Stellar limb darkening impacts a wide range of astronomical measurements. The accuracy to which it is modeled limits the accuracy in any covariant parameters of interest, such as the radius of a transiting planet. With the ever growing availability of precise observations and the importance of robust estimates of astrophysical parameters, an emerging trend has been to freely fit the limb darkening coefficients (LDCs) describing a limb darkening law of choice, in order to propagate our ignorance of the true intensity profile. In practice, this approach has been limited to two-parameter limb darkening laws, such as the quadratic law, due to the relative ease of sampling the physically allowed range of LDCs. Here, we provide a highly efficient method for sampling LDCs describing a more accurate three-parameter non-linear law. We first derive analytic criteria which can quickly test if a set of LDCs are physical, although naive sampling with these criteria leads to an acceptance rate less than 1%. We then show that the loci of allowed LDCs can be transformed into a cone-like volume, from which we are able to draw uniform samples. We show that samples drawn uniformly from the conal region are physically valid in 97.3% of realizations and encompass 94.4% of the volume of allowed parameter space. We provide Python and Fortran code (LDC3) to sample from this region (and perform the reverse calculation) at https://github.com/davidkipping/LDC3, which also includes a subroutine to efficiently test whether a sample is physically valid or not.
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