We have investigated the three-dimensional distribution of the polarization-brightness product (pB) and then quantitatively determined the electron density distribution relative to the inferred heliographic current sheet during the declining phase of solar cycle 20 (1973-1976). The current sheet is taken as the center of the bright, dense structures from combined synoptic pB data from ground-based K-coronameter and the white-light coronagraph aboard Skylab. Analyses of pB scans as a function of minimum distance from the current sheet (theta(min)) over the radial distance range 1.13 to 5.0 R. (from Sun center) led to the following new results: (1) a quantitative description of pB obtained around the inferred neutral line is given by the following equation:pB(rho, theta(min)) = pB(p)(rho) + [pB(cs)(rho) - pB(p)(rho)]e(-theta min2/w2(r)),where rho is the shortest distance to the line of sight from the Sun center, pB(cs)(rho) and pB(p)(rho) are the observed polarized brightness at the current sheet and the poles, respectively, and w(r) is the half-width of the distribution; (2) the electron density obtained by inverting the pB data is given byN(r, theta(mg)) = N-p(r) + [N-cs(r) - N-p(r)]e(-theta mg2/w2(r)),where N(r, theta(mg)) is the number of free electrons per cm(3), N-cs(r) and N-p(r) are the electron densities at the current sheet and the poles, respectively, and theta(mg) is the magnetic latitude. Here theta(mg) is given bytheta mg = sin(-1) [-cos theta sin alpha sin (phi -phi(o)) + sin theta cis alpha],where theta and phi are heliographic latitude and longitude, alpha is the tilt angle of the dipole axis with the rotation axis, and phi(o) is the intersecion of the heliomagnetic and heliographic equators; (3) during the period studied (the last third of the solar cycle), the mean pB at the current sheet and above the polar holes is approximately independent of the phase of the solar cycle; and (4) the organization of pB data about the neutral line allows inference of the boundary of the polar coronal holes.The usefulness of one-dimensional white-light density constraint in solar wind modeling has already been demonstrated by Habbal et al. The present three-dimensional model should prove very useful in better understanding of the global hydromagnetic structure of the corona and the solar wind, relating as it does to the magnetic structure of the corona, as opposed to heliocentric coordinates. For example, the density model could provide constraints on coronal temperature, flow velocity, and magnetic structure subject to a suitable analysis of geometric effects, which in turn would provide constraints on energy balance in the coronal expansion.
The discrete spectrum of the Schrodinger operator is studied for a system of three identical particles with short-range interactions in a homogeneous magnetic field. All the two-particle subsystems are supposed to be unstable. Finiteness of the discrete spectrum is established under some assumptions about the solutions of the corresponding two-particle Schrodinger equation.
