---
abstract: |
    We present 30GHz Sunyaev Zel’dovich observations of a statistically complete sample of galaxy clusters with OCRA-p. The clusters are the 18 most X–ray luminous clusters at $z > 0.2$ in the ROSAT Brightest Cluster Sample. We correct for contaminant radio sources via supplementary observations with the Green Bank Telescope, also at 30GHz, and remove a cluster that is contaminated by an unresolved X-ray source. All 17 remaining clusters have central SZ effects with Comptonisation parameter $y_0$ exceeding $1.9\times10^{-4}$, and 13 are detected at significance $\ge 3
    \sigma$. We use our data to examine scalings between $y_0$ and X-ray temperature, X-ray luminosity, and the X-ray mass proxy $Y_\mathrm{X}$, and find good agreement with predictions from self–similar models of cluster formation, with an intrinsic scatter in $y_0$ of about 25%. We also comment on the success of the observations in the face of the contaminant source population, and the implications for upcoming cm-wave surveys.
author:
- |
    Katy Lancaster$^1$, Mark Birkinshaw$^1$, Marcin P. Gawroński$^2$, Richard Battye$^3$, Ian Browne$^3$, Richard Davis$^3$, Paul Giles$^1$, Roman Feiler$^2$, Andrzej Kus$^2$, Bartosz Lew$^2$, Stuart Lowe$^3$, Ben Maughan$^1$, Abdulaziz Mohammad$^1$, Bogna Pazderska$^2$, Eugeniusz Pazderski$^2$, Mike Peel$^3$, Boud Roukema$^2$ and Peter Wilkinson$^3$\
    $^1$ H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL\
    $^2$ Toruń Centre for Astronomy, Nicolaus Copernicus University, ul. Gagarina 11, 87-100 Toruń, Poland\
    $^3$ Jodrell Bank Centre for Astrophysics, The University of Manchester, Alan Turing Building, Manchester M13 9PL\
bibliography:
- 'ref\_lancaster.bib'
date: 'Received \*\*insert\*\*; Accepted \*\*insert\*\*'
title: 'Sunyaev Zel’dovich observations of a statistically complete sample of galaxy clusters with OCRA-p'
---

\[firstpage\]

cosmology: observations – cosmic microwave background – galaxies: clusters: individual (A1835, ZWCL1953, A689, ZWCL3146, RXJ1532.9+3021, A2390, A2219, RXJ2129.6+0005, A2261, A781, A697, A1763, A68, A520, A267, RXJ0439.0+0715, ZWCL7160, A773) – methods: observational

INTRODUCTION {#intro}
============

The thermal Sunyaev Zel’dovich (SZ) effect ([@SZ]) is a spectral distortion of the Cosmic Microwave Background (CMB) radiation due to inverse Compton scattering by the hot gas in galaxy clusters. It has long been exploited in cosmological and cluster studies in order to derive, for example, the Hubble constant (e.g. [@Hughes1998], [@Mason_obs], [@Reese2002], [@Saunders2003], [@Bonamente2006]) and the gas mass fraction (e.g. [@Grego2001], [@Lancaster2005], [@LaRoque2006]). Thanks to well developed techniques, detections are becoming routine although signal–to–noise remains quite poor. However, we are entering an era of purpose–built instruments so this is set to improve dramatically, enabling SZ research to reach its evident potential.

The main focus of the SZ community at present is to utilise the redshift–independence of the SZ surface brightness in order to perform blind surveys for galaxy clusters. While other techniques suffer from large intrinsic biases and complex selection effects, SZ surveys will produce almost *mass–limited* catalogues and thus far superior datasets for constraining cosmological models. The dedicated SZ surveys, for example Planck ([@Ade2011]), SPT ([@Stan2009], [@Vanderlinde2010s]), ACT ([@Menanteau2010], [@Marriage2010]) and the SZA [@Muchovej2011] are now generating results. Many more are expected in the near future, e.g. from AMI [@Zwart2008s]. In order to fully exploit the results of these surveys, it will be necessary to improve understanding of both the ‘selection effect’ due to the presence of unsubtracted radio sources, and also the scalings between cluster SZ observables and various physical quantities, especially the cluster mass. Various groups have undertaken similar studies (e.g. [@Benson2004], [@Morandi2007], [@Bonamente2008], [@Huang2010]). To make further advances, reliable observations of large, well–selected samples are required.

The One Centimetre Receiver Array prototype, OCRA-p, is a two–element receiver mounted on the 32-m telescope at the Toruń Centre for Astronomy of the Nicolaus Copernicus University, Poland. OCRA–p proved its SZ capabilities by detecting four well known clusters at high significance [@Lancaster2007]. We present the first in a series of papers discussing an X–ray selected sample of 33 clusters with well understood selection effects. This paper contains details of our OCRA–p observations of a statistically complete subsample of 18 clusters. The structure of the paper is as follows. In Section 2 we give a brief description of the Toruń 32-m telescope and the OCRA-p receiver. Section 3 contains details of the cluster sample. The observations and data reduction are described in Section 4, and the problem of radio source contamination is detailed in Section 5. Section 6 gives a brief overview of our X-ray analysis, and Section 7 presents our results. We discuss these results and conclude in Section 8. Throughout the paper we adopt the following cosmological parameter values: $H_0 = 70 \, \mathrm{km} \, \mathrm{s}^{-1} \,
\mathrm{Mpc}^{-1}$, $\Omega_{\mathrm{m}0} = 0.3$, $\Omega_{\Lambda 0}
= 0.7$.

THE TORUN TELESCOPE AND OCRA-p
==============================

The Toruń observatory is located in Piwnice, 15km outside Toruń in northern Poland. The telescope consists of a 32-m parabolic dish and 3.2-m sub-reflector, with a fully steerable classical Alt–Az mount. It has receivers operating at 1.4-1.7, 5, 6.8, 12 and 30GHz. The telescope is used for a variety of studies including interstellar molecular spectroscopy (e.g. [@Blaszkiewicz2004]) and VLBI (e.g. [@Bartkiewicz2005]).

OCRA is a planned 30GHz 100–element continuum receiver (see [@Browne2000]). A prototype receiver, OCRA-p, funded by a grant from the Royal Society Paul Instrument fund, has been on the telescope since 2004. The instrument is described in detail in [@LowePhD] and [@Lowe2007]; here we present a short summary. The basic radiometer design is based on the prototype for the Planck Low Frequency Instrument (LFI, [@Mandolesi2000]), and is similar to the WMAP K-band receivers [@Jarosik2003]. OCRA-p provides only two horn–feeds, the beams of which are separated by $3'.1$ and have FWHM $1'.2$. As the beam separation is small, it is possible to reduce the effects of atmospheric and gain fluctuations by switching between the beams and taking the difference. This can be improved upon by further levels of switching. The full switching strategy for SZ observations is described in Section \[sec:obs\]. In addition to the SZ program, OCRA–p has been used to study planetary nebulae ([@Pazderska2009]) and for various radio source studies including flux density measurements to support the Very Small Array CMB program (see [@Gawronski2010] for details of the OCRA measurements, and Genova–Santos et al. (in preparation) for the details of the CMB work). The next phase of the project is an 8–element array, OCRA–Faraday ([@Peel2009]), which was mounted on the telescope in late 2009. This is still in the commissioning phase but is expected to be fully operational in autumn 2011, and can be upgraded to 16 elements in the future.

THE CLUSTER SAMPLE {#sec:clusters}
==================

In order to derive meaningful cosmological results, or indeed to comment on universal cluster properties, it is necessary to use a ‘fair’ sample, or at least one where the selection biases are clear and well understood. Large samples fulfilling these criteria are rare in SZ astronomy, largely due to the various practical challenges which observers must overcome. In this work, we attempt to select and observe such a sample, and turn to the ROSAT All–Sky Survey in the first instance.

The Brightest Cluster Sample (BCS, [@Ebeling1998]) is derived from a careful analysis of the ROSAT All-Sky Survey (RASS, [@Voges1999]) data at $\delta > 0^{\circ}$. The resulting catalogue is 90% flux complete to a limit of $f_{\mathrm{X}} = 4.4
\times 10^{-12}$ergcm$^{-2}$s$^{-1}$ in the 0.1-2.4keV energy band. Since the SZ surface brightness of a cluster is essentially unaffected by redshift, we select on luminosity, imposing a limit of $L_{\mathrm{X}} \geqslant 13.08\times10^{44}\mathrm{erg\,s^{-1}}$ (the luminosity of A773 in the BCS catalogue). In addition, we impose the criterion $z>0.2$ due to the $3'.1$ beam–throw of OCRA; observations at lower redshift are inefficient with this instrument. Our basic sample then consists of 18 clusters. We do not reject any clusters based on their radio–source environments at this stage, even though we appreciate the difficulties which this policy may introduce. This has the advantage of allowing us to investigate any correlation between gas properties and those of a central radio galaxy.

We reject one cluster from the basic 18–cluster sample, A689, as on further investigation we have found that the ROSAT data are contaminated by a bright central point source (Giles et al., in preparation). After removing this source from the *Chandra* data, the luminosity is just $(1.66\pm0.24)\times10^{44} \mathrm{erg
 \,s}^{-1}$, more than a factor 8 below the limiting luminosity of the cluster sample. The sample is summarised in Table \[tab:coord\]. These clusters appear not to be exceptional in their X–ray structures, and, indeed, follow the expected X-ray scaling relations (see Section \[sec:Xscale\]).

  ---------------- --------------- -------------- ------------ ------------- -------- ------------------------
  Cluster                RA             Dec        $\Delta$RA   $\Delta$Dec    $z$         $L_\mathrm{x}$
                       (J2000)        (J2000)       (arcmin)     (arcmin)              ($10^{44}$ergs$^{-1}$)
  A1835              14 01 01.99     02 53 12.8       0.02         0.53        0.25            38.53
  ZWCL1953           08 50 03.00     36 04 16.0      -0.80         -0.04       0.32            34.12
  *A689*            *08 37 25.01*   *14 59 40.9*      *-*           *-*       *0.28*       *30.41 (1.66)*
  ZWCL3146           10 23 35.98     04 11 56.0      -0.90         0.75        0.29            26.47
  RXJ1532.9+3021     15 32 58.99     30 21 11.2       1.13         0.20        0.35            24.40
  A2390              21 53 34.61     17 40 10.9      -0.50         -1.56       0.23            21.44
  A2219              16 40 21.80     46 42 47.8       0.24         0.30        0.23            20.40
  RXJ2129.6+0005     21 29 40.50     00 05 49.9       0.11         0.51        0.24            18.59
  A2261              17 22 27.00     32 07 04.1      -0.03         -0.88       0.22            18.18
  A781               09 20 25.99     30 29 57.8      -0.04         -0.11       0.30            17.22
  A697               08 42 58.01     36 21 45.0       0.03         -0.20       0.28            16.30
  A1763              13 35 23.21     41 00 04.0       0.04         0.08        0.22            14.93
  A68                00 37 05.30     09 09 10.8      -0.20         -0.38       0.26            14.89
  A520               04 54 19.01     02 56 48.1       0.00         0.00        0.20            14.44
  A267               01 52 42.00     01 00 25.9      -0.02         -0.24       0.23            13.71
  RXJ0439.0+0715     04 39 01.01     07 16 54.8       0.08         0.85        0.23            13.25
  ZWCL7160           14 57 19.99     22 20 35.9       1.13         0.03        0.26            13.19
  A773               09 17 51.00     51 43 19.9      -0.32         -0.34       0.22            13.08
  ---------------- --------------- -------------- ------------ ------------- -------- ------------------------

\[tab:coord\]

OCRA DATA
=========

Observing Strategy {#sec:obs}
------------------

The clusters were observed in good weather periods between September 2005 and January 2009. We aimed to achieve a uniform noise level $<0.5$mJy, which required $\sim 220$ minutes of data at each pointing centre. Trail fields, blank patches of sky separated from the target fields by 15–20m in RA, were observed over the same range of hour angle after each cluster observation, using the same position switching routine (described below) as for the cluster fields. The integration times for the target and trail fields are identical.

For each observation, whether the cluster or its corresponding trail field, we employ a position switching strategy to remove atmospheric contamination from our data. The feeds are positioned such that one beam, beam $B$, is coincident with the cluster centre and the other, beam $A$, provides a measure of the blank sky signal. (For extended sources, beam $A$ may measure a small signal itself. This must be properly accounted for - see Section \[sec:beta\].) We switch between the $A$ and $B$ beams at a rate of 277kHz, recording the $A-B$ difference every second. We integrate in this position for 25 seconds. The telescope then slews to reposition the beams such that beam $A$ measures the cluster and beam $B$ measures the sky background, and the differencing is repeated, this time integrating for 50 seconds. We then return to the starting position and integrate for another 25 seconds. The differenced signals from the first and third positions, $(A_1-B_1)$ and $(A_3-B_3)$ are summed, and then subtracted from the differenced signal from the second position $(A_2-B_2)$. This recovers twice the cluster signal relative to the background regions. The symmetric nature of the switching pattern ensures proper subtraction of most atmospheric emission and other contaminants such as ground pick-up, provided that the target region is not rapidly changing in azimuth and elevation.

Calibration
-----------

The data are calibrated against an internal noise source, which is itself calibrated via observations of the well–known bright radio source NGC7027 of flux density 5.37Jy. We observe significant changes in the level of the noise source in the form of: (a) sudden ‘jumps’, sometimes due to the telescope reconfiguration and (b) additional smaller amplitude fluctuations which appear random in nature. We note that other than the low–amplitude random fluctuations, the voltage level is generally stable on timescales of a few days. By performing regular calibration ($\approx$daily) we reduce the effects of the larger fluctuations. In addition, we adopt a conservative approach by comparing the total powers for the NGC7027 and cluster observations and rejecting any data for which they differ significantly, although a study suggests the calibration often stays virtually the same as the total power drifts. Residual uncertainties in the calibration of the system at the level of 5% may be expected.

The telescope pointing is calibrated via azimuth and elevation scans across a bright source close to each cluster. Where an acceptable pointing calibration does not exist within 60minutes of a cluster observation, the cluster data are rejected.

Statistical Data Analysis
-------------------------

After combination of the second-by-second average data into double–differenced measurements of the brightness of the sky, and calibration, the data are examined for periods of increased noise (which might arise from receiver instabilities or bad weather conditions) or individual anomalous points. The combination of the data into final averages was performed including statistical tests for outlier data. The fractions of the data points rejected by $3\sigma$ or $5\sigma$ cuts were small in all cases, and no cut-dependent changes larger than a small fraction of the error on the mean were seen. The distributions of data values in the double-difference data are close to Gaussian, with a slight tendency to show elevated wings in the distributions: the estimates of the error on the mean (Table \[tab:res\]) take account of the full distribution, and not the distribution after truncation of outliers.

RADIO SOURCE CONTAMINATION {#sec:src}
==========================

Radio source contamination remains a significant problem for observations of the Sunyaev Zel’dovich effect. If we know the positions and 30GHz flux densities of all sources near our observing locations, we can correct the data and establish the true magnitude of the SZ effect for each cluster. Ideally, we would consult a survey of the radio sky at a frequency close to 30GHz in order to identify contaminant radio sources. The only such survey currently available is the WMAP point source catalogue ([@Wright2009]), but this is complete only to the 2Jy level and so is three orders of magnitude too shallow for our purposes. We consult the all–sky NVSS ([@NVSS], 1.4GHz) and GB6 ([@GB6], 4.85GHz) catalogues and retain all sources within $5'$ of the pointing centre for each cluster. In addition, we include any sources found in similar projects and reported in the literature. The list of 58 contaminating sources is presented in Table \[table:sources\].

GBT observations
----------------

We observed the 58 sources over the course of several sessions at the Green Bank Telescope (GBT) in January 2008 (observations limited due to poor weather conditions), then January and May 2009 (re-scheduled time, again compromised in part by non-ideal conditions). We used the GBT Ka–band receiver and the Caltech Continuum Backend (CCB). For a detailed description of this configuration of equipment and the observing methods used, see [@Mason2009]; we will describe the salient points.

### Calibration

Every observing session included at least one flux calibration observation, using the most accessible of the GBT standard calibrators 3C 147, 3C 48 or 3C 286. We observed sources cluster by cluster, and thus they tend to lie in small regions. The GBT pointing is stable for up to 40minutes for night time observing, but we rarely required re-pointing near a field because slews tended to occur after only a few source integrations. We chose one pointing calibrator per cluster (and hence its associated sources). We took pointing calibrators from the GBT database, choosing the brightest available within 10degrees of the cluster centre. We also performed at least one sky dip per observing session to establish the sky temperature and opacity.

\[table:sources\]

  ---------------- ------- ----------------- ---------------- ---------- -------------- ------ ---------- ------- ----------- --------------- --------------- -----------
  Cluster          Src            RA               Dec                                                      Notes                                             
                   \#           (J2000)          (J2000)                                                                                                      
  A1835            1\*        14 01 02.1        02 52 41.0          39.3 $\pm1.6$            -               3.31 $\pm0.14$              2.93 $\pm0.18$       OB N
                                                                                                             2.88 $\pm0.07$                                   
                   2\*        14 01 00.5        02 51 53.0           1.6 $\pm0.1$            -               1.26 $\pm0.14$              1.33 $\pm0.07$       OB F
                                                                                                             1.36 $\pm0.08$                                   
  ZWCL1953         [1]{}    [08 50 07.8]{}    [36 04 21.7]{}    [19.4]{} ${\pm1.2}$          -                  -                    [0.79]{} [$\pm0.70$]{}   [GBT]{}
                   2          08 50 13.4        36 04 22.4          20.8 $\pm1.0$         19.0 $\pm4.0$      1.19 $\pm0.17$              1.28 $\pm0.16$       GBT OB
  ZW3146           1          10 23 37.2        04 09 06.4          31.5 $\pm1.0$            -               2.03 $\pm0.22$              2.35 $\pm0.81$       GBT OB
                                                                                                             2.12 $\pm0.15$                                   
                   2          10 23 39.6        04 11 15.4           7.1 $\pm0.5$            -               0.41 $\pm0.07$              0.33 $\pm0.07$       GBT OB
                   3          10 23 45.1        04 10 40.7          95.8 $\pm3.4$         42.0 $\pm7.0$      5.35 $\pm0.18$              5.27 $\pm0.11$       GBT OB
                                                                                                             5.70 $\pm0.10$                                   
                   4          10 23 45.2        04 11 39.7           3.6 $\pm0.4$            -               0.85 $\pm0.10$              0.80 $\pm0.10$       GBT OB
  RXJ1532.9+0005   1          15 32 47.4        30 18 46.0          18.0 $\pm1.0$            -                  -                        1.18 $\pm0.21$       GBT
                   2          15 32 50.7        30 19 47.6           7.9 $\pm0.5$            -               6.58 $\pm0.20$              6.56 $\pm0.21$       OB N
                   3          15 32 53.8        30 20 59.8          22.8 $\pm0.8$         20.0 $\pm4.0$      3.25 $\pm0.18$              3.27 $\pm0.12$       GBT OB
                   4          15 32 54.31       30 23 01.5           4.4 $\pm0.4$            -                  -                        0.92 $\pm0.21$       GBT
                   5          15 33 03.42       30 23 47.2           2.6 $\pm0.4$            -                  -                        1.12 $\pm0.50$       GBT
  A2390            1          21 53 32.4        17 42 19.8          20.9 $\pm1.6$            -                    0.23              $\pm0.12$ GBT             
                   2          21 53 36.8        17 41 44.8         235.3 $\pm8.3$          220 $\pm20$            45.77             $\pm0.10$ GBT             
                   *3*       *21 53 40.3*      *17 42 56.7*       *12.2* $\it \pm1.0$        -                    -*0.14*       $\it \pm0.09$ *GBT*           
  A2219            1          16 40 21.83       46 42 47.8         239.1 $\pm8.3$         84.8 $\pm8.0$     14.87 $\pm0.17$             13.74 $\pm1.00$       GBT OB
                   2\*        16 40 23.83       46 41 47.3           7.9 $\pm1.0$            -               0.97 $\pm0.17$              0.94 $\pm0.17$       OB N
                   3          16 40 14.98       46 42 28.7           6.1 $\pm0.5$            -                  -                        0.42 $\pm0.09$       GBT
                   4          16 39 58.07       46 40 37.2          14.1 $\pm0.5$            -                  -                        0.96 $\pm0.09$       GBT
  RXJ2129.6+0005   *1*       *21 29 36.61*     *00 02 35.4*        *2.2* $\it \pm0.4$        -                  -                     -*0.13* $\it \pm0.21$   *GBT*
                   2          21 29 40.00       00 05 22.9          25.4 $\pm1.2$            -               2.33 $\pm0.10$              2.04 $\pm0.06$       GBT OB
                   [3]{}    [21 29 40.14]{}   [00 01 44.4]{}     [4.5]{} $ \pm0.5$           -                  -                    [0.11]{} $ \pm0.13$      [GBT]{}
                   4          21 29 55.24       00 07 56.9          34.3 $\pm1.8$            -                  -                        2.82 $\pm0.11$       GBT
  A2261            1          17 22 16.9        32 09 10.5          23.0 $\pm1.5$            -               9.32 $\pm0.22$             15.58 $\pm3.89$       GBT OB(V)
                                                                                                            10.48 $\pm0.16$                                   
                   2          17 22 27.7        32 07 57.8           5.3 $\pm0.5$            -                  -                        0.13 $\pm0.08$       GBT
                   3          17 22 45.3        32 09 27.0          4.85 $\pm0.5$            -                  -                        0.14 $\pm0.10$       GBT
                   4          17 22 06.55       32 07 01.7           5.8 $\pm0.4$            -                  -                        0.14 $\pm0.08$       GBT
  A781             1          09 20 08.51       30 32 14.3          19.9 $\pm0.9$            -                  -                        1.42 $\pm0.60$       GBT
                   2          09 20 14.12       30 29 02.8          17.9 $\pm0.7$            -                  -                        1.22 $\pm0.53$       GBT
                   3          09 20 21.75       30 32 27.0           2.7 $\pm0.5$            -                  -                        0.11 $\pm0.18$       GBT
                   4          09 20 22.90       30 29 45.6          73.1 $\pm2.6$         32.0 $\pm5.0$      5.33 $\pm0.18$              5.35 $\pm0.30$       GBT OB
                   *5*       *09 20 30.83*     *30 28 02.1*       *15.8* $\it\pm1.6$         -                  -                     *-0.07* $\it\pm0.16$    *GBT*
                   6          09 20 47.35       30 28 21.3           5.4 $\pm0.6$            -                  -                        0.08 $\pm0.17$       GBT
  A697             1          08 42 40.22       36 19 16.4           5.4 $\pm0.5$            -                  -                        1.64 $\pm0.54$       GBT
                   2          08 42 59.67       36 17 43.7          32.5 $\pm1.4$            -                  -                        1.71 $\pm0.65$       GBT
  ---------------- ------- ----------------- ---------------- ---------- -------------- ------ ---------- ------- ----------- --------------- --------------- -----------

[llccr@lr@lr@lr@ll]{} Cluster &Src &RA &DEC & & & & &Notes\
&\# &(J2000) &(J2000) & & & &\
& & & & & & &\
A1763 &*1* &*13 35 15.60* &*41 00 25.8* &*7.9*&$\it\pm0.5$ &-& &-& &*-0.10*&$\it\pm0.14$ &*GBT*\
&2\* &13 35 19.86 &41 00 04.0 &859.2&$\pm29.5$ &225&$\pm20$ &31.30&$\pm0.41$ &29.49&$\pm0.38$ &OB G N\
A68 &1 &00 36 52.94 &09 05 21.7 &22.8&$\pm0.8$ &-& &-& &1.57&$\pm0.05$ &GBT\
&2 &00 37 05.21 &09 13 33.6 &4.4&$\pm0.4$ &-& &-& &0.60&$\pm0.05$ &GBT\
&3 &00 37 06.35 &09 07 30.4 &40.2&$\pm1.3$ &-& &1.20&$\pm0.12$ &1.21&$\pm0.06$ &GBT OB\
&4 &00 37 07.71 &09 08 24.0 &59.1&$\pm2.2$ &-& &1.60&$\pm0.10$ &1.65&$\pm0.05$ &GBT OB\
&*5* &*00 37 17.93* &*09 06 30.5* &*3.2*&$\it\pm0.6$ &-& &-& &*-0.05*&$\it\pm0.04$ &*GBT*\
&6\* &00 37 07.00 &09 07 58.7 &-& &-& &1.38&$\pm0.11$ &-& &OB\
A520 &1 &04 54 01.11 &02 57 45.6 &6.3&$\pm0.5$ &-& &7.83&$\pm0.25$ &5.75&$\pm1.74$ &GBT OB (V)\
&&&&&&& &4.42&$\pm0.88$ & &\
&2 &04 54 03.72 &02 56 01.9 &7.7&$\pm1.7$ &-& &-& &0.49&$\pm0.15$ &GBT\
&*3* &*04 54 12.41* &*02 57 52.3* &*7.1*&$\it\pm0.5$ &-& &-& &*-0.15*&$\it\pm0.16$ &*GBT*\
&4 &04 54 17.14 &02 55 33.5 &14.1&$\pm1.0$ &-& &0.84&$\pm0.11$ &0.90&$\pm0.08$ &GBT OB\
&&&&&&& &1.09&$\pm0.09$\
&5 &04 54 21.16 &02 55 01.4 &26.5&$\pm1.6$ &-& &1.00&$\pm0.13$ &0.63&$\pm0.10$ &GBT OB\
&&&&&&& &0.74&$\pm0.14$\
A267 &1 &01 52 29.46 &00 59 31.8 &30.0&$\pm1.0$ &-& &2.75&$\pm0.20$ &3.12&$\pm0.23$ &GBT OB\
&2 &01 52 54.58 &01 02 08.2 &4.2&$\pm0.5$ &-& &7.55&$\pm0.24$ &7.05&$\pm1.12$ &GBT OB(V)\
&&&&&&& &5.53&$\pm0.73$\
&3 &01 52 34.35 &01 01 20.5 &-& &-& &-& &0.59&$\pm0.32$ &GBT\
RXJ0439.0+0715 &1 &04 39 01.26 &07 15 42.6 &30.6&$\pm1.4$ &-& &1.18&$\pm0.16$ &0.89&$\pm0.22$ &GBT OB\
ZWCL7160 &1 &14 57 08.06 &22 20 11.2 &13.2&$\pm0.6$ &-& &0.95&$\pm0.05$ &0.90&$\pm0.06$ &GBT OB\
&2 &14 57 08.06 &22 20 11.2 &3.9&$\pm0.4$ &-& &0.99&$\pm0.07$ &0.91&$\pm0.08$ &GBT OB\
&3 &14 57 15.18 &22 20 36.0 &16.5&$\pm1.3$ &-& &0.96&$\pm0.04$ &0.90&$\pm0.06$ &GBT OB\
A773 &1\* &09 17 45.39 &51 43 11.2 &2.7&$\pm0.5$ &-& & &-& &*Negligible*\
&2\* &09 18 01.81 &51 44 11.4 &3.1&$\pm0.5$ &-& & &-& &*Negligible*\
\

### Differencing

We employed the standard GBT nodding strategy, which is similar to the OCRA differencing scheme, in order to remove atmospheric contamination. A complete nod (ie the sequence of beam 1 on source, beam 2 on source, beam 2 on source, beam 1 on source) takes 1.5minutes. Each source was observed for at least one complete nod. The number of nods was governed by telescope scheduling, though we tried to include extra nods for sources expected to have lower flux densities.

### GBT flux density determinations

The CCB has a 14GHz bandwidth, split into four frequency sub–bands centred at 27.75, 31.25, 34.75 and 38.25GHz. We can thus measure simultaneously four separate flux densities for each source, from which we can derive a 30GHz value by fitting a power law to the available data. We use a Monte–Carlo method to estimate the uncertainties on the interpolated flux densities. For poor–quality data, we were unable to derive accurate 30GHz flux densities from the GBT data alone, as discussed in Section \[sec:add\]

Determining the 30GHz flux densities {#sec:add}
------------------------------------

The quality of our GBT data is rather mixed due to the wide range of observing conditions experienced. As most sources were only observed once, we are also unable to constrain source variability with GBT data alone. In order to supplement our 30GHz measurements, we make use of the flux densities reported in [@Coble2007] who observed all of our clusters except Abell 2390 at 28.5GHz with the OVRO and BIMA interferometers. Such interferometric data not only measure the SZ effect but also enable identification of point sources via the longest baselines, where the SZ signal will be negligible. The uncertainties on the @Coble2007 flux densities are generally a few tenths of a mJy, so given the proximity of the two observing frequencies, we are able to place tighter constraints on our 30GHz source flux densities by fitting to both datasets, which is particularly useful for sources with poor GBT data. In addition, we are able to identify variable sources and estimate the effect on our SZ measurements.

Most source flux densities are well constrained by a combination of GBT and OVRO/BIMA data. We fit a power law to the measurements in the four GBT frequency sub-bands, plus the interferometer data at 28.5GHz. For sources with good GBT data, the @Coble2007 points serve as a valuable check for variability, but otherwise have minimal effect. In the three cases where significant variability is found, we take the two differing flux density measurements, and use half the difference as an estimate of the uncertainty in our measurement due to variability. We then add this in quadrature to the measurement error. The source exhibiting the greatest variability, source 1 in Abell 2261, has little effect on our results since it lies well away from the cluster centre and data where it lies close to a reference arc are flagged out (Section \[sec:corr\]). Where no 28.5GHz flux density is available, we proceed with the GBT data alone. Such cases are unlikely to be crucial, as the non detection by OVRO/BIMA suggests the source is either rather faint, or well away from the central regions of the cluster. Many sources have GBT data which are usable but rather noisy. In such cases, the 28.5GHz data are invaluable in tying down the 30GHz flux density due to their comparatively small uncertainties.

Seven sources have no usable GBT data: two in A1835, one in A2219, one in A1763, one in A68 and two in A773. We now discuss our method for each source in turn. Source 1 in A1835 was measured twice by OVRO/BIMA, and also appears in NVSS. We fit a power law to the three measurements. As our observing frequency is close to that of OVRO/BIMA, assuming a constant spectral index is unlikely to have a large impact. We rule out significant variability. Source 2 in A1835 was also measured twice at 28.5GHz, but does not appear in NVSS. We consult FIRST for a lower frequency measurement, and proceed as for source 1. For source 2 in A2219, we again fit a power law to the OVRO/BIMA and NVSS datapoints. Source 6 in A68 does not appear in any of the lower frequency catalogues but is well measured by OVRO/BIMA. It is outside the OCRA beams so its flux density has only a small effect on the SZ measurement. Both sources in A773 are faint in NVSS and are not detected by OVRO/BIMA. The noise on the OVRO map is $0.078$mJy, giving a $5 \sigma$ upper limit of $0.39$mJy. As the sources in A773 lie well away from the OCRA beams, we are confident that their effects on our data are negligible.

Source corrections {#sec:corr}
------------------

Due to OCRA-p’s position switching strategy, contaminant radio sources can affect the data by producing a *positive* signal when they lie close to the cluster centre, or a rogue *negative* signal when they lie in the reference beam of the telescope. We simulate the effects of the sources based on their measured positions and flux densities, and the OCRA beam response. We then use the resulting effective flux density to correct the SZ data. Contamination through sources at 30GHz is generally low for bright clusters, producing only small corrections to the SZ data, and thus introduces little bias as a result of the limitations of the method employed. Although we are confident that residual biases are below the level of our errors, we recognise that the absence of 30GHz data is a limitation in some cases, and there may be additional issues to do with source variability. For clusters where radio sources fall in the OCRA reference beams, we are able to perform an additional check by flagging out contaminated data based on the range of parallactic angles affected by the source in question. We compare the flagged data with the corrected data, and adjust our final measurement error accordingly.

We recognise that our strategy may miss sources which are important at 30GHz. The flux limit of NVSS is $\sim2.5$mJy. If we consider a source of flux density $\sim2.4$mJy at 1.4GHz, and assume a typical spectral index of $-0.7$, extrapolating to 30GHz gives a flux density of $0.34$mJy. Even if such a source were to be located in the central regions of a cluster, the effect on our measurement would still be small relative to random errors. Not all sources have typical spectral indices, but we note that in the 16 clusters additionally observed by [@Coble2007], only 1 ‘new’ source, i.e. a source not previously identified in NVSS, was found. However, the quality of our knowledge of the population of sources with mJy flux densities at 30GHz is poor. We were able to run an additional check for ‘new’ sources lying in OCRA–p’s reference arcs. We binned the data by parallactic angle to look for contaminating sources that might produce false SZ effects. Although this subdivides the data, and so leads to a noise level that is higher by a factor of $2 - 3$ than the central measured flux density averaged over all parallactic angles, no previously uncatalogued sources were detected.

X-RAY DATA
==========

Deriving cluster parameters {#sec:X}
---------------------------

Our sample has 15 clusters in common with an upcoming X-ray paper, Giles et al. (in prep), some results from which were made available in advance for this work. The remaining two clusters (ZWCL1953 and RXJ1532.9+3021, which lie outside Giles et al.’s specified redshift range of $0.15<z<0.30$) were analysed using an identical method. The authors use *Chandra* ACIS–I and ACIS–S imaging and spectroscopy for each cluster as available via the archive. For the data preparation, they follow closely the method outlined in @Maughan2008. The *Chandra* analysis package used was CIAO[^1] version 4.2 and calibration database CALDB 4.1.4. As in @Maughan2008, blank sky, rather than local, backgrounds are used because the clusters fill a large fraction of the field of view.

We fit a gas density model by converting the observed surface brightness profile into a projected emissivity profile, which was then modelled by projecting the modifed $\beta$–model of [@Vikhlinin2006] along the line of sight (see [@Maughan2008] for details). Gas parameters and errors were determined from Monte Carlo realisations of the projected emissivity profile. At each data point a new randomised point was drawn from a Gaussian distribution centered on the model value at that point, with a standard deviation equal to the fractional measurement error on the original data point, multiplied by the model value. The error bar on this new randomised point was set to the same fractional size as the error bar on the original observed point.

The cluster temperature, gas mass and $R_{500}$ (the radius enclosing a mean density of 500 times the critical density at the cluster’s redshift) were then determined iteratively. The procedure followed was to extract a spectrum from within an estimated $R_{500}$ (with the central $15\%$ of that radius excluded), integrate the gas density profile to determine the gas mass within the estimated $R_{500}$, and thus calculate $Y_\mathrm{X}$ which is the product of $kT$ and the gas mass, and a low scatter proxy for total mass [@Kravtsov2006]. A new value of $R_{500}$ was then estimated from the $Y_\mathrm{X}$-$M$ scaling relation of @Vikhlinin2009,

$$\begin{aligned}
\label{e.ym}
M_{500} & = & E(z)^{-2/5}\,A_\mathrm{YM}\left(\frac{Y_X}{3\times10^{14}
M_{\odot}\mathrm{keV}}\right)^{B_\mathrm{YM}},\end{aligned}$$

with $A_\mathrm{YM}=5.77\times10^{14}h^{1/2}M_{\odot}$ and $B_\mathrm{YM}=0.57$. Here, $M_{500}$ is the mass within $R_{500}$ (allowing $R_{500}$ to be trivially computed), and

$$E(z)=\sqrt{\Omega_{\mathrm{M}}(1+z)^3 +
(1-\Omega_{\mathrm{M}}-\Omega_\Lambda)(1+z)^2 + \Omega_\Lambda}
\label{eqn:E}$$

describes the redshift evolution of the Hubble parameter. This $Y_{\mathrm{X}}$–$M$ relation assumes self-similar evolution (as $E^{-2/5}$), which has been shown to be a good description of observed clusters to $z\approx0.6$ [@Maughan2007]. The process was repeated until $R_{500}$ converged. In these spectral fits, the source emission was modelled with an absorbed thermal plasma APEC model in the 0.6–9keV band, with the absorbing column fixed at the Galactic value. $N_\mathrm{H}$ values were taken from the HI map by @Kalberla2005.

Our final cluster parameters, with $T_\mathrm{X}$, $L_\mathrm{X}$ and $Y_\mathrm{X}$ determined within a radius of $R_{500}$ are given in Table \[tab:X\].

  ---------------- ------- ---------------------- ------ ------------- ------ ---------------------- ------
  Cluster                                                $R_{500}$                                   
                                                         Mpc                                         
  A1835               8.41 ${}^{+0.32}_{-0.26}$     2.14 $\pm0.01$       1.20 ${}^{+0.05}_{-0.04}$    1.40
  ZW1953              6.05 $^{+0.51}_{-0.50}$       1.45 $\pm0.05$       0.58 ${}^{+0.05}_{-0.05}$    1.19
  ZW3146              7.09 $^{+0.35}_{-0.37}$       1.62 $\pm0.02$       0.77 $^{+0.04}_{-0.04}$      1.27
  RXJ1532.9+3021      5.62 $^{+0.72}_{-0.58}$       1.23 $\pm{0.05}$     0.45 $^{+0.06}_{-0.06}$      1.11
  A2390              10.20 $^{+0.35}_{-0.35}$       2.87 $\pm0.05$       1.79 $^{+0.06}_{-0.07}$      1.53
  A2219              10.33 $^{+0.43}_{-0.44}$       3.32 $\pm0.04$       2.10 $^{+0.09}_{-0.09}$      1.58
  RXJ2129.6+0005      6.04 $^{+0.53}_{-0.52}$       1.03 $\pm0.04$       0.52 $^{+0.05}_{-0.06}$      1.20
  A2261               6.56 $^{+0.28}_{-0.28}$       1.46 $\pm0.02$       0.77 $^{+0.03}_{-0.03}$      1.31
  A781                5.32 $^{+0.66}_{-0.43}$       1.02 $\pm0.05$       0.53 $^{+0.06}_{-0.04}$      1.18
  A697                9.22 $^{+0.66}_{-0.65}$       2.70 $\pm0.05$       1.55 $^{+0.11}_{-0.11}$      1.45
  A1763               7.31 $^{+0.47}_{-0.47}$       1.48 $\pm0.03$       0.89 $^{+0.06}_{-0.06}$      1.34
  A68                 7.14 $^{+0.88}_{-0.78}$       1.09 $\pm0.04$       0.60 $^{+0.07}_{-0.07}$      1.23
  A520                6.78 $^{+0.29}_{-0.16}$       1.49 $\pm0.02$       0.74 $^{+0.03}_{-0.02}$      1.31
  A267                4.53 $^{+0.49}_{-0.38}$       0.75 $\pm0.05$       0.29 $^{+0.03}_{-0.03}$      1.08
  RXJ0439.0+0715      5.32 $^{+0.38}_{-0.29}$       1.58 $\pm0.04$       0.57 $^{+0.04}_{-0.04}$      1.23
  ZW7160              4.64 $^{+0.18}_{-0.18}$       0.69 $\pm0.01$       0.28 $^{+0.01}_{-0.01}$      1.06
  A773                6.75 $^{+0.40}_{-0.26}$       1.13 $\pm0.02$       0.65 $^{+0.03}_{-0.03}$      1.27
  ---------------- ------- ---------------------- ------ ------------- ------ ---------------------- ------

\[tab:X\]

$\beta$–model fitting {#sec:beta}
---------------------

As mentioned in Section \[sec:obs\], the small separation of the OCRA-p beams relative to the extent of a cluster’s SZ decrement implies that we subtract out a fraction of the SZ signal due to our differencing method. This significantly reduces the measured signal compared to the intrinsic central signal. This is further exaggerated in cases where we have adjusted the pointing centre to avoid bright radio sources. In order to remove this effect and recover the ‘true’ SZ signal, we model the cluster atmosphere and estimate the fraction of the signal ‘missed’. (Note: this analysis was conducted purely to correct the SZ measurements and is separate from the derivation of X–ray cluster parameters discussed in Section \[sec:X\]). In our previous work ([@Lancaster2007]), we assumed that the gas density distribution, $n_{\mathrm{e}}(r)$ was well fit by a standard spherical $\beta$–model [@beta1]:

$$n_{\mathrm{e}}(r) = n_{\mathrm{e0}}\left(1+ \frac{r^2}{r^2_c}
\right)^{-3\beta/2}
\label{eq:single}$$

Here, we improve on this approach by also considering the double $\beta$–model, which may provide a better fit to clusters which exhibit peaked central emission due to their cool cores. We follow @Bonamente2006 in adopting the form

$$n_{\mathrm{e}}(r) = n_{\mathrm{e0}} \left[p\left(1+
\frac{r^2}{r^2_{c1}}\right)^{-3\beta_1/2} +(1-p)\left(1+
\frac{r^2}{r^2_{c2}}\right)^{-3\beta_2/2} \right]
\label{eq:double}$$

although in our analysis, we allow different $\beta$ for the two components. Using the *SHERPA*[^2] modelling and fitting application for each set of cluster data, we fit the 1–D radial profiles of the X–ray surface brightness by three models:

1.  [The standard single $\beta$–model]{}, (\[eq:single\])

2.  [A double $\beta$–model, (\[eq:double\]), with $\beta_1 = \beta_2$]{}

3.  [A double $\beta$–model, (\[eq:double\]), allowing $\beta_1$ and $\beta_2$ to assume different values.]{}

Chi squared ($\chi ^2$) statistics are used to compare the fits, since each radial bin is based on a large number of X–ray counts. For each cluster, we accept the simplest model for which the fit statistic implies a probability of getting a greater $\chi ^2$ by chance, $Q \ge 6$ per cent. For some clusters, the gas distribution is sufficiently disturbed that $Q < 6 $ per cent for all models fitted: in such cases we adopt the best fit. This is not ideal, but we expect the additional error to be at a low level in comparison with the random errors on the SZ data. We do not observe any systematic effect on our results due to the type of $\beta$–model used to correct the SZ data.

$\beta$–model parameters are presented in Table \[tab:beta\]. We return briefly to the analysis of [@Bonamente2006], with whom we have eight clusters in common, for comparison of results. For A697, A68, A267 and A773, both groups find that a single $\beta$–model provides an acceptable fit, and the derived parameters are in broad agreement. For their double $\beta$–model fits, @Bonamente2006 fix the two values to $\beta$ to be equal, whereas ours were allowed to vary. For RXJ2129.6+0005 and A2261, we find that a model with $\beta_1=\beta_2$ gives the best fit, and the model parameters are again in broad agreement with @Bonamente2006 For ZWCL3146, our two $\beta$–values differ but are very similar, meaning the overall model fit is in broad agreement with @Bonamente2006, and indeed our $\beta_1=\beta_2$ fit is also consistent. The exception is A1835, where we find the best fitting model to be a double $\beta$–model with $\beta_1=0.675$, $\beta_2=1.059$, $\theta_1=35.6$arcsec and $\theta_2=13.2$arcsec, whereas @Bonamente2006 find $\beta=0.797$, $\theta_1=63.6$arcsec and $\theta_2=13.2$arcsec. Our model with $\beta_1=\beta_2$ is also slightly discrepant, as our best fit parameters are $\beta=0.695$, $\theta_1=92.1$arcsec and $\theta_2=19.1$arcsec. The source of discrepancy may well be in the use of differing data sets. We use ACIS-I observations totalling 200ks, whereas @Bonamente2006 only had access to 30ks of ACIS-S data which has a significantly smaller field of view.

We use the model gas distribution to simulate an OCRA observation. Many clusters were observed at positions slightly shifted from their X-ray maxima in order to avoid bright radio sources as far as possible. We consult our OCRA data for the range of parallactic angles sampled, and include the convolution with the OCRA beam response. We can thus estimate the fraction $F$ of the ‘true’ SZ signal measured in the real observations, and ‘correct’ our measurement (multiply by $1/F$) to recover the SZ surface brightness that would be measured by an ideal instrument with narrow beams at infinite separation. We note for clusters where the double $\beta$–model provides an improved fit, the value of F varies from the simpler single $\beta$–model by $\sim10$ per cent. This change is thus significant for the scientific interpretation of our results. Correction factors are presented in Table \[tab:res\]. Asymmetries in the SZ signal are assumed small in our analysis. Checks for such asymmetries by binning the SZ data by parallactic angle provide only a low signal–to–noise control against such effects which would cause our estimates of $F$ to be incorrect. The X–ray isophotes of these clusters are not always circular, but are most sensitive to asymmetry towards the cluster centre, where a relatively small fraction of the SZ effect originates. We estimate that the value of $F$ could be systematically uncertain by $20\%$ from this effect. This uncertainty dominates over that from varying the density model from $\beta$ or double-$\beta$ form to the [@Vikhlinin2006] form.

  ---------------- ------- ---------------------- ------- -------------------- ------- ---------------------- ------ ------------------ ------
  Cluster                                                                          $p$                                                  
                                                                                                                                        
  A1835              0.675 $^{+0.001}_{-0.001}$      35.6 $^{+0.6}_{-0.6}$       1.059 $^{+0.010}_{-0.010}$     13.2 $^{+0.1}_{-0.1}$    0.10
  ZWCL1953           0.972 $^{+0.147}_{-0.122}$     171.6 $^{+22.3}_{-21.9}$     0.972 $^{+0.147}_{-0.122}$     43.3 $^{+5.7}_{-5.1}$    0.04
  ZWCL3146           0.700 $^{+0.008}_{-0.007}$      23.6 $^{+0.9}_{-0.8}$       0.652 $^{+0.032}_{-0.023}$      4.8 $^{+0.3}_{-0.2}$    0.14
  RXJ1532.9+3021     0.650 $^{+0.011}_{-0.010}$      12.0 $^{+3.6}_{-1.7}$       0.650 $^{+0.011}_{-0.010}$      6.3 $^{+1.6}_{-1.8}$    0.40
  A2390              0.831 $^{+0.010}_{-0.009}$      92.9 $^{+1.6}_{-1.6}$       0.831 $^{+0.010}_{-0.009}$     18.5 $^{+0.3}_{-0.3}$    0.10
  A2219              0.725 $^{+0.015}_{-0.013}$      85.6 $^{+6.2}_{-5.0}$       0.725 $^{+0.015}_{-0.013}$     42.0 $^{+5.4}_{-5.6}$    0.58
  RXJ2129.6+0005     0.609 $^{+0.011}_{-0.011}$      24.4 $^{+2.9}_{-2.5}$       0.609 $^{+0.011}_{-0.011}$      5.0 $^{+0.8}_{-0.7}$    0.12
  A2261              0.598 $^{+0.003}_{-0.003}$      30.7 $^{+1.4}_{-1.4}$       0.598 $^{+0.003}_{-0.003}$     11.8 $^{+0.8}_{-0.8}$    0.33
  A781               0.706 $^{+0.041}_{-0.035}$      79.5 $^{+7.5}_{-6.6}$           -                             -                      -
  A697               0.648 $^{+0.008}_{-0.008}$      48.7 $^{+1.5}_{-1.5}$           -                             -                      -
  A1763              0.586 $^{+0.007}_{-0.006}$      48.0 $^{+1.6}_{-1.6}$           -                             -                      -
  A68                0.742 $^{+0.003}_{-0.002}$      54.5 $^{+3.5}_{-3.3}$           -                             -                      -
  A520               0.831 $^{+0.003}_{-0.003}$     117.5 $^{+0.6}_{-0.6}$           -                             -                      -
  A267               0.633 $^{+0.011}_{-0.011}$      34.4 $^{+1.5}_{-1.4}$           -                             -                      -
  RXJ0439.0+0715     0.735 $^{+0.022}_{-0.020}$      52.6 $^{+4.3}_{-3.9}$       0.735 $^{+0.022}_{-0.020}$     13.0 $^{+1.6}_{-1.5}$    0.23
  ZWCL7160           0.850 $^{+0.112}_{-0.077}$      22.2 $^{+4.3}_{-3.4}$       0.580 $^{+0.005}_{-0.006}$      7.2 $^{+0.3}_{-0.4}$    0.09
  A773               0.769 $^{+0.041}_{-0.035}$     129.6 $^{+14.3}_{-14.4}$     0.769 $^{+0.041}_{-0.035}$     48.2 $^{+2.7}_{-2.6}$    0.11
  ---------------- ------- ---------------------- ------- -------------------- ------- ---------------------- ------ ------------------ ------

\[tab:beta\]

RESULTS AND SCALING RELATIONS {#sec:res}
=============================

Sunyaev Zel’dovich Results
--------------------------

We present our measurements for the Sunyaev Zel’dovich effects in our 17-cluster sample in Table \[tab:res\]. For each cluster, we first give the raw OCRA flux density as measured directly by the telescope, followed by the revised figure once all radio source contamination has been corrected for. We then give our estimate of the fraction of total signal measured due to the instrumental response. Finally, we give the fully–corrected temperature decrement expressed in brightness temperature units, and the corresponding central Comptonisation parameter, $y_0 = -0.19 \times (\Delta T/K)$. We detect 13 clusters at significance $\ge 3 \sigma$.

Our trail fields serve as a valuable test of our observing strategy. We present differential and cumulative distributions for both the source corrected cluster target and trail fields in Figure \[fig:hist\]. The flux densities of the trail fields are distributed around zero with small scatter, and their distribution is clearly distinct from that of the SZ data, confirming that our observing strategy is effective.

![image](histogram.ps){width="0.6\linewidth"}

  ---------------- ------- ----------- ---------- ----------- -------------- ------- ----------- ---------------
  Cluster                                Fraction                 $y_0$                          
                                         Measured              $\times10^4$                      
  A1835              -2.05 $\pm0.46$        -3.82 $\pm0.47$       0.363        -4050 $\pm500$     7.70$\pm$0.95
  ZW1953             -1.65 $\pm0.37$        -1.65 $\pm0.37$       0.404        -1570 $\pm350$     2.98$\pm$0.67
  ZW3146             -1.06 $\pm0.39$        -0.78 $\pm0.41$       0.181        -1660 $\pm870$     3.15$\pm$1.65
  RX1532.9+3021      -1.41 $\pm0.33$        -1.11 $\pm0.34$       0.166        -2540 $\pm790$     4.83$\pm$1.50
  A2390              -0.78 $\pm0.20$        -1.02 $\pm0.22$       0.176        -2230 $\pm480$     4.24$\pm$0.91
  A2219              11.10 $\pm0.38$        -2.76 $\pm0.40$       0.379        -2810 $\pm400$     5.34$\pm$0.76
  RXJ2129.6+0005     -0.13 $\pm0.37$        -1.46 $\pm0.37$       0.329        -1710 $\pm430$     3.25$\pm$0.82
  A2261              -3.19 $\pm0.30$        -1.35 $\pm0.32$       0.321        -1610 $\pm390$     3.06$\pm$0.74
  A781                0.68 $\pm0.42$        -1.19 $\pm0.53$       0.330        -1390 $\pm620$     2.64$\pm$1.18
  A697               -3.01 $\pm0.46$        -3.01 $\pm0.46$       0.422        -2740 $\pm420$     5.21$\pm$0.80
  A1763              10.58 $\pm0.80$        -3.11 $\pm0.82$       0.296        -4040 $\pm1070$    7.68$\pm$2.03
  A68                -0.89 $\pm0.48$        -1.22 $\pm0.48$       0.451        -1040 $\pm410$     1.98$\pm$0.78
  A520               -1.97 $\pm0.47$        -1.80 $\pm0.48$       0.268        -2580 $\pm680$     4.90$\pm$1.29
  A267               -2.73 $\pm0.43$        -1.99 $\pm0.57$       0.471        -1620 $\pm470$     3.08$\pm$0.89
  RX0439.0+0715      -1.22 $\pm0.43$        -1.27 $\pm0.43$       0.351        -1390 $\pm470$     2.64$\pm0.89$
  ZWCL7160           -1.10 $\pm0.34$        -1.14 $\pm0.34$       0.188        -2330 $\pm700$     4.43$\pm$1.33
  A773               -1.20 $\pm0.32$        -1.21 $\pm0.32$       0.382        -1210 $\pm330$     2.30$\pm$0.63
  ---------------- ------- ----------- ---------- ----------- -------------- ------- ----------- ---------------

\[tab:res\]

SZ–X-ray scaling relations {#sec:scaling}
--------------------------

According to the self-similar model of galaxy cluster formation, which assumes evolution solely under gravity, we expect simple scalings between various cluster parameters. Following [@Morandi2007] we derive the following relations, each of which may be tested using OCRA data. We note that while it is advantageous to use the integrated Compton parameter for such studies (e.g. [@Benson2004], [@Bonamente2008]), deriving this from OCRA data would introduce further uncertainty since we have measured the cluster structures from the X–ray data alone. Thus, we restrict ourselves to discussing the central Comptonisation, $y_0$. The scalings of interest are first, the relation between central Comptonisation and X-ray temperature

$$y_0 \propto T_{\mathrm{X}}^{3/2} E(z),$$

second, that between the Comptonisation and X-ray luminosity

$$y_0 \propto L_{\mathrm{X}}^{3/4} E(z)^{1/4},
\label{eq:yL}$$

and finally, that between Comptonisation and $Y_{\mathrm{X}}$, the product of gas mass and X-ray temperature

$$y_0 \propto Y_{\mathrm{X}}^{3/5} E(z)^{8/5},$$

where $E(z)$ is as given in Equation \[eqn:E\]. Deviation from the expected scalings would suggest contributions from non-gravitational processes, and redshift dependence in the slope or amplitude of the scalings should provide evidence on the nature of these processes.

We present our scaling relations in figures \[yT\] to \[yY\]. For each plot, we colour–code the datapoints based on the reliability of the OCRA data. The most reliable points, i.e. those with no obvious issues, are shown as red circles. Points which have a signal to noise ratio of less than 3 are black squares. This is an arbitrary cut–off, and we see no reason to exclude these points from our analysis; however as an aid to the reader we choose to mark them clearly. Blue crosses represent clusters which may suffer from residual source contamination, i.e. those which contain a bright ($> 3$mJy) source, for which we have corrected, in their central regions. Due to the unavoidable 10 per cent uncertainty in the calibration of the GBT data, we cannot be confident that the effects of these sources have been accurately removed, and thus choose to ignore these three clusters in our subsequent analyses.

             Correlation              Expected slope  Data used                           Best-fit slope   Fit statistic  
  ---------------------------------- ---------------- ---------------------------------- ---------------- --------------- --
        $y_0, T_{\mathrm{X}}$              1.5        SZ, X-ray                           $1.82\pm0.72$        2.39       
                                                      SZ (no source correction), X-ray    $1.94\pm1.10$        6.64       
        $y_0, L_{\mathrm{X}}$              0.75       SZ, X-ray                           $0.77\pm0.21$        2.46       
                                                      SZ (no source correction), X-ray    $0.71\pm0.19$        6.86       
        $y_0, Y_{\mathrm{X}}$              0.6        SZ, X-ray                           $0.65\pm0.21$        2.50       
                                                      SZ (no source correction), X-ray    $0.55\pm0.21$        7.23       
   $L_{\mathrm{X}}, T_{\mathrm{X}}$        0.5        X-ray only                          $0.49\pm0.04$        3.40       
   $Y_{\mathrm{X}}, T_{\mathrm{X}}$        2.5        X-ray only                          $2.19\pm0.16$        1.85       
   $Y_{\mathrm{X}}, L_{\mathrm{X}}$        1.25       X-ray only                          $1.24\pm0.05$        4.29       
           \[tab:summary\]                                                                                                

The fitting was performed in the linear space of observables, rather than log–log space, and involved the minimisation of the squared and error–weighted residuals from a power law of the form $y=Ax^{\mathrm{B}}$, where $A$ and $B$ are to be estimated. The space of acceptable values of $A$ and $B$ was sampled by taking $10^5$ datapoint realisations, distributed according to their errors, assumed Gaussian. The best–fit slopes are summarised in Table \[tab:summary\], with the values of the fit statistic. Figure \[yT\] shows the scaling between $y_0$ and $T_{\mathrm{X}}$. For this relation, the space of acceptable $(A,B)$ parameters extends over a wide range in both parameters, so that the slope should be regarded as essentially unconstrained, although the best fit value $B = 1.82\pm0.72$ is in agreement with the predicted value $B = 1.5$ from self–similar models. The other two relations show modest covariances between $A$ and $B$ and are more robust. Figure \[yL\] shows the scaling between $y_0$ and $L_{\mathrm{X}}$. We obtain a slope of $0.77\pm0.21$ in good agreement with self–similarity, for which Equation \[eq:yL\] predicts 0.75. Finally, Figure \[yY\] depicts the scaling between $y_0$ and $Y_{\mathrm{X}}$. We derive a slope of $0.65\pm0.21$ which is in good agreement with the predicted $B = 0.6$. For all three cases, the intrinsic scatter in $y_0$ can be estimated by the additional error required to bring the fit statistic down to a statistically acceptable level $\sim1.0$ We find that the intrinsic scatter in $y_0$ is about 25 per cent. The errors on $B$ are large for all fits because of the modest range of cluster masses in our sample, but we see no evidence for departure from self–similar models of cluster evolution. Our results for the $y_0/L_{\mathrm{X}}$ and $y_0/T_{\mathrm{X}}$ scalings are consistent with those obtained by [@Morandi2007], although their gas masses and temperatures were defined over significantly different regions of the clusters (within $R_{2500}$), and their scalings have smaller slope errors because of the wider range of $T_{\mathrm{X}}$ of their clusters.

It is interesting to compare our scalings with those that we would have obtained if source corrections were not available to us, since blank–sky surveys for cluster SZ effects could attempt to constrain the scalings with inadequate source data. We find that the scatter induced by the radio sources degrades the fit quality for each correlation significantly, as illustrated by the fit statistics shown in Table \[tab:summary\]. Although the results broadly agree with self–similar predictions, the residual contamination tends to cause the scalings to appear too flat. For instance, in the high signal to noise sub–sample where the effect of sources is clearest, the slope of the $y_0/L_\mathrm{X}$ relation is found to be flatter by 0.27 if a fit is attempted before source corrections are made. A similar result is found for the $y_0/Y_{\mathrm{X}}$ scaling relation.

X-ray only scaling relations {#sec:Xscale}
----------------------------

We check the consistency of the set of clusters with the more usual X–ray scaling relations between $L_{\mathrm{X}}$ and $T_{\mathrm{X}}$, $Y_{\mathrm{X}}$ and $T_{\mathrm{X}}$, and $Y_{\mathrm{X}}$ and $L_{\mathrm{X}}$ (e.g. [@Morandi2007]) in the same way as for the SZ / X–ray scaling relations (results also summarised in Table \[tab:summary\]). The slopes that we measure, of $0.49 \pm 0.04$, $2.19 \pm 0.16$, and $1.24 \pm 0.05$, are consistent with the similarity expectations of 0.5, 2.5 and 1.25. As the slight discrepancy for the correlation between $Y_{\mathrm{X}}$ and $T_{\mathrm{X}}$ is not of high statistical significance, this study shows our sample to be representative of the population of hot ($T
\gtrsim 6$keV) clusters.

DISCUSSION AND CONCLUSION
=========================

We have observed a complete sample of galaxy clusters using OCRA–p, and studied the scaling of the central Compton parameter, $y_0$, with various X–ray quantities. For each relation, we find slopes in good agreement with the predictions from self–similar models.

Our study has similarities to that of [@Morandi2007], and the two samples have 7 clusters in common, although we imposed stricter initial selection criteria. @Morandi2007 consider scalings with $y_0$, so we are able to perform a direct comparison. They find that the $y_0/T_{\mathrm{X}}$ relation deviates by $\sim3\sigma$ from the self–similar prediction, in the sense of being steeper. While our data are consistent with this result, we note that they are essentially unable to constrain this relation at any level. Regarding the $y_0/L_{\mathrm{X}}$ scaling, they find a slope of $0.61\pm0.05$, which is in good agreement with our work. Thus we see no sign of the flattening that would have resulted from the presence of an undetected set of contaminating sources (Section \[sec:scaling\]).

More recently, [@Bonamente2008] studied a sample of 38 clusters observed with OVRO/BIMA and *Chandra*. They considered the integrated Comptonisation, $Y$, and in particular its scaling with $T_{\mathrm{X}}$, $M_{\mathrm{tot}}$ and $M_{\mathrm{gas}}$. They find good consistency with predictions from self–similar models for all scaling relations.

Our analysis has been limited by the narrow region of parameter space sampled, particularly for the scalings with $T_{\mathrm{X}}$. Based on our results, a sample double the size would provide a good test of the $y_0/Y_{\mathrm{X}}$ scaling; this will be presented in an upcoming paper in which we extend our sample to 33 clusters. To fully test the other relations, a sample four times larger is required. Thinking forwards to the prospects for blind cluster surveys, it is interesting to note that in this study we would have detected Sunyaev Zel’dovich effects for 14 out of 17 clusters in our sample with no source information. Of course, to accurately determine the strength of the SZ effect in each case, and indeed to derive any cluster parameters, good knowledge of the radio source environment is essential, and any attempt to test the scaling relations without taking sources into account will fail (see Section \[sec:scaling\]). That this is true for sub–mm sources, as well as cm–wave studies, is clear from the large population of sub–mm sources known to be lensed by massive clusters (e.g. [@Johansson2010]).

ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============

We acknowledge support for the design and construction of OCRA-p from the Royal Society Paul Instrument Fund, and funds for the data acquisition system and operation on the telescope from the Ministry of Science in Poland via grant number N N203 39043 who, along with STFC, also supported the scientific exploitation of the completed system. The GBT is a National Radio Astronomy Observatory (NRAO) instrument, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We extend special thanks to all the staff at the GBT, particularly Brian Mason and Carl Bignell, for their essential support and prompt re–scheduling.

\[lastpage\]

[^1]: http://cxc.harvard.edu/ciao/

[^2]: http://cxc.harvard.edu/sherpa/
