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February 03, 2011

Kepler Telescope related - what is the probability of locating a planet in relation to its major-axis (orbit radius) from various stars

Reader Goat guy used the Monte Carlo estimation method to determine for the NASA Kepler Space telescope:

What is the probability of locating a planet in relation to its major-axis (orbit radius) from various stars?

So first off - I can tell you that anyone who claims "1%" detection probability is pulling that number "statistically" out of his arse. It is just as good as 0.1%, 0.5%, 2% or 5%. The reason all comes down to the very strong link between detection probability and orbital radius. Here are the results:




   AU DISCOVERY P PERIOD(DAYS) PLANET EXAMPLE
0.026 1 : 20           2       
0.033 1 : 25           2       
0.041 1 : 31           3       
0.051 1 : 40           4       
0.064 1 : 50           6       
0.080 1 : 64           8       
0.100 1 : 82           12      Inner Kepler
0.125 1 : 104          16      
0.156 1 : 132          23      
0.195 1 : 173          32      
0.244 1 : 236          44      Outer Kepler
0.305 1 : 313          62      
0.381 1 : 428          86      Mercury
0.477 1 : 590         120      
0.596 1 : 846         168      
0.745 1 : 1,250       235      Venus
0.931 1 : 1,900       328      
1.000 1 : 2,280       365      Earth
1.164 1 : 3,000       458      
1.455 1 : 5,200       641      Mars
1.819 1 : 9,200       895      
2.274 1 : 18,000     1,251     
2.842 1 : 40,000     1,749     Ceres
3.553 1 : 90,000     2,444     
4.441 1 : 250,000    3,416     
5.551 1 : 800,000    4,774     Jupiter

Point is - the probability of detection is very heavily weighted toward planets that are closer to their parent stars. For a planet at 0.1 AU (with an orbital period of about 12 days), the probability of detection exceeds 1%. Since the Kepler study has so far turned up a LOT of planets that are "shockingly close" to their parent star, one might reasonably ask, "so is our system unusal then?"

I quite easily can now see: "no". It just wouldn't be an easily detected system, is all. There's a 1:400 chance of detecting Mercury, 1:1,200 for Venus and 1:2,300 for Earth. It gets much less probable for the outer planets. Essentially have a 1:1,000,000 chance of detecting Jupiter.

To address your point of my numbers being so different from others - it is most probably my "basis of detection", really. If one were to have a dedicated scope that could give "1 minute" to each patch - but once an hour, 24 hour a day, then there would be a 97% chance of catching the transits on all systems that are serendipitously inclined so that transits are observable from Earth. Then one only calculates the probability of a particular planet of being "serendipitously inclined" or "edge-on" relative to our system. Even then - you still need to estimate what you believe the spectrum of planetary orbital radii is going to be in order to say "1%" or any other number. And (by circular logic) you do by observing a bunch of stars (150,000 in this sample?), tallying up the transits, calcing out orbital dynamics for each (which is another 'assumption-based' miasma), and plotting them. Then, change all basis calculations so that for the sample, the stats more or less match. ("Build a Model')

The model at present - almost suspiciously! - found 1,200 planets under 150,000 stars (I still need to confirm that number). That would be 0.8% detection. My calcs for a spectrum of planets in geometrically uniform (arse) orbits from their parent star, with an average of 6 (arse) "inner" planets per star gives 5 planet detections for 250 stars. (1 in 50 stars) Or, about 1 planet out of every 250 planets.

The most significant "take-away" from this is that it is all but certain that virtually every star has planets., and not just "one or two", but many - on the order of our own system. If the planetary masses are essentially arbitrary, determined by protostellar accretion dynamics, "chance" and the clearing radius of influence of each planetessimal, then a number such as "10" might be perfectly reasonable as an average for a star. I mean, if we were just one dinky planet with no others - I really couldn't guess how other systems might be put together. But with 9 planets, a bunch of asteroids and so on ... it seems more likely that we're "ordinary", and not "special".

Further, tweaking the spectrum-of-planets factor, and I can easily get the "probability of detection" close to 1.0% with a bit of work. So - hey - 1%? Sure. It is just as convenient an arse-pull as any estimation of the probability of intelligences in the universe.

So. Virtually all stars have planets. About 10% of those will "statistically" have planet-masses in the Goldilocks zone. We have only a 1:1,500 chance of detecting most Goldlocks planets. Therefore, there are a LOT of Goldilocks planets out there. 68 detected out of 150,000 stars. ... equals about 100,000 Goldilocks planets, or 2 for every 3 stars. Note that "Venus, Earth, Mars" are all in our own Goldilocks zone - Venus just wasn't lucky enough to be smashed to smithereens and get a moon from it, and Mars appears to be too damned small to hold onto enough atmosphere. But clearly something only modestly larger than Earth at Mars' orbit might well have been a water-world. Again, this confirms the 1:10 hypothesis for Goldilocks systems. Pretty encouraging!

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