Magnetic Equilibria and Deformation

A magnetic neutron star is an equilibrium of fluid, spacetime, and Maxwell stress. Its field cannot be specified independently of the currents that support it, and its surface cannot be matched to an exterior without choosing whether that exterior is vacuum or a current-carrying magnetosphere. In axisymmetry the magnetic flux function organizes the problem, while the ratio of magnetic to binding energy estimates the deformation. Field topology and the current distribution can change even the sign of that deformation, so a quoted field strength alone does not determine the quadrupole.

Required background. Killing Fields and Isometry Groups supplies stationarity and axisymmetry and the associated conserved quantities.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies fluid and electromagnetic stresses; The Einstein Equation supplies the geometry–matter coupling; Relativistic Matter and Compact Objects supplies ideal-MHD closure; Formulations of General Relativity supplies constraint and gauge language; Computational Relativity supplies elliptic and divergence checks.

The Einstein–Maxwell equilibrium problem

The total stress tensor is

$$T^{\mu\nu}=T_{\rm fluid}^{\mu\nu} +F^{\mu\alpha}F^\nu{}_{\alpha} -\frac14 g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta},$$

in rationalized electromagnetic units. Ideal MHD imposes $F_{\mu\nu}u^\nu=0$, so the comoving electric field vanishes. Maxwell’s equations determine the field from a conserved current, while the Einstein equation responds to both material and electromagnetic energy and stress. A fully self-consistent strong-field model must solve all three parts; treating the field as a force on a fixed spherical background is only a perturbative approximation.

The comoving spatial Maxwell stress makes the mechanism concrete:

$$T_{\rm EM}^{\hat i\hat j} =\frac{B^2}{2}\delta^{ij}-B^{\hat i}B^{\hat j}.$$

Perpendicular to a field line it acts as pressure $B^2/2$; along the line it acts as tension $-B^2/2$. Deformation depends on where these directions lie inside the star, not just on the scalar energy density.

For a configuration with a nonzero poloidal component in stationary axisymmetry, the azimuthal vector-potential component can be used as a magnetic flux function, $\Psi=A_\phi$. Its level surfaces are poloidal magnetic surfaces. Integrability of the Euler and Maxwell equations restricts the toroidal current and magnetic force to prescribed functions of $\Psi$. The resulting relativistic Grad–Shafranov equation is coupled to the enthalpy and metric potentials. A purely toroidal field has no nontrivial poloidal flux surfaces and requires its own toroidal-field integrability prescription; it is not specified by freely choosing functions of a constant $A_\phi$. Bocquet and collaborators give a full Einstein–Maxwell construction for rotating poloidal-field stars in Bocquet et al. 1995, §§2–3, Eqs. (5)–(23).

Poloidal, toroidal, and mixed fields

A purely poloidal field threads meridional planes and generally extends outside the star. A purely toroidal field circles the symmetry axis and can be confined to the fluid. A mixed “twisted-torus” configuration has a global poloidal component and a toroidal component confined to a closed-flux region. These labels do not specify an equilibrium: one must also give the current functions, surface behavior, EOS, rotation, and exterior boundary condition.

Regularity requires the flux function to vanish with the appropriate power on the axis, the toroidal field to vanish where its supporting current prescription ends, and the normal magnetic flux to be continuous across the surface. A vacuum exterior disallows volume currents outside but may still expose an unphysical surface current if derivatives are mismatched. A force-free magnetosphere uses different exterior equations and cannot be replaced by the vacuum boundary silently.

Pili, Bucciantini, and Del Zanna construct purely poloidal, purely toroidal, and mixed equilibria and display how current choices redistribute magnetic energy in Pili, Bucciantini, and Del Zanna 2014, §§2–5 and Figs. 2–12. Their conformally flat spacetime condition is an approximation that must be distinguished from an unrestricted stationary metric when comparing high-precision multipoles.

Magnetic energy and deformation

For a weakly deformed Newtonian axisymmetric star, define the material ellipticity by $\epsilon_B=(I_{zz}-I_{xx})/I_0$, where $z$ is the symmetry axis and $I_0$ the spherical reference moment of inertia. An oblate mass distribution has $\epsilon_B\gt0$, while a prolate one has $\epsilon_B\lt0$. This is a leading weak-deformation diagnostic, not an invariant definition of a strongly relativistic quadrupole.

In the declared rationalized units, $E_B=\int B^2\,dV/2$ in a local weak-field description. Comparing magnetic and gravitational stresses gives

$$E_B\sim B^2R^3, \qquad |W|\sim\frac{GM^2}{R}, \qquad \epsilon_B\sim C_{\rm geom}\frac{E_B}{|W|} \sim C_{\rm geom}\frac{B^2R^4}{GM^2}.$$

Numerical factors depend on electromagnetic units, the definition of the representative $B$, the density profile, and the ellipticity convention. $C_{\rm geom}$ carries topology information. Under common current models, poloidal stress tends to make the star oblate, whereas toroidal stress tends to make it prolate. A mixed field can partially cancel the quadrupole even when its total magnetic energy is large.

This scaling is controlled only when $E_B/|W|\ll1$. At stronger fields, the density profile, surface, spacetime, and field configuration change together; one must solve the nonlinear equilibrium. The exterior multipole also depends on electromagnetic stress outside the material surface, so “stellar ellipticity” must specify whether it refers to matter, geometry, or a radiative quadrupole convention.

Checks and limitations

The zero-field sequence must recover the unmagnetized rotating or TOV model. The numerical magnetic field must satisfy the divergence constraint, current regularity, and surface matching. Global mass and angular momentum should be consistent with asymptotics, and relativistic virial residuals should converge under radial and angular refinement. Repeating the calculation with different current functions tests model dependence; it is not a numerical error bar.

Equilibrium existence does not imply stability. Purely poloidal and purely toroidal fields are susceptible to nonaxisymmetric instabilities in broad settings, while the stable range and nonlinear fate of mixed fields depend on stratification, superconductivity, rotation, resistivity, and evolution time. Nor does a static current prescription explain formation. These limitations are why deformation estimates should state topology and timescale alongside field strength.

Check your understanding

Two weak-field models have the same mass, radius, density profile, and magnetic geometry, but the characteristic field of model B is twice that of model A. What ratio of magnetic ellipticities does the scaling predict, and can it predict the sign without the geometry assumption?

Solution

With all other quantities and $C_{\rm geom}$ fixed, $\epsilon_B\propto B^2$, so

$$\frac{\epsilon_{B,\mathrm B}}{\epsilon_{B,\mathrm A}}=2^2=4.$$

The sign is inherited from $C_{\rm geom}$; field magnitude alone cannot provide it. If doubling $B$ makes $E_B/|W|$ non-negligible, even the factor of four requires a nonlinear equilibrium check.

References

• Bocquet, Marc, Silvano Bonazzola, Eric Gourgoulhon, and Jérôme Novak. “Rotating Neutron Star Models with Magnetic Field.” Astronomy and Astrophysics 301 (1995): 757–775. arXiv:gr-qc/9503044.
• Pili, Antonio G., Niccolò Bucciantini, and Luca Del Zanna. “Axisymmetric Equilibrium Models for Magnetized Neutron Stars in General Relativity under the Conformally Flat Condition.” Monthly Notices of the Royal Astronomical Society 439 (2014): 3541–3563. DOI.