Detection of exo-planets

One method used for detecting exo-planets is to look for a slight dip in the parent star's luminosity as the planet transits the stellar disc. Intuitively, it seems to me that if planetary systems in our galactic neighborhood are randomly oriented, there would have to be a very large proportion of them in which transits can never happen from Earth's viewpoint. Perhaps, however, the assumption of random orientation is incorrect, and there is some alignment of the axes of rotation of planetary systems, which would facilitate detection of planets in some preferred plane (the galactic plane?).

In popular presentations concerning the search for exo-planets, I have never seen this issue addressed. What observations and/or assumptions are used in arriving at a realistic estimate of the number of exo-planets in our region of the galaxy?

(There are related questions in this forum, but I haven't found one that asks about the possible alignment of the axes of rotation.)

It isn't usually an issue because most experiments are simply concerned with finding exoplanets. They are rarely designed in such a way that it is easy to estimate population statistics because of all sorts of biases that go into selecting the targets. Unfortunately the search for exoplanets has turned into a sport where discovery is everything.

If one assumes random orientation of orbits (and that is all it is, an assumption) then the probability of a transit scales roughly as $$P \simeq \frac{R_p+ R_s}{a}$$ where $R_p$ and $R_s$ are the radius of the planet and hot star respectively and $a$ the planet's orbital radius (with small modifications for non-circular orbits). The larger this is, the more likely a transit is to occur. Hence large exoplanets orbiting close to large stars are more likely to transit. In principle then, this effect can be corrected for when calculating the statistics and frequency of exoplanets.

So how good is the random orbital inclination assumption? I honestly think nobody knows at the moment. I have done work on the possible alignment of spin axes within the low-mass stars of clusters (Jackson & Jeffries 2010) finding consistency with the random hypothesis. More recent work using asteroseismology suggests that there may be alignment for more massive stars (Corsaro et al. 2017). However, even if the spin axes (and therefore presumably the majority of planet orbits) of stars in clusters line up, there is no obvious reason why each cluster should have the same angular momentum vector When the clusters eventually disperse into the field then they would, presumably, form a pseudo-random distribution?

Except, what if the Galactic tides or a large-scale Galactic magnetic field played a role in shaping the angular momentum direction of the clouds that formed the clusters. Might it be possible for some alignment to persist to old age? Corsaro et al. argue that interactions within a cluster are not sufficient to "scramble" the angular momenta after star formation has finished. Close interactions between stars become much less likely after they emerge from a cluster into the field. An intriguing piece of work by Rees & Zijlstra (2013) found that there was evidence for a non-random distribution of orientation for bipolar planetary nebulae towards the Galactic bulge. This suggested that the orbital angular momenta of binary systems responsible for the bipolar shape of the nebulae were oriented in the Galactic plane. The result is highly statistically significant but as far as I know has not been followed up despite its obvious implications for estimations of transit yields from exoplanetary surveys.

I think that there will be a much better answer to this question once we have all-sky exoplanet searches of the quality of the Kepler satellite (the main Kepler survey was in one particular direction). It should become very obvious if there are changes in the planet yields as a function of position of the sky (although you also have to control for the types of star being observed) associated with any large-scale alignment. Maybe there is enough information in the Kepler K2 fields that are taken at positions around the ecliptic - I have not seen any analysis. However, such data will surely become available with the launch of NASA's all-sky TESS satellite in 2018.

The assumption of random orientations is a reasonable one. One reason that exoplanets weren't detected in the 1980s was the expectation that most solar systems would be like ours, with large planets at a great distance, making transits rare, infrequent and hard to detect.

Hot Jupiters changed that. Most of the planets that Kepler detects are very close to their host star. This means that no great coincidence is required for the inclination of the axis of rotation relative to the solar system. An axial inclination of between 80 and 90 degrees would allow for a transit in many of the systems discovered.

This is taken into account when estimating the number of stars with planets, with the conclusion that nearly all sun-like stars have planetary systems. Kepler can only detect a fraction of these, but it surveys so many stars that it has found a good number of planetary systems. But most of the stars observed haven't shown a transit. Extrapolating from its discoveries, we have to conclude that the main reason that we don't detect planets around the other stars is due to the inclination of the exoplanetary systems.

For analysis of the probabilities involved in transiting exoplanets, you can consult Transit Probabilities for Stars with Stellar Inclination Constraints

• What is the evidence for your first sentence? – ProfRob Jun 3 '17 at 9:34
• Thank you for you fast and informative answer. I'd been wondering about this for some time but didn't know whom to ask until I found this cool website. – Clyde Jun 3 '17 at 10:09
• The linked article assumes random inclinations. "We begin by reviewing the transit probability for a single star under the assumption that the planet's orbital inclination is randomly and evenly distributed over all possible orientations." It seems a reasonable assumption, at least for the purposes of modelling. – James K Jun 3 '17 at 15:30
• It is just an assumption that everyone makes (including me in my work) because there is no other game in town. The assumption is not necessarily "a good one", it is something that is forced upon us. – ProfRob Jun 3 '17 at 17:28
• Ok, "reasonable one" is better. It's not forced on anyone, you can model the distribution of spin axes any way you want, providing it fits the data. The random model is simple, and the papers you cite don't seem to suggest that it is very wrong. So it is a reasonable model for estimating the population of stars with planetary systems. – James K Jun 3 '17 at 20:23