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

TL;DR

This paper derives the posterior distribution of sin(i) for exoplanets from radial velocity data, showing how intrinsic mass distributions influence observed inclination effects and emphasizing the importance of accounting for these in mass estimates.

Contribution

It provides a Bayesian framework to relate the true mass distribution of exoplanets to the observed sin(i) distribution, highlighting the impact on mass and inclination estimates.

Findings

Median sin(i) varies significantly with the assumed mass distribution.

Mass uncertainty due to sin(i) can be much larger than previously assumed.

Exoplanet mass reports should include confidence bounds based on intrinsic distributions.

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.