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I understand that we can count pulses in time, but how do we know that the magnetosphere is rotating along with the surface of a neutron star?

Can we measure the rotation of a neutron star in any other way like red-blue shifts? It should be pretty difficult due to the small radius and big distance but is it possible? Is it already done?

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    $\begingroup$ Differential rotation between crust and core is something that was been proposed to help explain observations of anti-glitches (a sudden decrease in observed spin rate). An equation of state for a neutron star remains an area of research, so exactly what the inside (core in particular) looks like and behaves is a bit of a mystery. I think all measurements of spin rates are based off of the magnetic field, or use orbital kinematics of binary systems that might only be enough to detect the averaged spin rate. But I'm not sure. $\endgroup$ – zibadawa timmy Dec 29 '17 at 4:18
  • $\begingroup$ @zibadawatimmy consider posting an answer of the form "we don't really, but..." and just add a bit about why one might not expect significant differential rotation. It should be straight forward to explain why the rates should be at least close, if not exactly the same. $\endgroup$ – uhoh Dec 31 '17 at 2:51
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How do we know how fast surface of neutron star is rotating?

I suppose we don't know the answer with total confidence, but I'd say we think we know with good confidence. Because we have robust evidence along with a model that matches that evidence.

I understand that we can count pulses in time, but how do we know that magnetosphere is rotating along with surface of neutron star?

I suppose we don't actually know for sure. But I can't think of any process whereby a stationary spherical body can emit regular pulses of radiation.

Can we measure rotation of a neutron star in any other way like red-blue shifts? It should be pretty difficult due to small radius and big distance but is it possible? Is it already done?

No, and no. At least as far as I know. But most of what I know came from Frozen Star by George Greenstein. The first half of the book is all about neutron stars. It dates from 1984 and it could be a little out of date. But as far as I know there's not much wrong with it, and it's a good read regardless. Note that you can buy a second hand copy fairly cheaply.

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    $\begingroup$ There are of course obvious ways of producing pulses from a non-rotating star; pulsations for one! Still, the question isn't asking for the evidence that pulsars are produced by rotating neutron stars. $\endgroup$ – Rob Jeffries Dec 29 '17 at 18:25
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    $\begingroup$ "...we have robust evidence along with a model that matches that evidence." You should actually name and describe this evidence, and add a supporting link or two. The question "How do we know..." needs a better stackexchange answer than evidence exists. $\endgroup$ – uhoh Dec 31 '17 at 2:23

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