The Posterior Distribution of sin(i) Values 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 (i.e. give information about) not only the planet's true mass M_T but also the value of sin(i) (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 uniform distribution between zero and unity (corresponding to an isotropic i distribution). The actual 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 0.25 and 0.71 (as compared to the 0.5 value for an isotropic i distribution) for indices of the power-law in the quite plausible range between -2 and -0.5, respectively. Over the same range of indicies, the 95% confidence upper bound on M_T ranges from 4.5 to 400 times larger than M_T sin(i), respectively, due to sin(i) uncertainty alone. More complex, but still simple and plausible, distributions of M_T yield still more complicated and less intuitive $sin(i)$ distributions. In particular, if the M_T distribution contains any characteristic mass scale M_c, the 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. (abridged)