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Why Phi? – Exoplanets Extra
‘An exoplanet, or extrasolar planet, is a planet outside the Solar System’ – Wikipedia.
At least 175 multiple planetary systems have been found as of 25 November 2013. Stuart Graham investigates.
Exoplanets are a mixed bunch. Some are 10 times the size of Jupiter, others seem more like moons and may orbit their star in less than 2 days. Here we’ll look first at a small planetary body in the solar system, see how it relates to its neighbours, and then see what similarities and/or differences can be found in a few selected exoplanet systems. There may even be a few surprises.
Less well-known than Pluto is its supposed twin Orcus, or 90482 Orcus to give its full name. It’s a trans-Neptunian object or maybe a dwarf planet. As it even has its own moon Vanth, it has the reputation of being the ‘anti-Pluto’. Its orbit looks like a mirror image of Pluto’s orbit (red: Pluto, blue: Orcus, grey: Neptune).
— Symmetric orbits of Orcus and Pluto – image credit Wikipedia —
In fact it completes 99 orbits in about the same time as Pluto does 98, which in terms of a ‘simple ratio’ is 1:1. We’ll see various examples of near-perfect simple ratios like this as we go along. In fact the two biggest planets in the solar system – Jupiter and Saturn – show it with an orbit ratio of 149:60 (150:60 = 5:2). Orcus also does 41 orbits to Neptune’s 61 (40:60 = 2:3) and 49 to every 143 Uranus orbits (48:144 = 1:3). By comparison Pluto:Neptune is 165:248 (166:249 = 2:3) and Pluto:Uranus is 19:56 (19:57 = 1:3). Of course even the larger figures are themselves only very close approximations (mostly greater than 99.9%), not 100% matches.
Now let’s take a short tour of a few exoplanet systems.
Ratios note: list shows the number required to match the other planet.
Planet name note: letters b,c,d etc. are in order of discovery date (largest planet first) NOT orbit period (letter a belongs to the star).
Gliese 876 or GJ 876 : distance from Earth = 15.33 light years
Originally thought to be a two-planet system with a near 2:1 orbit ratio.
359:179 orbit ratio for the 2 planets 876b and 876c (360:180 = 2:1).
Then two other planets were found: 876d nearly 7 times bigger than Earth with an orbit period under 2 days, and 867e just over twice the orbit period of 876c.
876b : 876e orbit ratio = 61:30 (60:30 = 2:1)
876d : 876c = 31:2 approx. (1149:74)
Since 360:180 = 30:15 it can be seen that all four planets are closely linked to 30/31 and 60/61 one way or another, and that the ratio of c:b:e is close to 4:2:1.
Studies have shown other interesting properties, e.g. ‘the two planets [b and c] are locked in a secular resonance where they librate around apsidal alignment with an amplitude of 34◦, and their joint line of apsides precesses at a rate of 41◦ every year.’
http://arxiv.org/pdf/1202.5865v1.pdf
Also the semi major axis ratios include the following:
d:b 10:1
b:e 1.605:1 (close to phi)
c:b 1.6075:1 ( ” ” ” )
HR 8799 : distance from Earth = 128.5 light years
This has four giant planets, all 7-10 times the mass of Jupiter and much further away from their star than it. Let’s look at the simple ratios first.
Orbits – neighbours:
b:c = 2:1 (19995:10000)
c:d = 2:1 (20009:10000)
d:e = 9:4 approx.
Orbits – others:
b:d = 4:1 (9662:2415) — 9660:2415 = 4:1
b:e = 9:1 approx.(73:8 exactly*) — 72:8 = 9:1
c:e = 9:2 approx.(575:126) — 576:128 = 9:2
* e = exactly 18000 days, and 73 x e = 8 x b exactly (b = 164250 days)
b:c:d orbit ratio = 4:2:1 (99.98%) as a close fit but not exactly.
Semi-major axis ratios:
e:d 2:1 approx.(54:29) — 54:27 = 2:1
d:c 3:2 approx.(143:90) — 145:90 = 3:2
c:b 5:8 approx.(200:317) — 200:320 = 5:8
Mass (the stated figures compared to Jupiter are 10, 9 and 7):
b:c = 10:7
b:d = 10:7
b:e = 7:9
c:d = 1:1
c:e = 10:9
d:e = 10:9
Planets c and d have the same mass, are neighbours and have a very close 1:2 orbital relationship. They are the heart of the system.
Synodic conjunctions per 2.16 million years approx.:
c-b:d-c:e-d = 4800:9611:24608 (9611 is just over double 4800)
Further reading:
‘Multiple mean motion resonances in the HR 8799 planetary system’
http://arxiv.org/abs/1308.6462
‘First Reconnaissance Of An Exoplanetary System’
HAT-P-17 : distance from Earth = 293.5 light years
In this two-planet system the nearest match to a whole number of orbits of the major planet with a whole number of the minor one seems to be:
HAT-P-17b = 10.338523 d x 2261 = 23375.4 days (64 years = 23376 d)
HAT-P-17c = 1798.0 d x 13 = 23374 days
2262 / 13 = 174
2262 / 6 = 377 (13 and 377 are Fibonacci numbers)
The ‘perfect ratio’ (2262:13) is avoided by 1 orbit of the minor planet (2261:13).
HD 82943 : distance from Earth = 89.56 light years
82943c and 82943b are a similar size and close to 2:1 orbit ratio. The third planet 82943d is much smaller and a lot further from the star.
c:b 2:1 approx. (117:58) — 116:58 = 2:1
b:d 2:5 approx. (158:385) — 156:390 = 2:5
c:d 1:5 approx. (12:59) — 12:60 = 1:5
47 Uma : distance from Earth = 45.56 light years
Three-planet system, outermost completing 3 orbits every 115 years (115.00615y)
b:d 13:1 (1169:90) — 1170:90 = 13:1
d:c 1:6 approx. (7:41) — 7:42 = 1:6
b:c 4:9 approx. (500:1109) — 496:1116 = 4:9
*****
This is a very small selection from a very large database, so can only give a flavour of what’s happening outside the solar system. Selection basis was partly random, partly to highlight some of the (perhaps) more obvious planetary relationships e.g. 1:2 orbit ratios.
Nevertheless it’s enough to let us find evidence of some similarities in the behaviour of the exoplanets, especially the larger ones, with what is seen in our own solar system.
Data from: http://exoplanet.eu/catalog
Source: https://tallbloke.wordpress.com/2013/11/26/why-phi-exoplanets-extra/
