A hydrodynamic model for asymmetric weak explosion of collapsing supernovae with rapid initial rotation

A two-dimensional hydrostatically equilibrium atmosphere of a neutron star with given differential rotation

“…As can be verified, only the factor sin 2 θ 0 appears in the first exponential term in (17) at other angles θ 0 ≠ π/2. However, choosing the boundary condition not at the equator for constant r ex (particularly at small θ 0 ) generally causes the total atmospheric mass to increase considerably, which seems unnatural.…”

“…Let us write some integrated quantities that are of interest in physically interpreting the solution (17) with allowance for a mirror symmetry. The total atmospheric mass M 0 for the given density distribution ρ(r, θ) ( 17) is…”

“…All integrals (18)-( 22) are written in this case of symmetry about the z = 0 equatorial plane; they imply the presence of a matter-free spatial region for which the expression in curly braces in (17) becomes a negative quantity without any physical meaning. Note that the (11) and (12) formally admit of an imaginary solution ρ, though the function φ [see relation (9)] in Eqs.…”

“…This is how the inner surface of the toroidal volume of an iron atmosphere is obtained, whereas its outer surface coincides with the external sphere of radius r = r ex , and the angle clearly increases from θ = θ ex to θ = π/2. In this case, ρ = ρ 0 at the equator by our definition of the constant F 0 described above when passing from ( 16) to (17); ρ = 0 at θ = θ ex , and, using (23), we explicitly derive at = r ex (24)…”

“…We now turn to the results of our numerical calculations of density distributions ρ(r, θ) [relation (17)] and of the corresponding integrated quantities (18)-( 22). In order for the validity conditions of the Roche approximation and the conditions of gravitational collapse with a residual iron gas mass M e (see above) of interest to us to be satisfied, we restrict the total mass of an equilibrium atmosphere M 0 (18) to the range (0.1-0.2)M ᭪ .…”