Astronomy & Celestial Mechanics
the sky, computed
Concept

Geomagnetic storm — where it appears

A worldwide depression of the Earth's surface magnetic field lasting a day or more. It is caused by a strengthened ring current, after the solar wind couples energy into the magnetosphere through reconnection. Coupling depends on the wind's field direction rather than its pressure, which is why the warning time is short.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

A boundary that a sixty-fourfold gust moves by a factor of two. The standoff distance of a dipole magnetosphere against a wind, in planetary radii, against the wind's proton density, for three wind speeds. The boundary sits where magnetic pressure equals ram pressure — the only balance available, since the field exerts no force on neutral gas and the wind touches nothing. Because a dipole falls as the cube of distance its pressure falls as the sixth power, so the standoff goes as the ram pressure to the −1/6: at 450 km/s a density of 6 per cubic centimetre puts the nose at 9.3 radii — against a measured average of ten to eleven — and a sixty-fourfold compression moves it only to 4.7. That exponent is the reason a magnetosphere is a stable thing to have. It is also the reason the boundary's position is a poor measurement of the field: a factor of two in the distance is a factor of sixty-four in what caused it, so the standoff measures the wind badly and the planet's moment hardly at all.

A boundary that hardly moves

The magnetopause sits where the planet's magnetic pressure equals the solar wind's ram pressure. Because a dipole falls as the cube of distance, its pressure falls as the sixth power — so a sixty-fourfold gust in the wind moves the boundary by a factor of two, and a boundary that will not move is what makes a magnetosphere a stable thing to have.

sky · Magnetosphere Three motions, three decades apart, and that is the point. The three periods of a trapped 1-MeV electron's motion against the shell it is trapped on, on a logarithmic time axis. It gyrates about a field line in 0.22 milliseconds, bounces between its mirror points in 0.33 seconds, and drifts right round the planet in 16 minutes — ratios of 1498 and 3015 at L = 4. Each motion carries a conserved quantity: the magnetic moment, the longitudinal invariant, and the magnetic flux the drift shell encloses. The separation is what makes them conserved. An invariant survives anything that changes slowly compared with its own period, so a disturbance lasting minutes destroys the third and leaves the first two untouched — and a particle that keeps its magnetic moment while being moved inward to a stronger field must gain energy. That is not a loophole; it is how the belts are filled.

Three clocks and nothing to fall onto

A charged particle in a dipole field gyrates, bounces and drifts, on timescales a millisecond, a second and a quarter of an hour. The three periods are three decades apart, and that separation is not a curiosity — it is the reason each motion has a conserved quantity, and the reason a magnetic storm can accelerate particles rather than merely stir them.

spaceflight · Radiation belts A density that spans a factor of 25 at 400 kilometres. Thermospheric density against altitude, for three levels of solar activity, with the model's own uncertainty band drawn around the middle curve. The extreme ultraviolet output of the Sun heats the upper atmosphere, so the scale height rises with activity and the density at a fixed altitude rises with it — by a factor of 25 at 400 kilometres between solar minimum and maximum. Superposed on that are a diurnal bulge of about a factor of two, semiannual variations, and geomagnetic storms that raise the density by tens of per cent within hours. The best empirical models reproduce past conditions to about 15 per cent, and orbital lifetime is inversely proportional to density, so a re-entry predicted a year ahead carries that error and the far larger one of not knowing what the Sun will do.

A density model wrong by a factor of two

Everything about a low orbit's future depends on the density of the air at four hundred kilometres, and that density varies by a factor of twenty-five over the solar cycle, by two within a day, and by tens of per cent during a storm nobody predicted. Every model of it is an empirical fit, and re-entry dates are quoted with the honesty that implies.

spaceflight · Atmospheric drag A factor of 100 in the wind, 8 degrees in the aurora. The latitude of the last closed field line against the solar wind's dynamic pressure, for a dipole of 0.31 gauss at the equator. The chain is short and every link is exact. Pressure balance puts the magnetopause at a distance going as the sixth root of the field pressure over the ram pressure; a dipole line reaching equatorial distance L returns to the surface at colatitude arcsin(1/√L); so the polar cap's edge is that, and the aurora sits just equatorward of it on the outermost closed lines. Across the 100-fold range of pressure drawn — which covers everything from a quiet wind to a severe storm — the magnetopause moves from 13.1 to 6.1 planetary radii and the oval from 74.0 to 66.1 degrees. The shaded band is where the oval is actually observed. The open magnetic flux is exactly the reciprocal of the standoff distance, so it rises by a factor of 2.15 over the same range — which is the quantity that actually matters for a storm, and it is the only one of the three that changes by much.

An aurora that is a sixth root inside an arcsine

The magnetopause distance fixes which field lines are open, the open ones map to a cap around each pole, and the cap's edge is where the aurora is. A hundredfold change in the solar wind moves that edge by eight degrees — and across planets whose fields differ by four orders of magnitude it moves by ten.

sky · Magnetosphere Two clocks for a radiation-belt electron, at Kp 2. Two times against drift shell L, on a logarithmic scale, for electrons of about 1 MeV. The first is how long radial diffusion takes to move an electron by one Earth radius, the inverse of an empirical diffusion coefficient that grows as the tenth power of L — at Kp 2, 550 years at L = 2, 10 years at L = 3, 21 days at L = 5. The second is how long the electron survives before it is scattered into the loss cone. Inside the plasmapause, here at L = 4.7, that is set by plasmaspheric hiss: years at L = 1.3, 14 days at L = 2.5, 5.2 days at L = 3; outside it a single lifetime of 30 days stands for every other loss. Inside L = 4.8 the electron is lost before diffusion can carry it further in, which is the shaded region; outside, diffusion wins and the belt is fed. The two curves cross at a single place, and that crossing is the inner edge of the outer belt.

The gap between the belts is a race between two clocks

Electrons reach the inner magnetosphere by diffusing inward from the outer edge, and they are lost there to waves that scatter them into the atmosphere. The diffusion slows as the tenth power of distance and the loss does not, so the two clocks cross at one sharp place. Inside it lies the slot, and at megaelectronvolt energies the inner electron belt is not a steady population at all but what the last great storm left behind.

spaceflight · Radiation belts Phase-space density during a storm fed from inside. Electron phase-space density at fixed first and second adiabatic invariants, against the drift-shell parameter L, before a two-day storm (grey) and 0.5, 1, 2 days after it begins, from a model of radial diffusion with a diffusion rate rising as L¹⁰ and a loss lifetime of 4 days. The boundary is held fixed and a source centred at L = 4.5, standing for acceleration by whistler-mode chorus waves, acts for the storm's two days; the density grows a peak at L = 4.5, well inside the boundary, and diffusion then spreads it both inward and outward. A peak in phase-space density inside the outer boundary cannot be produced by diffusion from outside while the boundary is rising — diffusion only smooths gradients — so its presence then is the signature of local acceleration, and its absence is consistent with transport.

A peak that says where the electrons were made

The outer radiation belt refills with electrons of millions of electronvolts within hours of a storm. Whether they were carried inward from the edge of the magnetosphere or accelerated where they are found shows in one place only — the density of electrons in the coordinates their motion conserves, which a detector does not measure directly.

spaceflight · Radiation belts A storm's depression at the equator, as an energy. The depression of the horizontal magnetic field at the Earth's surface against the total kinetic energy of the particles trapped in the ring current, on logarithmic axes. The Dessler–Parker–Sckopke theorem makes the two proportional — ΔB/B₀ = −⅔ E/Eₘ, with Eₘ = 8.4·10¹⁷ joules the energy of the Earth's dipole field above the surface — whatever the particles' energies, pitch angles or distribution in radius. The marked storms are read off the line: a moderate storm of 50 nT is 2·10¹⁵ J of trapped particles, an intense one of 100 nT 4·10¹⁵ J, the storm of March 1989 at 589 nT 2.4·10¹⁶ J, and the lowest estimate for the Carrington storm of 1859 3.4·10¹⁶ J — a few hours of the solar wind's power flowing past the magnetosphere.

A storm weighed by a compass needle

During a geomagnetic storm the horizontal field at the equator drops by tens or hundreds of nanotesla, and a theorem says the drop is proportional to the total energy of the particles trapped around the Earth — whatever their energies, their pitch angles or their distances. A ground magnetometer therefore weighs a population tens of thousands of kilometres up, in joules, and the shape of the record says what that population is made of.

sky · Magnetosphere How far inside the door a day of compression reaches. The innermost drift shell at which phase-space density has halved after a day of compression, against where the compression put the last closed drift shell, for diffusion at Kp = 4, 5, 6, 7. With the door at L = 5 and Kp = 6 the loss reaches L = 3.8; at Kp = 4 it reaches only 4.3. The diagonal is the door itself: everything above the curves and below the diagonal was emptied without ever being opened. A strong storm both brings the door in and speeds the diffusion that drains the shells behind it, and the two together carry the loss a full Earth radius or more inside the shells that were opened.

A belt drained through its outer door

When a storm's solar wind squeezes the magnetopause inward, the outermost drift shells of the radiation belt are cut open and their electrons leave within minutes. The shells inside are untouched by the boundary and by the atmosphere alike, and they empty anyway: the opened region is a sink, and the same diffusion that fills the belt from outside carries electrons out to it. A day of compression drains the belt a full Earth radius inside the shells that were actually opened.

spaceflight · Radiation belts A dropout that partly comes back. The flux of 1 MeV electrons at L = 4.5, relative to before the storm, through a storm whose ring current deepens to Dst = −150 nT over eight hours and decays with a one-day e-folding. Dashed: the adiabatic part alone, with phase-space density unchanged — the flux falls to 18 per cent at the storm's peak and returns completely. Solid: the same with a real loss of 60 per cent of the phase-space density during the main phase, as compression and outward diffusion would cause. The flux falls to 7 per cent and recovers only to 39. Seen in flux at fixed energy, the two causes look the same while the storm is on, and only the recovery — days later — tells how much of the dropout was electrons lost and how much was the field.

A dropout that is only the field

During a storm the ring current weakens the field the radiation belt sits in, and every electron, keeping its adiabatic invariants, loses energy. A detector counting electrons at a fixed energy and place sees the flux fall by half or more — and not one electron has gone anywhere. When the ring current decays, the electrons regain their energy and the flux comes back. Only in the coordinates the electrons themselves conserve can a storm's real losses be told from the field's sleight of hand, and the bigger the storm, the larger the sleight.

spaceflight · Radiation belts

Named alongside it

The objects these essays reach for when they reach for this one.

Adiabatic invariantDrift shellRadiation beltsMagnetopauseMagnetosphereRadial diffusionSolar windMagnetopause shadowingPhase-space densityDipole fieldLoss coneMagnetic moment

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