Stationary, axisymmetric neutron stars with meridional circulation in general relativity

We present the first stationary, axisymmetric neutron star models with meridional circulation in general relativity. For that purpose, we developed generally rotating neutron star, a new code based on a fixed-point iteration. We find a two-dimensional set of meridional circulation modes, which differ by the number of vortices in the stream lines of the neutron star fluid. For expected maximal meridional circulation velocities of about $1000\mathrm{km}/\mathrm{s}$, the vortices cause surface deformations of about a percent. The deformations depend on the shape of the vortices close to the surface and increase with the meridional circulation velocity. We also computed models of rotating neutron stars with meridional circulation, where neither the surface rotates nor does the rotation velocity exceed the circulation velocity.

Figure 1

Initial guess of basic field ${\chi }_0$. Each one of the four panels shows the distribution of the field ${\chi }_0$ inside of the neutron star for one choice of the pair $({\mathcal{M}}_r,{\mathcal{M}}_{\theta })$ according to Eq. (38). The top, left panel visualizes the case $({\mathcal{M}}_r,{\mathcal{M}}_{\theta })=(1,1)$. Proceeding to the right increases the value of the parameter ${\mathcal{M}}_{\theta }$, and we have to go down to raise the value of ${\mathcal{M}}_r$. For each panel, the maximal absolute field value is called max. The values max and –max are represented by the brightest red and green colors, respectively.

Figure 2

Basic fields of reference model. The four panels show the 3-lapse logarithm $\nu$, the 2-conformal factor logarithm $\alpha$, the 3-shift $N^a$ and the pressure for our reference model. The color coding of the three scalar plots is the same one as in Fig. 1, where the maximum absolute field value max is displayed at the top of the panels and the spacing of the contours is linear. For the vector plot, the quantity max denotes the maximum vector length (note that all vectors lie within the paper plane).

Figure 3

Circulation modes—Part 1. The panels show the stream lines of the first four meridional circulation modes of the reference model. The color coding and contour spacing is the same as in Fig. 1. Note that red and green refer to clockwise and counterclockwise motion, respectively. The kinks visible at the neutron star surface are a result of the finite numerical resolution and the use of contour plots.

Figure 4

Circulation modes—Part 2. The panel shows additional higher meridional circulation modes of the reference model (continuation of Fig. 3).

Figure 5

Surface deformations. The figure shows the radial coordinate $R(\theta )$ of the neutron star surface depending on the angle $\theta$. From thick to thin, the three lines refer to the upper left and upper right mode of Fig. 3, and the upper left mode of Fig. 4.

Figure 6

Constant $f(\psi )$. Stream function $\psi$ for the case $n=0$. The color coding and contour spacing is the same one as in Fig. 1.

Figure 7

Rotating neutron star. Specific angular momentum $l_{\varphi }$ and the 2-shift $M^m$ for the rotating model (61). The color coding and contour spacing of the contour plot is the same one as in Fig. 1. For the vector plot, the quantity max denotes the maximum vector length.

Figure 8

Specific angular momentum. The two panels show radial profiles of the specific angular momentum $j$ for two different modes of the rotating model (61) at angles $\theta =0$ (thinnest curve), $\theta =\frac{\pi }{4}$, and $\theta =\frac{\pi }{2}$ (thickest curve), respectively. The upper panel belongs to the fundamental mode, displayed in Fig. 7, and the lower one to one of the higher modes. The specific angular momentum $l_{\varphi }$ and the stream function $\psi$ of this higher mode exhibit a vortex distribution as in the upper, left panel of Fig. 4.

Figure 9

Convergence tests. Dependence of the convergence indicator $C_s$ defined in Eq. (g1) on the iteration step $s$ for a model similar to the reference model. The three solid lines refer to grid resolutions of $59\times 52$, $150\times 156$, and $501\times 507$ zones (the higher the resolution, the thinner the corresponding line). The blue dashed lines refer to the first higher mode.