Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

A belt emptied over one ocean

The Earth's magnetic field is not centred on the Earth. Its dipole sits five or six hundred kilometres towards the western Pacific, so the opposite side of the planet — the South Atlantic — has the weakest field at any altitude, and a trapped particle that mirrors safely above the atmosphere everywhere else dips into it there. The inner edge of the inner radiation belt is drawn by that one region, and it is moving.

Assumes Radiation belts, Magnetosphere and Atmospheric drag.

14 min read 6 figures Small perturbations, large consequencesScales that defeat intuitionMeasuring the unreachable

A charged particle trapped in the Earth’s magnetic field gyrates, bounces and drifts, and the bounce is a reflection: as the particle follows a field line towards either pole the field strengthens, and the particle’s magnetic moment — its first adiabatic invariant — forces its motion along the line to slow and reverse. It turns back where the field reaches a strength fixed by its pitch angle at the equator. If that strength is only reached inside the atmosphere, the particle does not turn back; it collides and is lost. The set of pitch angles for which that happens is the loss cone.

For a perfectly centred dipole that is a statement about latitude alone, the same at every longitude, and the loss cone is a single number for each field line. The Earth’s field is not a centred dipole. Its best-fitting dipole is displaced from the planet’s centre by about 570 kilometres, towards the western Pacific, and tilted by about ten degrees. On the far side of the planet from the displaced centre the field at any given altitude is weaker than anywhere else, and weaker field means mirror points closer to the ground.

The weak spot in the field. Contours of the strength of the Earth's magnetic field at the surface, in microtesla, over longitude and latitude, from an eccentric dipole: a dipole tilted to pass through the geomagnetic poles and displaced about 570 km from the Earth's centre towards the western Pacific. The field is strongest near the poles and weakest near the equator, as any dipole's is, but the displacement makes the side of the Earth furthest from the dipole's centre weaker still: the minimum, 22.9 µT, lies at 8° S, 40° W, over the South Atlantic. The measured anomaly sits further south and west, near 26° S and 55° W, and is slightly deeper, because the real field has higher harmonics that no dipole carries; the eccentric dipole gets the side of the planet and the size right and the exact place approximately. Trapped particles mirror where the field reaches a fixed strength, so wherever the field is weak they mirror closer to the ground — and over this region the inner belt reaches down into the upper atmosphere.
Fig. 1 Contours of the surface magnetic field strength from an eccentric dipole — tilted through the geomagnetic poles and displaced about 570 kilometres towards the western Pacific. The weakest field, inside the innermost contours, lies over the South Atlantic, on the side of the Earth furthest from the dipole’s centre.

Found by counters that went silent

The belts were discovered by accident, and the anomaly with them. The Geiger counter carried by the first American satellite, in 1958, recorded cosmic rays at the expected rate at low altitude and then, over parts of each orbit, stopped counting altogether. The silence was not an absence of particles but a flood of them: the counter was saturated. The follow-up missions, with instruments designed to cope, mapped a region of intense trapped radiation whose lower boundary was far lower on some longitudes than others, and the low boundary sat over the South Atlantic. The asymmetry was part of the belts’ first map, before anyone had a model of why.

A later measurement made the particles’ lifetimes visible directly. A high-altitude nuclear test in 1962 injected electrons onto drift shells near L = 1.2 to 1.5, creating an artificial belt far more intense than the natural one at those shells. Several satellites were disabled by it within months. Its decay was followed for years: electrons on the lowest shells, whose drift loss cones are widest, disappeared within weeks to months, while those a little higher persisted for years. The pattern of lifetimes against L is, in effect, a measurement of how the loss cones widen towards the Earth, taken with a tracer nobody would inject today.

A dipole that is not in the middle

Fitting the Earth’s measured field with a single dipole placed at the Earth’s centre captures about ninety per cent of it. Allowing the dipole to move away from the centre captures more, and the best position is displaced several hundred kilometres towards a point near 22° N, 141° E, in the western Pacific. The displacement is not a physical object — the field is generated by flows in the liquid outer core, and a dipole is only its largest harmonic — but it is a compact description of the field’s biggest departure from symmetry, and it accounts for the feature that matters most to spacecraft.

The field of a dipole weakens as the inverse cube of the distance from its centre. A point on the Earth’s surface on the Pacific side is five or six hundred kilometres nearer the dipole’s centre than the centre of the Earth is; a point on the opposite side is the same amount further away. Near the equator, where the dipole field is weakest anyway, the far side’s extra distance lowers the surface field from about 30 microtesla to about 23 in this model, and in reality to about 22. That region of weak field is the South Atlantic Anomaly.

The measured anomaly is not exactly where the eccentric dipole puts it. The real field has higher harmonics that pull the minimum further south and west, towards 26° S and 55° W, over southern Brazil and the adjacent ocean, and make it a little deeper. The dipole model gets the side of the planet and the size of the effect right, which is what the arguments below need.

The weak spot in the field, 500 km up. Contours of the strength of the Earth's magnetic field at an altitude of 500 km, in microtesla, over longitude and latitude, from an eccentric dipole: a dipole tilted to pass through the geomagnetic poles and displaced about 570 km from the Earth's centre towards the western Pacific. The field is strongest near the poles and weakest near the equator, as any dipole's is, but the displacement makes the side of the Earth furthest from the dipole's centre weaker still: the minimum, 18.6 µT, lies at 8° S, 40° W, over the South Atlantic. The measured anomaly sits further south and west, near 26° S and 55° W, and is slightly deeper, because the real field has higher harmonics that no dipole carries; the eccentric dipole gets the side of the planet and the size right and the exact place approximately. Trapped particles mirror where the field reaches a fixed strength, so wherever the field is weak they mirror closer to the ground — and over this region the inner belt reaches down into the upper atmosphere.
Fig. 2 The same field at an altitude of 500 kilometres, where low-orbiting spacecraft fly. The pattern is the same — weakest over the South Atlantic — and the whole field is weaker by the cube of the extra distance. What matters for trapped particles is the contrast between the anomaly and the rest of the sphere at a given altitude, and the displacement preserves it at every height.

A mirror point that dips on one side of the world

Where a particle turns round, and why it is not the same for all of them. A dipole field line at L = 1.5, with the path of a particle whose equatorial pitch angle is 45°. The first adiabatic invariant fixes the ratio of the perpendicular energy to the field strength, so as the particle moves toward the pole and the field strengthens, its motion tips further into the perpendicular — and where the field has grown by 1/sin²α, which here is a factor of 2.00, the parallel motion has gone entirely and it turns round. That happens at 23.1° of latitude. A particle with a smaller pitch angle mirrors deeper, and below 28.1° it would mirror inside the atmosphere and be absorbed instead — the loss cone, which at this shell is a narrow 28.1-degree hole in an otherwise full distribution. The gyration drawn along the path is the real one: the radius falls as the field rises, which is why the helix tightens toward each end.
Fig. 3 The bounce in a centred dipole, for comparison: a particle on shell L = 1.5 with a 45° equatorial pitch angle mirrors where the field has doubled, at 23° of magnetic latitude, and the loss cone on this shell is 28° wide. In a centred dipole those numbers are the same at every longitude; the rest of this essay is about what happens when they are not.

A trapped particle drifts around the Earth on a drift shell, labelled by L, the distance in Earth radii at which its field line crosses the magnetic equator. Its mirror points lie at a fixed field strength, which in the dipole’s own frame is at a fixed dipole latitude and a fixed distance from the dipole’s centre. Because that centre is displaced, the same set of mirror points is at different heights above the Earth’s surface at different longitudes.

Where a particle on shell L = 1.3 turns back, all the way round. The altitude of the northern and southern mirror points of a particle with an equatorial pitch angle of 45° on the drift shell L = 1.3, as it drifts once around the Earth, against the geographic longitude of the mirror point, in the eccentric-dipole field. In a centred dipole both would sit at one altitude all the way round. The displacement lifts them over the western Pacific, to 1197 km, and lowers them over the South Atlantic, to 71 km at 38° W — the southern mirror point dips furthest, because the anomaly lies in the southern hemisphere. The dashed line is 100 km, below which the atmosphere is dense enough to stop the particle in a single pass. A particle whose mirror point is safely above it for most of its drift can still be lost the moment its drift carries it over the anomaly.
Fig. 4 The altitude of the northern and southern mirror points of a particle on shell L = 1.3 with an equatorial pitch angle of 45°, through one drift around the Earth, against the longitude of the mirror point. Over the western Pacific they sit near 1,200 kilometres; over the South Atlantic the southern one dips to about 70, below the 100-kilometre level at which the atmosphere stops a particle in one pass.

The particle in the figure is trapped comfortably over most of the planet. Its mirror points are hundreds of kilometres up, in air so thin that a particle can bounce through it for years. But its drift carries it around the Earth in minutes to hours, depending on its energy, and on every circuit it passes over the South Atlantic, where its southern mirror point descends to about seventy kilometres. There it collides with the atmosphere and is lost. The anomaly acts as a drain on every drift shell that passes low enough over it.

Where a particle on shell L = 1.2 turns back, all the way round. The altitude of the northern and southern mirror points of a particle with an equatorial pitch angle of 60° on the drift shell L = 1.2, as it drifts once around the Earth, against the geographic longitude of the mirror point, in the eccentric-dipole field. In a centred dipole both would sit at one altitude all the way round. The displacement lifts them over the western Pacific, to 1352 km, and lowers them over the South Atlantic, to 212 km at 39° W — the southern mirror point dips furthest, because the anomaly lies in the southern hemisphere. The dashed line is 100 km, below which the atmosphere is dense enough to stop the particle in a single pass. A particle whose mirror point is safely above it for most of its drift can still be lost the moment its drift carries it over the anomaly.
Fig. 5 A particle on a lower shell, L = 1.2, but with a larger pitch angle of 60°, so that it mirrors nearer the equator. Its mirror points dip to about 210 kilometres over the South Atlantic and never reach the dense atmosphere; it survives. On the same shell, particles with smaller pitch angles do not.

Whether a given particle survives depends on both its shell and its pitch angle. A particle on a low shell with a large pitch angle mirrors near the magnetic equator, high above the ground everywhere, and survives even the anomaly. One on the same shell with a smaller pitch angle travels further along its field line before it mirrors, reaches lower, and is removed.

Two loss cones

The displacement therefore splits the single loss cone of a centred dipole into two.

Two loss cones, and the band between them emptied by one region. The equatorial pitch angle below which a trapped particle is lost to the atmosphere, against drift shell L, in the eccentric-dipole field. The lower curve is the bounce loss cone at the longitude where the field lines stand highest: particles below it hit the atmosphere within one bounce, wherever they are. The upper curve is the drift loss cone: particles below it are safe on some longitudes but hit the atmosphere when their drift carries them over the South Atlantic Anomaly, within one drift period. At L = 1.2 the two are 38° and 57°; at L = 2 the two are 15° and 19°; at L = 4 the two are 5° and 6°. The shaded band between them is the population that one region of the planet removes: at low L it is wide, which is why the inner edge of the inner belt is set by the anomaly rather than by the atmosphere in general, and why low-orbiting spacecraft receive most of their radiation dose in the few minutes of each orbit spent crossing it.
Fig. 6 The equatorial pitch angle below which particles are lost, against drift shell L. The lower curve is the bounce loss cone at the longitude where field lines stand highest; the upper curve is the drift loss cone, set by the South Atlantic. At L = 1.2 they are 38° and 57°; at L = 2, 15° and 19°; at L = 4 they have nearly merged.

The bounce loss cone is the local one: particles with pitch angles inside it reach the atmosphere within a single bounce wherever they are, in less than a second. The drift loss cone is the global one: particles with pitch angles inside it are safe at most longitudes but reach the atmosphere somewhere on their drift, and are lost within one drift period. The band between the two is populated only transiently — particles scattered into it survive until their drift next carries them over the South Atlantic — and in steady state it is nearly empty.

At high L the two cones nearly coincide, because far from the Earth the displacement of the dipole is small compared with the size of the field line, and the mirror altitudes vary little around the drift. At low L the band is wide: at L = 1.2 the drift loss cone is nineteen degrees wider than the bounce loss cone. The inner edge of the inner belt is therefore not where a centred dipole’s field lines would start to touch the atmosphere; it is where the anomaly’s field lines do, and the anomaly’s are the lowest. The bottom of the inner belt, at a few hundred kilometres over the South Atlantic and more than a thousand elsewhere, is drawn by one region of the planet.

What a spacecraft in low orbit sees

For a spacecraft in low Earth orbit, the consequence is that almost all of its radiation dose arrives in a few places. At four or five hundred kilometres, the altitudes of the space station and of most Earth-observing satellites, the orbit passes below the inner belt almost everywhere — except over the South Atlantic, where the belt reaches down to meet it. A spacecraft in a low, moderately inclined orbit crosses the anomaly on several of its fifteen or sixteen daily orbits, for a few minutes each time, and receives most of its trapped-proton dose in those minutes.

The protons of the inner belt have energies of tens of megaelectronvolts. They pass through thin shielding, deposit charge in electronics, and occasionally flip a bit in memory or trigger a false signal in a detector. Maps of such single-event upsets recorded by satellites in low orbit are maps of the anomaly: a cluster over South America and the South Atlantic, and almost nothing elsewhere at low latitude. Many instruments are switched off or put into a safe mode for the crossing — space telescopes in low orbit schedule their observations around it, losing a substantial fraction of their time to it — and astronauts’ dose on the space station comes largely from it.

The protons get there by a route that the loss-cone picture makes clear. They are made in the inner belt by cosmic rays striking the upper atmosphere, and they fill the belt down to its inner edge; their numbers are set by a balance between that source and the losses, which at the bottom of the belt are the anomaly’s. Below the drift loss cone at each shell the population is empty; above it, the particles have lifetimes of years, because nothing but the thin atmosphere at their mirror points slows them down. The high flux at low altitude over the anomaly is the flux of long-lived particles caught at the bottom of their bounce, where the magnetic geometry has brought them lowest.

Orbits chosen around a region

The anomaly shapes mission design as well as operations. An orbit’s exposure to it depends on its altitude and its inclination: an equatorial orbit at low altitude avoids it almost entirely, because the anomaly’s centre lies south of the equator; an orbit inclined by thirty to fifty degrees crosses it several times a day; a polar, sun-synchronous orbit crosses it on a few passes a day and the polar horns of the outer belt on every pass. Altitude matters most of all. Below about 400 kilometres the anomaly’s belt is thin, and a spacecraft there is shielded by the anomaly’s own weakness being higher than its orbit — at the price of the drag that thicker air brings. Above 600 or 700 kilometres the dose rises steeply as the orbit enters the belt over a wider region. The altitudes at which low-orbit spacecraft fly are a compromise between drag below and radiation above, and the radiation side of the compromise is set by one ocean.

An anomaly that drifts and deepens

The geomagnetic field changes, and the anomaly changes with it. The dipole has weakened by about nine per cent since the first measurements of its strength in the 1830s, and the eccentric dipole’s centre has been moving. The anomaly has correspondingly drifted westward, by about a third of a degree a year, and deepened. In recent decades it has also begun to develop two separate minima rather than one, a sign that the higher harmonics of the field — not captured by any dipole — are changing faster than the dipole itself.

For the radiation belts, a weaker field means that the belts reach lower everywhere and the anomaly lower still. The inner edge of the inner belt over the South Atlantic has moved down, and the region of the Earth’s surface below which low-orbiting spacecraft encounter the belt has grown. The effect on a spacecraft’s dose over a decade is measurable, and radiation models used for spacecraft design include the field’s secular variation for exactly that reason.

It also changes how a spacecraft uses the field for its own control. A satellite that brakes its tumbling or unloads its momentum wheels by pushing against the geomagnetic field has less to push against over the anomaly, and the same weakness that lowers the belt reduces the torque available there. The field is a single object with several consequences, and the anomaly is where all of them are most extreme.

Where the lost particles go

A particle scattered into the loss cone is not simply removed; it deposits its energy in the upper atmosphere. At high latitudes the particles precipitating from the outer magnetosphere make the aurora, whose position is set by which field lines are open to the solar wind. Over the South Atlantic the precipitation is from the inner belt, much fainter and at much lower latitude, and it has measurable effects on the ionosphere above the region: enhanced ionisation at night, and a population of energetic particles that ionospheric models have to include as a separate source. The anomaly is where the inner belt meets the atmosphere, and the atmosphere records the meeting.

The same geometry on other planets

The size of the effect is set by how far a planet’s dipole is displaced relative to its own radius, and on the Earth the displacement is a tenth of a radius. Other planets show both smaller and much larger cases. Saturn’s field is almost perfectly aligned with its rotation axis and almost exactly centred, and its radiation belts have no counterpart to the anomaly; their inner edge is set instead by the planet’s rings, which absorb particles on the drift shells that cross them. Jupiter’s field is tilted by about ten degrees and slightly offset, and its belts wobble with the planet’s rotation.

Uranus and Neptune are the extreme cases. Their dipoles are tilted by fifty to sixty degrees from their rotation axes and displaced by a third to a half of a planetary radius from their centres, so their fields at the surface vary by factors of ten from one side to the other. On such a planet the difference between the bounce loss cone and the drift loss cone is not a correction but the dominant feature of the belts, and particles trapped on one side are lost into the atmosphere on the other within a single drift. The spacecraft that flew past both planets found radiation belts much weaker than the planets’ field strengths alone would suggest, and the offset geometry is part of the reason: a field that dips into the atmosphere somewhere on every drift shell is a field that cannot hold particles for long.

What the model leaves out

The figures use an eccentric dipole, which reproduces the anomaly’s existence and its size approximately and misplaces its centre by twenty degrees or so. Real radiation-belt models trace particles through the full measured field, with its higher harmonics, and through the external field of the magnetosphere, which compresses the field on the dayside and stretches it into a tail on the nightside. At the low L of the inner belt the external field matters little; at higher L it dominates, and the drift shells are not the tidy surfaces of a dipole but distorted ones that split according to pitch angle.

They also take the atmosphere as a sharp boundary at a hundred kilometres. The real atmosphere thins gradually, and a particle mirroring at two hundred kilometres loses energy slowly on every bounce; its lifetime depends on the density at its mirror altitude, which changes by an order of magnitude with the solar cycle as the Sun’s ultraviolet heats and expands the upper atmosphere. The inner edge of the belt therefore breathes with the solar cycle, rising when the atmosphere is expanded at solar maximum and falling at solar minimum — which is why the dose at a given altitude is lower at solar maximum, the opposite of what a spacecraft’s exposure to solar storms would suggest.

Still open: a belt shaped by a field that is reorganising

The geomagnetic field has weakened steadily for as long as it has been measured, and the development of a second minimum within the anomaly shows its structure changing on timescales of decades. Whether the weakening continues — and whether it is the start of a larger reorganisation of the field, or a fluctuation of the kind the palaeomagnetic record shows happening often without leading to a reversal — is not known. For the radiation belts the question is practical. A field that keeps weakening lowers the inner belt towards the altitudes where most spacecraft fly, and the magnetosphere that shields the Earth from the solar wind shrinks with it. The anomaly over the South Atlantic is where those changes are felt first, and its drift and deepening are the most sensitive monitor of the field that anyone has.

Where this goes next

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. A tumble removed with a coil and a compass, at 500 km. The rotation rate of a small spacecraft — principal moments 0.0067, 0.041, 0.043 kg m², the proportions of a three-unit cubesat — tumbling at 8.8° a second after release, against orbits at 500 km, with nothing to control it but magnetic coils driven by the B-dot law: a dipole opposite to the rate of change of the field measured aboard, capped at 0.2 A m². The field is a dipole tilted 9.2° from the Earth's axis and turning with the Earth. In a polar, 97.4°, orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.58 orbits; in a 51.6° orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.32 orbits; in an equatorial orbit the rate settles at 2.44° a second over the last orbit drawn. The law needs no knowledge of the spacecraft's attitude: a tumbling body sees the Earth's field swing round in its own frame, and a dipole opposing that swing produces a torque that removes the part of the spin perpendicular to the field. The polar orbit does not reach zero. It settles at 0.94 of twice the orbital rate, 0.13° a second, and twice the orbital rate is how fast the field direction itself turns round a polar orbit: a body turning with the field sees little change to oppose. An equatorial orbit keeps the field pointing nearly the same way all the way round, so the spin about it is reached only through the dipole's tilt and the Earth's turning, and 28 per cent of the starting rate is still there at the end. A tumble stopped by the field it tumbles through A small satellite leaves its deployer tumbling, and the first thing most of them do is stop, using nothing but a magnetometer and three coils. The law they run needs no idea where the satellite is pointing. What it cannot do, at any instant, is touch the spin about the local field line — so how much tumble survives is decided by how much the field's direction changes along the orbit, and the stillness it reaches is defined by the field rather than by the stars. The shield that does not shield. Above: measured ion escape rates for three planets against their surface magnetic field. Venus and Mars have no dynamo at all and Earth has one, and the three rates lie within a factor of 9 — with the magnetised planet losing the most. The intuition that a magnetosphere protects an atmosphere is not a small correction away from being right; the measurement does not support it. Below: why. A dipole's field lines are not all closed. Those emerging within a polar cap reconnect with the wind's own field and lead straight to space, and the cap's area is set by how far the magnetosphere reaches — a boundary at ten planetary radii still leaves 5.1 per cent of the surface open. So a magnetosphere is both a shield and a funnel: it deflects the wind from most of the planet and collects ions from the whole ionosphere into the polar wind, which is exactly what an instrument above the poles measures leaving. Whether the net is protection depends on quantities nobody can compute from the field strength alone, and the three points above are the state of the evidence. The shield that is also a funnel A magnetic field is supposed to protect an atmosphere from the stellar wind. Venus and Mars have no dynamo and Earth has one, and their measured ion escape rates lie within a factor of a few — with the magnetised planet losing the most, because a dipole's polar field lines are open and lead straight to space.

The objects this essay names

Each one links to every other essay that touches it.

Drift loss coneEccentric dipoleGeomagnetic fieldLoss coneMagnetic momentMirror pointPitch angleRadiation beltsSingle event upsetSouth atlantic anomaly