SpaceSherpa Van Allen Beltinator — 3D Radiation Belt Explorer

Physics Parameters

L* (Drift Shell) 4.5

Energy (MeV) 1.0

Equatorial Pitch Angle α₀ (°) 45

D_L*L* (Radial Diffusion) 1.0e-6 ⟲

D_pp (Momentum Diffusion) 1.0e-7

D_α₀α₀ (Pitch-Angle Diffusion) 1.0e-5

D_α₀p (Mixed Diffusion) 0 (off)

Loss Timescale τ (days) 10.0

Geomagnetic Conditions

Kp Baseline 3

Dst Index (nT) -30

Solar Wind V_sw (km/s) 400

Quiet Conditions

Kp(t) — 3-Hour Cadence Profile

0h12h24h

Visualization

Radiation Belts

Satellite Orbits

Magnetic Field Lines

Particle Trajectories

Loss Cone

Include Mixed Terms (Eq.3)

Run Kp(t) Simulation

Sim Speed (×) 60

Flux Intensity (cm⁻²s⁻¹sr⁻¹MeV⁻¹)

10²10⁴10⁶10⁸

Presets

Satellites (Click in 3D to zoom)

Van Allen Probes (RBSP A/B)

THEMIS A / D / E

GOES (GEO)

LEO Satellites

Lomonosov

Computed Values

T(α₀) Bounce Factor 1.0802

Phase Space Density f 2.34e-10

Differential Flux j 4.56e+04

D_L*L* (Kp-driven) 1.0e-6

Kp(t) Current 3.0

Magnetopause L_mp 10.2 R_E

Last Closed Drift Shell 7.8 R_E

Bounce Loss Cone α_BLC 3.2°

Drift Loss Cone α_DLC 5.1°

Particle Status Trapped

Governing Equations

Eq.1 — Generalized Diffusion Equation

∂f/∂t = ∂/∂Jᵢ (Dᵢⱼ ∂f/∂Jⱼ)
i,j = 1,2,3

Phase space density f evolves via diffusion in the three adiabatic invariants (J₁, J₂, J₃), corresponding to gyration, bounce, and azimuthal drift motion.

Eq.2 — Fokker-Planck (no mixed terms)

∂f/∂t = L*² ∂/∂L*|_μ,J (D_L*L* L*⁻² ∂f/∂L*|_μ,J)
+ 1/p² ∂/∂p|_y,L (p² ⟨D_pp(y,p)⟩_ba ∂f/∂p|_y,L)
+ 1/(T(y)·y) ∂/∂y|_p,L (T(y)·y·⟨D_yy(y,p)⟩_ba ∂f/∂y|_p,L)
− f/τ

Separates radial diffusion (L*), momentum diffusion (p), and pitch-angle diffusion (y = sin α₀) with bounce-averaged coefficients. D_L*L* is now time-dependent via Kp(t).

Eq.3 — Fokker-Planck with Mixed Terms

∂f/∂t = L*² ∂/∂L*|_μ,J (D_L*L* L*⁻² ∂f/∂L*|_μ,J)
+ 1/p² ∂/∂p|_α₀,L p²(⟨D_pp⟩ ∂f/∂p + ⟨D_pα₀⟩ ∂f/∂α₀)
+ 1/(T(α₀)sin2α₀) ∂/∂α₀|_p,L T(α₀)sin2α₀
·(⟨D_α₀α₀⟩ ∂f/∂α₀ + ⟨D_α₀p⟩ ∂f/∂p) − f/τ

D_α₀p is the mixed energy–pitch angle diffusion coefficient. Cross-diffusion terms couple momentum and pitch-angle evolution.

Bounce Period Factor T(α₀)

T(α₀) = 1.3802 − 0.3198·(sin(α₀) + sin²(α₀))

Approximation for the normalized bounce period (Schulz & Lanzerotti 1974). Particles near the loss cone have T ≈ 1.38; those near 90° have T ≈ 0.74.

D_L*L*(Kp) — Brautigam & Albert 2000

D_L*L* = D₀ · 10^(0.506·Kp − 9.325) · L*¹⁰

Radial diffusion coefficient parameterized by Kp index and L-shell. Higher Kp dramatically increases radial transport, driving inward diffusion and energization during storms. Kp is updated every 3 hours in this simulation.

Radiation Belt Physics

Relativistic electrons (>500 keV) undergo three periodic motions: gyration (μ), bounce (J), and azimuthal drift (Φ = L*). ULF waves drive radial diffusion, chorus waves produce local acceleration, and magnetopause shadowing causes loss on the dayside.

Magnetic Field Notes

The local B-field determines energy from μ. Because electron spectra are steep, small ΔB errors produce large flux errors — especially at high L. At LEO, populations include Trapped, BLC, and DLC electrons requiring accurate magnetic mapping to classify.