Dissipation Efficiency of Reconfinement Shocks in Relativistic Jets
aa r X i v : . [ a s t r o - ph . H E ] O c t Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 20 November 2018 (MN L A TEX style file v2.2)
Dissipation Efficiency of Reconfinement Shocks inRelativistic Jets
Krzysztof Nalewajko ⋆ JILA, University of Colorado and National Institute of Standards and Technology, 440 UCB, Boulder, CO 80309, USANicolaus Copernicus Astronomical Centre, Bartycka 18, 00-716 Warsaw, Poland
20 November 2018
ABSTRACT
We calculate the dissipation efficiency of relativistic reconfinement shocks. Buildingon previous work (Nalewajko & Sikora 2009), we consider different distributions of theexternal pressure. The average dissipation efficiency ǫ diss is a function of the productof two parameters – the jet Lorentz factor Γ j and the reconfinement angle Θ r , whichis related to the opening angle Θ j and the external pressure index η . The spatial dis-tribution of the dissipation rate strongly depends on η . We discuss the significanceof these results for the properties of relativistic jets in gamma-ray bursts and activegalactic nuclei and propose that reconfinement shocks may explain a very high dis-sipation efficiency of the former and a moderate dissipation efficiency of the latter.Finally, we estimate the dissipation efficiency of the reconfinement shock associatedwith the quasi-stationary knot HST-1 in the jet of radio galaxy M87 and show that itis roughly consistent with the observational constraints. Key words: galaxies: individual: M87 – galaxies: jets – gamma-ray burst: general –shock waves.
Relativistic jets stand behind the brightest cosmic phenom-ena: gamma-ray bursts (GRBs) and blazars, a subclass ofactive galactic nuclei (AGNs). Their extreme isotropic lumi-nosities, up to ∼ erg s − for the former (GRB 080319B;Racusin et al. 2008) and up to ∼ erg s − for the latter(3C 454.3; Abdo et al. 2011), cannot be plausibly explainedwithout the relativistic Doppler effect. However, even takingthis into account, in order for this radiation to be producedin the co-moving reference frame, a substantial fraction ofthe jet mechanical power, including the particle rest energyflux, needs to be dissipated, then transferred into a popula-tion of ultra-relativistic particles in a non-thermal accelera-tion process and finally radiated away through non-thermalradiative mechanisms. The total efficiency of these processescan be estimated observationally if the total jet power isknown. In the case of GRBs, it can be well constrained byenergetics of the afterglow phase and the radiative efficiencyof the prompt phase has been claimed reach values up to ∼
90% (Zhang et al. 2007). In the case of blazars, these esti-mates are less certain, but typical values for their luminousclass of Flat Spectrum Radio Quasars (FSRQs) are ∼ ⋆ E-mail: [email protected] energy dissipation in relativistic jets must be at least com-parable to these observational constraints.The most widely discussed means of energy dissipationin relativistic jets are shock waves, magnetic reconnectionand instabilities. Shocks can arise within a jet when tworegions propagating with substantially different bulk veloc-ities collide with each other. Such internal shocks providedthe basic framework for theoretical models of blazars ( e.g. ,Blandford & K¨onigl 1979) and GRBs ( e.g. , Rees & Meszaros1994). However, these models have been questioned on thegrounds that they cannot account for required dissipationefficiencies. This is especially clear in the case of GRBs,for which several alternative models have been recently pro-posed, based on magnetic reconnection ( e.g. , McKinney &Uzdensky 2010; Zhang & Yan 2011) or relativistic turbu-lence ( e.g. , Narayan & Kumar 2009). In the case of blazars,detailed calculations showed that a substantial contrast ofinitial Lorentz factors must be assumed (Spada et al. 2001).However, the occurrence of such a velocity contrast cannotbe verified with current models of jet formation and accel-eration. Moreover, if the velocity modulations are related toprocesses at the black hole horizon scale, the internal shocksmodel predicts a particular length scale, a fraction of a par-sec, over which such shocks develop. There are now severalarguments for the bulk emission of luminous blazars beingproduced at much larger distances from the central blackhole ( e.g. , Sikora et al. 2008; Agudo et al. 2011). c (cid:13) Nalewajko
The other possibility for the shocks is that they resultfrom the interaction between the jet and its environment. Inthe case of GRBs, the jet is a transient phenomenon and hasto plough through its host star and the interstellar medium,forming an external shock that dominates during the af-terglow phase ( e.g. , Meszaros & Rees 1997). In the caseof blazars, the jet is relatively persistent and propagatesroughly along a tunnel drilled over time, so perpendicularexternal shocks are not usually considered. However, the ex-ternal medium can exert a substantial pressure on the jetboundary, forcing it to recollimate and triggering a recon-finement shock.Reconfinement shocks were first discussed by Sanders(1983) in the context of the kpc-scale jet of the radio galaxyNGC 315. The first analytical models were introduced byCant´o et al. (1989) in the non-relativistic regime applicableto the jets of young stellar objects (YSOs) and by Komis-sarov & Falle (1997) in the relativistic regime. A uniquesignature of reconfinement shocks is that, unless the jet orexternal medium parameters vary significantly on the dy-namical time scale, they would be observed as stationarypatterns. Daly & Marscher (1988) interpreted a stationaryknot in the pc-scale jet of radio quasar 4C 39.25 as a nozzle ofthe reconfinement shock. More recently, a stationary knotHST-1 has been discovered in the jet of radio galaxy M87at the 100 pc scale (Biretta et al. 1999) and subsequentlyit underwent a spectacular multiwavelength outburst ( e.g. Harris et al. 2006). Stawarz et al. (2006) showed that theassociation of this feature with a reconfinement shock is con-sistent with both the properties of the host galaxy and theestimated jet power. However, short variability time scalesrequired a very compact emitting region. Bromberg & Levin-son (2009) showed that efficient focusing of the shocked jetflow is possible, but requires substantial cooling of the post-shock plasma. Reconfinement shocks were also studied inthe context of GRBs (Bromberg & Levinson 2007).The problem of dissipation efficiency of relativistic re-confinement shocks was first studied in Nalewajko & Sikora(2009), hereafter Paper I. It was found that the dissipationefficiency ǫ diss depends strongly on the product of the jetLorentz factor Γ j and the opening angle Θ j . Here, we gener-alise this result, taking into account different distributionsof the external pressure. We also show how this result canbe applied to both GRBs and the jets of active galactic nu-clei. Because GRB jets are characterised by wide openingangles, reconfinement shocks provide a natural explanationof their high radiative efficiency in the prompt phase. InAGN jets, the efficiency of reconfinement shocks is muchlower, because collimation by a continuous medium limitsthe opening angle. We also estimate the efficiency of the re-confinement shock associated with the HST-1 knot in thejet of M87 and show that it is roughly consistent with theobserved luminosity of this radio galaxy.In Section 2, we present our simple model of the struc-ture of relativistic reconfinement shocks. The dependence ofthe dissipation efficiency on model parameters is discussedin Section 3. In Section 4, we discuss the applications of After the 2005 outburst this knot is no longer stationary andpropagates with an apparent velocity of ∼ . c (Giovannini et al.2010). Figure 1.
Geometric parameters of our reconfinement shockmodel. The jet is symmetric around the z axis and propagatesfrom its origin at z = 0 towards the reconfinement point at z = z r .The opening angle Θ j , the reconfinement angle Θ r and the max-imum jet width r m are indicated. these results to astrophysical relativistic jets. Conclusionsare given in Section 5. Reconfinement shocks result from the interaction betweena jet and its surrounding medium. The simplest model ofsuch a problem involves a cold, unmagnetized, sphericallysymmetric jet of Lorentz factor Γ j , opening angle Θ j andtotal power L j ; and a static medium of pressure distributiongiven by p e ( z ) ∝ z − η , where η < z is thecoordinate measured along the jet axis. Figure 1 shows thegeometric parameters of the reconfinement shock front of ra-dius r s ( z ), including reconfinement length z r , reconfinementangle Θ r and maximum jet width r m ; as well as the contactdiscontinuity of radius r c ( z ). Parameters measured immedi-ately upstream and downstream of the reconfinement shockare denoted with subscripts ’j’ and ’s’, respectively.The system of shock jump equations is β s cos( θ s − α s ) = β j cos( θ j − α s ) , (1) u s ρ s sin( θ s − α s ) = u j ρ j sin( θ j − α s ) , (2) u w s sin ( θ s − α s ) + p s = u ρ j c sin ( θ j − α s ) , (3)Γ s u s w s sin( θ s − α s ) = Γ j u j ρ j c sin( θ j − α s ) , (4)where β = v/c is the dimensionless velocity, u = Γ β is thedimensionless four-velocity, w = ρc + p + e is the relativisticenthalpy in the comoving frame, ρ is the mass density, e isthe thermal energy density, α s is the inclination of the shockfront with respect to the jet axis and θ j , s are the inclinationsof the velocity vectors. It is assumed that p j = 0. Given allthe parameters of the upstream plasma, this system can besolved when the post-shock pressure p s is given. In Paper I,we noted that the structure of the shocked zone, the regionbetween the shock front and the contact discontinuity, canbe quite complex . In principle, p s ( z ) < p e ( z ), so that thetransverse pressure gradient can focus the post-shock flow.However, since the results on the dissipation efficiency pre-sented in Paper I are not very sensitive to the treatment ofthe shocked zone, we use the simple ’Model 1’ from PaperI and assume that p s ( z ) = p e ( z ). The main improvement isthat we take a self-consistent equation of state p = ( γ − e with approximate adiabatic index γ = 12 p + 5 ρc p + 3 ρc , (5) For a comprehensive description of the shocked jet zone seeKohler et al. (2011). c (cid:13) , 000–000 he Efficiency of Reconfinement Shocks based on Ryu et al. (2006).The local dissipation efficiency is defined as ǫ diss ≡ f diss f kin , j ≡ f kin , j − f kin , s f kin , j , (6)where f kin = (Γ − β ⊥ ρc is the kinetic energy flux density, f diss is the dissipated energy flux density and β ⊥ is the di-mensionless velocity component perpendicular to the shockfront. Under the assumption of the cold upstream plasma,it can be simplified to ǫ diss = Γ j − Γ s Γ j − . (7) In Paper I, we studied the dependence of the average dis-sipation efficiency on Γ j and Θ j for the case of η = 0, i.e. uniform external pressure. We found that the efficiency de-pends sensitively on the product Γ j Θ j . For Γ j Θ j <
1, anapproximate scaling law ǫ diss ∼ j Θ j ) can be used. ForΓ j Θ j >
1, very high values can be achieved.Here, we have additionally calculated the average dis-sipation efficiency for different values of the pressure index η . The results are shown in Figure 2. Instead of the openingangle Θ j , the reconfinement angle Θ r is used, multiplied byΓ j , on the horizontal axis. As has been shown by Komis-sarov & Falle (1997), the relation between these two anglesis Θ r ∼ δ Θ j , where δ = 1 − η/
2. For η = 0 we have Θ r ∼ Θ j ,so these results are consistent with the findings of Paper I.We thus generalise the previous result and show that thedissipation efficiency is determined by a single parameter inthe three-dimensional space (Γ j , Θ j , η ). In Figure 2, we alsoplot a slightly different scaling law , ǫ diss = 8%(Γ j Θ r ) . Anoticeable discrepancy for Γ j Θ r < η = 1 . dz for different external pressure indices η . Theprofiles of dissipated energy depend very strongly on η . Forthe flat external pressure distribution ( η = 0), most ofthe energy is dissipated beyond the half of the reconfine-ment length. The amount of dissipated energy tends to 0 as z → z r , even though the dissipation efficiency increases with z , since the shock becomes less and less oblique and conse-quently Γ s decreases. But this increase in efficiency is muchslower than a decrease in the jet cross-section and hencea decrease in the jet kinetic energy flux per unit dz . Themain dissipation region shifts closer to the jet origin withincreasing η . The peak of energy dissipation rate z diss , max islocated at ∼ . z r for η = 0, at ∼ . z r for η = 0 . ∼ . z r for η = 1. For η = 1 .
5, the dissipation profilechanges to monotonically decreasing with z and the bulk ofthe dissipation takes place very close to the jet origin.Figure 4 shows the dependence of z diss , max on the jet The reason for the change of the normalising factor in thepower-law scaling is that in Paper I the value for Γ j Θ j = 1has been used, while here we require a good overall match for0 . . Γ j Θ r . ε d i ss Γ j Θ r η = 0 η = 0.5 η = 1 η = 1.58%( Γ j Θ r ) Figure 2.
Average dissipation efficiency as a function of the prod-uct of the jet Lorentz factor Γ j and the reconfinement angle Θ r .There are 4 families of models for the case of flat external pressure( η = 0), all plotted with red solid lines , with varying Θ j and Γ j fixed at values 5, 10, 20 and 40. Models for η > j = 10. A power-law scaling valid for Γ j Θ r < dashed black line . f d i ss z r d S / ( F d i ss d z ) z / z r η = 0 η = 0.5 η = 1 η = 1.5 Figure 3.
Energy dissipation rate per unit of jet length dz for different external pressure indices η . The length scale z isnormalised to the reconfinement length z r . The dissipation rateprofiles are normalised to unity, with F diss = R f diss d S , whered S = 2 πr s d z/ cos α s is the shock front surface area. opening angle and the external pressure index. For a given η ≤
1, the position of the dissipation peak with respect tothe reconfinement length z r is relatively stable for Γ j Θ j < j Θ j >
1. For η = 1 .
5, the dissipation peak is always lo-cated very close to z = 0. The black dashed line shows thedependence of the location of the dissipation peak on η forfixed Γ j and Θ j . We find that the z diss , max /z r ratio decreaseswith η , falling to the vicinity of 0 for η & .
3. It is remark-able that, despite the fact that for η > ǫ diss scales in the same way for all values of η . c (cid:13) , 000–000 Nalewajko z d i ss , m a x / z r Γ j Θ j η η = 0 η = 0.5 η = 1 η = 1.5 Figure 4.
Position z diss , max of the peak of the energy dissipationrate along the jet axis, relative to the reconfinement length z r , asa function of the external pressure index η and the jet openingangle Θ j . Solid colour lines show the dependence of z diss , max /z r on the product of Θ j and the jet Lorentz factor Γ j (plotted againstthe lower x -axis) for several values of η . Dashed black line showsthe dependence of z diss , max /z r on η (plotted against the upper x -axis) for Γ j = 10 and Θ j = 5 ◦ . The results of this work and Paper I show that relativistic re-confinement shocks can be very efficient means of energy dis-sipation. Their efficiency depends on a simple combinationof fundamental parameters of the jet and its environment.For many astrophysical jets, their Lorentz factors, openingangles and total powers can be measured or significantly con-strained. In such cases it is possible to test the hypothesisthat energy dissipation is dominated by the reconfinementshock.
Achromatic breaks detected by Swift in some afterglow lightcurves allow one to constrain GRB jet opening angles. Inseveral cases it has been found that Γ j Θ j ≫
1. It becamea challenge for numerists studying the initial accelerationand collimation of relativistic jets to reproduce such widejets (Komissarov et al. 2009). The solution was to interruptthe collimation at some point, as would be expected for ajet breaking out of its host star (Tchekhovskoy et al. 2010;Komissarov et al. 2010). Such a situation is unique for GRBjets and allows reconfinement shocks forming at larger dis-tances to be very efficient dissipators.A scenario for a long GRB involving a very efficient re-confinement shock has been investigated numerically by Laz-zati et al. (2009). The initial parameters adopted by themare Γ j = 400 and Θ j = 10 ◦ , which translates to Γ j Θ j = 70.Our model predicts in such a case an efficiency of ∼
90% for η = 0 and ∼
82% for η = 1. Their numerical result is thusconsistent with our scaling law.Observations of AGN jets indicate that they satisfy therelation Γ j Θ j . e.g. , Pushkarev et al. 2009), for whichwe predict at most a moderate dissipation efficiency. Thisis consistent with the initial jet collimation not being in-terrupted due to a change in the environment. The valueof ǫ diss ∼ j Θ r ∼
1, isin line with estimated radiative efficiencies of the brightest blazars. As we show below, the dissipation efficiency of jetsin low-luminosity AGNs, such as M87, can be much lower.
Stawarz et al. (2006) provided a thorough review of theproperties of the M87 jet and its environment. The Lorentzfactor is estimated at Γ j ∼ θ obs & ◦ . The HST-1 knot is located at deprojected dis-tance of z r ∼
180 pc. The pressure distribution of thehost galaxy within z B ∼
230 pc has been estimated as p ext ( z ) = p B ( z/z B ) − η , where p B = 1 . × − dyn cm − and η = 1 . δ = 0 . L j = (cid:18) πcp B µβ j (cid:19) (cid:18) z δ r z η B δ (cid:19) ∼ × erg s − , (8)where µ = 17 /
24. This value is a bit higher than the estimate10 erg s − obtained from the energetics of the radio lobesby Bicknell & Begelman (1996).This study can be complemented by an estimate of thedissipation efficiency. We already know the jet Lorentz fac-tor and we need to calculate the reconfinement angle. This iscomplicated by the fact that the jet region immediately up-stream the HST-1 knot is not visible on any high-resolutionradio maps. Using the 20 GHz VLBA map from Cheunget al. (2007), we measure the projected aspect ratio of the jetsection up to HST-1: (2 r m /z r ) proj ∼ . r m /z r ) = ( r m /z r ) proj sin θ obs ∼ . r ∼ (1 + δ ) /δ (cid:18) r m z r (cid:19) . (9)For η = 1 .
2, we obtain the value Θ r ∼ . ◦ . SinceΓ j Θ r ∼ . ≪
1, the dissipation efficiency can becalculated from the approximate scaling relation: ǫ diss ∼ × − . Multiplying it by the jet power estimated inEquation 8, we obtain the energy dissipation rate L diss = ǫ diss L j ∼ . × erg s − . If all of the dissipated en-ergy were radiated away, the observed luminosity wouldbe L obs ∼ ( D / Γ j ) L diss ∼ × erg s − . Here, D j =[Γ j (1 − β j cos θ obs )] − ∼ . D / Γ j ) is used in the formthat is valid for a stationary emitting region, rather thana co-moving one (see Sikora et al. 1997). At the distanceof d L ∼
16 Mpc, the observed bolometric flux would be f obs = L obs / (4 πd ) ∼ . × − erg s − cm − . This valueis a bit higher than the actually observed broad-band fluxof M87 (see Figure 4 in Abdo et al. 2009), which is domi-nated by the non-thermal emission from the inner jet. In-terestingly, the observed flux could be matched if the lowerestimate for the jet power by Bicknell & Begelman (1996)is used instead of the result of Equation 8. This indicatesthat the simple analytic model of relativistic reconfinementshocks overestimates the jet power, but predicts a correctdissipation efficiency.If the radio emission from the inner jet of M87 is pro-duced at the reconfinement shock, its spatial distributionshould be related to the distribution of the dissipation rate. c (cid:13) , 000–000 he Efficiency of Reconfinement Shocks The radio map from Cheung et al. (2007) shows that emis-sion peaks close to the galactic nucleus (see also Hada et al.2011) and decreases monotonically with the distance. Asshown in Figures 3 and 4, the dissipation rate behaves in asimilar manner for the external pressure index η & .
3. Theactual index inferred for M87, η = 1 .
2, is close to this range.Also, the radio map shows an edge-brightened jet, which is anatural consequence of dissipation at reconfinement shocks(Nalewajko 2009)
This work generalises the findings of Nalewajko & Sikora(2009, Paper I) on the dissipation efficiency of relativisticreconfinement shocks and sets these studies in a broaderastrophysical context.We find that the average dissipation efficiency dependson the product of the jet Lorentz factor Γ j and the reconfine-ment angle Θ r , which is equal to the opening angle Θ j for aflat distribution of external pressure ( η = 0). For Γ j Θ r < ǫ diss ∼ j Θ r ) can be used.This moderate-efficiency regime can be applied to the jets ofAGNs, while the high-efficiency regime (Γ j Θ r ≫
1) is char-acteristic for GRBs. The differences in radiative efficiencybetween these sources may be related to the different cir-cumstances of the initial jet collimation process. A similaridea for the unification of relativistic jets between GRBsand AGNs has been recently formulated by Nemmen et al.(2011).Our results have been applied to the jet of radio galaxyM87, hosting a peculiar knot HST-1. Emission from the in-ner jet of M87 is consistent with dissipation at a reconfine-ment shock extending upstream from HST-1 in two aspects:– the broad-band luminosity of M87 is consistent withthe product of the dissipation efficiency ǫ diss predicted byour model and the independently estimated jet power;– radio emission peaking close to the nucleus resemblesthe dissipation profile of reconfinement shocks for externalpressure index η = 1 . ACKNOWLEDGEMENTS
The author is grateful to Marek Sikora for his adviseand support. Mitch Begelman and Kris Beckwith readthe manuscript and provided helpful comments. This workhas been partly supported by the Polish MNiSW grantsN N203 301635 and N N203 386337, the Polish AS-TRONET grant 621/E-78/SN-0068/2007, the NSF grantAST-0907872 and the NASA Astrophysics Theory Programgrant NNX09AG02G.
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