Aurora and Magnetosphere (3D)

The 3D scene

The Earth and satellite are Blender models. The field lines, particles, magnetopause surface, solar-wind streaks and the auroral ovals (north and south) are drawn live from exactly the same model as the 2D page, so the numbers match. Oval colour and brightness come from the precipitated particles: energy sets the emission altitude and so the colour.

What you are looking at

Electrons or protons are injected on one magnetic L-shell of Earth's dipole field, near midnight (where the magnetotail releases energy). Each particle spirals along a field line, bounces between two mirror points, and drifts slowly around the planet. A particle whose mirror point would lie below 100 km altitude is not reflected: it hits the upper atmosphere, excites oxygen and nitrogen, and that is an auroral flash. The green dot is a test particle whose pitch angle you set; the red dots are particles inside the loss cone.

Dipole field and field lines

A field line of the dipole is r = L cos^2(lambda) (lambda = magnetic latitude, r in Earth radii). Along it the field strength is B(lambda) = (B0 / L^3) sqrt(1 + 3 sin^2 lambda) / cos^6 lambda with B0 = 31.2 microtesla at the surface equator. The invariant latitude where the line meets the ground is cos(Lambda) = 1/sqrt(L): L = 6 gives 65.9 degrees, the auroral oval; L = 2 gives 45 degrees.

Guiding-centre motion and the mirror force

While the field changes slowly over one gyration, the magnetic moment mu = p_perp^2 / (2 m B) is conserved. The mirror force is dv_par/dt = -(v_perp^2 / 2B) dB/ds with v_perp^2 = v^2 - v_par^2 and arc length ds = L RE cos(lambda) sqrt(1 + 3 sin^2 lambda) d lambda. The page integrates this with a fourth-order Runge-Kutta scheme. A particle with equatorial pitch angle alpha0 mirrors where B = B_eq / sin^2 alpha0.

Loss cone

Particles whose mirror field would exceed the field at 100 km altitude are lost: sin^2 alpha_lc = B_eq / B(100 km). At L = 6 this is 2.9 degrees. After one bounce the loss cone is empty, so the aurora needs a steady supply of fresh particles: that is the injection rate.

Scattering into the loss cone

An isotropic population loses only a tiny fraction per bounce, because the cone is a few degrees wide. In the real magnetosphere waves scatter particles in pitch angle and keep refilling the cone. The scattering slider is a stylised stand-in: at that rate per second a particle outside the cone is moved to a random pitch angle inside it (keeping its latitude and speed). Set it to zero to see how dark the sky would be without it.

Bounce and drift

Bounce period T_b = (4 L RE / v) T(alpha0) with T = integral cos(l) sqrt(1+3 sin^2 l) / sqrt(1 - sin^2 alpha0 B/B_eq) dl; Hamlin's fit T ~ 1.30 - 0.56 sin(alpha0) is shown in the readout and the exact integral is used for the check. The longitudinal drift (gradient and curvature) has period T_d = pi q B0 RE^2 / (3 L W (0.35 + 0.15 sin alpha0)): about 13 hours for a 10 keV electron at L = 6. Electrons drift east, protons west. The "drift time-compression" slider only speeds up this slow motion so you can see it; bounce motion always runs in real time times the time-speed slider.

Where the light comes from

The altitude where precipitating particles deposit their energy falls as the energy rises: roughly 240 km at 0.1 keV, 150 km at 1 keV, 104 km at 10 keV and 80 km at 100 keV for electrons. That altitude sets the colour: red (atomic oxygen, 630.0 nm) above about 200 km, green (557.7 nm) between about 105 and 200 km, blue-violet (N2+, 427.8 nm) below that. The altitude table is an approximation to published electron-transport results, not a transport calculation.

Magnetopause

The boundary of the magnetosphere uses the empirical Shue et al. (1997) shape r = r0 (2 / (1 + cos theta))^a with r0 = (10.22 + 1.29 tanh(0.184 (Bz + 8.14))) Pd^(-1/6.6) and a = (0.58 - 0.007 Bz)(1 + 0.024 ln Pd), where Pd = 1.6726e-6 n v^2 nPa is the solar-wind dynamic pressure (n in cm^-3, v in km/s). A stronger wind squeezes the nose inwards. The IMF Bz slider only affects the magnetopause size and, as a modelling assumption of this page rather than a fitted law, the injection rate (southward Bz multiplies it by 1 + |Bz|/5).

Numerical checks on this page

• The loss-cone angle from the formula is compared with the value found by bisection on the integrator (a particle either reaches 100 km or it does not).
• The RK4 bounce period is compared with a separate quadrature of the exact integral and with Hamlin's fit.
• The closed-form dipole field in latitude is compared with the vector formula B = B0 RE^3 (3 (m.r) r / r^2 - m) / r^3 at 80 points.
• A full Lorentz-force orbit is run with the Boris pusher (no guiding-centre assumption, hundreds of thousands of steps) and its magnetic moment, speed and bounce period are compared with the guiding-centre results.

What is simplified

The geomagnetic and rotation axes are the same, there is no electric field (no convection, no parallel potential drop), no pitch-angle scattering, no collisions inside the atmosphere, and the field is a pure dipole even where the real field is stretched into a tail. Real auroral electrons are accelerated along the field by a parallel electric field in the last few thousand kilometres, which this model leaves out. The model is a teaching tool for the motion and the geometry, not a space-weather forecast.

What to try

• Set L = 3.5 and a large Bz (Great storm): the oval moves to 57 degrees and the loss cone grows.
• Compare 0.5 keV (red, high altitude) with 80 keV (blue, low altitude).
• Raise the anisotropy: particles with large pitch angles never reach the atmosphere, so the oval fades and a trapped population builds up.
• Switch to protons and watch the drift reverse.