A Fast, Analytic Empirical Model of the Gaia Data Release 3 Astrometric Orbit Catalog Selection Function

In June 2022, the Gaia mission released a catalog of astrometric orbital solutions for 168,065 binary systems, by far the largest such catalog to date. The catalog's selection function is difficult to characterize because of choices made in its construction. Understanding the catalog's selection function is required to model and interpret its contents. We use a combination of analytic and empirical prescriptions to construct a function that computes the probability that a binary with a given set of properties would have been published in the Gaia Data Release 3 astrometric orbit catalog. This is a complementary approach to the more accurate but significantly more computationally expensive approach of El-Badry et al. (2024). We also construct a binary population synthesis model to validate our characterization of the selection function, finding good agreement with the actual Gaia NSS catalog, with the exception of the orbital eccentricity distribution. The NSS catalog suggests high-eccentricity orbits are relatively uncommon at intermediate periods $100 \lesssim P_{orb} \lesssim 1000$ days. As an example application of the selection function, we estimate the Gaia DR3 detection probabilities of the star + BH binaries Gaia BH1 and BH2, and find them to be 0.38 and 0.27, respectively. Compared to the values obtained by detailed modeling in El-Badry et al. (2024), the probabilities are identical for BH1, and within a factor of 2 for BH2. We also estimate the population of Sun-like star + BH binaries in the Galaxy to be $\sim 3000$ for $100 < P_{orb} < 400$ day, $< 800$ for $400 < P_{orb} < 1000$ day, and $< 12,000$ for $1000 < P_{orb} < 1500$ day.