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I mean, why do the spin angular momentum and the orbit angular momentum of a planet tend to have the same direction?

As we all know, a planetesimal $m$ orbiting a sun with mass $M_{sun}$ at $r$ will have a velocity $v=\sqrt{GM/r}$ since $m v^2/r=GMm/r^2$. For simplicity, I have assumed a circular orbit here.

Now consider two planetesimals with same mass $m$, one of them is orbiting the sun at $r-dr$, the other is orbiting the sun at $r+dr$. These two planetesimals move in the same dirction. Somehow, these two planetesimals collide (their sizes are larger than 2$dr$ or gravitation plays a role) and form a single large planetesimal. According to mass conservation, this single large planetesimal will have a mass $m_2=2m$. According to momentum conservation, it will orbit the sun with $v_2=1/2(\sqrt{GM/(r+dr)}+\sqrt{GM/(r-dr)})\sim\sqrt{GM/r}$. However, since the original planetesimal in the inner orbit $r-dr$ have a large velocity $\sqrt{GM/(r-dr)}$, the combined large planetesimal should spin, with a spin angular momentum equal to $J_{spin}=m(\sqrt{GM/(r-dr)}-\sqrt{GM/r})dr+m(\sqrt{GM/r}-\sqrt{GM/(r+dr)})dr\sim m \sqrt{GM/r} (dr)^2/r$.
What important is, this $J_{spin}$ is in the reverse direction with the orbital angular momentum!

As the accretion process continues, this large planetesimal grows larger. The materials in the inner orbit always have a larger velocity, and the materials in the outer orbit always have a smaller velocity. This will lead to accumulated reverse spin angular momentum, as the process described above happens over and over again.

However, in our solar system, only Venus spins in the reverse direction, which is very reasonable. All other planets spin in the same direction as their orbital angular momentum, which is very strange. Why?

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  • $\begingroup$ Axial tilt (or Obliquity to Orbit) describes exactly that. A tilt of zero means the planet spins on the same axis (direction) as it orbits. A tilt of 97.7° like Uranus means the two directions are almost perpendicular and Venus with 177° tilt means the two spin are almost in the opposite direction as the direction of orbit. $\endgroup$ Jul 27, 2023 at 16:01
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    $\begingroup$ Related question on our sister site: astronomy.stackexchange.com/q/6183/16685 $\endgroup$
    – PM 2Ring
    Jul 29, 2023 at 8:37

3 Answers 3

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Though the materials in the inner orbit always have a larger Kepler velocity, they actually have a lower energy (kinetic energy $E_k$ plus gravitational potential energy $E_p$). So they will slow down to a smaller velocity when they are attracted outward by a planetesimal. Though the materials in the outer orbit always have a smaller Kepler velocity, they actually have a higher total energy. They will be speeded up to a higher velocity when they are pulled inside. This will overturn the derivation in the question and result in spin angular momentum in the same direction as orbital angular momentum. The reason is so simple. I am such an idiot that it took me six hours to figure this out.

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Another effect that is important here is that if the cloud of dust and gas from which the solar system formed had any net angular momentum while it was still a cloud, then the condensing solar system strives to conserve that momentum as it clumps together to form the sun and the planets. Once the process has gone to completion, then each of the planets and the sun will left be rotating in the same direction, and the planets will all be orbiting the sun in the same direction.

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Consider 2 planetesimals with mass $m$, one at distance $r-\Delta r$ and one at $r+\Delta r$ orbiting with velocities $v+\Delta v$ and $v- \Delta v$ respectively. The total angular momentum is then

$$L = m(v+\Delta v)(r-\Delta r) +m(v-\Delta v)(r+\Delta r) =$$ $$= 2mvr +m\Delta v \:r- mv\:\Delta r -m\Delta v \: \Delta r -m\Delta v \:r+ mv\:\Delta r -m\Delta v \: \Delta r =$$ $$= 2mvr -2m\Delta v \: \Delta r$$

Since combining the 2 planetesimals into one moving with velocity $v$ at radius $r$ would however imply an orbital angular momentum $2mvr$, this means there must be an additional spin angular momentum of $-2m\Delta v \: \Delta r$ i.e. opposite to the orbital angular momentum. This would rule out the merging of freely orbiting objects as the reason for the forward rotation of planets. I think the actual reason is likely to be the further contraction of the already established protoplanet (i.e. a mass already dominated by its own gravity). In this case all parts of the mass would move together as one around the Sun and therefore the more distant parts would have a higher velocity than the closer parts (assuming an orbitally locked rotation, which is the equilibrium state in the 2-body problem, see this answer). So the angular momentum equation would read instead

$$L = m(v-\Delta v)(r-\Delta r) +m(v+\Delta v)(r+\Delta r) =$$ $$= 2mvr -m\Delta v \:r- mv\:\Delta r +m\Delta v \: \Delta r +m\Delta v \:r+ mv\:\Delta r +m\Delta v \: \Delta r =$$ $$= 2mvr +2m\Delta v \: \Delta r$$

So the further gravitational contraction of an already semi-established protoplanet should lead to a forward spin in order to account for angular momentum conservation. This will most likely happen already at the gas stage i.e. before solid matter is formed (which is also confirmed by the fact that all planets, and also their larger moons, are more less perfect spheres (which would be rather unlikely to form from the accretion of solid planetesimals).

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  • $\begingroup$ Wait.....you confuse me. The total angular momentum is $L$, orbital angular momentum is $2 m v r$. Shouldn't the spin angular momentum $J_{spin}=L_{total}-L_{orbit}=-2m\Delta v\Delta r$? It is in the opposite direction! $\endgroup$
    – Harry
    Jul 28, 2023 at 2:49
  • $\begingroup$ @Harry You are right. I seem to have confused myself. See my modified answer with a new interpretation. $\endgroup$
    – Thomas
    Jul 28, 2023 at 21:06
  • $\begingroup$ "all parts of the mass would move together as one around the Sun and therefore the more distant parts would have a higher velocity than the closer parts."——It seems that you imply something like tidal locking. Emmm..I highly doubt that it is a circular argument.. $\endgroup$
    – Harry
    Jul 29, 2023 at 5:51
  • $\begingroup$ @Harry Yes, sorry, I forgot to add that this assumes a locked rotation, which is the equilibrium state in the 2 body problem ( see astronomy.stackexchange.com/a/54308/35115 ). The point is that this rotation must speed up as the protoplanet contracts further because of angular momentum conservation. Once the final size is reached the rotation will go slowly down again because of tidal interaction. $\endgroup$
    – Thomas
    Jul 29, 2023 at 7:49
  • $\begingroup$ We're fairly certain that the large solid bodies did form through the accretion of solid planetesimals, although (of course) mass in the proto-solar system was dominated by hydrogen & helium. FWIW, we have lots of questions on planetesimals on Astronomy.SE: astronomy.stackexchange.com/… I've written a couple of answers there that are somewhat relevant to this question astronomy.stackexchange.com/a/26570/16685 & astronomy.stackexchange.com/a/49425/16685 When large planetesimals collide, the heat may be sufficient to melt solids. $\endgroup$
    – PM 2Ring
    Jul 29, 2023 at 8:47

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