Magnetopause 0 R_ESite |B| 0 µTDip 0°Decl. 0°Particle readyFPS 0

Drag to orbit, wheel or pinch to zoom, or use the view buttons. Blue lines: dipole field lines (r = L cos²λ), drawn only inside the magnetopause. Amber dots: solar wind (schematic flow, not MHD). The helix of the yellow particle is drawn ×25 wider than real; the line behind it is the guiding centre.

Earth's Dipole Field: L-Shells, Compass Dip Angle and Trapped-Particle Mirror Bounce

Read a compass anywhere on the globe, squeeze the magnetosphere with the solar wind, and follow a charged particle as it spirals, bounces between mirror points and drifts around the Earth.

Compass site (geographic)

Solar wind and dipole

Charged particle

Compass readout

Magnetosphere

Particle readout

Field strength along the field line, B / B_eq

The particle mirrors where B / B_eq reaches 1 / sin²α₀ (dashed line). Particles whose mirror point lies below 100 km altitude hit the atmosphere (loss cone).

Particle latitude vs time

Dip angle vs geomagnetic latitude

Experiment: loss cone and bounce period

Each row is a separate Boris-integrated run of the particle currently selected (same species, energy and L) with a different equatorial pitch angle. Measured = from the simulation; predicted = closed-form mirror condition and a numerical quadrature of the bounce integral.

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom. The view buttons jump to the compass on the Earth's surface or to the polar aurora ring.
Compass: latitude and longitude place the dip needle; the Earth model has a cut-away at longitudes 90° to 180°, where the compass is hidden; the tilt slider moves the dipole axis away from the rotation axis (about 10° today), which is why declination is not zero.
Solar wind density and speed set the magnetopause stand-off distance and the outermost closed field line drawn.
Particle: choose species, kinetic energy, L-shell and equatorial pitch angle; “Restart particle” re-launches it. Speed buttons scale simulated time; the integrator takes at most 25 000 steps per frame, so the fastest buttons may run slower than labelled.

The science

Dipole field. B = B₀ (R_E/r)³ √(1 + 3 sin²λ) with B₀ = 31.2 µT (typical equatorial surface value) and λ the geomagnetic latitude. Components: B_r = −2B₀(R_E/r)³ sin λ, B_λ = B₀(R_E/r)³ cos λ. A field line satisfies r = L R_E cos²λ (L-shell). The needle of a dip compass lies along B, so its inclination obeys tan I = 2 tan λ; declination appears because the dipole axis is tilted relative to the rotation axis. The true field has higher multipoles, a drifting pole and a secular variation; the dipole is about 80–90 % of the surface field energy.

Magnetopause. Pressure balance on the Sun line: ρ v² = (2B₀ (R_E/R)³)² / (2μ₀) (the field doubles at a perfectly conducting boundary), so R/R_E = (2B₀² / (μ₀ ρ v²))^(1/6), about 10 R_E for n = 5 cm⁻³ and v = 400 km/s. The shape is the empirical r = r₀ (2/(1 + cos θ))^0.6 (Shue-type). The pressure falls as the sixth root, so even a violent storm only compresses it to about 6 R_E. Field lines are drawn only where L is smaller than 0.95 r₀. The aurora ring marks the footprint of that outermost closed line, cos²λ = 1/L, which is a rough estimate, not a substorm model; the solar-wind flow is a schematic that slides around the boundary.

Charged particle. Relativistic Boris integrator for d(γm v)/dt = q v × B, with a fixed time step chosen from the field at the mirror point (about 20 steps per gyration there). Motion separates into gyration (period 2πγm/|q|B), bounce between mirror points and azimuthal drift. The first adiabatic invariant μ = p⊥²/2mB keeps sin²α/B constant, so the particle mirrors where B(λ_m)/B_eq = 1/sin²α₀. The loss cone follows from putting the mirror point at 100 km altitude. The bounce period is T_b = (4 L R_E/v) ∫ cos λ √(1+3 sin²λ) / √(1 − sin²α₀ √(1+3 sin²λ)/cos⁶λ) dλ (numerical quadrature) and the equatorial gradient drift period is T_d = 2π q B₀ R_E² / (3 W L) (non-relativistic, α₀ = 90°): protons drift westward, electrons eastward.

Simplifications. Pure dipole with no electric field, no atmosphere scattering and no rotation of the Earth; the guiding-centre line uses a first-order formula; heavy particles at large L have a large gyroradius compared with the field gradient (the adiabatic parameter is shown), so μ is conserved to about 0.1 % for the 100 keV electron but only to roughly 10 to 25 % for the proton presets. Energies, densities and speeds are typical values, not a forecast. The Earth model uses an invented continent texture, and the cut-away shows the core layers schematically (inner core radius 0.19 R_E, outer core 0.55 R_E). Models built with Blender, scene by Three.js.