Exploring the internal structure of a neutron star and the associated magnetic fields aided by the mass-radius relationship

TL;DR

The paper tackles whether ordinary and millisecond pulsars can sustain their magnetic dipoles solely through crustal neutron polarization, given strong density stratification and uncertain core physics. It develops a piecewise density meta-model anchored to robust mass–radius observations and representative EoS via the TOV framework, incorporating a core–crust density jump and inner-crust curvature to produce a physically motivated crust–core structure. By computing a realistic moment of inertia $I$, crust thickness $\Delta R$, and crustal neutron content, it estimates the crustal magnetic moment and defines the spin-polarization $P_{\rm spin}$ as $P_{\rm spin} = \mu_{NS}/(N_{n,crust} |\mu_n|)$. Across stars with measured $M$ and $R$ (notably MSPs), the study finds $P_{\rm spin} \lesssim 5.5\times 10^{-2}$ (99% C.L.), supporting crust-confined magnetism for non-magnetar neutron stars and providing a quantitative link between observable $M$/$R$ and the crustal magnetic properties, with caveats regarding core-field models and magnetar cases.

Abstract

Neutron stars exhibit magnetic fields and densities far beyond those achievable in terrestrial laboratories, offering a natural probe of strongly interacting matter under extreme conditions. Using observationally anchored mass-radius relations and a density profile consistent with established equations of state, we construct a piecewise model that explicitly integrates the neutron-drip line, nuclear-saturation, the electron-dominated halo, and core-crust interfaces. The resulting structure reproduces the stiffness and curvature behavior across the nuclear-pasta regime reported in the literature, validating our treatment of the crust-core transition. From this model, we derive updated moments of inertia, crustal mass fractions, and the effective number of neutrons contributing to the star's magnetic moment. Comparing these quantities with spin-down inferred magnetic dipole moments indicates that the observed magnetic fields of particularly millisecond pulsars can be sustained entirely by the crustal neutron polarization, requiring alignment of only about $\lesssim5.5\%$ ($99\%$ C.L.) of the neutrons in the crust. This finding supports a crust-confined magnetic-field origin for non-magnetar neutron stars, consistent with magneto-thermal evolution studies, and provides a quantitative framework for connecting neutron-star observables to its underlying structure.

Figures (5)

• Figure 1: Model of PSR J0437-4715 using the density model described in the text. This model uses a global charge neutrality condition, which can be seen with the mass discontinuity at the core-crust interface. Note the nuclear density $\rho_{nuc}$ marks the end of the core and the drip density $\rho_{drip}$ marks the end of the inner crust.
• Figure 2: Plot showing the period and the time-rate of change of the period for all stars in the ATNF catalog Manchester2005-vu with their respective categories, noting that the majority of neutron stars with documented mass values, denoted by markers with color correlating to the color scale on the right, are considered to be millisecond pulsars.
• Figure 3: Density model of J0437-4715 with documented mass $1.44\pm 0.07~M_{\odot}$ and radius $11.36\pm0.90$ km. The red line shows nuclear saturation, $\rho_{nuc}$ (ceiling core density) Douchin2001-es, and the green line shows neutron drip, $\rho_{drip}$ (floor crust density) Negele1973-hd. The dotted lines show the density model with the extremes of the central and average density values from the 8 different EoS models Haensel2006-uh. The grey shaded region includes the propagated uncertainties from $\{M,R\}$ and density choices. The blue strip marks $R_{\mathrm{core}}$ uncertainties within the $2\sigma$ range.
• Figure 5: Moment of inertia of a $11.36~$km star for masses varying from $0.5- 2~M_{\odot}$. This interval spans the full range of stable TOV solutions and includes all masses observed in real neutron stars, from the theoretical stability lower limit to the $\approx 2M_{\odot}$ maximal observed values. For reference, the canonical moment of inertia ($1\times 10^{38}~\rm kg~m^{-3}$ is shown in orange, as well as the moment of inertia for a solid and hollow sphere in green and red, respectively.
• Figure 6: Polarization for stars with documented mass, as well as all stars from the ATNF catalog Manchester2005-vu with an assumed canonical mass. The age was derived by cross-checking the pulsars' right ascension, declination, and distance. The color scale on the right is to denote the relationship between period and polarization.