NON-EQUILIBRIUM ELECTRONS IN THE OUTSKIRTS OF GALAXY CLUSTERS

We use a mass-limited sample of 65 galaxy clusters from a high-resolution cosmological simulation Omega500, the details of which can be found in Nelson et al. (2014b). The simulation box is $500{h}^{-1}\;\mathrm{Mpc}$ on each side, and has a peak spatial resolution of $3.8{h}^{-1}$ kpc. We neglect radiative cooling and star formation, which should have minimal effects on cluster outskirts.

Using the Adaptive Refinement Tree code (Kravtsov 1999; Kravtsov et al. 2002; Rudd et al. 2008), we performed the simulation with a modification to model electrons and heavier ions separately as described in Rudd & Nagai (2009). Electrons and ions are assumed to be in local thermodynamic equilibrium with separate respective temperatures T_e and T_i, and the relaxation process between the two components is explicitly calculated. The electron temperature time evolution is modeled as,

$$\begin{eqnarray}&&\displaystyle \frac{{{dT}}_{{\rm{e}}}}{{dt}}=\displaystyle \frac{{T}_{{\rm{e}}}-{T}_{{\rm{i}}}}{{t}_{\mathrm{ei}}}-(\gamma -1){T}_{{\rm{e}}}{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 1 }$$

where the second term accounts for heating and cooling from adiabatic accretion, $\gamma =5/3$ is the adiabatic index, ${\boldsymbol{v}}$ is the gas velocity, and t_ei is the equilibration timescale for a fully ionized medium (Spitzer 1962) comprised of electrons, protons, and He ii,

$$\begin{eqnarray}&&{t}_{\mathrm{ei}}=6.3\times {10}^{8}\;\mathrm{years}\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{10}^{7}\;{\rm{K}}}\right){\left(\displaystyle \frac{{n}_{{\rm{i}}}}{{10}^{-5}\;{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{40}\right)}^{-1},\end{eqnarray} \tag{ 2 }$$

where n_i is the ion number density, and $\mathrm{ln}{\rm{\Lambda }}$ is the Coulomb logarithm,

$$\begin{eqnarray}&&\mathrm{ln}{\rm{\Lambda }}=37.8+\mathrm{ln}\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{10}^{7}\;{\rm{K}}}\right)-\displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{{n}_{{\rm{i}}}}{{10}^{-5}\;{\mathrm{cm}}^{-3}}\right).\end{eqnarray} \tag{ 3 }$$

In this two-temperature model, the hydrodynamic equations are computed using the total thermal energy of the gas and the electron temperature is a passively evolving scalar field whose evolution is described by Equation (1). We do not keep track of potential temperature differences between heavier ions, such as hydrogen and helium ions, since the equilibration timescales between heavier ions are very short compared to that of the electrons; i.e., the equilibration timescale between He ii and protons is a factor of $\sqrt{{m}_{{\rm{e}}}/{m}_{\mathrm{He}\;{\rm{II}}}}$ of the equilibration timescale between electrons and protons.

Accretion shocks thermalize the electrons and ions accreting from the cosmic web. The heavier ions acquire most of the thermal energy, and heat the electrons through Colulomb collisions in the post-shock regions. However, in the diffuse cluster outskirts with gas temperatures around $T\gt {10}^{7}$ K and densities between 10 and 100 times the cosmic mean density, t_ei becomes comparable to the Hubble time, causing the electron temperatures to be lower than the ion temperatures.

The Spitzer timescale adopted in our simulations is an upper limit of the true equilibration time, as there are other physical mechanisms such as plasma instabilities (e.g., Bykov et al. 2008) that can couple the temperatures of the different ion species. However, the effects of plasma instabilities are expected to be small in the high Mach number accretion shocks, which are responsible for generating the non-equilibrium electrons in cluster outskirts. Plasma instabilities may provide additional non-adiabatic heating that shortens the equilibration timescale in gas with lower Mach numbers. Therefore, the results in this paper should be taken as a limit on the maximal effects of non-equilibrium electrons.

We create realistic mock X-ray photon maps from X-ray flux maps by convolving with Chandra response files. We generate two sets of X-ray flux maps: (1) the first uses a projected X-ray emissivity that has been generated using the mean gas temperature in the simulation ${T}_{\mathrm{gas}}=({n}_{{\rm{i}}}{T}_{{\rm{i}}}+{n}_{{\rm{e}}}{T}_{{\rm{e}}})/({n}_{{\rm{i}}}+{n}_{{\rm{e}}})$, where n_e is the electron density; and (2) the second uses the electron temperature, T_e.

Below we summarize the main elements of the mock Chandra analysis pipeline here. Further details of the pipeline can be found in Nagai et al. (2007) and Avestruz et al. (2014).

The X-ray emissivity for a given kth hydrodynamical cell with volume ${\rm{\Delta }}{V}_{k}$ in the simulation is given by,

$$\begin{eqnarray}&&{j}_{E,k}={n}_{e,k}{n}_{i,k}{{\rm{\Lambda }}}_{E}({T}_{k},{Z}_{k},z){\rm{\Delta }}{V}_{k},\end{eqnarray} \tag{ 4 }$$

where n_e and n_i are the respective number densities of electrons and ions, T_k is either the mean gas temperature (${T}_{\mathrm{gas}}$) or the electron temperature of that element (T_e), ${Z}_{k}=0.3{Z}_{\odot }$ is the assumed metallicity, and z is the redshift. We assume a constant metallicity that is representative of a plausible average metallicity for a galaxy cluster, since our simulation does not follow cooling and star formation. We compute the X-ray emissivity, ${{\rm{\Lambda }}}_{E}({T}_{k},{Z}_{k},z)$, using the MEKAL plasma code (Mewe et al. 1985; Kaastra & Jansen 1993; Liedahl et al. 1995). We multiply the plasma spectrum by the Galactic absorption corresponding to a hydrogen column density of ${N}_{{\rm{H}}}=2\times {10}^{20}$ cm⁻².

We convolve the emission spectrum with the response of the Chandra ACIS-I CCDs and draw photons from each position and spectral channel according to Poisson distribution. The resulting photon maps have an exposure time of 2.4 ms, similar to the deep Chandra observations of Abell 133 (A. Vikhlinin et al. 2015, in preparation), and comparable in photon counts to stacked galaxy cluster analyses. From images generated in the 0.7–2 keV band, we identify and mask out clumps using the wavelet decomposition algorithm described in Vikhlinin et al. (1998). We also exclude regions that correspond to overdense filamentary structures, the details of which are described in Avestruz et al. (2014).

The temperature bias from non-equilibrium electrons depends on the location of the accretion shock that generates the initial temperature difference between electrons and heavier ions and the relative age of the gas ($\sim M/\dot{M}$) compared with the equilibration timescale in Equation (2). The former depends on the MAR of the cluster, and the latter depends on the temperature and density (hence the mass) of the cluster.

The left panel of Figure 1 shows the profiles of the electron temperature bias ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ of the simulated clusters at $z=0,0.5,1,1.5$ averaged in four mass bins, with masses defined within ${R}_{200m}$ (black lines). Note that we have taken the profiles of the 65 clusters at z = 0, 0.5, 1, and 1.5 prior to binning the profiles by mass.

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In a companion paper (Lau et al. 2014), we show that the average location of the accretion shock occurs at a fixed fraction of ${R}_{200m},$ ⁵ independent of redshift. The location of the maximum temperature bias is mainly driven by the accretion shock, and therefore also exhibits the same redshift independence when scaled to ${R}_{200m}$. The ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ profile for each mass bin shows a consistent location for the maximum temperature bias, which is located at the accretion shock radius ${R}_{\mathrm{sh}}\approx 1.6\times {R}_{200m}$. At this radius, the shocked electrons are maximally out of equilibrium with the heavier ions. The extent of the bias is larger for more massive clusters with hotter gas and longer equilibration times (see Equation (2)).

The electron temperature bias also depends on the MAR of the cluster. The MAR shifts the location of the accretion shock that generates the non-equilibrium electrons. We use a proxy of the MAR of the cluster

$$\begin{eqnarray}&&{\alpha }_{200m}\equiv {V}_{r}^{\mathrm{DM}}(r={R}_{\alpha })/{V}_{\mathrm{circ},200m},\end{eqnarray} \tag{ 5 }$$

defined as the average mass-weighted dark matter radial velocity V_r^DM measured at radius ${R}_{\alpha }=1.25{R}_{200m}$ and normalized by the circular velocity ${V}_{\mathrm{circ},200m}\equiv \sqrt{{{GM}}_{200m}/{R}_{200m}}$ of the cluster (Lau et al. 2014). A more negative ${\alpha }_{200m}$ indicates higher MAR.

The value of ${R}_{\alpha }$ was chosen to minimize scatter relative to an alternative MAR proxy (Diemer & Kravtsov 2014),

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{200m}=\displaystyle \frac{{\mathrm{log}}_{10}({M}_{200m}(z=0)/{M}_{200m}(z=0.5))}{{\mathrm{log}}_{10}(a(z=0)/a(z=0.5))}\end{eqnarray} \tag{ 6 }$$

which only describes the MAR of a cluster at z = 0 (see Figure 5 in Lau et al. 2014). The choice of a fixed fraction of ${R}_{200m}$ to measure an accretion rate provides a consistent way to compare accretion rates of clusters at different epochs.

The right-hand panel of Figure 1 shows the profile of the temperature bias in bins of ${\alpha }_{200m}$. Following Lau et al. (2014), we decompose gas in a given radial bin into "filament" and "diffuse" components according to the density and radial velocity of the gas defined with respect to the cluster center. At large cluster-centric radii, a large fraction of the gas mass belongs to dense filaments and clumps, with net radial velocities that point toward the cluster center. The thick lines in the plot correspond to the bias in the diffuse ICM of the clusters, and the thin lines correspond to the bias in the filaments.

In the filament component of the gas (thin lines), the electron temperature is very close to the mean gas temperature. Gas in filaments also has higher densities and lower temperatures than gas in the diffuse component, leading to shorter equilibration times. In the diffuse component of the gas (thick lines), the location of maximum bias occurs closer to the cluster center for more rapidly accreting clusters. Clusters with higher accretion rates have more negative values of ${\alpha }_{200m}$. Accreted material in these clusters have higher momentum fluxes, causing the location of accretion shock to move inward.

The distinction between the diffuse ICM and the filaments is visible in Figure 2 where we show slices of ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ for a relaxed low mass cluster (left panel) and a merging massive cluster (right panel) from our simulation sample. The dashed circles indicate ${R}_{200m}$ for each cluster. Darker shades in the maps indicate regions where more electrons are out of equilibrium with the heavier ions, corresponding to a larger temperature bias. The regions of larger bias mostly correspond to gas in the diffuse ICM. The lighter shades indicate regions where the electron temperature is very close to the mean gas temperature. Near ${R}_{200m}$, the lighter shades correspond to gas in filaments, which experience shock heating at smaller radii than gas in the diffuse ICM.

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In the more massive and less relaxed cluster shown in the right panel, mergers drive more small scale shocks that lead to more non-equilibrium electrons. Additionally, the higher ICM temperature in the outskirts of the more massive cluster leads to longer equilibration times and therefore larger temperature biases.

The temperature bias in the left panel of Figure 1 uses the mass-weighted temperatures from all of the cluster gas. Since filaments are denser in cluster outskirts, the temperature bias is more heavily weighted toward the filament contribution, where the non-equilibrium effect is small.

Figure 3 shows ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ as a function of cluster mass, both measured at $r={R}_{200m}$. Different redshifts ($z\;=0,0.5,1.0,1.5$) are indicated by marker styles (respectively circles, squares, diamonds, and triangles). There is an apparent redshift dependence of ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$, with high-z clusters exhibiting less of a bias than low-z clusters. However, the redshift dependence is primarily due to cluster mass growth. At low redshift, there are more high mass clusters with a more pronounced bias. For a given cluster mass, there is no significant trend of ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ with redshift. There is, however, a systematic trend in ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ with MAR. Clusters with higher MAR (more negative values of ${\alpha }_{200m}$, as indicated by the color of the points in Figure 3) tend to a larger bias, with lower ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ values at ${R}_{200m}$. As discussed in Section 3.1, clusters with higher MAR have accretion shocks closer to the cluster center, leading to an inward shift of the location of maximum bias.

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We note that there can be additional redshift dependence in ${T}_{{\rm{e}}}/{T}_{\mathrm{gas}}$ due to any evolution in the distribution of cluster assembly history. At a given redshift, more recently assembled clusters have more recently accreted electrons with less time to equilibrate; we expect the fraction of recently assembled clusters to vary with redshift. The mass accretion timescale ${t}_{\mathrm{acc}}\equiv M/\dot{M}$ for a given cluster at a given redshift is an indicator of cluster age. Our MAR proxy, ${\alpha }_{200m}$ (defined in Equation (5)), is inversely proportional to ${t}_{\mathrm{acc}}$, as ${V}_{\mathrm{circ},200{\rm{m}}}$ is a proxy for cluster mass and ${V}_{r}^{\mathrm{DM}}(r={R}_{\alpha })$ describes the instantaneous MAR. However, in our sample, there does not appear to be any strong correlation between z and ${\alpha }_{200m}$ for a given mass. To make a more stringent statement on any additional redshift dependence, we would need a larger box that samples a wider mass range in a given redshift bin.

We provide a simple fitting function for the profile of the temperature bias:

$$\begin{eqnarray}&&{b}_{{\rm{e}}}(R)\equiv \displaystyle \frac{{T}_{{\rm{e}}}(R)}{{T}_{\mathrm{gas}}(R)}=\displaystyle \frac{{(x/{x}_{t})}^{-a}}{{\left(1+{(x/{x}_{t})}^{-b}\right)}^{a/b}},\end{eqnarray} \tag{ 7 }$$

where $x=R/{R}_{200m}$ and the three best-fit parameters that account for the dependence on cluster mass are given by,

$$\begin{eqnarray}{x}_{t} & = & 0.629\times {\left({M}_{200m}/{10}^{14}{h}^{-1}{M}_{\odot }\right)}^{-0.1798}\\ a & = & 0.086\times {\left({M}_{200m}/{10}^{14}{h}^{-1}{M}_{\odot }\right)}^{0.3448}\\ b & = & 2.851\times {\left({M}_{200m}/{10}^{14}{h}^{-1}{M}_{\odot }\right)}^{-0.006}.\end{eqnarray} \tag{ 8 }$$

In the left panel of Figure 1, we overplot profiles of the fitting function from Equation (7) in red. Each red line corresponds to the fitting function with an input mass that is the average cluster mass in each cluster mass bin. The fitting function adequately describe the averaged profiles of the bias, shown as black lines.

Here we do not decompose the ICM into diffuse and filament components, nor do we separately model dependence on ${\alpha }_{200m}$, since applications of the fitting function likely treat spherical averages of the halo profile and do not typically have the information necessary to calculate ${\alpha }_{200m}$. The dependence on ${\alpha }_{200m}$ comes from the fact that accretion rate shifts the location of the shock radius. We have characterized the relationship between the shock radius and accretion rate in Figure 10 of the companion paper by Lau et al. (2014), which shows that the scatter in the shock radius at a given ${\alpha }_{200m}$ is nearly a factor of two. This large scatter in the shock radius stems from highly aspherical and complex geometry at large radii. As a result, the location of the shock radius exhibits a relatively weak dependence (although systematic) on ${\alpha }_{200m}$, whereas the mass dependence of the temperature bias exhibits a stronger behavior with better predictive power. We therefore use the fitting function to describe the bias in a statistical manner, parameterizing by mass alone.

X-ray measurements of the ICM are only sensitive to the electron component of the plasma. A temperature bias due to non-equilibrium electrons propagates to ICM quantities inferred from the X-ray temperature.

We measure the X-ray temperature profiles from two sets of mock Chandra maps: one generated with the mean gas temperature, and the other with the electron temperature. This allows us to compare the temperature difference accounting for instrumental response, projection effects, and the spectroscopic weighting of the X-ray gas.

Figure 4 shows the bias in the X-ray projected temperature profile, measured from spectral fitting of photons from each of the two mock maps. The bias is then calculated as a ratio of measured X-ray electron temperature to the X-ray temperature that would be measured if electrons were in equilibrium with heavier ions. Red data points show the projected X-ray temperature bias for one of the less massive relaxed clusters with T_X = 5.96 keV (${M}_{200m}=7.2\times {10}^{14}{h}^{-1}{M}_{\odot }$) and ${\alpha }_{200m}=-0.05$. Blue data points show the same bias in one of our more massive unrelaxed clusters with T_X = 11.03 keV (${M}_{200m}=1.75\times {10}^{15}{h}^{-1}{M}_{\odot }$) and ${\alpha }_{200m}=-0.66$. These two representative clusters bracket the effects of non-equilibrium electrons on X-ray measurements in this mass range, but represent upper limits of the temperature bias in both cases. For the relaxed less massive cluster, the bias within $R\lt {R}_{500c}$ is less than 5%. In the unrelaxed, more massive cluster that has experienced rapid recent accretion, the bias is $\sim 10\%$ at ${R}_{500c}$, and increases precipitously at larger radii.

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We have also overplotted the ratio of the mass-weighted electron temperature profile to the mass-weighted mean temperature profile of the diffuse ICM, computed directly from the simulation. The agreement between the bias in mock X-ray measurements and the bias in the mass-weighted temperature shows that the bias is comparable between the two methods of measuring the ICM temperature.

Note that the upturn in the bias at $R\approx 0.8{R}_{200m}$ for the red mock X-ray data in the less massive cluster is due to line of sight (LOS) effects; the bias is not perfectly spherical, even in the bulk component. Some projections will have a bias in the projected X-ray temperature that is either below or above the bias calculated from the spherically averaged mass-weighted temperatures shown in the red dotted line. In this projection, there is slightly less shock heated material at this radius compared to other lines of sight.

Likewise, the drop in the X-ray temperature bias for the more massive less relaxed cluster at $R\approx 0.8{R}_{200m}$ is also due to the LOS; the LOS for the X-ray data has more shock heated material at that radius compared to other LOS, all of which have a mass-weighted average shown in the solid blue line.

To illustrate the scatter in projected temperature biases, we have also plotted, in lighter colors, the measured X-ray temperature bias along two other orthogonal projections for each cluster. In both clusters, the temperature biases are generally present in all three projections. However, the temperature measurements from X-ray mocks occasionally include photons from clumps and filaments that were not completely masked out; these photons can contribute to the projected annular average by driving the temperature ratio closer to unity, since the high-density gas in clumps and filaments has less of a temperature bias.

We calculate the corresponding effect of non-equilibrium electrons on hydrostatic mass estimates for our cluster sample in four temperature bins. In Table 1, we list the ratios between the hydrostatic mass estimates calculated with electron temperature to that with the mean gas temperature. The hydrostatic mass estimate from the electron temperature corresponds to the observationally inferred mass in the presence of non-equilibrium electrons, while the hydrostatic mass estimate from the mean gas temperature corresponds to the inferred mass with electrons fully in equilibrium with the heavier ions.

Table 1.  Biases in Hydrostatic Mass and Gas Temperature of Galaxy Clusters at z = 0

TX (keV)	$R={R}_{500c}$	$R={R}_{200c}$	$R={R}_{200m}$
(a) ${M}_{\mathrm{HSE}}(R;{T}_{{\rm{e}}})/{M}_{\mathrm{HSE}}(R;{T}_{\mathrm{gas}})$
4.2–6.1 (38)	0.993 ± 0.002	0.959 ± 0.007	0.889 ± 0.009
6.1–8.7 (24)	0.989 ± 0.005	0.926 ± 0.010	0.868 ± 0.012
8.7–12.5 (3)	0.961 ± 0.009	0.893 ± 0.016	0.803 ± 0.041
(b) Core excised ${b}_{{\rm{e}}}(\lt R)\equiv {T}_{{\rm{e}}}(\lt R)/{T}_{\mathrm{gas}}(\lt R)$
4.2–6.1 (38)	0.991 ± 0.001	0.985 ± 0.002	0.975 ± 0.001
6.1–8.7 (24)	0.985 ± 0.002	0.976 ± 0.003	0.967 ± 0.003
8.7–12.5 (3)	0.972 ± 0.003	0.962 ± 0.003	0.952 ± 0.004

Note. (a) Shows the mean biases in hydrostatic mass and their $1\sigma$ errors due to non-equilibrium electrons at the given radius R. (b) Shows the mean bias the mass-weighted temperature integrated between $0.15{R}_{500c}$ and R specified in the first row. The number in the parentheses indicate the number of clusters in each bin of T_X.

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In the most massive clusters (${T}_{{\rm{X}}}\gtrsim 8$ keV), the core-excised mass-weighted electron temperature is biased low by 3% at R_500c, and as much as 5% at ${R}_{200m}$. The corresponding bias in the observed hydrostatic mass for these clusters is 4% at ${R}_{500c}$, and 20% at ${R}_{200m}$. The bias in observed hydrostatic mass is larger than the bias in the temperature, b_e(R), at each of these radii, as the hydrostatic mass within a radius is proportional to both the temperature and the temperature gradient at that radius.

Table 1 also lists the resulting biases in mass-weighted cluster temperature, ${b}_{{\rm{e}}}(\lt R)\equiv {T}_{{\rm{e}}}(\lt R)/{T}_{\mathrm{gas}}(\lt R)$, measured out to three different values of R: R_500c, R_200c, and ${R}_{200m}$. ${b}_{{\rm{e}}}(\lt R)$ is a core-excised integrated quantity between the $0.15{R}_{500c}$ and the maximum given radius, R. ${b}_{{\rm{e}}}(\lt R)$ have a smaller bias than the temperature bias b_e(R) measured at R since the temperature bias monotonically increases until the shock radius. The bias in the core-excised temperature is $\lt 3\%$ at ${R}_{500c}$ and $\lt 5\%$ at ${R}_{200m}$, smaller than the biases in both the electron temperature at those radii and the observed hydrostatic mass.

Since the temperature bias due to non-equilibrium electrons has a larger effect on the hydrostatic mass estimate than on the integrated temperature, scaling relations such as the $M-{T}_{{\rm{X}}}$ and $M-{Y}_{{\rm{X}}}$ will have a shallower slope with increasing cluster mass, leading to deviations from self-similarity.