The Rotating Magnetic-Dipole Model

The rotating magnetic dipole is the cleanest baseline for pulsar spin-down. An oblique dipole in vacuum radiates at the rotation frequency and removes rotational energy, giving $\dot\Omega\propto-\Omega^3$ when the dipole, inclination, and moment of inertia are fixed. The model explains useful scalings, but its “inferred surface field” is conditional: a real pulsar fills its magnetosphere with plasma, carries currents, and can spin down even when the magnetic and rotation axes are aligned.

Required background. Frames, Coframes, and Anholonomy supplies the local orthonormal components used to distinguish measured electric and magnetic fields.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies electromagnetic energy flux; The Einstein Equation supplies the stellar exterior and redshift corrections omitted in the baseline; Relativistic Matter and Compact Objects supplies the stellar moment of inertia; Computational Relativity supplies field-evolution checks.

Dipole convention and radiation power

Assume flat spacetime, Gaussian electromagnetic units, and $c=1$. The analytic inputs are $I$, $R$, $\mu$, $\alpha$, and $\Omega$, all treated as constant except the evolving spin in the torque equation. Define the dipole moment $\boldsymbol\mu$ so that the quasistatic field is

$$\boldsymbol B(\boldsymbol r) =\frac{3\boldsymbol n(\boldsymbol n\mathbin{\cdot}\boldsymbol\mu) -\boldsymbol\mu}{r^3}.$$

Then the equatorial and polar surface fields are $B_{\rm eq}=\mu/R^3$ and $B_{\rm p}=2\mu/R^3$. If the dipole makes an angle $\alpha$ with the rotation axis and rotates at angular frequency $\Omega$, the wave-zone luminosity is

$$L_{\rm vac}=\frac23\mu^2\Omega^4\sin^2\alpha.$$

The origin of the angular factor is easy to see. Write $\boldsymbol\mu(t)=\mu(\sin\alpha\cos\Omega t,\sin\alpha\sin\Omega t,\cos\alpha)$. Only the component perpendicular to the spin axis varies, and $|\ddot{\boldsymbol\mu}|^2=\mu^2\Omega^4\sin^2\alpha$. The magnetic-dipole radiation formula $L=2|\ddot{\boldsymbol\mu}|^2/3$ then gives the result. In ordinary Gaussian units its right-hand side is divided by $c^3$, and $R_{\rm LC}=c/\Omega$. The quasistatic point-dipole approximation requires $R\Omega/c\ll1$.

This coefficient changes if $\mu$ or the electromagnetic unit system is defined differently; the convention must accompany the formula. Deutsch’s finite conducting-sphere solution shows how near, induction, and radiation fields join in Deutsch 1955, Appendix, pp. 9–10. The point-dipole expression retains the leading wave-zone power and omits finite-radius and relativistic corrections.

The light-cylinder radius

$$R_{\rm LC}=\Omega^{-1}$$

marks where rigid corotation would reach light speed. It is not a material cylinder or a singularity of flat spacetime. It is a critical scale for a rotating field pattern and becomes a regularity surface in plasma magnetosphere equations.

Spin evolution and the braking index

For a star with constant moment of inertia $I$, rotational energy is $E_{\rm rot}=I\Omega^2/2$. Equating its loss to the vacuum luminosity gives

$$I\Omega\dot\Omega=-L_{\rm vac}, \qquad \dot\Omega=-K\Omega^3, \qquad K=\frac{2\mu^2\sin^2\alpha}{3I}.$$

The braking index is

$$n=\frac{\Omega\ddot\Omega}{\dot\Omega^2}=3$$

only if $K$ is constant. Magnetic evolution, changing inclination, magnetospheric state changes, moment-of-inertia coupling, glitches, and timing noise all alter the measured value. Pacini’s early rotating-neutron-star energy argument and spin-down scaling appear in Pacini 1967, pp. 567–568.

Solving the torque law for the conventional equatorial field gives

$$B_{\rm eq} =\frac1{R^3} \left(\frac{3IP\dot P}{8\pi^2\sin^2\alpha}\right)^{1/2}.$$

This is a model-derived parameter, not a direct magnetometer reading. It requires assumed $I$, $R$, inclination, vacuum torque, and dipole geometry. Calling it simply “the magnetic field” hides those dependencies.

Where the vacuum model stops

A rotating conductor creates electric fields strong enough to pull charges from the surface or produce pairs. Goldreich and Julian show that plasma tends to screen the electric field along the magnetic field and derive the corotation charge-density scale in Goldreich and Julian 1969, §III, Eqs. (5)–(9). Currents then open magnetic flux across the light cylinder and form a current sheet. In that regime, the torque depends on global current closure, not just radiation from a prescribed vacuum moment.

The contrast is sharp for alignment. The ideal point-dipole formula gives $L_{\rm vac}=0$ at $\alpha=0$. A plasma-filled force-free aligned rotator has open flux and nonzero luminosity. The vacuum formula remains valuable as a limiting calculation and dimensional check, but it cannot decide pair supply, radio emission, current closure, or the observed torque coefficient.

General relativity adds frame dragging, curvature, redshift, and a relativistic stellar $I$; finite conductivity and multipolar surface fields add further scales. None changes the logical lesson: $P$ and $\dot P$ infer a field only through a declared torque model.

Check your understanding

In the fixed-parameter vacuum model, a pulsar’s angular frequency decreases from $\Omega_0$ to $\Omega_0/2$. By what factor do its luminosity and the magnitude of $\dot\Omega$ change?

Solution

The luminosity scales as $\Omega^4$, so

$$\frac{L_2}{L_1}=\left(\frac12\right)^4=\frac1{16}.$$

The torque law has $|\dot\Omega|\propto\Omega^3$, giving $|\dot\Omega_2|/|\dot\Omega_1|=1/8$. The different powers arise because power is torque times angular velocity.

References

• Deutsch, Armin J. “The Electromagnetic Field of an Idealized Star in Rigid Rotation in Vacuo.” Annales d’Astrophysique 18 (1955): 1–10. ADS record.
• Goldreich, Peter, and William H. Julian. “Pulsar Electrodynamics.” The Astrophysical Journal 157 (1969): 869–880. DOI.
• Pacini, Franco. “Energy Emission from a Neutron Star.” Nature 216 (1967): 567–568. DOI.