# Angular momentum of the sun and a white dwarf

• Fendergutt
In summary, the Sun rotates once every 26.0 days with a mass of 2.0E30 kg and a radius of 7.0E8 m. Astrophysicists predict that in the future, the Sun will collapse into a white dwarf with a higher density and a smaller radius of 5.8E6 m, while maintaining its mass. The Sun's angular momentum now is 1.1E42 kgm²/s, and as a white dwarf it will decrease to 7.5E37 kgm²/s. This is due to the conservation of angular momentum, which means that an external torque is needed to change the angular momentum. The white dwarf Sun will rotate once every 150 seconds.
Fendergutt

## Homework Statement

The Sun rotates about its own axis once every 26.0 days. Its mass is m_sun = 2.0E30 kg and radius is r_sun = 7.0E8 m. Assume that the Sun is a solid sphere with uniform density.

Astrophysicists predict that the Sun will collapse into a white dwarf in some billions of years. Its density will be high, and its radius 5.8E6 m. Assume the mass of the Sun to be unaltered (a good approximation for our calculation).

(i) What is the Sun's angular momentum now?
(ii) What is the Sun's angular momentum as a white dwarf?
(iii) How long will it take for the white dwarf Sun to rotate about its own axis?

## Homework Equations

I = (2/5)*m*r^2
L = Iω = Iωk (k being the z-axis unit vector)

## The Attempt at a Solution

(i)
ω_sun = (2pi)/(26 days) = 2.8E-6 rad/s

L = I_sun*ω_sun*k = (2/5)*m_sun*r_sun²*ω_sun*k = (2/5)*2.0E30 kg * (7.0E8 m)² * ω_sun * k = 1.1E42 kgm²/s

(ii)
I don't know the new spin angular velocity of the white dwarf, do I? If I leave ω_white = ω_sun the formula for L above with inserted new value for radius r_white gives appr. 7.5E37 kgm²/s k, ie. the angular momentum is lowered. Can this be correct?

(iii)
I don't know what to do here. I want to find the new period T, but I suspect there is something odd in (ii). Is there a relation I have overseen?

Thanks for any help.

Fendergutt said:
(i)
ω_sun = (2pi)/(26 days) = 2.8E-6 rad/s

L = I_sun*ω_sun*k = (2/5)*m_sun*r_sun²*ω_sun*k = (2/5)*2.0E30 kg * (7.0E8 m)² * ω_sun * k = 1.1E42 kgm²/s
I didn't check the arithmetic, but the method is correct.

(ii)
I don't know the new spin angular velocity of the white dwarf, do I? If I leave ω_white = ω_sun the formula for L above with inserted new value for radius r_white gives appr. 7.5E37 kgm²/s k, ie. the angular momentum is lowered. Can this be correct?
What's required to change the angular momentum? Does that apply here?

Doc Al said:
I didn't check the arithmetic, but the method is correct.

What's required to change the angular momentum? Does that apply here?

Ah, so you are saying to me: Angular momentum is a conserved quantity: a system's angular momentum stays constant unless an external torque acts on it.

Then I set up T_white = ( I_white * 2pi * k ) / L which gives a rotation period of 1.5E2 s or 150 s.

(Mathematically: L features numbers and the unit vector k; In the calculation above I mentally ignored the unit vector because it was a feature of both nominator and denominator -- I'm I right to do this?)

All good.

## 1. What is angular momentum and how is it related to the sun and a white dwarf?

Angular momentum is a measure of the rotational motion of an object around a fixed point. In the case of the sun and a white dwarf, it refers to the amount of rotational energy they possess due to their spinning motion.

## 2. Why is the angular momentum of the sun greater than that of a white dwarf?

The sun is a much larger and more massive object compared to a white dwarf, so it has a greater amount of angular momentum due to its larger size and faster rotation.

## 3. How does the angular momentum of the sun and a white dwarf affect their overall motion?

The angular momentum of an object determines its resistance to changes in its rotational motion. Therefore, the sun and a white dwarf will continue to spin at a constant rate unless acted upon by an external force.

## 4. Can the angular momentum of the sun and a white dwarf change over time?

Yes, the angular momentum of an object can change if external forces such as gravitational interactions or collisions with other objects affect its rotation. However, these changes are usually small and can take a long time to occur.

## 5. How does the angular momentum of the sun and a white dwarf impact their lifespan?

The angular momentum of an object does not directly impact its lifespan. However, it can indirectly affect the object's evolution and eventual fate, as changes in rotation can lead to changes in its internal structure and behavior.

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