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 $1.99 \times 10^{30} kg$, radius $6.96 \times 10^8 m$) 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

Angular Momentum Distribution in the Solar System

In astrophysical systems like our Solar System, there is a striking disparity between the distribution of mass and angular momentum. Despite the Sun containing most of the mass, the planets—especially the outer planets—carry most of the angular momentum. This observation is key in forming and testing theories about the formation of the Solar System, as it provides insights into the processes during its early evolution and accretion.

Angular Momentum

Angular momentum is a fundamental physical quantity that characterizes the amount of rotation an object has, taking into account both its mass distribution and its velocity. In astrophysics, it plays a crucial role in understanding the dynamics of systems, such as the distribution of spin in stars and the orbital motion of planets. The calculation for rotating bodies or orbiting systems typically involves parameters like mass, distance, and angular velocity.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on how the mass is distributed relative to the axis of rotation. For different shapes, such as a sphere or a disk, standard formulas are used (for example, I ? (2/5)MR² for a uniform sphere), and this concept is central in calculating the angular momentum of spinning objects like the Sun.

Angular Velocity and Rotational Period

Angular velocity is the rate at which an object rotates, and it is directly related to the rotational period by the relation ? = 2?/T. This concept is crucial when comparing the rotational contributions of objects in astrophysics—such as the Sun's spin—especially when the period changes the calculation of its angular momentum.

Orbital Angular Momentum

Orbital angular momentum refers to the momentum of bodies moving along orbital paths. For planets in near-circular orbits, it can be estimated using their mass, orbital speed, and orbital radius (typically using L = mvr for circular motions). This is particularly significant in a planetary system, as the enormous distances of planets from the central star usually result in much larger angular momentum compared to the star’s spin.

Transcript

00:01 So this problem is only tedious.

00:03 It's not very difficult.

00:06 However, we do need to do a lot of calculations, and there are a lot of room.

00:11 There's a lot of room for computational errors, so definitely just be careful.

00:16 Let's start the angular momentum.

00:20 The formula is going to be i omega.

00:23 And we can say that the angular momentum of the sun is going to be model.

00:31 Is going to be i omega, but i of the sphere omega.

00:36 So this is going to be equal to 2 times the mass of the sun, radius of the sun squared divided by 5 times 2 pi over the period of the sun in its orbit.

00:54 So we can say that l of the sun equals 2 times 1 .99 times 10 to the the 30th and these are all tabulated values so you simply need to look in any physics textbook in order to find these values times 2 pi i apologize to 5 we'll clean this up and this will be divided by 5 times 25 days times 24 hours per day times 3 ,600 seconds per hour and we have the angular momentum of the sun being 1 .12 times 10 to the 42 kilogram meter squared per second.

01:58 So in order to find the formula, the moment of inertia for all the planets, well, the planets are in orbit with the sun.

02:08 So the center, the, an axis going up going through the center of the sun, that would be the axis of rotation here.

02:16 So the distance between a planet to the axis of rotation would simply be the distance from the planet to the sun.

02:25 And these are tabulated.

02:28 So we can say that l of jupiter is going to be equal to the mass of jupiter times the radius of jupiter squared times 2 pi over the period of jupiter.

02:42 And again, this term right here, r subj, simply means the distance from jupiter to the sun.

02:51 And this is going to be equal to, rather, we don't need the parentheses.

03:00 So we can just say 2 pi 190 times 10 to the 25th kilograms, 778 times 10 to the 9th meters to the squared.

03:17 Squared divided by 11 .9 years times 365 days per year times 24 hours per day times 3 ,600 seconds per hour.

03:34 And this is going to give us 1 .92 times 10 to the 43 kilogram meters squared per second.

03:45 The same thing is going to happen for saturn.

03:48 So we'll say l of s, the angular momentum of saturn, equals the mass of saturn times the distance from saturn to the sun squared times 2 pi over the period of saturn.

04:00 And this is going to be equal to 2 pi, 56 .8 times 10 to the 25th, 1 ,427 times 10 to the 9th squared, all divided by the period of saturn is, 29 .5 years...

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