On the Magneto-Hydrostatic Theory of Sunspots.

Theoretical sunspot models are constructed which are a generalization of those considered by Schltiter and Temesvary (1958). A non-vanishing horizontal component of the pressure gradient, as required by magneto-hydrostatic equilibrium, is obtained by assuming that a magnetic field in the solar hydrogen convection zone (HCZ) inhibits the convective-energy transport. In the framework of Prandtl's mixing- length theory, this is achieved by making the ratio of the mixing length, t, and the pressure scale height, H, smaller in the magnetic-field region than in the surrounding undisturbed HCZ. The corresponding system of ordinary differential equations was solved numerically, and pressure, temperature, and magnetic-field strength along the axis of symmetry were obtained. When t/H ranges from zero to (t/H) cz a one-parameter family of solutions results; in the one extreme convecflve-energy transport is completely suppressed; in the other extreme there is no inhibition of convective-energy transport. From this family of models, relations between the coolness of a spot and its magnetic-field strength and between the magnetic-field strength and 1/H are derived. The former relation agrees with observations to within 50 per cent; the latter indicates a maximum possible field strength of about 5000 gauss.