The Kepler-42 system as compared to the Jovian system [credit: NASA/JPL-Caltech]

The headline was NASA’s joke about both the size and the short orbit periods (all less than two days) of the three planets in the Kepler-42 system.

The discovery of this system dates back to 2012, but there don’t seem to be any numbers on resonant periods, so we’ll supply some now.

Wikipedia says:
‘Kepler-42, formerly known as KOI-961, is a red dwarf located in the constellation Cygnus and approximately 131 light years from the Sun. It has three known extrasolar planets, all of which are smaller than Earth in radius, and likely also in mass.’

‘On 10 January 2012, using the Kepler Space Telescope three transiting planets were discovered in orbit around Kepler-42. These planets’ radii range from approximately those of Mars to Venus. The Kepler-42 system is only the second known system containing planets of Earth’s radius or smaller (the first was the Kepler-20 system). These planets’ orbits are also compact, making the system (whose host star itself has a radius comparable to those of some hot Jupiters) resemble the moon systems of giant planets such as Jupiter or Saturn more than it does the Solar System.’

The three planets in order of distance from their star (nearest first) are c,b and d. They all have very short orbit periods ranging from under half a day to less than two days, and the star has only 13% of the power of our Sun.

The resonances are a bit tricky but they do exist, even if it doesn’t look promising at first.
First we find the whole numbers of orbits that match (as closely as possible) the same period:
2011 c = 911.556 days
751 b = 911.539
491 d = 911.379
(planetary data: exoplanet.eu)

The number of synodic conjunctions (alignment of a planet pair with the star) is therefore:
2011 – 751 = 1260 c-b
751 – 491 = 260 b-d
2011 – 491 = 1520 c-d

These numbers are all divisible by 20, which means:
63 c-b = 13 b-d = 76 c-d (63+13)
Therefore 21*3 c-b = 13 b-d
3,13 and 21 are all Fibonacci numbers, so this tells us the synodic ratio of c-b:b-d is about 3*Phi:1.

As a cross-check we find:
63 c-b = 45.5783 days
13 b-d = 45.5921
76 c-d = 45.5807

Finally, if we say the orbit numbers at the top average about 911.5 days:
911.5 / 20 = 45.575 days, which fits well enough with the cross-check.

via Tallbloke’s Talkshop

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September 7, 2019 at 10:18AM