Analytic Estimates of the Achievable Precision on the Physical Properties of Transiting Planets Using Purely Empirical Measurements

We present analytic estimates of the fractional uncertainties on the mass, radius, surface gravity, and density of a transiting planet, using only empirical or semi-empirical measurements. We first express these parameters in terms of transit photometry and radial velocity (RV) observables, as well as the stellar radius $R_{\star}$, if required. In agreement with previous results, we find that, assuming a circular orbit, the surface gravity of the planet ($g_p$) depends only on empirical transit and RV parameters; namely, the planet period $P$, the transit depth $\delta$, the RV semi-amplitude $K_{\star}$, the transit duration $T$, and the ingress/egress duration $\tau$. However, the planet mass and density depend on all these quantities, plus $R_{\star}$. Thus, an inference about the planet mass, radius, and density must rely upon an external constraint such as the stellar radius. For bright stars, stellar radii can now be measured nearly empirically by using measurements of the stellar bolometric flux, the effective temperature, and the distance to the star via its parallax, with the extinction $A_V$ being the only free parameter. For any given system, there is a hierarchy of achievable precisions on the planetary parameters, such that the planetary surface gravity is more accurately measured than the density, which in turn is more accurately measured than the mass. We find that surface gravity provides a strong constraint on the core mass fraction of terrestrial planets. This is useful, given that the surface gravity may be one of the best measured properties of a terrestrial planet.