Pulsars: Cosmic Permanent 'Neutromagnets'?

We argue that pulsars may be spin-polarized neutron stars, i.e. cosmic permanent magnets. This would simply explain several observational facts about pulsars, including the 'beacon effect' itself i.e. the static/stable misalignment of rotational and magnetic axes, the extreme temporal stability of the pulses and the existence of an upper limit for the magnetic field strength - coinciding with the one observed in"magnetars". Although our model admittedly is speculative, this latter fact seems to us unlikely to be pure coincidence.


Introduction
We will assume that the simple model of a pulsar [1] as a rotating neutron star (NS) with a dipole magnetic field at an angle with respect to its orbital axis [2] is basically correct. The radiated power from the magnetic dipole is proportional to sin 2 θ [3], where θ is the angle between the dipole axis and and the rotational axis.
In order to make our point as simply as possible, we further assume that: • The NS is composed solely out of neutrons [4]. (Nearly true assuming that quark stars do not exist. There are observational indications [5] that NS indeed are composed out of normal nuclear matter.) • The density is constant throughout the NS and roughly the same as the density of normal nuclear matter. (In reality, the density is a few times higher in the NS core, and much less in its thin crust.) • The magnetic field is due to spin-alignment of the neutrons in the NS. This is motivated by the fact that aligned spins are energetically favored by the nuclear force, as evidenced e.g. by the deuteron, the more strongly so for the unusually small internucleon separation present in neutron stars [6]. We thus assume that the NS is a "neutromagnetic" material (in direct analogy to ferromagnetic materials). The orbital angular momentum does not contribute to the magnetic field as the neutrons are electrically neutral (no currents). We understand that this is far from the orthodox view, however, the extreme conditions inside neutron stars are not accessible to direct experimental tests, so some leeway seems reasonable. The Pauli principle, naively prohibiting parallel spin states for n − n (or p − p) may well be partially lifted by the extreme gravitational and magnetic interactions, so that some quantum numbers may differ. Also, isotopic triplet (I = 1) states allow spin triplet (S = 1) states for n − n (and p − p). There are also experimental observations of ferromagnet-like nuclear spin ordering phenomena in controlled laboratory experiments [7] (first example of "nuclear spin Ising system").

Origin of magnetic field
Magnetic fields generally can have two origins: i) charged particles in motion, ii) alignment of magnetic moments of the constituents. The observationally inferred magnetic field of neutron stars range from 10 4 T for millisecond radio pulsars to a few times 10 11 T for magnetars.
There is no general consensus about the microscopic origin of the magnetic field of a neutron star. If the "lighthouse"/"beacon" effect which produces the observed pulses in the assumed model [2] is correct, the magnetic field must be very strong and at the same time very stable to account for the fact that pulsars are extremely accurate "clocks". Any "wobbling" or dynamical behavior of the magnetic field would destroy the accurate pulsing. The magnetic field must also be oriented in a direction different from the rotational axis for any pulsar to exist.
In our model, we automatically get all these characteristics, as the neutron magnetic moments are "frozen" in the same direction by the requirement of lowest nuclear energy. In the orthodox model, it is hard to see how a coupled [superfluid neutron -superfluid and superconducting proton]-liquid can produce a simple, and misaligned, dipole field, as a superconductor will constrain the B-field into quantized vortex lines (and not give rise to them). The electrons (expected to be "normal") should be electromagnetically coupled to the proton "fluid" and hence all charged currents should co-rotate giving a magnetic field collinear with the angular momentum. It is also known that several dynamic magnetic instabilities may endanger the field itself. All in all it seems that a more orthodox model of neutron star interiors should give B-fields: i) collinear with L (θ = 0) and, ii) of highly dynamical complex non-dipole form.
In empirical nuclear potentials, e.g. [6], it can be seen that the spincontribution becomes increasingly attractive the smaller the separation. As the neutrons in a neutron star are more highly packed than in normal nuclei, due to gravitation, aligned spins are energetically favored configurations.
Also, in the presence of gravity bound neutrons are stable. It adds an additional, attractive background potential to the nuclear one, lowering the potential below the level required for bound states.
We take the attractive potential for aligned spins to be ≃ 10 percent of the total nuclear binding energy ∆mc 2 , as corroborated by calculations in various models. (Roughly 0.1 × 10 MeV = 1 MeV or 10 10 K.) The NS temperature, originally also roughly 10 10 K at birth in a supernova, rapidly cools via the neutrinos produced in (gravity driven) inverse beta-decay. When it falls below the neutron star "Curie-temperature" 10 10 K, the neutron star suddenly becomes magnetized, the mechanism being analogous to the case in a normal ferromagnetic material. If the temperature at creation happens to be less than 10 10 K the NS will be polarized from the outset, the global energy minimum of the NS will correspond to aligned neutron spins. In a NS the process is connected to the strong nuclear force (instead of the electomagnetic force in a ferromagnet). The NS can thus be labelled a "neutro-magnetic" material.
An independent way to motivate the numbers given above is to make a calculation of the classical dipole-dipole interaction. Their magnetic interaction energy is where, for neutrons, µ = −1.91µ N (the nuclear magneton), x ≃ 10 −15 m (1 fm), giving E ≃ 0.1 MeV, corresponding to a critical ("Curie") temperature of T ≃ 10 9 K. However, it is known that the above classical dipole-dipole calculation underestimates the real value for iron by almost four orders of magnitude, allowing for Curie temperatures, and interaction energies, for 'neutromagnets' to be substantially higher. Also, quantum mechanical entanglement effects should make the alignment much faster and more efficient, due to quantum correlation occurring even at macroscopic distances, as evidenced by laboratory experiments on the rate of macroscopic magnetization due to entangled quantum state of magnetic dipoles in salt [8]. As NS are expected to form at ∼ 10 10 K this could indicate that they become magnetized already at birth, which may help explain the supernova explosion itself, see below.
As all neutron stars seem to have very similar masses 1 M N S = 1.4 ± 0.08M ⊙ [9], where M ⊙ = 1.99 × 10 30 kg is the solar mass (and from general theoretical stability reasons cannot exceed M N S ∼ 4M ⊙ ), we get for the maximum attainable permanent magnetic field, corresponding to total, uniform polarization of the neutron magnetic moments B neutromagn ≤ 10 12 T. ( This coincides nicely with the largest measured magnetic fields of pulsars, in some so-called "magnetars" [10]. It seems strange that such a close match should be pure coincidence. acts as a "seed" for the final NS magnetic field (like the magnetizing field in normal ferromagnetism). However, the ("fossil") B-field of the original star is not conserved, and boosted through contraction of the field lines, as most of the star envelope is blown off. This is a problem in more orthodox models especially in trying to reproduce the extreme B-fields of magnetars [11], but not in our case as it is known that the magnetizing field can be a very small fraction (many orders of magnitude) of the resulting permanent magnetic field. (The other standard scenario, dynamo mechanism due to differential rotation during collapse, seems destined to produce magnetic fields collinear with the rotational axis, removing the 'beacon' altogether.) We know from the sun that the magnetic field is not a simple dipole, but has a more chaotic behavior (solar cycle, etc) and does generally not coincide with the rotational axis. The misalignment of the NS magnetic field will then be statistically distributed with respect to its orbital axis, according to the configuration at collapse. Also, the magnitude of the B-field will be dependent on how complete the spin-polarization will be. (Unless it always saturates, see section 6 below.) This, in turn, will depend on the deviation from simple dipole at the time of star collapse, differently polarized domains, etc.
In other models of neutron stars, where the interior is assumed to consist of superfluid neutrons and superconducting protons (roughly 1 percent of NS), it seems that the NS magnetic field must lie along the orbital axis, which would preclude pulsars. The superfluid neutron angular momentum vortices are strongly coupled to the protons, creating strong magnetic fields parallel to the orbital axis. If so, there would be no observable pulsars, as no "beacon effect" results. In such models, the magnetic field is believed to somehow arise in the highly (normal-) conducting crust, but it is hard to see how it could reach the strength [11], stability and misalignment needed.

Magnetic field -Period relation, and Glitches?
Very fast, millisecond pulsars, generically seem to have the weakest magnetic fields. In the orthodox view millisecond pulsars are supposed to be old pulsars that have been spun-up by accretion from a binary companion star. In our model one could imagine a different scenario. The magnetic field is proportional to the total spin of the neutrons, and only weakly dependent on other variables, However, the orbital angular momentum is strongly dependent on other variables, especially on the frequency of rotation, as the mass and composition of the NS can be assumed to be fairly generic, The maximum angular momentum of a NS arising from spin-polarization is whereas the orbital angular momentum is a function of the rotational angular frequency (or rotational period, P ) The total angular momentum of the NS is then For a solitary (radio) pulsar, as there is no outside torque, One could then speculate that pulsar glitches, sudden speed-ups of ∆P/P ∼ 10 −8 , may be due to rearrangement of L and S through L-S coupling, Tensor coupling or relaxation (small amount of S ↔ L). However, as pulsars exhibiting glitches are very rare, the data set at present may be too small to test such a hypothesis, and we will refrain from further analysis here.

Supernova 'bounce'
In our model the NS (pulsar) could be the cause of the supernova (SN) explosion, and not an effect of it. (At the very least it will augment the explosion.) The 'bounce' which halts and reverses the infall of material may be due to electromagnetic shock in the very dense plasma. This results when the rapidly rotating NS and its dipole magnetic field is suddenly born as a consequence of energy minimization. As v sound is the speed of multinucleon interaction, and v sound ∼ c in the dense NS, this process takes only R N S /c ≃ 10 −5 s, quickly releasing the energy producing the huge magnetic field.
The electromagnetic forces can be seen to be more than enough for the purpose of 'bounce': For protons at the surface of a NS with B ≃ 10 12 T and rotational period P ∼ 1 s and for electrons the ratio is ∼ 10 3 higher. (Even if combining the longest periods known, P ∼ 10 s, with the weakest inferred fields, B ∼ 10 4 T, one obtains F EM /F grav ≃ 10 4 .) In the conventional core-collapse scenario for a SN, the infall is expected to "bounce" (compression and rebound) when the inner core exceeds nuclear densities and the, at very short distances, repulsive potential of the nuclear force "stiffens" the core. It is known that the outgoing shock-wave which results is insufficient to disrupt the star. The shock stalls and the material falls back onto the core. The usual way to remedy this is by neutrino heating of the shock. However, even in this case it is difficult to reproduce a star that actually explodes in simulations [12]. A pulsar-driven/augmented, electromagnetic 'bounce' would automatically give the non-spherically symmetric explosion needed in core collapse SN scenarios [12]. Non-spherical SN ejecta are also seen in observations [13].

Total explosion due to electromagnetic shock?
Even though it is well known that 99 percent of the energy in a SN is deposited in the neutrinos produced in inverse beta decay driven by the gravitational potential during collapse, they may have little to do with the explosion of the star, or at least not be the dominating factor.
Energetically, the actual explosion of a SN is thus a minor phenomenon (∼ 1 percent of total). dB/dt is very large when neutron spins align. This, coupled with the extremely dense plasma result in highly nonlinear plasma interactions. The extremely strong, complexly entangled electromagnetic field, give rise to tur-bulence and shocks. It also rids angular momentum from the progenitor so that the central core can rotate slower than break-up speed.
The theoretical break-up rotational period is ∼ 5 × 10 −4 s, but could be smaller during collapse when infalling material stabilizes the core by external dynamical pressure, also the proto-neutron star can then not be considered an isolated object, invalidating the theoretical calculation of the limit (gravitational radiation leaking angular momentum away).
The automatic deviation from spherical symmetry is also necessary for explaining the observations of very high peculiar velocities of many NS [14] Furthermore, the unexplained blow-off of the envelope of non-massive stars [15] can be due to the same (universal) mechanism. It is less violent because the driving force originates in a white dwarf with much lower B and ω.

Energy balance
The energy released as the neutron spins align, assuming for now 100 percent (saturated) spin-polarization, is where the number of neutrons in a generic neutron star. (The actual number is somewhat higher as the attractive potential lowers the effective mass.) So we get for the energy release due to spin-alignment ∆E spin ≃ 10 51 erg This energy is thus in principle capable of powering the whole SN explosion.
A canonical SN has a total energy output of roughly 10 51 erg, but only 1 percent of this goes into the kinetic energy of the ejecta, the rest (99 percent) escapes as neutrinos.
Magnetic (dipole) field strengths of pulsars are indirectly inferred from observed spin-down rates, or, in Tesla, where P is measured in seconds andṖ = dP/dt is dimensionless. In a normal ferro-magnet below the Curie-temperature the spin alignment is near 100 percent. In a neutron star the process should be at least equally efficient, and most likely also faster, as it is driven by the strong nuclear force instead of electromagnetism.
If we assume that the same (but with much higher effective binding forces) applies for neutron stars, they will all be almost identical permanent magnets. NS will then be extremely simple, all having almost the same mass (1.4 ± 0.08M ⊙ from observations [9]) and the same magnetic field (∼ 10 12 T). This loss of individuality is well in line with the next step on the cosmic compact object ladder, the black hole, which is very simple and is totally described by only three numbers (its mass M, angular momentum L and charge Q).
If now B is constant, the power of dipole radiation dE/dt depends on angle and period only, where const = 32π 4 R 6 B 2 /3c 3 In cases where B is parallel to L (θ = 0) no pulsar appears, if they are almost aligned (θ ∼ 0) a "weak" B is inferred, and for large misalignment (θ ∼ π/2) a huge "magnetar" B is inferred.
Assuming the pulsar-driven SN mechanism above, no SN should appear in the θ = 0 case. The massive star will then quietly settle to a black hole as the energy dissipates, and gravity overtakes everything else. The relative "violence" and spatial structure of a SN then depends only on the angle θ, and P .

Conclusions
Even though the presented model of a neutron star being a "giant polarized nucleus" is overly simplified, it nevertheless has an attractive simplicityin the vein of Zwicky, who together with Baade originally introduced the very concepts of NS, SN and their interconnections [4] -and explains several unresolved properties of pulsars: i) The origin of the magnetic field is simple and unavoidable. In other models it is a complication which has to be addressed separately. That a completely polarized neutron star automatically gets a magnetic field comparable to that of magnetars seems, to us, too compelling to be pure coincidence.
ii) The non-zero angle of the magnetic field to the rotational axis is explained. The direction is triggered by the original magnetic field of the massive star at time of collapse, and then "frozen in" by the nuclear force.
iii) We get a natural maximum limit for the magnetic field, B ≃ 10 12 T, corresponding to the field in 'magnetars'. The model also predicts that no pulsars (or neutron stars) will have a B-field greater than this, as all measured neutron star masses are highly peaked around 1.4 solar masses, and from general stability arguments their maximum masses cannot be more than a few times higher than this. In that sense our model is directly falsifiable; if any neutron star with B > 10 12 T is detected some other mechanism for generating the magnetic field must apply.
iv) The fact that pulsars are extremely exact "clocks" means that their magnetic fields must be very stable. As the neutrons align their spin akin to the atoms in a normal ferromagnet, we get this property for free. v) Glitches may possibly be caused by relaxation, due to L-S coupling, to a state with lower energy. This should then be accompanied by a (minute) decrease in the B-field, which in principle could be measured.
vi) If only the small proton admixture, of order 1 percent in the orthodox scenario, contributes to permanent magnetization through quantum mechanical (n − p) pairing, B max ∼ 10 10 T, with only marginal alteration in Curie temperature.
One should remember that the nuclear physics at these extreme circumstances and densities is not known a priori, so several unexpected properties (such as "neutromagnetism") might apply. The "proof is in the pudding", and from our back-of-the-envelope calculations the model is at least not immediately ruled out. The fact that there also exists a huge "seeding" external magnetizing field from the collapsing star at the moment of neutron star creation makes neutro-magnetization plausible.