Cold Fronts from Shock Collisions
aa r X i v : . [ a s t r o - ph . C O ] J un Mon. Not. R. Astron. Soc. , 1–5 (2010) Printed 9 November 2018 (MN L A TEX style file v2.2)
Cold Fronts from Shock Collisions
Yuval Birnboim , Uri Keshet ⋆ & Lars Hernquist Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge MA, USA
Accepted —. Received —; in original —
ABSTRACT
Cold fronts (CFs) are found in most galaxy clusters, as well as in some galaxiesand groups of galaxies. We propose that some CFs are relics of collisions betweentrailing shocks. Such a collision typically results in a spherical, factor ≈ . − . Key words: galaxies: clusters: general – galaxies: haloes – X-rays: galaxies: clusters– shock waves
Recent X-ray observations of galaxy clusters reveal variousphenomena in the gaseous haloes of clusters, such as merg-ers, cavities, shocks and cold fronts (CFs). CFs are thoughtto be contact discontinuities, where the density and tem-perature jump, while the pressure remains continuous (upto projection/resolution effects). They are common in clus-ters (Markevitch & Vikhlinin 2007) and vary in morphology(they are often arcs but some radial or filamentary CFs ex-ist), contrast (the density jump is scattered around a valueof ∼ &
500 kpc) is currently unknown because of observa-tional limitations.In what follows, we propose that some CFs are producedby collisions between shocks. When two trailing shocks col-lide, a CF is always expected to form. Its parameters canbe calculated by solving the shock conditions and corre-sponding Riemann problem. Most of the scenarios in which ⋆ Einstein fellow shocks are produced (quasars, AGN jets, mergers) predictthat shocks will be created at, or near, a cluster center,and expand outwards (albeit not necessarily isotropically).When one shock trails another, it always propagates faster(supersonically with respect to the subsonic downstreamflow of the leading shock), so collisions between outgoingshocks in a cluster are inevitable if they are generated withina sufficiently short time interval.In § §
3, using 1D spherical cluster simulation that collisions be-tween the virial shock and a secondary shock are expectedto produce a distinct pattern of CFs. In § § In this section we examine two planar shocks moving inthe same direction and calculate the contact discontinuitythat forms when the second shock overtakes the leading one.This problem is fully characterized (up to normalization) bythe Mach numbers of the first and second shocks, M and M . At the instant of collision, a discontinuity in velocity,density and pressure develops, corresponding to a Riemannproblem (second case in Landau & Lifshitz 1959, § c (cid:13) Birnboim Keshet & Hernquist rated by a contact discontinuity which we shall refer to asa shock induced CF (SICF). The density is higher (and theentropy lower) on the SICF side closer to the origin of theshocks. This yields a Rayleigh-Taylor stable configurationif the shocks are expanding outwards. Now, we derive thediscontinuity parameters.
We consider an ideal gas with an adiabatic constant γ. Be-fore the shocks collide, denote the unshocked region as zone0, the region between the two shocks as zone 1, and the dou-bly shocked region as zone 2. After the collision, regions 0and 2 remain intact, but zone 1 vanishes and is replaced bytwo regions, zone 3 o (adjacent to zone 0; the outer regionfor outgoing spherical shocks) and zone 3 i (adjacent to 2; in-ner), separated by the SICF. We refer to the plasma density,pressure, velocity and speed of sound respectively as ρ , p , u and c . Velocities of shocks at the boundaries between zonesare denoted by v i , with i the upstream zone. We rescale allparameters by the unshocked parameters ρ and p .The velocity of the leading shock is v = u + M c , (1)where c i = (cid:18) γp i ρ i (cid:19) / (2)for each zone i . Without loss of generality, we measure ve-locities with respect to zone 0, implying u = 0.The state of zone 1 is related to zone 0 by the Rankine-Hugoniot conditions, p = p γM − γ + 1 γ + 1 , (3) u = u + p − p ρ ( v − u ) , (4) ρ = ρ u − v u − v . (5)The state of the doubly shocked region (zone 2) is relatedto zone 1 by reapplying equations 1- 5 with the subscripts0 , ,
2, and M replaced by M .Across the contact discontinuity that forms as theshocks collide (the SICF), the pressure and velocity are con-tinuous but the density, temperature and entropy are not;the CF contrast is defined as q ≡ ρ i /ρ o . Regions 0 and 3 o are related by the Rankine-Hugoniot jump conditions acrossthe newly formed shock, p = p ( γ + 1) ρ o − ( γ − ρ ( γ + 1) ρ − ( γ − ρ o , (6) u − u = (cid:20) ( p − p ) (cid:18) ρ − ρ o (cid:19)(cid:21) / . (7)The adiabatic rarefaction from pressure p down to p = p i = p o is determined by u − u = − c γ − " − (cid:18) p p (cid:19) ( γ − / γ . (8)The system can be solved by noting that the sum of eqs. (7)and (8) equals u − u , fixing ρ o as all other parameters areknown from eqs. (1-6). M M f M M Figure 1.
Contrast q of a CF generated by a collision betweentwo trailing shocks with Mach numbers M and M , for γ = 5 / M f normalized by M M . Finally, the Mach number of the new shock and therarefacted density are given by M f = 2 ρ o /ρ ( γ + 1) − ( γ − ρ o /ρ , (9) ρ i = ρ ( p /p ) /γ . (10)Figure 1 shows the discontinuity contrast q for vari-ous values of M and M , for γ = 5 / . Typically q ∼ , ranging from 1 .
45 for two Mach 2 shocks, for example, to q max . The maximal contrast q max depends only on the adi-abatic index; for γ = 5 / q max = 2 . M ≫ M ≃ .
65. The possible contrast rangeof SICFs is in good agreement with the CF contrasts ob-served (Markevitch & Vikhlinin 2007; Owers et al. 2009).The figure colorscale shows the dimensionless factor f de-fined by M f = fM M , ranging between 0 .
75 and 1 for1 < { M , M } < The stability of CFs limits the time duration over whichthey are detectable, and so is important when comparingCF formation models with observations. Various processescan cause a gradual breakup or smearing of the CF. Theyact in CFs regardless of their formation mechanism.Thermal conduction and diffusion of particles acrossthe CF smears the discontinuity on a timescale that is setby the thermal velocity and the mean free path of theprotons. The Spitzer m.f.p. λ in an unmagnetized plasmawith typical cluster densities is a few kpc, and depends onthe thermal conditions on both sides of the discontinuity(Markevitch & Vikhlinin 2007). Taking λ ∼
10 kpc and athermal velocity ¯ v ∼ − , and assuming that aCF is visible if it is sharper than L obs ∼
10 kpc , we get acharacteristic timescale for CF dissipation, c (cid:13) , 1–5 hock Induced Cold Fronts m Figure 2.
Fractional decline (contours) in initial CF contrast q induced by a collision with a shock of Mach number m arrivingfrom the dense CF side, for γ = 5 / . The vertical dashed linecorresponds to the maximal SICF contrast, q max . t ∼ L D ∼ L ¯ vλ ∼ yr . (11)This result indicates that in order for shock induced CFs tobe observable, either (i) they are formed frequently (e.g.,by a series of AGN bursts; see Ciotti & Ostriker 2007;Ciotti et al. 2009); or (ii) magnetic fields reduce the m.f.p.considerably, as some evidence suggests (see the discussionin Lazarian 2006).Note that the heat flux driven buoyancy instability(HBI) tends to preferentially align magnetic fields perpen-dicular to the heat flow in regions where the temperaturedecreases in the direction of gravity. This effect could re-duce radial diffusion within the inward cooling regions inthe cores of cool core clusters (Quataert 2008; Parrish et al.2009). Across a CF transition there is a sharp temperaturedrop towards the inner side, in the direction of gravity. Heatflow could induce HBI on the scale width of the CF, aligningthe magnetic fields parallel to the discontinuity and dimin-ishing radial diffusion. This possibility needs to be addressedfurther by detailed numerical magneto-hydrodynamic simu-lations.CFs could be degraded by subsequent shocks that sweepoutwards across the CF. However, the CF contrast is onlyslightly diminished by this effect, by <
4% for q < m <
2, as shown in fig. 2. Suchpassing shocks seed Richtmyer-Meshkov instabilities whichcould cause the CF to break down. Although less efficientthan Rayleigh-Taylor instabilities, these instabilities oper-ate regardless of the alignment with gravity. The outcomedepends on m and on any initial small perturbations in theCF surface. However, the instability is suppressed if the CFbecomes sufficiently smoothed.It is worth noting that KHI, which could break downCFs formed through ram pressure stripping and sloshing, does not play an important role in SICFs because there islittle or no shear velocity across them. On the other hand,the stabilizing alignment of magnetic fields caused by suchshear (Markevitch & Vikhlinin 2007) is also not expectedhere. At the edge of galaxy clusters there are virial shocks withtypical Mach numbers ∼ − , heating the gas from ∼ K to ∼ K by converting kinetic energy into ther-mal energy. The rate of expansion of a virial shock is set onaverage by the mass flux and velocity of the infalling ma-terial (see for example Bertschinger 1985). We find in oursimulations that while this expansion is steady on average,the shock sometimes oscillates, moving faster or slower thanthe steady state expansion. This creates periods dominatedby alternating halo compression and expansion. During peri-ods of enhanced compression in the outer halo regions, whenthe shock slows down, compression is sent inwards and is re-flected through the center, gradually steepening into an out-going shock. When this shock collides with the virial shock,a situation corresponding to that described in §
2, bouncingthe virial shock into another cycle of reverberation, creat-ing an SICF and sending inwards a steep rarefaction . Thisrarefaction marks the end of the compression period of thenext cycle setting the oscillation period to approximatelythe sound crossing time, in and out across the cluster. Thecombined effect is a long-lived, coherent “breathing” mode.Such “breathing” oscillations are observed in the 3D galacticand cluster halo simulations of Kereˇs & Hernquist (2009) .Using 1D spherical simulations of the formation of clus-ters from initial cosmic perturbations (Birnboim & Dekel2003), we test the formation and evolution of associatedSICFs. The simulation includes baryonic shells, dark mat-ter shells, and angular momentum support. The oscillatorymode in the simulation has not been intentionally excited,and results from stochastic inner core ( ∼ . R vir ) vibra-tions as external dark matter shells with low angular mo-mentum interact gravitationally with the core. Physically,any waves, shocks or gravitational perturbations (mergers orAGNs) will contribute to the mode, as well as non-smoothaccretion rates that would cause the virial shock to vibrate.A more detailed examination of this is beyond the scope ofthis letter (note, however, that the strength of the SICF de-pends only weakly on the parameters of the secondary shockand is always close to 2 . )Fig. 3 shows the evolution of a 3 × M ⊙ halo un-til redshift z = 0 (an adiabatic version of those describedin Birnboim & Dekel 2009, in prep. ). The expanding virialshock is traced by a jump in the velocity of the Lagrangianshells (red, thin lines), and by the shock-finding algorithmthat is based on entropy increase along Lagrangian shells(blue region). The outgoing, secondary shocks that reach thevirial shock periodically are also clearly visible. The virialshock bounces outwards each time it is hit by a secondary. A A movie of the evolution of radial profiles in time is availableat Duˇsan Kereˇs, (private communication).c (cid:13) , 1–5
Birnboim Keshet & Hernquist -12 -10 -8 -6 -4 -2 08 5 3 2 1.4 1 .5 .2 R ad i u s [ kp c ] Time [ Gyr ] z Figure 3.
Evolution of a galaxy cluster from cosmological ini-tial perturbations. The final mass of the cluster at z = 0 is3 × M ⊙ . The radius and time of Lagrangian shells (every25 th shell) are plotted in red thin lines. Shocks are traced bylarge Lagrangian derivatives of the entropy ( d ln S/dt > . − ,blue dots), and CFs are traced by their large entropy gradients( ∂ ln S/∂ ln r > .
5, green dots). Rarefaction waves can be seen assmall motions of the Lagrangian shells (illustrated by the dashedmagenta curve, manually added based on a time series analysis).Their trajectory is approximately the reflection of the precedingoutgoing compression, time inverted about the last secondary-virial shock collision. rarefaction wave is reflected inwards (illustrated by a dashedcurve), and a spherical SICF is left behind. The SICFs arevisible through the entropy gradients (green region), and, inthe absence of diffusion and instabilities, persist until z = 0 . As expected, the interaction of the discontinuity with sub-sequent shocks ( § z = 0 . Aseries of SICFs (marked with blue vertical lines) is visible,seen as entropy jumps with continuous pressure. The den-sity jumps by roughly a factor of q ∼
2, as expected, withthe temperature dropping accordingly.The SICFs formed in this simulation are almost static,and roughly logarithmically spaced. The calculations pre-sented in Figures 3-4 are adiabatic. When cooling is turnedon in the absence of feedback, this simulation suffers fromovercooling, creating an overmassive BCG of 2 × M ⊙ andstar formation rates of 100 M ⊙ yr − , and the luminosity ex-ceeds the L x − T relation. The excitation of the basic mode,and periodic CFs, are observed in all these simulations. Thesimulation presented here was performed with 2 ,
000 bary-onic and 10 ,
000 dark matter shells. When the resolution islowered to 250 baryonic shells, the reverberation amplitudeand periods are essentially unchanged, indicating that theresults are well converged. l og S [ k e V × c m ] -13-12-11-10 l og P [ e r g × c m − ] Radius [ kpc ] -6-5-4-3-2 l og n H [ c m − ] T [ k e V ] Figure 4.
Thermodynamic profile of the simulated cluster shownin Fig. (3) at z = 0 . Top: entropy (red, solid, left axis) and pres-sure (green, dashed, right axis).
Bottom: density (red, solid, leftaxis) and temperature (green, dashed, right axis). CFs are markedwith blue dashed vertical lines.
Owers et al. (2009) present high quality Chandra observa-tions of 3 relaxed CFs. They all seem concentric with re-spect to the cluster center, and spherical in appearance.Owers et al. (2009) interpret these as evidence for sloshing,and find spiral characteristics in 2 of them. We argue herethat some CFs of this type could originate from shock col-lisions. In the SICF model, CFs are spherical, unlike thetruncated CFs formed by ram pressure stripping or slosh-ing, so no special projection orientation is required. On theother hand, SICFs do not involve metallicity discontinuities,observed in some of these CFs. The observed contrasts ofthese CFs are quite uniform, with best fit values in the range2 . − . ∼
20% uncertainty) in all three, as expectedfor SICFs .The SICF formation model is entirely distinct from rampressure stripping and sloshing CFs, and it makes many dif-ferent predictions. Morphology.
SICFs are quasi-spherical around thesource of the shocks. A galactic merger defines an orbitplane, and the corresponding perturbation (stripped mate-rial or center displacement) will create CFs parallel to thisplane. In some sloshing scenarios (Ascasibar & Markevitch2006) the CFs extend considerably above the plane, butwould never appear as a full circle on the sky; an observedCF “ring” would strongly point towards an SICF. The sta-tistical properties of a large CF sample could thus be usedto distinguish between the different scenarios.
Amplitude.
SICFs have distinct entropy and densitycontrasts that depend weakly on shock parameters; q is typ- Note that collisions with outgoing shocks could diminish thecontrast towards q ∼
2, regardless of CF origin. A larger, morecomplete sample of CFs is needed to determine the origin/s ofthe relaxed population. c (cid:13) , 1–5 hock Induced Cold Fronts ically larger than ∼ . M >
2) and is alwayssmaller than q max = 2 . Extent.
If the cluster reverberation mode is excited,SICF radii should be approximately logarithmically spaced.Any shock that expands and collides with the virial shockwill create SICF at the location of the virial shock, far be-yond the core. The external SICFs are younger, and wouldappear sharper owing to less diffusion and collisions withsecondary shocks. Deep observations capable of detectingCFs at r ∼ Plasma diagnostics.
Shocks are known to modifyplasma properties in a non-linear manner, for exampleby accelerating particles to high energies and amplify-ing/generating magnetic fields. The plasmas on each sideof an SICF may thus differ, being processed either by twoshocks or by one, stronger shock. This may allow indirect de-tection of the CF, in particular if the two shocks were strongbefore the collision. For example, enhanced magnetic fieldsbelow the CF may be observable as excess synchrotron emis-sion from radio relics that extend across the CF, in nearbyclusters, using future high-resolution radio telescopes (LO-FAR, SKA).
Our study consists of two parts. The first is a general discus-sion about CFs that form as a result of a collision betweentwo trailing shocks. The second part describes a specific re-verberation mode of galaxy and cluster haloes that has beenidentified in 1D simulations, and also seen in 3D. One as-pect of these reverberations is periodic collisions betweenthe virial shock and outgoing secondary shocks, resulting inSICFs – a special case of the mechanism discussed in thefirst part of our paper.We have shown that when shocks move in the samedirection they collide and generate a CF. The density con-trast across the CF is calculated as a function of the Machnumbers of the two shocks. It is typically larger than 1 . M & q max = 2 .
65. The dis-continuity is smeared over time by diffusion, at a rate thatdepends on the unknown nature and amplitude of magneticfields. CFs are susceptible to heat-flux-driven buoyancy in-stability (HBI), which could align the magnetic field tangentto the CF and potentially moderate further diffusion. SICFs,like all other CFs, are subject to Richtmyer-Meshkov insta-bilities from subsequent shocks passing through the cluster.Such collisions reduce the CF contrast until it reaches q ∼ §
3, using a 1D spherical hydrodynamic code toevolve cosmological perturbations, we demonstrate that areverberation mode exists in the haloes of galaxies and clus-ters and causes periodic collisions between the virial shocksand secondary shocks that produce SICF. The simulatedSICF contrast is consistent with the theoretical predictionsof §
2. A more thorough investigation of this potentilly im-portant mode, including its stability in 3D is left for futureworks. We use it here as a specific scenario for SICF forma-tion.The predicted properties of SICFs are presented in §
4, and reproduce some of the CF features discussed inOwers et al. (2009). In particular, we suggest that CFs inrelaxed clusters, with no evidence of mergers, shear, or chem-ical discontinuities, may have formed by shock collisions. Welist the properties of SICFs that could distinguish them fromCFs formed by other mechanisms. The SICF model predictsquasi-spherical CFs which are concentric about the clustercenter, with contrast q ∼ , and possibly extending as farout as the virial shock. An observed closed (circular/oval)CF could only be an SICF. In the specific case of clusterreverberation, a distinct spacing pattern between CFs is ex-pected. It may be possible to detect them indirectly, for ex-ample as discontinuities superimposed on peripheral radioemission.Shocks originating from the cluster center naturally oc-cur in feedback models that are invoked to solve the over-cooling problem. They are also formed by mergers of sub-structures with the BCG. Thus, SICFs should be a naturalphenomenon in clusters. Further work is needed to assesshow common SICFs are with respect to other types of CFs,and to characterize inner SICFs that could result, for exam-ple, from collisions between offset AGN shocks. The proper-ties of reverberation in 3D will be pursued in future work. ACKNOWLEDGEMENTS
We thank M. Markevitch for useful discussions. YB ac-knowledges the support of an ITC fellowship from the Har-vard College Observatory. UK acknowledges support byNASA through Einstein Postdoctoral Fellowship grant num-ber PF8-90059 awarded by the Chandra X-ray Center, whichis operated by the Smithsonian Astrophysical Observatoryfor NASA under contract NAS8-03060.
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