Basic properties of sunspots: equilibrium,stability and long-term eigen oscillations

A number of fundamental questions as regards the physical nature of sunspots are formulated. In order to answer these questions, we apply the model of a round-shaped unipolar sunspot with a lower boundary consisting of cool plasma and with strong magnetic field at the depth of about 4 Mm beneath the photosphere, in accordance with the data of local helioseismology and with certain physically sound arguments (the shallow sunspot model). The magnetic configuration of a sunspot is assumed to be close to the observed one and similar to the magnetic field of a round solenoid of the appropriate size. The transverse (horizontal) and longitudinal (vertical) equilibria of a sunspot were calculated based on the thermodynamic approach and taking into account the magnetic, gravitational, and thermal energy of the spot and the pressure of the environment. The dependence of the magnetic field strength in the sunspot center, B ₀, on the radius of the sunspot umbra a is derived theoretically for the first time in the history of sunspot studies. It shows that the magnetic field strength in small spots is about 700 Gauss (G) and then increases monotonically with a, tending asymptotically to a limit value of about 4000 G. This dependence, B ₀(a) includes, as parameters, the gravity acceleration on the solar surface, the density of gas in the photosphere, and the ratio of the radius of the spot (including penumbra), a _p, to the radius of its umbra a. It is shown that large-scale subsurface flows of gas in the sunspot vicinity, being the consequence but not the cause of sunspot formation, are too weak to contribute significantly to the pressure balance of the sunspot. Stability of the sunspot is provided by cooling of the sunspot plasma and decreasing of its gravitational energy due to the vertical redistribution of the gas density when the geometric Wilson depression of the sunspot is formed. The depth of a depression grows linearly with B ₀, in contrast to the quadratic law for the magnetic energy. Therefore, the range of stable equilibria turns out to be limited: large spots, with radius a larger than some limit value (about 12–18 Mm, depending on the magnetic field configuration), are unstable. It explains the absence of very large spots on the Sun and the appearance of light bridges in big spots that divide the spot into a few parts. The sunspots with B ₀≈2.6÷2.7 kilogauss (kG) and a≈5 Mm are most stable. For these spots, taken as a single magnetic structure, the period of their vertical eigen oscillations is minimal and amounts, according to the model, to 10–12 hours. It corresponds well to the period derived from the study of long-term oscillations of sunspots using SOHO/MDI data.