On the Equilibrium of Magnetic Stars.

Abstract

The necessary condition for the equilibrium of a liquid magnetic star is derived and the solution in a particular case given. It is shown, as was first pointed out by Chandrasekhar and Fermi, 1 that a magnetic field has the same effect as rotation, i.e., it tends to produce a flattening of the poles of the star. A formula for the ellipticity is derived. 1. Introduction.—The mechanical equilibrium of an ionized medium in which electric currents flow is of interest in connection with problems of cosmical magnetism. Chandra-sekhar and Fermi 1 first pointed out that we must expect a magnetic field to produce the same effect as a rigid-body rotation. They showed that for a spheroidal liquid sphere, in which the magnetic field was uniform inside and like a dipole field outside, the ellipticity to be expected is given by the formula (2 = (7/2)3K/20, where denotes the magnetic energy and 2B the gravitational energy of the star. We shall here consider the problem of the equilibrium of a rotating liquid star in which electric currents flow, restricting the discussion to the case of symmetry about the axis of rotation. It will be shown that the magnetic field in the star must satisfy a certain condition for equilibrium similar to that found by Stokes 2 for steady rotational motion in an incompressible fluid. No account is taken of the decay of currents through elec-trical resistance, since Cowling 3 has shown that this is extremely slow, and we can there-fore consider the currents as effectively steady. Provided that the magnetic energy of the currents is small compared with the gravitational energy, the surface of the liquid will be a spheroid of small ellipticity whose value is given by a formula similar to that found by Chandrasekhar and Fermi. It is hoped to discuss the stability of the equilibrium in a future note. 2. The equation of the problem.—We consider a finite mass of liquid of density p rotating as a rigid body with constant angular velocity co about a fixed axis. Let this be taken as the s-axis of a system of cylindrical polar co-ordinates (gj,