The particle content of low-power radio galaxies in groups and clusters
MMon. Not. R. Astron. Soc. , 1– ?? (2009) Printed 27 September 2018 (MN L A TEX style file v2.2)
The particle content of low-power radio galaxies in groups andclusters
J. H. Croston (cid:63) and M.J. Hardcastle School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, UK
27 September 2018
ABSTRACT
The synchrotron-radiating particles and magnetic fields in low-power radio galaxies (in-cluding most nearby cluster-centre sources), if at equipartition, can provide only a small frac-tion of the total internal energy density of the radio lobes or plumes, which is now well con-strained via X-ray observations of their external environments. We consider the constraints onmodels for the dominant energy contribution in low-power radio-galaxy lobes obtained from adetailed comparison of how the internal equipartition pressure and external pressure measuredfrom X-ray observations evolve with distance for two radio galaxies, 3C 31 and Hydra A. Werule out relativistic-lepton dominance of the radio lobes, and conclude that models in whichmagnetic field or relativistic protons/ions carried up the jet dominate lobe energetics are un-likely. Finally, we argue that entrainment of material from the jet surroundings can providethe necessary pressure, and construct a simple self-consistent model of the evolution of theentrainment rate required for pressure balance along the 100-kpc scale plumes of 3C 31. Sucha model requires that the entrained material is heated to temperatures substantially above thatof the surrounding intra-group medium, and that the temperature of the thermal component ofthe jet increases with distance, though remaining sub-relativistic.
Key words: galaxies: active – X-rays: galaxies: clusters
Low-power (FRI: Fanaroff & Riley 1974) radio galaxies are com-monly found in the centres of rich galaxy groups and clusters,where they are thought to play an important role in regulating thecentral gas properties and galaxy evolution via a (currently poorlyunderstood) feedback process (e.g. McNamara & Nulsen 2007;Fabian 2012, and references therein). Among the many uncertain-ties about the way in which this feedback process operates, onelong-standing problem is the unknown nature of the dominant par-ticle or field component within the radio lobes, which are importantas the lobes are the means of energy transfer to the surrounding gasvia their expansion. The radio synchrotron emission from the lobesprovides only a combined constraint on electron density and mag-netic field strength, and so it has been common to assume equiparti-tion of energy in field and radiating particles (e.g. Burbidge 1956),which corresponds roughly to the minimum total energy the sourcerequires in order to produce the observed radio emission. But whilethe lobes of powerful FRII radio galaxies appear to be close toequipartition (e.g. Croston et al. 2005; Kataoka & Stawarz 2005),it has been known for some time that the energy content of FRI ra-dio galaxies must be distributed differently to that of FRIIs, as the (cid:63)
Email: [email protected] radiating particles and magnetic field, if at equipartition, cannot inthe vast majority of cases provide sufficient pressure to balance themeasured external pressures surrounding FRI lobes (e.g. Morgantiet al. 1988; Worrall & Birkinshaw 2000).The external pressure acting on the jets and lobes can now beconstrained tightly on scales of a few to several hundred kpc formany low-power radio galaxies, using X-ray observations of thesurrounding group or cluster gas with
Chandra and
XMM-Newton (e.g. Hardcastle et al. 2002; Croston et al. 2003, 2008). If it is as-sumed that the jets and lobes are close to pressure equilibrium withthe surrounding medium (likely to be true on kpc – hundred kpcscales for low-power sources), then the external pressure profilemust correspond closely to the run of internal pressure along thejet as it evolves into a lobe or plume. The internal pressure can-not be measured directly from the radio observations of the source;however, the internal pressure in some combination of radiatingparticles (electrons and positrons) and magnetic field can be mea-sured by modelling the radio emission. This type of comparison hasnow been carried out for many low-power radio galaxies, includ-ing large samples of cavity sources in galaxy clusters (includingso-called “ghost” cavities in which any radio emission is weak orabsent), and, as mentioned above, typically shows that the radiatingparticles and magnetic field cannot dominate the internal energy of c (cid:13) a r X i v : . [ a s t r o - ph . H E ] D ec J. H. Croston & M. J. Hardcastle the source if they are at equipartition (Croston et al. 2003, 2008;Dunn & Fabian 2004; Dunn et al. 2005, 2006; Bˆırzan et al. 2008).Given that the lobes of low-power radio galaxies cannotbe dominated by an equipartition electron-positron plasma, othermodels for the energetically dominant component of the radio lobecontents must be considered. The two most obvious explanationsare that the dominant internal pressure is provided by a departurefrom equipartition or by a significant population of non-radiatingparticles. There is evidence from X-ray inverse Compton observa-tions that powerful FRII radio galaxies may deviate from equiparti-tion by a small amount in the direction of electron dominance (e.g.Isobe et al. 2002; Croston et al. 2005); however, electron domi-nance by large factors would be expected to produce detectablelevels of X-ray inverse-Compton emission in at least some FRI ra-dio galaxies, which are inconsistent with observations (Hardcastleet al. 1998; Croston et al. 2003). Recently, detailed models of mag-netically dominated jets and lobes have been developed (e.g. Liet al. 2006; Nakamura et al. 2006); however, they are difficult toreconcile with observations, e.g. of radio jet polarization proper-ties and geometry (see later discussion). Proton-dominated modelshave been discussed by a number of authors (e.g. De Young 2006;Bˆırzan et al. 2008; McNamara & Nulsen 2007), but it is energeti-cally difficult to supply the proton population required by transportfrom the inner jet (e.g. De Young 2006).There are several reasons to favour instead a model in whichentrainment of material as the jet expands leads to an energeticallydominant proton population on scales of tens to hundreds of kpc.Entrainment of the ISM and ICM is thought to be the means bywhich FRI jets decelerate from relativistic to transonic speeds onkpc scales (e.g. Bicknell 1994). There is growing observational ev-idence that entrainment is occurring (e.g. Hardcastle et al. 2003,2007), as well as strong support for its importance from detailedkinematic modelling of FRI jets (e.g. Laing & Bridle 2002; Lainget al. 2006). A model in which entrainment accounts for the appar-ent “missing” pressure in FRI radio galaxies also has the advantageof explaining the observed difference in the energetics of the FRIand FRII populations (the former being massively underpressuredif at equipartition, while the latter appear close to equipartitionboth from IC observations and pressure comparisons) without theneed to invoke differences in the intrinsic particle content of the in-ner jets, which might require different jet production mechanisms:since FRII jets do not decelerate or interact with their environmentssignificantly, they would not in general be expected to entrain sig-nificant amounts of material. Finally, we have previously found arelationship between FRI source structure and particle/energy con-tent, suggesting that sources likely to be undergoing strong entrain-ment have a larger contribution from non-radiating material thanthose likely to be weakly entraining (Croston et al. 2008). This pro-vides further support for an entrainment-dominated model.In this paper we investigate in detail the observational con-straints on models for the particle and energy content of low-powerradio galaxies, by considering how the non-radiating and radiatingcomponents of the jets in the well-studied radio galaxy 3C 31 mustevolve with distance in order to maintain pressure balance and pro-duce the observed radio emission. We use new deep X-ray data andhigh-resolution radio data to place tight constraints on the exter-nal pressure and internal pressure from radiating particles and fieldwithin the radio jets and plumes of 3C 31 . We consider in detail theconstraints this result provides for what particle population or mag-netic field structure dominates the source energetics, and also carryout a pressure comparison for the cluster-centre source Hydra A asa preliminary test of the generality of our results. Throughout the paper we use a cosmology in which H = 70 km s − Mpc − , Ω m = 0 . and Ω Λ = 0 . . At the redshifts of 3C 31( z = 0 . ) and Hydra A ( z = 0 . ), this gives luminosity dis-tances of D L = 73 . Mpc and D L = 244 . Mpc, respectively, andangular scales of . kpc/arcsec (3C 31) and . kpc/arcsec(Hydra A). Spectral indices α are defined in the sense S ν ∝ ν − α .Reported errors are 1 σ for one interesting parameter, except whereotherwise noted. We used new
XMM-Newton data to obtain a radial profile of the ex-ternal pressure surrounding the radio jets and plumes in 3C 31. Weobserved 3C 31 on 2008 July 1st for ∼ ks (ObsID 0551720101).The data were processed in the standard way using XMM-Newton
SAS version 11.0.0, and the latest calibration files from the
XMM-Newton website. The pn data were filtered to include only singleand double events (PATTERN ≤ ), and FLAG==0, and the MOSdata were filted according to the standard flag and pattern masks(PATTERN ≤ and XMM-Newton data using the closed-filterdouble-background method described by Croston et al. (2008). The
Chandra surface brightness profile of Hardcastle et al. (2002) wasalso used to help constrain the inner profile shape. The combined
XMM-Newton (MOS1, MOS2 and pn) profile and
Chandra pro-file were jointly fitted with a projected double beta model (Cros-ton et al. 2008), convolved with the appropriate point-spread func-tion (PSF) for each telescope, using the Markov-Chain Monte Carlo(MCMC) method described by Ineson et al. (2013). The resultingmodel was used to obtain a gas density profile for the environment.A corresponding temperature profile was obtained by extract-ing spectra from six annular regions, and using the background fit-ting method described by Croston et al. (2008), which correctlyaccounts for both particle and X-ray background, to obtain (pro-jected) temperature measurements. For each region, the spectrafrom the three
XMM-Newton cameras were fitted jointly with an apec model (using the energy range 0.3 – 7.0 keV, but excluding theregion between 1.4 – 1.6 keV, which is affected by an instrumen-tal line). The normalizations for the three cameras were allowed tovary, but the temperatures were tied together. A free abundance fitled to unphysically large values for the abundance, and so we fixedthe abundance to the best-fitting abundance from a global spectralfit ( Z = 0 . ). The results of spectral fitting are given in Table 1.For the inner regions of the group, we used the Chandra temper-ature profile of Hardcastle et al. (2002) in order to obtain moreaccurate pressure constraints. We used the deprojected temperaturevalues, although the effect of deprojection on the temperature pro-file is small. In the outer regions of the group the temperature variesonly by ∼ per cent, so that any uncertainty from not correctingfor projection is small, and less than the statistical uncertainty onthe outer temperature.In order to obtain a gas pressure profile with high resolution,we fitted the measured temperature profile with the analytic model c (cid:13) , 1–, 1–
XMM-Newton cameras were fitted jointly with an apec model (using the energy range 0.3 – 7.0 keV, but excluding theregion between 1.4 – 1.6 keV, which is affected by an instrumen-tal line). The normalizations for the three cameras were allowed tovary, but the temperatures were tied together. A free abundance fitled to unphysically large values for the abundance, and so we fixedthe abundance to the best-fitting abundance from a global spectralfit ( Z = 0 . ). The results of spectral fitting are given in Table 1.For the inner regions of the group, we used the Chandra temper-ature profile of Hardcastle et al. (2002) in order to obtain moreaccurate pressure constraints. We used the deprojected temperaturevalues, although the effect of deprojection on the temperature pro-file is small. In the outer regions of the group the temperature variesonly by ∼ per cent, so that any uncertainty from not correctingfor projection is small, and less than the statistical uncertainty onthe outer temperature.In order to obtain a gas pressure profile with high resolution,we fitted the measured temperature profile with the analytic model c (cid:13) , 1–, 1– ?? he particle content of low-power radio galaxies in groups and clusters Table 1.
Results of spectral fitting for the environment of 3C 31. Spectralfits were obtained for annular regions between the radii listed, using an
APEC model, were in the energy range . − keV, assuming N H = 5 . × cm − . The abundance was fixed to the best-fitting value from a globalspectral fit, as free abundance fits led to unphysical values (likely due to theadditional free parameters of the background model).Region kT Z χ (d.o.f.)60 – 80 arcsec . ± . . +0 . − . . +0 . − . . ± . . +0 . − . . +0 . − . of Vikhlinin et al. (2006), and obtained a finely binned look-up ta-ble for Λ( T ) , the conversion factor between volume emission mea-sure and gas density (obtained from XSPEC ). The resulting tablewas used together with the analytic temperature model to obtain agas pressure profile, which is shown in Fig 1.
Although the majority of this paper focuses on 3C 31, using ournew X-ray data, we also carried out a pressure comparison for Hy-dra A as a preliminary test of whether our findings are likely toapply widely to FRI radio galaxies. For Hydra A, we did not re-analyse the archival
Chandra and
XMM-Newton observations, butmade use of previously published gas density and temperature pro-files. For the gas density profile we used the double beta model ofWise et al. (2007), normalised to the density profile published byDavid et al. (2001). We interpolated over the (projected) tempera-ture profile of David et al. (2001) to obtain a cluster pressure profileover the radial ranges of interest, which is also shown in Fig. 1.
The radio emission from the sources does not place a constraint onthe total internal pressure of the lobes, as the radiating plasma couldbe far from equipartition; however, it does place constraints on theinternal pressure of the radiating particles and magnetic field. Weused high-resolution radio data to obtain profiles of synchrotronemissivity along the jets.For 3C 31 we used the combined 1.4-GHz map of Laing et al.(2008), which has a resolution of 5 arcsec, and a 330-MHz mapmade in the standard way from VLA archival data in the B andC configurations (Program AL597), with a resolution of 21.3 arc-sec × γ = 10 and maximum energy of γ = 10 .This correctly describes the radio spectra of the two sources in theinner regions (in the GHz radio regime), but is a somewhat flatterspectral index than is measured in the outer parts of the source. Anysystematic effects of spectral steepening at GHz frequencies on ourpressure results for the outer parts of the sources will be small, asthe total electron energy content is dominated by the low-energyelectron population. Allowing the spectral index to vary based onthe observed spectral index at GHz frequencies would introducelarge systematic uncertainty in the low-energy electron density. Wetherefore used a single electron distribution for all regions, normal-ized to the measured radio flux density for that region from theappropriate radio map (in the case of Hydra A flux densities at 5GHz were used in order to have sufficient spatial resolution out toa distance of 40 kpc, with the 330-MHz map used beyond that dis-tance). In future work we will make use of new low-frequency datafrom the Low-Frequency Array (LOFAR) to improve our spectralmodel.In order to investigate the variations in internal conditionsalong the source, a power law was fitted to the emissivity distribu-tion so as to provide a smooth model for the variation with distance.Although there is some small systematic deviation of the observedemissivity about the model, the measured profile is never more than ∼ percent different from the model (and typically within 10 per-cent).Using the smoothed emissivity profiles for 3C 31, we first de-termined the internal pressure as a function of position along the jet,under the assumptions of equipartition of energy between particlesand magnetic field, and no non-radiating particles ( κ = 0 , where κ = U NR /U R , i.e. the ratio of energy density in non-radiatingparticles to that in synchrotron-emitting particles). The internal,equipartition pressure profiles for are shown in Fig. 1, together withthe external pressure profiles determined from the X-ray observa-tions. For Hydra A, we simply calculated an internal pressure pro-file for the existing radio bins, under the same assumptions. Thisprofile is shown in Fig. 1, illustrating a strong qualitative similarityto the behaviour of the 3C 31 jets. Fig. 2 shows the ratio of external pressure to internal, equipartitionpressure (with no protons) for 3C 31, determined from the exter-nal and internal pressure profiles described in the previous section.As seen in previous work (e.g. Worrall & Birkinshaw 2000), theinternal equipartition pressure is significantly below the externalpressure at all radii. It also is readily apparent that the apparentpressure “deficit” increases with distance, apart from in the inner c (cid:13) , 1– ?? J. H. Croston & M. J. Hardcastle
Figure 1.
External and internal (equipartition) pressure profiles for the two sources 3C 31 (l) and Hydra A (r). The external pressures derived from X-raymeasurements are shown as shaded regions, which indicate the 1 σ errors, and the internal, equipartition pressures with the assumption of no protons aregiven by the solid black (3C31 north, Hydra A north) and dashed red (3C 31 south) lines. The statistical uncertainties on the internal pressures are negligiblecompared to model assumptions and so are not plotted. ∼ kpc. This figure illustrates clearly that on scales > kpcthe contribution of the radiating material to the total internal pres-sure of the radio source, in the equipartition case, must decreasesubstantially as the jet evolves out into the group or cluster envi-ronment. Alternatively, if equipartition between radiating particlesand magnetic field does not hold, then there must be a systematicdeparture from this condition that increases with distance from thenucleus and group/cluster centre. Such an effect was first observedin ROSAT environmental studies (e.g. Hardcastle et al. 1998; Wor-rall & Birkinshaw 2000), and is also seen in our combined Chandra and
XMM-Newton analysis of NGC 6251 (Evans et al. 2005) and3C 465 (Hardcastle et al. 2005); however, the higher quality of theX-ray and radio pressure constraints in the new work we presenthere places the result on a much firmer footing.We have considered in detail the possible effects of projectionon this conclusion (see Section 3.4). Neither 3C 31 or Hydra A isthoughout to be highly projected, and for plausible jet orientationsthe plots in Fig. 1 and 2 do not alter significantly as the two ef-fects of projection act in the same direction: the internal pressuredecreases with θ los since the jet volume at a given projected dis-tance increases, and the external pressure acting on the jet at thisprojected distance decreases because it is further out in the X-rayatmosphere whose pressure is dropping off.Although such detailed pressure profile comparisons have notbeen carried out previously, it is interesting to note that a similarbehaviour can be seen at a statistical level in the sample of clustercavities studied by Dunn et al. (2005), where the so-called “ghost”cavities are typically at much larger distances from the cluster cen-tre than the active lobes, which are systematically closer to pressurebalance assuming κ = 0 .The pressure constraints shown in Fig. 2 can be used to testa range of models that have been proposed for the particle or fieldcontent dominating the energy budget of low-power radio lobes. In the following section we consider four models for the dominantenergy content of the lobes: • Model I – lepton dominance: the jets and lobes are out ofequipartition, but the contribution from protons remains negligibleand it is the radiating electrons and positrons that dominate the in-ternal pressure • Model II – magnetic field dominance: the jets and lobes areout of equipartition, but the contribution from protons remains neg-ligible and the magnetic field dominates the internal pressire. • Model III – relativistic proton or ion dominance: the jetsand lobes are in equipartition, with relativistic protons (and/or ions)dominating the internal pressure (i.e. κ >> ) • Model IV – thermal gas dominance (entrainment): the jetsand lobes are in equipartition, with thermal material, likely en-trained from the surrounding intragroup medium, dominating theinternal pressure (i.e. κ >> )It is clear that more complex models are possible – in particular,it is plausible that non-radiating particles are present, but the jetsand lobes are not at equipartition in all locations along the jet. Suchmodels are harder to test, and so we begin by considering the foursimpler models listed above. As previously stated, it is clear from Figs 1 and 2 that in order fora departure from equipartition to be the explanation for the appar-ent “missing” pressure in FRI lobes, the jets must evolve furtherand further from the equipartition condition as the source expands(apart from in the very inner parts – we will consider the implica-tions of the differing behaviour in the inner ∼ kpc of 3C 31 inSection 3.6)The energy densities and magnetic field strengths requiredin order that the total energy density in the synchrotron-emitting c (cid:13) , 1–, 1–
XMM-Newton analysis of NGC 6251 (Evans et al. 2005) and3C 465 (Hardcastle et al. 2005); however, the higher quality of theX-ray and radio pressure constraints in the new work we presenthere places the result on a much firmer footing.We have considered in detail the possible effects of projectionon this conclusion (see Section 3.4). Neither 3C 31 or Hydra A isthoughout to be highly projected, and for plausible jet orientationsthe plots in Fig. 1 and 2 do not alter significantly as the two ef-fects of projection act in the same direction: the internal pressuredecreases with θ los since the jet volume at a given projected dis-tance increases, and the external pressure acting on the jet at thisprojected distance decreases because it is further out in the X-rayatmosphere whose pressure is dropping off.Although such detailed pressure profile comparisons have notbeen carried out previously, it is interesting to note that a similarbehaviour can be seen at a statistical level in the sample of clustercavities studied by Dunn et al. (2005), where the so-called “ghost”cavities are typically at much larger distances from the cluster cen-tre than the active lobes, which are systematically closer to pressurebalance assuming κ = 0 .The pressure constraints shown in Fig. 2 can be used to testa range of models that have been proposed for the particle or fieldcontent dominating the energy budget of low-power radio lobes. In the following section we consider four models for the dominantenergy content of the lobes: • Model I – lepton dominance: the jets and lobes are out ofequipartition, but the contribution from protons remains negligibleand it is the radiating electrons and positrons that dominate the in-ternal pressure • Model II – magnetic field dominance: the jets and lobes areout of equipartition, but the contribution from protons remains neg-ligible and the magnetic field dominates the internal pressire. • Model III – relativistic proton or ion dominance: the jetsand lobes are in equipartition, with relativistic protons (and/or ions)dominating the internal pressure (i.e. κ >> ) • Model IV – thermal gas dominance (entrainment): the jetsand lobes are in equipartition, with thermal material, likely en-trained from the surrounding intragroup medium, dominating theinternal pressure (i.e. κ >> )It is clear that more complex models are possible – in particular,it is plausible that non-radiating particles are present, but the jetsand lobes are not at equipartition in all locations along the jet. Suchmodels are harder to test, and so we begin by considering the foursimpler models listed above. As previously stated, it is clear from Figs 1 and 2 that in order fora departure from equipartition to be the explanation for the appar-ent “missing” pressure in FRI lobes, the jets must evolve furtherand further from the equipartition condition as the source expands(apart from in the very inner parts – we will consider the implica-tions of the differing behaviour in the inner ∼ kpc of 3C 31 inSection 3.6)The energy densities and magnetic field strengths requiredin order that the total energy density in the synchrotron-emitting c (cid:13) , 1–, 1– ?? he particle content of low-power radio galaxies in groups and clusters Figure 2.
The fraction of required internal pressure that can be providedby the synchrotron-emitting components of the jets if at equipartition, as afunction of distance from group/cluster centre for 3C 31, showing that thiscomponent can provide a decreasing fraction of the jet pressure on scales oftens to hundreds of kpc. Line styles are as for Fig. 1. plasma should match the measured external pressure were deter-mined by modelling the electron energy distribution using the pa-rameters discussed in Section 2.2. Fig. 3 shows the evolution of therequired energy ratio between magnetic fields and leptons requiredto maintain pressure balance with the surrounding hot gas for Mod-els I and II.In Model I (lepton domination) the particle energy dominatesby a factor ∼ in the inner regions, then, after an initial de-crease, increases to ∼ at hundred-kpc distances. For the largeelectron densities required in this model, the predicted level of X-ray inverse-Compton radiation from the radio jets and lobes wouldbe significant, and can be ruled out in a number of individual cases(e.g. Croston et al. 2003; Hardcastle & Croston 2010). In particu-lar, Hardcastle & Croston (2010) have examined in detail the con-straints on inverse Compton emission from Hydra A, and concludethat relativistic electrons (and positrons) can contribute at most ∼ per cent of the internal pressure of the radio lobes. We can thereforeconclusively rule out this explanation. For 3C 31, we considered theoutermost region of our profile, and calculated the predicted levelof X-ray inverse Compton emission at 1 keV using the SYNCH codeof Hardcastle et al. (1998) under the assumptions of Model I. Wefind that the observed residual level of X-ray flux in this regionafter background subtraction is a factor ∼ times lower thanthe prediction of this model, consistent with results for other FRIsources.In Model II (magnetic field domination), the energy ratio U B /U E evolves similarly to Model I, with the factor by whichthe magnetic field dominates increasing from around 30 to ∼ by hundred kpc scale distances. Fig. 4 shows the magnetic fieldstrengths required as a function of distance to achieve pressure bal-ance in this model. The magnetic field strengths required are high( ∼ − µ G), decreasing by a factor of a few from the inner partsto hundred kpc scale distances. This model requires the generationof magnetic field energy density along the source. The dashed anddotted lines in Fig. 4 show the expected evolution of magnetic fieldstrength due to adiabatic expansion for the case of a predominantly
Figure 3.
The evolution of the energetically dominant component of the3C 31 jets with distance, showing the ratio of lepton to magnetic field en-ergy density for Model I, the inverse ratio for Model II, and the proton/ioncontent κ for Models III and IV. Line styles are as for Fig. 1. radial/toroidal and predominantly longitudinal field structure, re-spectively (e.g. Baum et al. 1997). A constant velocity profile wasassumed, which is conservative, as a decreasing velocity wouldsteepen the losses for the perpendicular components of B . Hence apassively evolving magnetic field component is inconsistent withthe observations. The results shown in Fig. 4 are not consistentwith previously proposed models for cylindrical jets with helical B fields (e.g. Nakamura et al. 2006), but such models are also in-consistent with FRI jet geometries and polarization structures (e.g.Laing 1981; Laing et al. 2008). The requirement for a slow decreasein B along the jets (despite lateral expansion of the jet) could beconsistent with a model in which turbulence increasingly amplifiesthe magnetic field on large scales; however, this would need to takeplace with no appreciable particle acceleration for consistency withthe radio constraints, and turbulent amplification of magnetic fieldsbeyond equipartition values is challenging (De Young 1980). Ourresults show that energy would have to be being transferred fromthe particle population to the magnetic field to a greater and greaterextent at larger distances. This model cannot be ruled out directly,but from the constraints on the model given above we conclude thatmagnetic domination of the jets and lobes is highly unlikely. The question of whether or not the inner jets of radio galaxies con-sist of an electron-positron or electron-proton plasma is a long-standing one, which has not yet been resolved satisfactorily, despitesubstantial efforts over the past couple of decades (e.g. Ghiselliniet al. 1992; Celotti & Fabian 1993; Wardle et al. 1998; Homanet al. 2009). On kpc scales, there is an obvious additional sourceof non-radiating particles in the form of material entrained into thejets from the surroundings: there is substantial evidence for entrain-ment in FRI jets, and the standard picture of FRI dynamics relieson entrainment to decelerate the jets from relativistic to transonicspeeds on scales of a few kpc (e.g. Bicknell 1994; Laing & Bri- c (cid:13) , 1– ?? J. H. Croston & M. J. Hardcastle
Figure 4.
The magnetic field strength required as a function of distancein the case where magnetic field energy dominates the internal pressure(shown for the northern jet of 3C 31). The dotted and dashed lines show theexpected evolution of magnetic field strength due to adiabatic expansion forthe case of a predominantly tangential and predominantly longitudinal fieldstructure, respectively. Line styles are as for Fig. 1. dle 2002). In this section we consider models in which relativisticprotons (and/or ions (Model III) or thermal gas entrained from thesurroundings (Model IV) dominate the internal pressure.The contribution from heavy particles (protons/ions) requiredto achieve pressure balance can be determined straightforwardlyunder the assumption of equipartition of energy between all parti-cles (radiating and non-radiating) and magnetic field. Details of thiscalculation for the cases of relativistic and thermal gas are providedin Appendix A. In Fig. 3 we plot the required ratio of energy den-sity in non-radiating particles to radiating particles for these twomodels.Fig. 5 shows the run of energy density in relativistic protons(or ions) required to balance the external pressure for Model III, as-suming equipartition of particles (both radiating and non-radiating)with magnetic field. If the electron population suffers significant ra-diative losses (which do not affect the proton/ion energy density), itmight be expected that the relative energy density in protons (and/orions) would increase with distance, as required by the external pres-sure data. However, if the energy is carried by relativistic protonsinjected in the jets’ inner regions, then their energy density wouldbe expected to evolve adiabatically with distance, in the absenceof significant radiative losses or particle acceleration. Fig. 5 showsthat the simplest version of Model III in which protons are injectedonly in the inner jet is not viable, because the proton energy den-sity in this model decreases much less steeply with distance thanexpected as a result of adiabatic losses (calculated from 10 kpc out-wards). We can therefore rule out a model in which protons injectedin the inner regions evolve passively along the jet. For relativisticprotons and/or ions to dominate the jets and lobes over their entirelength, significant particle acceleration is required on scales of tensto hundreds of kpc (which must not significantly affect the leptonpopulation).A model in which entrainment of surrounding material leadsto an increasing thermal gas content as the jets evolve (whether ornot they initially contain relativistic protons), such as Model IV, is
Figure 5.
The energy density in relativistic protons and/or ions requiredto balance the external pressure (filled squares), shown for the northern jetof 3C 31. The dashed and dotted lines indicate the expected evolution ofenergy density with distance along the source assuming adiabatic losses.Note that the flattening of the adiabatic model between 20 and 70 kpc iscaused by the jet’s cylindrical geometry in that region (see also Fig. 6) more consistent with the data as it provides a simple explanationfor the decreasing energetic importance of the radiating particles asthe jet evolves. Fig. 3 shows how the ratio of energy density in non-radiating particles to radiating particles must evolve along the jet inthis model. This evolution of energy density could occur either byincreasing entrainment (via an increasingly large boundary layer),or by increased heating/acceleration of entrained thermal gas. Therequired entrainment rate for Model IV can be obtained by consid-eration of mass, momentum and energy flux conservation along thejet. In the following section we develop a toy model to investigatethis scenario.
We model the region of jet between 12 kpc and 140 kpc, whichis where the X-ray constraints are tight while the uncertainties onjet geometry are acceptable (beyond this distance further jet bendsand flaring making it difficult to constrain the geometry). The in-ner boundary is chosen to be beyond the initial deceleration re-gion according to the model of Laing & Bridle (2002), so thatrelativistic effects can be neglected. We assume Model IV, above,i.e. the following assumptions hold: (1) the jet internal pressure, P int , balances the external pressure ( P ext , as measured from theX-ray observations) at each radius; (2) the internal pressure hascontributions from magnetic field ( P B ), synchrotron radiating lep-tons ( P E ), and thermal gas entrained from the environment ( P th );and (3) the magnetic field strength and energy density are assumedto be in equipartition with the total particle energy density (fromsynchrotron-radiating and non-radiating particles). We later discussthe effects of relaxing the final assumption.By making use of the (non-relativistic) equations for conser-vation of momentum and energy flux along the jet, the density andtemperature of the ‘missing’ thermal component of the jet can beobtained, as described in detail in Appendix B. We require initialconditions of density and velocity at the inner boundary. We take c (cid:13) , 1–, 1–
We model the region of jet between 12 kpc and 140 kpc, whichis where the X-ray constraints are tight while the uncertainties onjet geometry are acceptable (beyond this distance further jet bendsand flaring making it difficult to constrain the geometry). The in-ner boundary is chosen to be beyond the initial deceleration re-gion according to the model of Laing & Bridle (2002), so thatrelativistic effects can be neglected. We assume Model IV, above,i.e. the following assumptions hold: (1) the jet internal pressure, P int , balances the external pressure ( P ext , as measured from theX-ray observations) at each radius; (2) the internal pressure hascontributions from magnetic field ( P B ), synchrotron radiating lep-tons ( P E ), and thermal gas entrained from the environment ( P th );and (3) the magnetic field strength and energy density are assumedto be in equipartition with the total particle energy density (fromsynchrotron-radiating and non-radiating particles). We later discussthe effects of relaxing the final assumption.By making use of the (non-relativistic) equations for conser-vation of momentum and energy flux along the jet, the density andtemperature of the ‘missing’ thermal component of the jet can beobtained, as described in detail in Appendix B. We require initialconditions of density and velocity at the inner boundary. We take c (cid:13) , 1–, 1– ?? he particle content of low-power radio galaxies in groups and clusters Figure 6.
Jet properties vs. distance for our entrainment model, assuming an initial temperature for the thermal component of 100 keV. Top row: cross-sectionalarea (l) and velocity (r), middle row: gas density (l) and temperature (r) for the thermal component, bottom row: mass entrained per unit length (l) and ratio ofkinetic to jet internal energy flux (r), all shown for the northern (black solid) and southern (red dashed) jets of 3C 31. the jet velocity of 3C 31 at 12 kpc from the model of Laing & Bri-dle (2002) as our inner boundary condition, and assume a range ofinitial gas temperatures. The choice of temperature for the thermalcomponent at 12 kpc sets a boundary condition on the gas density(via the pressure constraints), and hence determines the jet power.As discussed later, we can therefore use the jet power as a consis-tency check on the most appropriate choice of initial temperature.Fig. 6 shows some illustrative results, with initial conditionschosen to obtain jet powers matched for the two jets, and in broad agreement with the model of Laing & Bridle (2002) (this requiresinitial temperatures at 12 kpc of 100 keV and 230 keV for the north-ern and southern jets, respectively. In this model the behaviour ofthe two jets is broadly similar, but with some differences driven byvariation in how the jet geometry evolves. The northern jet can bedivided into several regions on scales of tens to hundreds of kpc inwhich its geometry differs. As shown in the top left panel of thefigure, the cross-sectional area initially increases steeply with dis-tance, the jet then becomes cylindrical between around 20 kpc to c (cid:13) , 1– ?? J. H. Croston & M. J. Hardcastle
Figure 7.
The evolution of energy flux with distance along the jet, for models with matched jet power, showing kinetic energy (black), magnetic fieldenergy (green), internal energy of thermal particles (red) and of relativistic leptons (blue), with left and right panels indicating the northern and southern jets,respectively.
60 kpc; and then the jet radius increases again to 100 kpc scalesand beyond. These geometrical features show an interesting corre-spondence with bends in the jet (occuring at both of those transitionpoints), and with the external pressure gradient, as the pressure pro-file flattens at around 20 kpc (plausibly moving from a galaxy-scalehalo to a group-scale atmosphere) and then steepens again between50 and 100 kpc. The density profile that results from an assump-tion of constant temperature along the jet shows features that cor-respond to this geometry, with an inner region of increasing density,followed by a region of constant density and then a decreasing den-sity in the outer region as the jet/plume widens. Finally the bottomleft panel shows that in this model the entrainment must be fairlylocalized, with large amounts of material ingested at the two transi-tion points of ∼ and kpc (note that these are distances alongthe jet centre-line, rather than radial distances in the group atmo-sphere). At other times the entrainment rate is low. The southern jetexpands more smoothly, and somewhat faster, consequently requir-ing entrainment to be spread out over larger distances. At large dis-tances the cross-sectional area expands significantly more steeplythan for the northern jet, which leads to higher entrainment, decel-eration, and thermalization of kinetic energy.The conservation-law analysis of Laing & Bridle (2002) leadsto an entrainment rate at 12 kpc of ∼ kg s − kpc − . Themodels shown in Fig 6 are consistent with this level of entrainment;however, it is also possible that the entrainment in our model resultsfrom fairly localised disruption of the jet at its bends, which maybe unconnected to the steadier entrainment implied by the model ofLaing & Bridle (2002), in which case consistency with their mea-surement of entrainment rate is not required.In our model the energy flux is primarily in the form of kineticenergy in the inner parts of the jet, but is increasingly converted intointernal energy of the thermal (and presumably relativistic) parti-cles, as shown in Fig 7. The temperatures required by our model,for realistic jet powers, are much higher than the temperature ofthe surrounding gas, indicated that the entrained material must beheated fairly rapidly by tapping the jet’s kinetic energy. We are as-suming that all of the thermal material at a particular distance inthe jet has a single temperature, which is simplistic; however, at any given position the majority of material will have been in the jetfor some time with recently entrained gas comprising only a smallfraction. We note that temperatures of > keV for entrained gasare consistent with the limits on the presence of thermal materialin cluster cavities obtained from limits on the thermal X-ray emis-sion due to this gas (e.g. Blanton et al. 2003; Sanders & Fabian2007). The ‘thermal’ component, although very hot, remains (pre-dominantly) sub-relativistic in this model, although a non-thermal,relativistic tail cannot be ruled out.Hence we conclude that our simple entrainment model is qual-itatively consistent with providing the dominant energetic contribu-tion to the jets and plumes of 3C 31 on scales from 10 to 100 kpc.Most interestingly, if entrainment does drive the source energetics,then much of the mass ingestion appears to be localised, and coin-cide with regions where the jets bend and/or spread. In particular,the two regions of the northern jet where entrainment takes place inour model coincide with the flattening and steepening of the exter-nal pressure profile, it is clear that the gas distribution in the groupenvironment determines the energetic evolution of the radio-galaxyplasma on these scales.Our assumption of equipartition of energy density betweenparticles and magnetic field may not be correct. We argue in Sec-tion 3.1 that non-equipartition models with no protons are unlikelyto be correct, but a model where thermal and relativistic particlestogether dominate the energetics, with a lower magnetic field en-ergy density, cannot be ruled out. However, such a model would notstrongly differ in the qualitative picture for the evolution of thermalcontent of the jet – the radio synchrotron constraints mean that if themagnetic field strength contributes a lower fraction of the internalenergy flux then the electron contribution must increase. Significantentrainment would still be required at the locations seen in Fig 6,but the quantities of mass entrained and the required temperatureprofile could be somewhat different. Uncertainty in the geometry of the radio jets, and in particular howthe jet orientation changes relative to the plane of the sky at the ob- c (cid:13) , 1–, 1–
60 kpc; and then the jet radius increases again to 100 kpc scalesand beyond. These geometrical features show an interesting corre-spondence with bends in the jet (occuring at both of those transitionpoints), and with the external pressure gradient, as the pressure pro-file flattens at around 20 kpc (plausibly moving from a galaxy-scalehalo to a group-scale atmosphere) and then steepens again between50 and 100 kpc. The density profile that results from an assump-tion of constant temperature along the jet shows features that cor-respond to this geometry, with an inner region of increasing density,followed by a region of constant density and then a decreasing den-sity in the outer region as the jet/plume widens. Finally the bottomleft panel shows that in this model the entrainment must be fairlylocalized, with large amounts of material ingested at the two transi-tion points of ∼ and kpc (note that these are distances alongthe jet centre-line, rather than radial distances in the group atmo-sphere). At other times the entrainment rate is low. The southern jetexpands more smoothly, and somewhat faster, consequently requir-ing entrainment to be spread out over larger distances. At large dis-tances the cross-sectional area expands significantly more steeplythan for the northern jet, which leads to higher entrainment, decel-eration, and thermalization of kinetic energy.The conservation-law analysis of Laing & Bridle (2002) leadsto an entrainment rate at 12 kpc of ∼ kg s − kpc − . Themodels shown in Fig 6 are consistent with this level of entrainment;however, it is also possible that the entrainment in our model resultsfrom fairly localised disruption of the jet at its bends, which maybe unconnected to the steadier entrainment implied by the model ofLaing & Bridle (2002), in which case consistency with their mea-surement of entrainment rate is not required.In our model the energy flux is primarily in the form of kineticenergy in the inner parts of the jet, but is increasingly converted intointernal energy of the thermal (and presumably relativistic) parti-cles, as shown in Fig 7. The temperatures required by our model,for realistic jet powers, are much higher than the temperature ofthe surrounding gas, indicated that the entrained material must beheated fairly rapidly by tapping the jet’s kinetic energy. We are as-suming that all of the thermal material at a particular distance inthe jet has a single temperature, which is simplistic; however, at any given position the majority of material will have been in the jetfor some time with recently entrained gas comprising only a smallfraction. We note that temperatures of > keV for entrained gasare consistent with the limits on the presence of thermal materialin cluster cavities obtained from limits on the thermal X-ray emis-sion due to this gas (e.g. Blanton et al. 2003; Sanders & Fabian2007). The ‘thermal’ component, although very hot, remains (pre-dominantly) sub-relativistic in this model, although a non-thermal,relativistic tail cannot be ruled out.Hence we conclude that our simple entrainment model is qual-itatively consistent with providing the dominant energetic contribu-tion to the jets and plumes of 3C 31 on scales from 10 to 100 kpc.Most interestingly, if entrainment does drive the source energetics,then much of the mass ingestion appears to be localised, and coin-cide with regions where the jets bend and/or spread. In particular,the two regions of the northern jet where entrainment takes place inour model coincide with the flattening and steepening of the exter-nal pressure profile, it is clear that the gas distribution in the groupenvironment determines the energetic evolution of the radio-galaxyplasma on these scales.Our assumption of equipartition of energy density betweenparticles and magnetic field may not be correct. We argue in Sec-tion 3.1 that non-equipartition models with no protons are unlikelyto be correct, but a model where thermal and relativistic particlestogether dominate the energetics, with a lower magnetic field en-ergy density, cannot be ruled out. However, such a model would notstrongly differ in the qualitative picture for the evolution of thermalcontent of the jet – the radio synchrotron constraints mean that if themagnetic field strength contributes a lower fraction of the internalenergy flux then the electron contribution must increase. Significantentrainment would still be required at the locations seen in Fig 6,but the quantities of mass entrained and the required temperatureprofile could be somewhat different. Uncertainty in the geometry of the radio jets, and in particular howthe jet orientation changes relative to the plane of the sky at the ob- c (cid:13) , 1–, 1– ?? he particle content of low-power radio galaxies in groups and clusters served jet bends, is potentially a major limitation of our analysis.As discussed in Section 3, our main observational result – that thesynchrotron-emitting components of the jet contribute a decreas-ing fraction of the jet pressure, if at equipartition – is not affectedby uncertainties in projection. In a geometry with high inclination,the resulting larger synchrotron emitting volume and lower externalpressure acting on a particular region due to larger radial distancein the cluster counteract each other, which means that the overallresult is largely unaffected. The evolution of the non-radiating par-ticle energy fraction in Models III and IV (or of U E and U B inModels I and II) are therefore qualitatively similar in any plausi-ble geometry, even though the numerical values will change some-what. We do not attempt here to derive precise constraints on thejet energy content at a particular radius, but rather to develop arobust qualitative understanding of how the components of the jetplasma evolve. Therefore, while we acknowledge that the geometryis poorly constrained, our general conclusions are robust. A further uncertainty comes from our lack of knowledge of thelow-energy electron distribution as a function of distance along thejets. This will soon be remedied by ongoing work with LOFAR(Heesen et al., in prep); however, at present we can only extrapo-late to the lowest frequencies from the radio spectrum at 330 MHz.As discussed in Section 2.2 we assumed a low-frequency spectralindex of α = 0 . (e.g. Laing & Bridle 2013) and a value of γ min = 10 . Evidence for γ min (cid:29) comes from the broad-bandspectra of hotspots (e.g. Meisenheimer et al. 1997; Carilli et al.1999); however, the situation in FRI jets remains unknown. For theelectron distribution assumptions to significantly alter our resultswe would require an evolution in the low-frequency properties ofthe jet with distance from the nucleus. If γ min is determined bythe particle acceleration process that occurs in the inner jet, thenit is plausible that it could evolve to lower energies (e.g. via adia-batic losses) at the plasma is advected downstream. Alternatively,the low-frequency spectral index could evolve to become steeper atlarger distances, but there is no indication that this is the case in theexisting 330-MHz data (e.g. the spectral index between 330 MHzand 608 MHz is ∼ . for the outermost region we consider in thenorthern jet).We investigated the electron energy distribution that wouldbe required to achieve pressure balance in the outermost regionof the 3C 31 northern plume, assuming equipartition (the non-equipartition cases having been considered previously). Simply re-ducing γ min to 1, while extrapolating from the observed spectralindex of 0.55, is inadequate to achieve pressure balance. It wouldnecessary for the radio spectrum to steepen significantly below 330MHz, to α > . , and to have a low-energy cut-off of γ min = 1 in order for the synchrotron emitting components to provide allof the pressure within the lobes at this distance. As the radiationfrom such a component is currently unobservable with existing ra-dio data, this scenario is effectively indistinguishable from ModelsIII and IV, above; however, it is difficult to reconcile with particleacceleration models, and would require a second relativistic parti-cle population that has previously been undetected. Such a domi-nant lepton population with γ < , emitting below 330-MHz,cannot currently be ruled out by existing radio or X-ray inverseCompton constraints. We also cannot at this stage rule out morecomplex models in which the spectral index (and γ min ) vary whilethe contribution from thermal gas also changes with distance, but we look forward to being able to test such models in the near futurewith LOFAR data. We have focused mainly on the region of the jet beyond 10 kpc,where it is thought to be subrelativistic and evolving into the groupgas environment. As shown in Figs. 1 and 2, the evolution of thejet plasma appears to be different in the region inside 10 kpc. Wehave made no attempt to correct for the effects of relativistic beam-ing in calculating our radio emissivity profile as our focus is on theouter regions, but the effect of “de-beaming” the synchrotron emis-sivity [assuming the velocity model of Laing & Bridle (2002)] isa small decrease in the pressure of the synchrotron components ofthe northern jet, and an increase in their contribution for the south-ern jet. Hence this does not qualitatively alter the behaviour of thenorthern jet, though it brings the southern jet to have a roughly con-stant ratio of P ext /P synch in the inner region.If Model IV above is the correct explanation for the evolutionof the jet plasma on scales beyond 10 kpc, then other effects mustbe more important in the inner region. One possibility is that the jetis initially significantly electron (or relativistic electron and proton)dominated (e.g. due to substantial particle acceleration in the in-ner jet) before evolving towards equipartition between particles andmagnetic field, with entrainment taking over as an important mech-anism affecting the overall energetics from around 10 kpc. Sucha model is somewhat speculative, however, with the microphysicsof energy transfer between jet components poorly understood anddifficult to test. We have shown that X-ray and radio measurements of externalpressure and internal pressure from radiating material as a functionof distance along the source can be used to distinguish betweenmodels for the contents of radio lobes. Considering in detail thecases of 3C 31 and Hydra A, we have shown that: • The fractional contribution to the total energy budget fromsynchrotron-emitting components (relativistic leptons and mag-netic field), if at equipartition, must decrease with distance fromthe central AGN. • A model in which the energetics are dominated by relativisticleptons can be ruled out by inverse-Compton limits. • Magnetic domination requires the magnetic field strength toremain close to constant along the jet, which is implausible giventhe jet geometry, due to the need to convert an increasing fractionof the jet energy into magnetic field as the jet evolves, without pro-ducing significant particle acceleration. • A model in which relativistic protons/ions injected in the innerjet dominate the jet energetics and evolve adiabatically along the jetis ruled out. • Finally we have demonstrated that a simple entrainment modelis consistent with the external pressure constraints and the evolu-tion of radio emissivity, with regions of entrainment correspondingto locations of jet bending/disruption and changes in the externalpressure profile. Such a model requires a high temperature for theentrained component, and an increasing temperature with distance,consistent with a rapidly decreasing kinetic energy flux of the jetbeing converted to particle and magnetic field internal energies. c (cid:13) , 1– ?? J. H. Croston & M. J. Hardcastle
The results presented here are based on consideration of a sin-gle object, for which the highest quality radio and X-ray data on thescales of interest are available. Our detailed pressure comparisonfor Hydra A, as well as indications from less well constrained com-parisons for other objects (Evans et al. 2005; Hardcastle et al. 1998;Worrall & Birkinshaw 2000) and circumstantial evidence from ob-servations of cluster cavities, mean that it is plausible that our con-clusion that an entrainment-dominated model is favoured in 3C 31can be generalised to low-power radio galaxies in general. In fu-ture work we will apply these analysis methods to other systemswith high-quality X-ray and radio data, as well as incorporatingnew low-frequency radio measurements to minimise uncertaintiesfrom extrapolation of the electron energy distribution.
ACKNOWLEDGMENTS
JHC acknowledges support from the South-East Physics Network(SEPNet) and from the Science and Technology Facilities Council(STFC) under grant ST/J001600/1. We would like to thank RobertLaing for providing the 1.4-GHz map of 3C 31. We would also liketo thank the referee, Geoff Bicknell, for a helpful report, which hasenabled us to improve the paper.
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APPENDIX A. METHOD FOR CALCULATING κ (RATIOOF NON-RADIATING PARTICLES TO RELATIVISTICLEPTON ENERGY DENSITY) We assume: • The jet internal pressure, P int , balances the external pressureat each distance, P ext , which is measured from the X-ray observa-tions. • The internal pressure has contributions from magnetic field, P B , synchrotron-radiating relativistic particles, P E , and thermalgas entrained from the environment, P th . • Pressure balance along the jet is described by the followingrelation between the external pressure and the internal energy den-sities of magnetic field, relativistic and non-relativistic (thermal)particles: P ext = 13 U E + 13 U B + fU P (1) c (cid:13) , 1–, 1–
APPENDIX A. METHOD FOR CALCULATING κ (RATIOOF NON-RADIATING PARTICLES TO RELATIVISTICLEPTON ENERGY DENSITY) We assume: • The jet internal pressure, P int , balances the external pressureat each distance, P ext , which is measured from the X-ray observa-tions. • The internal pressure has contributions from magnetic field, P B , synchrotron-radiating relativistic particles, P E , and thermalgas entrained from the environment, P th . • Pressure balance along the jet is described by the followingrelation between the external pressure and the internal energy den-sities of magnetic field, relativistic and non-relativistic (thermal)particles: P ext = 13 U E + 13 U B + fU P (1) c (cid:13) , 1–, 1– ?? he particle content of low-power radio galaxies in groups and clusters where f is / for relativistic protons/ions and f = 2 / for thermalgas. The ratio of energy densities in non-radiating particles andsynchrotron-emitting particles (relativistic leptons) is κ , i.e. U P = κU E , leading to the following equations for the relativistic case: P ext = (1 + κ rel ) U E + U B (2)and the thermal case: P ext = (2 κ th + 1) U E + U B (3)If we assume that the distribution of electron energy density isdescribed by a power law with index p (cid:54) = 2 (i.e. N ( E ) = N E − p ),then the electron energy density is given by: U E = (cid:90) E max E min EN ( E )d E = N E − pmax − E − pmin − p (4)where N is the electron energy density normalization, p is theelectron energy index, E min and E max are the lower and uppercut-offs of the electron energy distribution.We assume equipartition between magnetic field and all parti-cles (relativistic and non-relativistic), i.e. U B = (1 + κ ) U E , whichleads to the standard expression for the equipartition magnetic fieldstrength: B eq = (cid:34) µ (1 + κ ) J ( ν ) ν ( p − c (2 − p ) (cid:0) E − pmax − E − pmin (cid:1)(cid:35) p +5 (5)where J ( ν ) is the synchrotron emissivity at a frequency ν , givenby J ( ν ) = c N ν − ( p − B ( p +1)2 (6)where c is a constant (Longair 1994): c = k ( p ) e (cid:15) cm e (cid:18) m e c e (cid:19) − ( p − (7)where k ( p ) is 0.050407 for p = 2 , 0.039484 for p = 2 . , and0.031547 for p = 2 . . We can now make use of the pressure con-straints derived earlier for the relativistic proton/ion and thermalgas cases (Eqs, 2 and 3, applying to Models III and IV, respec-tively) to get a second expression for B . For Model III, substitutingin for U E in Eq. 2 gives: B µ = 3 P ext − (1 + κ ) N − p (cid:2) E − pmax − E − pmin (cid:3) (8)and substituting in Eq. 6 gives: B µ = 3 P ext − (1 + κ ) J ( ν ) ν ( p − B − ( p +1)2 c (2 − p ) (cid:2) E − pmax − E − pmin (cid:3) (9)We can now substitute in our previously derived expression for theequipartition B field (Eq. 5): (cid:34) µ (1 + κ rel ) J ( ν ) ν ( p − c (2 − p ) (cid:0) E − pmax − E − pmin (cid:1)(cid:35) p +5 (2 µ ) − =3 P ext − (1 + κ rel ) J ( ν ) ν ( p − c (2 − p ) (cid:2) E − pmax − E − pmin (cid:3)(cid:34) µ (1 + κ rel ) J ( ν ) ν ( p − c (2 − p ) (cid:0) E − pmax − E − pmin (cid:1)(cid:35) − ( p +1)( p +5) (10) This expression can be simplified to: κ rel = (cid:18) P ext (cid:19) p +54 (2 µ ) p +14 c − (11)where c = J ( ν ) ν p − c (2 − p ) (cid:2) E − pmax − E − − pmin (cid:3) (12)For the thermal case (Model IV), a similar expression can bederived from Eq 3, with a slightly difference dependence on κ : (cid:34) µ (1 + κ th ) J ( ν ) ν ( p − c (2 − p ) (cid:0) E − pmax − E − pmin (cid:1)(cid:35) p +5 (2 µ ) − =3 P ext − (1 + κ th ) J ( ν ) ν ( p − c (2 − p ) (cid:2) E − pmax − E − pmin (cid:3)(cid:34) µ (1 + κ th ) J ( ν ) ν ( p − c (2 − p ) (cid:0) E − pmax − E − pmin (cid:1)(cid:35) − ( p +1)( p +5) (13)which simplifies to: P ext = (2 µ ) − p +1 p +5 c p +5 (cid:104) (1 + κ th ) p +5 + (2 κ th + 1)(1 + κ th ) − p +1 p +5 (cid:105) (14)For both models we have now have an equation that contains onlyone unknown, κ . For an observed S ( ν (cid:48) ) and P ext , the value of κ can therefore be obtained (numerically, in the thermal case), whichalso allows B , U E , and U p , and finally the thermal gas density inthe jet for a given assumed temperature, to be obtained. APPENDIX B. DETAILS OF ENTRAINMENTCALCULATIONS
For a steady-state jet, the evolution of the dynamics and energycontent of the jet can be described by the equations of conservationof momentum flux and energy flux. We assume that the jet velocityis non-relativistic, which is appropriate for the region of jet consid-ered in this analysis. We consider a region of jet between distance l and l from the nucleus. The conservation of momentum flux, Π = ρv A , is described by: ρ v A = ρ v A + Π buoy (15)where ρ , , A , and v , are the gas density, cross-sectional areaand velocity of the jet, respectively, and Π buoy is the change inmomentum flux due to the buoyancy force acting on the jet (seebelow).The conservation of energy flux can be described by (cf. Bick-nell 1994): (cid:18) ρ v + U + P (cid:19) v A = (cid:18) ρ v + U + P (cid:19) v A (16)where P , is the total internal pressure (assumed to match the ex-ternal pressure at the given radius), and the internal energy terms, U , are given by: U i = 32 P i + U e + U B , (17)i.e. including terms for the internal energy carried by thermal par-ticles, relativistic leptons and magnetic field, respectively. c (cid:13) , 1– ?? J. H. Croston & M. J. Hardcastle
With suitable initial conditions, the run of external pressureand of κ determined from the analysis in Section 3.2, Equations 15and 16 can be solved for the unknowns ρ and v (where themean particle mass µ − . , as appropriate for entrained ICM gas).The gas temperature of the thermal material is also determined via P therm,i = ( ρ i /µm H ) kT i ) where P thermal = (2 / U thermal =(2 κ i / U E,i is determined from the analysis in Section 3.2.As discussed in Section 3.3 as initial conditions we assume v kpc = 6 × m s − , and test a range of initial temperaturevalues, which together with P kpc determine ρ . Equation 15 isrearranged for v , and then substituted into Equation 16. A stan-dard root-finding algorithm can then be used to solve for ρ . Thetemperature of the thermal material in region 2 is then determinedas explained above.The buoyancy term Π buoy in Equation 15 is determined asfollows. The buoyant force acting on the jet material between l and l is given by: F buoy = − ∆ mg (18)where ∆ m is the mass of the surrounding material displaced bythe chunk of jet material between l and l , i.e. ∆ m = ∆ m env − ∆ m jet , and g is the acceleration due to gravity: g = Gm ( l ) l (19)where m ( l ) is the enclosed gas mass within the galaxy cluster atradius l . In this case we can assume that ∆ m env >> ∆ m jet , andso we take ∆ m = ∆ m env = ρ env ( l ) A ( l ) δl , where ρ env ( l ) is theexternal gas density at distance l .The change in momentum is given by: ∆ p = F buoy ∆ t (20)where ∆ t is the interval during which material travels from l to l . Hence ∆ l = v ( l )∆ t .The change in momentum flux due to buoyancy is therefore: Π buoy = ∆ pv = (cid:90) l l F buoy d l (21)Using the equation of hydrostatic equilibrium, F buoy can be ex-pressed in terms of the external pressure gradient: F buoy = − Gm ( l ) ρ env ( l ) A ( l )∆ ll = d P ext d l A ( l )∆ l (22)So Π buoy = (cid:90) l l d P ext d l A ( l )d l (23)Hence the buoyancy term in Equation 15 can be evaluated using ourmeasured external pressure gradient (e.g. Fig 1) and jet geometry.Finally, the mass entrainment rate can be determined fromconservation of mass flux as follows: ρ v A = ρ A v + Ψ (24)where Ψ is the mass entrained per unit time in the region between l and l . c (cid:13) , 1–, 1–