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) = PP(ρ) + Pth(ρ,ε)

For the cold part of the pressure, PP, we adopt the following form:

PP(ρ) = K1ρΓ1,   ρ ≤ ρnuc
K2ρΓ2,   ρ ≥ ρnuc

where K1, K2, Γ1, and Γ2 are constants, and ρnuc ≈ 2x1014 g/cm3 is nuclear density. We set K2 = K1ρnucΓ12 for continuity at &rho = &rhonuc. The parameters Γ1 = 1.3, Γ2 = 2.5, and K1 = 5x1014 (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, Pth, plays an important role when shocks occur. For it, we adopt the simple form:

Pth = (Γth-1)ρεth

where εth is the thermal specific internal energy. The value of Γth determines the efficiency of converting kinetic energy to thermal energy at shocks. To conservatively account for shock heating, we set Γth = Γ1.

The pre-collapse stellar cores are modeled as rotating polytropes with central density chosen to be ρc = 1010 g/cm3, and equation of state given by:

P = K0ρΓ

Here, K0 is set to be 5x1014 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:

utuφ = ϖd2c - Ω)

where Ωc is the angular velocity along the rotation axis, and ϖd is a constant. Here we set ϖd/R = 0.5, where R is the initial equatorial radius. In the Newtonian limit, this reduces to the so-called 'j-constant' law:

Ω = Ωcϖd2/(ϖ2d2)

In star A2, we then introduce a weak poloidal magnetic field to the equilibrium model with a vector potential:

Aϕ = Abϖ2 max[(ρ - ρcut), 0]

where the cutoff density is set to 10-4 times the central density, and the constant Ab is chosen so that the z-component of the magnetic field is 7x1012 - 4x1013 G.

last updated 06 apr 07 by svw

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