# 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. – userLTK Apr 24 '17 at 0:08
• 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". – PM 2Ring Apr 24 '17 at 6:59
• – Rob Jeffries Apr 24 '17 at 11:10
• 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. – kingW3 Apr 24 '17 at 11:28

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. – Rob Jeffries Apr 23 '17 at 19:36
• 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. – StephenG Apr 23 '17 at 20:27
• 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. – Rob Jeffries Apr 23 '17 at 21:07
• Still not getting to the point. What rotation period corresponds to "typically"? – Rob Jeffries Apr 24 '17 at 11:11
• @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. – StephenG Apr 24 '17 at 12:05

Current understanding is that star is an oblate spheroid. An extreme example is shown below.

For a neutron star, the difference between the polar diameter and the equatorial diameter is around 10% and would look more like this:

• 10% sounds extreme. What rotation period would that be for? – Rob Jeffries Apr 24 '17 at 18:21
• @Rob Jeffries: I will chase the reference down and send it. Would love another opinion. – dantopa Apr 24 '17 at 19:05

Logically speaking, they should be spherical as things with higher gravity tend to collapse into spheres. Neutron stars are extremely dense, and have high gravity. However, they are as far as we know rotating extremely fast as well (e.g. Pulsars). It should be that the faster they rotate, the more disc-like they would become (like an ellipse or slight possibility of more of a disc in extreme cases). So, depending on rotation speed, sphere for no to fairly high rotation speed, ellipse for high rotation speed, or possibly even a disc for extremely high rotation speed. There is room for debate here, but this is how I logically see it.

Edit: By ellipse, I mean a 3 dimensional ellipse, like an egg, but "squished the other way". Basically a sphere that has been stretched out on its equator. The faster it spins, the more it should be deformed (stretched along the equator). Dantopia's answer shows the shape I am describing.

• This answer adds nothing. – theonlygusti Apr 23 '17 at 20:24
• You're never going to see a disk shaped neutron star. I'm quite sure a neutron star would be rotating faster than the speed of light at that point and it would have long been flung apart before it got close to that point. – zephyr Apr 23 '17 at 22:48
• @zephyr You may be right, hence why I said possibly disc shaped (not sure if they ever can or do form that shape. Spheres and becoming more of an ellipse, wider on the equator with higher rotation speeds should be what is observed. Would be interesting to know if they can rotate fast enough to form a disc or if that would exceed the speed of light as you mention. – Jonathan Apr 24 '17 at 1:39
• And, in the opposite direction, you're not going to see a neutron star with no rotation, since the neutron star still has the angular momentum of the core of the collapsed star, which will always result in extremely rapid spin. – David Richerby Apr 24 '17 at 9:14
• @Rob Jeffries I meant basically a 3D ellipse, good catch! I edited my answer accordingly. – Jonathan Apr 24 '17 at 19:42