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 ) can not 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 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 .
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 more accurately reflect the mass uncertainties associated with RV studies .