Astrometry from {} DR3 has produced a sample of $$170,000 Keplerian orbital solutions, with many more anticipated in the next few years. These data have enormous potential to constrain the population of binary stars, giant planets, and compact objects in the Solar neighborhood. But in order to use the published orbit catalogs for statistical inference, it is necessary to understand their selection function: what is the probability that a binary with a given set of properties ends up in a catalog? We show that such a selection function for the {} DR3 astrometric binary catalog can be forward-modeled from the {} scanning law, including individual 1D astrometric measurements, the fitting of a cascade of astrometric models, and quality cuts applied in post-processing. We populate a synthetic Milky Way model with binary stars and generate a mock catalog of astrometric orbits. The mock catalog is quite similar to the DR3 astrometric binary sample, suggesting that our selection function is a sensible approximation of reality. Our fitting also produces a sample of spurious astrometric orbits similar to those found in DR3; these are mainly the result of scan angle-dependent astrometric biases in marginally resolved wide binaries. We show that {} sensitivity to astrometric binaries falls off rapidly at high eccentricities, but only weakly at high inclinations. We predict that DR4 will yield ∼ 1 million astrometric orbits, mostly for bright ( G ≲ 15 ) systems with long periods ( d). We provide code to simulate and fit realistic {} epoch astrometry for any data release and determine whether any hypothetical binary would receive a cataloged orbital solution.