The name neutron star is a bit of a misnomer. The interiors of neutron stars are expected to contain neutrons (about 80%), protons and electrons (each 10%). It is these charged particles that support the magnetic fields. The origin of the fields has two possibilities. One is that the magnetic field that originally threaded the core of the massive star is frozen in the collapsing plasma that forms the neutron stars. This flux-freezing hypothesis can account for the fields of most neutron stars. Some neutron stars (the magnetars) have very strong magnetic fields. The fossil fields may not be strong enough to account for these. Here it is thought that a dynamo operates in the differentially rotating, convective proto neutron star. This stretches and folds the magnetic field, strengthening it dramatically.

Thanks for your answer. Now, about the first mechanism, and I quote you

'the magnetic field that originally threaded the core of the massive star is frozen in the collapsing plasma that forms the neutron stars';

is there a kind of conservation law involved here? And if it is, of which kind? I can imagine a kind of magetic lines of flux conservation during the process of collapsing, but I do not have a clear picture for it. For example, what is the form of the thermodynamic internal energy for this system?

Am I thinking correctly? (The problem is far from my field.)

The electrical conductivity is very high so that over the timescale of the collapse of the star the magnetic field lines cannot move far relative to the collapsing plasma, so the magnetic flux threading the core is conserved Flux ~ B R^2 so as R decreases the field increases.

The magnetic field of a neutron star comes from the intrinsic magnetic dipole fields of aligned neutrons. A formula is derived from my cosmological model to calculate the magnetic flux density of a neutron star in section 5.5 of my paper.

Article A Cosmological Model with Variable Constants (Functions of t...

I don't know what is the cause aligning the neutrons, but they (neutrons) have their own magnetic moment.  On the other hand neutron stars are perfectly spherical.  This makes possible, in my opinion, to have a neutron star magnetized uniformly.  Electric conductivity seems irrelevant.  Unfortunately, I haven't found any information about the (average) density of neutron star.  Having it we should be able to estimate their magnetization, and, consequently, the magnetic field generated there.  Maybe it was done already?

Generally, it is believed that the strong magnetic field of neutron stars originated from the collapse of the core of a supernova with the conservation of magnetic flux.

From the conservation of magnetic flux, We have

ΦF=∫B dS=const,

where ΦF is the magnetic field flux, B is the magnetic field strength, and dS is the elemental closed surface; then, this integral is constant through the surface.

Consider the star surface over which take the integral, then S=4πR2, where R is the star radius, together with the magnetic flux conservation law, we have:

Bf=Bi(RiRf)2

where i and f are the indices for initial and final stages. We know that the star implodes from a whatever star size to about10 km. So the radii ratio is huge. You just need a starting magnetic field of 10−100 G, to get a final magnetic field of the order of 10^12 G, that is typical in neutron stars.

Dear Issam Sinjab,

Your explanation looks spectacular but is seriously flawed.  Where from did you get "the conservation of magnetic flux"?  Your equation is perfectly valid for electric field, E, but doesn't make much sense for magnetic induction.  Imagine a simple magnetic dipole as a source of B, located at the center of a ball.  It is easy to see that your integral is exactly equal to zero in this case, as B(-x)=-B(x).  Consequently, your second expression contains the term "0/0".

J.D. Jackson in his "Classical Electrodynamics" (John WIley & Sons, Inc., 1962, 1975) presents the formula 5.62 stating that integral Bd3x = 8\pi m/3, where B and m are 3D vectors, and m is the magnetic moment of stationary currents located completely within the integration area. Of course, this formula is useless for our purposes, but at least it expresses some conservation law, unfortunately not the magnetic flux is conserved here.

Besides, the intergalactic magnetic field is rather weak, it is estimated to be 4-8 orders of magnitude weaker than your "starting field", while the magnetars produce 1012-1015 G, (http://en.wikipedia.org/wiki/Magnetar).

Dear Marek Wojciech Gutowski 

Sorry for my late reply as I am on holiday and I should explain better my earlier post and point out some of the difficulties that come with the conservation of magnetic flux assumption especially in connection with magnetars.

You asked:

"Where from did you get "the conservation of magnetic flux"? "

Well, it was first suggested by Manchester & Taylor (1977) and by Smith (1977). See for example:

Manchester, R. N., & Taylor, J. H. 1997, Pulsars (San Francisco: Freeman, W.H. & Company).

Also, please refer to Jeremy Heyl's post dated Dec 4 2012:

"The electrical conductivity is very high so that over the timescale of the collapse of the star the magnetic field lines cannot move far relative to the collapsing plasma, so the magnetic flux threading the core is conserved Flux ~ B R^2 so as R decreases the field increases."

There are however serious problems with this assumption as we shall see below.

A large value of the magnetic field near the neutron star's surface is customarily assumed to be associated with magnetic flux conservation during gravitational collapse, also called fossil field hypothesis. The simplest and most popular hypothesis, magnetic flux conservation, is that neutron stars magnetic fields are simple remnants of their main sequence (MS) progenitors. For a star of radius R and surface magnetic field strength B, Conservation of magnetic flux in gravitational collapse require the magnetic flux BR^2=constant. Consider a star of  B=100G and R=7x 10^8 m. If a star of this surface magnetic field strength and radius collapsed to a neutron star of radius R(ns)=10 km, it would have a surface magnetic field of B(ns) equal to about 10^11 G. Admittedly, this is unrealistically optimistic estimate because the neutron star contains only some 15% of the progenitor's mass! The other major problem of course, as you rightly indicated, is that the magnetic field strength 10^11 G is at least 1-4 orders of magnitude weaker to explain magnetars with magnetic field strengths 10^12-10^15 G. The fossil hypothesis is therefore not an attractive one, at least not for magnetars.

One could argue that some MS stars have much higher fields, the strongest fields known in MS stars are around 10^4 G. Magnetic flux conservation during gravitational collapse of these stars would imply  B(ns) equal to about 10^13. Not only is this still insufficient for magnetar but there is also a problem of statistics. Only a very small fraction of progenitors has magnetic fields as large as 10^4 G. Magnetars, on the other hand, are born frequently. Their birth rate is probably comparable to that of normal neutron stars-Woods(2008).

One other serious problem of the magnetic flux conservation is that not only the neutron star contains only some 15% of the progenitor's mass but also the core of a typical progenitor MS star occupy only some 2-3% of the star's cross section!

Here's a research paper that attempts to explain the phenomena using an approach for induced magnetic moments in neutron superfluids:

The Physics of Strong Magnetic Fields in Neutron Stars:http://arxiv.org/ftp/arxiv/papers/0706/0706.0060.pdf

Origin of neutron star magnetic fields. Possible origins of the magnetic fields of neutron stars include inheritance from the main sequence progenitor and dynamo action at some stage of evolution of progenitor. Once formed in core collapse, the field is in danger of decaying again by magnetic instabilities.

For magnetars ( H ~ 103 Gauss ) there is an explanation, Dr. Ignacio Cortese.

I just read several posts as a fan of the use of statistical physics in neutron stars:

https://imagine.gsfc.nasa.gov/science/objects/neutron_stars1.html

https://en.wikipedia.org/wiki/Magnetar