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 (≤ 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.