How can i apply conservation of angular momentum to protostar rotation problem?

[SofiaMM]

I'm really struggling to understand how to apply the concept of conservation of angular momentum to a real-world problem I'm working on. My textbook example involves a spinning ice skater pulling in their arms, but my assignment asks about the rotational speed of a collapsing protostar, and I can't quite bridge the gap between the two.

[Violet53]

Same idea as the ice skater, but with caveats. If you treat the protostar as a body with negligible external torque during collapse, its L = I ω stays roughly constant. The moment of inertia I depends on how mass is distributed; as the core contracts, the radius shrinks and I drops, so ω has to rise. A simple estimate uses I ≈ k M R^2. If M stays roughly fixed during a phase, ω ∝ 1/R^2. In reality the mass inside the contracting region may change and the outer envelope can interact via magnetic fields, so you don’t get a perfect 1/R^2. Also accretion from the disk can add angular momentum or remove it via jets. If you’re stuck, try a back‑of‑the‑envelope: pick representative M and R, assume no torques, compute ω_now = L/I and compare with initial ω. Then check how big external torques would have to be to explain a given observed spin.

[Logan91]

I tried something similar last week. I took a protostellar core with M about one solar mass and R a few thousandths of an AU and then shrank it by a factor of two. With I ∝ M R^2, ω jumped by a factor four if L stayed the same. It felt too big, and I kept thinking about mass growth from accretion. If the mass doubles during collapse, L might go up and ω might not change as fast. Also the core isn’t rigid; different layers spin differently, so the spin budget is redistributed inside and can leak outward.

[ElizabethFC]

Nice rough rule of thumb: ω ∝ 1/R^2 if M is constant. Halving the radius means roughly four times faster rotation. But in reality there are jets, magnetic torques, and ongoing accretion.

[PaulQW]

Is the real issue that the problem assumes an isolated collapse, while disk processes and magnetic braking dominate the spin budget?