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We report a new measurement of the velocity dispersion profile within $1'$ (3 pc) of the center of the globular cluster M15 (NGC 7078), using long-slit spectra from the 4.2-m William Herschel Telescope at La Palma Observatory. Spectra for a total of 23 slit positions were obtained during two observing runs. Each run used a set of parallel positions to map out the central region of the cluster; the position angle used during the second run was orthogonal to that used for the first. The spectra are centered near the Ca~II infrared triplet at 8650~\AA, with a spectral range of 400~\AA.
We determined radial velocities by cross-correlation techniques for 130 cluster members. A total of 32 stars were observed more than once. Internal and external comparisons indicate a velocity accuracy of about 4~\mbox{km~s$^{-1}$}. The velocity dispersion profile rises from about $\sigma=8~\mbox{km~s$^{-1}$}$ near $1'$ from the center of the cluster to about $\sigma=12~\mbox{km~s$^{-1}$}$ at $20''$. Inside of $20''$ the dispersion remains approximately constant with no evidence for a sharp rise near the center. This last result stands in contrast with that of Peterson et al. \ (1989, ApJ, 347, 251) who found a central velocity dispersion of $25\pm7~\mbox{km~s$^{-1}$}$, based on a line broadening measurement. Our velocity dispersion profile is in good agreement with those determined in the recent studies of Gebhardt et al. \ 1994 (ApJ, in press) and Dubath~\& Meylan (1994, A\&A, in press).
The behavior of the central velocity dispersion profile of M15 is consistent with the predictions of Fokker-Planck models developed by Grabhorn et al. \ (1992, ApJ, 392, 86) and Phinney (1993, ASP Conf.\ Ser., 50, 141) for globular clusters undergoing core collapse. These models predict the presence of central populations of a few $\times 10^4$ degenerate remnants with masses exceeding about $1~M_\odot$, of which a few $\times 10^3$ are $1.4~M_\odot$ neutron stars. There appears to be no need to invoke the presence of a massive central black hole in M15.
