The Occurrence Rate of Quiescent Radio Emission for Ultracool Dwarfs using a Generalized Semi-Analytical Bayesian Framework

We present a generalized analytical Bayesian framework for calculating the occurrence rate of steady emission (or absorption) in astrophysical objects. As a proof-of-concept, we apply this framework to non-flaring quiescent radio emission in ultracool ($\leq$ M7) dwarfs. Using simulations, we show that our framework recovers the simulated radio occurrence rate to within 1-5% for sample sizes of 10-100 objects when averaged over an ensemble of trials and simulated occurrence rates for our assumed luminosity distribution models. In contrast, existing detection rate studies may under-predict the simulated rate by 51-66% because of sensitivity limits. Using all available literature results for samples of 82 ultracool M dwarfs, 74 L dwarfs, and 23 T/Y dwarfs, we find that the maximum-likelihood quiescent radio occurrence rate is between $15^{+4}_{-4}$ - $20^{+6}_{-5}$%, depending on the luminosity prior that we assume. Comparing each spectral type, we find occurrence rates of $17^{+9}_{-7}$ - $25^{+13}_{-10}$% for M dwarfs, $10^{+5}_{-4}$ - $13^{+7}_{-5}$% for L dwarfs, and $23^{+11}_{-9}$ - $29^{+13}_{-11}$% for T/Y dwarfs. We rule out potential selection effects and speculate that age and/or rotation may account for tentative evidence that the quiescent radio occurrence rate of L dwarfs may be suppressed compared to M and T/Y dwarfs and phenomenon. Finally, we discuss how we can harness our occurrence rate framework to carefully assess the possible physics that may be contributing to observed occurrence rate trends