# Angular momentum of the sun and a white dwarf

## 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.

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Doc Al
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(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?

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?)

Doc Al
Mentor
All good.