### Shape of neutron stars

• I've heard that the more an object spins, the less of a true sphere it is.
Using this logic most of neutron stars would be far from spherical,in general what shape are most neutron stars?

Consider that as it shrinks it's gravitation also increases so there are two effects that work against each other, the faster rotation wants to stretch the equator, the increase in gravition works against that.

Bear in mind that neutron stars generally have strong magnetic fields, and it appears that in extreme cases the magnetic field can induce shape distortion. According to Kazuo Makishima et al the distortion can be sufficient to "deform the magnetar into a prolate shape, like a football, which wobbles as it spins".

So in general neutron stars aren't that far from a sphere since gravity overpowers rotation,so most of the deformation comes from relativity i.e by length contraction.

5 years ago

I don't think you'll find a single agreed shape for a rotating neutron star, not least because we don't have an agreed single model for the equation of state of the material in a neutron star (which is more complex than the name suggests).

I found one openly available paper (I'm sure there are more) which will give you a rough flavor for the complexity of modeling the shape of neutron stars. As you'll see the difficulty of no single model for an equation of state (EOS is the shorthand typically used) is just one issue.

I think "ellipsoid" should be considered as an approximation, although it's not something I'd consider written in stone.

Remember that to be useful a paper has to provide not just a model for what the shape might be, but also someone has to provide a way to measure this, which is challenging. I think one of the hopes for the new era of gravitational wave astronomy is to be able (eventually) to make more useful and measurements that help us investigate the interior of neutron stars.

So this is an open question, I think.

@Rob-Jeffries asked a question in comment about typical numbers for the deformation, and I answered in comment but comments can be removed by the system, so I'm adding that information as an edit :

In the first section of the paper I linked to they do quote fractional deformations as being typically \$10^{−5}\$, perhaps \$10^{−4}\$ in special cases and in extreme cases up to \$10^{−3}\$. However another paper gives an analysis based on crustal rigidity and a very small deformation for a particular neutron star. The paper I initially liked to describes an upper limit based on gravitational wave considerations, I think, rather than a general analysis.

Better if you gave an order of mgnitude estimate for how important these considerations are. The fastest rotating neutron star has a period of 1.4 ms.

In the first section of the paper I linked to they do quote fractional deformations as being typically \$10^{-5}\$, perhaps \$10^{-4}\$ in special cases and in extreme cases up to \$10^{-3}\$. However another paper gives an analysis based on crustal rigidity and a very small deformation for a particular neutron star. The paper I initially liked to describes an upper limit based on gravitational wave considerations, I think, rather than a general analysis. I'd be interested in hearing more on this from better informed members.

I think that was the point I wanted you to make. A maximally rotating neutron star has a rotation period of like 0.3 ms. Even the fastest known rotating neutron stars are a lot slower than this. So to look at, they would be spherical. The change in shape is very subtle.

Still not getting to the point. What rotation period corresponds to "typically"?

@rob-jeffries : I've never seen a distribution for the periods of rotation of neutron stars, so I'd be loath to give a "typical" value. I'd be interested in seeing such a distribution, in fact.