We report the results of a detailed analysis of the temperature structure of the X-ray emitting plasma halo of M~87, the cD galaxy of the Virgo Cluster. Using the MEKAL model, the data provide strong indications that the intracluster medium has a single phase structure locally, except the regions associated to the radio structures. The deprojected spectrum at each radius is well fitted by a single temperature MEKAL model, except for the very central region ($<$ 2 arcmin) which seems to be affected by the jet and radio lobe structure. The temperature of the intracluster plasma is 1 keV at the center and gradually increases to 2.5 keV at 80 kpc. We have also fitted spectra using the APEC code. Although the large changes of the strength of K$alpha$ lines causes a discrepancy between the Fe-L and Fe-K lines for the APEC results, the overall temperature structure has not changed. There is no sign of excess absorption in the spectral data. The single-phase nature of the intracluster medium is in conflict with the standard cooling flow model which is based on a multi-phase temperature structure. In addition, the signature of gas cooling below 0.8 keV to zero temperature is not observed as expected for a cooling flow. The gravitational mass profile derived from the temperature and density distribution of the intracluster gas shows two distinct contributions that can be assigned to the gravitational potential of the cD galaxy and the cluster. The central temperature of the intracluster medium agrees well with the potential depth and the velocity dispersion of the cD galaxy. The latter result implies that the central region of the intracluster medium is equivalent to a virialized interstellar medium in M 87.
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