Neutron stars compress a solar mass into a sphere barely twenty kilometers across, spinning hundreds of times per second and anchoring surface magnetic fields exceeding (10^{12} ext{ Gauss}). Such extreme rotating magnets generate immense electrostatic fields capable of ripping charges straight out of solid iron crusts, surrounding the dead star with a dense, relativistic plasma shroud. In this installment, we explore the electrodynamics of Pulsar Magnetospheres, derive the Goldreich-Julian Density, and analyze particle acceleration at the light cylinder.
1. Unipolar Induction and the Quadrupolar Electric Field
Consider a neutron star of radius (R_* sim 10 ext{ km}) rotating at angular velocity (boldsymbol{Omega} = Omega hat{mathbf{z}}) with a dipolar magnetic field (mathbf{B}). In the interior of the highly conductive star, Ohm’s law requires the co-rotating electric field to satisfy:
[
mathbf{E}_{ ext{in}} + (boldsymbol{Omega} imes mathbf{r}) imes mathbf{B} = 0 quad implies quad mathbf{E}_{ ext{in}} = -(boldsymbol{Omega} imes mathbf{r}) imes mathbf{B}
]
If the exterior were a true vacuum, matching boundary conditions at the surface (r = R_*) generates a quadrupolar vacuum exterior electric field with a large parallel component:
[
E_parallel = rac{mathbf{E} cdot mathbf{B}}{B} sim rac{Omega R_*}{c} B_* sim 10^{14} ext{ V/m}
]
This electric field exerts an upward force on surface electrons that exceeds surface gravity (g = G M / R_*^2 sim 10^{12} ext{ m/s}^2) by more than nine orders of magnitude: (e E_parallel / (m_e g) sim 10^9). The vacuum breaks down spontaneously as electrons and positrons are stripped from the crust and multiplied through electromagnetic cascades.
2. Derivation of the Goldreich-Julian Charge Density
In 1969, Peter Goldreich and William Julian proved that the magnetosphere fills with plasma until the parallel electric field is screened everywhere: (mathbf{E} cdot mathbf{B} = 0). Throughout this force-free co-rotating plasma, the electric field is locked to rotation:
[
mathbf{E} = -(boldsymbol{Omega} imes mathbf{r}) imes mathbf{B}
]
Taking the divergence of (mathbf{E}) and applying Gauss’s law (
abla cdot mathbf{E} =
ho / arepsilon_0):
[
rac{
ho_{ ext{GJ}}}{arepsilon_0} = –
abla cdot left[ (boldsymbol{Omega} imes mathbf{r}) imes mathbf{B}
ight]
]
Using the vector identity (
abla cdot (mathbf{A} imes mathbf{B}) = mathbf{B} cdot (
abla imes mathbf{A}) – mathbf{A} cdot (
abla imes mathbf{B})) with (
abla imes (boldsymbol{Omega} imes mathbf{r}) = 2boldsymbol{Omega}) and (
abla imes mathbf{B} = mu_0 mathbf{J}):
[
ho_{ ext{GJ}} = -arepsilon_0 left[ mathbf{B} cdot (2boldsymbol{Omega}) – (boldsymbol{Omega} imes mathbf{r}) cdot (
abla imes mathbf{B})
ight]
]
In the non-relativistic near-zone ((Omega r ll c)), the second term is negligible. Thus, we obtain the fundamental Goldreich-Julian Charge Density:
[
ho_{ ext{GJ}}(mathbf{r}) = -2arepsilon_0 boldsymbol{Omega} cdot mathbf{B} = -rac{boldsymbol{Omega} cdot mathbf{B}}{2pi c}
]
The number density of charges is (n_{ ext{GJ}} = rac{|
ho_{ ext{GJ}}|}{e} approx rac{Omega B}{2pi e c} approx 7 imes 10^{10} left(rac{B}{10^{12} ext{ G}}
ight) left(rac{P}{1 ext{ s}}
ight)^{-1} ext{cm}^{-3}).
3. The Light Cylinder and Polar Cap Cascade
Co-rotation cannot extend indefinitely. At a radial distance where the co-rotation speed reaches the speed of light ((v_phi = Omega R_{LC} = c)), we define the Light Cylinder Radius:
[
R_{LC} quiv rac{c}{Omega} = rac{c P}{2pi}
]
Magnetic field lines that close within (R < R_{LC}) form the Closed Magnetosphere, where plasma remains permanently trapped in co-rotation. Field lines that cross the light cylinder cannot close without violating causality; they bend backward, forming the Open Field Line Bundle that vents into the interstellar medium.
The boundary on the stellar surface demarcating open from closed lines defines the Polar Cap, with angular radius ( heta_{pc} approx sqrt{R_* / R_{LC}}). Above the polar cap, charge starvation creates an accelerating electric gap ((E_parallel
eq 0)). Primary electrons are accelerated to (gamma sim 10^7), emitting curvature gamma-ray photons ((hbaromega > 2 m_e c^2)) that convert in the strong magnetic field into secondary electron-positron pairs ((gamma + B o e^- + e^+ + B)), generating the dense pair plasma that powers coherent radio pulses.
