The Posterior Distribution of sin(i) for Exoplanets with M_T sin(i) Determined from Radial Velocity Data

Abstract

Radial velocity (RV) observations of an exoplanet system giving a value of M_T sin(i) condition (ie. give information about) not only the planet's true mass M_T but also the value of sin(i) for that system (where i is the orbital inclination angle). Thus the value of sin(i) for a system with any particular observed value of M_T sin(i) cannot be assumed to be drawn randomly from a distribution corresponding to an isotropic i distribution, i.e. the presumptive prior distribution . Rather, the posterior distribution from which it is drawn depends on the intrinsic distribution of M_T for the exoplanet population being studied. We give a simple Bayesian derivation of this relationship and apply it to several "toy models" for the (currently unknown) intrinsic distribution of M_T. The results show that the effect can be an important one. For example, even for simple power-law distributions of M_T, the median value of sin(i) in an observed RV sample can vary between 0.860 and 0.023 (as compared to the 0.866 value for an isotropic i distribution) for indices (alpha) of the power-law in the range between -2 and +1, respectively. Over the same range of indicies, the 95% confidence interval on M_T varies from 1.002-4.566 (alpha = -2) to 1.13-94.34 (alpha = +1) times larger than M_T sin(i) due to sin(i) uncertainty alone. Our qualitative conclusion is that RV studies of exoplanets, both individual objects and statistical samples, should regard the sin(i) factor as more than a "numerical constant of order unity" with simple and well understood statistical properties. We argue that reports of M_T sin(i) determinations should be accompanied by a statement of the corresponding confidence bounds on M_T at, say, the 95% level based on an explicitly stated assumed form of the true M_T distribution in order to more accurately reflect the mass uncertainties associated with RV studies.