The local Galactic escape speed

Abstract

Statistical techniques for estimating the local Galactic escape speed from a uniform sample of high-velocity stars are presented for the cases of exact and uncertain observations. The former case is appropriate for high-precision radial velocities, and the latter for tangential and space velocities, which are usually derived from uncertain distances and proper motions. These two techniques have been applied to the Carney-Latham sample of high-velocity stars. It is found that the sample is too small to estimate the escape speed precisely without prior knowledge of the form of the high-velocity tail of the velocity distribution. For plausible assumptions regarding this distribution, the radial velocities of Carney-Latham stars suggest that the escape speed lies in the range from 450 to 650 km s-1 (90% confidence). The tangential and space velocities are too uncertain to improve this estimate. The strongest constraint on the escape speed is that it must exceed 430 km s" 1 based on the radial velocity of the star G166-37, assuming this star is bound to the Galaxy. Of course, even if the escape speed were known exactly, the total mass of the Galaxy would remain uncertain, since it is not known how the matter exterior to the solar circle is distributed. The approach presented in this paper is to use simple models for the velocity and error distributions to produce statistical models for the distribution of stars near the escape speed. We shall use these models to estimate v e from the Carney-Latham (1987) sample of high-velocity stars. We will conclude with a discussion of the relation between the local escape speed and the total mass of the Galaxy. II. THE STATISTICAL APPROACH a) Assumed Velocity Distribution Before applying statistical methods to the distribution of high-velocity stars, we must choose an analytic representation of the velocity distribution. We shall assume that the distribution of space velocities in the Galaxy's rest frame v for a sample of high-velocity stars near the Sun is f(v\v e , where v e is the local Galactic escape speed, and k is a constant. The distribution function / is defined so that /dv is the probability of observing a star with v in the range vtov + dv given v e and k. The assumed form of/needs only to be valid near v e9 since we are only interested in extreme high velocity stars. In fact, equation (1) can be thought of as the first term of a Taylor series expansion off near v e. The distribution of stellar radial velocities in the Galaxy's rest frame v r corresponding to equation (1) is given by fr(V r I V e, ty = j/M V e , k)ô(V r-V * H)dv , (2) where Ö is the Dirac delta function, and Ä is a unit vector along the line of sight. The product f r dv r is the probability of observing a star with v r in the range v r to v r + dv r , given v e and k. Note that equation (2) implicitly assumes that either the distribution function is isotropic [i.e.,/(t>) = or that the lines of sight h are isotropically distributed. Also note that v 2 = v? + vf, where v t is the velocity tangential to h in the Galaxy's rest 486 v) k , V v" (1)