific thermal energy e th , eff . As we mention before,we cannot directly compare this effective temperature withthe kinetic temperature in the frame of this model.
5. Application II - Model for FRII spine jets
Conversely to the case of FRI, the environment of FRIIradio-loud galaxies is relatively poor. Thus, jets from FRIIgalaxies should interact only slightly with the ambientmedium. In these outflows, the velocity increases contin-uously until the asymptotic region. In fact, unlike FRI out-flows, the velocity in FRII jets is relativistic both on theparsec and kilo-parsec scales. Besides, FRII jets are so wellcollimated on large scales that they are very likely to havean asymptotic cylindrical shape. In our model these typesof jets correspond to a solution associated with an efficientcentral magnetic rotator. Therefore, the value of the free pa-rameter η is probably smaller, i.e. the rotation is closer toKeplerian velocity than in the case of the solution for FRI.This increases the available Poynting flux at the base ofthe jet. Therefore, the jet will be collimated by the toroidalmagnetic pinching without any oscillations.The parameters for this specific solution are: µ = 0 . ,ν = 0 . ,λ = 1 . ,δ = 1 . , (31) . Meliani, et al.: Relativistic spine jets 9 Fig. 5.
In (a) we plot the temperature profile and in (b)the density profile for the FRI-type jet solution. Densityis normalized to n such that the mass loss rate is ˙ M = n − ˙ M Edd . κ = 0 . ,(cid:15) = 0 . . This is an Efficient Magnetic Rotator. It corresponds toa slightly lower value of η = 0 .
86. This shows that theefficiency is very sensitive to the variations of the rotationfrequency.This FRII-type jet solution is characterized by a contin-uous expansion up to a distance of about 100 Alfv´en radiiagain, but the outflow after this distance remains cylindri-cal, slightly expanding further out. This expansion is re-lated to the high magnetic pressure at the base, togetherwith the strong gravity, in addition to some non negligiblecontribution of the force of charge separation. However, thecollimation of the jet in the asymptotic region is exclusivelyof magnetic origin. It is induced by the toroidal magneticpinching force and the transverse magnetic pressure. Thesetwo forces balance the centrifugal and charge separationforces. Oscillations in the jet are very weak, due to the rel-atively small contribution of the thermal confinement, asexpected.The opening angle of the last open streamline of thesolution at a distance of one parsec is only 0 . ◦ , which israther small compared to our initial guess. This definitelyrules out the possibility to describe the whole jet uniquelywith this model. Instead, we prefer to see it as the spineor inner part of the jet that carries away the angular mo-mentum of the central black hole. The situation is similarto what happens in Young Stellar Objects where the stel-lar jet is responsible for the spinning down of the protostarwhile the outer disk wind is responsible for the observedmass loss. Fig. 6.
The same as in Fig. 3 for the second solution cor-responding to a FRII-type spine jet.
The acceleration of this solution is continuous. First, thereis a small but effective thermal acceleration in the lower re-gion of the corona. In this region, the high thermal energyboth accelerates the fluid up to 0 .
4c at a distance of 6 r G andenables it to escape from the deep gravitational potential.A second stronger thermal acceleration of the jet occursbeyond the Alfv´en surface up to the collimation region. Inthis region, the pressure drops rapidly and asymptoticallygoes to negligible values. Therefore, the thermal energy is Fig. 7.
In (a) we plot the energetic fluxes normalized tothe mass energy of the second solution for a ”FRII-type”spine jet. In (b) we plot for the same solution, the Lorentzfactor γ along four different streamlines. The solid line cor-respond to the polar axis and the dotted line to the laststreamline connected to the central corona. Other lines areintermediate ones.transformed into kinetic energy more effectively. In this in-termediate regime the velocity in the flow increases from0 .
4c to 0 .
92c on a scale of the order of 200 r G . The light cylinder in the asymptotic region of the jet (Fig.6) is roughly vertical and asymptotically parallel to thepoloidal streamlines, that remain inside the light cylinder.However, in this solution conversely to the previous one,the light cylinder is not at infinity but at a distance ofabout 20 r G from the polar axis. This reduces the domainof validity of the solution around its axis (cf. Meliani et al.2006a) where the effects of the light cylinder can be ne-glected. In fact self similar disk wind models can producesolutions crossing the light cylinder (Vlahakis & K¨onigl,2003a,b). Such solutions undergo a strong magnetic accel-eration ideal to obtain high Lorentz factor in GRBs forinstance. This is of course not necessarily the case for thespine jet which can be accelerated by other means than themagnetocentrifugal process. The effective temperature profile of this second solutiongoes through four different regimes.First, in the lower corona, the effective temperature in-creases extremely rapidly (Fig. 8a) from about 10 K at thebase up to about 3 × K. This increase is due to somestrong initial heating in the expanding corona. The large
Fig. 8.
In (a) we plot the temperature profile and in (b)the density profile for the FRII-type jet solution. On theleft the vertical lines delimitate the various domains of thetemperature profile. On the right, density is normalized to n such that the mass loss rate is ˙ M = n − ˙ M Edd .expansion induces a strong decrease of the density, but be-cause of the heating, the pressure decreases less rapidly.Second, in the intermediate region, the effective tem-perature still increases up to its maximum value of about3 × K, after a relatively small decrease, because of theglobal expansion and drop of the density.Third, we have a transition region after the maximumand the asymptotic part. The effective temperature de-creases again to attain values around 10 K. This decreaseis induced by the magnetic compression of the jet thatbrakes the density decreasing (Fig. 8b).The effective temperature obtained in this solution, isof course high compared to observed temperatures in AGNjets which is typically of the order of 10 K in the asymptoticregions. As in the case of FR I this indicates that thecontribution of non thermal energies to the acceleration andthe heat of the jet play a relevant role.
6. Conclusions
We have applied exact GRMHD solutions from Melianiet al. (2006a) to the canonical classification of AGN jets(Urry & Padovani 1995) according to their morphology.Our model is constructed in the frame of general relativityusing the metric of a central Schwarzschild black hole. Itvalidates the classification of AGN proposed in Sauty et al.(2001) where the classical MHD solutions were used withsome additional features due to relativistic effects. In thisstudy it was proposed that the inner regions of jets (spines)are collimated by an external denser medium in FRI andby the force of magnetic pinching in FRII. . Meliani, et al.: Relativistic spine jets 11
We first proposed a method to estimate the free pa-rameters of the model from the known properties of AGNjets. In particular, the departure from Keplerian rotationat the footpoints of the fieldlines is measured by an extraparameter η . It turns out that we get very different classesof solutions by slightly changing this parameter.First for η = 0 .
90 we obtained a recollimating solu-tion. On a small spatial scale the outflow expands and theLorentz factor reaches a maximum of γ = 2 . γ = 2 in the asymptotic region. Thisdecrease is related to thermal compression of the jet in theasymptotic region by the outer medium. We insist on thefact that this recollimation occurs on a scale smaller thanone parsec and that the external pressure is certainly thepressure of the surrounding disk wind rather than the exter-nal gas from the host galaxy. However if we assume that theextra pressure of the host galaxy can enhance the pressurein the disk wind, it is striking to note that our simple toymodel for the spine jet shows a typical feature of FRI jets.Indeed FRI jets show a deceleration on the kiloparsec scaledown to non relativistic speeds sometimes though they areusually highly relativistic on smaller scales. Besides thatthey are also known to have a rich ambient medium andthat the external gas pressure is important as seen in theX-ray (Capetti et al. 2002). Finally the fact that the FRI jetradius is larger on the kiloparsec scale can be due preciselyto this recollimation effect which enhances the density suchthat probably a larger part of the radio jet is emitting assuggested in the double component jet of Sol et al. (1989).In the case of the FRII jets, the optimal value found is η = 0 .
86, which also gives a maximum Lorentz factor of γ (cid:39)
3. The fact that η changes very little when we passfrom the ”FRI”-type solution to the ”FRII”-type, showsthat the outflow properties are highly sensitive to the rota-tion velocity of the corona. The solution obtained is how-ever very different as the gas expands monotonically and re-mains highly relativistic up to large distances, as observedin FRII jets. Moreover it is self collimated by its own mag-netic field throughout the length of the jet, something againthat is characteristic of FRII jets wherein there is evidencethat the host galaxy gas is rather underdense.On the other hand, the fact that in these two cases, η is slightly smaller than 1 shows that the central launchingregion of the spine jet has to be slightly sub-Keplerian.In the present paper we did not discuss the case ofSeyfert galaxies which are known to have outflows, thoughtheir winds are not very well collimated and are not rela-tivistic with velocities of the order of 30 ,
000 km/s, as ob-served by the HST. Again, such flows can be understood inthe frame of this simple self-similar model as non collimatedsolutions which are radial and exist only if the velocity doesnot reach relativistic values (Meliani et al. 2006a). Such so-lutions are obtained if the magnetic rotator efficiency is very low, i.e. for (cid:15) very negative, something similar to thesolar wind.Altogether then, we may conclude that using a simpletoy model for spine jets, the usual classification of radiosources can be understood on one hand by projection ef-fects and Doppler boosting and on the other by consider-ing the efficiency of the central magnetic rotator. Of coursethis does not exclude any of the other explanation such asthe role of the external confinement or shear instabilities.Indeed our FRI-type solution is at least partially confinedby the disk wind which may be a signature of external pres-sure confinement as well. This idea needs to be further ex-plored with more sophisticated models or simulations com-bining the central coronal jet with an external disk wind,something worth to pursue in another study.Next step under consideration is the extension of themodel to the Kerr metric. This is a unique chance to con-struct the first analytical models for a jet around a rotat-ing black hole. One issue of this extension is to test theBlandford-Znajek mechanism (Blandford & Znajek 1977).
Acknowledgements.
Z. Meliani acknowledges financial support fromthe FWO, grant G.027708. The authors thank Nektarios Vlahakis forhelpful discussions and suggestions.
References
Aibeo, A., Lima, J. J. G., & Sauty, C. 2007, A&A, 461, 685Allen, S. W., Dunn, R. J. H., Fabian, A. C., Taylor, G. B., & Reynolds,C. S. 2006, MNRAS, 372, 21Aloy, M.A., Ib´a˜nez, J. M., Mart´ı, J. M., & M¨uller, E. 1999, ApJS,122, 151Asada, K., Inoue, M., Uchida, Y., et al. 2002, PASJ, 54, L39Bardeen J. M., Berger B. K. 1978, ApJ, 221, 105Baum, S. A., Zirbel, E. L., & O’Dea, C. P. 1995, ApJ, 451, 88Bicknell, G. V. 1995, ApJS, 101, 29Biretta, J., Junor, W., & Livio, M. 2002, New Astron. Rev., 46, 239Begelman, M. C., & Celotti, A. 2004, MNRAS, 352, L45Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433Bridle, A. H. 1992, Testing the AGN paradigm, AIP Conf. Proc., 254,386Camenzind, M. 1986, A&A, 156, 137Canvin, J. R., Laing, R. A., Bridle, A. H., & Cotton, W. D. 2005,MNRAS, 363, 1223Capetti, A., Trussoni, E., Celotti, A., et al. 2002, New AstronomyReviews, Volume 46, Issue 2-7, p. 335-337.Celotti, A. 2003, Ap&SS, 288, 175Contopoulos, J. 1994, ApJ, 432, 508Das, S., & Chakrabarti, K. 2002, JApA, 23, 143De Young, D. S. 1993, ApJ, 405, L13Duncan, C., & Hughes, P. 1994, ApJ, 436, 119Fabian, A.C., & Rees, M.J. 1995, MNRAS, 277, L55Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31Fendt, C. 1997, A&A, 319, 1025Fendt, C. 2009, ApJ, 692, 346Ferreira, J. 1997, A&A, 319, 340Ferreira, J., Dougados, C., & Cabrit, S. 2006, A&A, 453, 785Gabuzda, D. C., Mullan, C., Cawthorne et al. 1994, ApJ, 435, 140Gabuzda, D. C. 2003, Ap&SS, 288, 39Giovannini, G., Taylor, G. B., Feretti, L., et al. 2005, ApJ, 618, 635G´omez J. L., Marscher A. P., Jorstad S. G., Agudo I., Roca-SogorbM. 2008, ApJ, 681, L69Gopal-Krishna, & Wiita, P. J. 2000, A&A, 363, 507Gopal-Krishna, & Wiita, P. J. 2002, New Astr. Rev., 46, 357Gracia J., Vlahakis N., Tsinganos K.2006, MNRAS, 367, 201Gracia J., Vlahakis N., Agudo I., Tsinganos K., Bogovalov S. V. 2009,ApJ, 695, 503Hardcastle, M. J., & Worrall, D. M. 2000, MNRAS, 319, 562Hardee, P., Mizuno, Y., & Nishikawa K.-I. 2007, Ap&SS, 311, 281Henri, G., & Pelletier, G. 1991, ApJ, 383, L7Horiuchi, S., Meier, D. L., Preston, R. A., & Tingay S. J. 2006, PASJ,58, 2112 Z. Meliani, et al.: Relativistic spine jets