The Orbital Period Prior for Single Transits

The finite baseline of any photometric survey means that transiting planets (or stars) can have a sufficiently long period to impart just a single transit in our observations. With multiple consecutive transits, the determination of the object’s orbital period, P, is trivial—given by the time difference between events. In such cases, the photometric data, ${ \mathcal D }$, provides a highly precise and unambiguous period, which means that choice of period prior, $\Pr (P)$, has negligible influence on the results. With a single transit, however, the period is only weakly constrained by the data, via the transit duration, stellar density and (unknown) eccentricity (Kipping 2010). Accordingly, the choice of period prior has a large influence on the a posteriori period.

While there are a only few dozen examples of single transits observed by Kepler among the 4000+ planetary candidates (Wang et al. 2015; Foreman-Mackey et al. 2016; Uehara et al. 2016), TESS is expected to find over a thousand (Villanueva et al. 2018) among a similar sized total (Barclay et al. 2018), bringing greater attention to this issue.

The prior on P should reflect the expected distribution of orbital periods as seen by the survey in question. This does not equal the intrinsic distribution of periods, $\Pr (P)$, because the survey’s own observational effects sculpt the distribution. For example, it would be impossible to ever record a single transit in a continuous photometric time series of baseline W with $P\lt W/2$ (else multiple transits would occur), even though the intrinsic prior $\Pr (P)$ can have support here. Thus, the fact that only a single transit occurred represents a piece of information which we can condition our prior upon, which we represent by the symbol ${\hat{n}}_{1}$. See Figure 1 for more details.

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In addition to ${\hat{n}}_{1}$, we also know the observational window, W, and the fact the planet has the correct geometry to undergo transits, $\hat{b}$, which biases us toward short periods more likely to display transits. Finally, we know the mid-transit time minus the start of the observational baseline, L, which we might expect to provide a useful lower limit on the period. Ignoring detection bias, which is only relevant for low signal-to-noise transits (Kipping & Sandford 2016), the overall prior on a single transit period may be expressed as

$$\begin{eqnarray}&&\Pr (P| {\hat{n}}_{1},W,\hat{b},L)\propto \Pr (L| {\hat{n}}_{1},W,\hat{b},P)\Pr (P| {\hat{n}}_{1},W,\hat{b})\end{eqnarray} \tag{ 1 }$$

where the right-hand side expands the left using Bayes’ theorem. The final term can expanded in a similar way as

$$\begin{eqnarray}&&\Pr (P| {\hat{n}}_{1},W,\hat{b})\propto \Pr ({\hat{n}}_{1}| P,W,\hat{b})\Pr (P| W,\hat{b})\end{eqnarray} \tag{ 2 }$$

where $\Pr (P| W,\hat{b})$ has no dependency on W and thus becomes $\Pr (P| \hat{b})$. Expanding $\Pr (P| \hat{b})$ once more with Bayes’ theorem and then combining all the terms we have

$$\begin{eqnarray}\begin{array}{rcl}\mathop{\underbrace{\Pr (P| {\hat{n}}_{1},W,\hat{b},L)}}\limits_{\mathrm{overall}\,\mathrm{prior}} & \propto & \mathop{\underbrace{\Pr (L| {\hat{n}}_{1},W,\hat{b},P)}}\limits_{\mathrm{phase}\,\mathrm{constraint}}\mathop{\underbrace{\Pr ({\hat{n}}_{1}| P,W,\hat{b})}}\limits_{\mathrm{window}\,\mathrm{effect}}\\ & & \times \,\mathop{\underbrace{\Pr (\hat{b}| P)}}\limits_{\mathrm{geometric}\,\mathrm{bias}}\mathop{\underbrace{\Pr (P)}}\limits_{\mathrm{intrinsic}\,\mathrm{prior}}.\end{array}\end{eqnarray} \tag{ 3 }$$

Let us write the intrinsic prior as ${P}^{\alpha }$, where α = −1 returns a log-uniform distribution. The geometric bias is well-understood and scales as ${P}^{-2/3}$ after applying Kepler’s Third Law to Equation (2) of Kipping (2014). The window bias is more subtle and most easily calculated by defining it as

$$\begin{eqnarray}\begin{array}{rcl}\Pr ({\hat{n}}_{1}| P,W,\hat{b}) & = & \Pr (n\geqslant 1| P,W,\hat{b})\\ & & \times \,(1-\Pr (n\geqslant 2| P,W,\hat{b})),\end{array}\end{eqnarray} \tag{ 4 }$$

where n is the number of transits occurring in the window W. The first term is easy to compute by considering that $P\lt W$ is guaranteed to provide at least one transit and $P\gt W$ will drop off thereafter as $1/P$:

$$\begin{eqnarray}\Pr (n\geqslant 1| P,W,\hat{b})\propto \left\{\begin{array}{ll}1 & \mathrm{if}\ P\leqslant W,\\ \tfrac{W}{P} & \mathrm{if}\ P\gt W.\end{array}\right.\end{eqnarray} \tag{ 5 }$$

Similarly, $n\geqslant 2$ is guaranteed if $P\leqslant W/2$ but impossible if $P\gt W$. Between these two extremes one should expect a ${P}^{-1}$ scaling as before, and so normalizing this intermediate regime to connect to the cases into a continuous function yields:

$$\begin{eqnarray}\Pr (n\geqslant 2| P,W,\hat{b})=\left\{\begin{array}{ll}1 & \mathrm{if}\ P\leqslant \tfrac{W}{2},\\ \tfrac{W-P}{P} & \mathrm{if}\ \tfrac{W}{2}\lt P\leqslant W,\\ 0 & \mathrm{if}\ P\gt W.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

Finally, turning to the phase constraint, for $P\gt W$ we expect that L is uniformly distributed between 0 and W—no phase is more likely than any other. But, if P was slightly larger than $W/2$, then L could only take on a narrow range around $W/2$ in order for the transit to avoid multiple transits. This can be encoded by writing that

$$\begin{eqnarray}&&\begin{array}{l}\Pr (L| P,W,\hat{b})\\ =\,\left\{\begin{array}{ll}\tfrac{{\mathbb{H}}[L+P-W]-{\mathbb{H}}[L-P]}{2P-W} & \mathrm{if}\ \tfrac{W}{2}\lt P\leqslant W,\\ \tfrac{{\mathbb{H}}[L]-{\mathbb{H}}[L-W]}{W} & \mathrm{if}\ P\gt W,\end{array}\right.\end{array}\end{eqnarray} \tag{ 7 }$$

where ${\mathbb{H}}$ is the Heaviside Theta function. Combining all of these components leads to the simplified result that

$$\begin{eqnarray}&&\begin{array}{l}\Pr (P| {\hat{n}}_{1},W,\hat{b},L)\\ \quad \propto \,\left\{\begin{array}{ll}0 & \mathrm{if}\ P\leqslant \max (W-L,L),\\ {P}^{\alpha -5/3} & \mathrm{if}\ \max (W-L,L)\lt P\leqslant {P}_{\max },\\ 0 & \mathrm{if}\ P\gt {P}_{\max }.\end{array}\right.\end{array}\end{eqnarray} \tag{ 8 }$$