How is the angular momentum of the solar system divided among its different components, the Sun, the planets, and the moons?
Angular momentum is the measure of the tendency of a rotating body to remain rotating. Angular momentum is always conserved. It’s often illustrated by a spinning ice-skater, who pulls in their arms, and thereby increases their rate of rotation, because they have reduced the radius of their outer parts, so the overall speed of their spinning increases.
The value of the angular momentum of an object swinging around in a circle is something like the mass times the speed at which the mass is moving, times the radius, squared.
Angular momentum is usually stated in kg m2 / sec, whereas the data is in km and days.
To change days into seconds, multiply by 24 · 60 · 60. To change km to meters, multiply by 1000. To change orbital radius into distance, multiply by 2π.
The angular momentum L of an object of mass m moving in a circle of radius r, with linear speed p is given by
Using this formula, calculate the L column of this table from the given orbital data. (Data from The Nine Planets)
| body | orbit radius (km) | orbit period (days) | mass (kg) | L (kg m2/s) |
|---|---|---|---|---|
| Mercury | 58.e6 | 87.97 | 3.30e23 | 9.1e38 |
| Venus | 108.e6 | 224.70 | 4.87e24 | 1.8e40 |
| Earth | 150.e6 | 365.26 | 5.97e24 | 2.7e40 |
| Mars | 228.e6 | 686.98 | 6.42e23 | 3.5e39 |
| Jupiter | 778.e6 | 4332.71 | 1.90e27 | 1.9e43 |
| Saturn | 1429.e6 | 10759.50 | 5.68e26 | 7.8e42 |
| Uranus | 2871.e6 | 30685.00 | 8.68e25 | 1.7e42 |
| Neptune | 4504.e6 | 60190.00 | 1.02e26 | 2.5e42 |
| Pluto | 5914.e6 | 90800 | 1.27e22 | 3.6e38 |
| 3.1e43 | ||||
| Moon | 384e3 | 27.32 | 7.35e22 | 2.9e34 |
| Io | 422e3 | 1.77 | 8.93e22 | 6.5e35 |
| Europa | 671e3 | 3.55 | 4.80e22 | 4.4e35 |
| Ganymede | 1070e3 | 7.15 | 1.48e23 | 1.7e36 |
| Callisto | 1883e3 | 16.69 | 1.08e23 | 1.7e36 |
The rotational angular momentum of a solid homogeneous sphere of mass m and radius r with rotational rate p is given by
When applied to gaseous bodies such as the Sun or Jupiter, this will yield an overestimate, because the interiors of such gaseous bodies are denser than their outer layers. But this is also true of “rocky” bodies like the Earth, to a lesser degree.
| body | radius (km) | rotational period (days) | mass (kg) | L (kg m2/s) |
|---|---|---|---|---|
| Sun | 695000 | 24.6 | 1.99e30 | 1.1e42 |
| Earth | 6378 | 0.99 | 5.97e24 | 7.1e33 |
| Jupiter | 71492 | 0.41 | 1.90e27 | 6.9e38 |
So the rotational angular momentum of the Sun, which is 1.1e42, is less than 4% that of the total orbital angular momentum of the planets, which is 3.1e43.
Based on this calculation, Jupiter’s orbital angular momentum alone accounts for over 60% of the total angular momentum of the Solar system!
The orbital angular momentum of the Moon 2.9e34 is about four times that of the rotational angular momentum of the Earth, which is 7.1e33.
However, the total orbital angular momenta of the largest moons of Jupiter is less than a hundredth the rotational angular momentum of the planet.
The fact that most of the angular momentum in the solar system resides in the planets says something.
Could there be an explanation for the solar system being this way? I suggest that in order for a gas cloud, carrying a lot of angular momentum, to collapse into a relatively compact star, the cloud has to somehow shed angular momentum. There is more than one way that this might occur, but the above calculation indicate that a lot of angular momentum can end up in planets orbiting the central star.
If our solar system is typical, at least typical of single-star systems, this strongly suggests that most individual stars should have planets. Of course, this is just what modern observations show.
The argument doesn’t work so well for binary stars, because the orbital momentum of the two stars is huge... and that alone might account for much of the excess angular momentum in a diffuse cloud, and because often, tidal forces might eject smaller objects from multi-star systems.
By the way, another mechanism by which a nascent star could shed angular momentum is by way of a jet. A dense beam of very energetic particles streaming from the poles of the star can also remove a great deal of angular momentum.
Nobody would have invented jets for this purpose, but they are now seen to be typical of the formation of at least some types of stars.
P.S.: Frank Potter has written in to point out that I should take into consideration the angular momentum of the Oort as well. He reckons that its angular momentum could be comparable to the total angular momentum of the planets.
I have some doubts about the figures — the Wikipedia page currently quotes 5 Earth masses for the outer Oort cloud. But the Oort cloud is nebulous by nature and description, besides being very scantily observed.
I have argued that the dynamics of the early solar system was highly non-conservative, losing a great deal of energy by friction. But also, much of the material belonging to the early solar system has been completely lost from the system: having been ejected far into interstellar space. This also amounts to a loss of energy to the system, besides momentum.
The last observation brings up another metaphysical question: what to take as the boundary of a physical system? The notion of a physical system is a practical idealization, but a broader view of physical reality is that everything interacts (up to the limits of causality set by relativity); there is no completely closed system.
The Oort cloud occupies an intermediate situation, between material completely lost to the solar system, and material still to some degree in orbit of the Sun.
Regarding this lost material as still being part of the system, there is no loss of momentum — this is the simplification usually adopted as principle. But in the practical restriction to the solar system — the stuff that remains in orbit of the Sun, that stuff ejected from the solar system is really gone — a loss of much energy and momentum to the solar system.
The loss of energy and momentum by the combined masses of the Oort cloud and the material that is completely lost to the system can play a role in a story about gradual modification (or migration) of planetary orbits.