Galaxy Cluster Pressure Profiles as Determined by Sunyaev Zel'dovich Effect Observations with MUSTANG and Bolocam. II. Joint Analysis of 14 Clusters

Our cluster sample is based primarily on the Cluster Lensing And Supernova survey with Hubble (CLASH) sample (Postman et al. 2012). The CLASH sample has 25 massive galaxy clusters, 20 of which are selected from X-ray data (from Chandra X-ray Observatory, hereafter Chandra), and 5 are based on exceptional lensing strength. These clusters have the following properties: $0.187\lt z\lt 0.890$, $5.5\lt {k}_{B}T$ (keV) $\lt \,15.5$, and $6.7\times {10}^{44}\lt {L}_{\mathrm{bol}}$ (erg s⁻¹) $\lt \,90.8$. Thus, these clusters are large enough that we should expect to detect them with MUSTANG with a reasonable amount of time on the sky (on average, <25 hr per cluster).

Of the 25 clusters in the CLASH sample, 4 are too far south to be observed with MUSTANG from Green Bank, WV. Of the remaining 21, we were able to observe 14 given the available good weather and their limited visibility during the observational campaign from 2009 to 2014. Abell 209 was observed, but was relatively noisy and showed no trace of any detection. Our final sample includes 13 CLASH clusters. We also include Abell 1835, a cluster of similar mass and redshift as the CLASH clusters, which was observed under the program GBT/09A-052. These clusters (see Table 1 and Figure 1) were also observed with Bolocam, and have been analyzed in Sayers et al. (2012, 2013a) and Czakon et al. (2015). The Archive of Chandra Cluster Entropy Profile Tables (ACCEPT Cavagnolo et al. 2009) and Bolocam centroids are indicated in Figure 1 with red and blue asterisks, respectively, and their separations (${\rm{\Delta }}{r}_{X,\mathrm{SZ}}$) are also listed in Table 1. The total integration times of MUSTANG and Bolocam observations, along with detection significances, of our sample are listed in Table 2. Bolocam and MUSTANG significances, A10_B and A10_M, respectively, are taken as the significance of the fitted spherical A10 (Arnaud et al. 2010) profile (see Section 3.2.1) based on the amplitude of the fit (${P}_{0}/{\sigma }_{{P}_{0}}$) to the respective data set (separately). Aside from fixing the pressure profile shape, the fits are performed as described in Section 3, with relevant (point source and/or residual) components fit simultaneously. This calculation of cluster significance is better than a peak surface brightness measure because it incorporates signal, even if weak, within the entire fitted region. As this metric is intended to measure the strength of an overall cluster detection, negative values are permitted. Null detections with MUSTANG set upper limits on the slope of the inner pressure profile, which are stronger than those from Bolocam data.

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Table 1.  Cluster Properties

Cluster	z	M500	P500	R500	R500	Txa	Txb	Tmg	Dynamical	${\rm{\Delta }}{r}_{X,\mathrm{SZ}}$
		(${10}^{14}\,{M}_{\odot }$)	(keV cm−3)	(kpc)	(')	(keV)	(keV)	(keV)	state	('')
Abell 1835	0.253	12	0.00594	1490	6.30	9.0	10.0	7.49	CC	6.8
Abell 611	0.288	7.4	0.00445	1240	4.75	6.8	⋯	6.71	⋯	18.7
MACS 1115	0.355	8.6	0.00545	1280	4.28	9.2	9.14	7.04	CC	34.8
MACS 0429	0.399	5.8	0.00448	1100	3.41	8.3	8.55	5.56	CC	18.7
MACS 1206	0.439	19	0.01059	1610	4.73	10.7	11.4	10.0	⋯	6.9
MACS 0329	0.450	7.9	0.00596	1190	3.44	6.3	5.85	5.64	CC & D	14.8
RXJ1347	0.451	22	0.01171	1670	4.83	10.8	13.6	9.86	CC	9.6
MACS 1311	0.494	3.9	0.00399	930	2.56	6.0	6.36	5.18	CC	27.7
MACS 1423	0.543	6.6	0.00612	1090	2.85	6.9	6.81	5.50	CC	19.8
MACS 1149	0.544	19	0.01228	1530	4.01	8.5	8.76	7.70	D	6.0
MACS 0717	0.546	25	0.01490	1690	4.40	11.8	10.6	9.06	D	32.4
MACS 0647	0.591	11	0.00923	1260	3.17	11.5	12.6	8.06	⋯	6.9
MACS 0744	0.698	13	0.01199	1260	2.96	8.1	8.90	6.85	D	4.9
CLJ1226	0.888	7.8	0.01184	1000	2.15	12.0	11.7	11.3	⋯	15.3

Note. z, M₅₀₀, R₅₀₀, and T_X^a are taken from Mantz et al. (2010): T_X^a is calculated from a single spectrum over $0.15{R}_{500}\lt r\lt {R}_{500}$ for each cluster. T_X^b is from Morandi et al. (2015), and is calculated over $0.15{R}_{500}\lt r\lt 0.75{R}_{500}$. T_mg is a fitted gas mass weighted temperature, (Section 6.2) determined by fitting the ACCEPT2 (Baldi 2014) temperature profiles to the gas mass weighted profile found in Vikhlinin et al. (2006). The dynamical states: cool core (CC) and disturbed (D) are taken from (and defined in) Sayers et al. (2013a). ${\rm{\Delta }}{r}_{X,\mathrm{SZ}}$ denotes the offset between the ACCEPT and Bolocam centroids.

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Table 2.  Bolocam and MUSTANG Observational Properties

Cluster	z	R.A.	Decl.	${t}_{\mathrm{obs},B}$	NoiseB	A10B	${\tilde{\chi }}_{B}^{2}$	${t}_{\mathrm{obs},M}$	NoiseM	A10M	${\tilde{\chi }}_{M}^{2}$
		(J2000)	(J2000)	(hr)	$\mu K$ a	(${P}_{0}/{\sigma }_{{P}_{0}}$)		(hr)	μJy/bm	(${P}_{0}/{\sigma }_{{P}_{0}}$)
Abell 1835	0.253	14:01:01.9	+02:52:40	14.0	16.2	28.9	1.05	8.6	53.4	10.0	0.99
Abell 611	0.288	08:00:56.8	+36:03:26	18.7	25.0	13.9	0.97	12.0	46.2	1.73	1.03
MACS 1115	0.355	11:15:51.9	+01:29:55	15.7	22.8	16.3	1.08	10.0	56.4	8.66	1.04
MACS 0429	0.399	04:29:36.0	−02:53:06	17.0	24.1	13.2	1.05	11.6	47.2	−0.02	1.03
MACS 1206	0.439	12:06:12.3	−08:48:06	11.3	24.9	28.7	0.97	13.3	42.5	8.89	1.02
MACS 0329	0.450	03:29:41.5	−02:11:46	10.3	22.5	17.4	1.09	13.1	39.9	8.63	0.98
RXJ1347	0.451	13:47:30.8	−11:45:09	15.5	19.7	45.3	1.04	1.9	276.	8.90	0.98
MACS 1311	0.494	13:11:01.7	−03:10:40	14.2	22.5	11.3	1.06	10.6	64.5	0.71	1.00
MACS 1423	0.543	14:23:47.9	+24:04:43	21.7	22.3	11.8	0.88	11.2	35.7	6.15	1.00
MACS 1149	0.544	11:49:35.4	+22:24:04	17.7	24.0	22.0	0.99	13.9	32.7	−1.47	1.01
MACS 0717	0.546	07:17:32.1	+37:45:21	12.5	29.4	31.3	1.09	14.6	27.1	3.05	1.05
MACS 0647	0.591	06:47:49.7	+70:14:56	11.7	22.0	24.1	1.03	16.4	20.3	11.3	1.01
MACS 0744	0.698	07:44:52.3	+39:27:27	16.3	20.6	17.8	1.19	7.6	48.5	7.67	1.01
CLJ1226	0.888	12:26:57.9	+33:32:49	11.8	22.9	13.7	1.20	4.9	85.6	9.43	1.00

Note. Subscripts _B and _M denote Bolocam and MUSTANG properties respectively. Noise_B and ${t}_{\mathrm{obs},B}$ are those reported in Sayers et al. (2013a). Noise_M is calculated on MUSTANG maps with 10'' smoothing, in the central arcminute. t_obs are the integration times (on source) for the given instruments. A10_B and A10_M are the Bolocam and MUSTANG significances, respectively. The quality of the fits is respectable, as indicated by the ${\tilde{\chi }}^{2}$ values being close to 1.

^a $\mu K$ is more precisely $\mu {K}_{\mathrm{CMB}}$-amin.

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MUSTANG is a 64 pixel array of Transition Edge Sensor (TES) bolometers arranged in an 8 × 8 array located at the Gregorian focus on the 100 m GBT. Operating at 90 GHz (81–99 GHz), MUSTANG has an angular resolution of 9'' and pixel spacing of $0.63f\lambda$ resulting in an FOV of 42''. More detailed information about the instrument can be found in Dicker et al. (2008).

Our observations and data reduction are described in detail in Romero et al. (2015), and we briefly review them here. Absolute flux calibrations are based on the planets Mars, Uranus, and Saturn, or the star Betelgeuse (${\alpha }_{\mathrm{Ori}}$). At least one of these flux calibrators was observed at least once per night, and we find that our calibration is accurate to a 10% rms uncertainty. We also observe bright point sources every half hour to track our pointing and beam shape. To observe the target galaxy clusters, we employ Lissajous daisy scans with a $3^{\prime}$ radius and in many of the clusters we broadened our coverage with a hexagonal pattern of daisy centers (with $1^{\prime}$ offsets). For most clusters, the coverage (weight) drops to 50% of its peak value at a radius of 13.

Processing of MUSTANG data is performed using a custom IDL pipeline. Raw data is recorded as time ordered data (TOD) from each of the 64 detectors. An outline of the data processing for each scan on a galaxy cluster is as follows.

• 1.  
  We define a pixel mask from the nearest preceding CAL scan; unresponsive detectors are masked out. The CAL scan provides us with unique gains to be applied to each of the responsive detectors.
• 2.  
  A common mode template, polynomial, and sinusoid are fit to the data and then subtracted. The common mode is calculated as the arithmetic mean of the TOD across detectors.
• 3.  
  After the common mode and polynomial subtraction, each scan is subjected to spike (glitch), skewness, and Allan variance tests and are flagged according to the following criteria. Glitches are flagged as $4\sigma$ excursions based on the median absolute deviation; the skewness threshold for flagging is 0.4. Flags based on Allan variance require the variance over a two second interval to be greater than nine times the variance between each integration. Typical scan integration times were 150 s.
• 4.  
  Individual detector weights are calculated as $1/{\sigma }_{i}^{2}$, where ${\sigma }_{i}$ is the rms of the non-flagged TOD for that detector.
• 5.  
  Maps are produced by gridding the TOD in 1'' pixels in right ascension (R.A.) and declination (decl). A weight map is produced in addition to the signal map.

The effect of the MUSTANG data processing results in the transfer function shown in Figure 2. Specifically, it is the average across our sample. This transfer function is very stable because little scatter is seen across our sample.

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Bolocam is a 144-element camera that was a facility instrument on the Caltech Submillimeter Observatory (CSO) from 2003 until 2012. Its field of view is 8' in diameter, and at 140 GHz it has a resolution of 58'' FWHM (Glenn et al. 1998; Haig et al. 2004). The clusters were observed with a Lissajous pattern that results in a tapered coverage dropping to 50% of the peak value at a radius of roughly 5', and to 0 at a radius of 10'. The Bolocam maps used in this analysis are $14^{\prime} \times 14^{\prime}$. The Bolocam data¹⁵ are the same as those used in Czakon et al. (2015) and Sayers et al. (2013a); the details of the reduction are given therein, along with Sayers et al. (2011). The reduction and calibration is similar to that used for MUSTANG, and Bolocam achieves a 5% calibration accuracy and 5'' pointing accuracy.

The joint map fitting technique used in this paper is described in detail in Romero et al. (2015). We review it briefly here. The general approach follows that of a least squares fitting procedure, which assumes that we can make a model map as a linear combination of model components.

This linear combination can be written as

$$\begin{eqnarray}&&{{\boldsymbol{d}}}_{\mathrm{mod}}={{\boldsymbol{Aa}}}_{\mathrm{mod}},\end{eqnarray} \tag{ 1 }$$

where d_mod is the total model, each column in ${\boldsymbol{A}}$ is a filtered model component (Section 3.2), and ${{\boldsymbol{a}}}_{\mathrm{mod}}$ is an array of amplitudes of the components. There are up to four types of components for which we fit: a bulk component, point-source(s), residual component(s), and a mean level. From these, we produce a sky model for the bulk component and point source to be filtered. The residual component is calculated directly as a filtered component.

We wish to fit ${{\boldsymbol{d}}}_{\mathrm{mod}}$ to our data, ${\boldsymbol{d}}$, and allow for a calibration offset between Bolocam and MUSTANG data. To accomplish this, we define our data vector as

$$\begin{eqnarray}&&{\boldsymbol{d}}=[{{\boldsymbol{d}}}_{B},k{{\boldsymbol{d}}}_{M},k],\end{eqnarray} \tag{ 2 }$$

where ${{\boldsymbol{d}}}_{B}$ is the Bolocam data, taken as the provided map (14' sides), ${{\boldsymbol{d}}}_{M}$ is the MUSTANG data, taken as the inner (radial) arcminute of MUSTANG maps. k is the calibration offset of MUSTANG relative to Bolocam, to which we apply an 11.2% Gaussian prior derived from the MUSTANG and Bolocam calibration uncertainties.

We use the ${\chi }^{2}$ statistic as our goodness of fit:

$$\begin{eqnarray}&&{\chi }^{2}={(\overrightarrow{d}-{\overrightarrow{d}}_{\mathrm{mod}})}^{T}{{\boldsymbol{N}}}^{-1}(\overrightarrow{d}-{\overrightarrow{d}}_{\mathrm{mod}}),\end{eqnarray} \tag{ 3 }$$

where ${\boldsymbol{N}}$ is the covariance matrix; however, because we wish to fit for k in addition to the amplitude of model components, we no longer have completely linearly independent variables, and thus we employ MPFIT (Markwardt 2009) to solve for these variables. Confidence intervals are derived from ${\chi }^{2}$ values over the parameter space searched (Section 3.3), and adjusted based on Monte Carlo simulations (Romero et al. 2015).

Our approach is to fix the shape and position of point sources and residuals (if any), fitting only their amplitudes. We explore the shape of the bulk ICM component parametrically, where each point in the parametric space may be forward modeled (Section 3.3). At each point in the parameter space, we do a linear least squares fit followed by a nonlinear minimization over k, the Bolocam pointing, and ${{\boldsymbol{a}}}_{\mathrm{mod}}$.

In order to produce component maps, it is necessary to account for the response of both instruments and imaging pipeline filter functions. For Bolocam, we use the transfer function provided. For MUSTANG, we perform simulated observations, processing the sky models in the same manner that real data is processed.

3.2.1. Bulk ICM

As in Romero et al. (2015), the bulk component is taken to be a spherically symmetric 3D electron pressure profile as parameterized by a generalized Navarro, Frenk, and White profile (hereafter, gNFW Navarro et al. 1997; Nagai et al. 2007):

$$\begin{eqnarray}&&\tilde{P}=\displaystyle \frac{{P}_{0}}{{({C}_{500}X)}^{\gamma }{[1+{({C}_{500}X)}^{\alpha }]}^{(\beta -\gamma )/\alpha }}\end{eqnarray} \tag{ 4 }$$

where $X=R/{R}_{500}$, and C₅₀₀ is the concentration parameter; one can also write (${C}_{500}X$) as ($R/{R}_{s}$), where ${R}_{s}={R}_{500}/{C}_{500}$. $\tilde{P}$ is the electron pressure in units of the characteristic pressure P₅₀₀. This pressure profile is integrated along the line of sight to produce a Compton y profile, given as

$$\begin{eqnarray}&&y(r)=\displaystyle \frac{{P}_{500}{\sigma }_{T}}{{m}_{e}{c}^{2}}{\int }_{-\infty }^{\infty }\tilde{P}(r,l){dl}\end{eqnarray} \tag{ 5 }$$

where ${R}^{2}={r}^{2}+{l}^{2}$, r is the projected radius, and l is the distance from the center of the cluster along the line of sight. Once integrated, y(r) is gridded as $y(\theta )$ and is realized as two maps with the same astrometry as the MUSTANG and Bolocam data maps (pixels of 1'' and 20'' on a side, respectively). In each case, we convolve the Compton y map by the appropriate beam shape. For Bolocam, we use a Gaussian with FWHM = 58'', and for MUSTANG we use the double Gaussian, representing the GBT main beam and stable error beam (Romero et al. 2015). Subsequently, we account for the filtering effects of data processing for each instrument, as described in Romero et al. (2015).

3.2.2. Point Sources

3.2.3. Residual Components

3.2.4. Mean Level

As in Romero et al. (2015), we fix MUSTANG's centroid, but allow Bolocam's pointing to vary by ±10'' in R.A. and decl. with a prior on Bolocam's radial pointing accuracy with an rms uncertainty of 5''. Our approach to find the absolute calibration offset between Bolocam and MUSTANG is the same as in Romero et al. (2015; see also Section 3.1).

In Romero et al. (2015), we performed a grid search over γ and C₅₀₀, marginalizing over P₀, where α and β are fixed to values determined from A10. To determine the impact of our choice of fixed α and β, we explored how the profile shapes change when different, fixed, values of α and β are adopted. In all cases, we find that the pressure profile shapes are in very good agreement with one another and that the differences in ${\chi }^{2}$ values are statistically consistent. Thus, our fits are not sensitive to the exact choice of α and β.

We adopt R₅₀₀ from Mantz et al. (2010) and we search over $0\lt \gamma \lt 1.3$ in steps of $\delta \gamma =0.1$, and over $0.1\lt {C}_{500}\lt 3.3$ in steps of $\delta {C}_{500}=0.1$. This choice of parameter space searched is determined by computation requirements (largely in filtering maps) and covering a sufficient range of values. Our choice to limit $\gamma \geqslant 0$ is motivated by its implications to hydrostatic equilibrium under thermal pressure support. We revisit this choice in Section 4.1. To create models in finer steps than $\delta \gamma$ and $\delta {C}_{500}$, we interpolate filtered model maps from nearest neighbors from the grid of original filtered models.

All of the gNFW parameters (P₀, C₅₀₀, α, β, and γ) have some degeneracy with each other. C₅₀₀ relates the scaling radius, R_s, which is directly constrained by the SZ data, to R₅₀₀ as ${R}_{s}={C}_{500}{R}_{500}$. Because we take R₅₀₀, α, and β from A10, which used X-ray data and numerical simulations to derive their values of α and β, the values constrained by our SZ data in this analysis are not entirely independent of X-ray data. However, given the insensitivity to α and β found in Romero et al. (2015), and the independent nature of R_s, the profile shapes themselves should be considered approximately independent from X-ray data, if not the constrained shape parameter values as well.

3.3.1. Centroid Choice

Our goodness of fits are tabulated as reduced ${\chi }^{2}$ in Table 5. The residual MUSTANG and Bolocam maps indicate that a spherically symmetric gNFW pressure profile provides an adequate description of the data. In several residual MUSTANG maps, especially for those clusters with ${\tilde{\chi }}^{2}\gt 1.02$, some $3\sigma$ features remain (within the fitted region). MUSTANG residuals in MACS 1115 and MACS 0717, which have significances beyond $4\sigma$, are well away from the X-ray cluster centroid and nearly outside of the fitted region. Thus, these residuals will not impact the fitted cluster profiles, but can still elevate the overall ${\tilde{\chi }}^{2}$.

Table 5.  Summary of Fitted Pressure Profiles

Cluster	R500a	${Y}_{\mathrm{cyl}}({R}_{500})$	${Y}_{\mathrm{sph}}({R}_{500})$	${10}^{3}\,{P}_{500}$ a	P0	C500	α	β	γ	k	${\tilde{\chi }}^{2}$	dof
	(Mpc)	(10−5 Mpc2)	(10−5 Mpc2)	keV cm−3
Abell 1835	1.49	${26.75}_{-6.15}^{+6.05}$	${21.81}_{-4.49}^{+4.12}$	5.94	2.15 ± 0.07	${0.77}_{-0.17}^{+0.23}$	1.05	5.49	${0.78}_{-0.13}^{+0.12}$	1.08	0.99	12880
Abell 611	1.24	${9.67}_{-2.57}^{+4.85}$	${8.73}_{-2.21}^{+3.68}$	4.45	35.43 ± 2.46	${2.00}_{-0.30}^{+0.40}$	1.05	5.49	${0.00}^{+0.15}$	0.96	1.02	12882
MACS 1115	1.28	${30.28}_{-6.30}^{+7.32}$	${20.10}_{-3.52}^{+3.84}$	5.45	0.67 ± 0.04	${0.35}_{-0.10}^{+0.15}$	1.05	5.49	${0.87}_{-0.27}^{+0.18}$	1.11	1.04	12875
MACS 0429	1.10	${30.41}_{-6.88}^{+7.72}$	${19.57}_{-3.74}^{+4.00}$	4.48	11.01 ± 0.77	${0.59}_{-0.09}^{+0.11}$	1.05	5.49	${0.00}^{+0.15}$	1.00	1.03	12875
MACS 1206	1.61	${61.52}_{-12.63}^{+12.49}$	${48.16}_{-8.27}^{+8.19}$	10.59	2.39 ± 0.10	${0.74}_{-0.14}^{+0.16}$	1.05	5.49	${0.51}_{-0.16}^{+0.14}$	1.09	1.01	12874
MACS 0329	1.19	${13.38}_{-2.99}^{+3.83}$	${11.86}_{-2.37}^{+2.93}$	5.93	9.30 ± 0.50	${1.18}_{-0.28}^{+0.72}$	1.05	5.49	${0.41}_{-0.41}^{+0.19}$	1.03	0.99	12876
RXJ1347	1.67	${42.47}_{-6.81}^{+8.29}$	${37.80}_{-5.11}^{+5.78}$	11.71	3.24 ± 0.08	${1.18}_{-0.48}^{+1.02}$	1.05	5.49	${0.80}_{-0.70}^{+0.30}$	1.15	0.99	12880
MACS 1311	0.93	${17.18}_{-3.49}^{+3.80}$	${10.08}_{-1.73}^{+1.79}$	3.99	2.75 ± 0.22	${0.35}_{-0.05}^{+0.15}$	1.05	5.49	${0.41}_{-0.41}^{+0.34}$	0.98	1.00	12881
MACS 1423	1.09	${10.35}_{-2.73}^{+4.00}$	${8.89}_{-2.07}^{+2.53}$	6.12	22.39 ± 1.71	${1.58}_{-0.48}^{+0.22}$	1.05	5.49	${0.00}^{+0.35}$	1.04	0.98	12876
MACS 1149	1.53	${56.87}_{-9.00}^{+8.04}$	${41.62}_{-5.67}^{+4.99}$	12.28	5.50 ± 0.25	${0.83}_{-0.03}^{+0.07}$	1.05	5.49	${0.00}^{+0.05}$	0.87	1.00	12876
MACS 0717	1.69	${54.16}_{-8.72}^{+9.56}$	${48.06}_{-6.95}^{+7.71}$	14.90	21.88 ± 0.68	${2.00}_{-0.20}^{+0.20}$	1.05	5.49	${0.00}^{+0.05}$	0.49	1.03	13583
MACS 0647	1.26	${34.06}_{-7.76}^{+10.21}$	${26.33}_{-4.72}^{+5.37}$	9.23	2.78 ± 0.11	${0.70}_{-0.20}^{+0.30}$	1.05	5.49	${0.60}_{-0.20}^{+0.15}$	1.14	1.01	12876
MACS 0744	1.26	${15.10}_{-3.01}^{+4.50}$	${13.20}_{-2.29}^{+3.18}$	11.99	13.15 ± 0.81	${1.71}_{-0.21}^{+0.29}$	1.05	5.49	${0.00}^{+0.15}$	0.90	1.02	12875
CLJ1226	1.00	${10.50}_{-1.94}^{+2.65}$	${9.46}_{-1.60}^{+2.03}$	11.84	19.29 ± 1.25	${1.90}_{-0.50}^{+0.60}$	1.05	5.49	${0.29}_{-0.29}^{+0.36}$	0.92	1.03	12875
All	⋯	⋯	⋯	⋯	8.58 ± 2.37	${1.3}_{-0.1}^{+0.1}$	1.05	5.49	${0.3}_{-0.1}^{+0.1}$	⋯	⋯	⋯
Cool Core	⋯	⋯	⋯	⋯	3.55 ± 1.53	${0.9}_{-0.1}^{+0.1}$	1.05	5.49	${0.6}_{-0.1}^{+0.1}$	⋯	⋯	⋯
Disturbed	⋯	⋯	⋯	⋯	13.81 ± 1.55	${1.6}_{-0.1}^{+0.1}$	1.05	5.49	${0.0}^{+0.1}$	⋯	⋯	⋯
All (A10)	⋯	⋯	⋯	⋯	$8.403{h}_{70}^{-3/2}$	1.18	1.05	5.49	0.31	⋯	⋯	⋯
Cool core (A10)	⋯	⋯	⋯	⋯	$3.249{h}_{70}^{-3/2}$	1.13	1.22	5.49	0.78	⋯	⋯	⋯
Disturbed (A10)	⋯	⋯	⋯	⋯	$3.202{h}_{70}^{-3/2}$	1.08	1.41	5.49	0.38	⋯	⋯	⋯

Note. Results from our pressure profile analysis. Y_sph is calculated using the tabulated value of R₅₀₀. We have assumed A10 values of α and β. The findings from A10 are reproduced in the last three rows. The h₇₀ dependence is included for explicit replication of A10 results; all P₀ values have this dependence (the assumed cosmologies are the same).

^aValues of R₅₀₀ and P₅₀₀ are taken from Mantz et al. (2010) and Sayers et al. (2013a) respectively.

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Another potential source of noise worth considering is the primary CMB anisotropies. Bolocam accounts for CMB anisotropies in their noise model by adding astronomical sky realizations based on Keisler et al. (2011) and Reichardt et al. (2012) SPT measurements (Sayers et al. 2013a). In the MUSTANG data, we are not concerned with the primary CMB anisotropies because these are negligible beyond ${\ell }\gtrsim 6000$ ($\theta \lesssim 2$') (George et al. 2015). Given the MUSTANG transfer function, we estimate that the expected CMB contamination will fall below 4 μJy/beam, which is well below our noise level, and therefore negligible.

Our pressure profile fits do not change significantly between the chosen Bolocam or X-ray centroids. As seen in Figure 9, MACS 0647 is the only cluster to show significant residuals near the centroid, indicative of a centroid offset. However, given MUSTANG's sensitivity to substructure and potential degeneracy, it is possible that this apparent centroid offset could be due to substructure that is not well separated from the cluster core. We note that the reduced ${\chi }^{2}$ (Table 5) indicates that MACS 0647 is still well fit.

3.4.1. Impact of MUSTANG Mean Level on the Pressure Profile

3.4.2. Impact of Potential Substructure on the Pressure Profile

The impact of residual components and point sources is heterogeneous given the varying relevancy of these components. The residual maps and ${\chi }^{2}$ suggest that the point-source components are appropriate models for the sources we see in our clusters. MUSTANG maps, removing the fitted point sources, are shown in Figure 3. In Section 4.1, we revisit the impact of point sources on the pressure profile. The residual components generally appear to be sufficient, despite the simplicity of a 2D Gaussian. In the case of MACS 0717, the structure is not well modeled by a 2D Gaussian, but its modeling is minimally impactful because it is sufficiently far from the center. In contrast, if a residual component is not fit in RXJ 1347 and MACS 0744, we find a maximum of 20% and 100% increase in the pressure profile, occurring toward the center (at 45 radius: MUSTANG's half width at half maximum, HWHM). This increase drops to 3% and 60% at 240'' (half of Bolocam's FOV). These two clusters exhibit this strong dependence on the treatment of substructure due to the substructures' proximity to the core, where azimuthal averaging does not dilute the signal. In the other clusters, MACS 1206 and MACS 0717, the omission of a residual component results in a difference in fitted pressure profiles by less the 10%. Residual MUSTANG maps, i.e., maps with all components (including residual components) subtracted, are shown in Figure 9.

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In 6 of our 14 clusters, we find best-fit pressure profiles with $\gamma =0$, the limit we impose as a prior. There is no clear segregation based on dynamical state or presence of the central point source. Here, we consider two effects that could spuriously bias the cluster central pressures: our choice of centroid and the mis-subtraction of a central point source.

As it stands, finding slopes in the cores of galaxy clusters that are fit with $\gamma =0$ is not unprecedented; A10 find 6 of their 31 analyzed clusters in the REXCESS sample have $\gamma =0$, where all gNFW parameters except β were fit for individual clusters. They find a similar range in C₅₀₀ as we do. They fit for α, which is fit by the range $0.3\lt \alpha \lt 2.5$ over their sample. While A10 is a local ($z\lt 0.2$) sample, Mantz et al. (2016) find $\gamma =-0.01$, using Chandra data, for their sample of 40 galaxy clusters of $0.07\lt z\lt 1.10$. Moreover, in an analysis of X-ray (Chandra) data from observations of 80 clusters, McDonald et al. (2014) find $\gamma =0$ for their low redshift ($0.3\lt z\lt 0.6$), non-cool core clusters, and similarly shallow inner pressure profile slope ($\gamma =0.05$) for the high-redshift ($0.6\lt z\lt 1.2$) non-cool core clusters. Although these previous studies indicate that $\gamma =0$ is relatively common, we explore whether systematics related to either our data or analysis methods may produce these results in our fits.

Shallow slopes in the cores of clusters could be suggestive of a centroid offset either between MUSTANG and Bolocam or between SZ and X-ray data. Given the MUSTANG and Bolocam pointing accuracies (2'' and 5'', respectively), it is unlikely that the centroid offsets between MUSTANG and Bolocam are driving the fits to shallow slopes. The difference between SZ (Bolocam) centroids and ACCEPT centroids (Table 1) are large relative to pointing accuracies and thus potentially more important. However, when we adopt Bolocam's centroid, we find negligible change to the SZ pressure profile as compared to adopting the ACCEPT centroid.

Individually, the Bolocam and MUSTANG data sets yield consistent fits with each other, where changes in best-fit parameters generally occur along the shallow gradient in confidence intervals (i.e., along the degeneracy between C₅₀₀, and γ).

Additionally, we consider the impact of the assumed flux densities of point sources in the Bolocam maps. There are four clusters (Abell 1835, MACS 0429, RXJ 1347, and MACS 1423) where it was necessary to extrapolate a 140 GHz flux density from lower frequency measurements in order to analyze the Bolocam data (Sayers et al. 2013a). For point sources other than those in MACS 0429 and RXJ 1347, their uncertainty is less than the noise in the Bolocam maps. Moreover, in all but MACS 0429, the point-source uncertainty is considerably less than the peak Bolocam decrement and mis-estimations of the point-source flux densities in these clusters will not significantly change our results. Therefore, we are left with only MACS 0429, where we believe that the treatment of the point source may affect our results non-trivially.

If we utilize the same low-frequency point-source flux densities in Sayers et al. (2013b) and add in the MUSTANG data, we can recalculate the expected flux densities of point sources at 140 GHz, still assuming one power law. We find that the current Bolocam estimates, with reported uncertainties, are within $1\sigma$ of this recalculated value, except for MACS 1423, whose current value falls $1.3\sigma$ below the recalculated expectation. This does not address the potential for a break in the power law, which appears to be the case for RXJ 1347.

In RXJ 1347, flux densities of ${S}_{86}=4.16\pm 0.03\pm 0.25$ mJy and ${S}_{98}=3.96\pm 0.03\pm 0.24$ mJy have been reported from ALMA (Kitayama et al. 2016) and ${S}_{86}=4.9\pm 0.1$ mJy from CARMA (Plagge et al. 2013). The flux densities reported in Kitayama et al. (2016) come from a baseline cutoff to separate the point source from signal beyond roughly 5''. Additionally, we note that Adam et al. (2014) used an extrapolated flux of ${S}_{140}=4.4\pm 0.3$, deduced from the power law shown in Pointecouteau et al. (2001), which was calculated from data between 1.4 GHz and 300 GHz.

Only two (MACS 0429 and MACS 1423) of these four clusters are fit by notably low γ values. In addition, for the remaining four clusters in which MUSTANG detects a point source, the Bolocam maps assume no point-source contamination. Of these remaining clusters, only MACS 0717 is fit by a notably low γ, and that is best attributed to the dynamics of the cluster (Appendix B.11).

We also consider that our treatment of point sources in the MUSTANG maps may leave residual emission from point sources, either due to our fitting procedure or the assumption that our assumed point source has a non-trivial extent. Our point-source treatment was designed and extensively tested (e.g., Romero et al. 2015) to accurately remove point sources. In the case of Abell 1835, Romero et al. (2015) find good agreement between the MUSTANG and ALMA (McNamara et al. 2014) point-source flux density.

The geometry of a cluster along the line of sight can be calculated by comparing SZ and X-ray pressure profiles. If we assume azimuthal symmetry in the plane of the sky with scale radius ${\theta }_{\mathrm{proj}}$ and a scale radius along the line of sight of ${\theta }_{\mathrm{los}}$, then we denote the elongation/compression along the line of sight with an axis ratio of $c={\theta }_{\mathrm{los}}/{\theta }_{\mathrm{proj}}$, where $c\gt 1$ implies that the cluster is longer along the line of sight than in the plane of the sky. The X-ray surface brightness is proportional to $\int {n}_{e}^{2}{\rm{\Lambda }}{(T,Z){dl}\propto \int (P/T)}^{2}{\rm{\Lambda }}(T,Z)\,{dl}$, where Z is the abundance of heavy elements and Λ is the X-ray cooling function, while the SZ signal is proportional to $\int {Pdl}$ (Equation (5)). The temperature T can be derived from X-ray. Initially, we will assume that the cluster is spherically symmetric, and derive the pressure profile from the X-ray observations (giving P_X), and from the SZ observations (giving P_SZ). If the pressure profiles disagree, one explanation would be the elongation of the cluster along the line of sight. In this case, the elongation is given by

$$\begin{eqnarray}&&c={({P}_{\mathrm{SZ}}/{P}_{X})}^{2}.\end{eqnarray} \tag{ 9 }$$

To estimate the ellipticity of clusters, we wish to compare the amplitudes, as fit to X-ray and SZ data, of a given pressure profile shape per cluster. Thus, we fit the ACCEPT2 pressure profiles with a gNFW pressure profile, with α and β fixed at their A10 values: 1.05 and 5.49, respectively. The resultant gNFW profile is then integrated along the line of sight (LOS) to create a Compton y map, and then filtered as discussed in Romero et al. (2015). We refer to this filtered map, per cluster, as the "ACCEPT2 model". Allowing the amplitude to vary, we take P_SZ as the amplitude (renormalization) of this ACCEPT2 model when fit to the SZ data, whereby we have effectively set P_X to 1 in Equation (9). Similarly, we define P_B as the fitted amplitude (renormalization) of the ACCEPT2 model to just Bolocam data.

The axis ratio is calculated as $c={P}_{\mathrm{SZ}}^{2}$, and its associated uncertainty is calculated as ${\sigma }_{c}^{2}=4{P}_{\mathrm{SZ}}^{4}(({\sigma }_{\mathrm{SZ}}/{P}_{\mathrm{SZ}}{)}_{\mathrm{tot}}^{2}\ +({\sigma }_{X}/{P}_{X})2)$, where $({\sigma }_{\mathrm{SZ}}/{P}_{\mathrm{SZ}}{)}_{\mathrm{tot}}^{2}=({\sigma }_{\mathrm{SZ}}/{P}_{\mathrm{SZ}}{)}_{{st}.}^{2}+{0.11}^{2}$ includes the total statistical and calibration uncertainties of Bolocam and $({\sigma }_{X}/{P}_{X})=0.10$ is the calibration uncertainty of ACCEPT2. Table 6 presents relevant fitted gNFW parameters used in calculating the cluster geometry.

Table 6.  ACCEPT2 gNFW Fitted Parameters and Comparison to SZ data

Cluster	P0	C500	γ	PSZ	k	PB	c	${\sigma }_{c}$	${\rm{\Delta }}{P}_{\mathrm{SZ},B}/{\sigma }_{{P}_{\mathrm{SZ}}}$
Abell 1835	10.7	1.4	0.44	0.83 ± 0.03	1.15	0.82 ± 0.03	0.69	0.16	0.48
Abell 611	3.3	0.9	0.62	1.31 ± 0.09	0.92	1.35 ± 0.10	1.71	0.45	0.44
MACS 1115	13.7	1.5	0.35	0.84 ± 0.05	1.14	0.80 ± 0.05	0.70	0.17	0.71
MACS 0429	3.8	1.0	0.71	1.30 ± 0.11	0.64	1.48 ± 0.11	1.70	0.47	1.56
MACS 1206	3.7	1.0	0.49	1.12 ± 0.04	1.01	1.11 ± 0.04	1.24	0.29	−0.03
MACS 0329	4.7	1.2	0.59	1.61 ± 0.09	0.90	1.64 ± 0.09	2.58	0.64	0.41
RXJ 1347	22.8	2.4	0.40	0.95 ± 0.02	1.18	0.94 ± 0.02	0.89	0.20	0.37
MACS 1311	19.2	1.6	0.26	1.28 ± 0.12	0.85	1.40 ± 0.12	1.64	0.47	0.96
MACS 1423	11.2	1.8	0.51	1.26 ± 0.11	0.81	1.39 ± 0.12	1.58	0.44	1.12
MACS 1149	3.3	0.9	0.23	1.24 ± 0.06	0.70	1.28 ± 0.06	1.54	0.37	0.77
MACS 0717	10.2	1.5	0.00	1.36 ± 0.04	0.71	1.39 ± 0.04	1.85	0.43	0.54
MACS 0647	3.6	0.9	0.54	1.29 ± 0.05	1.09	1.27 ± 0.05	1.67	0.40	−0.38
MACS 0744	0.6	0.6	0.93	1.04 ± 0.06	0.94	1.05 ± 0.06	1.08	0.27	0.24
CLJ 1226	20.6	1.3	0.04	0.64 ± 0.04	1.15	0.60 ± 0.04	0.41	0.11	0.97

Note. P₀, C₅₀₀, and γ as determined by fitting the ACCEPT2 pressure profiles. P_SZ denotes the fitted amplitude (renormalization) of the ACCEPT2 model to the SZ data. P_B denotes the fitted amplitude (renormalization) of the ACCEPT2 model to just Bolocam data. We fix the gNFW parameters $\alpha =1.05$ and $\beta =5.49$. The elongation c is the ratio between the scale radius along the line of sight and the projected scale radius (taken to be azimuthally symmetric in the plane of the sky). Positive values in the column ${\rm{\Delta }}{P}_{\mathrm{SZ},B}/{\sigma }_{{P}_{\mathrm{SZ}}}$ indicate that the core is more spherical than the extended cluster.

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Table 7.  Normalized Gas Mass Weighted Temperatures

Cluster	${T}_{\mathrm{mg},\mathrm{SZ}}$	${\chi }^{2}$	DOF	${T}_{\mathrm{mg},X}$
Abell 1835	7.29 ± 0.17	29.70	9	7.49
Abell 611	8.56 ± 0.31	2.65	7	6.71
MACS 1115	6.35 ± 0.19	31.55	7	7.04
MACS 0429	3.71 ± 0.25	96.30	5	5.56
MACS 1206	8.44 ± 0.22	31.10	7	10.00
MACS 0329	8.52 ± 0.27	15.67	7	5.64
RXJ 1347	10.73 ± 0.16	14.32	9	9.86
MACS 1311	4.68 ± 0.24	76.77	6	5.18
MACS 1423	6.77 ± 0.27	35.39	6	5.50
MACS 1149	9.04 ± 0.26	30.10	7	7.70
MACS 0717	11.35 ± 0.27	212.62	9	9.06
MACS 0647	10.00 ± 0.28	7.91	6	8.06
MACS 0744	7.30 ± 0.26	5.70	7	6.85
CLJ 1226	8.78 ± 0.39	13.27	4	11.30

Note. The gas mass-weighted temperature proxies, ${T}_{\mathrm{mg},\mathrm{SZ}}$ and ${T}_{\mathrm{mg},X}$, are calculated by fitting a fixed profile shape, T_mg (Equation (11)), to T_SZ and T_X. The ${\chi }^{2}$ and degrees of freedom for the ${T}_{\mathrm{mg},\mathrm{SZ}}$ fits are tabulated. We find that ${T}_{\mathrm{mg},\mathrm{SZ}}$ is generally larger than ${T}_{\mathrm{mg},X}$.

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This investigation has made the assumption that the geometry of a given cluster is globally consistent. That is, one ellipsoidal geometry applies to all regions of the cluster. However, a cluster should appear more spherical toward the center, where baryons have condensed (e.g., Kravtsov & Borgani 2012, and references therein). Also, the DM and baryonic distributions need not align (one need only look at the Bullet cluster (Markevitch et al. 2004) for a dramatic example). This is not a particular concern to this analysis because we are comparing quantities based on the baryonic distribution, but would be more of a concern when including lensing.

Across our sample, we find an average pressure ratio $\langle {P}_{\mathrm{SZ}}\rangle =1.14\pm 0.09$, where we have included the calibration uncertainties in this calculation. We note that the cluster-to-cluster scatter in the pressure ratios is 0.25, which is larger than our uncertainty. That average pressure ratio corresponds to $\langle c\rangle =1.31\pm 0.22$, where again, our cluster-to-cluster scatter is quite large (0.58) compared to our uncertainty. Using cosmological smooth particle hydrodynamics (SPH) simulations, Battaglia et al. (2012) find average 2D (random projection) minor-to-major axis ratios $\simeq 0.95$ based on gas pressure distributions at ∼R₅₀₀ over all cluster masses at z = 0. This ratio has some dependence on cluster mass and redshift, where in both cases the deviations from unity grow with increasing mass and with increasing redshift.

Working with a smaller sample than that in Battaglia et al. (2012; 16 clusters) and higher resolution, Lau et al. (2011) use a cosmological simulation with an adaptive mesh refinement (AMR) code to investigate the shape of gas and dark matter, assuming different baryonic physics in two separate runs: a radiative (CSF) and non-radiative (NR) run. While comparable 2D projections of the gas density or pressure are not tabulated in Lau et al. (2011), they find smaller 3D minor-to-major axis ratios of the gas density than in Battaglia et al. (2012). We may conclude that simulations support average elongation values $0.9\lt c\lt 1/0.9$, which is in reasonable agreement with our derived average elongation $\langle c\rangle =1.31\pm 0.22$.

Observationally, using SZ and X-ray data on a sample of 25 clusters, De Filippis et al. (2005) find a median projected elongation of 1.24 ± 0.09, and median elongation along the line of sight (c) of 1.08 ± 0.17, where two clusters have $c\gt 2.0$, and three clusters have $1.5\lt c\lt 2.0$. Accounting for our uncertainties, only MACS 0329 and CLJ 1226 are outside (by $1\sigma$) of the range of elongations found in the literature. While this is true, our investigations here are only concerned with elongations along the line of sight, for which we are dominated by clusters with $c\gt 1$. This could be due to a systematic bias of P_SZ high, or P_X low, or even a selection bias within the CLASH sample. The CLASH sample contains X-ray (20) and lensing (5) selected clusters and was not explicitly designed to be orientation unbiased. It is, therefore, not too surprising that we find indications that many of the clusters in our sample are elongated along the line of sight ($c\gt 1$). Abell 1835 is not in the CLASH sample, but is a notably well-studied cool core cluster, i.e., it is the subject of many studies on the basis of its cool core.

We take the difference, ${\rm{\Delta }}{P}_{\mathrm{SZ},B}=({P}_{\mathrm{SZ}}-{P}_{B})$ to be indicative that the gas in the core has a different elongation than ICM at moderate to large radii. In particular, for ${P}_{B}\lt 1$, then ${P}_{\mathrm{SZ}}\gt {P}_{B}$ is indicative of a more spherical core and for ${P}_{B}\gt 1$, then a more spherical core will have ${P}_{\mathrm{SZ}}\lt {P}_{B}$. As P_SZ and P_B are not independent, we approximate the uncertainty in ${\rm{\Delta }}{P}_{\mathrm{SZ},B}$ as ${\sigma }_{{P}_{\mathrm{SZ}}}$ and report the pseudo-significances $({\rm{\Delta }}{P}_{\mathrm{SZ},B}/{\sigma }_{{P}_{\mathrm{SZ}}}$) of core sphericity (relative to the region outside the center) in Table 5. While none of our determinations individually are above $3\sigma$, it is nonetheless interesting to note the tendency for core sphericity.

If we assume a given geometry (known ellipticity), then instead of solving for the ellipticity, we can derive a temperature profile, making use of the direct pressure constraints from SZ observations and the electron density constraints from X-ray observations. That is, we calculate

$$\begin{eqnarray}&&{T}_{\mathrm{SZ}}=\displaystyle \frac{{P}_{\mathrm{SZ}}}{{n}_{e,X}},\end{eqnarray} \tag{ 10 }$$

where P_SZ is the pressure derived from pressure profile fits to the SZ data (Section 4) and ${n}_{e,X}$ is the deprojected electron density derived from X-ray data by the ACCEPT2 collaboration. For each bin, we assign radial values as the arithmetic mean of its radial bounds. Binned values of P_SZ are then calculated from the fitted gNFW profile for each radial value for the corresponding bins used for ${n}_{e,X}$.

Our SZ and X-ray derived temperature profiles (Figure 8) reveal, on average, larger temperatures than the spectroscopically derived temperatures from ACCEPT2. As an additional means of comparison, we fit an average profile derived in Vikhlinin et al. (2006) for the gas mass weighted temperature:

$$\begin{eqnarray}&&\displaystyle \frac{T(r)}{{T}_{\mathrm{mg}}}=1.35\displaystyle \frac{{({x}_{r}/0.045)}^{1.9}+0.45}{{({x}_{r}/0.045)}^{1.9}+1}\displaystyle \frac{1}{{(1+{({x}_{r}/0.6)}^{2})}^{0.45}},\end{eqnarray} \tag{ 11 }$$

where ${x}_{r}=r/{R}_{500}$. Thus, since we take R₅₀₀ as known, the shape of the profile is fixed. The values fit to the ACCEPT2 temperatures are reported in Table 1. We fit T_mg to our T_SZ profiles and T_X profiles from ACCEPT2, which we take as respective gas mass temperature proxies. The results of these fits are reported in Table 7. We compute the ratio ${T}_{\mathrm{mg},\mathrm{SZ}}/{T}_{\mathrm{mg},X}$ of the two fitted gas mass weighted temperature proxies. We find that $\langle {T}_{\mathrm{mg},\mathrm{SZ}}/{T}_{\mathrm{mg},X}\rangle =1.06$ with an rms scatter of 0.23.

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From Figure 8, we see that the shape of T_mg is generally quite consistent with the spectroscopic X-ray temperatures, while it is, in some cases, not reflective of the shape of T_SZ. Despite the difference in shapes between T_X and T_SZ, it is of moderate surprise that the shape of T_mg fits similar temperatures between ${T}_{\mathrm{mg},X}$ and the ${T}_{\mathrm{mg},\mathrm{SZ}}$.

In contrast to our results, which indicate, on average, higher values of Tsz than Tx, we note that Rumsey et al. (2016) find the opposite trend when comparing SZ data from the Arcminute Microkelvin Imager (AMI) with Chandra X-ray data for a subsample of the CLASH clusters (10 of 25), 7 of which overlap with our sample. However, Rumsey et al. (2016) use a much different technique to constrain T_SZ, based solely on the SZ data with strong priors on cluster parameters such as f_gas. In addition, the potential systematics in the 15 GHz interferometric SZ data used by Rumsey et al. (2016) are largely distinct from those related to our higher frequency bolometric SZ images. As a result, it is not possible to make a direct comparison of the results to better ascertain the cause of the discrepancy.

Given the typically long exposure times (our sample has total Chandra exposure times, $19.5\lt {t}_{\exp }\lt 134.1$ ks) required to derive spectroscopic X-ray temperatures, it is likely that deriving temperatures from SZ pressure profiles and X-ray electron densities will be more commonplace as SZ instruments have progressed rapidly in recent years. We consider the how the uncertainties of two temperature derivations (${\sigma }_{{T}_{\mathrm{SZ}}}$ and ${\sigma }_{{T}_{X}}$) compare within our sample. We find that ${\sigma }_{{T}_{\mathrm{SZ}}}$ is generally about twice as large as ${\sigma }_{{T}_{X}}$. Furthermore, ${\sigma }_{{T}_{\mathrm{SZ}}}$ is dominated by both the statistical and systematic uncertainties associated with P_SZ, where the fractional uncertainties in our SZ pressure profiles are roughly a factor of three larger than the fractional uncertainties in the X-ray electron densities.

Thus far, we have not taken into consideration systematic errors within ${\sigma }_{{T}_{X}}$. The systematic errors on T_x are not well quantified, despite evidence that these systematic errors can be notable (e.g., Donahue et al. 2014). We are thus interested in finding for what fractional systematic error do the uncertainties in ${\sigma }_{{T}_{\mathrm{SZ}}}$ and ${\sigma }_{{T}_{X}}$ become comparable and find that a systematic uncertainty of 20% on spectroscopic X-ray temperatures results in ${\sigma }_{{T}_{\mathrm{SZ}}}\sim {\sigma }_{{T}_{X}}$ over our sample.

We should additionally revisit our uncertainties on T_SZ to consider the impact of the uncertainty of the cluster geometry on ${\sigma }_{{T}_{\mathrm{SZ}}}$. With consideration for elongation along the line of sight, ${T}_{\mathrm{SZ}}=({P}_{\mathrm{SZ}}/{n}_{e,X}){c}^{1/2}$, which will results in an additional fractional error term: $({\sigma }_{c}/2c)$ to be added in quadrature. From Battaglia et al. (2012) and Lau et al. (2011), we can likely expect this term to be of the order of 0.1, which is the same as the statistical and systematic uncertainties on P_SZ.

We find overall agreement in ensemble constraints of the pressure profile between our SZ pressure profiles and those fitted to ACCEPT2 B14 pressure profiles (Figure 5). When calculating elongation along the line of sight, we find an average axis ratio $\langle c\rangle =1.38\pm 0.58$. In our temperature analysis, we find the average gas mass weighted temperature ratio $\langle {T}_{\mathrm{mg},\mathrm{SZ}}/{T}_{\mathrm{mg},X}\rangle =1.06\pm 0.23$.

While these average values show consistency between the SZ and X-ray quantities, the SZ pressure is, on average, generally larger than the X-ray pressure, especially at larger radii. In our elongation analysis, this pressure difference is manifest as the majority of our clusters showing elongation along the line of sight, which we find is largely consistent with numerical simulations (Section 6.1). Alternatively, in our temperature analysis, we find that T_SZ is generally larger than T_X, especially at larger radii. Differences in temperatures could indicate a bias of spectroscopic X-ray temperatures to lower temperatures, as emission will be dominated by the cooler (denser) regions. Moreover, Chandra is not sensitive to higher energy photons and therefore constraints on gas hotter than ${k}_{B}T\gtrsim 10\,\mathrm{keV}$ are generally poor.

In two clusters, MACS 0717 and CLJ 1226, we attribute the differences in SZ and X-ray pressure profiles to be primarily driven by differences in temperature. The triple merging cluster MACS 0717 (Appendix B.11) does not present a clear shock in SZ or X-ray within the central region, but it may be that the merger activity is primarily along the line of sight. The notable enhancement of SZ-to-X-ray spectroscopic temperature in the center is undoubtedly due to merger activity, and bears credence as other studies have found hot (roughly 20 keV in Sayers et al. 2013a; Adam et al. 2016, and 34 keV in Mroczkowski et al. 2012) gas in the region about subcluster C, which would contribute to temperature enhancements at small radii. In CLJ 1226, the average temperature values we derive are not significantly different than those in ACCEPT2, but the slope is reversed. In particular, the ACCEPT2 temperature in CLJ 1226 rises from 10 keV in the core to 15 keV at $r\sim 200\,\mathrm{kpc}$. In contrast, T_SZ shows a more characteristic, declining temperature profile. It is unclear what would cause this difference. We believe that this difference in slope accounts for a non-trivial change in the fitted pressure profile to ACCEPT2 (Section 6.1), which drives the corresponding SZ-fitted normalizations (P_SZ and P_B) low.

A third cluster with notable differences in the pressure profiles is MACS 0429. The SZ and X-ray pressure profiles have considerably different shapes . While this may be due, in part, to an increase in temperature with radius, we do not contend that this increase is as dramatic as that shown in Figure 8. It is possible that the intrinsic weakness of the decrement of this cluster, combined with the unusual strength of the central source (${S}_{90}=8$ mJy), has exceeded the capabilities of our point-source treatment (Section 4.1) and could thus be biasing the results on this particular cluster. However, this should primarily affect the inner pressure profile, and Bolocam is constraining the pressure at moderate to large radii to be well above that found in ACCEPT2.

Clumping may also be responsible for raising T_SZ relative to T_X at larger radii. Clumping is expected to increase with radius, and thus may account for some of the discrepancy between our inferred temperature and the X-ray spectroscopically derived temperatures. Battaglia et al. (2015) find that clumping is more pronounced for more massive clusters. For the most massive bin of clusters considered, which is most applicable to our sample, the density clumping (${C}_{2,\rho }=\langle {\rho }^{2}\rangle /\langle \rho {\rangle }^{2}$) at R₅₀₀ is roughly 1.2. Some SZ/X-ray constraints (e.g., Morandi et al. 2013; Morandi & Cui 2014) find clumping factors ${C}_{2,\rho }\sim 2$ at ${R}_{200}\sim 1.6{R}_{500}$, are within agreement with simulations. A clumping factor of 1.2 can account for biasing the T_X low relative to T_SZ by roughly ∼5%.

Abell 1835 is a well-studied massive cool core cluster. The cool core was noted to have substructure in the central 10'' by Schmidt et al. (2001) and identified as being due the central AGN by McNamara et al. (2006). Abell 1835 has also been extensively studied via the SZ effect (Reese et al. 2002; Benson et al. 2004; Bonamente et al. 2006; Sayers et al. 2011; Mauskopf et al. 2012). The models adopted were either beta models or generalized beta models, and tend to suggest a shallow slope for the pressure interior to 10''. Previous analysis of Abell 1835 with MUSTANG data (Korngut et al. 2011) detected the SZ effect decrement, but not at high significance, which is consistent with a featureless, smooth, broad signal. Our updated MUSTANG reduction of Abell 1835, shown in Figure 1, has the same features as in Korngut et al. (2011).

The MUSTANG map (Figure 1) shows an enhancement south of the X-ray centroid, and the Bolocam map shows elongation toward the south–southwest. Weak lensing maps are suggestive of a southwest–northeast elongation (Newman et al. 2009; Zitrin et al. 2015). Using the density of galaxies, Lemze et al. (2013) find a core and a halo that align with the elongation seen in the SZ. We note that AMI (AMI Consortium et al. 2012) and De Filippis et al. (2005) also see an elongation in the same direction in the plane of the sky. However, more recent AMI observations (Rumsey et al. 2016) show almost no elongation. Along the line of sight, De Filippis et al. (2005) calculate an elongation $c=1.05\pm 0.37$. Despite these notions of elongation, within the sample investigated in AMI Consortium et al. (2012), Abell 611 is the most relaxed cluster in their sample and that the X-ray data presented from LaRoque et al. (2006) is very circular and uniform. Despite being relaxed, Abell 611 is not listed as a cool core cluster (nor disturbed; Sayers et al. 2013a).

In an analysis of the dark matter distribution, Newman et al. (2009) find that the core (logarithmic) slope of the cluster is shallower than an NFW model, with ${\beta }_{\mathrm{DM}}=0.3$, where the dark matter distribution has been characterized by yet another generalization of the NFW profile:

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{0}}{{(r/{r}_{s})}^{{\beta }_{\mathrm{tot}}}{(1+r/{r}_{s})}^{3-{\beta }_{\mathrm{tot}}}}.\end{eqnarray} \tag{ 12 }$$

They find the distribution of dark matter within Abell 611 to be inconsistent with a single NFW model.

MACS 1115 is listed as a cool core cluster (Sayers et al. 2013a). It is among seven CLASH clusters that show unambiguous ultraviolet (UV) excesses attributed to unabsorbed star formation rates of 5–80 M_⊙ yr⁻¹ (Donahue et al. 2015). MUSTANG detects a point source in MACS 1115, which is coincident with its BCG. MACS 1115 is fit by a fairly steep inner pressure profile slope to the SZ data (Table 5). Adopting the Bolocam centroid, the inner pressure profile slope is notably reduced, yet the goodness of fit is not significantly changed. In particular, the Bolocam image shows a north–south elongation (particularly to the north of the centroids). In contrast, weak and strong lensing (Zitrin et al. 2015) show a more southeast–northwest elongation.

MACS 0429 has been well studied in the X-ray (Comerford & Natarajan 2007; Schmidt & Allen 2007; Allen et al. 2008; Maughan et al. 2008; Mann & Ebeling 2012) MACS 0429 is identified as a cool core cluster (cf. Mann & Ebeling 2012; Sayers et al. 2013a). The bright point source in the MUSTANG image is the cluster BCG, which is noted as having an excesses UV emission (Donahue et al. 2015). Of the point sources observed by MUSTANG, this has the shallowest spectral index between 90 GHz and 140 GHz of ${\alpha }_{\nu }=0.55$.

Despite MACS 0429's stature as a cool core cluster, its pressure profile (Table 5) is surprisingly shallow in the core, and shows elevated pressure relative to X-ray derived pressure at moderate radii. The offset between the Bolocam centroid (Sayers et al. 2013a) and ACCEPT (Cavagnolo et al. 2009) centroid is 100 kpc, which is notably larger than the X-ray-optical separations of the cluster peaks and centroids reported in Mann & Ebeling (2012) of 12.8 and 19.5 kpc respectively. Siegel et al. (2016) report an excess in SZ pressure (Bolocam) relative to X-ray (Chandra) pressure at moderate to large radii.

MACS 1206 has been observed extensively (e.g., Ebeling et al. 2001, 2009; Gilmour et al. 2009; Umetsu et al. 2012; Zitrin et al. 2012b; Biviano et al. 2013; Sayers et al. 2013a). It is not categorized as a cool core or a disturbed cluster (Sayers et al. 2013a). Using weak lensing data from Subaru, Umetsu et al. (2012) find that the major–minor axis ratio of projected mass is $\gtrsim 1.7$ at $1\sigma$. They infer that this high ellipticity and alignment with the BCG, optical, X-ray, and LSS shapes are suggestive that the major axis is aligned close to the plane of the sky. In Young et al. (2015), substructure is identified that corresponds to an optically identified subcluster, which may either be a merging subcluster or a foreground cluster. In this analysis, the SZ signal observed by MUSTANG is well modeled by a residual component (coincident with the subcluster) and a spherical bulk ICM component. We note that the Bolocam contours of MACS 1206 do not exhibit much ellipticity. We do find that MACS 1206 has a major–minor axis ratio of 1.24 ± 0.29 (Table 6), where the major axis is along the line of sight.

MACS 0329 has the distinction of being listed as both a cool core and disturbed cluster. Although it has been classified as relaxed (Schmidt & Allen 2007), substructure has been noted (Maughan et al. 2008), and it earns its cool core and disturbed classifications based on central weighting of X-ray luminosity and comparing centroid offsets between optical and X-ray data (Sayers et al. 2013a). The elongation of the weak lensing and strong lensing are toward the northwest and southeast of the centroid.

MACS 0329 has two systems with multiple images: one at z = 6.18 and the other at z = 2.17. The Einstein radii for these two systems are r_E = 34'' and r_E = 28'', respectively (Zitrin et al. 2012a), which is noted as being typical for relaxed, well-concentrated lensing clusters.

RXJ1347 is one of the most luminous X-ray clusters, and has been well studied in radio, SZ, lensing, optical spectroscopy, and X-rays (e.g., Schindler et al. 1995; Pointecouteau et al. 1999; Komatsu et al. 2001; Allen et al. 2002; Kitayama et al. 2004; Gitti et al. 2007a; Bradač et al. 2008; Miranda et al. 2008; Ota et al. 2008). X-ray contours have long suggested that RXJ1347 is a relaxed system (e.g., Schindler et al. 1997), and it is classified as a cool core cluster (e.g., Mann & Ebeling 2012; Sayers et al. 2013a).

Indeed, the first sub-arcminute SZ observations (Komatsu et al. 2001; Kitayama et al. 2004) saw an enhancement to the southeast of the cluster X-ray peak, which was suggested as being due to shock heating. This enhancement was confirmed by MUSTANG (Mason et al. 2010). Further measurements were made with CARMA (Plagge et al. 2013), which find that 9% of the thermal energy in the cluster is in sub-arcminute substructure. Most recently, Kitayama et al. (2016) has observed this cluster with ALMA to a resolution of 5''. At low radio frequencies (Ferrari et al. 2011, 237 MHz and 614 MHz), Gitti et al. (2007b, 1.4 GHz) find evidence for a radio mini-halo in the core of RXJ1347. The cosmic-ray electrons are thought to be reaccelerated because of the shock and sloshing in the cluster (Ferrari et al. 2011).

We observe a point source (coincident with the BCG) with a flux density of 7.40 ± 0.58 mJy. A previous analysis of the MUSTANG data found the point-source flux density to be 5 mJy (Mason et al. 2010). The difference in the flux densities is likely accounted for by the different fitting approaches, primarily that (1) we filter the beam shape (a double Gaussian), (2) we simultaneously fit the components, and (3) we assume a steeper profile in the core than the beta model assumed in Mason et al. (2010). Lower frequency radio observations found the flux density of the source to be 10.81 ± 0.19 mJy at 28.5 GHz (Reese et al. 2002), and 47.6 ± 1.9 mJy at 1.4 GHz (Condon et al. 1998). The BCG is observed to have a UV excess (Donahue et al. 2015).

Despite the classification of being a cool core cluster, it is also observed that there are hot regions, initially constrained as ${k}_{B}T\gt 10\,\mathrm{keV}$ (e.g., Allen et al. 2002; Bradač et al. 2008), and more recently constrained to even hotter temperatures (${k}_{B}T\gt 20\,\mathrm{keV}$ Johnson et al. 2012), indicative of an unrelaxed cluster. Johnson et al. (2012) also interpret the two cold fronts as being due to sloshing, where a subcluster has returned for a second passage.

Several previous studies have found similar evidence for compression along the line of sight in this cluster (e.g., Plagge et al. 2013, and references therein). However, the compression we find in this study is less severe as in Plagge et al. (2013).

MACS 1311 is listed as a cool core cluster (e.g., Sayers et al. 2013a), and appears to have quite circular contours in the X-ray and lensing images, yet has evidence for some disturbance, given its classification in Mann & Ebeling (2012). However, the SZ contours from Bolocam show some enhancement to the west, and has a notable centroid shift (277, 167 kpc) westward from the X-ray centroid. When fitting pressure profiles to this cluster, it appears that the enhanced SZ pressure at moderate radii ($r\sim 100$'') is due to this enhancement, especially when noting that we use the X-ray centroid. Adopting the Bolocam centroid does not change the pressure profile much, and we still observe a pressure enhancement at moderate radii. In contrast, in their analysis, Siegel et al. (2016) find that X-ray (Chandra) and SZ (Bolocam) data are in good agreement with a spherical ICM model, which is supported primarily with thermal pressure.

MACS 1423 is a cool core cluster (Mann & Ebeling 2012; Sayers et al. 2013a). While the Bolocam contours are quite concentric and suggestive of a relaxed cluster, the centroid is still offset from the X-ray peak by an appreciable angle (198, 126 kpc). While AMI (Rumsey et al. 2016) shows a perturbation/extension to the southwest of the cluster, their analysis is supportive of MACS 1423 being a relaxed cluster. Similar to MACS 1311, the pressure is slightly less than the ACCEPT2 X-ray derived pressure in the core and slightly greater at moderate radii. While this is expected for a centroid offset, we find that adopting the Bolocam centroid again yields no substantial difference in the SZ pressure profile. Both our analysis and that of Siegel et al. (2016) find good agreement between SZ and X-ray pressure profiles. We observe a point source (the cluster BCG) with a flux density of 1.36 ± 0.13 mJy, which is also observed to have a UV excess (Donahue et al. 2015).

MACS 1149 is classified as a disturbed cluster (e.g., Mann & Ebeling 2012; Sayers et al. 2013a), and lensing studies have found that a single DM halo does not describe the cluster well, but rather at least four large-scale DM halos are used to describe the cluster (Smith et al. 2009). A large radial velocity dispersion (1800 km s⁻¹ Ebeling et al. 2007) is observed, indicative of merger activity along the line of sight. X-ray, SZ, and lensing (particularly strong lensing) all show elongation in the northwest–southeast direction. More recently, Golovich et al. (2016) investigate the dynamics of the cluster, identifying three subclusters, with merger activity (velocities) primarily in the plane of the sky. SZ data from AMI (Rumsey et al. 2016) does not strongly indicate merger activity, arguably because the mass of the primary halo is much greater than the subhalos. Morphologically, the SZ map from AMI does show minor asphericity.

While the Bolocam map of this cluster shows a modest elongation in the northwest–southeast direction, it is well modeled as a spherical cluster. Our SZ derived pressure profile roughly matches the shape of the X-ray derived pressure profile, with the SZ pressure consistently greater than the X-ray pressure. We calculate that the axis along the line of sight is 1.54 ± 0.37 (Section 6.1) times greater than the axes in the plane of the sky. In the MUSTANG map, we see a $3\sigma$ feature to the east of the centroids, but it is not clear that this is associated with any particular feature.

Despite MACS 1149's impressive merging activity, MACS 0717 is thought to be the most disturbed massive cluster at $z\gt 0.5$ (Ebeling et al. 2007), which appears to be accreting matter along a 6-Mpc-long filament (Ebeling et al. 2004), and has the largest known Einstein radius (${\theta }_{e}\sim 55$''; Zitrin et al. 2009). Four distinct components are identified from X-ray and optical analyses (Ma et al. 2009), and the lensing analyses (Zitrin et al. 2009; Limousin et al. 2012) find agreement in the location of these four mass peaks with those from the X-ray and optical. While the complex X-ray morphology is not evident in AMI (Rumsey et al. 2016) or Bolocam SZ maps, there is still asphericity in the maps.

There are four identified subclusters (labeled A through D Ma et al. 2009). They find that subcluster C is the most massive component, while subcluster A is the least massive, and subclusters B and D are likely remnant cores. The velocities of the components from spectroscopy are found to be (v_A, v_B, v_C, v_D) = $(+{278}_{-339}^{+295}$, $+{3238}_{-242}^{+252}$, $-{733}_{-478}^{+486}$, $+{831}_{-800}^{+843})$ km s⁻¹ (Ma et al. 2009). The first indication of detection of the kSZ signal toward these subclusters was presented in Mroczkowski et al. (2012), with a subsequent paper from Sayers et al. (2013a) having the first significant detection and derived cluster velocities. Most recently, Adam et al. (2016) has mapped the kSZ signal and derived model-dependent subcluster velocities.

MACS 0717 has also been observed at 610 MHz with the Giant Metrewave Radio Telescope (GMRT), which reveals both a radio halo and a radio relic (van Weeren et al. 2009). This is interpreted as likely being due to a diffuse shock acceleration (DSA). More recently, however, deep, higher resolution JVLA data have found a connection between a central radio source and the diffuse emission, and favor re-acceleration as the source of the relativistic electrons (van Weeren et al. 2017).

We observe a foreground radio galaxy, well outside the cluster centered, which we model as a point source here, with a flux density of 2.08 ± 0.25 mJy at 90 GHz. This was previously reported with an integrated flux density of 2.8 ± 0.2 mJy and an extended shape of 144 × 161 (Mroczkowski et al. 2012). However, improved beam modeling has allowed us to model the foreground galaxy given a known beam shape. It is also worth noting that the MUSTANG data itself has been processed slightly differently from that presented in Mroczkowski et al. (2012); in this work, the map is produced with a common calculated as the mean across detectors, whereas, in Mroczkowski et al. (2012), the common mode was calculated as the median across detectors.

MACS 0647 is at z = 0.591 and is classified as neither a cool core nor a disturbed cluster (Sayers et al. 2013a). It was included in the CLASH sample due to its strong lensing properties (Postman et al. 2012). Gravitational lensing (Zitrin et al. 2011), X-ray surface brightness (Mann & Ebeling 2012), and SZ effect (MUSTANG, see Figure 1, and Bolocam) maps all show elongation in an east–west direction. Rumsey et al. (2016) find a circular SZ morphology with AMI, and take the discrepancies in the SZ and X-ray temperatures as an indication of a recent head-on merger. In the joint analysis presented here, we see that the spherical model provides an adequate fit to both data sets, as evidenced in Table 5. Still, from Figure 9, it appears that some elongation of the bulk ICM or residual feature would better describe the cluster center.

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MACS 0744 is neither classified as a cool core cluster nor a disturbed cluster (Mann & Ebeling 2012; Sayers et al. 2013a), but qualifies as a relaxed cluster (Mann & Ebeling 2012). There is a dense X-ray core, and a doubly peaked red sequence of galaxies, as found by Kartaltepe et al. (2008). The gas is also found to be rather hot: ${k}_{B}T={17.9}_{-3.4}^{+10.8}$ keV, as determined by combining SZ and X-ray data (LaRoque et al. 2003). AMI observations (Rumsey et al. 2016) show some elongation in the plane of the sky in their SZ map of this cluster, but they otherwise find that the cluster is in a relaxed state.

The data presented here is the same as in Korngut et al. (2011), but has been processed differently: again, the primary difference is in the treatment of the common mode. Additionally, Korngut et al. (2011) optimize over the low-pass filtering of the common mode and do not implement a correction factor for the SNR map. The surface brightness significance of the shock feature is the same, but is perhaps less bowed than the kidney bean shape seen previously. The excess found in Korngut et al. (2011) marked the first clear detection of a shock in the SZ that had not been previously been known from X-ray observations. Korngut et al. (2011) reanalyze the X-ray data with the knowledge of the shocked region from MUSTANG, and calculate the Mach number of the shock based on (1) the shock density jump, (2) stagnation condition between the pressures at the edge of the cold front and just ahead of the shock, and (3) temperature jump across the shock, and find Mach numbers between 1.2 and 2.1, with a velocity of ${1827}_{-195}^{+267}$ km s⁻¹. The shocked region (region II in Korngut et al. 2011) is well modeled with 19.7 keV gas.

CLJ 1226 is a well-studied high-redshift cluster (e.g., Mroczkowski et al. 2009; Bulbul et al. 2010; Adam et al. 2015). Adam et al. (2015) find a point source at R.A. 12:27:00.01 and decl. +33:32:42 with a flux density of $6.8\pm 0.7\ (\mathrm{stat}.)\pm \,1.0\ (\mathrm{cal}.)$ mJy at 260 GHz and $1.9\,\pm 0.2\ (\mathrm{stat}.)$ at 150 GHz. This is not the same point source seen in Korngut et al. (2011), which is reported as a point source with $4.6\sigma$ significance in surface brightness, and can be fit in our current analysis as a point source with a flux density of 0.33 ± 0.13 mJy. A short VLA filler observation (VLA-12A-340, D-array, at 7 GHz) was performed to follow-up this potential source. To a limit of $\sim 50\,\mu {Jy}$ nothing is seen, other than the clearly spatially distinct radio source associated with the BCG at the cluster center (1 mJy at 7 GHz and 3.2 mJy in NVSS). Rumsey et al. (2016) find a point source of weak significance with a flux density of ∼0.18 mJy in CLJ 1226 (at 15 GHz); however, coordinates are not provided and the location indicated on the maps would be consistent with either the point source found by Adam et al. (2015) or Korngut et al. (2011). In contrast, the point source found in Adam et al. (2015) is fit to our data with a flux density of 0.36 ± 0.11 mJy. Given the slight increase in significance of the point source from Adam et al. (2015), we adopt that point-source location for our pressure profile analysis of CLJ 1226.

In the previous analysis of the MUSTANG data, Korngut et al. (2011) find a ridge of significant substructure after subtracting a bulk SZ profile (N07, fitted to SZA data). They find that this ridge, southwest of the cluster center, alongside X-ray profiles, are consistent with a merger scenario. Rumsey et al. (2016) also take the descrepancy that they find between SZ and X-ray temperature as indicative a merger scenario. When comparing to merger simulations, they find CLJ 1226 could be consistent with a head-on minor merger. Adam et al. (2015) found evidence for a disturbed core, but relaxed on large scales. However, in this work, we do not find any significant substructure after fitting a bulk component, or other indication of merger activity.