Goldreich–Julian Charge Density Calculator
Overview
This calculator estimates the Goldreich–Julian (GJ) charge and number density and the associated polar-cap potential drop for a rotating, magnetised neutron star. By supplying the spin period P, surface magnetic field B, and stellar radius R, you can explore how pulsar magnetospheres are filled with plasma and how strongly they can accelerate particles.
The tool is aimed at advanced undergraduates, graduate students, and researchers working on pulsars, neutron stars, and high-energy astrophysics, as well as educators who need a quick way to generate realistic example numbers for coursework and demonstrations.
Physical background
A rotating, magnetised neutron star cannot remain an electrovacuum. As first argued by Goldreich and Julian (1969), the rotation of the star with angular velocity Ω in the presence of a magnetic field B induces an electric field that, in vacuum, would have a component parallel to the magnetic field lines. This parallel electric field extracts charged particles from the stellar surface until their collective charge density just cancels the parallel component. The resulting configuration is often described as a corotating or force-free magnetosphere.
In this state, the charge density adjusts so that Gauss’s law is satisfied with the corotation electric field. The required charge density is the Goldreich–Julian charge density, usually denoted by ρGJ. Dividing by the elementary charge e gives the corresponding number density nGJ of charges needed to enforce corotation.
Key formulas
For a star rotating with period P, the angular velocity is
In the simplest case of an aligned dipole rotator 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
ρGJ ≈ −(Ω B) / (2 π c),
where c is the speed of light and B is the local magnetic field strength at the surface. The corresponding number density of charges is
nGJ = |ρGJ| / e.
In cgs units, a widely used approximate expression at the magnetic pole is
nGJ ≈ 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. Even for modest values of B and P, these densities are many orders of magnitude larger than those achieved in terrestrial plasmas.
The rotation also induces a strong potential difference across the bundle of open magnetic field lines emerging 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 typical radio pulsars, this potential can reach 1012–1015 V, sufficient to accelerate particles to ultra-relativistic energies and trigger pair cascades.
Interpreting the results
The calculator outputs the Goldreich–Julian number density and the polar-cap potential drop for the pulsar parameters you provide. Here is how to interpret these quantities.
- Goldreich–Julian number density nGJ (cm−3): this is the minimum density of charges (electrons or positrons) needed locally to screen the parallel electric field and maintain corotation. A higher nGJ means that the magnetosphere must contain more plasma.
- Polar-cap potential drop ΔΦ (V): this characterises the maximum potential difference available to accelerate particles along open field lines above the magnetic poles. Larger values of ΔΦ generally imply higher achievable particle energies and more vigorous pair production.
For context, laboratory plasmas often have densities around 1010–1014 m−3, whereas typical Goldreich–Julian densities near pulsar surfaces are around 1016–1020 m−3 (or 1010–1014 cm−3), depending on B and P. Polar-cap potentials of 1012 V and above are far beyond what can be attained in human-made accelerators.
Worked examples
Canonical radio pulsar
Consider a typical radio pulsar with period P = 1 s, surface magnetic field B = 1012 G, and radius R = 10 km.
- Goldreich–Julian density: with B12 = 1 and P = 1, the cgs estimate gives nGJ ≈ 7 × 1010 cm−3 near the polar cap.
- Polar-cap potential: inserting these values into the potential-drop expression yields a characteristic ΔΦ of order 1012 V. Particles accelerated through this potential quickly reach Lorentz factors of 106 or more.
Such conditions are typical of “normal” radio pulsars detected in the Galaxy and are sufficient to power coherent radio emission through pair cascades in the magnetosphere.
Millisecond pulsar
Now take a millisecond pulsar with P = 5 ms, B = 3 × 108 G, and R = 10 km.
- The reduced magnetic field is compensated by the much shorter period. Here B12 = 3 × 10−4 and P = 5 × 10−3 s, so the rough scaling gives nGJ ≈ 7 × 1010 (3 × 10−4) / (5 × 10−3) ≈ 4 × 109 cm−3 near the surface.
- The much larger Ω still produces substantial potential drops along open field lines, often supporting high-energy gamma-ray emission even when the radio emission is weak or absent.
You can reproduce and further explore these examples by entering the above parameter sets into the calculator and comparing the resulting densities and potentials.
Parameter dependence and comparison
Both nGJ and ΔΦ are sensitive to the spin period and magnetic field strength:
- nGJ ∝ B / P (for fixed geometry): denser magnetospheres arise for stronger fields and shorter periods.
- ΔΦ ∝ B R3 / P2: faster rotation and larger radii increase the potential, while stronger fields also help.
| Object type | Typical P | Typical B (G) | nGJ scale (cm−3) | ΔΦ scale (V) |
|---|---|---|---|---|
| Canonical radio pulsar | 0.1–1 s | 1011–1013 | 1010–1012 | 1012–1014 |
| Millisecond pulsar | 1–10 ms | 108–109 | 108–1010 | 1011–1013 |
| Magnetar | 2–10 s | 1014–1015 | 1011–1013 | 1013–1015 |
While these ranges are approximate, they show how different neutron star populations occupy distinct regions in the space of magnetospheric densities and acceleration potentials.
Assumptions and limitations
The calculator is based on an idealised version of the Goldreich–Julian model and makes several simplifying assumptions:
- Aligned dipole geometry: the underlying formulas assume that the large-scale magnetic field is a centred dipole roughly aligned with the rotation axis, evaluated near the magnetic pole. Strongly inclined or multipolar fields can modify both nGJ and ΔΦ.
- Corotating, force-free magnetosphere: the model presumes that the magnetosphere is filled with plasma such that the electric field parallel to B is screened almost everywhere. In reality, gaps where the charge density falls below the Goldreich–Julian value can form and support unscreened acceleration.
- Simple unit conventions: the input magnetic field is taken in Gauss, the period in seconds, and the radius in kilometres. The internal implementation uses a mix of cgs-based expressions and standard constants; these estimates are suitable for order-of-magnitude work rather than precision modelling.
- No general relativistic corrections: effects of strong gravity (such as frame dragging and redshift) near the neutron star surface are ignored. These can modify the effective charge density by factors of a few in detailed models.
- Surface values only: the calculator characterises conditions near the stellar surface and at the polar caps. The structure of the magnetosphere further out, especially near the light cylinder, 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. Extremely large or small inputs may yield unphysical results.
Within these limitations, the calculator is well suited to exploring how changes in P, B, and R affect magnetospheric charge densities and acceleration potentials, for use in teaching, problem sets, and quick back-of-the-envelope checks in research.