X-ray Properties of B2 1028+313: A Quasar at the Center of the Abell Cluster A1030

X-ray observations with the ROSAT HRI and with ASCA are presented for the nearby radio quasar B2 1028+313, which is located in the cD galaxy at the center of the Abell cluster A1030. We also analyze archival ROSAT PSPC observations. We find that the X-ray emission is dominated by the quasar. The flux varied by a factor of about two between the ROSAT HRI and ASCA observations, which were about one year apart. The X-ray spectrum of the quasar is fit by a single power-law, except at low energies where there is a soft excess. Although the shape of the soft excess is not strongly constrained, it can be fit by a blackbody with a temperature of about 30 eV. There was evidence for extended X-ray emission, which contributed about 25% of the total flux. However, this emission does not appear to be normal X-ray emission from intracluster gas or a central cooling flow. The extended X-ray emission appears to be quite soft; if its spectrum is modeled as thermal emission, the temperature is ~0.2 keV, rather than the 5-10 keV expected for ICM emission. The radial surface distribution of the emission was not fit by either the beta model which usually describes ICM emission, or by a cooling flow model. The ASCA and ROSAT spectra showed no convincing evidence for a thermal component with a cluster-like temperature, either in the overall spectral shape or in emission lines. In addition, the ROSAT PSPC image showed that the extended X-ray emission was highly elongated to the NNW and SSE, in the same direction as the extended radio emission from the quasar. We suggest that the extended emission is inverse Compton emission from the extended radio lobes.