Generator

The magnetic-pressure generator

The height at which the gas stops being in charge

Gas pressure and magnetic pressure through the solar atmosphere, against height above the photosphere, on a logarithmic vertical axis. The gas curve falls by about twelve orders of magnitude between the photosphere and the corona because it is held up by its own weight and the scale height is small; the field falls by four, because a flux tube can only spread. The two cross at 1.64 thousand kilometres, and that crossing is the boundary of two different subjects. Below it the field is carried by the gas and does what the convection tells it; above it the gas is carried by the field, which is why a corona has a shape at all, why it is structured into loops that outline no density gradient, and why a flare can release in minutes an energy the gas at that height could not store in a year. The field itself is invisible in both regimes — what is plotted is the pressure it exerts, inferred from a splitting measured below and from the shape of what the gas does above.

13 essays call magnetic-pressure. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Stars

The darkness a field pays for

A sunspot is not cool because something is missing. It is cool because a three-thousand-gauss field supplies part of the pressure that holds the column up, the gas therefore supplies less, and the level at which that gas becomes opaque sits some hundreds of kilometres deeper than the surface around 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.

The observed sky

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.

Two effects that are blind in opposite directions. The sensitivity of two magnetic diagnostics against field strength, on a logarithmic axis spanning five decades. The Zeeman effect measures the line-of-sight component and adds it up along the path, so a field tangled into 100 independent cells inside one resolution element averages down by a factor of 10 and reports almost nothing — which is exactly the situation in a chromosphere or a turbulent cloud. The Hanle effect is a different device altogether: a field precesses the atom between absorption and re-emission, so a scattering line's polarisation is rotated and reduced, and the amount depends on how far the precession gets in one radiative lifetime. That makes it sensitive around 1 gauss for a 100-nanosecond level, and — the useful part — it does not care about sign, so a tangled field does not cancel. Above the crossing at 27.7 gauss the Hanle signal has saturated and carries no strength information, and the Zeeman effect is the instrument. Neither is a measurement of the field; each is a measurement of what the field did to something else.

Starlight

Two instruments blind in opposite directions

The Zeeman effect measures the component of a field along the line of sight and adds it up, so a field tangled into a hundred cells reports a tenth of one cell's strength. The Hanle effect measures how far an atom precesses between absorbing and re-emitting, does not care about sign, and saturates just where the Zeeman effect becomes useful.

The observed sky

The flare that arrives from somewhere else

The solar wind blows radially outward and the Sun rotates, so the field frozen into the wind is wound into a spiral making forty-five degrees to the radius at the Earth. Energetic particles follow the field rather than the line of sight, which is why the flares that deliver particle storms are the ones on the western limb rather than the ones facing the planet.

Stars

A threshold with no free parameter in it

Divide a cloud's mass by the magnetic flux threading it. Gravity and the magnetic force both fall as the inverse square of the radius, so the ratio cannot change during a collapse — and the critical value that separates a cloud which must collapse from one that never can is one over two pi root G, a pure constant.

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.

The observed sky

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.

The observed sky

A substorm timed by webers and volts

The solar wind opens the Earth's magnetic field at a rate that can be quoted as a voltage — tens of kilovolts, which is webers of flux per second — and the tail closes it again. Half a gigaweber loaded at fifty kilovolts takes three hours, and three hours is how often substorms recur. The aurora's position is the open flux, its surges are the unloading, and above a certain driving the ionosphere starts to limit the circuit that drives it.

Plasma beta from the corona's base outward. The ratio β of gas pressure to magnetic pressure against distance from the Sun's centre, one to ten solar radii on logarithmic scales, for a static isothermal corona in a dipole field. quiet Sun, 5 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.41 and crosses one again at 3.46 solar radii; quiet Sun, 10 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.1 and crosses one again at 5.00 solar radii; active region, 30 G at the base, with 3·10⁹ electrons per cubic centimetre at 2.5 million kelvin: β starts at 0.058 and crosses one again at 2.97 solar radii. The gas pressure falls steeply at first and then levels off, because gravity weakens with distance; the field's pressure, falling as the sixth power, never levels. Every corona therefore has a second place where the gas takes over, and it sits at two to four solar radii — where the closed loops of the corona end and the streamers' tips begin.

Stars

A corona held by its field and opened by its gas

A thousand kilometres above the Sun's surface the gas pressure has fallen so far that the magnetic field takes charge, and the corona's loops are the shape the field gives it. But the gas pressure of a hot atmosphere stops falling a few solar radii out, while the field's keeps falling as the sixth power of distance. So there is a second place where the gas takes over — and it is where the loops end, the streamers begin, and the solar wind is let out.

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.

The observed sky

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.

Ions picked up from Mars and swept off on loops. The paths of a hydrogen and an oxygen ion created at rest just above the dayside of Mars and picked up by the solar wind, flowing left to right at 420 km/s through an interplanetary field of 3 nT perpendicular to the page. In the planet's frame each moves on a cycloid: it is accelerated by the wind's electric field, turns in the magnetic field, comes momentarily to rest and repeats, drifting downstream at the wind's speed and reaching twice it at the top of each loop — four times the wind's energy per unit mass. The loop height is twice the gyroradius: 2923 km for hydrogen, 46774 km for oxygen, against a planetary radius of 3390 km. The oxygen is thrown clear of the planet in a single plume to one side, which is how a planet without a magnetic field loses its heavy atmosphere to the wind.

The observed sky

A planet that borrows the wind's field

Venus and Mars have no magnetic field of their own, yet the solar wind does not reach their atmospheres. It is stopped by their ionospheres, whose thermal pressure balances the wind's at an altitude that moves logarithmically with every gust, and the interplanetary field piles up and drapes around the obstacle to make a magnetosphere on loan. The arrangement holds only while the ionosphere out-pushes the wind, and it leaks heavy ions on loops wider than the planet.

Whether a field helped, depending on what limits the escape. The ratio of the atmospheric loss of a magnetised Earth to that of the same Earth without a field, against the Sun's age, both on logarithmic scales, set at 2 today — the magnetised planet losing more — for two ways an unmagnetised planet's loss can scale. If both planets lose atmosphere in proportion to the solar-wind power they intercept (blue), the magnetosphere's cross-section shrinks as the young Sun's denser wind compresses it while the bare planet's does not, and the ratio falls to 0.46 at 0.7 billion years: the field protected the young Earth, on balance from before 1.9 billion years. If instead the bare planet's loss is set by the ions the Sun's ultraviolet makes above it (orange), the ultraviolet rises more slowly into the past than the wind's pressure does, and the ratio grows to 3.7: the field cost the young Earth more than it saves today. The answer to whether a magnetic field protects an atmosphere is not a property of fields; it is a statement about which process limits the escape, and the young Sun amplifies the difference.

The observed sky

A shield that depends on what limits the leak

Today the Earth, with its magnetic field, loses ions to space about as fast as Venus and Mars without one — a little faster, by most measures. Whether that was also true under the young Sun, whose wind pressed eighty times harder, depends on something no measurement of the present can settle. If escape is limited by the wind's power, the field protected the young Earth; if it is limited by the ions the Sun's ultraviolet makes, the field cost it more than it does now. The young Sun does not decide the question. It amplifies whichever answer is right.

The observed sky

Two exponents a solar cycle cannot pull apart

Whether a magnetic field protected the early Earth's air depends on what limits the escape of ions from a planet without one — the ultraviolet that makes them or the solar wind that carries them off. Measured at Mars through a solar cycle, the two causes rise and fall together, and the fit that tries to separate them leaves each exponent uncertain by half its value. Remarkably, the young Sun's loss is predicted well anyway. What spoils the prediction is not the confusion but a switch between the two limits that the present-day data cannot see coming.

Elements sorted by how easily they lose an electron. The abundance of each element in the solar wind relative to the photosphere, normalised to oxygen, against its first ionisation potential, logarithmic: in the slow wind (filled) and the fast wind from coronal holes (open). Representative values. Elements easy to ionise — K, Na, Ca, Mg, Fe, Si, all below about 9 eV — are about three times over-abundant in the slow wind and about one and a half in the fast; those harder to ionise sit near one. The step comes at about 10 eV, which is the energy of a photon from the Lyman-α line of hydrogen: the sorting happens where hydrogen's own light sets the ionisation.

Stars

A wind that carries the ledger of its loops

The slow solar wind is richer in iron, magnesium and silicon than the Sun's surface is — about three times richer, relative to oxygen — and the elements that are enriched are exactly the ones that lose an electron easily. That sorting happens low in the chromosphere, inside closed magnetic loops, and it takes days. The fast wind from coronal holes barely shows it. So the composition of the wind arriving at the Earth is a record of where it was held, and for how long, before the field let it go.

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