Available in: http://www.redalyc.org/articulo.oa?id=57102710 Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica Arieh Königl MHD Driving of Relativistic Jets Revista Mexicana de Astronomía y Astrofísica, vol. 27, marzo, 2007, pp. 91-101, Instituto de Astronomía México How to cite Complete issue More information about this article Journal's homepage Revista Mexicana de Astronomía y Astrofísica, ISSN (Printed Version): 0185-1101 [email protected]Instituto de Astronomía México www.redalyc.org Non-Profit Academic Project, developed under the Open Acces Initiative
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Available in: http://www.redalyc.org/articulo.oa?id=57102710
Red de Revistas Científicas de América Latina, el Caribe, España y Portugal
Sistema de Información Científica
Arieh Königl
MHD Driving of Relativistic Jets
Revista Mexicana de Astronomía y Astrofísica, vol. 27, marzo, 2007, pp. 91-101,
Instituto de Astronomía
México
How to cite Complete issue More information about this article Journal's homepage
izRevMexAA (Serie de Conferencias), 27, 91–101 (2007)
MHD DRIVING OF RELATIVISTIC JETS
A. Konigl1
Received 2005 December 26; accepted 2006 November 3
RESUMEN
Paulatinamente se ha ido reconociendo que los campos magneticos juegan un papel dominante en la producciony colimacion de chorros astrofısicos. Demostramos aquı, usando soluciones semianalıticas exactas para lasecuaciones de MHD ideal en relatividad especial, que un disco de acrecion altamente magnetizado (con un campomagnetico principalmente poloidal o azimutal) alrededor de un agujero negro es capaz de acelerar un flujo deprotones y electrones a los factores de Lorentz y energıas cineticas asociadas a fuentes de destellos de rayos gamay nucleos activos de galaxias. Tambien se discuten las contribuciones a la aceleracion provenientes de efectostermicos (por presion de radiacion y pares electron-positron) y de MHD no ideal. Notamos que la aceleracionpor MHD se caracteriza por ser extendida espacialmente, y esta propiedad se manifiesta mas claramente enflujos relativistas. Las indicaciones observacionales de que la aceleracion de movimientos superlumınicos enchorros de radio ocurre sobre escalas mucho mas grandes que las del agujero negro propiamente, apoyan la ideade que la produccion de chorros es principalmente un fenomeno magnetico. Presentamos resultados preliminaresde un modelo global que puede utilizarse para probar esta interpretacion.
ABSTRACT
There is a growing recognition that magnetic fields play a dominant role in driving and collimating astrophysicaljets. Using exact semianalytic solutions of the special-relativistic ideal-MHD equations, it is demonstrated thata strongly magnetized accretion disk (with a dominant poloidal or azimuthal magnetic field) around a blackhole can efficiently accelerate a proton-electron outflow to the Lorentz factors and kinetic energies inferred ingamma-ray burst sources and in active galactic nuclei. The possible contributions of thermal driving (by thepressure of radiation and electron-positron pairs) and of nonideal-MHD effects to the acceleration of the floware also discussed. It is pointed out that MHD acceleration is distinguished by being spatially extended andthat this property should be most noticeable in relativistic flows. It is argued that observational indicationsthat “superluminal” radio jets are accelerated over distances that far exceed the scale of the central black holesupport the magnetic-driving picture. Preliminary results of a comprehensive model that could be used to testthis interpretation are presented.
The acceleration and collimation of jets in a va-riety of astronomical objects are often attributed tothe action of magnetic fields (e.g., Livio 2000; Konigl& Pudritz 2000). It is commonly envisioned thatmagnetic field lines threading a rotating compact ob-ject or its surrounding accretion disk can efficientlytap the rotational energy of the source and accel-erate gas to high speeds through centrifugal and/ormagnetic pressure-gradient forces. It is argued thatthe hoop stresses of the twisted field lines can ac-count for the narrowness of many jets and that, atleast in some cases, alternative production mecha-
1Dept. of Astronomy & Astrophysics, U. Chicago, IL, USA.
nisms (such as thermal driving) can be excluded onobservational grounds.
Until fairly recently, much of the theoretical workon jets concentrated on the nonrelativistic regime.This has been motivated by the fact that most of theoutflows that were studied observationally (primar-ily large-scale extragalactic radio jets and jets fromyoung stellar objects) were inferred to move at non-relativistic speeds, and that even in the case of theblazar class of active galactic nuclei (AGNs), whererelativistic bulk flows were indicated by apparentsuperluminal motions and rapid Stokes-parametervariability, the implied (terminal) bulk Lorentz fac-tors γ∞ were typically not much greater than 1. Thissituation has now changed on account of the follow-
ing developments: (1) In the case of AGNs, apparentsuperluminal speeds that imply values of γ∞ as highas ∼ 40 c have been recorded (e.g., Jorstad et al.2001). (2) Observations of long-duration gamma-ray bursts (GRBs) and their afterglows have beenconvincingly interpreted in terms of ultrarelativistic(γ∞ ∼ 102 − 103) jets (e.g., Piran 1999). (3) Ap-parent superluminal motions of radio features withimplied values of γ∞ that can exceed ∼ 10 have beenmeasured also in Galactic black-hole and neutron-star binaries (e.g., Fender et al. 2004). These find-ings have highlighted the strong similarities amongthe various types of relativistic jet sources and havefocused attention on the question of their origin. Al-though the physical scales of the outflows can be verydifferent— AGN jets are inferred to emanate fromthe vicinity of a supermassive (∼ 108 − 1010 M)black hole, whereas long-duration GRB outflows areevidently associated with a newly formed stellar-mass black hole or rapidly rotating neutron star—itis believed that magnetic driving is still the under-lying driving mechanism in all of these cases (e.g.,Blandford 2002b). The current challenge for theo-rists is to devise quantitative models for the rela-tivistic regime that can be tested by the new obser-vations.
Recent developments in the modeling of relativis-tic magnetohydrodynamic (MHD) outflows are pre-sented in § 2 of this contribution. Applications toGRB and AGN sources are discussed in §§ 3 and 4,respectively. The conclusions and a brief outline ofwork in progress are given in § 5.
2. EXACT RELATIVISTIC-MHD SOLUTIONS
This section describes the derivation of exactrelativistic-MHD jet solutions representing initiallyPoynting-dominated configurations that convert alarge fraction of their electromagnetic energy intorelativistic bulk motion of baryons. For definiteness,it is assumed that the jet originates in a disk arounda central compact object (e.g., a stellar-mass blackhole in the case of a GRB source or a supermassiveblack hole in an AGN). It is further assumed that thedisk is threaded by a large-scale, well-ordered mag-netic field, although flow acceleration and collima-tion could conceivably be produced even if the fieldwere small-scale and tangled (e.g., Heinz & Begel-man 2000; Li 2002). Instead of tapping the rota-tional kinetic energy of the disk, the outflow mightharness the rotational energy of the central object—through the stellar magnetic field in the case of aneutron star (e.g., Usov 1994; Kluzniak & Ruder-
man 1998) or a field supported by the disk in thecase of a black hole (e.g., Blandford & Znajek 1977;van Putten & Levinson 2003)—but this should notaffect the qualitative nature of the outflow at largedistances from the origin. The quantitative detailscould, however, change from source to source as theydepend on the boundary conditions (magnetic fluxdistribution, angular velocity distribution, and massloading) at the origin. The basic properties of theoutflow are modeled on the assumption that it canbe properly described by the equations of special-relativistic, ideal MHD. While the exact behavior ofthe flow in the immediate vicinity of the compact ob-jects requires a fully general-relativistic treatment, itis shown below that the bulk of the jet accelerationgenerally occurs on scales where these effects can beneglected. The application of the ideal MHD equa-tions to the problem of highly relativistic outflowshas been questioned by Blandford (2002a; see alsoLyutikov & Blandford 2002), who adopted instead aforce-free electromagnetic description (which has thebenefit of being computationally more tractable); itis, however, worth noting that a force-free behaviorcan be recovered from the relativistic MHD formula-tion as a limiting case of negligible particle mass andpressure. Although finite-conductivity effects (asso-ciated with magnetic reconnection) might play a rolein the jets (see § 5), the formal validity conditions forideal MHD are typically well satisfied for the solu-tions presented here (e.g., Vlahakis & Konigl 2001).
The initial (subscript i) field amplitude at thebase of the flow that is required for driving the out-flow can be inferred from an estimate of the injectedenergy. For example, in the case of GRBs, Ei =(Poynting flux) × (surface area) × (burst duration).For a jet element with an initial cylindrical radius $i
and radial width (∆$)i, Ei ≈ cEiBφ,i$i(∆$)i∆t,where the electric field is given by E = BpVφ/c −BφVp/c (with the subscripts p and φ denoting thepoloidal and azimuthal components, respectively).For characteristic parameters of long-duration GRBs(Ei ≈ 1052 ergs, $i ∼ (∆$)i ≈ 106 cm, ∆t ≈ 10 s),one infers Bi ∼ 1014 − 1015 G. This field is mostplausibly generated by differential-rotation amplifi-cation of a much weaker poloidal field componentthat was originally present in the disk (see discus-sion in Vlahakis & Konigl 2003b). If |Bp,i/Bφ,i| >1, a trans-Alfvenic outflow is produced, whereas if|Bφ,i/Bp,i| > 1, the outflow is super-Alfvenic fromthe start. The latter situation may correspond toamplified toroidal flux loops that have been discon-nected by magnetic reconnection and escape fromthe disk surface in a nonsteady fashion.
The system of equations of special-relativistic,ideal MHD consists of the Maxwell and Eulerequations together with the mass and specific-entropy conservation relations. Assuming axisym-metry [∂/∂φ = 0 in spherical (r , θ , φ) and cylin-drical ($ ,φ , z) coordinates] and a steady state, thefull set of equations can be partially integrated toyield several field-line constants (e.g., Lovelace etal. 1986). The assumption of a steady state isapplicable if the magnetic flux distribution at thesource is approximately constant on the time scaleof interest and, in the case of distinct ejections asin GRBs, if the Lorentz factor of the poloidal mo-tion of the ejected shell satisfies γp 1 (Vlahakis& Konigl 2003a). The field-line constants are thetotal specific angular momentum L(A), the field an-gular velocity Ω(A), the “magnetization parameter”σM(A) (with the mass-to-magnetic flux ratio givenby AΩ2/σMc3), the adiabat Q(A) ≡ P/ρΓ
0 (withthe index Γ being either 4/3 or 5/3, correspond-ing to relativistic and nonrelativistic temperatures,respectively), and the total energy-to-mass flux ra-tio µ(A)c2 = ξγc2 + (c/4π)(E|Bφ|/γρ0Vp). HereA = (1/2π)
∫ ∫Bp·dS is the poloidal magnetic flux
function (which identifies the field line), V is thebulk velocity, γ is the Lorentz factor, ρ0 and Pare the comoving matter density and pressure, andξc2 = c2 + 5P/2ρ0 is the specific enthalpy. Thevariable ξ makes it possible to incorporate thermaleffects into the model. Although purely hydrody-namic driving of relativistic jets can probably beruled out (see, e.g., Di Mateo et al. 2002 and Daigne& Mochkovitch 2002 for the GRB case, and Vla-hakis & Konigl 2004 for AGNs), thermal forces maynonetheless dominate the initial acceleration of mag-netic outflows (e.g., Meszaros, Laguna, & Rees 1993;see § 3). An initial value ξi 1 could correspond toa “hot” electron-positron/radiation component thatdominates the thermal pressure.
Vlahakis & Konigl (2003a,b; hereafter VK) ob-tained exact solutions of the above equations by in-tegrating the two remaining relations (the Bernoulliand transfield force-balance equations) under the as-sumption of radial self-similarity of the form r =F1(A)F2(θ) (see Figure 1). With this ansatz it ispossible to separate the (A , θ) variables if the follow-ing relations hold (cf. Vlahakis & Tsinganos 1998):F1(A) ∝ A1/F , L(A) ∝ A1/F , Ω(A) ∝ A−1/F ,Q(A) ∝ A−2(F−2)/3, µ(A) = const , and σM(A) =const (see Li, Chiueh, & Begelman 1992 and Con-topoulos 1994 for the “cold” limit of this model).The parameter F controls the distribution of thepoloidal current I: 2|I|/c = $|Bφ| = A1−1/FF(θ).
Fig. 1. Sketch of r self-similar field lines in the meridionalplane. For any two field lines A1 and A2, the ratio ofcylindrical distances for points corresponding to a givenvalue of θ is the same for all the cones θ = const.
Close to the origin the field is force-free: F(θ) ≈const , and hence $|Bφ| ∝ A1−1/F . Thus, the pa-rameter regime F > 1 corresponds to a current-carrying jet, with the poloidal current density be-ing antiparallel to the magnetic field (J‖ < 0; seeFigure 2). In this case the current tends to zeroas the symmetry axis is approached, so such solu-tions should provide a good representation of theconditions near the axis of a highly collimated flow.Conversely, solutions with F < 1 correspond to thereturn-current regime (in which the poloidal currentdensity is parallel to the field, J‖ > 0) and are mostsuitable at larger cylindrical distances. Although thedetailed global current distribution cannot be mod-eled using the self-similarity approach, one can nev-ertheless generate “hybrid” flow configurations thatcombine a current-carrying solution for low valuesof $ and a return-current solution for high valuesof $ (see Figure 4). Initially Poynting-dominatedflows that attain a rough equipartition between thekinetic and Poynting energy fluxes at large distancesfrom the origin have F close to 1. When F > 1 theLorentz force can collimate the flow to cylindricalasymptotics. For F < 1 the collimation is weakerand the flow only reaches conical asymptotics; how-ever, the acceleration is more efficient in this casein that a larger fraction of the Poynting flux is con-verted into kinetic energy.
The original radially self-similar MHD jet modelconstructed by Blandford & Payne (1982) in the non-relativistic regime corresponds to F = 0.75. Thus, in
Fig. 2. Sketch of two meridional field lines (solid) andthree meridional current lines (dashed). The meridionalcurrent lines represent the loci of constant total poloidalcurrent (I = const). The magnetic and electric forcesare shown for the current-carrying (J‖ < 0, left fieldline) and return-current (J‖ > 0, r ight field line) cases.
contrast with the relativistic solutions derived by VK(which have F > 1 near the origin), it has a singu-larity in I at $ = 0, a feature for which it has some-times been critiqued (although note that the vicinityof the axis can typically be excluded in practical diskapplications). A basic difference between the rela-tivistic and nonrelativistic self-similar models is theexistence of a characteristic speed, c, in the relativis-tic case, which precludes the incorporation of gravityinto the self-similar equations and a simple matchingof the outflow solution to a particular (e.g., Keple-rian) disk rotation law, as was done in the Blandford& Payne (1982) solution. [VK were able to mitigatethis limitation by allowing the height zc of the diskto vary (see Figure 1); for zc = 0, Ω ∝ 1/$ along theconical surface of the disk, whereas for zc > 0 the de-crease of Ω with $ is steeper.] The relativistic-MHDregime is further complicated by the fact that thedisplacement current and the charge density cannotbe neglected in the formulation.
The variable separation transforms the originalP.D.E.’s into O.D.E.’s, and one seeks a solutionby integrating these equations. The general prob-lem requires the specification of seven constraints:four associated with the boundary conditions at thesource and three determined by the regularity re-quirements at the critical points of the joint solution
of the Bernoulli and transfield equations. Represen-tative solutions that illustrate the general propertiesof magnetically driven relativistic jets are presentedin the next two sections.
3. APPLICATION TO GRB OUTFLOWS
This section is divided into two parts. First, ageneric solution that demonstrates the ability of themagnetic driving model to account for the inferredbasic properties of GRB outflows is presented andanalyzed. In the second part, the “baryon loading”problem for disk-driven GRB jets is addressed inthe context of an initially neutron-rich MHD outflowmodel, and additional observational implications ofthis picture are outlined.
3.1. Poynting Flux-Dominated MHD Jet Solutions
Long-duration (∼
> 2 s) GRBs have been inferredto arise in ultrarelativistic, highly collimated (open-ing half-angle θj ∼ 2 − 20) outflows of typical ki-netic energy EK ∼ 1051 ergs. Early models of GRBoutflows have interpreted them in terms of thermallydriven “fireballs” powered by neutrino emission ormagnetic field dissipation at the source. However,the current view is that magnetic fields provide themost plausible means of extracting the inferred en-ergy on the burst time scale (e.g., Meszaros & Rees1997; Di Mateo et al. 2002). VK verified that mag-netic fields can also guide, collimate, and acceleratethe flow. In particular, their derived semianalyticsolutions demonstrate that Poynting flux-dominateddisk outflows (either trans- or super-Alfvenic) cantransform
∼
> 50% of their magnetic energy into ki-netic energy of γ∞ ∼ 102 − 103 baryons.
A representative solution is shown in Figure 3.It describes a trans-Alfvenic flow in the current-carrying regime. The outflow is “hot” (ξi 1) andcorresponds to a “fast rotator” (µ ξi). Initially,the acceleration is predominantly thermal and themagnetic field only guides and collimates the flow.The behavior of the flow in this regime is completelyanalogous to that of a classical fireball, except thatthe spherical radius r in the scaling relations is re-placed by the cylindrical radius $ (see Vlahakis &Konigl 2001). The thermal acceleration zone ter-minates when ξ decreases to ∼ 1, at which pointγ ≈ ξi. The bulk of the acceleration, however, takesplace on larger scales in the magnetic accelerationregion, where the Poynting energy (represented bythe top curve in Figure 3a) is converted into kineticenergy (represented by the γ curve). The acceler-ation by the Lorentz force corresponds to the de-crease of |$Bφ| along the poloidal streamlines. The
Fig. 3. Illustrative relativistic-MHD solution of a GRBoutflow. (a) Poynting (top) and enthalpy (ξ) energyfluxes, normalized by the mass flux × c2, and the Lorentzfactor (γ) as functions of height along a fiducial magneticfield line. (b) Meridional projections of the innermost andoutermost field lines are shown on a logarithmic scale,along with a sketch of the black-hole/debris-disk system.The vertical lines mark the positions of various relevanttransition points along the innermost field line.
large value of γ∞ attained by the flow can be at-tributed to the large extent of the acceleration re-gion. In fact, the acceleration continues all the wayto the modified fast-magnetosonic surface (the “eventhorizon” for the propagation of fast-magnetosonicwaves), which is situated well beyond the classicalfast-magnetosonic surface (Figure 3b). The acceler-ation terminates when the flow collimates to a cylin-der (after which time $Bφ no longer varies along thestreamlines).
An extended region over which the magnetic“spring” uncoils and drives the flow is characteristicof MHD acceleration models and distinguishes themfrom purely hydrodynamic scenarios. This mech-anism operates also in nonrelativistic MHD flows(Vlahakis et al. 2000), but the effect should be easierto discern in relativistic jets. The relatively high effi-ciency of Poynting-to-kinetic energy conversion thatcan be attained in this model makes it possible to at-tribute at least some of the γ-ray emission in GRBsto colliding shells (the “internal shock” scenario; e.g.,Piran 1999). On the other hand, the fact that a sig-nificant fraction of the Poynting energy may remain
untapped makes it in principle possible to utilizethe magnetic energy directly in the emission process(e.g., Spruit, Daigne, & Drenkhahn 2001) and mightalso have implications to the subsequent afterglowemission (e.g., the relative weakness of the emissionfrom the reverse shock driven into the deceleratingejecta).
3.2. Neutron-Rich MHD Jets
If one compares the estimated mass of protonsin a typical long-duration GRB jet [Mproton = 3 ×10−6(EK/1051 ergs)(γ∞/200)−1 M] with the min-imum mass of the debris disk from which the jetcould plausibly originate (obtained under the as-sumption that at most ∼ 10% of the disk gravita-tional potential energy is converted into outflow ki-netic energy), one finds that the outflow can com-prise at most ∼ 10−4 of the disk mass. However,disk outflow models that utilize a large fraction ofthe disk potential energy typically also entail sub-stantial mass loading—this is the essence of theGRB “baryon loading” problem. One approach tothis issue has been to postulate that the outflowemerges along magnetic field lines that thread theblack-hole event horizon and not the disk, but thenthe converse problem — how to avoid having toofew baryons — must be addressed (e.g., Levinson &Eichler 2003). A possible resolution of the problemin the context of disk-fed jet models was proposedby Fuller, Pruet, & Abazajian (2000), who notedthat such outflows are expected to be neutron-rich[the initial neutron/proton ratios could be as highas (n/p)i ∼ 20− 30]. Since only the charged outflowcomponent couples to the electromagnetic field, theneutrons could potentially decouple from the pro-tons before the latter attain their terminal Lorentzfactor. In this picture, the inferred value of Mproton
may represent only a small fraction of the total bary-onic mass ejected from the disk, which would allevi-ate the loading problem. However, it can be shownthat, for purely hydrodynamic outflows, the Lorentzfactor γd at decoupling is at least a few times 102.This implies that γd/γ∞ ∼ 1 and hence that the pro-tons end up with only a small fraction of the injectedenergy, which is not a satisfactory resolution of theproblem.
Vlahakis, Peng, & Konigl (2003) argued that theincorporation of magnetic fields makes it in principlepossible to attain γd γ∞ and thereby reclaim thepromise of the Fuller et al. proposal. They mod-eled the pre-decoupling region using a “hot” super-Alfvenic outflow solution. A general property of suchsolutions is that, during the initial thermal acceler-
Fig. 4. Illustrative relativistic-MHD solution of aneutron-rich outflow. (a) Components of the total en-ergy flux, normalized by the mass flux × c2, as functionsof height along a fiducial magnetic field line. The Poynt-ing and enthalpy curves are discontinuous at the decou-pling point, reflecting the decrease in the mass flux offield-coupled gas above that point. (b) Components ofthe proton–neutron drift velocity.
ation phase, a fraction of the enthalpy flux is con-verted into Poynting flux . This reduces the accel-eration rate, so at the point of decoupling (whenVproton−Vneutron ∼ c) the Lorentz factor is still com-paratively low. The energy deposited into the Poynt-ing flux is returned to the matter beyond the de-coupling point as kinetic energy, thereby enhancingthe acceleration efficiency of the proton component.The end result is a large γ∞/γd ratio and comparableterminal kinetic energies in the proton and neutroncomponents, in clear contradistinction to the purelyhydrodynamic solutions.
An illustrative solution [with (n/p)i = 30] isshown in Figure 4. This is a “hybrid” config-uration (see § 2) in that the pre-decoupling andpost-decoupling regions correspond to the current-carrying (F = 1.05) and return-current (F = 0.1)regimes, respectively. In this case the enthalpy fluxinitially dominates the Poynting flux(“slow rotator”regime, ξi ≈ µ), but the charged component never-theless collimates from an initial opening half-angleof 55 to θj ≈ 7. The thermal acceleration effec-
tively terminates at a height z ≈ 109 cm above thedisk, and the neutrons decouple from the protonsat zd ≈ 1013 cm with γd ≈ 15. By the time ofdecoupling the neutrons have acquired ∼ 2/3 of theinjected energy, with the remainder residing predom-inantly in the electromagnetic field. The latter por-tion is then transferred with almost 100% efficiencyinto proton kinetic energy, so that, ultimately, theprotons have γ∞ = 200 and EK,proton ≈ 1051 ergs ≈0.5EK,neutron. The proton jet thus carries ∼ 1/3 ofthe injected energy but only ∼ 3% of the injectedmass. Figure 4b shows that, even though the decou-pling in this case is initiated by the growth of then–p drift velocity along the poloidal magnetic field,there is also a transverse drift component (inducedby the ongoing magnetic collimation), which at thetime of decoupling is |Vneutron⊥| ∼ 0.1 c.
The decoupled neutrons undergo β decay intoprotons at a distance ∼ 4 × 1014 (γd/15) cm. Incontrast with the situation in purely hydrodynamicoutflow models (Pruet & Dalal 2002; Beloborodov2003), there may well be no interaction between thetwo decoupled components in the MHD case sincetheir motions are not collinear. The latter scenariothus gives rise to a 2-component outflow: an outer(wider) component (comprising the decoupled neu-trons) that carries most of the energy and may beresponsible (after the neutrons decay) for the bulkof the optical/radio afterglow, and an inner (nar-rower) component (comprising the original protons)that accounts for the prompt γ-rays and possibly alsofor some of the X-ray afterglow. A 2-component out-flow of this type was inferred in GRB 030329 (Bergeret al. 2003; Sheth et al. 2003). General implicationsof such a model to the afterglow lightcurves and tothe energetics of GRB and X-ray flash sources wereconsidered by Peng, Konigl, & Granot (2005; see Fig-ure 5). A 2-component jet interpretation could po-tentially also help to account for some of the appar-ent peculiarities of the early afterglow emission thatwere revealed by recent Swift observations (Granot,Konigl, & Piran 2006).
4. APPLICATION TO BLAZAR JETS
The MHD acceleration model also provides a nat-ural interpretation of “superluminal” AGN jets. Incontrast with GRB outflows (or jets from Galactic X-ray binaries, for that matter), the acceleration zonein blazars is evidently resolvable by radio interferom-etry. This makes relativistic AGN jets prime candi-dates for testing and constraining this model. Theevidence for parsec-scale acceleration in blazars andan illustrative model fit are presented in the first
Fig. 5. R-band afterglow lightcurves for a two-component jet with representative parameters. Resultsare given for different ratios of the kinetic energies ofthe two components, with the total outflow energy fixedat 1051 ergs. The contribution of the narrow compo-nent (θj,n = 0.05), wide component (θj,w = 0.15), andtheir sum is shown by the dashed, dash-dotted, and solidcurves, respectively. The nominal deceleration and jet-break times for the two components are also indicated.
part of this section. Further observational tests areconsidered in § 4.2.
4.1. Parsec-Scale Accelerations
A growing body of data indicates that relativis-tic AGN jets undergo the bulk of their accelerationon parsec (more generally, ∼ 0.1 − 10 pc) scales.In particular, the absence of bulk-Comptonizationspectral signatures in blazars has been argued to im-ply that Lorentz factors
∼
> 10 must be attained onscales
∼
> 1017 cm (Sikora et al. 2005). In the case ofthe quasar 3C 345, Unwin et al. (1997) combined aVLBI proper-motion measurement of the jet compo-nent C7 with an inference of the Doppler factor froman X-ray emission measurement (interpreted as SSCradiation) to deduce an acceleration from γ ∼ 5 toγ
∼
> 10 over r ∼ 3−20 pc. Piner et al. (2003) inferredan acceleration from γ = 8 at r < 5.8 pc to γ = 13at r ≈ 17.4 pc in the quasar 3C 279 jet using a sim-ilar approach. Extended acceleration in the 3C 345jet has been independently indicated by the increasein apparent component speed with separation from
the nucleus (Lobanov & Roland 2005) and by theobserved luminosity variations of the moving com-ponents (Lobanov & Zensus 1999). Similar effects inother blazars (e.g., Homan et al. 2001a) suggest thatparsec-scale acceleration may be a common featureof AGN jets.
The inferred large-scale accelerations in AGN jetsare very hard to interpret naturally in purely hydro-dynamic terms since in the latter class of models theacceleration generally saturates on the much smallerscale of the central mass distribution, which sets thesize of the sonic “nozzle.” Extended acceleration is,however, a signature of MHD driving, as discussed in§ 3.1, and one can reproduce the observed behaviorusing the semianalytic model described in § 2 (Vla-hakis & Konigl 2004). Figure 6 shows an illustrativefit to the 3C 345 data presented by Unwin et al.(1997). It is worth bearing in mind, however, thatthe kinematic data used in obtaining this fit do notuniquely determine the solution: exactly the sameflow speeds and field-line shape are obtained if thedensity, particle pressure, and squared amplitudesof the magnetic field components are rescaled by thesame factor.
Panels (c)–(g) in Figure 6 show various quan-tities as functions of $/$A (which, in turn, is afunction of the polar angle θ) along the outermostfield line. (Here $A is the Alfven lever arm, and$A,out = 150$A,in = 4.1×10−2 pc in this example.)Panel (c) depicts the force densities in the poloidaldirection, showing that thermal and centrifugal ef-fects are important only near the origin, with themagnetic pressure-gradient force rapidly becomingthe dominant driving mechanism. Panel (d), in turn,shows that, asymptotically, an approximate equipar-tition between the kinetic and Poynting fluxes is at-tained (γ∞ ≈ µ/2). This panel also demonstratesthat the model fit reproduces well the inferred accel-eration of component C7. For the adopted fiducialparameters, this component is predicted to continueaccelerating up to γ∞ ≈ 35. Interestingly, Lorentzfactors of this order have been inferred in the moredistant components (in particular C3 and C5) of the3C 345 jet (Lobanov & Zensus 1999).
Panels (e) and (f) depict the bulk velocity com-ponents and the temperature, respectively. Even ifthe initial temperature is as high as ∼ 1012 K, ther-mal effects are overall insignificant—this is thus aneffectively “cold” (ξi ≈ 1) outflow. Panel (g) showsthat the magnetic field is primarily poloidal near theorigin of the flow but becomes predominantly az-imuthal further downstream. Asymptotically, Bz ∝$−2 ,−Bφ ∝ $−1, and also B$ << Bz—a signature
Fig. 6. r self-similar solution describing the superluminal jet in 3C 345. (a) Poloidal field-line shape on a logarithmicscale. (b) Mass-loss rate as a function of $out/$in, the ratio of the outermost and innermost disk radii. The remainingpanels are discussed in the text.
of cylindrical collimation. As in the case of the GRBoutflow solutions presented in § 3, the results pre-sented in Figure 6 demonstrate that MHD drivingimplies that jet collimation (and not just accelera-tion) takes place over an extended region (althoughthe rate of field-line bending is reduced with increas-ing Lorentz factor as the effective inertia goes up andthe electric force becomes nearly as large as — andalmost cancels out — the transverse magentic force).This predicted behavior is also supported by obser-vations of relativistic jets (e.g., Junor, Biretta, &Livio 1999). It is conceivable that a slower wind fromthe outer regions of the accretion disk that feeds thecentral source could aid in the collimation of the rel-ativistic outflow that emanates from the innermostregion (e.g., Bogovalov & Tsinganos 2005).
4.2. Comprehensive Modeling of Parsec-Scale Jets
The VLBI-traced paths of superluminal compo-nents in blazar jets are typically curved and are oftenwell approximated by helical trajectories (see, e.g.,Steffen et al. 1995 for the case of 3C 345). A mag-netically driven jet from a circumnuclear accretiondisk could exhibit a helical flow pattern if it wereejected from a localized region (in both r and φ) onthe disk surface. Physically, this would correspondto mass loading of only an isolated flux bundle thatthreads the disk (see Camenzind & Krockenberger1992). Figure 7 shows a realization of this possibil-ity corresponding to the 3C 345 model fit presented
0
2·1019
4·1019
6·1019z(cm)
-1·101801·1018
x(cm)
-1·101801·1018
y(cm)
0
2·1019
4·1019
6·1019z(cm)
-1·101801·1018
y(cm)
Fig. 7. Visualization of the 3C 345 jet model of Figure 6,showing the shape of magnetic field lines anchored nearthe fiducial outermost disk radius (thin line) and an iso-lated fluid streamline emerging from that radius (thickline).
in Figure 6. In particular, this figure depicts theshape of a streamline that originates near the outerboundary of the model disk (along which the motionclosely reproduces the acceleration data for compo-nent C7). Also shown is the shape of the magneticfield line on the surface of an axisymmetric outflowfrom the same disk radius—the difference betweenthese two curves can be understood from the factthat, whereas the poloidal velocity is parallel to thepoloidal field under ideal-MHD conditions, the mat-ter angular velocity is close to that of the field lineonly near the base of the flow and becomes muchsmaller further out. Figure 8 demonstrates that the
Fig. 8. Top: projection of the streamline shown in Fig-ure 7 on the plane of the sky, with the x axis correspond-ing to the projected jet axis. Bottom: data points forcomponent C7 in the 3C 345 jet (from Lobanov 1996).
model fit to the projected trajectory on the plane ofthe sky is in good agreement with the observations.
The incorporation of observational input on thejet’s kinematic properties could help constrain themodel parameters and might be useful in distinguish-ing a “minimal” model, wherein magnetic field ef-fects account for the observed acceleration as wellas for the helical trajectories, from alternative inter-pretations in which the jet’s shape is due to othereffects (such as current-driven or Kelvin-Helmholtzinstabilities—e.g., Hardee 2000, or gravitationallyinduced precession at the source—e.g., Scheuer 1992;Kaastra & Roos 1992; Katz 1997). In fitting themotion of a distinct jet component along an isolatedstreamline, one needs to specify the initial disk ra-dius r0 and azimuthal angle φ0 at the disk surface aswell as the angle θobs between the line of sight andthe jet axis. One can attempt to obtain approximatevalues for these parameters by trying to optimize thefit to the time evolution of the apparent speed andDoppler factor [from which the evolution of γ and θcan be derived by using βapp = β sin θ/(1 − β cos θ)and D−1 = γ(1 − β cos θ)] together with the fit tothe shape of the projected trajectory.
Figure 9a depicts a tentative realization of thisprocedure for component C7 in 3C 345, which hasyielded r0 ≈ 2 × 1016 cm, φ0 ≈ 180, and θobs ≈ 9.
1991 1992 1993 1994t (yr)
viewing angle (degr)
5 x (Doppler factor)
10
20
30
40
50
60
0.2 0.4 0.6 0.8 1 1.2(mas)
Lorentz factor
apparent velocity / c
2
4
6
8
10
Fig. 9. Preliminary fits to the evolution of componentC7 in 3C 345, using the magnetic outflow model shownin Figure 6. Top: (a) Doppler factor and viewing angle(thin and thick curves, resp.) as functions of time inthe observer’s frame. Bottom: (b) Apparent speed andLorentz factor (thin and thick curves, resp.) as functionsof distance along the projected jet axis.
Figure 9b demonstrates that a superluminal compo-nent moving along a helical trajectory may exhibitan apparent deceleration during an early phase ofits evolution even as its Lorentz factor continues toincrease. This behavior has been observed in knotC7 of 3C 345 (e.g., Zensus 1997) as well as in othersuperluminal jet components.
5. SUMMARY AND FUTURE PROSPECTS
The main results discussed in this contributioncan be summarized as follows:
• Magnetic fields likely play a prominent role inthe extraction of rotational energy at the sourceas well as in the guiding, acceleration, and col-limation of relativistic outflows from compactastronomical objects (GRB sources, AGNs, andGalactic black-hole and neutron-star binaries).
• Exact semianalytic solutions of special-relativistic ideal MHD can account for GRBand AGN jets.
• An extended acceleration region is a distinguish-ing characteristic of MHD driving of relativisticoutflows. In the case of AGN jets the acceler-ation zone can probably be resolved by radiointerferometry, which may make it possible totest and constrain this model.
It would be desirable to complement the semiana-lytic approach that has been used to establish the ba-
sic properties of relativistic MHD outflows with nu-merical work. By suitably modifying the self-similarsolutions to apply to a finite grid, one could employthe semianalytic results for initiating and testingnumerical calculations. The envisioned simulationscould, in turn, be used to test the generality and sta-bility of the semianalytic solutions; they should alsobe useful in exploring new parameter-space regimesand the behavior of nonsteady outflows. There arealready several existing relativistic-MHD numericalcodes that could potentially be applied to this prob-lem, and some of them even incorporate general rel-ativity and can be used to model the details of theoutflow launching process (e.g., McKinney & Gam-mie 2004; De Villiers et al. 2005; Komissarov 2005).It is worth noting, however, that no existing numer-ical calculation spans the range in scales that is re-quired for explicitly demonstrating the accelerationto a high Lorentz factor; in fact, the high termi-nal Lorentz factors inferred in some of these simula-tions (e.g., McKinney 2006) only represent estimatesof the maximum achievable (rather than determina-tions of the actually attained) values (see also Komis-sarov 2005). In this respect, the self-similar solu-tions are currently still the best indicators of theglobal behavior of relativistic MHD outflows. Inas-much as these solutions do not incorporate gravity,they do not accurately model the launching pro-cess, although this deficiency could be mitigated byemploying non–self-similar semianalytic solutions forthe base of the jet (e.g., Levinson 2006) to locate theslow-magnetosonic surface of the outflow.
Even under the assumption that ideal MHD isa good approximation for treating the dynamics ofrelativistic jets, departures from this state could con-ceivably lead to a direct conversion of magnetic en-ergy into nonthermal radiation. This possibility hadbeen considered in the context of pulsar-type scenar-ios for GRBs (e.g., Usov 1994; Thompson 1994) andwas also discussed in connection with AGN jets (e.g.,Choudhuri & Konigl 1986; Romanova & Lovelace1997.) The dissipation of magnetic energy natu-rally results in a decrease in the azimuthal magneticfield component along the outflow, and the magneticpressure gradient established in this fashion couldcontribute to the flow acceleration (Drenkhahn &Spruit 2002). The incorporation of this effect into asemianalytic jet-acceleration model is currently un-der study.
The potential importance of a two-componentGRB jet of the type that arises naturally in the ini-tially neutron-rich MHD outflow model (§ 3.2) pro-vides a motivation for further studies of this scenario.
The original dynamical treatment of the neutron-rich outflow (Vlahakis et al. 2003) was based onthe single-fluid equations. A more general treat-ment, which considers the neutrons and the charged-particle component separately, would give a moreaccurate representation of the neutron decouplingprocess. Further insight into this process could begained by combining the dynamical model with de-tailed thermal-structure and nuclear-reactions calcu-lations.
To fully capitalize on the potential of superlu-minal jet sources for testing the basic magnetic ac-celeration model, one can supplement the kinematicconstraints considered in § 4.2 with constraints pro-vided by the radiative properties of the outflow.Since the dominant emission process in these jetsis evidently synchrotron radiation, which is inti-mately tied to the intrinsic magnetic field structure,one could in principle gain valuable insights froma comparison of the emitted flux density, the lin-ear and circular polarizations, and the Faraday ro-tation measure with the model predictions. Workon these generalizations has already begun. Theresults of linear-polarization and Faraday-rotationmeasurements in relativistic parsec-scale AGN jetsare consistent with the presence of a pervasive helicalmagnetic-field configuration in these sources (e.g.,Gabuzda 2003; Gabuzda, Murray, & Cronin 2004;Lyutikov, Pariev, & Gabuzda 2005). Circular polar-ization has also been detected in many blazars, andthere is a clear indication that these sources exhibit apreferred handedness (or sign) that may persist fordecades (e.g., Homan, Attridge, & Wardle 2001b).Perhaps the most natural interpretation of this find-ing is that it reflects the twist imprinted on the or-dered magnetic field in the jet by the rotation of thesource (e.g., Ensslin 2003). Given that model fits tohelical component trajectories in a superluminal jetcan in principle yield an independent determinationof the sense of rotation of the source (see Figs. 7–9),a measurement of the sign of circular polarization insuch a jet might provide a check on the underlyingkinematic model.
I am grateful to Nektarios Vlahakis, Fang Peng,and Jonathan Granot for their contributions to thework reported in this article. This research was sup-ported in part by NASA Astrophysics Theory Pro-gram grant NAG5-12635.
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