AN ANALYTICAL STUDY OF THE GLOBULAR-CLUSTER LUMINOSITY FUNCTION

Following Harris and Pudritz (1993, ApJ, submitted), it is assumed that a globular-cluster system (GCS) has a mass spectrum of the form N(m)similar to m(-alpha), where alpha may take on different characteristic values in selected mass ranges. From this, an analytical globular-cluster luminosity function (GCLF) is then derived. It is shown that this GCLF is non-Gaussian, but will be peaked, unimodal, and symmetric (as is observed) if the mass spectral slope changes in a very specific way: there must be exactly one transition in alpha from >1 to <1, and the relation \1-alpha\ = const. must hold. Similarly, the luminosity-weighted luminosity function (LWLF), which is a useful observable for the distant, very large GCSs in Virgo and beyond, is shown to be peaked, unimodal; and symmetric if there is exactly one transition from alpha>2 to alpha<2, such that \2-alpha\=const. The Gaussian approximation to the analytical GCLF is considered, and it is shown that the dispersion sigma(G) is directly related to the mass spectral slope alpha. Implications for the use of the GCLF as a standard candle are briefly discussed.