# Initial Configuration

We consider two cases of equilibrium neutron stars, one with slow and the other with rapid rotation. The neutron star obeys a polytropic equation of state, P = Kρ0 1+1/n, where P is the pressure, K is the polytropic gas constant, n is the polytropic index and ρ0 is rest-mass density. We chose n = 0.5, corresponding to a moderately "stiff" nuclear equation of state.

For these simulations, the matter and gravitational field equations are not evolved. Rather, for sufficiently weak magnetic fields, the matter and velocity profiles and spacetime metric can be determined to high accuracy by solving the Einstein equations for a stationary gravitational field in axisymmetry, coupled to the equation of hydrostatic equilibrium. In this work we assume that the magnetic fields are weak with $\mathcal{M}/|\mathcal{W}|<<1$, where $\mathcal{M}$ is the magnetic energy and $\mathcal{W}$ is the gravitational potential energy. The background fluid and metric fields are kept fixed and correspond to stationary, axisymmetric, uniformly rotating neutron stars. We thus only need to evolve the electromagnetic fields in these stationary background matter and gravitational fields. In the stellar interior we solve the full GRMHD equations for the magnetic field, while in the exterior we treat the field in the force-free limit of GRMHD. The exterior field is a dipole.

### Slowly Rotating Neutron Star

For the slowly rotating neutron star case, the simulations shown here have been executed in flat spacetime. The light cylinder is located at RLC = c/Ω = 10 R where R is the radius of this star. This star is assumend to be spherical as $\mathcal{T}/|\mathcal{W}|<<1}$, where $\mathcal{T}$ is the rotational kinetic energy.

The luminosity of the this spherical star given by $L_0=1.02\mu^2\Omega^4\simeq10^{43}B^2_{12}R^6_{10}P^{-4}_{ms}$ erg/s

where $B_{12}=B/10^{12} G$, $R_{10}=R/10 \mathrm{km}$ and $P_{ms}=P/1 \mathrm{ms}$.

### Rapidly Rotating Neutron Star

For the rapidly rotating star, the simulations were performed in general relativity. The light cylinder is located at approximately RLC = 2.9 R. The rotation of the star has an angular frequency of MΩ = 0.057. The shape of the star is characterized by its eccentricity of e = (1 - Rp/Re)   1/2 = 0.799, where Rp is the polar radius and Re is the equatorial radius.

The magnetic field inside the neutron star is determined by solving the ideal GRMHD equations for the electromagnetic field. Magnetic fields in the exterior of the neutron star are evolved according to the force-free approximation for GRMHD fields.