Evolutions of magnetized and rotating neutron stars

We study the evolution of magnetized and rigidly rotating neutron stars within a fully general relativistic implementation of ideal magnetohydrodynamics with no assumed symmetries in three spatial dimensions. The stars are modeled as rotating, magnetized polytropic stars, and we examine diverse scenarios to study their dynamics and stability properties. In particular, we concentrate on the stability of the stars and possible critical behavior. In addition to their intrinsic physical significance, we use these evolutions as further tests of our implementation, which incorporates new developments to handle magnetized systems.

Figure 1

Demonstration of the effectiveness of divergence cleaning at handling deviations from a divergenceless magnetic field. A calculation of the divergence of the magnetic field, $x^2[\overrightarrow{\nabla }·\overrightarrow{B}]$, along the $x$ axis for 3 times is shown for each of three cases: (top) No divergence cleaning; (middle) Cleaning with $c_h=0.1$ and $c_r=0.01$; and (bottom) Cleaning with $c_h=1.0$ and $c_r=0.1$. These runs were otherwise identical for a magnetized, rotating star of coordinate equatorial radius of 10 with spin along the $z$ axis with three levels of refinement (at $\pm 50$, $\pm 25$, and $\pm 12.5$) and a perturbation to the magnetic field to introduce an explicit deviation from divergenceless. Concentrating on the MHD equations, the metric terms were frozen at their initial values, e.g. the Cowling approximation. The bottom row shows clear wavelike behavior as it “cleans” the divergence.

Figure 2

Diagram of solution space of stars generated with $\mathtt{M}\mathtt{a}\mathtt{g}\mathtt{s}\mathtt{t}\mathtt{a}\mathtt{r}$ (using its units). The total gravitational mass is plotted versus the central enthalpy. The upper curve (purple, 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 $1000\mathrm{GT}={10}^{16}\mathrm{G}$. 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 ($M_B=3.6M_{\mathrm{sol}}$) 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 a nonvanishing magnetic field.

Figure 3

The residuals of two constraint equations, namely, the Hamiltonian constraint and the $y$ component of the momentum constraint. A slice along the $x$ axis of the computed residuals for various unigrid resolutions. Top: That the Hamiltonian constraint residual converges to zero with increasing resolution is taken as evidence that the initial data are properly constructed and read into the code. Bottom: That the momentum constraint residual improves with the first increase 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 truncation error but instead with inherent errors in the numerical 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.

Figure 4

Convergence results for the evolution of a stable, nonmagnetized, rotating star (the particular initial solution is shown in Fig. 2 as a green solid square). Three fixed mesh refinement evolutions are shown, each with a multiple of the coarsest resolution resolving the equator of the star with the following: (magenta, dot-dashed line) 30 points/star, (blue, dotted line) 60 points/star, and (black, solid line) 120 points/star. Also shown is a run with the same stellar resolution of (red, long-dashed line) 60 points/star but with a coarse grid that extends twice as far in all directions. The two highest resolution runs were terminated early because of the computational cost, not because of any robustness problems. The insets show the same data but in finer detail for the first rotational period. The top frame shows the fractional change in maximum density versus rotational period. With increasing resolution the solution better approximates a stationary solution. The upper middle frame shows the fractional change in the baryon mass, Although our scheme, using vertex centered AMR, an atmosphere, and full general relativity with sources, is not strictly conservative, the plot shows that deviations from strict conservation are small and decrease with more resolution. The lower middle frame shows the fractional change in integrated angular momentum. Again, the code demonstrates convergence to conservation. The bottom frame shows the maximum of the norm of the $Z^a$ constraint residuals as a function of time that also converge.

Figure 5

Collapse to black hole of an unstable, unperturbed, magnetized star for multiple of a base resolution. The initial datum is an unperturbed $\mathtt{M}\mathtt{a}\mathtt{g}\mathtt{s}\mathtt{t}\mathtt{a}\mathtt{r}$ solution with central enthalpy $H_c=0.8$ rotating at a frequency $f=835\mathrm{Hz}$ and with polar magnetic field of $1000\mathrm{GT}={10}^{16}\mathrm{G}$. The top frame shows the maximum density, which increases with time as the star collapses. The upper middle frame shows a norm of the divergence of the magnetic field. The lower frames show the distortion parameters of the $D$ field as functions of time.

Figure 6

Density ${\rho }_0$ and magnetic field strength on the $y=0$ plane for the same star as shown in Fig. 5. The density is shown with respect to the color map while the contours denote the magnetic field, which is equally spaced from 0 to $5\times {10}^{15}\mathrm{G}$. The top left corresponds to $t=0.5P$ while the top right to $t=0.9P$ (the circle shown represents the apparent horizon.) The bottom plot illustrates the normalized mass as a function of time, along with its rate of change.

Figure 7

The maximum density ${\rho }_{\max }$ as a function of time for an unstable, perturbed [$m=3$ in Eq. (53)], nonmagnetic, rotating star. The (unperturbed) initial star has central enthalpy 0.8 and rotates at the mass shedding limit. Solutions resulting from tuning the amplitude of the perturbation $A_p$ to roughly one part in ${10}^6$ show the two disparate outcomes (as discussed in 53): collapse to black hole or violent oscillations about a stable stellar solution with equivalent mass. With more tuning, the star resembles the initial, unstable solution for a longer time.

Figure 8

Results of three critical searches for a magnetized, rotating star with central enthalpy of 0.8 and a radial magnetic field at the pole of 1000 GT. One search varied the amplitude of the pressure perturbation $A_p$ with $m=7$ (black and cyan lines), another with $m=3$ (blue and magenta lines), and another varied the density perturbation $A_{\rho }$. In solid lines are shown the supercritical evolutions that collapse to black holes, while subcritical solutions for the two searches are shown with dotted lines. The search over pressure with $m=7$ achieved tuning to about one part in ${10}^3$, that with $m=3$, one part in ${10}^4$, and that over density achieved about one part in ${10}^5$.