7.5 Magnetic field of pulsars

*low-frequency magnetic "dipole" radiation*
__Didactic purpose:__ An example of approximations that retain the essential physics.

__Physics needed:__ Energy density and propagation speed of an electromagnetic wave.

__The setting:__ The Crab nebula needs a prodigious source of energy, about 1.2x10^{5}Lo. Indeed, the central rotating neutron star is slowing down and provides the energy needed for the nebula (problem 3.4). But how is the energy carried from the pulsar to the nebula? The rotating pulsar causes the surrounding magnetic field to vary in time. Thus the pulsar emits electromagnetic radiation at the pulsar rotation period. The pulsar radiates as much energy as the kinetic energy lost by the pulsar's slow-down if the magnetic field at the pulsar's surface has the appropriate value. This problem leads the student through the appropriate (approximate) steps to evaluate the necessary pulsar magnetic field.

__The problem:__ A neutron star (of mass M, radius R, rotation period P) carries a dipole-like magnetic field with strength Bp at the surface, tilted with respect to the rotation axis. Radio radiation is emitted in a cone centered on the magnetic axis. As the cone sweeps past Earth, we observe a pulse of radiation. The low-frequency radiation at period P removes rotational energy from the pulsar and causes a slow-down in the rotation rate, dP/dt > 0. Evaluate the low-frequency radiation loss by the pulsar, as follows: Near the pulsar, write B(r)=Bp(r/R) ^{n}. What is n for a dipole-like field? Assume that this relation is valid out to the "light cylinder", that value of r where an object circling the neutron star with period P moves at the speed of light, c. Evaluate B at the light cylinder, roughly at the equator, in terms of Bp, R, P, and c.

Now assume that, exactly at the light cylinder, B makes a transition from a static dipole inside the light cylinder to a propagating electromagnetic field outside. The electromagnetic wave consists mostly of magnetic field and very little electric field. The wave moves from the light cylinder into space at speed c. What is the energy density of the electromagnetic field at the light cylinder, expressed in terms of Bp, R, P, and c? What is the energy flux (i.e. energy crossing unit area per second)?

Suppose this radiation is emitted from the light cylinder all around the equator and for a distance R above and below the equator. What is the total energy, dE/dt, emitted over this surface, expressed in terms of Bp, R, c, and P?

Now equate your formula for dE/dt to the kinetic energy lost by the neutron star as its rotation slows down (derived in problem 3.4), dE/dt = -8/5 ^{2}MR^{2} (dP/dt) / P^{3}. Derive an equation for Bp involving M, R, c, P and dP/dt. Evaluate Bp using the same numerical values appropriate for the Crab Nebula pulsar : P = 0.0333 sec , dP/dt = 4.21 x 10^{-13} sec/sec, R = 10 km and, as used in problem 3.4, M = 1.4 Mo .

__The solution:__ n=3. The radius of the light cylinder is Pc/2. B at the light cylinder is Bp(2R/Pc) ^{3}. The energy density there is [Bp(2R/Pc) ^{3}]^{2}/2, the flux is energy density x c, the area is 4(Pc/2)^{2}, and so dE/dt = -(2c/)Bp^{2} (2/Pc) ^{4} R^{6}.

On equating the two expressions for dE/dt, Bp^{2} = (/20^{3})(Mc^{3}P dP/dt /R^{4}). The numbers for the Crab pulsar yield Bp = 4.6x10^{8}Tesla.

__Interpretation:__ The space surrounding the neutron star is clearly not a vacuum but is permeated by a plasma, possibly electrons and protons. After a few theoretical improvements for the radiation loss and plasma flow, the most widely accepted value for the Crab pulsar is Bp = 4x10^{8}Tesla, remarkably close to our estimate. Magnetic fields estimated in this way for various radio pulsars range from 3x10^{4}to 4x10^{9}Tesla. The lower values are for the very fast (millisecond) pulsars that have acquired gases from a companion star.

One obvious simplification here is the assumed dipole magnetic field inside the light cylinder. A dipole field involves no electrical current, it cannot do work, and thus it cannot emit radiation. It is more accurate, but much more complicated, to describe the field as nearly dipole near the neutron star and becoming gradually more distorted, current-carrying, time-dependent and electromagnetic at larger distances from the pulsar. If one attempts this, one must promptly ask: "What is "near", relative to what distance?". The electromagnetic nature of the desired waves then implicates the distance of the light cylinder, and that distance is then chosen for the assumed sudden transition.

Even the surface of the neutron star is by no means free of electrical current. Because of the rapid rotation, the electric field there is so strong that it pulls either ions or electrons from the surface. The details of the resulting magnetosphere, including the electrical currents, are not yet solved self-consistently. (MHD is not adequate.) Certainly, particle energies are high enough for emission of gamma rays, which then can create electron-positron pairs. Probably an electron-positron wind, aided by the low-frequency magnetic waves, permeates the entire Crab Nebula (except the filaments for which the Crab is named). Perhaps shocks in this wind accelerate a fraction of the electrons and positrons to sufficiently high energies so that they emit the observed synchrotron radiation. There is not yet any observational evidence whether the synchrotron radiation is emitted by only electrons or by both electrons and positrons. The low-frequency waves, being mostly magnetic, somehow also maintain the magnetic field of the Crab Nebula.

__Didactics:__ A great art in frontier physics, and especially in astrophysics, is to make those simplifications that retain the important physics. The example here concerns the assumed sudden transition from the dipole to the radiation field. There have been many other examples earlier. These simplifications allow some understanding of new phenomena and allows asking questions for new observations. Of course, afterwards it is important to verify the simplifications in detail. Sometimes, the simplifications turn out to be wrong. But not in this example of the Crab pulsar's energy loss.