long-period planets

I went off the grid for the weekend, but that didn't stop me from working out a probabilistic approach to understanding the population of long-period planets given low signal-to-noise radial-velocity data. This problem was given to me by Heather Knutson (Caltech), who has a great data set, in which many stars show no companion at high detection confidence. The problem is hairy if you permit any star to have any number of long-period planets. It becomes much more tractable if you assume (by fiat) that every star has exactly one or zero. However this assumption is clearly unphysical!

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