Selection Functions in Astronomical Data Modeling, with the Space Density of White Dwarfs as a Worked Example

Statistical studies of cataloged object properties are central to astrophysics. But one cannot model those objects' population properties without the sample's selection function, the quantitative understanding of which objects could have ended up in such a catalog. As didactic introductions to this topic are scarce in the astrophysical literature, we provide one here, addressing the following questions: What is a selection function? On what arguments q should it depend? Over what domain must a selection function be defined? What simplifying approximations can be made? And, how is a selection function used in modeling? We argue that volume-complete samples, limited by the faintest objects, reflect a highly suboptimal selection function, needlessly reducing the number of bright and usually rare sample members. We illustrate these points by a worked example: github.com/gaia-unlimited/WD-selection-function, deriving the space density of white dwarfs (WDs) in the Galactic neighborhood as a function of their luminosity and color, phi(0)(M-G, (B - R)) in [mag(-2) pc(-3)]. We construct a sample. of 105 presumed WDs through straightforward selection cuts on the Gaia EDR3 catalog in magnitude, color, and parallax, q = (G, (B - R), pi). We then combine a simple model for phi(0) with this selection function's S-c (q) effective survey volume to estimate phi(0)(M-G, (B - R)) precisely and robustly against the detailed choices for S-c (q). This resulting WD luminosity-color function phi(0)(M-G, (B - R)) differs dramatically from the initial number density distribution in the luminosity-color plane: by orders of magnitude in density and by four magnitudes in density peak location.

Automated summary

Statistical studies of object properties in astrophysics depend heavily on understanding the sample's selection function, which determines which objects end up in a catalog. The selection function should be carefully defined and considered in modeling to ensure accurate and robust results. In this study, a worked example involving white dwarfs illustrates how the selection function affects the resulting distribution of objects in a luminosity-color plane.