Initial Stellar Model
Stars A, B1, and B2 obey a polytropic equation of state: P=Kρ0Γ, where P is the pressure, K the polytropic constant, and ρ0 the rest-mass density. We choose Γ = 2 to mimick stiff nuclear matter. The physical mass of the stars may be scaled to any desired value by adjusting K (M ∝ K1/2). In addition, the rotation law is given by
utuφ = A2 (Ωc - Ω),
where Ω ≣ uφ/ut is the angular velocity of the fluid and Ωc is Ω on the rotation axis. The constant A has units of length and determines the steepness of the differential rotation. A is set equal to the coordinate equatorial radius Req. In the Newtonian limit, it reduces to
Ω = Ωc / (1 + ϖ2/A2)
(ϖ is the cylindrical radius.) We add a weak poloidal magnetic field to the equilibrium model by introducing a vector potential
Aϕ = ϖ2 max[Ab(P - Pcut), 0]
where the cutoff Pcut is 4% of the maximum pressure, and Ab is a constant which determines the initial strength of the magnetic field. We characterize the strength of the initial magnetic field by C ≣ max(b2/P), where b2 = B2/8π. We choose Ab such that C ∼ 10-3-10-2. We have verified that such small initial magnetic fields introduce negligible violations of the Hamiltonian and momentum constraints in the initial data.