Will a radioactive ball conserve its angular velocity?

Question

At first sight, the following question may sound trivial, but I believe that it is more subtle than it seems. This problem is inspired (but not equivalent) to the possible spin-down of a newly formed neutron star via neutrino emission.

A sphere in vacuum should spin forever because of angular momentum conservation. However, assume that the sphere is made of radioactive material, or that its material is emitting radiation like neutrinos or photons. Since the body is emitting its mass-energy is decreasing. If radiation is emitted isotropically, then it is clear that the sphere will always spin at the same angular velocity. One may expect that radiation is isotropic in the material's local frame of reference: if this is the case, then there should be no change in the angular velocity.

However, is this even true? In fact, the local frame of a matter element in the sphere is not inertial: there could be a bit of bias in the direction of emission if the angular velocity is such that the local elements feel a very high acceleration!

Question: Will the radioactive sphere experience a torque and spin down while emitting? This depends on whether or not there is a bias in the direction of the emission in the local non-inertial frame.

Answer 1

In your linked paper on pulsar spindown, the mechanism seems to be scattering while the stellar matter is optically thick from the neutrinos' perspective. In that case, you have neutrinos which are isotropically emitted from the center of the collapsing star. As the neutrinos escape the star, they are scattered by material in the star's outer layers, which are rotating "to the east." The neutrinos will preferentially be scattered to the east, stealing angular momentum from these outer layers and carrying it off to infinity.

To reproduce this effect in your spinning radioactive sphere, you would want the diameter of the sphere to be a few mean free paths for your most energetic radiation species. Too small of a sphere and your radiation won't scatter before exiting; too thick and the radiation will thermalize completely before escaping. For instance, imagine your sphere is a pure alpha emitter. If there is any substantial thickness, the majority of the alphas will turn all of their kinetic energy into heat, steal two electrons from the material bulk, and their escape into the vacuum will be an outgassing process.

The parity-violation physicist in me wants to tell you that, if you radioactive sphere is a beta emitter, then the matter lepton is more likely to go out its south pole and the antimatter lepton is more likely to go out its north pole. If I weren't tired tonight, I'd write an amusing paragraph about the Einstein-de Haas experiment on electron spin, and Beth's use of circular-polarized light to drive a torsion pendulum. But the spin-orbit coupling between the rotational angular momentum of an entire crystal and the spin of a nucleus is too tiny for this to be interesting — even for someone like me, with a history of part-per-billion measurements.

Answer 2

In the frame of reference of a point on the surface, nearby radiation will be isotropic. But in an inertial frame of reference, the surface is moving. Photons in one direction will be blue shifted and the other red shifted. So it is a reasonable question to ask.

But as @CarlWitthoft says, it is the same as spitting out a blob. If the surface just lets go of a blob, the blobs will all have one direction. This carries away angular momentum without slowing down the rest of the sphere that remains behind.

If you spit out blobs in random directions, the blobs in one direction will have a higher velocity. On average they will not change the angular velocity of what remains behind. The blobs (fragments of nuclei) will have higher energy because of radioactive decay. But it doesn't change the total momentum.

Answer 3

A sphere in vacuum ... should spin forever because of angular momentum conservation. However, assume that the sphere is made of radioactive material, or that its material is emitting radiation like neutrinos or photons. Since the body is emitting radiation its mass-energy is decreasing.

If radiation is emitted isotropically, then it is clear that the sphere will spin at increasing angular velocity. Because the radiation has no angular momentum, the angular momentum must stay in the sphere, whose rotational inertia is decreasing, so angular velocity must be increasing.

However, if radiation is isotropic in the local frame of reference of the material, then there is no change in the angular velocity. Because recoil force of radiation being shot outwards is directed in the direction normal to the surface of the sphere. So there is no torque causing force. And the angular momentum lost by the sphere is in the radiation.