3.5. Neutron Stars (~ 10 km)
When immersed in a strong magnetic field B 1012 Gauss, the atom takes a cigar-shaped structure, since the Coulomb force becomes more effective for binding electrons in the direction parallel to the magnetic field axis while the electrons are extremely confined in the direction transverse to the magnetic field axis. Using the atomic unit a0 = 0.5 × 10-10 m, one finds that the mean transverse separation of the electron and proton in the hydrogen atom is L ~ [2m + 1]0.5 / [426 B12]0.5 atomic units, where m = 0, 1, 2, 3, ..., and where B12 is the magnetic field in units of 1012 Gauss. The ionization energy of the atom becomes EH = 161 eV for B12 = 1 and rises as EH 161 [ln(426B12) / ln(426)]2 eV (e.g., Lai & Salpeter 1997).
3.5.1. Normal Pulsars (~ 1012 Gauss)
When a normal star with radius R ~ 106 km collapses to form a rotating neutron star or pulsar with radius ~ 10 km, the magnetic flux B will be conserved (B ~ R-2) and the surface magnetic field will increase from ~ 100 Gauss to ~ 1012 Gauss. A previously dipolar shaped magnetic field will remain dipolar shaped. The magnetic field shape of pulsars is generally dipolar, and a magnetosphere is created around the pulsar (e.g., Radhakrishnan & Cooke 1969; Navarro et al. 1997). Neutron stars typically have a gas density decreasing from 1042 cm-3 at the center to 1012 cm-3 at the surface.
The data for normal pulsars are consistent with some kind of passive frozen-in magnetic dipole, of strength ~ 1024 Gauss m3. The radius of a normal pulsar is ~ 10 km, with a mass ~ 1 solar mass, and a rotation period ~ 0.1 sec, giving an angular momentum ~ 5 × 1039 kg. m2 s-1 (not far from the dipolar dynamo law for other bodies).
The basic radio emission process is essentially the same in millisecond-period pulsars and in slower pulsars. The nearby ~ 100 pc PSR J0437-4715 pulsar has a 5.8 millisecond period, a characteristic age of 2.5 × 1010 years, and a dipole magnetic field of 2 × 108 Gauss. This pulsar shows evidence of inertial dragging of its magnetic field lines in the outer magnetosphere, with the low frequency radio emission coming from higher altitudes in the pulsar magnetosphere (e.g., Navarro et al. 1997).
In recent models, the pulsar magnetosphere is divided into closed magnetic field lines (where particles are trapped for long periods of time) and open magnetic field lines (where particles are not confined and are eventually lost to the interstellar medium). In addition, recent models have added a secondary magnetospheric shell, having 1% of the number of particles in the primary shell (e.g., Fig. 4 in Eastlund et al. 1997).
The time evolution of the magnetic field with time in a pulsar is still controversial, a major issue in compact object astrophysics. While some theories favor no magnetic field decay over a long time (e.g., Romani 1990), most theories favor a magnetic field decay of some sort (e.g., Wang 1997). In the decay theories, the pulsar's evolution is divided into three phases: (i) the dipole phase, in which the pulsar spins down through magnetic dipole radiation, ending when the ambient material's ram pressure overcomes the pulsar's wind pressure; (ii) the propeller phase, in which ambient material fills the corotating magnetosphere in a shell above the Alfvénic radius, ending when the shrinking Alfvénic radius becomes smaller than the corotation radius; (iii) the accretion phase, in which the matter accretes directly on the polar cap of the neutron star. In the first (dipolar) and second (propeller) phases, the magnetic field decays either as a law
with td 108 years, and B0 1012 Gauss. In the third (accretion) phase, there is no magnetic field decay and the field strength remains steady. The 8.4-second pulsar RX J0720.4-3125 may be in the third or accretion phase (e.g., fig. 1 in Wang 1997).
Mukherjee & Kembhavi (1997) used statistics to obtain a lower limit on the decay timescale of pulsar magnetic fields (> 160 millions years).
The distance to a pulsar is best determined when one uses a detailed model of the distribution of free electrons in the disk of the Galaxy. Taylor & Cordes (1993) have provided such a distribution model, with electron density enhancements in four spiral arms and near the Gum nebula (their equ. 11 and Fig. 1, using a value of 8.5 kpc for the Sun-Galactic Center distance). From the electron density distribution, one can compute the plasma frequency and the group velocity at which a radio signal at a frequency can propagate in the interstellar medium. Comparing time arrivals at two or more frequencies, yields the dispersion measure DM, an integral of the free electron density over the distance to the pulsar. Thus the observed DM and the use of the distribution model for the free electrons will yield the pulsar distance, to an accuracy of 25%.
3.5.2. Magnetars (~ 1015 Gauss)
Extreme pulsars are called magnetars. They typically have a dipolar magnetic field strength ~ 1015 Gauss and a mean gas density ~ 1040 cm-3 (e.g., Thompson & Duncan 1996; Frail et al. 1997). The data for extreme pulsars (magnetars) are consistent with some kind of passive frozen-in crustal magnetic dipole, of strength ~ 1027 Gauss m3. The radius of an extreme pulsar is ~ 10 km. Rotating with a period of a few seconds, with a mass ~ 1.4 solar masses, magnetars have an angular momentum ~ 1038 kg m2 s-1. When a magnetic field stronger than 1 - 14 Gauss is dragged through the pulsar crust by the diffusive motions in the pulsar core, Hall turbulence is excited and leads to multiple fractures tha will release enough energy to power soft gamma ray bursts.
The physical upper limit to the neutron star's magnetic field strength is the virial equilibrium value ~ 1018 Gauss (Lai & Salpeter, 1997).