Particle acceleration and magnetic field amplification in the jets of 4C74.26
aa r X i v : . [ a s t r o - ph . H E ] J un Particle acceleration and magnetic field amplification in the jetsof 4C74.26
A.T. Araudo , A.R. Bell and K.M. Blundell ABSTRACT
We model the multi-wavelength emission in the southern hotspot of the radio quasar 4C74.26.The synchrotron radio emission is resolved near the shock with the MERLIN radio-interferometer,and the rapid decay of this emission behind the shock is interpreted as the decay of the amplifieddownstream magnetic field as expected for small scale turbulence. Electrons are accelerated toonly 0 . ∼
100 TeV.
Subject headings: galaxies: active — galaxies: jets — quasars: individual(4C74.26) — acceleration ofparticles — radiation mechanisms: non-thermal — shock waves
1. Introduction
Diffusive shock acceleration (DSA) is an estab-lished mechanism to convert bulk kinetic energyinto a non-thermal distribution of relativistic par-ticles with a maximum energy much larger thanthe average energy of particles in the plasma. Thistheory explains well the spectrum of Galactic cos-mic rays (CR) with energies up to ∼ r g of particles being accelerated which precludesCR acceleration to EeV energies unless other pro- University of Oxford, Astrophysics, Keble Road, Ox-ford OX1 3RH, UK University of Oxford, Clarendon Laboratory, ParksRoad, Oxford OX1 3PU, UK cesses can be found to amplify the magnetic fieldon larger scales.Hotspots are usually detected at the jet termi-nation region in type II Fanaroff-Riley (FR) radio-galaxies (Fanaroff & Riley 1974). The location ofthe hotspot is coincident with the downstream re-gion of the jet termination shock, where particlesaccelerated by the shock emit synchrotron radi-ation. Therefore, hotspots are suitable places tostudy DSA in high velocity shocks.We model the emission from radio to X-rays inthe southern hotspot of the FR II source 4C74.26using data provided in Erlund et al. (2010). Wedetermine that the compact radio emission tracesout the location of the shock where the magneticfield B is amplified by plasma instabilities up to ∼ µ G, and it damps rapidly downstream of theshock. The turnover in the synchrotron spectrumbetween infrared (IR) and optical wavelengths im-plies that the CR electron scattering length ismuch longer than the Larmor radius and consis-tent with the amplified magnetic field being struc-tured on very small scales and comparable withthe ion skin-depth c/ω pi . Cosmic ray accelerationis consequently very slow and electrons are accel-erated to only E e, max ∼ . ∼
100 TeV and preclude proton acceleration to1eV energies.
2. The giant FR II galaxy 4C74.26
The FR II galaxy 4C74.26 is located at red-shift z = 0 .
104 ( ∼ . TwoX-ray sources separated by ∼ ′′ were detectedwith Chandra (by Erlund et al. 2010) at the ter-mination region of the southern jet, as shownin Figure 1 (upper). In the present work westudy the southern X-ray source, with a luminosity L x ∼ erg s − at 2 keV and called “the south-ern arc”. The shape of this emission is arc-likewith a characteristic size l x ∼ ′′ , and encloses acompact radio source. Compact radio emission from the southernarc was detected with the MERLIN high reso-lution interferometer ( ν r = 1 .
66 GHz) with a flux f r ∼ L r ∼ × erg s − per unit logarithmic bandwidth δν = ν . Thisemission is located in a region of width l r < ′′ onthe plane of the sky. In addition, faint and diffuseradiation was detected at IR ( ν ir = 1.36 × Hz)and optical ( ν opt = 6.3 × Hz) bands, withfluxes ∼ × − and 2.82 × − Jy, respectively,and located in a region of width & l r . However,there is a linear structure (in both bands) thattraces the brightest edge of the MERLIN radioemission, and seems to be cupped within it.Two factors indicate that the southern arc ofX-ray emission is not synchrotron. First, l x > l r is inconsistent with the X-ray emitting electronsbeing more energetic and therefore cooling morerapidly as they advect away from the shock. Sec-ond, the steep spectrum between IR and optical(see Fig. 13 in Erlund et al. 2010) indicates themaximum energy of (synchrotron) emitting elec-trons. We note that similar characteristics are ob-served in other sources (e.g. Orienti et al. 2012).Erlund et al. (2010) suggested that the multi-wavelength emission from the southern arc is pro-duced by non-thermal electrons, emitting syn-chrotron radiation from radio to optical, andup-scattering the cosmic microwave background Throughout this paper we use the cosmology H =71 km s − Mpc − , Ω = 1 and Λ = 0 .
73. One arcsecondrepresents1 .
887 kpc on the plane of the sky at z = 0 . 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ShockSouthern arcX−ray peak
Jet S yn c h r o t r on r a d i o t oop ti ca l e m itt e r M ag n e t i c f i e l d d a m p i n g emitterIC X−ray S ho c k l ~ 0.1" l ~ " x r Fig. 1.—
Upper: the hotspot(s) at the southernjet of the FR II galaxy 4C74.26 (adapted and ro-tated by ∼ ◦ from Erlund et al. 2010). Whiteand yellow contours are X-rays and radio data,respectively. Red and green correspond to IRand optical, respectively. Lower: sketch of ourmodel of the southern arc (not to scale). Thesynchrotron radio-to-optical radiation is locatedwithin the compact MERLIN emitter, whereas ICX-ray emission is produced in a more extendedregion.2CMB) photons (with energy ∼ × − eV andenergy density U cmb =6 × − erg cm − ) to theX-ray domain. In this scenario, the radio-to-IRspectral index is α = 0 .
75 (the synchrotron fluxdensity at frequency ν is f ν ∝ ν − α ), which cor-responds to p = 2 .
3. The hotspot as a magnetic field damp-ing region
In this work we consider the same emissionmechanisms as in Erlund et al. (2010), but allowthe synchrotron and Inverse Compton (IC) emis-sion to be produced in regions with different spa-tial extents. In particular; electrons accelerated atthe shock emit synchrotron radiation from radioto optical in a compact region behind the shock,whereas the IC X-ray emission is located in anextended region. In Fig. 1 (lower) we sketch ourmodel.The synchrotron (s) and IC cooling lengthof electrons with Lorentz factor γ is l s , ic ( γ ) = t s , ic ( γ ) v sh /r , where t s , ic ( γ ) is the cooling timescale.The shock velocity is approximately the same asthe jet velocity which we take characteristicallyto be v sh = 10 cm s − ( ∼ c/ sh ∼ .
06) in line with observations of similarobjects (see Steenbrugge & Blundell 2008, and ref-erences therein). We use r = 7 as the shock com-pression ratio for a non-relativistic shock whosedownstream thermal pressure is dominated by rel-ativistic electrons, although r ∼ r . The IC X-ray emission is produced by electronswith γ x ∼ and l ic ( γ x ) ∼ ( v sh / cm s − ) arc-sec, which is much larger than l x . The synchrotroncooling length l s ( γ x ) is also greater than l x , unlessthe magnetic field in the X-ray emitting region is ∼ µ G. However, such a large magnetic fieldwould produce synchrotron radio radio emissionmuch brighter than L r (see next section). Further-more, we show below that the amplified magneticfield, of the order of ∼ µ G, is confined to a small volume close to the shock. Therefore, adia-batic expansion is probably the dominant coolingmechanism as the particles flow out of the hotspot.Unless Γ sh &
10, X-ray emitting electrons arenon-thermal and follow a power law energy distri-bution ∝ γ − p (e.g. Giannios & Spitkovsky 2009).Assuming that the X-ray emitting volume is V x ∼
300 arcsec , the energy density of these non-thermal electrons is ∼ − ( γ min / − . erg cm − where the power law terminates at a minimumLorentz factor γ min . The magnetic field with thesame energy density is ∼ γ min / − . µ G.These results correspond to the case where p = 2 . µ G as a fiducial magnetic field sinceit represents equipartition between magnetic andrelativistic electron energy densities and is typicalof other hotspots (e.g. Godfrey & Shabala 2013).
Considering that γ ( ν ) ∼ . × − ( ν/B ) . is the Lorentz factor of electrons emitting syn-chrotron radiation at ν in a magnetic field B , l s can be written as l s ( ν )[ ′′ ] ∼ (cid:16) ν GHz (cid:17) − . (cid:18) B µ G (cid:19) − . (cid:16) v sh cm s − (cid:17) . (1) MERLIN emitting electrons have γ r ≡ γ ( ν r ) ∼ γ x ( B/ µ G) − . . If both radio and X-ray emis-sion are produced by non-thermal electrons thatfollow the same power-law energy distribution, L x /L r ∼ ( γ x /γ r ) − p ( U cmb /U mag ) V x /V r and B ∼ µ G corresponds to V x /V r ∼ × , where U mag = B / π and V r is the volume of thesynchrotron emitter . Such a large ratio be-tween emitting volumes is not implausible pro-vided the magnetic field is inhomogeneous in theshock downstream region and the synchrotronemitter consists of features smaller than the MER-LIN point spread function (FWHM 0 . ′′ ) as seenin parts of the MERLIN data.The synchrotron cooling length of MERLINemitting electrons is l s ( ν r ) ∼ . ′′ ( B/ µ G) − / ( v sh / cm s − ),and a very large magnetic field of ∼ . v sh / cm s − ) / mG Note that if V x = V r , an unrealistically small magnetic fieldof 0 . µ G would be needed to explain the observed fluxes. l s ( ν r ) = 0 . ′′ ∼ l r .This result suggests that the downstream extentof the compact emission detected at ν r is not theresult of fast synchrotron cooling, as we can con-firm when we take into account the IR emission. The synchrotron cooling length of IR emittingelectrons is l s ( ν ir ) ∼ . ′′ ( B/ µ G) − / ( v sh / cm s − ),indicating that these particles radiate most of theirenergy within l r . (This angular distance is not re-solved by the IR observations.) This is consistentwith a radio-to-IR electron energy spectral indexof p ∼ . . Note that if theemitting volume were determined by synchrotroncooling, l s ( ν ) ∝ ν − . giving p = 2 α = 1 . α is measured to be 0 .
75. This very hard spectrumis unlikely since it diverges toward high energyand would be remarkable in hotspots, supportingthe conclusion that the downstream radio extent l r must be determined by factors other than syn-chrotron cooling. As we discuss in Sect. 4 thismay be the result of the damping of the magneticfield (see Schure et al. 2012, for a review). Optical emission produced by synchrotron ra-diation of electrons with γ ( ν opt ) ∼ γ ( ν ir ) is al-most co-spatial with the IR emission, and thisexplains the linear structure cupped within l r4 .The synchrotron turnover ν c between ν ir and ν opt indicates that the maximum energy of non-thermal electrons is E e, max ∼ γ ( ν c ) m e c ∼ . ν c /ν ir ) . ( B/ µ G) − . TeV.
4. Magnetic field amplification
The amplification of the magnetic field atstrong shocks in supernova remnants was demon-strated by Vink & Laming (2003) and Berezhko et al.(2003), deriving the magnetic field from l s down-stream of the shock. A theoretical explanation Note that the relationship p = ( r + 2) / ( r −
1) breaks downfor mildly relativistic shocks (Kirk et al. 2000; Bell et al.2011) or when non-linear feedback is important (e.g.Amato & Blasi 2005). The faint diffuse IR and optical emission may be the resultof CMB photons up-scattered by electrons with γ ∼ was provided by Bell (2004) showing that non-resonant hybrid instabilities are capable of en-hancing the magnetic field by orders of magnitude.Magnetic field amplification is also responsiblefor B ∼ µ G in the southern arc in 4C74.26since it is much larger than the expected valuein the jet upstream of the termination shock (e.g.Hardcastle & Krause 2014).Bohm diffusion (electron mean free path λ ∼ r g ) in a ∼ µ G magnetic field would be ex-pected to accelerate electrons with synchrotron X-ray emitting energies as seen in supernova rem-nants (Stage et al. 2006). However, ν c ∼ ν ir , opt determined by a competition between shock ac-celeration and synchrotron cooling indicates thatacceleration is slow and therefore that the elec-tron diffusion coefficient D is much larger thanthe Bohm value D Bohm : DD Bohm ∼ (cid:16) v sh cm s − (cid:17) (cid:18) ν ir ν c (cid:19) , (2)independent of B (see e.g. Casse et al. 2002). Sucha large diffusion coefficient in an amplified mag-netic field is expected if it is structured on a scale s much smaller than the Larmor radius of the elec-trons being accelerated. Small angle scattering bymagnetic field randomly orientated in cells of size s produces D ∼ ( r g /s ) D Bohm and then s cm ∼ (cid:18) ν c ν ir (cid:19) . (cid:18) B µ G (cid:19) − . (cid:16) v sh cm s − (cid:17) − . (3)In comparison the ion skin-depth is c/ω pi ∼ . × ( n/ − cm − ) − . cm, where n is the particledensity downstream of the shock (assumed to be7 times the jet density), and sc/ω pi ∼ . (cid:16) ν c ν ir (cid:17) . (cid:0) v sh cm s − (cid:1) − (cid:16) B µ G (cid:17) − . (cid:0) n − cm − (cid:1) . . (4)Given the uncertainties in the parameter val-ues, the approximate nature of the theoreti-cal models, and the wide range of the spatialscales ( s , r g , l r ), it is not significant or surpris-ing that our estimate of s/ ( c/ω pi ) differs fromunity by a factor of ∼ .
01. The order of magni-tude similarity of s and c/ω pi supports the con-tention that shock-generated small-scale turbu-lence scatters non-thermal electrons during diffu-sive shock acceleration. This is consistent with4ironi & Spitkovsky (2011) who discuss the var-ious processes related to the Weibel instabilitythat excite turbulence on the characteristic scaleof c/ω pi . Simulations show that magnetic fieldgenerated by the Weibel instability decays down-stream of the shock because of its relatively smallscalelength (Sironi & Spitkovsky 2011; Bret et al.2013; Sironi et al. 2013). This would account forthe cut-off of synchrotron emission in 4C74.26 farshort of the synchrotron cooling distance of radio-emitting electrons ( l r ≪ l s ( ν r )). These electronscontinue up-scattering CMB photons, thus pro-ducing IC X-ray emission downstream of the shockafter the MERLIN radio emission has ceased. Since the response of highly relativistic ions issimilar to that of electrons with the same energy ina tangled amplified magnetic field, we can expectprotons to have a similar ratio of
D/D
Bohm . Pro-tons can be accelerated to higher energies thanelectrons because their radiative losses are min-imal, but the maximum energy to which theyare accelerated is limited because their acceler-ation time is increased by the ratio
D/D
Bohm .The Hillas parameter v sh B R (Hillas 1984), where R ∼ ′′ ( ∼ . ∼
100 EeV in the termination shock of4C74.26, but the maximum energy is reduced toonly ∼
100 TeV if D ∼ D Bohm since the Hillasparameter assumes D ∼ D Bohm and is otherwisereduced by the factor (
D/D
Bohm ) − . Anotherperspective on the same effect is that the meanfree path for scattering by small-scale turbulence λ ∼ r /s is larger than the size of the system if s ∼ c/ω pi and r g is the Larmor radius of an EeVproton. This result suggests that the mildly rel-ativistic termination shock in 4C74.26 is a pooraccelerator of UHECR.
5. Conclusions
We model the radio to X-ray emission in thesouthern hotspot of the FR II galaxy 4C74.26.Our study is based on three key observational fea-tures: 1) Compact MERLIN emission region: it istoo thin to be the result of fast synchrotron cool-ing (Sect. 3.2.1). 2) The radio to IR spectrum( α = 0 .
75) is too flat for the emitting volume to be determined by synchrotron cooling through thiswavelength range (Sect. 3.2.2). 3) The turnover ofthe synchrotron spectrum at IR/optical frequen-cies requires λ ≫ r g for any reasonable shock ve-locity (Sect. 4). These three features fit well ina scenario in which the MERLIN radio emissiontraces out the region where the magnetic field isamplified by plasma instabilities with small lengthscale (e.g. Weibel).The magnetic field in equipartition with non-thermal electrons in the MERLIN emission re-gion is ∼ µ G and similar to the values ob-tained by other authors (e.g. Godfrey & Shabala2013). An unrealistically large magnetic field ∼ . v sh / cm s − ) / mG would be neededto explain the compact radio emission in terms ofsynchrotron cooling. If B ∼ µ G in the syn-chrotron emission region, the maximum energy ofnon-thermal electrons is ∼ ∼ λ ≫ r g . If λ is similarly larger than the Larmorradius at higher proton energies, then the maxi-mum proton energy at the termination shock of4C74.26 is only 100 TeV instead of the 100 EeVindicated by the Hillas parameter. This may haveimportant implications for the understanding ofthe origins of UHECR.We thank the referees for the constructive re-ports. The research leading to this article has re-ceived funding from the European Research Coun-cil under the European Community’s SeventhFramework Programme (FP7/2007-2013)/ERCgrant agreement no. 247039. We acknowl-edge support from the UK Science and Tech-nology Facilities Council under grant numberST/K00106X/1. REFERENCES
Amato, E., & Blasi, P., 2005, MNRAS, 364, L76Bell, A. 2004, MNRAS, 353, 550Bell, A., Schure, K., & Reville, B., 2011, MNRAS,418, 1208Bell, A. 2014, Braz. J. Phys., 44, 415Berezhko, E., Ksenofontov, L., & Volk, H. 2003,A&A, 412, L115ret, A., Stockem, A., Fiuza, F., Ruyer, C.,Gremillet, L., Narayan, R., & Silva, L. O., 2013,Physics of Plasmas, 20, 42102Casse, F, Lemoine, M, & Pelletier, G. 2002,PhRvD, 65, 023002Erlund, M., Fabian, A., Blundell, K., Crawford,C., & Hirst, P. 2010, MNRAS, 404, 629Fanaroff, B., & Riley, J. 1974, MNRAS, 167, 31PGallant, Y. & Achterberg, A. 1999, ApJ, 305, L6Giannios, D., & Spitkovsky, A. 2009, MNRAS,400, 330Godfrey, L., & Shabala, S. 2013, ApJ, 767, 12Hardcastle, M., & Krause, M. 2014, MNRAS, 443,1482Hillas, A. 1984, ARA&A 1984, 425Kirk, J., Guthmann, A., Gallant, Y., & Achter-berg, A., 2000, ApJ, 542, 235Lemoine, M., & Pelletier, G. 2010, MNRAS, 402,321Murase, K., Dermer, C., Takami, H. & Migliori,G. ApJ, 749, 63Orienti, M., Prieto, M., Brunetti, G., et al. 2012,MNRAS, 419, 2338Pelletier, G., Lemoine, M. & Marcowith, A. 2009,MNRAS, 393, 587Reville, B., & Bell, A. 2014, MNRAS, 439, 2050Schure, K., Bell, A., O’C Drury, L., & Bykov, A.2012, Space Sci. Rev., 173, 491Sironi, L., & Spitkovsky, A. 2011, ApJ, 726, 75Sironi, L., Spitkovsky, A. & Arons, J. 2013, ApJ,771, 54Stage, M., Allen, G., Houck, J., & Davis, J. 2006,Nature Physics, 2, 614Steenbrugge, K., & Blundell, K. 2008, MNRAS,388, 1457Vink, J., & Laming, J. 2003, ApJ, 584, 758