Analytical and Numerical Investigation of Magnetic Field Evolution in Neutron Stars

Abstract

We conducted a comprehensive study on the evolution of the magnetic field on the crust of an isolated neutron star. Our analysis reveals that the magnetic field evolution is influenced by three key physical processes: ambipolar diffusion, Ohmic dissipation, and the Hall drift. Among these processes, Ohmic dissipation and the Hall drift dominate in the crust region. Ohmic dissipation smoothens the magnetic field profile by inducing diffusion, while the non-linear Hall drift generates steep gradients over time. We investigate the intricate interplay between these two dominant processes and observe how the field profile evolves when one process is more dominant than the other.

For the initial configuration, which consists of a purely toroidal magnetic field, the field either migrates towards the pole and dissipates due to boundary conditions or converges at the equator, resulting in a sharp magnetic field gradient, leading to efficient Ohmic dissipation and energy release.

In this study, we explore a mechanism where the penetration velocity does not transport matter towards the reconnection site.

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Unlike typical magnetohydrodynamic scenarios where reconnection rates are limited due to matter transport, we examine a scenario where matter transport is negligible, allowing for sustained magnetic field movement towards the current sheet, promoting further reconnection. Our analytical findings are substantiated by numerical simulations where feasible, providing quantitative insights into the field evolution of neutron stars.

Hall MHD Equation

1.1 Equation of Motion

The interior of a neutron star is primarily composed of lightly ionized plasma consisting of protons (p), electrons (e), and neutrons (n).

To derive the equation of motion for these charged particles in the presence of a magnetic field within a neutron star, we adopt the mathematical formalism developed by Goldreich and Reisenegger. Our analysis focuses on normal protons and neutrons while excluding cases of proton superconductivity and neutron superfluidity. However, the solutions derived for the former cases can be extended to the latter.

The motion of charged particles within the plasma is governed by the Lorentz force acting on them, resulting in the equation:

$$mfrac{partial v}{partial t} = q(E + v times B)$$

Here, (v) represents the particle velocity, (E) denotes the electric field, and (B) is the magnetic field. Subsequent sections will provide specific formulations for each particle type.

We consider weak interactions among neutrons, protons, and electrons, leading to the creation and annihilation of particles. Any deviations from chemical equilibrium among these particles are swiftly rectified. We also neglect thermal contributions due to the exceptionally high thermal conductivity within the neutron star's interior. The equation of state for each particle species is approximated as that of an ideal, fully degenerate gas, with each particle's state defined by its internal chemical potential ((µ_i)), including rest mass.

We assume that neutrons form a fixed background in diffusive equilibrium. The magnetic field exerts minimal influence on the overall fluid motion because the magnetic field-to-stress pressure ratio for charged particles is small. Consequently, we neglect the contributions of electrons and protons to the gravitational potential (Φ), along with corrections arising from general relativity. With these assumptions, the equation of motion for protons can be expressed as:

$$mp frac{partial vp}{partial t} + mp (vp cdot nabla)vp = -nablaµp - mpnablaΦ + e(E + vp times B) - mpvpτpn - mp(vp - ve)τep$$

Similarly, the equation of motion for electrons is given by:

$$m^∗e frac{partial ve}{partial t} + m^∗e (ve cdot nabla)ve = -nablaµe - e(E + ve times B) - m^∗e veτen - m^∗e(ve - vp)τep$$

Here, (m^∗e = µe/c^2) describes the effective inertia of electrons, and (B) and (E) represent the magnetic and electric fields, respectively. Additionally, (vi) represents the mean velocity of particle (i), and (τij) represents the relaxation time between collisions of particles of species (i) and (j).

We impose the condition that the average velocity of neutrons ((vn)) is zero, and we neglect the gravitational force acting on electrons. By assuming equal mass for protons and neutrons, we can write (mp/τpe = m^∗e/τep).

We initiate our formalism by considering a highly conductive fluid comprising neutrons, protons, and electrons, existing in both chemical and magnetostatic equilibrium. The evolution of the magnetic field over time is described by Faraday's induction law:

$$frac{partial B}{partial t} = -cnabla times E$$

In a highly conductive fluid, the electric field is expected to vanish. However, the fluid in question is in motion, and due to the presence of the magnetic field, the electric field only vanishes in the moving frame of reference. Thus, we obtain the electric field by combining Equation (1.1) and Equation (1.2), neglecting the inertial term:

$$E = Jσ_0 - frac{v}{c} times B + frac{mp/τpn - m^∗e/τen}{mp/τpn + m^∗e/τen} J times B + frac{τpn/mp}{τpn/mp + τen/m^∗e} nablaµp - frac{τen/m^∗e}{τpn/mp + τen/m^∗e} nablaµe$$

Where (σ_0 = nce^2 left( frac{1}{τep/m^∗e} + frac{1}{τpn/mp + τen/m^∗e} right)) represents the electrical conductivity in the absence of a magnetic field. To derive the equation for the magnetic field's evolution, we substitute Equation (1.4) into Equation (1.3) and obtain:

$$frac{partial B}{partial t} = -cnabla times left( frac{J}{4πσ_0} right) + nabla times (v times B) - frac{mp/τpn - m^∗e/τen}{mp/τpn + m^∗e/τen} nabla times left( frac{J times B}{nce} right)$$

Here, (J) is related to (B) through Ampere's law:

$$J = frac{cnabla times B}{4π}$$

The right-hand side of Equation (1.6) encompasses Ohmic decay, ambipolar diffusion, and the Hall drift. Our primary focus is on the Hall drift, allowing us to simplify Equation (1.6) by taking the limit (τpn → 0) and (τen → ∞). This immobilizes protons and results in all currents being carried solely by electrons, eliminating ambipolar diffusion. Thus, we obtain the reduced form of Equation (1.6):

$$frac{partial B}{partial t} = -cnabla times left( frac{cnabla times B}{4πσ_0} right) - cfrac{1}{4πnce} nabla times left( (nabla times B) times B right)$$

Substituting the expression for (J) from Equation (1.7) into Equation (1.8) yields:

$$frac{partial B}{partial t} = -cnabla times left( frac{c}{4πnce} [(nabla times B) times B] right) - cnabla times η(nabla times B)$$

Where (η = frac{c^2}{4πσ_0}). Equation (1.9) further leads to an energy balance equation:

$$frac{1}{2} frac{partial}{partial t} int B^2 dV = -int η(nabla times B)^2 dV$$

1.3 Decay Timescale

Previous observations have indicated significant variations in the characteristic timescales of magnetic field evolution due to the Hall effect and Ohmic dissipation for isolated neutron stars with magnetic field strengths on the order of (10^{13}) and higher. Building on the work of Baym, Pethick, Pines, Goldreich, Reisenegger, and Cumming, it has been established that Ohmic dissipation typically occurs over very long timescales, sometimes exceeding the age of the universe.

Goldreich and Reisenegger proposed that the non-linear Hall term drives the cascade of magnetic energy to smaller scales, ultimately leading to its dissipation through Ohmic diffusion. In this section, we calculate the Ohmic and Hall timescales and examine their combined impact on magnetic field evolution.

Starting with Equation (1.9), we can express the Ohmic timescale as:

$$t_ohm ∼ (frac{4πσ₀L²}{c²})$$

Where (σ₀) represents the electrical conductivity, (L) is the characteristic length scale, and (c) is the speed of light. Notably, both Ohmic and Hall timescales are highly sensitive to the length scale (L) over which the magnetic field varies.

It is evident that (t_ohm) is proportional to (L²) and independent of magnetic field strength. We calculate the electrical conductivity of the fluid core as:

$$σ₀ = 4.2 × 10²⁸ T^-28 (left(frac{ρ}{ρ_nuc}right)³) s^-1$$

Where (T₈) represents the temperature in units of (10⁸) K, and (ρ_nuc equiv 2.8 × 10¹⁴) g cm^-3. Substituting this into the Ohmic timescale expression (1.11), we obtain:

$$t_ohm ∼ 2 × 10¹¹ (frac{L²}{T²}) (left(frac{ρ}{ρ_nuc}right)) yr

Several conclusions can be drawn from Equation (1.11). Firstly, the magnetic field of a neutron star, supported by current in the fluid core, does not experience significant Ohmic decay when the core matter is normal. Secondly, Ohmic decay would occur much faster if the crustal current supported the magnetic field, which could explain the observed magnetic field strength declines.

Next, we calculate the Hall timescale by simplifying Equation (1.9) when neglecting the Ohmic term:

$$frac{partial B}{partial t} = -nabla times (frac{c}{4πnce}) (left[(nabla times B) times Bright])$$

This results in the following expression for the Hall timescale:

$$t_hall = (frac{4πnceL²}{cB})$$

Here, (L) represents the timescale over which (B), (J), and (nc) vary. Unlike Ohmic dissipation, the Hall term is unaffected by the state of matter in neutron stars and is prevalent both in the solid crust and the fluid core, with a stronger presence in the crust. We proceed to calculate the Hall timescale using the value of (nc) given by Goldreich and Reisenegger:

$$nc ∼ 5 × 10^-2 (frac{ρ}{mn}) ∼ 8 × 10³⁶ (frac{ρ}{ρ_nuc}) cm^-3$$

Substituting this into the Hall timescale expression (1.13), we obtain:

$$t_hall ≈ 5 × 10⁸ (frac{L²}{B¹²}) (left(frac{ρ}{ρ_nuc}right)) yr

The Hall drift, unlike Ohmic dissipation, conserves energy and is not directly responsible for magnetic field decay in neutron stars.

However, the Hall drift is thought to play a crucial role in transferring magnetic energy from large to small scales, thereby enhancing Ohmic dissipation. Let's summarize the key findings regarding the decay timescales:

Ohmic Timescale (t_ohm):

The timescale for Ohmic dissipation is primarily dependent on the electrical conductivity (σ₀), the characteristic length scale (L), temperature (T₈), and density (ρ). Importantly, it is proportional to L² and independent of magnetic field strength. This suggests that the magnetic field of a neutron star, supported by current in the fluid core, experiences minimal Ohmic decay if the core matter is in a normal state. However, it could decay more rapidly if crustal currents support the magnetic field, potentially explaining observed magnetic field strength declines.

Hall Timescale (t_hall):

The Hall timescale is primarily influenced by electrical conductivity (σ₀), characteristic length scale (L), magnetic field strength (B), and density (ρ). Unlike Ohmic dissipation, the Hall term is unaffected by the state of matter in neutron stars and occurs in both the solid crust and fluid core. While it does not directly cause magnetic field decay, the Hall drift is implicated in transferring magnetic energy from large to small scales, which can enhance Ohmic dissipation.

These timescales provide crucial insights into the mechanisms governing the evolution of magnetic fields in neutron stars. They help us understand why some neutron stars maintain strong magnetic fields over cosmic timescales, while others exhibit a gradual decline. Further investigation and numerical simulations are needed to validate these theoretical predictions and provide a comprehensive understanding of magnetic field evolution in neutron stars.

Conclusion

In conclusion, our analytical and numerical investigation sheds light on the intricate processes governing the evolution of magnetic fields in neutron stars. By examining the interplay between ambipolar diffusion, Ohmic dissipation, and the Hall drift, we gain valuable insights into the timescales and mechanisms involved. These findings contribute to our understanding of the long-term behavior of magnetic fields in neutron stars, which is essential for unraveling the mysteries of these astrophysical objects.