Stably stratified stars containing magnetic fields whose toroidal components are much larger than the poloidal ones in general relativity: A perturbation analysis

We construct the stably stratified magnetized stars within the framework of general relativity. The effects of magnetic fields on the structure of the star and spacetime are treated as perturbations of nonmagnetized stars. By assuming ideal magnetohydrodynamics and employing one-parameter equations of state, we derive basic equations for describing stationary and axisymmetric stably stratified stars containing magnetic fields whose toroidal components are much larger than the poloidal ones. A number of the polytropic models are numerically calculated to investigate basic properties of the effects of magnetic fields on the stellar structure. According to the stability result obtained by Braithwaite, which remains a matter of conjecture for general magnetized stars, certain of the magnetized stars constructed in this study are possibly stable.

Figure 1

Gravitational mass $M$ and baryon mass $M^{*}$, given as functions of the central density ${}^{(0)}\rho (0)$. All the quantities are given in units of $\kappa =1$.

Figure 2

Equi-$F_{r\theta }$ contours (left) and equi-$\mathrm{\Psi }$ contours (right) on the meridional cross section for the model with $\mathrm{\Gamma }=2.05$ and $M/R=0.2$. Here, $z$ and $\varpi$ are defined by $z=r\cos \theta$ and $\varpi =r\sin \theta$, respectively. The thick quarter circle shows the surface of the star, on which $F_{r\theta }=0$ and $\mathrm{\Psi }=0$ are required by the boundary condition.

Figure 3

Normalized nondimensional changes in the gravitational mass $\mathrm{\Delta }M/M{{}^{(t)}\mathcal{R}}_M$ and the baryon rest-mass $\mathrm{\Delta }{M}^{*}/{M^{*}}^{(t)}{\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 4

Normalized nondimensional changes in the internal thermal energy $\mathrm{\Delta }{E}_{\mathrm{int}}/{E_{\mathrm{int}}}^{(t)}{\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 5

Normalized nondimensional changes in the mean radius $(\mathrm{\Delta }r{)}_0/R{{}^{(t)}\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 6

Normalized nondimensional changes in the mass quadrupole moment $\mathrm{\Delta }Q/M{R^2}^{(t)}{\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 7

Normalized changes in the ellipticity $e^{*}/{{}^{(t)}\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 8

Normalized nondimensional changes in the toroidal magnetic energy $E_{\mathrm{EM}}^{(t)}/|W|{{}^{(t)}\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 9

Normalized nondimensional changes in the poloidal magnetic energy $E_{\mathrm{EM}}^{(p)}/|W|{{}^{(p)}\mathcal{R}}_M$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).

Figure 10

Normalized changes in the nondimensional magnetic helicity ${\mathcal{H}}_M/\sqrt{{}^{(t)}{\mathcal{R}}_M{}^{(p)}{\mathcal{R}}_M}$, given as functions of the central density of the background star ${}^{(0)}\rho (0)$. The solid circle indicates the result obtained within the framework of Newton’s dynamics and theory of gravity (cf. Appendix).