A method for deriving stellar space densities

Instituto Argentino de Nivología y Glaciología (IANIGLA), C.C. 330, 5500 Mendoza, Argentina

Corresponding author: rlb@lanet.com.ar

Received: 16 July 2002
Accepted: 28 January 2003

Abstract

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 offers 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, log  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.