7 where This mass is related to the expected baryon number and should be strictly conserved as long as mass is not leav- ing the computational domain. Many HRSC schemes explicitly conserve this quantity, but here it is not a pri- ori conserved. Our use of an atmosphere entails adding a small amount of mass in regions that would otherwise become evacuated. Another reason is that we are using a finite difference based AMR which does not accom- modate as readily a strictly conservative treatment as a finite volume method would [51]. Finally, the presence of source terms in the evolution equations for the other conservative variables also breaks perfect conservation of these other quantities. We also compute the angular momentum of the fluid. Since our stars rotate about the z -axis, we need only compute a single such quantity The use of Cartesian coordinates is typically avoided when one expects angular momentum to be conserved since cylindrical coordinates are better adapted to the respective Killing vector. However, because this project is part of a more general effort to model a variety of systems with different natural coordinates and no sym- metries, we simply monitor the extent to which this is conserved by computing a fractional change as Finally, we also monitor the extent to which our nu- merical solution satisfies the Einstein equations as ex- pressed in Eq. (1). That is, we compute the so-called residual of these equations, expressed as the four-vector Z a as in Eq. (9). In particular, we monitor the norm of this vector Analytical solutions to Einstein equations must have a vanishing residual and thus we monitor this residual for solutions obtained at various resolutions to check for con- vergence to a consistent solution. We use initial data generated with the program Magstar , part of the Lorene software package. These solutions are described in [19] and are rigidly rotating, magnetized neutron stars. They are generated as fully consistent solutions of Einstein’s equations as opposed to taking nonmagnetized stars and adding a seed mag- netic field without re-solving the constraints. The initial where C is constructed by physical constants and h$_{e}$ is evaluated at the center of the star. To get a handle on the solutions generated, we first turn off the magnetic field and compute the “usual” two- parameter solution space as described in [52]. Using the terminology of [19], we compute solutions based on the two parameters of central enthalpy and frequency. Ex- amination of Eq. (13) of [19] shows that the log-central enthalpy H$_{c}$ is related to the entropy used here ( h$_{e}$ as defined in Eq. (14)) by magnetic field is dipolar and aligned with the rotation axis, produced by a current function f ( A$_{φ}$ ) = constant where A$_{φ}$ is the toroidal component of the electromag- netic potential vector. FIG. 2: Diagram of solution space of stars generated with Magstar (using its units). The total gravitational mass is plotted versus the central enthalpy. The upper curve (pur- ple, open triangles) represents stars with no magnetic field at the mass shredding limit. Very close to this are shown (green, open circles) stars rotating at the same frequencies but with a radial magnetic field at the pole of 1000 GT = 10 $^{16}$G . The masses of these stars are barely changed with respect to their nonmagnetic counterparts. The lower curve (blue, open squares) represents the static limit with no rotation. A sequence of constant baryon mass stars ( M$_{B}$ = 3 . 6 M sol ) is shown (red stars). This sequence is also shown in the inset along with their angular momentum (yellow open circles) and frequency of rotation (green stars). The location of the initial data used in the initial data convergence test of Fig. 3 and the long term, stability test of Fig. 4 is also shown (green, solid square) although that solution strictly does not belong here because it has non-vanishing magnetic field. M$_{baryon}$ ≡ ∫ D dV. (48) J$_{z}$ = ∫ ( xn $^{j}$T$_{jy}$ − yn $^{j}$T$_{jx}$ ) √$_{h d}$ $^{3}$x. (49) Δ J$_{z}$ ≡ J$_{z}$ ( t ) − J$_{z}$ (0) J$_{z}$ (0) . (50) | ⃗ Z | ≡ || Z $^{a}$| | $_{2}$. (51) V. INITIAL DATA H$_{c}$ = ln ( h$_{e}$ ) + C (52)