SIMULATING ASTRO-H OBSERVATIONS OF SLOSHING GAS MOTIONS IN THE CORES OF GALAXY CLUSTERS

The moment maps in the previous section indicate a number of interesting locations associated with sloshing cold fronts that produce line shifts and broadening that will be observable by Astro-H. We will use these maps to choose locations for Astro-H pointings and elliptical regions within these pointings from which spectra will be extracted. We will take advantage of the velocity structure revealed by Figures 1–6 from the simulation to decide where to locate the pointings. Of course, we will not know a priori in real observations where to point in this fashion, but the moment maps show that the regions with the most significant line shifting and broadening are located just underneath the cold front surfaces, the locations of which will be available from higher-resolution imaging of the same clusters from Chandra and XMM-Newton.

The chosen pointings and regions for the inviscid simulation are shown in Figures 1–3. Each elliptical region on the maps in 2D defines a cylindrical region in 3D, extended along the line of sight across the cluster. To get a sense of the underlying velocity distribution sampled by these regions, we examine the velocity field within these cylinders. Figures 7 through 10 show the results of this exercise. In the remainder of this work, we label each region by the axis of projection and the number of the pointing, as identified in Figures 1–3. For example, pointing 1 on the z-axis projection is labeled "z1."

Zoom In Zoom Out Reset image size

We take Figure 7 as an example and describe the different panels of this figure in detail; the descrption will apply to all of the related figures. Figures 7 and 8 examine the z-component of the gas velocity. The upper panels show slices through the x–z plane of the temperature (left) and z-component of the velocity (right). The slice is taken at the y coordinate of the center of the cylindrical region in Figure 1 (where lines showing the location of the slice planes for each projection are shown). The center and edges of the cylinder within the slice plane are indicated by black dot-dashed and dashed lines, respectively, in Figure 7. The lower left panel of Figure 7 shows a plot of the velocity distribution phase space in the cylinder, where the color indicates the amount of helium-like iron line emission as a function of both the position along the line of sight and the line-of-sight velocity. The solid black line indicates the emission-weighted average velocity along the length of the cylinder. Finally, the lower right panel of Figure 7 shows the shape of a "toy" helium-like iron line, in the absence of nearby lines or continuum, computed from the total emission within the cylindrical region. The blue lines show the line shape due to thermal broadening without velocity broadening, whereas the green lines show the combined effects of thermal broadening and plasma motions.

Zoom In Zoom Out Reset image size

In Figure 7, region "z1" is situated within the cold fronts enveloping the cluster center. The z-velocity slice plot and phase plot indicate that the gas motion is mostly random on scales smaller than the core size within this region, with no significant velocity shift, but with a range of gas motion from v ∼ −400 km s⁻¹ to v ∼ +400 km s⁻¹. The line emission plot confirms this, showing a symmetric Gaussian-looking line that has been slightly broadened by the turbulent velocity field. Figure 8 shows an entirely different velocity field in region "z2." In this case, the elliptical region is centered on the outermost part of the spiral to the south, and along the z-direction the symmetry of the sloshing motions results in two oppositely directed gas flows of approximately v_z ∼200 km s⁻¹ on either side of the x–y-plane, shown prominently in both the z-velocity slice plot and the phase plot. Though the velocity distribution here is very different from that of region "z1," the shape of the line emission is nevertheless nearly identical—both are approximately Gaussian in shape.

In Figure 9, now examining the x-component of the velocity, the slice plots in the upper panels are taken through the x–y plane of the two subclusters' mutual orbit and cut through the center of both the "x1" and "x2" cylindrical regions. These regions align with the cold gas component of the sloshing cold fronts, which are regions of significant bulk motion along the x-axis. The phase-space plots (middle panels) demonstrate that in both regions there is a substantial portion of gas moving along the line of sight in each region, causing the line shifts seen in Figure 2. There is also a considerable amount of variance in the velocity field, both along the line of sight (indicated by the large variation in the black line) and in the plane perpendicular to the line of sight along the length of the cylinder. For example, at the x = 0 position in both regions, there is a Δv ∼ 200 km s⁻¹ spread in velocity from the average value. In the lower panels of Figure 9, the large bulk motions along the line of sight in each region create line centroid shifts of Δv ∼ 300 km s⁻¹ (ΔE ∼ 7 eV) in either direction, with a fairly significant line broadening of FWHM ∼ 400–500 km s⁻¹ in both cases. Each line also appears slightly skewed in the direction of the velocity shift. These features of the lines may be identified with features of the phase plots, which both show a broad distribution of gas in velocity space centered around a nonzero mean velocity in either direction.

Zoom In Zoom Out Reset image size

Lastly, Figure 10 shows the same plots for the single region "y1," for the y-component of the velocity. The slice is taken through the x–y plane at z = 0. Within this region, there is a large bulk motion in the +y direction. The phase plot shows that within the cold front region the bulk of this gas moves with an average y-velocity of v ∼ +400 km s⁻¹, with a significant spread of v ∼ +200–800 km s⁻¹, seen in the central portion of the panel. A smaller fraction of the X-ray-emitting gas is moving in the −y direction with a speed of v ∼ −150 km s⁻¹, with much smaller variation, seen at a distance of ∼200 kpc on either side of the center. The shape of the emission line in the lower right panel of Figure 10 shows evidence of both of these components.

Zoom In Zoom Out Reset image size

We also perform the same examination of the viscous simulation. Our results are very similar to those in the previous section, so for brevity we only show the results for the x-component of the velocity here in Figure 11, and we refer the reader to Appendix B for the figures showing the other projections.

Zoom In Zoom Out Reset image size

Figure 11 shows the slice, phase-space, and line-shape plots for the viscous simulation for the x-component of the velocity. Overall, owing to the damping of turbulence and instabilities by viscosity, the velocity distribution within each region is smoother, though the large-scale motions remain essentially the same. The slice plots are almost devoid of small-scale features, and though the average value of the velocity along the length of each cylindrical region is roughly the same as in Figure 9 (the black lines in the phase plots), the emission is not quite as spread out in phase space as in the inviscid simulation. This manifests itself in a slight narrowing of the velocity distribution at any given position along the axis in the phase-space plots. Nevertheless, the line profiles from the viscous simulation are very similar to those from the inviscid case. Figure 12 demonstrates this by comparing the line shapes from both simulations directly along all three principal axes. This indicates that in both cases the gas motion on large scales is not only responsible for the shift in the spectral line but also largely responsible for the line broadening, owing to its smooth variation along the line of sight. As a result of this similarity, in the next section we will only generate and fit synthetic spectra from the inviscid simulation, for brevity.

Zoom In Zoom Out Reset image size

A thermally broadened spectral line with line flux profile F₀(E) in the rest frame of the gas that is Doppler broadened to the new line profile F(E) can be represented by

$$\begin{eqnarray}&&F(E)={\displaystyle \int }_{-\infty }^{\infty }{F}_{0}\left[E\left(1-\displaystyle \frac{v}{c}\right)\right]\;f(v)dv,\end{eqnarray} \tag{ 7 }$$

which is essentially a convolution with the velocity distribution function f (v). We choose to model the thermal spectrum and the velocity distribution and fit these models to our synthetic spectra in XSPEC, using the bapec ⁷ thermal plasma model, which accounts for thermal broadening of spectral lines and allows for velocity broadening to be accounted for with a single parameter σ that represents the standard deviation of a Gaussian velocity distribution. We constrain the velocity shift μ of the spectral lines using the difference of the fitted redshift parameter z of the bapec model to the cosmological redshift of the cluster model, μ = −(z − z_cosmic)c, where the sign reflects our convention that positive velocities are toward the observer, and z_cosmic = 0.05.

We test three different models for the velocity distribution f(v). The first model for f (v), "Model 1," is a single-valued velocity distribution at the shift parameter μ:

$$\begin{eqnarray}&&f(v)=\delta (v-\mu )\quad {\rm{(Model\; 1)}}.\end{eqnarray} \tag{ 8 }$$

This corresponds to a bapec model with the velocity broadening parameter set to zero and frozen. This model serves as a test of whether the data are consistent with the assumption of no velocity broadening, even with Astro-H's improved spectral resolution. The second model, "Model 2," incorporates velocity broadening by thawing the σ parameter and fitting it along with the others:

$$\begin{eqnarray}&&f(v)=G(v;\mu ,{\sigma }^{2})\quad {\rm{(Model\; 2)}}\end{eqnarray} \tag{ 9 }$$

where G is a normalized Gaussian distribution. The third model, "Model 3," is a weighted sum of two "Model 1" components:

$$\begin{eqnarray}&&f(v)={w}_{1}\delta (v-{\mu }_{1})+{w}_{2}\delta (v-{\mu }_{2})\quad {\rm{(Model\; 3)}}.\end{eqnarray} \tag{ 10 }$$

The total Model 3 therefore represents a single-temperature and single-metallicity plasma that is a mixture of two single-valued velocity components (the redshift parameters) with different normalizations. In XSPEC, this is achieved by adding two bapec models, tying together the temperature and abundance parameters of each component and allowing the redshift and normalization parameters of each component to vary separately. In this model, the velocity broadening parameter of each component is frozen at zero. Each of our three models also incorporates a tbabs absorption model, where the N_H parameter is held fixed at the value of 2 × 10²⁰ cm⁻². All other parameters are free to vary, unless noted above.

We minimize the C-statistic (Cash 1979) to determine model parameter values. Though the C-statistic does not have utility as a goodness-of-fit test as the χ²-statistic does, is works properly with Poisson statistics and does not require rebinning the spectrum to approximate Gaussian statistics. This enables us to use the full spectral resolution of Astro-H in our model fits. The spectra are fit within a broad energy band of 0.3–10.0 keV.

We also must ensure that our observations are long enough so that our estimates of the line shift and width have reasonable statistical accuracy. Ota et al. (2015) showed that ≳200 counts in the He-like iron line complex are required to achieve 20% accuracy on the measurement of the turbulent velocity for values of σ ∼ 200 km s⁻¹. Table 1 shows the number of counts in this line complex for each of our spectra, all of which easily exceed this requirement. In real clusters, the metallicity in the core region will be nearly solar, instead of the 0.3 Z_⊙ assumed in this work, so this requirement will be fulfilled for even shorter exposure times in the centers of clusters.

It should also be noted that our procedure does not take into account other systematic uncertainties that will be important. These include the systematic uncertainty on the line shift due to uncertainty in the SXS gain stability, and the uncertainty on the line width due to uncertainty in the line-spread function. The gain uncertainty is of particular concern. Its required accuracy is 2 eV, with a goal of 1 eV (Mitsuda et al. 2014; Takahashi et al. 2014). These values would correspond to an accuracy on the line shift of Δv_sys ∼ 90 (45) km s⁻¹ at 6.7 keV. In what follows, we will see that this uncertainty will dominate the statistical uncertainty on the line shift. Since we are focused in this work on identifying the systematic effects arising from projection and non-Gaussian intrinsic line shapes, we refer the reader to Kitayama et al. (2014) for a thorough discussion of the instrumental and calibration uncertainties.

Another consideration that must be taken into account is the broadness of the SXS PSF, with a width of roughly ∼1'. Most of our cold front pointings are offset from the cluster center, which has a much higher flux, and inspection of Figures 1 though 6 shows that the velocity structure of the core is generally very different from that of the cold front regions. A number of photons will be scattered by the PSF from the central region into our offset pointings, biasing the line shift and width. To account for this effect, for each region we performed a second set of otherwise identical spectral simulations, except that we have turned off the effect of vignetting and artificially reduced the FWHM of the SXS PSF to 001, to compare to our uncorrected spectra. For spectral analysis of real observations, such an artifical reduction of the PSF scattering is obviously unavailable, so the core region and the cold front regions will have to be modeled concurrently. We show an example of such an analysis in Section 3.3.3.

The value of the reduced C-statistic for each model fit is tabulated in Table 2, for spectral simulations with and without PSF scattering.⁸ Figure 13 shows example spectra and fitted models for the "x1" region, within the E ∼ 6.2–6.8 keV band, which surrounds the Fe–K lines (at our redshift of z_cosmic = 0.05). We will describe each model in turn, referring to the figure as needed.

Zoom In Zoom Out Reset image size

From Table 2, we see that the fit to Model 1 is generally poor (except in the "z2" case). The upper left panel of Figure 13 shows the spectrum for region "x1" with its fitted Model 1, showing significant fit residuals around the Fe emission lines. For most of the fits to Model 1 for the various regions, we are able to recover the correct temperature, since it is strongly constrained by the continuum emission, but the metallicity is always underestimated by approximately 10%–15%. Taking the "x1" region as an example, we find that we can recover the expected temperature from the simulation of kT_sim = 6.18 keV, with kT_fit = ${6.23}_{-0.06}^{+0.05}$ keV, but the metallicity is underestimated, with Z_sim = 0.3 Z_⊙ and Z_fit = ${0.266}_{-0.009}^{+0.008}$ Z_⊙. We find similar results if we ignore PSF scattering. The fitted value of the velocity shift is strongly biased if PSF scattering is included, with ${\mu }_{\mathrm{fit}}=-{145}_{-12}^{+14}$ km s⁻¹, far away from the expected value of μ_sim = −221 km s⁻¹. Without PSF scattering, we find μ_fit = $-{238}_{-14}^{+13}$ km s⁻¹, a much more accurate value. This indicates that modeling of the bright core component will be necessary to accurately measure line shifts of cold fronts.

As a sanity check, if we turn off the effects of Doppler shifting and broadening from gas velocity when creating the synthetic spectrum, Model 1 produces a much better fit to the data than in the case where these effects were included (see the last row of Table 2 and the lower right panel of Figure 13) despite the fact that we are fitting a single-temperature model to a spectrum from gas with a range of temperatures. In this case, the metallicity is correctly recovered with ${Z}_{{\rm{fit}}}={0.292}_{-0.008}^{+0.009}$ Z_⊙. Since Model 1 provides a good fit to this spectrum, but not the broadened one, this confirms that Astro-H is sensitive enough to detect the effects of Doppler shifting and broadening on the emission lines from the subsonic sloshing motions.

Fitting the synthetic spectra with Model 2 results in a significant reduction in the fit statistic (except in the "z2" case, and in the case of "x2" only a modest improvement is achieved), as seen in the upper right panel of Figure 13. Table 3 shows the fitted parameters from Model 2, for spectra with PSF scattering (top rows) and those without (bottom rows). For nearly all the spectra, the fitted temperature T_fit agrees with the "real" temperature T_sim, with the exception of the "x2" region, where photons scattered from the core by the PSF bias the temperature upward. With PSF scattering turned off, we recover the correct temperature for "x2." The metallicity parameters are much closer to the correct value of Z_sim = 0.3 Z_⊙, though they are all still biased lower.

Table 3.  Model 2 Parameters and Simulation Values

Spectrum	Tfit	Tsim	Zfit	Zsim	μfit	μsim	σfit	σsim
	(keV)	(keV)	(Z⊙)	(Z⊙)	(km s−1)	(km s−1)	(km s−1)	(km s−1)
With PSF Scattering
x1	${6.20}_{-0.05}^{+0.05}$	6.18	${0.293}_{-0.009}^{+0.009}$	0.3	$-{156}_{-12}^{+12}$	−221	${203}_{-12}^{+12}$	197
x2	${5.41}_{-0.06}^{+0.07}$	5.21	${0.292}_{-0.012}^{+0.012}$	0.3	${156}_{-14}^{+13}$	206	${157}_{-14}^{+14}$	149
y1	${6.12}_{-0.04}^{+0.04}$	6.16	${0.296}_{-0.007}^{+0.007}$	0.3	${231}_{-14}^{+13}$	290	${291}_{-10}^{+12}$	254
z1	${6.35}_{-0.02}^{+0.03}$	6.40	${0.295}_{-0.004}^{+0.004}$	0.3	${15}_{-4}^{+4}$	16	${134}_{-4}^{+4}$	139
z2	${5.64}_{-0.07}^{+0.06}$	5.73	${0.292}_{-0.008}^{+0.014}$	0.3	$-{9}_{-11}^{+13}$	−10	${118}_{-15}^{+14}$	110
Without PSF Scattering
x1	${6.12}_{-0.05}^{+0.05}$	6.18	${0.297}_{-0.009}^{+0.009}$	0.3	$-{240}_{-13}^{+13}$	−221	${203}_{-12}^{+9}$	197
x2	${5.19}_{-0.06}^{+0.05}$	5.21	${0.299}_{-0.013}^{+0.011}$	0.3	${224}_{-13}^{+14}$	206	${147}_{-13}^{+17}$	149
y1	${6.10}_{-0.03}^{+0.05}$	6.16	${0.298}_{-0.008}^{+0.009}$	0.3	${322}_{-15}^{+14}$	290	${266}_{-9}^{+12}$	254
z1	${6.35}_{-0.03}^{+0.03}$	6.40	${0.295}_{-0.005}^{+0.004}$	0.3	${14}_{-3}^{+4}$	16	${133}_{-4}^{+4}$	139
z2	${5.63}_{-0.07}^{+0.09}$	5.73	${0.292}_{-0.011}^{+0.013}$	0.3	$-{6}_{-18}^{+9}$	−10	${122}_{-16}^{+15}$	110

Download table as:  ASCIITypeset image

With PSF scattering included in the simulated observation, the velocity shifts of the "x1," "x2," and "y1" regions, all within the sloshing plane, are biased in the direction of zero velocity from their expected values by ∼60–70 km s⁻¹, owing to the mostly unshifted photons from the bright core scattering into the FOV. The velocity shifts for the spectra without PSF scattering are much closer to the true values, but they still exhibit a small bias of ∼20–30 km s⁻¹ from the true line shift, in the direction toward zero velocity. In Section 3.3.2, we will see that this is due to the fact that the true velocity distributions are not strictly Gaussian. The effect of PSF scattering on the velocity broadening parameter σ is not quite as severe. The fitted line shift parameters for the "z1" and "z2" regions exhibit no bias due to PSF scattering, but this is expected, since the "z1" region is centered on the core itself, and the "z2" region has nearly the same line shift, owing to the symmetry of the sloshing motions. In all cases, however, the velocity bias from PSF scattering is comparable to or even less than that expected from instrumental uncertainties, if Δv_sys ∼ 45 (90) km s⁻¹ as noted above.

Fitting the spectra with Model 3 also provides a good fit in most cases (see the lower left panel of Figure 13 for the fit for region "x1"). The reduction in the C-statistic is approximately the same as in Model 2. Though it might not be expected that such a crude velocity distribution function (Equation (10)) would provide a good fit to complex velocity distributions, this is made possible because of the blending together of the individual line components of complexes (such as Fe near 7 keV in the rest frame) due to the finite spectral resolution of SXS (∼5 eV) and line broadening. Table 4 shows the fitted parameters from Model 3, with no PSF scattering. In general, the temperature parameters in each spectrum are correctly recovered, and the two velocity components μ₁ and μ₂ have comparable normalization, with a typical difference of ∼250–400 km s⁻¹. Though the model provides a good fit, its utility as a physical description of the velocity distribution is limited.

Table 4.  Model 3 Parameters and Simulation Values

Spectrum	Tfit	Tsim	Zfit	Zsim	μ1,fit	μ2,fit	w1,fit	w2,fit
	(keV)	(keV)	(Z⊙)	(Z⊙)	(km s−1)	(km s−1)
x1	${6.13}_{-0.05}^{+0.05}$	6.18	${0.292}_{-0.009}^{+0.009}$	0.3	$-{372}_{-28}^{+19}$	$-{7}_{-41}^{+31}$	${0.60}_{-0.05}^{+0.04}$	${0.40}_{-0.04}^{+0.05}$
x2	${5.19}_{-0.06}^{+0.05}$	5.21	${0.298}_{-0.013}^{+0.011}$	0.3	${318}_{-20}^{+22}$	${2}_{-43}^{+48}$	${0.68}_{-0.09}^{+0.06}$	${0.32}_{-0.06}^{+0.09}$
y1	${6.12}_{-0.03}^{+0.05}$	6.16	${0.290}_{-0.007}^{+0.008}$	0.3	${492}_{-15}^{+22}$	${37}_{-31}^{+37}$	${0.61}_{-0.03}^{+0.03}$	${0.39}_{-0.03}^{+0.03}$
z1	${6.35}_{-0.03}^{+0.03}$	6.40	${0.293}_{-0.005}^{+0.004}$	0.3	${153}_{-12}^{+18}$	$-{101}_{-12}^{+13}$	${0.47}_{-0.05}^{+0.03}$	${0.53}_{-0.03}^{+0.05}$
z2	${5.63}_{-0.07}^{+0.09}$	5.73	${0.291}_{-0.010}^{+0.013}$	0.3	$-{123}_{-51}^{+253}$	${87}_{-180}^{+40}$	${0.5}_{-0.16}^{+0.10}$	${0.50}_{-0.10}^{+0.16}$

Download table as:  ASCIITypeset image

How well do our models reproduce the actual velocity distributions within the regions from which the spectra are extracted? Figure 14 shows the predicted velocity distribution function f (v) from Model 2 (green curves) for our five regions, compared to the actual f (v) for the same regions (black curves). For the "x1," "x2," and "y1" regions, the velocity distributions are complex and exhibit deviations from Gaussianity, though the Model 2 fits do approximately capture the width of the distribution. This explains why the Model 2 line shifts from Table 3 (without PSF scattering) do not always agree precisely with the line shift measured from the simulation (though this disagreement is small compared to the expected systematic error due to uncertainty in the line-spread function and gain); the mean of the best-fit Gaussian component and the mean of the true distribution are not the same if the underlying distribution is non-Gaussian. The velocity distributions for the "z1" and "z2" regions are well fit by Gaussian distributions, and as a result both the line shift and width are correctly recovered by Model 2.

Zoom In Zoom Out Reset image size

Figure 14 indicates that the velocity distribution of cold fronts may be better modeled by multiple components. Shang & Oh (2012) used a mixing-model approach to fit velocity distribution models to Doppler-broadened spectral lines. They found that they were able to fit velocity distributions from a variety of simulated velocity fields (including those from core gas sloshing) using a sum of Gaussian models. They also found that for the Astro-H spatial and spectral resolution, typically only two Gaussian components can be constrained. Compared to Shang & Oh (2012), we are limited by statistics. For each of their spectra, they assumed 10⁴ counts in the single He-like iron line at 6.7 keV, corresponding to roughly a megasecond exposure of the entire SXS FOV pointed at the cluster core for a number of nearby clusters (see their Table 2). As noted above, our observations of 200 ks exposure typically have ∼10³ counts in the entire He-like iron line complex (see Table 1). Also, we extract spectra from smaller regions underneath cold front surfaces, which are fainter than the cluster core.

Though we do not employ a mixing-model approach, we can fit a sum of two Gaussian models by extending our Model 3 (Equation (10)); we thaw the velocity broadening parameters of the separate bapec components, allowing them to vary in the fit. Along with the temperature and metallicity parameters, we have six additional parameters: a normalization, line centroid, and line width for each Gaussian velocity component. In all cases, we are unable to constrain all of these parameters uniquely with our simulated spectra. However, we do find that we can constrain one velocity broadening parameter. This defines our "Model 4," a simple generalization of Model 3:

$$\begin{eqnarray}&&f(v)={w}_{1}\delta (v-{\mu }_{1})+{w}_{2}G(v;{\mu }_{2},{\sigma }_{2}^{2})\quad {\rm{(Model\; 4)}}.\end{eqnarray} \tag{ 11 }$$

Predicted velocity distributions from Model 4 fits to the "x2" and "y1" spectra are shown in Figure 14, as red curves with arrows showing the positions of the line shift parameter along with its 1σ error bar. We are only able to constrain parameters for these particular regions, as these cases have the clearest evidence for two well-separated velocity components. In these two cases, Model 4 resolves both components well.

To compare Model 3 to the actual velocity distribution, it is more instructive to examine the cumulative distribution function (CDF) of the velocity, F(v). For Model 3, the CDF of the velocity distribution is

$$\begin{eqnarray}&&F(v)={w}_{1}H(v-{\mu }_{1})+{w}_{2}H(v-{\mu }_{2})\quad {\rm{(Model\; 3)}}\end{eqnarray} \tag{ 12 }$$

where H is the Heaviside step function. Figure 15 shows F(v) for the five different regions, with the Model 3 prediction overlaid. From this figure, we see that Model 3 represents a crude, piecewise-constant approximation to the cumulative distribution of the velocity, which may provide a rough sense of the location and relative importance of the two most dominant velocity components, but in general a Gaussian velocity model or (if possible) a sum of two models is to be preferred, as they will provide a more accurate physical representation of the velocity field for a comparable number of parameters.

Zoom In Zoom Out Reset image size

As mentioned above, in SXS observations of cold fronts a non-negligible number of photons may be scattered into the FOV from the nearby core region. Therefore, to achieve accurate measurements of model parameters, it will be necessary to model the spectrum of a cold front region simultaneously with the spectrum of the cluster core. Examples of such analyses can be found in the Astro-H cluster white paper (Kitayama et al. 2014), where Chandra images were used in conjunction with simulated spectra to estimate the effects of PSF scattering. In this section, we perform such a joint analysis using the "x1" and "x2" regions as examples.

The left panel of Figure 16 shows maps of the X-ray surface brightness and the line-of-sight velocity, with the "x1" and "x2" regions chosen for spectral extraction (which are the same as before), with an additional "core" region that contains the bright cluster core. We then perform separate simulations including only the photons originating from this core region that are scattered into regions "x1" and "x2." We create spectra from each region that contain only photons originating in the core. We find that the photon flux from the core scattered into the "x1" and "x2" regions is about ∼20% of the total flux of each region. We performed a joint fit in XSPEC of the two spectra using a sum of two bapec models, where one component models the photons from the core region spectrum and the other models those from the cold front spectrum. The right panel of Figure 16 shows example spectra for the "x1" region and the fitted models. Table 5 shows the fitted parameters for the "x1" region resulting from the joint fit, where we only show the parameters for the cold front model component. The joint fit results in an improvement of the estimation of the line shift of the cold front component in both regions.

Zoom In Zoom Out Reset image size

Table 5.  Accounting for PSF Scattering: Joint Fit Parameters and Simulation Values

Spectrum	Tfit	Tsim	Zfit	Zsim	μfit	μsim	σfit	σsim
	(keV)	(keV)	(Z⊙)	(Z⊙)	(km s−1)	(km s−1)	(km s−1)	(km s−1)
x1	${6.22}_{-0.07}^{+0.07}$	6.18	${0.293}_{-0.013}^{+0.013}$	0.3	$-{192}_{-14}^{+17}$	−221	${196}_{-16}^{+15}$	197
x2	${5.30}_{-0.09}^{+0.06}$	5.21	${0.294}_{-0.017}^{+0.017}$	0.3	${176}_{-20}^{+20}$	206	${161}_{-22}^{+20}$	149

Download table as:  ASCIITypeset image

This effect may be slightly larger than our simulations indicate. In reality, clusters have central metallicities of ∼Z_⊙ or higher, and the metallicity decreases with radius. In our simulations we have assumed a spatially uniform Z = 0.3 Z_⊙. Depending on the difference in metallicity between the core and the cold front for a given cluster, the PSF-scattered flux from the core into the cold front region may be somewhat larger.