Ambipolar and Hall Field Decay Powering Magnetars

What field decay means inside a neutron star

A neutron star's magnetic field is not attached to empty space — it is frozen into charged matter. In the fluid core, that matter is a mixture of neutrons, protons and electrons (with protons likely superconducting); in the solid crust, it is a rigid ion lattice permeated by a degenerate electron gas. A magnetic field can only decay if the currents supporting it are allowed to move relative to the matter or to dissipate against resistance.

Peter Goldreich and Andreas Reisenegger, in a landmark 1992 Astrophysical Journal paper, identified the three channels that let this happen: Ohmic decay (currents dissipating against electrical resistivity), Hall drift (the field being carried along by the electron current without dissipating), and ambipolar diffusion (the charged fluid drifting bodily through the neutral neutron background).

• Ordinary pulsars (B ≈ 10¹² G) barely evolve — these processes are far too slow.
• Magnetars (B ≈ 10¹⁴–10¹⁵ G) have fields strong enough that Hall drift and ambipolar diffusion become fast, so decay becomes their engine, not a footnote.

The mechanism: three coupled channels

Ohmic decay is the simplest: a current density J dissipates energy at a rate J²/σ against conductivity σ, and the field diffuses on a timescale τ_Ohm = 4πσL²/c², where L is the field's length scale. Because τ ∝ L², short-wavelength structure decays fastest.

Hall drift arises because in the crust only the electrons are mobile; the field is advected at the electron drift velocity v ≈ J/(n_e e c). This is a nonlinear, non-dissipative term (∇×[(∇×B)×B]). It doesn't destroy energy by itself, but it stirs the field into ever-smaller scales — a Hall cascade — where Ohmic decay finishes the job. This coupling is why magnetar crust fields dissipate far faster than pure Ohmic estimates predict.

Ambipolar diffusion operates in the fluid core: the proton–electron charged fluid, tied to the field, drifts through the neutron sea. Its drift velocity scales as B², so the associated decay time scales as roughly 1/B² — making it the dominant core channel for the strongest fields. It comes in two modes: an irrotational part damped by weak-interaction (beta) reactions that convert particles, and a solenoidal part limited by inter-particle friction.

Characteristic numbers and a worked estimate

The Hall timescale can be written τ_H = 4π n_e e L²/(c B). Plugging in crust values — electron density n_e ≈ 10³⁶ cm⁻³ near the core–crust boundary, L ≈ 1 km, and B = 10¹⁴ G — gives

• τ_H ≈ 7.9 × 10⁵ yr × (B / 10¹⁴ G)⁻¹. So at 10¹⁵ G it drops to ~10⁴–10⁵ yr, and the field cascades even faster.
• Ohmic: τ_Ohm ≈ 10⁶ yr for large-scale crustal fields, shorter once the Hall cascade builds small scales.
• Ambipolar: ~10³–10⁵ yr in the core at 10¹⁵ G, thanks to its 1/B² scaling.

The energy budget is what makes this observable. A 10¹⁵ G field stores E_B ≈ B²R³/6 ≈ 10⁴⁷ erg. Decaying over ~10⁴–10⁵ yr, this supports a quiescent X-ray luminosity of order 10³⁵ erg s⁻¹ — precisely the level seen in magnetars, and far above what their slow rotation could supply. The decaying field literally heats the crust.

How we observe it: SGRs, AXPs and cooling curves

Field decay is not observed directly — it is inferred from what a decaying field produces. The magnetar population, about 30 confirmed objects, appears in two historically distinct guises now understood as the same class:

• Soft Gamma Repeaters (SGRs) — sources of repeating short (~0.1 s) gamma-ray bursts and, rarely, giant flares (SGR 1806−20 released ~10⁴⁶ erg on 27 December 2004).
• Anomalous X-ray Pulsars (AXPs) — steady X-ray pulsars whose luminosity (~10³⁵ erg s⁻¹) exceeds their spin-down power.

Three observational fingerprints point to internal field decay: (1) quiescent X-ray luminosities above the spin-down budget, matched by crustal heating from decaying currents; (2) surface temperatures hotter than passively cooling neutron stars of the same age; and (3) population statistics — magnetars are young (kyr ages) and become undetectable as their fields fade, implying a decay time of ~10³–10⁴ yr for the strongest fields. Magnetothermal simulations (Pons, Viganò, Aguilera, and others) that couple Hall+Ohmic crust evolution to heat transport reproduce the observed L_X–B_dip correlation.

How it differs from related regimes

It helps to place these channels against their cousins:

• vs. ordinary pulsar spin-down: A canonical pulsar loses rotational energy through magnetic-dipole braking; its field is essentially fossil and stable for ~10⁷–10⁸ yr. Magnetar activity is powered by field decay, a fundamentally different energy source.
• Hall vs. Ohmic: Hall drift conserves magnetic energy (it only rearranges it), while Ohmic decay and ambipolar diffusion genuinely destroy it. Hall's crucial role is to feed Ohmic dissipation by generating small scales.
• Crust vs. core: Hall+Ohmic dominate the solid crust; ambipolar diffusion dominates the fluid core. Because the core holds most of the flux, ambipolar diffusion sets the long-term reservoir, while the crust governs the flashy surface activity.
• vs. terrestrial ambipolar diffusion: In molecular clouds, ambipolar diffusion means neutrals slipping past ions to shed field during star formation — the same name and family of physics, but many orders of magnitude different in density and timescale.

Significance and open questions

Field decay is the organizing idea of magnetar astrophysics. It explains why these objects are luminous, transient and short-lived, and it links a single parameter — the internal field — to bursts, flares, thermal spectra and timing anomalies. It also underpins the leading model in which some magnetars power Fast Radio Bursts, after the Galactic magnetar SGR 1935+2154 produced an FRB-like radio burst in April 2020.

Several questions remain genuinely open:

• Core vs. crust dominance: How much of the flux (and decay) lives in the superconducting core, where flux-tube dynamics and ambipolar physics are still uncertain, versus the crust?
• The Hall attractor: Simulations suggest the Hall cascade may relax toward a quasi-stationary state; whether real magnetars reach it is debated.
• Weak-field magnetars: Sources like SGR 0418+5729 show magnetar-like bursts despite dipole fields of only ~10¹³ G, implying strong hidden toroidal fields — a direct test of internal decay models.
• Superfluid/superconducting effects on ambipolar diffusion and beta-reaction rates remain among the largest theoretical uncertainties.

The three magnetic-field decay channels in a neutron star interior and where each dominates

Mechanism	Where it acts	Dissipative?	Timescale (at 10¹⁴–10¹⁵ G)	Field scaling
Ohmic decay	Crust (lattice resistivity)	Yes (→ heat)	~10⁶ yr at large scale; faster at small scale	τ ∝ L²/η, no B dependence
Hall drift	Crust (electron fluid)	No (advective, but cascades energy)	~10³–10⁵ yr	τ_H ∝ 1/B
Ambipolar diffusion	Fluid core (charged particles vs neutrons)	Yes (via weak reactions / friction)	~10³–10⁵ yr	velocity ∝ B², so τ ∝ 1/B²
Combined (real magnetar)	Core + crust coupled	Yes	~10³–10⁴ yr initial rapid phase	strongly nonlinear