A measurement of H_0 from Ryle Telescope, ASCA and ROSAT observations of Abell 773

We present new Ryle Telescope (RT) observations of the Sunyaev Zel'dovich (SZ) decrement from the cluster Abell 773. The field contains a number of faint radio sources that required careful subtraction. We use ASCA observations to measure the gas temperature and a ROSAT HRI image to model the gas distribution. Normalising the gas distribution to fit the RT visibilities returns a value of H_0 of 77 (+19,-15) km/s/Mpc (1-sigma errors) for an Einstein-de-Sitter universe, or 85 (+20,-17) km/s/Mpc for a flat model with Omega_Lambda = 0.7. The errors quoted include estimates of the effects of the principal errors: noise in the SZ measurement, gas temperature uncertainty, and line-of sight depth uncertainty.


INTRODUCTION
We have previously reported the detection of a Sunyaev-Zel'dovich (SZ) decrement (Sunyaev & Zel'dovich 1972) towards the z = 0.217 cluster Abell 773 using the Ryle Telescope (RT) (Grainge et al 1993). (The SZ effect in this cluster has also been mapped by the millimeter array of the Owens Valley Radio Observatory (Carlstrom, Joy & Grego 1996).) The RT observations of Abell 773 form part of a continuing programme to observe an X-ray luminosity-limited sample of rich, intermediate-redshift clusters in order to measure H0 by combining SZ and X-ray observations (Jones et al.2001). Such programmes (e.g. Reese et al., 2002;Mason, Myers and Readhead, 2001; see also Birkinshaw, 1999 for a review) are direct measurements of H0 free from distanceladder arguments.
In Grainge et al. 1993 we did not calculate an estimate of H0 because no suitable X-ray image of A773 and no esti-mate of its gas temperature existed. A ROSAT HRI image and ASCA spectroscopic data have since become available, and we have also made additional RT observations. These now enable us to make an estimate of the Hubble constant from this cluster, which, when combined with other clusters from the sample, will give an estimate of H0 unbiased by the individual shapes and orientations of the clusters.

RYLE TELESCOPE OBSERVATIONS AND SOURCE SUBTRACTION
The RT (Jones 1991) is an east-west synthesis telescope of 13-m antennas with a bandwidth of 350 MHz and an average system temperature for these observations of 65 K at an observing frequency of 15.4 GHz. We used five antennas in a compact configuration, giving two baselines of 18 m, three of 36 m, and five more out to 108 m. The short baselines alone are sensitive to the SZ signal; the longer ones are used to recognize and subtract the radio sources in the field that would otherwise mask the SZ decrement. We have made a total of 30 12-h observations of A773, each with the pointing centre RA 09 h 17 m 51 s .91, Dec. +51 • 43 ′ 32 ′′ (J2000). Phase calibration using 0859+470 and flux calibration using 3C 48 and 3C 286 were carried at as described in Grainge et al (1993). Similarly, we used the Postmortem package (Titterington 1991) to flag the data for interference and antenna pointing errors, and to weight them in accord with the continuously monitored system temperature of each antenna. As a standard check, we used the Aips package to make a map of each 12-h run and then combined the data. We removed radio sources from the data by a simultaneous maximum-likelihood fit to several point sources and the SZ effect using a technique described by Grainger et al (2002). We use a model for the SZ signal as a function of baseline that is based on the β-model fit to the X-ray image described below (Section 3). We simultaneously fit flux densities for trial sources whose initial positions are determinied both from a map made from just the long-baseline data (> 2 kλ), and from a VLA 1.4-GHz image of the cluster field ( Figure 1). This allows us to fit the optimum flux densities of sources whose existence we know of from the VLA image but which would not give a significant detection from the RT data alone. The postitions and fitted flux densities are given in Table 1. The image made from the long (> 2kλ) source-subtracted baselines is consistent with noise ( Figure  2).
To image the decrement, we removed the sources in Table 1 from all the visibilities and made a short-baseline map from baselines shorter than 1 kλ, and CLEANed this. The resulting image is shown in Figure 3. The decrement is of −527 µ Jy beam −1 with a noise (1-σ) of 60 µJy beam −1 ; the beam is 152 × 119 arcsec FWHM. Also shown is the X-ray image of the cluster; it can be seen that the alignment with the X-ray image is very good. The extension of the SZ image to the north-east is of marginal significance. The magnitude of the decrement is consistent with that of −590 ± 116 µJy, in the same beam, reported in Grainge et al (1993).
An alternative way of looking at the data is shown in Figure 4, which shows the real part of the source-subtracted visibilities binned radially, along with the best-fitting model based on the X-ray data. These data have the advantage, unlike the image pixels, of having independent gaussian noise on each point; it is these that are used in the fitting for H0.

X-RAY OBSERVATIONS AND FITTING
We measure the gas temperature from ASCA observations on 1994 April 29 of 46240 s (GIS) and 39904 s (SIS), using standard XSPEC tools. Times of high background flux were excluded and both GIS and SIS data were used. We took the Galactic absorbing column density predicted by Dickey and Lockman (1990) in the direction of A773 of 1.3 × 10 24 H atoms m −2 . Using a Raymond-Smith model, we find a temperature of 8.7 ± 0.7 keV (90%-confidence error bounds) and a metallicity of 0.25 solar. The 2-10 keV flux from A773 is 6.7±1.0 x 10 −13 W m −2 . Our temperature estimate is consistent with that of Allen and Fabian (1998) who find a temperature of 9.29 +0.69 −0.60 keV (90%-confidence error bounds).
For the X-ray surface-brightness fitting we used a ROSAT HRI image of A773 with an effective exposure of 16518 s obtained on 13-15 April 1994 and analysed using standard ASTERIX routines. We calculate the ROSAT HRI count rate, given our estimates of metallicity and Galactic column and with the K-correction appropriate to the redshift of A773, to be 1.53(±0.08) × 10 −69 counts s −1 from a 1 m 3 cube of gas of electron density 1 m −3 at the temperature of A773 and at a luminosity distance of 1 Mpc.
We then fitted an ellipsoidal King profile to the X-ray image. Since the high spatial resolution of the HRI leads to a low count rate per pixel, we use Poisson rather than Gaussian statistics to fit for the measured count in each pixel. For ci counts measured at position xi, and for a mean number f (xi|a) of counts predicted by the model given parameters a (such as core radius), the probability of obtaining ci counts is and the most likely value of a can be obtained in a computationally efficient way by maximizing We fitted an ellipsoidal King profile to the HRI data with θ1 and θ2 as the perpendicular angular sizes in the plane of the image, assuming that the length along the line of sight is the geometric mean of the other two. We find θ1 = 60 ′′ and θ2 = 44 ′′ , with the major axis at position angle 16 • , β = 0.64, and central electron density n0 = 6.80×10 3 h −1/2 50 m −3 where H0 = 50 h50 km s −1 Mpc −1 . Fig 5 shows the HRI image, the model, and the residual image with the best model subtracted. To assess the goodness of fit, we made 50 realisations of the image with the appropriate Poisson noise added, and calculated the mean and standard deviation of their Poisson likelihoods. The likelihood of the observed HRI image is 0.32 standard deviations from the mean; we therefore conclude that the fit is good and the cluster is well represented by a β model.
There is a strong degeneracy in the fit between β and θ1,2; however this has little effect on the comparison with the SZ data and the derived value of H0. Figure 6 shows the likelihood contours for the fit in the β-θ1 plane, marginalised over n0 and using the best-fit value of the axial ratio (which is very well constrained). Overlaid are the contours of predicted mean observed SZ flux density on the shortest RT baseline. It can be seen that despite the degeneracy between β and θ1, the range of SZ flux densities corresponding to the 1-σ limits of the model fit is only ±3%. Since the SZ flux density varies as H −2 0 , this corresponds to a 6% error in H0 due to the model fitting. This lack of sensitivity to the β-θ degeneracy is characteristic of observations that are sensitive to spatial frequencies around the cluster core size (see eg Reese et al (2000)) and contrasts with the sensitivity to the model fitting of measurements that measure only lower spatial frequencies (eg Birkinshaw & Hughes(1994)).

H0 ESTIMATION
To measure H0, we compared the real SZ data with a simulation of the SZ effect from the X-ray gas model. We use the expression of Challinor & Lasenby (1998) to provide a relativistic correction to the standard non-relativistic SZ expression; in the case of A773, the effect is to increase our estimate of the y-parameter by 2.4%. We then simulated RT observations of the SZ effect due to the model gas distribution and to compared these with the real source-subtracted RT visibilities on the same baselines, and adjusted H0 to get the best fit. Using our temperature of 8.7±0.7 keV we find H0 = 77 +13 −11 km s −1 Mpc −1 , assuming an Einstein-de-Sitter universe. The 1-σ error quoted is that due solely to noise in the SZ data. For the best fit β, θ1,2 model, the corresponding central density n0 is 8.44 × 10 3 m −3 and the central decrement 737 ± 85µK.  consider at some length the contributions to error in the H0 determination from A1413. The situation in A773 is very similar. The dominant contributions to the error in H0 in A773 are ±16% from noise in the SZ measurement, ±12% from our estimation of the gas temperature and a likely error of ±14% from the uncertain line-of-sight depth. This is obtained by considering the range of axial ratios of simulated clusters that is needed to reproduce the projected axial ratio distribution observed in clusters with redshift similat to that of A773 (Grainger 2001). Clearly this estimate is rather uncertain for a single object, but can be significantly reduced by averaging a sample of clusters with random orientations. Table 2

CONCLUSIONS
Using ASCA, ROSAT HRI, and RT observations of A773, we find: (i) there are eight radio sources detectable in the field of the cluster that we have removed from the data, which would otherwise contaminate the measurement of the SZ effect; (ii) the correlated fitting errors on the shape parameters β and θ have negligable effect on the derived value of H0, a feature characteristic of observations on the scale of the cluster core size; (iii) the estimated value of H0 is 77 +19 −15 km s −1 Mpc −1 if (Ωm, ΩΛ) = (1.0, 0.0) or 85 +20 −17 km s −1 Mpc −1 if (Ωm, ΩΛ) = (0.3, 0.7) , where the 1-σ error bars include estimates from the main sources of error-noise in the SZ data, X-ray temperature uncertainty, and uncertain line-of-sight depth.      Overlaid are dotted contours of the predicted mean flux on the shortest RT baseline for the given model parameters (using the most likely value of n 0 ); the contour interval is 40µJy. Since the normalization of the SZ flux relative to the X-ray is what controls the fit for h, this Figure shows the insensitivity of our H 0 determination to the degeneracy in the X-ray model fit.