The luminosity function for the CfA redshift survey slices

We calculate the luminosity function for two complete slices of the extension of the Center for Astrophysics (CfA) redshift survey. The shape of the luminosity function can be approximated by a Schechter function with M^*B (0) _ = -19. 2 +/- 0. 1 and α = -1. 1 +/- 0. 1. Because the size of the large-scale inhomogeneities is comparable with the extent of the survey, the shape of the luminosity function must be derived with an inhomogeneity- independent method. We show that the nonparametric technique derived by Lynden-Bell and Turner can yield an unbiased estimate of the form of the luminosity function. Because of the large-scale inhomogeneities in the sample, the amplitude φ^*^ of the luminosity function is poorly determined: the fluctuations in ph^*^ (φ^*^ = 0. 020 +/- 0. 005h³^ Mpc^-3^) are consistent with the 25% uncertainty predicted by comparison of the size of the largest inhomogeneities with the sample depth (de Lapparent, Geller, and Huchra). We also show that random errors in the magnitudes significantly affect the spatial density contrast of large-scale structures. Assuming rms errors of ~0. 35 mag (Huchra) in the Zwicky magnitudes for the CfA slices, the density of the structures at 10, 000 km s^-1^ is enhanced by a factor ~1. 13 relative to structures at 5000 km s^-1^. This differential enhancement in the spatial structures causes a bias in the determination of the luminosity function. We show analytically that the parameters of the Schechter function fitted to the CfA slices might thus be systematically biased by DELTAM^*^ ~ -0. 3 and DELTAα = -0. 1. Of all the sources of error we consider (rms fitting errors, Virgo infall, velocity distortions, and Malmquist bias), the Malmquist bias has the largest effect on the determination of the luminosity function parameters. Systematic errors in the Zwicky et al. magnitudes could well be comparable with the random errors and could seriously affect the determination of the luminosity function; however, insufficient data are available to model this problem in a reliable way. We take the Hubble constant H₀_ = 100h km s^-1^ Mpc^-1^ with h = 1 unless otherwise specified.