The Posterior Distribution of sin(i) for Exoplanets with MT sin(i) Determined from Radial Velocity Data
Abstract
Radial velocity (RV) observations of an exoplanet system giving a value of MT sin(i) condition (ie. give information about) not only the planet's true mass MT 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 MT 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 MT 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 MT. The results show that the effect can be an important one. For example, even for simple power-law distributions of MT, 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 MT varies from 1.002-4.566 (alpha = -2) to 1.13-94.34 (alpha = +1) times larger than MT 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 MT sin(i) determinations should be accompanied by a statement of the corresponding confidence bounds on MT at, say, the 95% level based on an explicitly stated assumed form of the true MT distribution in order to more accurately reflect the mass uncertainties associated with RV studies.