Question

1.)

So we know that the planet Jupiter and its big Galilean moons are sometimes termed as a ‘mini-solar system’ because Jupiter is composed of mainly hydrogen and helium like stars and the Galilean moons seem like planets revolving around it. The total angular momentum of a system is contributed by the sum of orbital and rotational angular momenta of the central body and the bodies orbiting around it.

Note: For parts a and also b you can go ahead and ignore the contribution of all satellites, asteroids, dwarf planets for the Solar System and the smaller Jovian satellites for the Jupiter system

a. Go ahead and derive an equation for rotational angular momentum and calculate the same for the Solar System and the Jupiter system.  

b. Also go ahead and also derive an equation for orbital angular momentum and calculate the same for the Solar System and the Jupiter system.

c. Please give a good explanation on how this analogy of the ‘mini-solar system’ breaks down.

Answer 1

Answer:

(a)

Rotational angular momentum of body is defined as the product of (I) moment of inertia X(w) angular velocity of body about axis

for sphere of Mass M and radius R rotating about its diameter

2 I=-MR

So the total rotational angular momentum of the whole solar system will be sum of rotational momentum of individual Planets and Sun

$\mathbf{\left ( P_{R} \right )_{T}=(P_{R})_{s}+(P_{R})_{1}+(P_{R})_{2}+(P_{R})_{3}+(P_{R})_{4}+(P_{R})_{5}+(P_{R})_{6}+(P_{R})_{7}+(P_{R})_{8}}$

  $=I_{1}W_{1}+I_{2}W_{2}+I_{3}W_{3}+.....+I_{9}W_{9}$

9 (Pa), = ΣΙΝ, 1=1

M, RW, 1=1

Where $(P_{R})_{T}$ : Total Rotational Angular Momentum

While for Jupiter the total rotational angular momentum will be only due to Jupiter and its big Moon

$(P_{R})_{J}=\frac{2}{5}M_{J}R_{J}^{2}W_{J}+\frac{2}{5}M_{B}R_{B}^{2}W_{B}$

Integral the contribution of Sun in rotational angular momentum of solar system is only about 3% as W is very low while the contribution of jupiter lies 60% almost as its M and W both are high

(b)

Orbital angular momentum is defined as radial componet of m linearmomentum along radius

i.e

where

M = Mass of the body

V = Velocity of body

R = Radius of body

So total orbital momentum of whole solar system will be = Sum of orbital momentum of individual planets keeping sum of its centre

$(P_{0})=\sum_{m=1}^{8}M_{i}V_{i}R_{i}$

Respesent mass of the planet

Corresponding radius of Planet about Sun

Velocity which its orbits around Sun

While incase of Jupiter

(c)

Only due to its big moon hence we can say that

Orbital momentum of solar