Initial Stellar Model
In evolving each case, we adopt a parametric hybrid equation of state, consisting of a cold part and a thermal part:
P(ρ,ε) = P_{P}(ρ) + P_{th}(ρ,ε)
For the cold part of the pressure, P_{P}, we adopt the following form:
P_{P}(ρ) = K_{1}ρ^{Γ1}, ρ ≤ ρ_{nuc}
K_{2}ρ^{Γ2}, ρ ≥ ρ_{nuc}
where K_{1}, K_{2}, Γ_{1}, and Γ_{2} are constants, and ρ_{nuc} ≈ 2x10^{14} g/cm^{3} is nuclear density. We set K_{2} = K_{1}ρ_{nuc}^{Γ1-Γ2} for continuity at &rho = &rho_{nuc}. The parameters Γ_{1} = 1.3, Γ_{2} = 2.5, and K_{1} = 5x10^{14} (in cgs units) are chosen so that this simplified cold EOS will mimic the behavior of a more sophisticated cold nuclear EOS. The thermal part of the pressure, P_{th}, plays an important role when shocks occur. For it, we adopt the simple form:
P_{th} = (Γ_{th}-1)ρε_{th}
where ε_{th} is the thermal specific internal energy. The value of Γ_{th} determines the efficiency of converting kinetic energy to thermal energy at shocks. To conservatively account for shock heating, we set Γ_{th} = Γ_{1}.
The pre-collapse stellar cores are modeled as rotating polytropes with central density chosen to be ρ_{c} = 10^{10} g/cm^{3}, and equation of state given by:
P = K_{0}ρ^{Γ}
Here, K_{0} is set to be 5x10^{14} in cgs units, and Γ is set to be 4/3, so that this EOS corresponds to the degenerate pressure of ultra-relativistic electrons. The rotation law is given by:
u^{t}u_{φ} = ϖ_{d}^{2} (Ω_{c} - Ω)
where Ω_{c} is the angular velocity along the rotation axis, and ϖ_{d} is a constant. Here we set ϖ_{d}/R = 0.5, where R is the initial equatorial radius. In the Newtonian limit, this reduces to the so-called 'j-constant' law:
Ω = Ω_{c}ϖ_{d}^{2}/(ϖ^{2}+ϖ_{d}^{2})
In star A2, we then introduce a weak poloidal magnetic field to the equilibrium model with a vector potential:
A_{ϕ} = A_{b}ϖ^{2} max[(ρ - ρ_{cut}), 0]
where the cutoff density is set to 10^{-4} times the central density, and the constant A_{b} is chosen so that the z-component of the magnetic field is 7x10^{12} - 4x10^{13} G.
last updated 06 apr 07 by svw
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