A method for deriving stellar space densities

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The fundamental integral equation of stellar statistics represents a direct, model-independent approach to calculating stellar densities. Many techniques exist for its solution, but some of these require assumptions, such as a Gaussian luminosity function or a specific form for the density function, that may be unrealistic. To solve the equation as an undeterdetermined total least squares system with Tikhanov regularization recognizes that the problem is ill-posed and generally ill-conditioned as well and oers decided advantages: it is unnecessary to assume a Gaussian luminosity function nor a specific form for the density function; discretization error in the kernel of the integral equation as well as the Poisson error in the star counts are accounted for; mean errors for the densities are calculated; the densities are constrained to be both continuous and positive. The greatest drawback to the method comes from the selection of the ridge parameter, but the drawback becomes surmountable. The method is first applied to three examples, general star counts, the distribution of K0 giants, and the distribution of M 2-M 4 dwarfs, and compared with densities calculated from methods such as Malmquist's and the ( m ,l og) table. Regularized total least squares competes well with these methods. Then the method is applied to a new data set from the AC2000.2 catalog to calculate the densities of M giants and supergiants in the directions of the north and south galactic poles. The densities decrease exponentially to near zero at 2000 pc, with half-density points near 550 pc: No evidence for asymmetry between the two hemispheres can be seen.