8 quences it has been shown that there is a stability change at the minimum in the angular momentum [52]. Looking at the inset of Fig. 2, one therefore expects the solutions on the right to be unstable while those on the left should remain stable. Note that the addition of a non-vanishing magnetic field adds another dimension to this diagram although the effect of the magnetic field on the initial data is not so dramatic. Certainly there are many ways in which to “add” a magnetic field to a stellar solution. In [19] a non- vanishing current is assumed and a solution is obtained with the same baryon mass but now with nonvanishing magnetic field. In particular, solutions at the mass shed- ding limit with no magnetic field are shown in Fig. 2 by open triangles. Almost indistinguishable from these solu- tions are the those magnetized such that the radial mag- netic field at the pole of the star is 1000 gigatesla ( GT ) or 10 $^{16}$G . shown with open circles. We consider different perturbations. To perturb the pressure, we decrease it according to real parameters A$_{p}$ and m in terms of the unperturbed pressure p$_{0}$ depending on the azimuthal angle ϕ We also consider perturbations to the rest mass density decreasing it everywhere by a fraction A$_{ρ}$ We verify the initial data in our evolution code in a number of ways. We examine convergence of the data to a unique solution which solves the constraints. In Fig. 3 we show the residual of the Hamiltonian and y -component of the momentum constraint for three different resolutions. The Hamiltonian constraint is a good measure of the fi- delity of the solution to the Einstein equations, and, as is clear from the figure, its residual decreases rapidly with resolution. Furthermore, we confirm that the divergence of the magnetic field is around machine precision. Before addressing the effects of the magnetic field, we verify that the code reproduces the expected behavior as described, for example, in [53]. First, we consider evo- lutions of stable, rotating stars and find that the code evolves such a star as long as desired while maintain- ing a stationary solution. This a demanding test as it depends on the balance between gravitational and hy- drostatic forces in a rotating configuration with both a non-conforming grid and variables not adapted to the symmetries of the problem. One example of the the behavior of the numerical so- lution is shown in Fig. 4. As apparent in the top frame of the figure, the fractional change in the maximum of the density oscillates with a slow overall increase. This oscil- lation is characteristic of quasinormal ringing of the star These nonmagnetized solutions are diagrammed in a plot of total gravitational mass versus central enthalpy in Fig. 2. The lower curve shows the static limit for stars which are not rotating. The upper curve is the mass shedding limit, represented by the largest frequency for which Magstar returned a solution. These curves serve as the upper and lower bounds on the solution space of stationary, unmagnetized stars. It should be noted that Magstar generates only rigidly rotating stars, and there- fore the evolutions are far from the fast rotating regime expected to excite large and growing nonaxisymmetric modes. We also compute and show a sequence of stars at con- stant baryon mass. Such sequences are important be- cause real stars are expected to conserve baryon mass as they evolve and thus the sequences are expected to ap- proximate their evolution. Furthermore, along such se- FIG. 3: The residuals of two constraint equations, namely the Hamiltonian constraint and the y -component of the mo- mentum constraint. A slice along the x -axis of the com- puted residuals for various unigrid resolutions. Top: That the Hamiltonian constraint residual converges to zero with in- creasing resolution is taken as evidence that the initial data is properly constructed and read into the code. Bottom: That the momentum constraint residual improves with the first in- crease in resolution is similar evidence. However, the next increase in resolution fails to bring down this residual. We suspect that this remaining error is associated not with trun- cation error but instead with inherent errors in the numeri- cal transformation and interpolation from Lorene ’s spherical basis to our Cartesian one. Note the spikes that arise at the stellar boundaries due to discontinuities in the fluid variables, these are not expected to converge to zero. p = p$_{0}$ [ 1 − A$_{p}$ sin 2 ( mϕ ) ] . (53) ρ = ρ$_{0}$ [1 − A$_{ρ}$ ] . (54) VI. RESULTS Stable, Rotating Star Test: