NASA quip: ‘Honey I Shrunk the Planetary System’

Posted: September 7, 2019 by oldbrew in Astrophysics, Fibonacci, Maths, moon
Tags: , ,

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.

Comments
  1. p.g.sharrow says:

    The size of the star, the size of the planets and their distance from their star. and the size and speed of their orbits all lead me to believe BS, Bad Science is involved here. This thing can not be stable or even come into existence in this configuration,..pg.

  2. oldbrew says:

    pg – to see it requires a 8m telescope, as it says here…

    ‘The star can not be seen by the naked eye, you need a telescope to see it.
    . . .
    For Kepler-42, the location is 19h 28m 53.00 and 44° 37` 10.00 .’

    https://www.universeguide.com/star/kepler42

    The star is a red dwarf.
    https://en.wikipedia.org/wiki/Red_dwarf
    https://en.wikipedia.org/wiki/List_of_red_dwarfs#List_of_red_dwarf_firsts

  3. oldbrew says:

    Life Among the Dwarfs
    By Corey S. Powell | June 30, 2019

    More than 90 percent of the stars in the Milky Way are cooler and dimmer than the Sun. A full three-fourths are red dwarfs, the smallest of the small. Furthermore, dwarf stars seem to be especially likely to have rocky planets. Together, those statistics indicate that there are a trillion planets around red dwarfs in our galaxy, including at least 100 billion potentially habitable Earth-size worlds. The big unknown is whether those planets are actually habitable — that is, if the genuinely typical Earth-size worlds out there are really anything at all like our own.

    http://blogs.discovermagazine.com/outthere/2019/06/30/dwarf-stars-trappist-1-proxima-centauri/
    – – –
    One problem re. habitability is that planets close to their star are likely to be tidally locked, like the Moon is to the Earth, so the same side would face the heat all the time.

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