Image shows 4 equipotentials (the surface of the ocean is an equipotential) for a VERY rapidly rotating planet. The corners reach orbital velocity. The vertical Y axis is the axis of rotation.
The equation assumes that all of the mass is concentrated in a point at the exact center but even if the mass were uniformly distributed, the outer layers (which is the part we are most interested in) would behave as though nearly all of the mass of the inner layers was concentrated in a point in the exact center.
The corners are, mathematically at least, sharp 120 degree angles.
Y= ±0.6666 at x=0
If the planet were Jupiter sized then the lines are about 2000km apart at the top and bottom.
https://www.wolframalpha.com
plot [ -(x^2/2) -1/sqrt(x^2+y^2)= -1.5, -(x^2/2) -1/sqrt(x^2+y^2)= -1.55, -(x^2/2) -1/sqrt(x^2+y^2)= -1.6, -(x^2/2) -1/sqrt(x^2+y^2)= -1.65] for x= -1 to 1
All planets from 1 JM (Jupiter Mass) to 64 JM are known to have approximately the same radius. If hydrogen only comes in two forms, nondegenerate liquid hydrogen and degenerate metallic hydrogen, then it's very hard to see how a planet could keep the same radius as its mass increases from 1 to 64 JM. It is necessary therefore to suppose that metallic hydrogen comes in multiple forms which are arranged in layers with the lightest at top and the densest at the bottom.
Liquid hydrogen/helium
Metallic hydrogen/helium
Double degenerate metallic hydrogen/helium
Triple degenerate metallic hydrogen/helium
If the sun began as a rapidly growing and rapidly rotating Jupiter sized, 64 JM (Jupiter mass) brown dwarf then it was at that time almost entirely double degenerate metallic hydrogen/helium. When the pressure in the core of the 64 JM protosun became high enough the double degenerate metallic hydrogen core finally collapsed (creating triple degenerate metallic hydrogen and marking the beginning of nuclear fusion.
Gen 1:3 Let there be light.) and the protosun would have begun to spin faster due to conservation of angular momentum. If it spun fast enough then its outermost layers of liquid hydrogen and part of its metallic hydrogen would have been thrown out into space where it would have coalesced into a moon which would have immediately begun receding from the rapidly rotating protosun due to tidal interactions.
One orbit would be completed in only 25 minutes.
2 * pi * sqrt((76000 km)^3/(G * 60 * 317 earth masses)) = 25 min
If it continued to spin faster and faster then more of its metallic hydrogen would have been thrown out into space and would have formed a second moon which would also have immediately begun receding due to tidal interactions. Perhaps the reason all of the metallic hydrogen didnt end up in the 1st moon was because some of it was solid at that time. Perhaps later the heat released by the core collapse liquified all the remaining metallic hydrogen and this, now liquid metallic hydrogen, became a second moon.
Maybe, just maybe, Saturn=1st moon and Jupiter=2nd moon (I'm just throwing it out there as a possibility)
This would explain why hot Jupiters are so common
Jupiter's orbital momentum is far larger than the suns rotational angular momentum.
The moons orbital momentum is 4 times the Earths rotational angular momentum.
Jupiter would have been 1/60th of the mass of the protosun.
The Moon is 1/80th of the mass of Earth.
The depth of the outer layer of the protosuns metallic hydrogen that became jupiter might be the depth at which the metallic hydrogen transitioned to double degenerate metallic hydrogen.
60 * 2.528(9.8m/s^2) * (saturn mass + Jupiter mass) / (Jupiter surface area) in bar = 601.9 million bar
60 * 2.528(9.8m/s^2) * (5.7*10^26kg + 1.9 * 10^27kg) / (61 * 10^9 km^2) in bar = 601.9 million bar
Neptune and Uranus could have been created much earlier in a nearly identical process when the pressure in the metallic hydrogen core of the then 8 JM protosun reached 600 million bar and the core collapsed creating double degenerate metallic hydrogen and the outer 300 km of metallic hydrogen was thrown off into space. However it is not clear what would have happened to the 8 EM of liquid hydrogen/helium. Either it became planet nine or it simply went back to being gas in the disk orbiting the protosun.
After Jupiter was created and receded from the protosun, the Protosun's rotation would have temporarily slowed and gas and dust from the circumstance disc would once again have begun to accumulate on it. If the rotation increased again then a planet of rock and iron could have formed. 1 EM would form a layer 20km deep. At 150 g thats equivalent to 3000km at 1 g. Earth contains 0.25 EM of iron. 250 EM of gas would have to fall on to the protosun to bring that much iron. Since the rock obviously wasnt metallic that gas must somehow have been boiled off of the rapidly rotating protosun. Perhaps superheated double degenerate metallic hydrogen from the core erupted onto the surface forming a pancake dome (farra) and boiled it off.
Gen 1:6 Let it separate gas from liquid.
The mass of the ice moons of Jupiter add to 1/300th the mass of Jupiters hydrogen ocean.
The mass of the ice moons of Saturn add to 1/200th the mass of Saturns hydrogen ocean.
It would not be surprising therefore if the 8 Jupiter mass protosun already had 8 Earth masses of ice moons. (Pluto, Triton, whatever retrograde moon Uranus captured, Eris, 2007 OR10, Makemake and Haumea)
Had Saturn and Jupiter continued to grow into Stars then one of them would have been ejected from the system (a 3-body system is unstable) and formed its own star system.
The planet that was not ejected would have grown until it also had 2 large moons. At least one of these 2 moons would have been ejected and formed its own star system.
In this way, each star system spawns 2 more star systems until the parent molecular cloud is destroyed by a supernova. (This explains the rarity of Population III stars). The circumstellar disk dissipated naturally once the molecular cloud that was feeding it was destroyed.
https://en.wikipedia.org/wiki/Brown_dwarf
http://www.science20.com/robert_inve...155538
For reference:
https://plus.google.com/100636757164...CcPgBX
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