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Given that we are at a random orientation to any remote star system, it seems to me that there is only a narrow angle at which transits of exoplanets can be observed.

Imagine a large mathematical sphere with the remote system at the center, and the Earth on the surface. There would be a belt on that sphere where transits could be observed. Assuming an even probability of our position on that sphere, the chance of us being able to observe a transit would be the ratio of the area of that "transit-observing" belt and the total area of the sphere.

enter image description here

A quick calculation tells me that the probability of being in position to see a transite $P_t$ is almost equal to radius of the star / radius of the planet's orbit ($R_{orbit}=a$).

$$ P_t \approx \frac {R_{star}}{R_{orbit}} $$

Assumptions:

  1. Distance to star >> $R_{star}$
  2. Exoplanet is a point. In reality it is non-zero, but this just simple blurs the edge case.

If true, the chance of far observers seeing Earth transits of the sun: $\frac {0.696 \space million \space km}{149.6 \space million \space km} \approx \frac12\% $

The coincidence that we are in position to see the planets in the Trappist 1 system: $\frac {0.114 \cdot 0.696 \space million \space km}{0.011 \cdot 149.6 \space million \space km} \approx 5\% $

Michael
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    Are we at a random orientation? Consider the angular momentum of our solar system with respect to the galaxy. I'm not sure that other systems will be truly randomly oriented. – Jon Custer Feb 24 '17 at 19:34
  • You've neglected a couple of important things. First the radius of the habitable zone depends on the radiated power and is proportional to both the radius squared of the star and to the fourth power of temperature of the star. Secondly that the radius of a star is not a linear function of it's mass (even assuming you compare main-sequence stars for which $R \approx M^{4/5}$). Both of these things work in favor of finding planets around small cool stars. – dmckee --- ex-moderator kitten Feb 24 '17 at 20:32
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    All this means (with the caveats mentioned in other comments) is that there are probably many more systems with planets that we do not detect. (Silly prediction: in 30 years it will be assumed that almost all stars have planets.) –  Feb 24 '17 at 21:05
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    Detection statistics are a hot topic in exoplanet searches, see for example https://arxiv.org/abs/1301.0842 – astronat Feb 24 '17 at 23:07
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    @JonCuster, I do believe it is rather random from this answer. http://physics.stackexchange.com/questions/205738/do-solar-systems-typically-spin-in-the-same-direction-as-their-galaxy/205741#205741 – Michael Feb 24 '17 at 23:18
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    @dmckee... Neither habitable zone radius, nor star mass are relevant points. I'm not trying to make a statistical statement about the geometries of all systems, I have no idea what that might be.

    I'm ONLY talking about the probability of seeing transits given the geometry of a system... and gave 2 examples.

     – Michael Feb 24 '17 at 23:25
  • @Michael - very interesting, and thanks for pointing that out. Kind of curious at some level though... – Jon Custer Feb 24 '17 at 23:25
  • @JonCuster I do not understand your first comment. The ecliptic plane and galactic plane are clearly not aligned by some 63 degrees. – ProfRob Feb 25 '17 at 00:50

2 Answers2

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You are right, the probability of a priori being able to see transits around any star is low. But you made a mistake. The inner planet in the Trappist-1 system is 0.011au from the star. Thus the transit probability is actually about $0.114*6.96\times 10^8/0.011*1.5\times 10^{11} = 0.048$. (I also think you got the radius of the Sun wrong).

Thus unlikely, but not astoundingly so. If you observed 20 such stars and they all had planetary systems like this, then you would expect to see one with transits. I will bet a lot of cash that Trappist-1 is not the only star that was monitored by the Trappist experiment. In fact I'd give you odds of 10-1 on that. (Or if you prefer, evens that they observed at least 10).

ProfRob
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This being a rare event dosen't stop astronomers from finding them. They don't look for the planets directly. Observing the motion of a star in detail, reveals whether it has a planetary system. Once this is confirmed for some star, various other factors are studied in detail over long periods of time, like the change in apparent brightness(luminosity) of the star, to decide whether its an exoplanet requires further study, but to find planetary systems, is not a very rare occurrence, given the technological advancements we have made so far.

Lelouch
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