Evolution of a buried magnetic field in the central compact object neutron stars

The central compact objects are a newly-emerging class of young neutron stars near the centre of supernova remnants. From X-ray timing and spectral measurements, their magnetic fields are determined to be ~ 10^10-10^11 G, which is significantly lower than that found on most pulsars. Using the latest electrical and thermal conductivity calculations, we solve the induction equation to determine the evolution of a buried crustal or core magnetic field. We apply this model of a buried field to explain the youth and low observed magnetic field of the central compact objects. We obtain constraints on their birth magnetic field and depth of submergence (or accreted mass). Measurement of a change in the observed magnetic field strength would discriminate between the crustal and core fields and could yield uniquely the birth magnetic field and submergence depth. If we consider the central compact objects as a single neutron star viewed at different epochs, then we constrain the magnetic field at birth to be ~ (6-9)x10^11 G. A buried magnetic field can also explain their location in an underpopulated region of the spin period-period derivative plane for pulsars.

💡 Research Summary

The paper addresses the puzzling low surface magnetic fields (∼10¹⁰–10¹¹ G) observed in central compact objects (CCOs), a class of young neutron stars located near the centers of supernova remnants. The authors propose that these objects were born with typical pulsar‑like magnetic fields (∼10¹² G) but that the fields were buried beneath the stellar surface by a brief episode of hypercritical fallback accretion shortly after the supernova explosion. The buried field then diffuses outward on a timescale of 10³–10⁴ yr, producing the weak surface fields observed today.

To test this hypothesis, the authors solve the magnetic induction equation
∂B/∂t = −∇ × (η ∇ × B)
with η = c²/4πσ, where σ is the electrical conductivity. They adopt a spherically symmetric dipolar geometry and reduce the problem to a one‑dimensional radial diffusion equation for the scalar function s(r,t) that encodes the magnetic field. Two initial configurations are considered: (1) a field confined to the crust (the core is assumed superconducting, so the field does not penetrate) and (2) a field that extends into the core with a constant value below the burial depth. The burial depth is parameterized by a density ρ_sub, which corresponds to a specific amount of accreted mass ΔM ≈ 10⁻⁴ M⊙ for the densities relevant to CCOs.

A crucial ingredient is the electrical conductivity σ(ρ,T). The authors employ the modern CONDUCT08 code, which includes electron‑ion, electron‑phonon, and electron‑electron scattering, and they verify results with the independent SFITTING package. Conductivity depends strongly on temperature, so a realistic cooling history is required. They adopt a standard modified Urca cooling law (T ∝ t⁻¹⁄⁶) with an initial core temperature of 1.1 × 10⁹ K, which yields an isothermal core after a few hundred years and a temperature gradient through the heat‑blanketing envelope.

The neutron‑star structure is modeled using the SLy equation of state, solving the Tolman‑Oppenheimer‑Volkoff equations for a 1.63 M⊙ star with radius 11.5 km. This provides the radial density profile needed for both the conductivity and the burial depth calculations. The authors note that variations in the EOS would affect the crust thickness and thus the diffusion timescale, but they keep the EOS fixed to isolate the effects of burial depth and initial field strength.

Numerical integration of the diffusion equation yields the time evolution of the surface dipole field B_p(t) for a grid of (B*, ρ_sub) values. For shallow burial (ρ_sub ≈ 10¹² g cm⁻³), the surface field rises from essentially zero to ∼10¹⁰ G within a few thousand years, matching the ages of known CCOs (∼10³–10⁴ yr). Deeper burial (ρ_sub ≈ 10¹³–10¹⁴ g cm⁻³) leads to much longer diffusion timescales, inconsistent with observations unless the objects are significantly older.

The authors then apply the model to three well‑studied CCOs: PSR J0821‑4300 (Puppis A), 1E 1207.4‑5209, and PSR J1852+0040 (Kes 79). Observational constraints on their spin periods, period derivatives, and, where available, cyclotron line energies give surface fields of 4 × 10¹¹ G, 6 × 10¹¹ G, and 6 × 10¹⁰ G, respectively. By fitting the model curves to these data, they infer a common birth field B* ≈ (6–9) × 10¹¹ G and a burial density ρ_sub ≈ 10¹² g cm⁻³, corresponding to an accreted mass of order 10⁻⁴ M⊙. The two initial configurations (crust‑only vs. core‑penetrating) produce subtly different growth rates: the crust‑only case predicts a more rapid increase in B_p, while the core‑penetrating case yields a slower, smoother rise. The authors argue that precise long‑term timing (measuring changes in Ṗ over decades) could discriminate between these scenarios, thereby providing a direct probe of the internal magnetic geometry.

An important implication is that CCOs occupy an underpopulated region of the P–Ṗ diagram because their surface fields are temporarily suppressed. As the buried field resurfaces, a CCO would migrate upward in Ṗ, eventually joining the normal pulsar population. This evolutionary pathway offers a natural explanation for the apparent scarcity of objects with both low P and low Ṗ.

The paper also discusses uncertainties. Impurity scattering, which becomes important at high densities and low temperatures, is neglected; magnetic field effects on conductivity are ignored because the fields are weak enough that electrons are not in the quantizing regime (T > T_B). The authors note that faster cooling processes (e.g., direct Urca) would lower the temperature more quickly, increase σ, and thus lengthen diffusion timescales, but their results show that standard modified Urca cooling suffices to reproduce the observations. They also acknowledge that variations in the EOS could change the crust thickness and hence the diffusion timescale, but such effects are secondary to the burial depth.

In summary, the study provides a self‑consistent, physics‑based model for the evolution of buried magnetic fields in young neutron stars. By combining up‑to‑date microphysical inputs (conductivities, EOS) with realistic cooling, the authors demonstrate that a modest amount of fallback accretion can hide a typical pulsar‑strength field for a few thousand years, explaining the low magnetic fields of CCOs. The model makes clear, testable predictions: (i) a measurable increase in surface B over decades, (ii) a correlation between inferred burial depth and the amount of fallback mass, and (iii) a natural evolutionary track from CCOs to ordinary pulsars in the P–Ṗ diagram. Future high‑precision timing and spectroscopic observations will be crucial to confirm or refute this buried‑field scenario.