What is the new period of rotation?

New Period Rotation Astronomy Question?

• When a molecular cloud collapses to form a star, it ``spins up'' due to conservation of angular momentum. The angular momentum of a spherical system is given by: L = (2 M R^2)/(5 T) where M is the mass of the system, R is its radius, and T is its period of rotation (that is, how long it takes the object to turn around once completely). Currently, the sun rotates on its axis once every 30 days. What was the rotation of the cloud that originated the sun when its radius was 1 pc (parsec), assuming the angular momentum is conserved? If the sun collapsed into a neutron star, its radius (R) would shrink by a factor of 30,000, and its angular momentum would also remain the same. What would its new period of rotation be?

• Answer:

  L = 2/5 ( M R^2 ) / T *L and M are constant* in this as you said, so by re-arranging we get 2/5 M R^2 / L = T if the radius *increased* to 1pc, the *ratio* of the two radii is R(1pc) / R(current) = 4.44 x10(7) R^2 = 2 x 10(15) so the R^2 term increases by a *factor* of 2 x 10(15), and by plugging this in you can calculate the *factor* by which the time will increase. currently, 2/5 M R^2 / L = T 2/5 M 1 / L = T (using R^2 = 1 becasue we're dealing with factors) 2/5 M / L = 30 days at 1pc 2/5 M R^2 / L = T 2/5 M 2x10(15) / L = T the left hand side has increased by 2x10(15), so the right hand side has too as well 2/5 M 2x10(15) / L = 30 days x 2x10(15) use same process for when radius decreases where the ratio woud be 1/30000 instead