Goldreich–Julian Charge Density Calculator

Goldreich–Julian formulas used by this calculator

For this Goldreich–Julian model, the starting point is the star's rotation rate. If the pulsar spins with period P, its angular velocity is

In the simplest aligned-dipole picture, where the magnetic field is parallel to the rotation axis at the pole, the Goldreich–Julian charge density near the stellar surface can be written (in SI units) as

ρ_GJ ≈ −(Ω B) / (2 π c),

with c the speed of light and B the local surface field strength. Dividing by the elementary charge gives the corresponding number density,

n_GJ = |ρ_GJ| / e.

In cgs units, a widely used approximation at the magnetic pole is

n_GJ ≈ 7 × 10¹⁰ B₁₂ P⁻¹ cm⁻³,

where B₁₂ is the surface magnetic field in units of 10¹² G and P is the spin period in seconds. The calculator uses this scaling to highlight how rapidly the required charge density rises when the star spins faster or carries a stronger magnetic field.

The rotation also produces a substantial potential difference across the open magnetic field lines that emerge from the polar caps. A commonly used estimate for the polar-cap potential drop is

ΔΦ ≈ (Ω² B R³) / (2 c²),

where R is the neutron star radius. For a pulsar, this potential can reach 10¹²–10¹⁵ V, which is why the Goldreich–Julian framework is so often used when discussing particle acceleration and pair cascades in the magnetosphere.

Interpreting Goldreich–Julian results

The Goldreich–Julian calculator returns two linked outputs for the pulsar parameters you enter: the minimum charge-carrier density needed to support corotation, and the open-field-line potential available to accelerate particles above the magnetic pole.

• Goldreich–Julian number density n_GJ (cm⁻³): this is the local charge density needed to screen the electric field parallel to the magnetic field and keep the plasma corotating with the star. Larger values mean the pulsar magnetosphere has to be filled more densely.
• Polar-cap potential drop ΔΦ (V): this is the approximate voltage available along open field lines above the polar cap. A larger ΔΦ usually means particles can reach higher energies and the magnetosphere can more readily support pair creation.

As a rough scale, laboratory plasmas often have densities around 10¹⁰–10¹⁴ m⁻³, while a pulsar surface can demand Goldreich–Julian densities around 10¹⁶–10²⁰ m⁻³ (or 10¹⁰–10¹⁴ cm⁻³) depending on B and P. The corresponding polar-cap potentials are so large that they sit far beyond anything used in terrestrial accelerators.

Goldreich–Julian worked examples

Canonical radio pulsar example

For a canonical Goldreich–Julian radio pulsar, take a period of P = 1 s, a surface magnetic field of B = 10¹² G, and a radius of R = 10 km.

• Goldreich–Julian density: with B₁₂ = 1 and P = 1, the cgs estimate gives n_GJ ≈ 7 × 10¹⁰ cm⁻³ near the polar cap.
• Polar-cap potential: inserting the same values into the potential-drop expression gives a characteristic ΔΦ of order 10¹² V. In a Goldreich–Julian picture, that is ample voltage for particles to be accelerated to very large Lorentz factors.

These values are representative of ordinary radio pulsars in the Galaxy. They are high enough to sustain pair cascades, which is one reason the Goldreich–Julian model remains central to basic pulsar phenomenology.

Millisecond pulsar example

For a millisecond Goldreich–Julian pulsar, use P = 5 ms, B = 3 × 10⁸ G, and R = 10 km.

• The smaller field is partly offset by the much shorter period. Here B₁₂ = 3 × 10⁻⁴ and P = 5 × 10⁻³ s, so the rough scaling gives n_GJ ≈ 7 × 10¹⁰ (3 × 10⁻⁴) / (5 × 10⁻³) ≈ 4 × 10⁹ cm⁻³ near the surface.
• The much larger Ω still produces a substantial polar-cap potential in the Goldreich–Julian framework, which helps explain why some millisecond pulsars remain bright gamma-ray sources even when their radio beams are comparatively faint.

You can reproduce both examples by entering those parameter sets into the calculator and comparing the resulting Goldreich–Julian density and polar-cap voltage outputs.

Goldreich–Julian parameter dependence and comparison

The Goldreich–Julian outputs move in predictable ways as P, B, and R change:

• n_GJ ∝ B / P (for fixed geometry): stronger fields and shorter spin periods require denser magnetospheres.
• ΔΦ ∝ B R³ / P²: faster rotation and larger radii increase the open-field potential, and stronger fields also raise it.

These ranges are only approximate, but they show how different neutron-star populations occupy different parts of the Goldreich–Julian density and acceleration-potential landscape. The calculator makes those comparisons easier to inspect one pulsar at a time.

Goldreich–Julian assumptions and limitations

This Goldreich–Julian calculator uses an idealised pulsar-magnetosphere model, so the outputs should be read as order-of-magnitude guidance rather than a full numerical simulation:

• Aligned dipole geometry: the formulas assume the large-scale magnetic field is a centred dipole roughly aligned with the rotation axis and evaluated near the magnetic pole. Strongly inclined or multipolar fields can change both n_GJ and ΔΦ.
• Corotating, force-free magnetosphere: the model assumes that plasma fills the magnetosphere well enough to screen the electric field parallel to B almost everywhere. Real pulsars can develop charge-starved gaps where the Goldreich–Julian density is not met and unscreened acceleration becomes possible.
• Simple unit conventions: the input magnetic field is taken in Gauss, the period in seconds, and the radius in kilometres. The internal implementation mixes cgs-based expressions with standard constants, so the results are best used for quick scaling checks rather than precision modelling.
• No general relativistic corrections: effects of strong gravity, including frame dragging and gravitational redshift near the neutron-star surface, are ignored here. In detailed treatments, those effects can shift the effective charge density by factors of a few.
• Surface values only: the calculator describes conditions near the stellar surface and polar caps. Farther out in the magnetosphere, especially near the light cylinder, the field structure and plasma distribution can differ substantially from these simple scalings.
• Applicable parameter ranges: the formulas are most meaningful for neutron stars with P between roughly 1 ms and 20 s, B between about 10⁸ and 10¹⁵ G, and radii of order 8–15 km. Extreme values may push the calculator outside the domain where the Goldreich–Julian approximations are sensible.

Within those limits, the calculator is a practical way to explore how changes in P, B, and R reshape the Goldreich–Julian density and the available acceleration potential, whether you are preparing a class example, checking a homework problem, or doing a quick back-of-the-envelope estimate.

Goldreich–Julian physical background

A rotating, magnetised neutron star cannot stay an electrovacuum for long. In the Goldreich–Julian picture introduced by Goldreich and Julian (1969), the star's rotation with angular velocity Ω in the presence of a magnetic field B induces an electric field that would have a component parallel to the field lines if no charges were present. That parallel electric field quickly pulls charge from the stellar surface until the plasma density adjusts and the corotation field is largely cancelled. The resulting state is usually described as a corotating or force-free magnetosphere.

In this state, the charge density settles to the value required by Gauss's law for the corotation electric field. That required charge density is the Goldreich–Julian charge density, written ρ_GJ. Dividing by the elementary charge e gives the corresponding number density n_GJ of charges needed to enforce corotation. This is the quantity the calculator turns into a concrete number for the pulsar you enter.