POLARIZATION OF ASTRONOMICAL MASER RADIATION .2. POLARIZATION MODES AND UNSATURATED GROWTH

Maser polarization is analyzed in the limit of overlapping Zeeman comp onents (gv(B) much less than DELTAv(D), where gv(B) is the Zeeman spli tting and DELTAnu(D) is the Doppler width). All the discrepancies amon g the conflicting conclusions of previous studies that identified mase r polarizations in this limit are fully resolved. In the case of m-ind ependent pumping, proper application of the eigenvalue technique of Go ldreich, Keeley, & Kwan shows that the polarization eigenvectors are t he same for saturated and unsaturated masers and are independent of sp in for pure spin states, in agreement with the results of the first pa per in this series. Stable eigenvectors correspond to the peak of the polarization mode distribution of self-amplified radiation at any degr ee of saturation. But the distribution average, the actually measured polarization, does not necessarily coincide with its peak. The mode di stribution starts with a rectangular shape, because the seed radiation generated in spontaneous decays is unpolarized. and evolves toward a sharply peaked profile whose average, and not just its peak, coincides with the eigenvector solution because of the following two effects. F irst, interaction with the maser molecules induces rotation of the pol arization vectors of individual modes, similar to Faraday rotation. Th e rotation rate is different for different modes, and the polarization eigenvectors correspond to stationary modes that do not rotate. Start ing from unpolarized radiation generated by the source terms and conta ining an equal mix of all modes, all individual polarization vectors r otate into the stationary stable modes, resulting in a radiation field polarized according to the solution of the eigenvalue problem. As a r esult of this rotation the ensemble-averaged Stokes parameters reach t he eigenvector solution when J greater than or similar to J(s), where J(s) is the angle-averaged intensity and J. is the saturation intensit y, i.e., only after the maser saturates. This explains the results of numerical studies of the maser polarization problem presented in the l iterature. Second, and more important, maser growth is highly unstable during the unsaturated phase for any polarization configuration excep t for that of the eigenvector solution. The Stokes parameters of all o ther polarization structures include terms proportional to exp \aI\, w here I is the intensity and a not-equal 0, and thus are highly unstabl e against arbitrarily small intensity perturbations. Such perturbation s induce runaway divergence of the ensemble-averaged Stokes parameters away from their initial values, a divergence that stops only when the polarization settles into the appropriate eigenvector solution. The e -folding growth rate of the instabilities increases with J and reaches unity when J approximately J(s)/tau(s), where tau(s) is the optical d epth of the maser when it saturates; pumping schemes of astronomical m asers typically produce tau(s) approximately 12-17. Instabilities impo se an upper bound on the intensity of radiation whose polarization dif fers from that of the eigenvector solution and are the dominant factor in narrowing the polarization mode distribution around its peak. Only radiation whose ensemble-averaged polarization corresponds to the eig envector solution can grow to saturation and beyond. Furthermore, all polarization configurations are unstable for propagation at 0 < theta < theta0, where theta is measured from the magnetic axis and sin2 thet a0 = 1/3. One eigenvector solution, corresponding to fully polarized r adiation, is stable in this region during the unsaturated growth phase against perturbations that rotate the polarization at a fixed intensi ty, but not against intensity perturbations. As a result, stable build up of maser radiation in a magnetic field with gv(B) much less than DE LTAv(D) is possible only for theta greater-than-or-equal-to 0,; propag ation directions too close to the field axis, corresponding to a fract ional volume of approximately 0.09, are excluded. Propagation along th e axis, theta = 0, is allowed, but the corresponding radiation is unpo larized.