DYNAMICS AND MAGNETIZATION IN GALAXY CLUSTER CORES TRACED BY X-RAY COLD FRONTS

In the past decade, high-resolution X-ray observations have revealed an abundance of density and temperature discontinuities known as cold fronts (CFs). They are broadly classified as merger CFs (not discussed here) and cool core CFs; for a review, see Markevitch & Vikhlinin (2007).

Core CFs are observed in more than half of the otherwise relaxed, cool core clusters (CCs; Markevitch et al. 2003), in distances up to ∼400kpc from the center. They are usually nearly concentric or spiral, and multiple CFs are often observed in the same cluster. The temperature contrast across such a CF is q ≡ T_o/T_i ∼ 2 (Owers et al. 2009), where inside/outside subscripts i/o refer to regions closer to/farther from the cluster center, or equivalently below/above the CF. The plasma beneath the CF is typically denser, colder, lower in entropy, and higher in metallicity than the plasma above it. Usually, little or no pressure discontinuity is identified across these CFs, suggesting negligible CF motion.

Such CFs were modeled as associated with large-scale "sloshing" oscillations of the intracluster medium (ICM) driven, for example, by mergers (Markevitch et al. 2001, henceforth M01), possibly involving only a dark matter subhalo (Tittley & Henriksen 2005; Ascasibar & Markevitch 2006), or by weak shocks/acoustic waves displacing cold central plasma (Churazov et al. 2003; Fujita et al. 2004).

We argue that a tangential shear flow across and beneath CFs is ubiquitous in CCs. One line of argument relies on thermodynamic profiles across CFs found in the literature. We show that all these profiles reveal a distinct deviation from hydrostatic equilibrium and argue that this can be naturally interpreted only as shear. An independent indication for the presence of shear across CC CFs is their remarkable sharpness and stability, which implies strong magnetic fields along the CF plane. In the absence of radial CF motion, the most natural explanation for the persistence of such fields is shear across the CF.

In Section 2, we show that deviations from hydrostatic equilibrium, previously reported in two CC CFs, are common among such CFs and argue in Section 3 that this directly gauges bulk tangential flows and shear along and beneath these CFs. In Section 4, we show that such shear can magnetize the core and produce near-equipartition fields along CFs, thus stabilizing them. We summarize and discuss the implications of our model in Section 5. We assume a Hubble constant H = 70kms⁻¹Mpc⁻¹. Error bars are 1σ confidence levels.

The mass density ρ(r) and pressure P(r) radial profiles near CFs in RX J1720+26 and A1795 were reported to be inconsistent with hydrostatic equilibrium (M01; Mazzotta et al. 2001). Indeed, assuming such equilibrium, a_r = −ρ⁻¹∂_rP − GM_g/r² = 0, where a_r is the radial acceleration and G is Newton's constant, would imply an unphysical gravitating mass profile M_g(r) that becomes abruptly larger just outside the CF. In RX J1720, M_g derived from the above assumption jumps by a factor of 5 (at a 2.5σ confidence level), and there is marginal evidence for a pressure discontinuity which translates into radial CF motion with Mach number ${\mathcal {M}}=0.4_{-0.4}^{+0.7}$ (Mazzotta et al. 2001). In A1795, M_g jumps by a factor of 2 (2.5σ) and the pressure appears continuous across the CF (Figure 1, inset; see M01).

Zoom In Zoom Out Reset image size

To investigate the prevalence of this phenomenon, we calculate the pressure acceleration discontinuity

$$\begin{eqnarray} &&\Delta a_r \equiv a_{r,i}-a_{r,o} = \Delta \left(-\frac{\partial _r P}{\rho }\right)=-\frac{k_B}{\mu r}\Delta ({\lambda }T) \end{eqnarray} \tag{ 1 }$$

across various CFs for which deprojected thermodynamic profiles have appeared in the literature. Here, we defined ΔA ≡ A_i − A_o (for any quantity A) and λ ≡ ∂ln P/∂ln r (typically λ < 0), with k_B being the Boltzmann constant, μ ≃ 0.6 m_p being the average particle mass (assumed constant), and m_p being the proton mass. The gravitating mass M_g(r) is assumed to be continuous everywhere.

We extract all the thermodynamic profiles across CFs in cluster cores from the literature. We select only profiles that have been deprojected along the line of sight, where at least two radial bins exist on each side of the CF such that ∂_rP can be estimated. When possible, we omit radial bins that are very far (>4 times in radius) from the CF, to avoid excessive contamination. We end up with six CF profiles in which λT can be calculated based on 2–3 radial bins on each side of the front, supplemented by three CFs for which only the density has been deprojected. We present the (dimensionless) normalized acceleration discontinuity δ ≡ (c²_i/r)⁻¹Δa_r in Figure 1, along with the literature references. Here, c = (ΓP/ρ)^1/2 is the sound velocity with Γ = 5/3 being the adiabatic index.

All the Δa_r values we find are negative and inconsistent with zero at confidence levels ranging between 1.1σ and 5.4σ. The CF data points shown in Figure 1 are consistent with a best fit δ = −0.78 ± 0.14, i.e., nonzero at 5.5σ (the average is not necessarily physically meaningful). As a sanity check, we apply the exact same procedure to parts of the same deprojected thermodynamic profiles where no CF is present. This is possible in a few cases, where the profiles extend sufficiently far from the CF, shown in Figure 1 as empty cyan symbols. These continuum δ values are consistent with zero (1σ) or are slightly positive (within 2.5σ from zero; a resolution effect explained below). Overall, they are best fit by δ = +0.07 ± 0.08. This indicates that the a_r discontinuity at CFs is real.

Furthermore, the effect is qualitative and robust. As T jumps by a factor of q ≃ 2 as one crosses outside the CF, hydrostatic equilibrium would require a slower radial decline of pressure outside the CF, λ_i/λ_o ≃ q. However, the pressure profile typically steepens outside the CF. Hence, two phenomena contribute to the Δa_r < 0 discontinuity: the density drop and the pressure steepening. Neither of these effects can be inverted by spherical deprojection. The differentiation procedure used to estimate λ removes long wavelength errors, so the significance of δ is probably better than that estimated above. Finally, note that resolution effects introduce an opposite bias, toward positive Δa_r, as (−ρ⁻¹∂_rP) typically declines with increasing r. This accounts for the slight tendency of continuum regions toward Δa_r > 0.

We conclude that deviations from hydrostatic equilibrium are common in CC CFs, with less pressure support below the CF, Δa_r < 0, such that if the plasma above the CF is in equilibrium, the plasma below should be falling. This phenomenon, previously reported in RX J1720 and in A1795, has been interpreted as evidence that the core is sloshing, with plasma near the CF radially accelerating in some phase of the oscillation cycle. For the apparently stationary CF in A1795, cool plasma would then be sloshing at its maximal radial displacement, where it has zero radial velocity but nonzero radial acceleration (M01).

We note, however, that in this picture, without additional effects, the discontinuity in radial acceleration Δa_r ≡ a_r,i − a_r,o < 0 would cause an unphysical gap to open along the CF. A different interpretation of the observations is therefore required. Below we argue that the most natural interpretation of the data involves a noncontinuous (within the observational resolution) centripetal acceleration of flow along the curved CF.

Consider an infinitesimal, two-dimensional piece ${\mathcal {F}}_2$ of the CF plane, and an inertial "CF frame" S' instantaneously comoving with ${\mathcal {F}}_2$ perpendicular to the CF plane, as illustrated in Figure 2. Let ${\mathcal {F}}_1$ be an arc in ${\mathcal {F}}_2$ parallel to the local flow beneath the CF. We may approximate ${\mathcal {F}}_1$ as a segment of a large circle with radius R. Consider the spherical coordinate system S' = (r', θ, ϕ) with ${\mathcal {F}}_1$ lying at radius r' = R and colatitude θ = π/2. Its origin, the center of the circle, is typically close but not necessarily coincident with the cluster center.

Zoom In Zoom Out Reset image size

The r' component of the velocity u measured in S' must vanish at ${\mathcal {F}}_2$, $u_{{r^{\prime }}}=0$, as $\mathbf {\hat{r}}^{\prime }$ is perpendicular to the discontinuity. The r' component of the acceleration is continuous across ${\mathcal {F}}_2$, $\Delta (\partial _t u_{{r^{\prime }}})=0$. Hence, taking the difference in the r' component of the Euler (momentum) equation between the two sides of ${\mathcal {F}}_2$ yields

$$\begin{eqnarray} &&\Delta (u^2)/{r^{\prime }}= \Delta (\rho ^{-1}\partial _{{r^{\prime }}} P) \mbox{,} \end{eqnarray} \tag{ 2 }$$

where the pressure P = P_th + P_nt may include thermal and nonthermal components, u² ≡ u · u = u²_ϕ + u²_θ, and we assumed slow changes in the CF pattern, $\mathbf {u}\;{\bm \cdot}\; {\bnabla }{u_{r^{\prime }}}\ll u^2/{r^{\prime }}$. Note that although u is confined to the CF plane, it does not have to be parallel on the two sides of ${\mathcal {F}}_2$.

Equation (2) indicates that the observed discontinuity $\Delta (\rho ^{-1}\partial _{r^{\prime }}{P_{{\rm th}}})>0$ must be balanced by a discontinuously larger centripetal acceleration u²/r' beneath the CF. The discontinuity cannot be plausibly attributed to P_nt, because the continuous P_th, noncontinuous ${\bnabla }{{P_{{\rm th}}}}$ combination observed would then require predominantly nonthermal pressure at large, sometimes r > 50kpc radii, which is unlikely (e.g., Churazov et al. 2008). Henceforth we neglect P_nt.

We conclude that shear flow across CFs is the most natural interpretation of the observations. Thus

$$\begin{eqnarray} &&u_{i}^2-u_{o}^2 = \Delta (u^2) \simeq (k_B/\mu)\Delta ({\lambda }^{\prime } T) \simeq -r \Delta a_r > 0 \mbox{,} \end{eqnarray} \tag{ 3 }$$

where ${\lambda }^{\prime }\equiv \partial \ln P/\partial \ln {r^{\prime }}\simeq ({r^{\prime }}/r)({\hat{\bm r}}\; {\bm \cdot }\; {\hat{\bm r}}^{\prime }){\lambda }\simeq {\lambda }$. Hence, CFs are tangential discontinuities, with shear and faster tangential flow beneath the CF, u_i > u_o. In particular, Δu ⩽ Δu₂ ⩽ u_i, where Δu₂ ≡ [Δ(u²)]^1/2 is measurable.

In terms of the local sound velocity, Equation (3) becomes

$$\begin{eqnarray} &&\Gamma \Delta (u^2) = (q^{-1}{\lambda }^{\prime }_{{i}}-{\lambda }^{\prime }_{{o}})c_{{o}}^2 = ({\lambda }^{\prime }_{{i}}-{\lambda }_{{o}}^{\prime }q)c_{{i}}^2 \mbox{.} \end{eqnarray} \tag{ 4 }$$

Typically the P slopes are λ_i ≳ −0.7, steepening to λ_o ≲ −1 above the CF, so a subsonic flow in the CF frame (u_i < c_i) requires q ≲ 2.4. Indeed, q ≲ 2.5 is observed in CC CFs (Owers et al. 2009). Assuming the subsonic flow and continuous pressure, Equation (4) also imposes an upper limit on the velocity above the CF, ${\mathcal {M}}_{{o}}^2 \equiv (u_{o}/c_{o})^{2}<{q}^{-1}+{\lambda }^{\prime }_{{o}}/\Gamma - {\lambda }^{\prime }_{{i}}/(\Gamma {q})$.

As an example, consider the CF in A1795. The M_g discontinuity was previously interpreted as reflecting radial acceleration, a ∼ 2 × 10⁻⁸cms⁻². Interpreted as a tangential flow, this corresponds to Δu₂ ≃ (ar)^1/2 ∼ 750kms⁻¹, uncertain to within a factor of ∼2. This velocity difference constitutes a considerable fraction of the sound velocity, Δu₂ ≃ 0.67 c_i ≃ 0.55 c_o. It places a lower limit ${\mathcal {M}}_{{i}}>0.67$ on the (tangential) Mach number beneath the CF and an upper limit Δu < 0.67 c_i on the shear across it. Assuming ${\mathcal {M}}_{{i}}<1$, it also imposes an upper limit ${\mathcal {M}}_{{o}}<0.61$ on the flow above the CF.

Note that the tangential flow inferred from Δa_r ≠ 0 may coexist with perpendicular (radial) motion of the CF through the ICM, inferred from an apparent pressure discontinuity, for example, in RX J1720+26 (Mazzotta et al. 2001).

The M_g(r) profile near a CF can be compared to other, equidistant regions in the core, preferably on the opposite side of the cluster, that do not harbor discontinuities. In both cases examined, A1795 (Figure 1, inset) and RX J1720+26, M_g(r > r₀) is similar on both sides of the cluster above the CF radius r₀, but inertia appears to be missing (M_g is too small) in most of the volume beneath the CF. In the tangential flow picture, the missing inertia region beneath the CF corresponds to an extended tangential bulk flow. Consider again the CF lying r₀ ≃ 85kpc south of the center of A1795. Here, inertia appears to be missing in a considerable, 0.7 ≲ r/r₀ < 1 region, involving ∼1/2 of the gas in that sector. There is no evidence for inertia discrepancy below ∼0.6 r₀. Hence, plasma within the discrepant radii range cannot be uniformly flowing; shear must be present.

The existence and extent of the shear flow region are also gauged by nearly radial optical (and sometimes X-ray) filaments that lie below some CFs. In A1795, a pair of filaments extends out to ∼0.6 r₀ (McDonald & Veilleux 2009, and references therein), just beneath the shear region. In M87, the southwest filaments (Forman et al. 2007) extend out to ∼0.6 r₀. They are nearly radial out to ∼0.4 r₀, but have some curvature at 0.4r₀ ≲ r ≲ 0.6 r₀ which may be driven by shear. The (sometimes curved) termination of filaments just below the shear region suggests that they may have been eroded by the shear.

An independent argument implying shear across CFs stems from their remarkable sharpness and stability. Stationary CFs are stable against Rayleigh–Taylor instabilities, as the inside plasma has lower entropy. However, particle diffusion, heat conduction, and instabilities such as Kelvin–Helmholtz (KHI) and Richtmyer–Meshkov, could potentially broaden and deteriorate the CF on short, subdynamical time scales. In contrast, observations indicate that CFs are not only stable over cosmological time scales, but in fact the transition in thermodynamical properties subtends a small fraction of the Coulomb mean free path of protons (Vikhlinin et al. 2001a; Markevitch & Vikhlinin 2007).

This suggests that transport across CFs is magnetically suppressed. Such suppression is expected in CFs that move through the ICM, due to draping of magnetic fields and possibly the formation of a magnetic barrier (plasma depletion layer), as found in observations of planetary magnetospheres and in simulations (Lyutikov 2006, and references therein). However, the processes that suppress transport in the more common, stationary type of CFs were thus far unknown. Here, we argue that shear magnetic amplification near CFs naturally sustains the strong magnetic fields parallel to the CF needed to suppress radial transport.

Consider first the limit where CFs are infinitely sharp tangential discontinuities. The magnetic field perpendicular to such a CF must vanish, and there is no shear magnetic amplification. However, in this limit the stability against KHI inferred from observations requires a strong magnetic field B, as shown by Vikhlinin et al. (2001b) for the merger CF in A3667. Namely, the shear velocity difference must be smaller than an effective Alfvén velocity (Landau & Lifshitz 1960, Section 53),

$$\begin{eqnarray} &&\left(\frac{\Delta u}{2} \right)^2 < \frac{\bar{B}^2}{4\pi \bar{\rho }} \simeq \frac{3(1+{q})}{5\bar{\beta }_i} c_i^2 \mbox{,} \end{eqnarray} \tag{ 5 }$$

and in addition

$$\begin{eqnarray} &&(\mathbf {B}_{{i}} \times \mathbf {B}_{{o}})^2 \ge 2\pi \bar{\rho }[ (\mathbf {B}_{{i}} \times \Delta {\mathbf {u}})^2 +(\mathbf {B}_{{o}} \times \Delta {\mathbf {u}})^2 ] \mbox{,} \end{eqnarray} \tag{ 6 }$$

where $\bar{B}^2\equiv (B_{{i}}^2+B_{{o}}^2)/2$, $\bar{\rho } \equiv 2\rho _{{i}}\rho _{{o}}/(\rho _{{i}}+\rho _{{o}})$, and $\bar{\beta }\equiv {P_{{\rm th}}}/P_{\bar{B}}=8\pi nk_B T/\bar{B}^2$, with n being the particle number density.

Equation (5) imposes a lower limit on $\bar{B}$,

$$\begin{eqnarray} \bar{B} > B_{\rm min} & \equiv & \sqrt{\frac{2\pi \rho _i}{1+{q}}} |\Delta u| = \sqrt{\frac{5}{12(1+q)}} \frac{|\Delta u|}{c_i} B_{{\rm eq},{i}} \nonumber \\ & \simeq &17 {\mathcal {M}}_i \sqrt{ \frac{3n_i k_B T_i/(1+q)}{50\mbox{ eV}\mbox{ cm}^{-3}} } \mbox{ $\mu $G}\mbox{.} \end{eqnarray} \tag{ 7 }$$

The large shear velocities derived above thus imply large $\bar{B}$, up to ∼40% of its equipartition value B_eq for which β = 1. An asymmetry B_i/B_o ≠ 1 would require B well aligned with Δu. In the case B_i = B_o = B, α_i ≠ α_o,

$$\begin{eqnarray} &&B > \sqrt{\frac{2(\sin ^2\alpha _{{i}}+\sin ^2\alpha _{{o}})}{\sin ^2(\alpha _{{i}}-\alpha _{{o}})}} B_{\rm min} \mbox{,} \end{eqnarray} \tag{ 8 }$$

with α being the angle between B and Δu, so stronger fields are needed if misaligned with the shear.

A magnetic, plasma-depleted barrier may form. As long as its temperature does not greatly exceed T_o, it remains strongly magnetized. Note that the corresponding density drop could possibly be observed more easily here than in merger-type CFs, as there is no confusion with a stagnation region. As shown below, equipartition fields are expected only very close to the CF, so P_nt can still be neglected at ≳kpc distances, as assumed in Section 3.

How did strong magnetic fields form along stationary CFs, and how are they sustained over cosmological time scales? Perpendicular transport suppression could preserve such fields practically indefinitely in the absence of large perturbations (e.g., mergers) and disruptive kinetic plasma effects. Hence, they could be remnants of an early stage where the CF was formed. But if such initial fields are absent or decay, shear-driven magnetic amplification can naturally generate or replenish B_ϕ, by stretching $B_{r^{\prime }}\ne 0$ magnetic structures advected by the flow via line freezing, d(B/ρ)/dt ≃ [(B/ρ) · ∇]u.

Consider a weakly magnetized, B ≪ B_eq stage where the kinematic viscosity ν is a fraction f_ν of its Spitzer value. The velocity discontinuity across such a CF is gradually smoothed out, forming a transition layer of thickness l ∼ (νt)^1/2 ≃ 7f^1/2_νT^5/4₄n^−1/2₋₂(t/Gyr)^1/2kpc beneath the CF at time t, where T₄ ≡ (T_i/4keV) and n₋₂ ≡ (n_i/10⁻²cm⁻³). This enables rapid shear magnetic amplification, on an e-fold time scale τ ∼ l/u. Equating the viscous and amplification times suggests that a magnetization layer of thickness

$$\begin{eqnarray} &&{l}\sim N\nu /u \simeq 50 N f_\nu {\mathcal {M}}_i^{-1} T_4^2 n_{-2}^{-1} \mbox{ pc} \end{eqnarray} \tag{ 9 }$$

develops over a very short time scale

$$\begin{eqnarray} &&t \sim N^2\nu /u^2 \simeq 4\times 10^4 N^2 f_\nu {\mathcal {M}}_i^{-2}T_4^{3/2}n_{-2}^{-1}\mbox{ yr}\mbox{,} \end{eqnarray} \tag{ 10 }$$

with N being the growth factor. Typically Nf_ν < 1; N ∼ 1 may suffice in the case of a weakening magnetic barrier.

The high B estimates above hold only in the vicinity of CFs, where fast shear is measured. As mentioned in Section 3, evidence suggests that the associated shear involves a fair fraction of the plasma beneath the CF. Shear amplification may thus effectively magnetize the entire core, at least beneath the outermost CF. For a simple model where the shear is constant in a layer of thickness Δ beneath the CF, the energy of the frozen magnetic field is amplified in proportion to $(B_\phi /B_{r^{\prime }})^2$, where

$$\begin{eqnarray} &&\frac{B_{\phi }}{B_{r^{\prime }}} \sim t \partial _{r^{\prime }}u \sim 10 {\mathcal {M}}_i T_4^{1/2} \left(\frac{\Delta }{10\mbox{ kpc}}\right)^{-1} \left(\frac{t}{10^8\mbox{ yr}}\right) \mbox{.}\,\,\, \end{eqnarray} \tag{ 11 }$$

Saturation of this process depends on the flow details. For example, in a spiral flow there is a characteristic cutoff time when magnetic structures leave the shear region.

We have seen that CFs are strongly magnetized and that the associated shear flow can magnetize much of the plasma beneath them. Magnetic processes should therefore be enhanced below, and in particular near CFs. This naturally explains the recently discovered coincidence between CFs and the edges of radio minihalos (MH; Mazzotta & Giacintucci 2008; see Keshet & Loeb 2010).

We interpret deviations from hydrostatic equilibrium previously observed near CFs as evidence for centripetal acceleration. This reveals a tangential shear flow, nearly sonic at the CFs and extending well beneath them. Such behavior is shown to be common among CC CFs, as illustrated in Figure 1. The measured discontinuity Δ(u²), given by Equations (3) and (4), constrains the shear and inside flow, Δu ⩽ Δu₂ ⩽ u_i, and imposes an upper limit on ${\mathcal {M}}_{o}$. The apparent CF stability against KHI suggests near-equipartition magnetic fields along the CF (Equations (7) and (8)); this could manifest in a shear-amplified, ≲50pc magnetic layer. Shear magnetic amplification could rapidly magnetize the entire core up to the CFs and explain some of the observed properties of radio MHs.

In our model, core CFs are part of extended tangential discontinuities seen in projection. They directly trace both the orientation and approximate magnitude of bulk flows in the ICM. CFs are usually nearly concentric, so B should be nearly tangential and the polarization radial. Note that this magnetic configuration differs from the predictions of magnetic compression models, where B is typically isotropic (e.g., Gitti et al. 2002) or radial (Soker & Sarazin 1990), but is similar to the saturation of heat flux-driven buoyancy instabilities in the core (Quataert 2008). In MHs, we predict detailed morphological correlations with X-ray gradients, which trace projected CFs, and radio polarization parallel to these gradients.

Such fast, extended flows would modify the energy budget of the core and could alter its structure, for example, by distributing heat and relaxing the cooling problem. Churazov et al. (2003) pointed out that the absence of resonant features in the X-ray spectrum in Perseus indicates motions with at least ${\mathcal {M}}=1/2$ in the core. Although initially interpreted as turbulent motion, this may reflect the bulk motions discussed here. The presence of such flows could be tested, for example, using high-resolution spectroscopy and future X-ray polarization measurements (e.g., Sazonov et al. 2002).

We thank W. Forman and O. Cohen for useful discussions. U.K. acknowledges support from NASA through Einstein Postdoctoral Fellowship grant number PF8-90059. This work was supported in part by NASA contract NAS8-39073 (M.M.), by an ITC fellowship from the Harvard College Observatory (Y.B.), and by NSF grant AST-0907890 (A.L.).