Gas Sloshing in Abell 2204: Constraining the Properties of the Magnetized Intracluster Medium

Although A2204 has a regular morphology on large scales, the cluster is highly disturbed in the core (Figure 3), which looks like a tapered water drop with a "tail" to the east. A spiral pattern is prominent in the core, which starts from the southeast at a radius of $10^{\prime\prime}$, then wraps outward clockwise to southwest, reaching a radius of $25^{\prime\prime}$, and grows fuzzy in the south. Careful examination reveals a gap in the southwest, and a small "bite" in the northwest.

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Two distinct surface brightness edges, which delineate the spiral at a radius of $25^{\prime\prime}$ in the west and a radius of $14^{\prime\prime}$ in the east, are readily seen (Figure 3). The western edge is very sharp at position angles between 320° and 370°, but blurs further northward and southward. The eastern edge is prominent in the direction between 140° and 180°, but loses its clear trace southward as it merges into the "rim," i.e., the innermost core.

We produced a residual image (right panel in Figure 3), using Proffit, a custom surface brightness fitting software (Eckert et al. 2011). To do so, we generated the radial surface brightness profile centered on the X-ray peak ([R.A., decl.] = [16:32:47.0, +5:34:31]), fitted the profile with a one-dimensional β-model (Cavaliere & Fusco-Femiano 1976) and subtracted the model image. The residual image shows a bright core corresponding to the rim, with a small depression immediately northwest of it. There is an excess at the position of the western edge, and a depression immediately outside the eastern edge (between 14'' and 25''), which is understandable because the western edge increases the overall model surface brightness at those positions. There is also an excess corresponding to the tail.

The morphology and location of the two edges strongly suggest that they are the result of gas sloshing due to a recent gravitational perturbation of the ICM. In the Appendix, we discuss the perturber candidates. Previous work (Sanders et al. 2005) has shown that the two edges are CFs. Here, using the deeper exposure, we perform an in-depth study of their physical properties. For each of the two edges, we extract spectra from six concentric sectors (Figure 3(b)) and fit them with the wabs*apec model to obtain the projected temperature profiles across the CFs. We can see that the temperature jump at the western CF is as large as by a factor of ∼2 (Figure 4(a)). The temperature jump at the eastern CF is not as abrupt as at the western CF (Figure 5(a)). We also used dsdeproj (Sanders & Fabian 2007; Russell et al. 2008), which assumes spherical symmetry, to obtain the deprojected temperature profile (red points in Figures 4 and 5). This method subtracts photons emitted from the outer region, therefore, leading to a lower temperature in the inner regions, with larger error bars. It is noteworthy that since the cluster core is sloshing predominantly in a plane, the spherical symmetry assumed by this method is not exact and is a source of systematic error. Nevertheless, the deprojected temperature profiles at both edges support the conclusion that they are CFs.

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Next, we characterize the gas density profile, for which we adopt the following form,

$$\begin{eqnarray}{n}_{e}(r)={n}_{e0}\left\{\begin{array}{ll}J{\left[1+{\left(\tfrac{{r}_{j}}{{r}_{c}}\right)}^{2}\right]}^{-3\beta /2}{\left(\tfrac{r}{{r}_{j}}\right)}^{-\alpha }, & r\lt {r}_{j}\\ {\left[1+{\left(\tfrac{r}{{r}_{c}}\right)}^{2}\right]}^{-3\beta /2}, & r\,\geqslant \,{r}_{j}\end{array}\right.\end{eqnarray} \tag{ 1 }$$

to account for the density jump (J) across the edge (at the break radius r_j). Here, the density profile is described by a power law with slope α and a β-model inside and outside r_j, respectively.

The above parameters, except n_e0, are constrained by fitting the surface brightness profile across the two CFs (Table 2). The western CF is well fitted by this model (${\chi }^{2}$/d.o.f. = 109.4/110), and the very small error ($\sim 0\buildrel{\prime\prime}\over{.} 1$) on r_j indicates that the position of the edge is well-determined. To test this, we artificially increase r_j by a value of 05 (i.e., the ACIS pixel size) greater than the best-fit value (248), and redo the fit, fixing r_j. This results in ${\chi }^{2}$/d.o.f. = 139.6/111.

Table 2.  Fitted Parameters of the Surface Brightness Profiles across the Edges

Region	Azimuthal Range	α	rc (arcmin)	β	rj (arcmin)	J	${\chi }^{2}$ (d.o.f)
West	320°–10°	${0.953}_{-0.021}^{+0.021}$	${0.506}_{-0.023}^{+0.023}$	${0.561}_{-0.006}^{+0.006}$	${0.414}_{-0.002}^{+0.002}$	${2.05}_{-0.05}^{+0.05}$	109.391 (110)
East	120°–180°	${1.013}_{-0.055}^{+0.055}$	${0.393}_{-0.012}^{+0.013}$	${0.570}_{-0.006}^{+0.006}$	${0.222}_{-0.001}^{+0.001}$	${1.96}_{-0.05}^{+0.05}$	162.877 (124)

Note. The fitting radial range is 01–40 for the western edge and 01–20 for the eastern edge.

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The fit to the eastern CF is poorer (${\chi }^{2}$/d.o.f. = 162.9/124), which is apparently due to the variation in the break radius with the azimuthal angle, while our fit necessarily assumes the same r_j within the fitted range of 120°–180°. Therefore, we further divide the eastern CF into three smaller sectors (120°–140°, 140°–160°, and 160°–180°) and redo the fit. The resultant r_j varies from 129 to 141, with significantly improved the reduced ${\chi }^{2}$ (1.08, 1.05, and 1.21).

Finally, we determine the normalization parameter, n_e0, in Equation (1), relating it to the emission measure, which is derived from the APEC model fitted to the spectrum of the sector immediately outside each CF (the fourth bin in Figures 4 and 5). By definition of the emission measure (EM), we have

$$\begin{eqnarray}\mathrm{EM} & = & \displaystyle \int {n}_{e}{n}_{H}{dV}=\displaystyle \frac{{n}_{e0}^{2}}{{n}_{e}/{n}_{H}}{r}_{c}^{3}{\theta }_{r}\tilde{B}(3\beta -1/2,1/2)\\ & & \times {\displaystyle \int }_{\tfrac{{b}_{l}}{{r}_{c}}}^{\tfrac{{b}_{h}}{{r}_{c}}}b{(1+{b}^{2})}^{-3\beta +1/2}{db},\end{eqnarray} \tag{ 2 }$$

where ${\theta }_{r}$ is the opening angle of the CF, and b_h, b_l are the outer and inner radii of the sector. $\tilde{B}$ is the beta function, obtained by integrating density square along the line of sight. The APEC norms (equivalent to EM) are $(4.34\pm 0.15)\times {10}^{-4}\,\,{\mathrm{cm}}^{-5}$ and $(4.34\pm 0.10)\times {10}^{-4}\,\,{\mathrm{cm}}^{-5}$ for western and eastern regions, respectively. The calculated densities are listed in Table 3.

Table 3.  Gas Properties across the Edges

Region	${n}_{\mathrm{in}}$	${n}_{\mathrm{out}}$	${T}_{\mathrm{in}}$ (dsdeproj)	${T}_{\mathrm{out}}$(dsdeproj)
	(cm−3)	(cm−3)	(keV)	(keV)
West	${0.032}_{-0.001}^{+0.001}$	${0.0158}_{-0.0004}^{+0.0004}$	${6.16}_{-0.32}^{+0.32}({5.14}_{-0.39}^{+0.41})$	${11.74}_{-1.52}^{+1.53}({9.60}_{-1.89}^{+3.63})$
East	${0.061}_{-0.002}^{+0.002}$	${0.0310}_{-0.0006}^{+0.0006}$	${5.64}_{-0.30}^{+0.34}({4.93}_{-0.42}^{+0.44})$	${8.16}_{-0.61}^{+0.68}({6.18}_{-1.01}^{+1.32})$

Note. Density and temperature immediately inside and outside the edges.

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We also calculated the pressure ${P}_{e}={n}_{e}{k}_{{\rm{B}}}T$ and entropy ${S}_{e}={k}_{{\rm{B}}}{{Tn}}_{e}^{-2/3}$. The density at the temperature bin center is used to compute the pressure and entropy. The errors in density were calculated using a Monte Carlo method, assuming Gaussian errors in the fitting parameters. The errors in pressure and entropy were calculated in a similar way. Blue data are calculated using projected temperature and red data are calculated using temperature obtained by dsdeproj. As shown in Figure 4, the western edge has a continuous pressure profile and an abrupt entropy jump. The pressure profile at the eastern edge is not as continuous as in the western edge.

The lack of significant KHI at the cold fronts suggests that they are suppressed by some physical process. The simplest possible explanation is that the fronts are stabilized against KHI by gravitational acceleration. The condition for stability against sinusoidal perturbations at the cold front interface is (Vikhlinin & Markevitch 2002)

$$\begin{eqnarray}&&\displaystyle \frac{g}{k}\gt {({v}_{\mathrm{in}}-{v}_{\mathrm{out}})}^{2}\displaystyle \frac{{\rho }_{\mathrm{in}}{\rho }_{\mathrm{out}}}{{\rho }_{\mathrm{in}}^{2}-{\rho }_{\mathrm{out}}^{2}}\end{eqnarray} \tag{ 3 }$$

where ρ is the gas mass density, v is the velocity tangential to the front surface, g is the magnitude of the gravitational acceleration, and k is the wavenumber. This equation implies that there is a minimum wavelength of ${\lambda }_{\min }=2\pi /{k}_{\max }$ below which KHI perturbations are not suppressed by gravity.

For this calculation, we may use the measured values of the densities ${\rho }_{\mathrm{in}}$ and ${\rho }_{\mathrm{out}}$ from Table 3, but since we are unable to directly measure the velocity of the gas above and below the front surface, we must use estimates of the velocity shear in such systems from simulations and theoretical arguments. Keshet (2012) formulated a theoretical model for spiral sloshing flows in cluster cool cores, arguing that the overall pattern of motion can be separated into a cold, "fast" flow and a hot, "slow" flow, implying that $| {v}_{\mathrm{in}}| \gg | {v}_{\mathrm{out}}|$, which is also seen in simulations (Ascasibar & Markevitch 2006; ZuHone et al. 2010; Roediger et al. 2011, 2012). Simulations also show that the cold flow is subsonic, with typical Mach numbers of ${{ \mathcal M }}_{\mathrm{in}}\sim 0.3-0.5$ (Ascasibar & Markevitch 2006; Roediger et al. 2011). Given this fact from the simulations and the measured value of the sound speed ${c}_{{\rm{s}},\mathrm{in}}$ underneath the front surface, we may make a rough estimate for the tangential velocity of the cold gas ${v}_{\mathrm{in}}$, which we show in Table 4.

Table 4.  Derived Properties of the Cold Fronts

Region	${c}_{{\rm{s}},\mathrm{in}}$ (km s−1)	${v}_{\mathrm{in}}$ (km s−1)	${\lambda }_{\min }$ (kpc)	${t}_{\mathrm{KH}}$ (Myr)	${B}_{\mathrm{in}}$ (μG)
West	1180 ± 50	470 ± 120	43 ± 18	190 ± 90	24 ± 6
East	1150 ± 50	460 ± 120	38 ± 16	170 ± 80	32 ± 8

Note. Sound speeds are calculated from deprojected temperature.

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Determining the gravitational acceleration g from the X-ray data is not straightforward, since the gas in the core is not in an equilibrium state, as evidenced by the sloshing motions, and hydrostatic equilibrium cannot be assumed in this region. However, N-body/hydrodynamics simulations of sloshing (Ascasibar & Markevitch 2006; ZuHone et al. 2010) indicate that the DM core of the main cluster should be relatively stable against the influence of the subcluster, and that hydrostatic equilibrium is a good approximation outside the sloshing region. These two facts together indicate that fitting a mass profile to these outer regions and extrapolating it inward should give a reasonable estimate of the mass inside the CFs.

For this purpose, we use the deprojected density and temperature profiles in the radial range of $r\sim 100\mbox{--}1000\,\mathrm{kpc}$ to compute a mass profile assuming hydrostatic equilibrium, and fit this profile to an NFW profile (Navarro et al. 1996):

$$\begin{eqnarray}&&{\rho }_{\mathrm{DM}}(r)=\displaystyle \frac{{\rho }_{s}}{\tfrac{r}{{R}_{s}}{\left(1+\tfrac{r}{{R}_{s}}\right)}^{2}}.\end{eqnarray} \tag{ 4 }$$

The parameters we obtain are ${\rho }_{s}=4.1\pm 0.4\times {10}^{6}\,{M}_{\odot }\,{\mathrm{kpc}}^{-3}$ and ${R}_{s}=256\pm 13\,\mathrm{kpc}$. The implied gravitational accelerations at the western and eastern CFs are $6.8\pm 1.4\ \times {10}^{-8}\,\mathrm{cm}\,{{\rm{s}}}^{-2}$ and $7.8\pm 1.5\times {10}^{-8}\,\mathrm{cm}\,{{\rm{s}}}^{-2}$, respectively.

In Table 4, we show the minimum wavelength ${\lambda }_{\min }$ of KHI modes that can be suppressed by gravity. At either the western or eastern front, ${\lambda }_{\min }$ is still quite large compared to the size of the cold fronts (despite the uncertainties), indicating that gravity is incapable of suppressing perturbations that should still be visible.

We note that Richard et al. (2010) used strong gravitational lensing to derive the total mass in the cluster core, which we find to be nearly twice our derived mass at a radius of ∼65 kpc. In this case, the minimum wavelength that can be stabilized becomes half that quoted in Table 4.

One must also consider the possibility that, although the cold front discontinuity is indeed unstable to KHI, there simply has not been enough time for perturbations to grow. To determine whether this is the case, we calculate the KHI growth timescale via

$$\begin{eqnarray}&&{t}_{\mathrm{KH}}=\displaystyle \frac{2\pi }{{\omega }_{\mathrm{KH}}}=\displaystyle \frac{{\rho }_{\mathrm{in}}+{\rho }_{\mathrm{out}}}{\sqrt{{\rho }_{\mathrm{in}}{\rho }_{\mathrm{out}}}}\displaystyle \frac{\lambda }{| {v}_{\mathrm{in}}-{v}_{\mathrm{out}}| }\end{eqnarray} \tag{ 5 }$$

and tabulate the timescales ${t}_{\mathrm{KH}}$ corresponding to ${\lambda }_{\min }$ in Table 4. The timescales are rather long (with substantial uncertainty), indicating that for perturbations at these wavelengths, they may simply have not had enough time to grow since the formation of the cold fronts. However, perturbations at smaller wavelengths will grow faster. For example, a perturbation with a wavelength of 10 kpc will grow with a timescale of ${t}_{\mathrm{KH}}\sim 45\,\mathrm{Myr}$ for both CFs, so perturbations of these scales may be expected to be observed.

This indicates that such perturbations may be suppressed by the action of a strong magnetic field layer aligned with the front surface. Such layers should arise naturally at sloshing cold fronts (Keshet et al. 2010; ZuHone et al. 2011), which are produced by the shear amplification of an existing magnetic field, stretching the magnetic field lines parallel to the cold front surface and increasing the magnetic field strength. The minimum magnetic field in the layer necessary to suppress KHI is given by (Vikhlinin & Markevitch 2002; Keshet et al. 2010)

$$\begin{eqnarray}&&{B}_{\mathrm{in}}^{2}+{B}_{\mathrm{out}}^{2}\gt 4\pi \displaystyle \frac{{\rho }_{\mathrm{in}}{\rho }_{\mathrm{out}}}{{\rho }_{\mathrm{in}}+{\rho }_{\mathrm{out}}}{({v}_{\mathrm{in}}-{v}_{\mathrm{out}})}^{2}\end{eqnarray} \tag{ 6 }$$

where ${B}_{\mathrm{in},\mathrm{out}}$ are the magnetic field strengths on either side of the front. The simulations of ZuHone et al. (2011) showed that sloshing motions amplify the magnetic field strength predominantly below the front, so ${B}_{\mathrm{in}}^{2}+{B}_{\mathrm{out}}^{2}\approx {B}_{\mathrm{in}}^{2}$, giving an estimate for the minimum magnetic field strength in the cold flow. These estimated magnetic field strengths, on the order of 10 μG, are listed in Table 4. Such field strengths could be easily produced by shear amplification of the magnetic field at the cold front surfaces, as demonstrated in simulations by ZuHone et al. (2011).

In the previous subsection, we drew on the results of simulations to infer the probable velocity of the sloshing cold gas underneath the cold fronts. To the best of our knowledge, with the exception of Keshet et al. (2010), no works have attempted to measure the tangential velocity underneath a sloshing cold front. Previous works (e.g., Vikhlinin et al. 2001a) have attempted to estimate the velocity in the free-stream region ahead of a merger cold front, but the sloshing scenario here is distinct from a major merger. Therefore, it is worth making the velocity estimate to see if it is consistent with what we expect from simulations.

Markevitch et al. (2001) noted an unphysical jump in the mass profile derived from the X-ray data within an angular sector containing a cold front edge in the relaxed cluster A1795. In such a case, although the gas pressure itself is continuous, its first derivative is discontinuous. They argued that this was evidence for radial acceleration of the gas underneath the front. Keshet et al. (2010) later noted that this would cause an unphysical gap to open along the cold front. Instead, they formulated an argument that the difference in gravitational acceleration is compensated by the centripetal acceleration of the tangentially moving gas underneath the cold front surface. Under suitable assumptions, they derived the following expression for the velocity shear (their Equation (2)):

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{v}^{2}}{r}=-{\rm{\Delta }}{a}_{r}={\rm{\Delta }}\left[\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial P}{\partial r}\right]\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Delta }}{v}^{2}={v}_{\mathrm{in}}^{2}-{v}_{\mathrm{out}}^{2}\approx {v}_{\mathrm{in}}^{2}$. We will use this expression to make an estimate for the velocity ${v}_{\mathrm{in}}$ of the cold component of the western cold front, since it exhibits the properties of pressure continuity with a pressure derivative discontinuity. To compute the pressure derivative, we assume that, in the immediate vicinity of the cold front, the temperature is constant with radius on both sides and that the behavior of the density on either side of the front is given by the β-model and power-law fits from above (Equation (1)). We further assume for simplicity that the motion of the cold front is in the plane of the sky, which is not unreasonable according to its spiral-like morphology. We determine that ${v}_{\mathrm{in}}=510\pm 320$ km s⁻¹, corresponding to a Mach number of ${{ \mathcal M }}_{\mathrm{in}}={v}_{\mathrm{in}}/{c}_{{\rm{s}},\mathrm{in}}\ =0.43\pm 0.27$, which is very similar to the value taken from the simulations discussed in Section 4.1. We note that the value we obtain for the dimensionless pressure discontinuity parameter from Keshet et al. (2010) $\delta =(r/{c}_{{\rm{s}},\mathrm{in}}^{2}){\rm{\Delta }}{a}_{r}\ =-0.19\pm 0.23$ is marginally consistent with their upper limits for this parameter for the western cold front of A2204 (see their Figure 2).

The relatively large error bars on our estimates for the velocity of the gas underneath the front and the pressure discontinuity point to the need for further study. Despite the fact that our deep exposure of A2204 and the apparent smoothness of its western cold front provides the opportunity to attempt such an estimate, it is still difficult to measure this number accurately, primarily due to the uncertainty in the gas temperature on either side of the front. Additionally, this estimate depends on the assumptions that the gas moving underneath the front can be approximated as having circular motion and that this motion is in the plane of the sky. The best way to quantify the uncertainty associated with these assumptions would be to analyze mock X-ray observations of simulations of cold fronts, where we could carry out the same analysis as for A2204, but would be able to project the cold front along slightly different lines of sight and compare our estimate for the velocity directly to the velocity field in the simulation. We leave this for future work.