Most of our Solar System's mass is contained in the Sun, and...

Most of our Solar System's mass is contained in the Sun, and the planets possess almost all of the Solar System's angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar System's total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass $1.99 \times 10^{30} \mathrm{~kg}$, radius $6.96 \times 10^{8} \mathrm{~m}$ ) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planet's spin about its own axis.

Key Concepts

Simplified Model Assumptions

A simplified model in this context involves focusing only on the major contributors to the system’s total angular momentum, specifically the large outer planets. By ignoring less significant spin contributions (for example, the spin of the planets about their own axes) and concentrating on the dominant factors, one can more easily estimate the fraction of angular momentum present in the planetary orbits relative to the total angular momentum of the Solar System.

Rotational Angular Momentum and Moment of Inertia

Rotational angular momentum of an object, such as the Sun, depends on its moment of inertia and its rotational angular velocity. The moment of inertia reflects how the mass is distributed relative to the axis of rotation, and for celestial bodies like the Sun, this distribution significantly affects the computed angular momentum. Understanding this concept is essential for comparing the intrinsic spin of the Sun to the orbital angular momentum of the planets.

Angular Momentum

Angular momentum is a fundamental quantity in mechanics that measures the rotational motion of an object or system. In the context of celestial mechanics, it is the product of an object's mass, velocity, and the distance from the axis of rotation, and it is conserved in isolated systems. This conservation law plays a crucial role in understanding the formation and evolution of the Solar System.

Orbital Angular Momentum

Orbital angular momentum refers to the angular momentum a body possesses due to its motion along an orbit. For planets in nearly circular orbits, it can be estimated using the product of the planet’s mass, orbital radius, and orbital velocity. In many Solar System formation theories, the vast majority of the angular momentum is stored in the orbital motions of the planets rather than in the Sun's rotation.

Transcript

00:01 So we will calculate the angular velocities, omega equaling 2 pi over t, the period.

00:10 And so we can calculate the angular momentum of the sun.

00:18 This would be equally to the moment of inertia of the sun, multiplied by the angular velocity of the sun.

00:25 This is equaling two -fiths times the mass of the sun, modeling it as a sphere times the radius of the sun squared, and then multiplied by 2 pi times the period of the sun.

00:41 And so we can actually solve this, and the angular momentum of the sun is equaling 2 5ths times the mass of the sun, 1 .99 times 10 to the 30th kilograms, multiplied by 6 .96 times 10 to the 8th meters quantity squared.

01:04 We're going to multiply this by 2 pi divided by 25 days because it takes 25 days for the sun to rotate about about its own axis and then we're going to multiply this by one day for every 86 ,400 seconds and we find that the angular momentum of the sun is equaling 1 .1217 times 10 to the 30th or rather 10 to the 4 .4 .4 .5.

01:39 Second my apologies kilograms meters per second and so we can then calculate the angular momentum of all of the planets so we're going to be doing the exact same thing however we're going to be treating the planets as a a point mass orbiting around the sun and then we're going to do this for jupiter saturn uranus and neptune so so we can say that first we can say that the angular momentum of jupiter would be equaling the mass of jupiter.

02:24 We'll just say mass sub j.

02:27 Radius of jupiter squared.

02:31 And now this is not the radius of jupiter itself, but the radius of jupiter's orbit.

02:37 Because if we're treating jupiter as a point mass, we're not going to be taking the moment of inertia of jupiter itself.

02:44 Or are there, yeah, we're not taking the moment of jupiter itself.

02:47 We're taking the moment of inertia of jupiter rotating about the sun.

02:52 So we're going to have to say the radius or not really the radius, but rather more appropriately the distance from the center of jupiter to the center of the sun squared.

03:04 And then multiplied by 2 pi over again.

03:09 Following the same lines, this would be the period, not of jupiter.

03:15 Rotation about its own axis but jupiter's rotation jupiter's period about the about the sun so when it makes a full revolution around the sun and so this would be the period of jupiter's orbit and so we can say that this is going to be equaling 190 times 10 to the 25th kilograms multiplied by 7778 times 10 to the 9th meters.

03:49 Again, this is obviously much larger than the radius of jupiter because this is, again, the distance between the center of jupiter and the center of the sun.

04:00 And then it takes 2 pi over 11 .9 years.

04:07 So it takes 11 .9 years for jupiter to create one revolution around the sun.

04:14 And then we're going to multiply this by for every one year.

04:23 There are 3 .156 times 10 to the 7th seconds in one year.

04:30 And so the angular momentum of jupiter would be 1 .9 -240 times 10 to the 43 kilograms meters per second.

04:49 And so keep track of this number, keep track of this value...

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