THE POSTERIOR DISTRIBUTION OF sin(i) VALUES FOR EXOPLANETS WITH MT sin(i) DETERMINED FROM RADIAL VELOCITY DATA

Radial velocity (RV) observations of an exoplanet system giving a value of M-T sin(i) condition (i.e., 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 intrinsic distribution of M-T, on which we have significant information from available RV data in some mass ranges but little or none in others. 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 of the power law in the range between -2 and +1, respectively. Over the same range of indices, the 95% confidence interval on M-T varies from 1.0001-2.405 (alpha = -2) to 1.13-94.34 (alpha = +2) times larger than M-T sin(i) due to sin(i) uncertainty alone. More complex, but still simple and plausible, distributions of M-T yield more complicated and somewhat unintuitive posterior sin(i) distributions. In particular, if the M-T distribution contains any characteristic mass scale M-c, the posterior sin(i) distribution will depend on the ratio of M-T sin(i) to M-c, often in a non-trivial way. 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 reflect more accurately the mass uncertainties associated with RV studies.