6.6 Magnetic fields of white dwarfs and neutron stars

*Faraday's law of induction*
__Physics Introduction:__ In the laboratory, if a wire heats up because of electrical resistance, one uses a thicker wire. Cosmic magnetic fields imply electrical currents flowing in extremely thick gaseous "wires", so thick that dissipation of the currents into heat is negligible over time scales of interest. Effectively, there is no electrical resistance. Therefore, the electrical field in the frame of the ions vanishes. The electrical field seen in any other frame of reference is __E__ = -__v__x__B__, and Faraday's law of induction becomes __B__/t = -curl__E__ = curl(__v__x__B__). The integral form of Faraday's law is /t__B__d__s__ = __v__x__B__d__l__ where the integrals are over any fixed surface and around the edge of that surface. On interchanging and x in the line integral, one can show geometrically that Faraday's law reduces to d/dt__B__d__s__ = 0 , where now the integral is over any surface moving with the gas. The magnetic flux through any surface moving with the gas is conserved.

Imagine two elements of gas connected by a field line at some time. Draw a surface, generally not flat, which includes this field line and which also includes neighboring field lines. This surface has zero magnetic flux through it. Let this surface move with the gas. By Faraday's law without current dissipation, the surface must continue to have zero magnetic flux through it. Therefore, the same field line still connects the two elements of gas. In effect, the field lines are attached to the gas. One says the magnetic field is "frozen into" the gas. Although magnetic field lines are mathematical constructs, in MHD they take on a real physical property. That is why, for instance, the vibration of field lines and gas participating in an Alfvén wave is completely analogous to the vibration of a string with tension as in a violin.

__The problem:__ Imagine that the Sun, in its interior, contains a magnetic field encircling the axis of the Sun, parallel to the solar equator, much as if a giant solenoid were stretched around the Sun at its equator. Now imagine making the Sun smaller, keeping the same structure of density and frozen-in magnetic field. If the field strength B changes in proportion to R^{n}, what is n? If the field strength within the Sun is 10 Tesla, what is it after the Sun shrinks to the size of a white dwarf, and after the Sun shrinks to the size of a neutron star? For the stellar radii, use R = 10^{6}km, 10^{4}km, and 10 km, respectively.

__The solution:__ It is simplest to consider any part of a plane through the axis of the star, that is, normal to the magnetic field, for instance a circle within that plane. Follow the motion of the gas on the boundary of that circle as the circle shrinks. The magnetic flux through the circle must remain constant. But the area of the circle, attached to the gas, shrinks as R^{-2}. Thus n = -2, and B/Bo = (Ro/R) ^{2}. The white dwarf then is expected to have B = 10^{5}Tesla, the neutron star B = 10^{11}Tesla.

__Interpretation:__ The strongest magnetic field observed on a white dwarf is 5x10^{4}Tesla. Magnetic fields of neutron stars observed as radio pulsars have values up to 4x10^{9}Tesla. The highest field deduced in an x-ray pulsar is 1.6x10^{11}Tesla. So the estimates are quite satisfying. One can really expect only order-of-magnitude agreement because the solar B = 10 Tesla is estimated only rather indirectly, from the manner how sunspots emerge from the deep solar interior; because the white dwarf fields are left in the star after the star has shed its outer layers; and because the neutron star fields are left after the star exploded as a supernova. Moreover, it is possible that powerful convection during the supernova explosion amplified the field in the neutron star. Nevertheless, it seems reasonable that the observed fields on white dwarfs and neutron stars derive from the compression of magnetic fields that were within the stars when these stars were main sequence stars.

Exceptions to the frozen-in approximation occur at very powerful events, such as solar flares (whose explosive energy must derive from powerful localized electric currents suddenly dissipated in the solar corona), the turbulence during a supernova explosion, the interaction of the Crab Nebula with the pulsar at its center (since the magnetic fields in the nebula now are far too strong as to have expanded in frozen-in form from the original supernova and its progenitor star), and the turbulence of the gases between the stars (which must have created the observed interstellar galactic magnetic field), and at singular geometries such as the front and back of our magnetosphere (diagram in problem 6.4).

__Didactics:__ When electrical currents are negligibly dissipated into heat, this is often interpreted as "zero resistance" or "infinite conductivity", inspiring the analogy with superconductors. However, the conductivity of many astrophysical gases is within an order of magnitude of the conductivity of copper in Earth's laboratories. Negligible dissipation in astrophysics is literally due to the extreme thickness of the "wires" (mathematically, the very large value of the length B/|curl__B__|).

Possible students' problem. (Students should discuss each part and agree on the answer before moving on). Suppose the magnetic field now in the Crab Nebula, B = 4x10^{-8}Tesla, were truly frozen into the expanding nebular gas, approximately a sphere of radius r = 3 light years. Earlier, r was smaller and B was higher. With what exponent does B depend on r? (Answer: B is proportional to r^{-2}, as for the stars.) How do the magnetic energy density and the total magnetic energy in the nebula depend on r? (Answer: r^{-4} and r^{-1}, respectively). Estimate backwards in time: what was the total magnetic energy when r equaled the radius of an about-to-explode star, say r = 150Ro = 10^{11}m ? (Answer: Now the magnetic energy density = 2/ x 10^{-9}j m^{-3}, the volume = 3.6 x10^{49}m^{3}, the total energy = 7.2x10^{40}j; scaled by a factor 3x10^{16-11}, the total magnetic energy then = 2x10^{46}j.) Compare to the gravitational energy of the original star, assuming M = 10Mo, and interpret. (Answer: GM^{2}/R 2.6x10^{41}j 2x10^{46}j. No stable star can exist with negligible gravitational energy.) [The Virial theorem extended to MHD has the total kinetic energy K replaced by K + total magnetic energy.]