## The Ferraro Law of Isorotation

The law of isorotation was found by Ferraro [183,184]. This law or theorem says the following: "The magnetic field of a star can only remain steady if it is symmetrical about the axis of rotation and each line of force lies wholly in a surface which is symmetrical about the axis and rotates with uniform angular velocity". To derive this law, we separate the magnetic field into poloidal and toroidal components  in spherical coordinates (r, ft, p),

Several simplifications are made. The fluid is assumed to have no other motions than a rotation with velocity component vp = r sin ftQ in the direction of the unity vector ep. It is incompressible which implies V • v = 0 and perfectly conducting (n = 0). The solution is stationary and axisymmetric. The induction equation (13.4) for the component Bp becomes

'dp = [V x (v x B)] • ep = [(V • B)v - (V • v) • B] • ep. (13.45)

The operator V acts as a derivative on the subsequent terms so that one has dB

[w(V • B) + (B • V)v - B(V ■ v) - (v ■ V)B] ■ ep. (13.46)

The first and third terms are zero, because of the Maxwell equations (13.3) and because of the incompressibility. Accounting for the fact that the only motion is rotation with a velocity component = r sin VQ, one gets

= [(Bp • V)v^ev - (v9e9 • V)B] • e^ = (Bp • V ■ (13.47)

The second term in the bracket is zero, because there is no q gradient of Bq due to the axial symmetry. This equation shows that a toroidal field can be produced from a poloidal field in the presence of differential rotation. The stationarity imposes

This expression states that there is no gradient of Q along a poloidal field line. Thus, the Ferraro theorem is that stationarity, together with the other mentioned conditions, imposes that the poloidal field lines have a constant Q. Since the loop described by the poloidal field lines go from the center to the surface, it is a common extension to say that magnetic fields produce solid body rotation. In practice, this requires some time during which diffusion may adjust the fluid velocity to that of the field lines.