# Probability of Transit

Transiting Planets. Credit: NASA

Transiting planets are valuable items to explore the properties of planetary atmospheres. Planet searches like Kepler that focous on fields of sky tend to reap rewards amongst dimmer stars simply because there are many more dim stars in a given patch of the sky than bright ones. Transiting planets around bright stars are of particular value, though, as the increased brightness makes the system easier to study.

Radial velocity surveys tend to monitor brighter stars since spectroscopy is even more severely limited by stellar brightness than photometry, but it is not limited to observing patches of sky – telescopes performing Doppler spectroscopy tend to observe a single object at a time due to technical and physical limitations. Radial velocity surveys are also much less sensitive to the inclination angle of a planet orbit with respect to the plane of the sky. The planet doesn’t have to transit to be spectroscopically detectable. As such, radial velocity surveys tend to generate discoveries of planet candidates with unknown inclinations and true masses, but around much brighter stars than those planets discovered by the transit method.

As such, planet candidates discovered by radial velocity, especially planet candidates in short orbital periods are excellent targets for follow-up observations to attempt to detect transits. Transiting planets that have been discovered first through radial velocity have been of great scientific interest due to their host stellar brightness and thus ease of study. If more such systems are found, it would be of great benefit to understanding extrasolar planet atmosphere. While only a hand-full of transiting planets have been discovered first through radial velocity, they all orbit bright stars and are some of the best-characterised planets outside our solar system.

The probability that a planet will transit is, as has been discussed previously, given by
$\displaystyle P_{tr} = \frac{R_*}{a}$
where a is the semi-major axis of the planet orbit. This is the distance between the centre of the star and the centre of the planet. However, due to the inclination degeneracy – the reoccurring evil villain constantly plaguing radial velocity science – the star-planet separation is unknown. Remember that the period of the RV curve gives only the orbital period of the planet. If the orbital period is held constant, increasing the mass of the planet increases the star-planet separation. An increase in the total system mass requires greater separation between the two bodies to preserve the same orbital period.

For example, if radial velocity observations of a star reveal the presence of a mp sin i = 1 ME planet candidate, but the inclination is actually extremely low such that the true mass of the companion is in the stellar regime, then because the mutual gravitational attraction between the two stars will be much greater than the mutual gravitational attraction between the star and an Earth-mass planet at the same period, the two stars must have a wider separation, otherwise their orbital period would be smaller.

Mathematically, the true semi-major axis is given by
$\displaystyle a = \left(\frac{G[M_*+M_{\text{pl}}(i)]}{4\pi^2}\right)^{1/3}T^{2/3}$
Where G is the gravitational constant, and Mpl(i) is the mass of the planet at a given inclination i, and T is the period of the system. It is worth noting that the true semi-major axis is not significantly different from the minimum semi-major axis as long as the mass of the star is much greater than the mass of the planet – which is typically the case.

The fact that the true semi-major axis is a function of the unknown inclination makes for an interesting clarification: The probability that a planet of unknown inclination will transit is not simply given by Rstar/a, but is only approximated by it. If we assume that the distribution of planet masses is uniform (and extending through into the brown dwarf mass regime), then you would expect a planet with a minimum mass equal to Earth to have a much greater chance of being a bona-fide planet than a planet with a minimum-mass of 10 MJ, simply because there is a greater range of inclinations the former planet can be while still remaining in the planetary mass regime. Taking this a step further, even if both the Earth-mass planet candidate and the 10 Jupiter-mass planet candidate have the same orbital period, the probability that the latter planet transits ends up being less than the Earth-mass planet simply because of its high mass. Since its inclination is unknown, the probability that its mass is so high that the true semi-major axis is noticeably larger than the minimum semi-major axis is much higher, resulting in a likely lower transit probability.

Except it turns out that the mass distribution of planets and brown dwarfs isn’t constant. Earth-sized planets are significantly more common than Jupiter-sized planets, and super-Jupiters appear rare. It isn’t clear yet what the mass distribution planets actually is, with significant uncertainty in the sub-Neptune regime, but it is clear that for a highly accurate estimate of the transit probability, the inclination distribution cannot be thought of as completely random as it is fundamentally tied to the planet mass distribution.

Planet Mass Distribution given by Ida & Lin (Left) and Mordasini (Right)

Consider the case of a super-Jovian planet candidate, perhaps with a minimum mass of 7 or 8 Jupiter-masses. Because a significant fraction of physically allowable inclinations would place the true mass planet into a mass regime that is in reality sparsely populated, it is less likely that the planet candidate’s orbit is in those inclinations. It is thus more likely that the planet candidate’s orbit is edge-on than would be expected from the probability function of randomly oriented orbits. As such, the transit probability of a super-Jovian planet is actually boosted by ~20 – 50% over what you would expect from Ptr = Rstar/a. If this is the case, then we would expect to find an excess in the fraction of transiting planets in this mass regime then would be expected purely from the standard transit probability function. Indeed this is what we see.

Candidate planets with masses in the terrestrial planet regime are similarly affected, with broadened transit probabilies owing to the fact that terrestrial planets are more common than higher mass planets, arguing in favour of a higher inclination than the random inclination distribution function.

On the other hand, planet or brown dwarf candidates of minimum masses in the most sparsely populated region of the mass distribution are unlikely to truly have that mass. They are quite likely in orbits with low inclinations and with much higher true masses. The transit probability for companion candidates with minimum masses in this mass regime are actually reduced from the standard transit probability function.

Geometric and a posteriori transit probabilities

In the table above, taken from this preprint, we see that the geometric transit probability, Ptr,0, can be much less than the a posteriori transit probability, Ptr. The transit probability for 55 Cnc e, for example, jumps up from 28% to 36%. With these higher a posteriori transit probabilities, these short-period low-mass planets should be followed-up for transits. If transits are found, it would be of significant benefit to the extrasolar planet field.

In summary, there are various additional effects that can cause the a posteriori transit probability to be significantly different from the geometric transit probability. Planets with only minimum masses known can be more accurately assigned a transit probability when taking into account the uneven planetary mass distribution. Low-mass planets and super-Jupiters are more likely to transit than their geometric transit probability because a significant range of the inclination space is consumed by planets of masses that are simply rare. These planet candidates are more promising targets for transit follow-up than, for example, Jupiter-mass planets or intermediate-mass brown dwarfs.