Dynamics of rising magnetized cavities and UHECR acceleration in clusters of galaxies
MMon. Not. R. Astron. Soc. , 1–9 (-) Printed 29 October 2018 (MN L A TEX style file v2.2)
Dynamics of rising magnetized cavities and UHECRacceleration in clusters of galaxies
Konstantinos N. Gourgouliatos (cid:63) & Maxim Lyutikov
Department of Physics, Purdue University,525 Northwestern Avenue, West Lafayette, IN 47907-2036
Accepted -. Received -; in original form -
ABSTRACT
We study the expansion of low density cavities produced by Active Galactic Nuclei jetsin clusters of galaxies. The long term stability of these cavities requires the presence oflinked magnetic fields. We find solutions describing the self-similar expansion of struc-tures containing large-scale electromagnetic fields. Unlike the force-free spheromak-likeconfigurations, these solutions have no surface currents and, thus, are less suscepti-ble to resistive decay. The cavities are internally confined by external pressure, withzero gradient at the surface. If the adiabatic index of the plasma within the cavity isΓ > /
3, the expansion ultimately leads to the formation of large-scale current sheets.The resulting dissipation of the magnetic field can only partially offset the adiabaticand radiative losses of radio emitting electrons.We demonstrate that if the formation of large-scale current sheets is accompaniedby explosive reconnection of the magnetic field, the resulting reconnection layer canaccelerate cosmic rays to ultra high energies. We speculate that the enhanced flux ofUHECRs towards Centaurus A originates at the cavities due to magnetic reconnection.
Key words: magnetic reconnection; cosmic rays: ultra-high energies; galaxies: clus-ters: general; acceleration of particles
Active galactic nuclei (AGNs) produce relativistic jets ema-nating from the central black hole that shock the intraclustermedium (ICM), and create over-pressurized lobes. As theselobes expand to achieve pressure equilibrium, they form lowdensity, high entropy cavities, also referred to as “bubbles”.Most bubbles are not expanding supersonically and, thus,are in pressure balance with their surroundings. Their risein the cluster potential is controlled by buoyancy, and theyappear as depressions in the X-ray emissivity (Fabian et al.2000; Churazov et al. 2001; Bˆırzan et al. 2004; Diehl et al.2008). The jets that create the bubbles contain both plasmaand magnetic fields. During the active stage the location ofthe jet termination is seen as a radio lobe, while cavitiesare likely to be the outcome of disconnected lobes duringthe quiescent stage. Jets are active for typical timescalesof ∼ years, with luminosities of 10 erg s − . During thequiescent stage bubbles rise in the intergalactic medium.It is quite likely that the magnetic field plays a cru-cial rˆole, however there is no direct measurement yet. Cavi-ties survive on timescales much longer than the timescale ofbuoyant rise. Thus, rising bubbles in the ICM must remain (cid:63) E-mail: [email protected] coherent, and avoid being shredded by the Rayleigh-Taylorinstability (RT), Richtmyer-Meshkov instability (RM) andKelvin-Helmholtz instability (KH). On the other hand, sim-ple hydrodynamic models “face multiple failures” (Reynoldset al. 2005) when addressing bubble dynamics: in fluid sim-ulations the RT and KH instabilities quickly disrupt thebubbles on approximately one rise time ( e.g.
Kaiser et al.2005). One of the principal reasons, perhaps, for the fail-ure of fluid codes is that the importance of magnetic fieldshas been generally underestimated in the ICM, based onthe notion that magnetic fields are dynamically subdomi-nant. In fact, even dynamically subdominant magnetic fieldcan have drastic effects on bubble evolution, stability andentropy production . First, magnetic fields in the ICM drapearound expanding cavities forming a layer of nearly equipar-tition magnetic field, which stabilizes the bubble againstthe Kelvin-Helmholtz and Rayleigh-Taylor instability (Dursi2007; Dursi & Pfrommer 2008). Secondly, if the AGN jetwhich blew the bubble carries large-scale magnetic fields,it will also stabilize the bubble. Stable magnetic modelsfor cavities have been proposed by Benford (2006); Dong& Stone (2009); Gruzinov (2010); Braithwaite (2010), whofind the fields inside the bubble and suppress instabilities.Examples of magnetic fields in the exterior of the bubblethat contribute to the stability of the cavities via the mag- c (cid:13) - RAS a r X i v : . [ a s t r o - ph . H E ] M a y K. N. Gourgouliatos & M. Lyutikov netic draping effect have been presented by Lyutikov (2006).Nevertheless, it has been argued that unmagnetised bubblescan be stable provided a dense shell forms around it duringthe inflation stage (Sternberg & Soker 2008).Motivated by the above astrophysical structures we ad-dress and solve the following physical problem. Let us as-sume that a magnetic structure containing plasma and mag-netic field in equilibrium is confined by some external pres-sure. This pressure is taken to be constant on the boundaryof the cavity. As the bubble rises in the gravitational poten-tial of a cluster, the external pressure decreases. How willthe internal structure of the bubble adjust to the changingexternal conditions? We find that the evolution of the cavitydepends on the adiabatic index of the material: for Γ = 4 / > / < / > / × erg s − (Alexander &Leahy 1987). There is a mildly relativistic jet, detected inboth radio and at high energies, emitting ∼ × erg s − in keV – MeV range (Kraft et al. 2002; Hardcastle et al.2003). The orientation θ of the jet of Centaurus A with re-spect to our line of sight is 35 ◦ < θ < ◦ ( e.g. Skibo et al.1994). The jet is launched by a central black hole of mass M BH ∼ − × M (cid:12) . The galaxy is surrounded by aset of AGN-blown cavities, inner ( ∼ − ∼ −
20 kpc in size) and largescale of severalhundred kiloprsecs, (see, e.g.
Israel 1998, for review).A crucial question is how these cosmic rays are acceler-ated. Giannios (2010) has suggested that reconnection canaccelerate UHECRs up to the essential energies in AGN jets;Benford & Protheroe (2008) have suggested that this processtakes place in fossil AGN jets through slow reconnection.Our suggestion in this paper is that the change of the struc-ture of the cavity leads to the formation of current sheets,where explosive reconnection is triggered and accelerationof UHECRs actually takes place therein. An important is-sue with cosmic ray acceleration through reconnection is tohave slow enough magnetic field dissipation at early stagesso that a strong field builds, and then a physical mechanismto trigger reconnection and accelerate them. The descriptionof the cavity evolution we propose provides these properties.They start without currents sheets, thus the magnetic fieldgradient is low enough to have a slow dissipation rate, butas they expand their plasma β drops low enough to createa magnetically dominated region and a current sheet.The structure of the paper is the following: We startfrom dimensional arguments about the magnetic and plasmapressure evolution during the expansion of a magnetized cav-ity; then we solve explicitly the partial differential equationsdescribing the problem: we find an exact dynamical solutionand a series of quasi-static solutions which correspond to awider range of parameters. Finally we apply our results forthe bubble evolution in the acceleration of UHECRs. Let us start with a sphere of radius R containing plasma oftotal mass M and and some magnetic flux Ψ, the plasmapressure p is related to the density ρ through an adiabaticindex Γ. The density of the cavity shall be ρ ∝ MR − , themagnetic field B ∝ Ψ R − , the magnetic pressure p mag ∝ Ψ R − and the plasma pressure p ∝ ρ Γ ∝ ( MR − ) Γ . Letus consider some change by ∆ R of the radius of the cavity,while the total mass and flux are conserved, and estimatethe change caused to the plasma pressure, to the magneticpressure and the plasma β = p/p mag .∆ pp = −
3Γ ∆
RR , ∆ p mag p mag = − RR , ∆ ββ = (4 − RR . (1)From the above result we expect that if Γ < / c (cid:13) - RAS, MNRAS , 1–9 agnetized cavities and UHECR acceleration sphere expands, whereas if Γ > / / > /
3, decreasing the plasma β implies that atsome point during expansion, the minimal pressure insidethe sphere becomes zero. Further expansion leads to the ap-pearance of current sheets. Stability arguments require that the internal structure ofmagnetic fields should be a combination of toroidal andpoloidal components, which, in order to be in equilib-rium with a constant external pressure, must vanish onthe boundary. These conditions lead to an overdeterminedmathematical problem: We seek solutions of an ellipticalpartial differential equation, the Grad-Shafranov equation(Shafranov 1966), with a given value of the function we aresolving for and its derivative on the boundary. Thus the so-lution must satisfy both Newman and Dirichlet boundaryconditions simultaneously. However, two more functions ap-pear in the elliptical partial differential equation: a functionrelated to the poloidal current and a function related to theplasma pressure. The only constraint they must satisfy is tobe functions of the flux. As the pressure and the poloidalelectric current are not determined a priori , they can bechosen so that the solution satisfies the over-constrainingboundary condition. This problem has been solved in GBL,and we use this result as the initial state for the rising mag-netic cavity.In general the magnetic field can be expressed in termsof poloidal and toroidal components: B = ∇ Ψ × ∇ φ + 2 I (Ψ) ∇ φ . (2)The poloidal flux and current passing through a sphericalcap of radius r and opening angle θ are 2 π Ψ( r, θ ) and cI (Ψ)respectively. A solution satisfying the Grad-Shafranov equa-tion J × B = ∇ p isΨ = sin θ (cid:110) c (cid:104) α cos( αr ) − sin( αr ) r (cid:105) − F α r (cid:111) , (3) I = α , (4) p = F Ψ4 π + p , (5) B r = 2 cos θr (cid:110) c (cid:104) α cos( αr ) − sin( αr ) r (cid:105) − F α r (cid:111) ,B θ = sin θr (cid:110) c (cid:104) a cos( αr ) r − sin( αr ) (cid:16) r − α (cid:17)(cid:105) + 2 F rα (cid:111) ,B φ = α sin θr (cid:110) c (cid:104) α cos( αr ) − sin( αr ) r (cid:105) − F α r (cid:111) . (6)This solution has been studied in detail in GBL, and corre-sponds to a linear relation between the physical quantities:poloidal magnetic flux, poloidal electric current and plasmapressure. Given an appropriate choice of α and F the fieldis confined inside a spherical cavity (Figure 1) and the sur-face currents can be eliminated, while the whole structure Figure 1.
Plot of the the poloidal field lines (thin red) and thecentral toroidal field line (thick blue). is stable . In the static form of the Grad-Shafranov equa-tion the pressure appears through its derivative, while p isa constant which is used so that there is no negative energydensity inside the cavity, thus it has a minimum permittedvalue, but no upper limit. The cavity is a dip in plasma pres-sure and density compared to the external medium, whereasthe magnetic field inside the cavity is higher. The choice of p affects β , the ratio of the plasma pressure over the mag-netic pressure in the cavity. The value of β is not constantwithin the cavity, we can evaluate an average value¯ β = (cid:82) p mag dV (cid:82) pdV . (7)The value for ¯ β that corresponds to the minimum permittedvalue of p is 0 .
8, however this is mainly due to the contri-bution of the area near the boundaries of the cavity, if westop the integration at 0 . β drops to 0 .
5, while if we stop at 0 . β is as lowas 0 . α and F are given subject to the condi-tions that Ψ( r c ) = 0 and d Ψ dr | r c = 0, where r c is the radiusof the sphere, these conditions lead to zero fields on thesurface and a smooth transition from the cavity to the ex-ternal medium. For instance, setting r c = 1, and normalisingthe maximum value of Ψ to unity, we find that α = 5 . c = − .
134 and F = − . c (cid:13) - RAS, MNRAS000
134 and F = − . c (cid:13) - RAS, MNRAS000 , 1–9 K. N. Gourgouliatos & M. Lyutikov
In this section we allow α = α ( t ) and we introduce time-dependence through self-similar expansion of concentricshells with velocity v = − r ˙ αα ˆ r . The evolution of the struc-ture is given by the momentum equation, while it mustsatisfy Maxwell equations, the continuity equation and theentropy equation. Given that we are working in a non-relativistic context we shall use the Maxwell equations keep-ing only the appropriate terms. ∇ · B = 0 , (8) ∇ × E = − c ∂ B ∂t , (9) ∇ × B = 4 πc j , (10)We shall not consider the dynamical effects of the electricfield, thus we do not take into account Gauss’ law for theelectric field and in the Amp`ere-Maxwell equation we shallnot take into account the displacement current. The electricfield through the ideal MHD approximation is E = − v c × B . (11)The continuity equation is ∂ρ∂t + ∇ · ( ρ v ) = 0 . (12)The momentum equation is ρ (cid:16) ∂∂t + v · ∇ (cid:17) v = −∇ p + j × B c . (13)Finally the entropy equation is (cid:16) ∂∂t + v · ∇ (cid:17)(cid:16) pρ Γ (cid:17) = 0 . (14)Flux and helicity conservation under self-similar expansionlead to the following form for ΨΨ = sin θα (cid:110) c (cid:104) sin( α ( t ) r ) α ( t ) r − cos( α ( t ) r ) (cid:105) − F ( α ( t ) r ) (cid:111) . (15)Note that Ψ = Ψ( α ( t ) r, θ ). The boundary conditions, sim-ilarly to the static case, are that the magnetic flux andits derivative are both zero at the boundary of the cavity r c , while the boundary of the cavity now depends on time r c = α r α ( t ) . The field is expressed in terms of a poloidal anda toroidal component. B = ∇ Ψ × ∇ φ + α ( t )Ψ ∇ φ , (16)Constraining our interest to uniform expansion without ac-celeration we take from the left-hand-side of the momentumequation (13) that α ( t ) = ( v t + r ) − , where v is the ex-pansion velocity of the boundary of the cavity. The plasmapressure becomes p = p + F α ( t ) Ψ4 πα . (17)From the baryonic mass conservation equation we find thatthe density is ρ ( r, θ, t ) = ˜ ρ ( α ( t ) r, θ )( α ( t ) r ) , (18)where ˜ ρ is a function of the self similar variable and θ . Fromthe entropy equation (14) it is p = Kρ Γ and combining thiswith equation (18) we find for Γ = 4 / p and α arerelated through p ( t ) ∝ α ( t ) . In the previous section we have found an exact solution forthe evolution of the cavity. However, this solution constrainsthe value of Γ and the time dependence of p . In this sec-tion we investigate the evolution of the cavity when the ex-ternal parameters change slowly compared to the dynamictimescale of the cavity itself. Under this assumption we candescribe the evolution as a series of static solutions whichare subject to the changing boundary conditions (Lynden-Bell 2003), without constraining Γ, or requiring any timedependence for p . Such solutions need to satisfy the righthand side of equation (13), and as the evolution is slow iner-tia terms are negligible; practically we find Grad-Shafranovequilibria.From equations (17), (18) and the adiabatic relation( p ∝ ρ Γ ) we find that p + F α Ψ4 πα = K (˜ ρr α ) Γ . (19)Let us express p = ˜ p α n , where ˜ p is constant. Takingthe logarithm of the above equation we find that there issome relation between n and Γ, which changes as the cavityexpands and cannot be expressed analytically, except for thespecial case of Γ = 4 /
3, where n = 4 and the radius of thecavity is r c ∝ p − / which leads us back to the solution ofsection 3.2.To find out the approximate form of n when Γ (cid:54) = 4 /
3, westart from a structure of a given ¯ β and then we decrease thepressure, demanding that the system remains in equilibriumwhile the mass, the flux and the helicity are conserved. Thusthe solution shall be of the form of equation (15). Doing sofor various values of Γ we find how the radius increases as p decreases. In accordance with the dimensional discussionof section 2, there are three distinct regimes depending onwhether Γ is greater, smaller or equal to 4 / /
3, ¯ β does not change with expansion. Thisis not the case when Γ > /
3, since the expansion causesa faster decrease of the plasma pressure over the magnetic,leading to smaller ¯ β . At some critical point, the gas pressureat the minimum becomes zero so that the evolution underthe self-similar scheme is impossible. At this stage we ex-pect that reconnection becomes important. If Γ < /
3, asthe cavity expands its magnetic pressure becomes weakercompared to the gas pressure while ¯ β increases, and fi-nally the magnetic field becomes diluted in the background,see Figures (2)-(4). The inverse process takes place if westudy the effect of compression instead. From this discus-sion we can determine the dependence of r c on p . The gen-eral result is plotted in Figure (5), we find r c ∝ p n , where n ≈ − (3Γ) − + (cid:15) n , Γ appears because of the plasma contentof the cavity, while (cid:15) n is related to the contribution of themagnetic pressure in the system and its value is such that n will tend towards − /
4, with greater contribution when themagnetic pressure is important in the cavity. In a similar waywe evaluate the evolution of ¯ β under expansion for variousvalues of Γ. From Figure (6) we find that ¯ β obeys a powerlaw with ¯ β ∝ r mc , where m = 4 −
3Γ + (cid:15) m ; (cid:15) m is a param-eter so that m will tend towards 0 for strongly magnetizedstructures. In general (cid:15) n and (cid:15) m will be small compared to n and m . Finally we conclude that ¯ β ∝ r − / (3Γ)+1+ (cid:15) p c where (cid:15) p = (4 − (cid:15) n − (cid:15) m . c (cid:13) - RAS, MNRAS , 1–9 agnetized cavities and UHECR acceleration Figure 2.
The pressure for the expanding cavity with Γ = 1,presented as a paradigm of pressure with smaller adiabatic indexthan 4 /
3. The cavity expands while p drops. As it expands thegas pressure dominates over the magnetic one and finally themagnetic cavity gets diluted. Figure 3.
The pressure for the expanding cavity with Γ = 4 / p drops, but it does not become dilutednor does it reach zero density at the centre. In this section we shall do some order of magnitude cal-culations to show that it is possible to reach the essentialpotentials for the UHECR acceleration. For the numericalestimates below we assume that the intrinsic Centaurus Ajet power is L j = 10 L ergs s − , while during flares it canreach 10 ergs s − . We also assume that the central black Figure 4.
The pressure for an expanding cavity with Γ = 2,presented as a paradigm for plasma with Γ > /
3. The cavity ex-pands while p drops, after some amount of expansion the densityat the centre becomes zero, while the field there is no longer zero.At this stage the field in the centre of the cavity tries to reach aforce-free equilibrium while the rest of the cavity has a pressure-magnetic field equilibrium. This state cannot be described by thesolution presented in this paper. Figure 5.
The dependence of the radius of the cavity r c on p ,for various values of Γ, empty circles are for Γ = 1, asterisksΓ = 6 /
5, empty squares Γ = 4 /
3, cross Γ = 3 /
2, solid circleΓ = 5 / r c on p canbe approximated by a power-law. For Γ > / (cid:13) - RAS, MNRAS000
2, solid circleΓ = 5 / r c on p canbe approximated by a power-law. For Γ > / (cid:13) - RAS, MNRAS000 , 1–9 K. N. Gourgouliatos & M. Lyutikov
Figure 6.
The plasma ¯ β as a function of r c , for various values ofΓ, empty circles are for Γ = 1, asterisks Γ = 6 /
5, empty squaresΓ = 4 /
3, cross Γ = 3 /
2, solid circle Γ = 5 / > / β ∼ . hole mass is M BH ∼ M (cid:12) and jet activity lasted for ∼ β > > /
3, after someexpansion will have parts with zero plasma pressure andcannot evolve further in this regime. At this stage some-thing must happen, e.g. formation of current sheets, whichbecome resistive and form reconnection layers.
The jet ejects material into the intergalactic medium withparticle density 10 − cm − . The cavity formed by ejectedmaterial is confined within a radius r c , further out is theforward shock with radius r FS formed from the compressedIGM material. From the Sedov-Taylor solution the forwardshock radius is r FS ∼ L / t / ρ / IGM ∼ L / t / n − / − . (20)As the cavity is in equilibrium with the shocked IGM, thepressure of the cavity ρ Γ c and that of the shocked IGM ρ IGM ( r FS /t ) are equal. Assuming that most of the energycomes in the form of mildly relativistic ejecta L ∼ ˙ Mc , wefind for Γ = 4 / r c ∼ (cid:16) L t c ρ IGM (cid:17) / ∼ L / t / n − / − , (21)and the ratio of the two radii is r c r FS = 0 . L / t − / n − / − . (22)which is a very weak function of the parameters, thus thecavity shall be about half the size of the forward shock, unless there is an extremely drastic change of the physicalparameters of the problem. Let us assume that a fraction σ j (cid:54) σ j Lc ) / . At thesame time magnetic flux is destroyed by reconnection at arate Ψ t r , where t r is the reconnection timescale. Thus theevolution of the flux in the cavity obeys˙Ψ = ( σ j Lc ) / − Ψ t r . (23)We expect that the magnetic field within the cavity willrelax to a minimum energy state on an Alfv´en time scale,which as we show below can be very short. Such a minimumenergy state will include a toroidal-poloidal configuration ofmagnetic field as a purely toroidal or poloidal field is un-stable and the co-existence of plasma will make the solutiondescribed in the previous sections appropriate. As the AGN-blown cavity expands, it does work on the shocked IGM, soits energy is not conserved; on the other hand, in the absenceof resistivity the magnetic flux is conserved. Neglecting theresistive decay of magnetic flux, the magnetic field in thecavity is B ∼ ˙Ψ tr c ∼ √ σ j (cid:18) Lc ρ IGM t (cid:19) / = 60 µ G √ σ j n / − . (24)The density in the cavity is very low ρ c = ( L/c ) tr c = (cid:18) L ρ IGM c t (cid:19) / = 3 × − cm − n / − . (25)By equating the shocked IGM pressure to ρ c v T , we can findthe typical thermal velocity of the cavity particles v T c = (cid:18) Lc ρ IGM t (cid:19) / = 0 . . (26)If the jet is ion-dominated, the corresponding ion tempera-ture is T i ∼ v T m p ∼
100 MeV, while for pair-dominated jet,the electron temperature is 45 keV.When jet ions pass through a reverse shock they areheated to T i ∼ m p c ∼ eV. If electrons are coupledto ions effectively, they are heated to the same temper-ature. This justifies the use of adiabatic index Γ = 4 / > /
3, so that the mag-netic pressure eventually dominates the cavity. In addition,both species can be accelerated non-thermally.The Alfv´en velocity is v A c = Bc √ πρ c = √ σ j (cid:18) ρ IGM c t L (cid:19) / = 3 √ σ j . (27)Let η A be the reconnection efficiency, a charge e shall beaccelerated to energyeΦ = η A v A c Br c = η A σ j c / L / t / ρ / IGM =2 × eV η A σ j n / − . (28) c (cid:13) - RAS, MNRAS , 1–9 agnetized cavities and UHECR acceleration We can neglect resistive decay provided that the reconnec-tion timescale, t r ∼ r c / ( η A v A ) = 1 η A √ σ j (cid:18) L t c ρ IGM (cid:19) / , (29)is longer than the AGN activity time, t r > t . Thus, t ∼ t r > √ L ( η A √ σ j ) / c / √ ρ IGM . (30)We should also make sure that the Alfv´en velocity, whichincreases with time, Eq. (27) does not exceed the speed oflight t (cid:54) √ Lc / σ / j √ ρ IGM . (31)The most effective acceleration will occur right before theresistivity becomes important. Under the constraint that theAlfv´en velocity at this point is smaller than the speed oflight, Eqns (30-31) give σ j = η A . The resistive time thenbecomes t r = √ Lη / A c / √ ρ IGM . (32)and η A = (cid:18) L c t AGN ρ IGM (cid:19) / = 10 − L / t − / n − / − . (33)This value leads to a jet where about 10 − of its energybudget comes in as magnetic field, thus it is consistent withthe total luminosity of the system. Using this estimate of η A and the condition σ j = η A , the available potential shallaccelerate a charge e to energyeΦ = (cid:18) L c ρ IGM t AGN (cid:19) / = 2 × eV . (34)The main reason the potential (34) is fairly small is thateffective reconnection requires high reconection speeds (Φ ∝ v rec ), but if reconnection is too efficient the magnetic fieldis destroyed too early (Φ ∝ B ). Still, it might be a viablemechanism. First, our order-of-magnitude estimates can eas-ily miss a factor of a few; in addition, accelerated particlescan be of CNO (Centaurus A is located sufficiently close)or Fe type. In that case, the acceleration shall take place inthe area where the field reverses through slow reconnectionas it has been proposed by Benford & Protheroe (2008). In section 4.2 we have shown that steady reconnection can-not lead to sufficient particle acceleration for UHECRs sincethe magnetic field constantly decays, reducing the availablepotential. However, reconnection in astrophysics is rarely asteady state process. For instance, solar flares can develop ontimescales of minutes, while being driven, one might argue,on the solar dynamo of 22 years.What we have shown in section 3.3 is that quasi-staticevolution with the appropriate Γ leads to the formation ofa region with very high magnetization which shall give ahigh value for η A ∼ η A v A c Br c = η A c / L / t / ρ / IGM == 2 × L / t / n − / − eV . (35)As the bubble with Γ > / The presence of magnetic fields in cavities leads to syn-chrotron radio emission; indeed some X-ray cavities haveenhanced radio emission (McNamara et al. 2000; Fabianet al. 2000) which is stronger when the cavity is closer tothe AGN. During the explosive reconnection stage, it is pos-sible to have additional ongoing acceleration within the cav-ities, offsetting the effects of adiabatic and radiative cooling.However, as we shall show below, this effect averaged overa longer time does not change significantly the radio profileof the cavities.The scenario we consider is the following: When theminimum plasma pressure inside the cavity reaches zero theideal expansion cannot proceed any more, current sheetsform which are susceptible to resistive decay. Resistive de-cay proceeds slowly at the beginning, with η A (cid:28)
1, untilthe formation of current sheets with η A ∼
1. The timescalefor reconnection at the current sheets can be evaluated fromequation (32); it is t r ∼ . The flux available for reconnec-tion is provided with a rate directly related to the expansionof the cavity which is longer than the reconnection timescale.Indeed, in 10 years the cavity shall move with respect tothe AGN a few kpc or less; assuming a pressure dependencein distance from the AGN to be p ∝ d − , the expected ex-pansion shall be of the order of 2 × − of its current radius.Demanding that the pressure be non-negative in the cavityand repeating the calculations of section 3, we find that the c (cid:13) - RAS, MNRAS , 1–9 K. N. Gourgouliatos & M. Lyutikov magnetic field must decrease. Assuming mass conservationwe conclude that about 5 × − of the magnetic flux will bereconnected. Thus a timescale for reconnection to convert asignificant fraction of the magnetic field into heat and thusradiation is 10 years, longer by two orders of magnitudecompared to the rising time.A simple model describing the radio luminosity evo-lution is the following. Let us assume that the luminosityof the cavity is L rad = − Nm e c ˙ γ rad , where N is the to-tal number of emitting electrons in the cavity, and ˙ γ rad israte of change of the Lorentz factor, γ , because of radia-tive losses. Synchrotron emission depends on the magneticfield ˙ γ rad ∝ γ B . γ changes due to three factors: recon-nection, which injects energy from the magnetic field in theplasma through heating, adiabatic loses to the intraclustermedium through adiabatic expansion and radiative losses:˙ γ = ˙ γ rec − ˙ γ exp − ˙ γ rad . The timescale of expansion is 10 years, the timescale for radiative losses is 10 years and aswe have shown above the reconnection timescale is evenlonger. We conclude that the radio luminosity evolution willbe mainly affected by the adiabatic expansion, first through γ and second through the magnetic field B . These two com-bined give the radio luminosity profile shown in Figure (7).The radio luminosity will evolve as L rad ∝ t − , where t isthe rising timescale. The frequency where synchrotron radi-ation peaks is ω max ∝ γ B , from the evolution of cavitieswe find that ω max ∝ t − , Figure (8), making them harderto detect in radio at great distances from the AGN. Thisis consistent with the fact that the cavities that have a sig-nificant radio luminosity are the ones close to the AGN ordirectly connected to the jet, which is more evident whenthe evolution is studied in cavities of a given cluster (Dunnet al. 2005).Thus, although explosive reconnection can producelarge potentials that accelerate cosmic rays, it does not pro-vide an efficient way of converting magnetic energy into heatand consequently radio emission. The main reason is thatwhile reconnection per se is fast, the flux available to recon-nect is limited and determined by the expansion rate whichis much slower. From a different viewpoint this result can beunderstood in the following terms. If the fraction of the fluxcomprising the reconnection layer over the total flux is δ andthe volume of this area is V rec the energy available for radioemission is proportional to V − / rec δ , whereas the potentialthat accelerates cosmic rays is proportional to V − / rec R rec δ .In addition even if the volume of the reconnection layer issmall, one of its dimensions R rec can be large, i.e. of theorder of the circumference of the cavity, thus the energyavailable for acceleration is 2 × eV derived in the previ-ous section and therefore sufficient to accelerate UHECRs. In this study we have found analytical solutions for expand-ing cavities, both for fully dynamical evolution and also inthe quasi-static regime. The cavity is confined by some pres-sure that decreases with time, but remains constant aroundthe cavity. In a more realistic environment and given thatthe cavity size is comparable with the distance the pressureof the IGM drops noticeably, we expect a deviation from
Figure 7.
The radio luminosity expected by the cavity. While thecavity expands by a factor of a few, the luminosity drops almostthree orders of magnitude, since the evolution of radio luminosityis L rad ∝ t − . Figure 8.
The peak frequency for synchrotron emission. We ex-pect that it will move to much shorter frequencies as it movesaway from the AGN, since its evolution is ω ∝ t − spherical geometry. In addition, the drag force on the topsurface may cause further deformation. The parametrizationof such factors can be treated more appropriately througha numerical model and is beyond the scope of an analyti-cal model such as ours. Another issue is the stability of therising cavity. GBL have shown that static cavities are sta-ble using both analytical and numerical methods. In caseof strong drag forces on the rising cavity, we expect moresignificant perturbations. c (cid:13) - RAS, MNRAS , 1–9 agnetized cavities and UHECR acceleration While the absence of surface currents makes less vulner-able to resistive decay, for appropriate values of Γ currentsheets will form and their equilibrium will not be describedby that solution. This allows us to propose that rising mag-netized cavities are UHECR accelerators. Expansion leadsto lower density, and if the gas pressure of the cavity dropsfaster than the magnetic pressure the final outcome will bea region containing only magnetic field and confined by cur-rent sheets. It is those currents sheets that trigger resistiveinstabilities and allow the available electric potential to ac-celerate cosmic rays. Order of magnitude estimations sug-gest that this mechanism is viable.
ACKNOWLEDGEMENTS
This study is supported by NASA NNX09AH37G. The au-thors are grateful to Dimitrios Giannios for insightful com-ments and to Eric Clausen-Brown for comments on themanuscript.
REFERENCES
Alexander P., Leahy J. P., 1987, MNRAS, 225, 1Benford G., 2006, MNRAS, 369, 77Benford G., Protheroe R. J., 2008, MNRAS, 383, 663Bˆırzan L., Rafferty D. A., McNamara B. R., Wise M. W.,Nulsen P. E. J., 2004, ApJ, 607, 800Braithwaite J., 2010, MNRAS, 406, 705Churazov E., Br¨uggen M., Kaiser C. R., B¨ohringer H., For-man W., 2001, ApJ, 554, 261Dalakishvili G., Rogava A., Lapenta G., Poedts S., 2011,A&A, 526, A22+Diehl S., Li H., Fryer C. L., Rafferty D., 2008, ApJ, 687,173Dong R., Stone J. M., 2009, ApJ, 704, 1309Dunn R. J. H., Fabian A. C., Taylor G. B., 2005, MNRAS,364, 1343Dursi L. J., 2007, ApJ, 670, 221Dursi L. J., Pfrommer C., 2008, ApJ, 677, 993Fabian A. C., Sanders J. S., Ettori S., et al., 2000, MNRAS,318, L65Giannios D., 2010, MNRAS, 408, L46Gourgouliatos K. N., Braithwaite J., Lyutikov M., 2010,MNRAS, 409, 1660Gourgouliatos K. N., Lynden-Bell D., 2008, MNRAS, 391,268Gourgouliatos K. N., Vlahakis N., 2010, ArXiv e-printsGruzinov A., 2010, ArXiv e-printsHardcastle M. J., Cheung C. C., Feain I. J., Stawarz (cid:32)L.,2009, MNRAS, 393, 1041Hardcastle M. J., Worrall D. M., Kraft R. P., FormanW. R., Jones C., Murray S. S., 2003, ApJ, 593, 169Israel F. P., 1998, A&A Rev., 8, 237Kaiser C. R., Pavlovski G., Pope E. C. D., Fangohr H.,2005, MNRAS, 359, 493Kraft R. P., Forman W. R., Jones C., Murray S. S., Hard-castle M. J., Worrall D. M., 2002, ApJ, 569, 54Lazarian A., Vishniac E. T., 1999, ApJ, 517, 700Low B. C., 1982, ApJ, 254, 796Lynden-Bell D., 2003, MNRAS, 341, 1360 Lyutikov M., 2006, MNRAS, 373, 73Lyutikov M., Gourgouliatos K. N., 2010, ArXiv e-printsMcNamara B. R., Wise M., Nulsen P. E. J., et al., 2000,ApJ, 534, L135Prendergast K. H., 2005, MNRAS, 359, 725Reynolds C. S., McKernan B., Fabian A. C., Stone J. M.,Vernaleo J. C., 2005, MNRAS, 357, 242Shafranov V. D., 1966, Reviews of Plasma Physics, 2, 103Skibo J. B., Dermer C. D., Kinzer R. L., 1994, ApJ, 426,L23Sternberg A., Soker N., 2008, MNRAS, 389, L13Takahashi H. R., Asano E., Matsumoto R., 2011, ArXive-printsThe Pierre Auger Collaboration, Abraham J., Abreu P.,et al., 2007, Science, 318, 938The Pierre Auger Collaboration: J. Abraham, Abreu P.,Aglietta M., et al., 2009, ArXiv e-printsTsui K. H., Serbeto A., 2007, ApJ, 658, 794 c (cid:13) - RAS, MNRAS000