Astronomy & Celestial Mechanics
the sky, computed
Concept

Space-weather — where it appears

The state of the Sun's output and of the near-Earth plasma environment, and its effects on spacecraft and on the upper atmosphere. It drives thermospheric density by more than an order of magnitude, and it is monitored rather than predicted.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

A radial wind from a rotating star draws a spiral. The interplanetary field out to 5 astronomical units, drawn as the Archimedean spiral it is. Nothing here rotates: the plasma moves radially outward at 400 kilometres a second and the field is frozen into it, so each parcel remembers the longitude it left from and the pattern winds up while the material does not. The pitch angle is arctan(Ωr/v), which is 47° at one astronomical unit — the radius where the star's rotation has carried the footpoint through one radian in the time the wind takes to arrive. The practical consequence is a matter of hours: a flare's particles follow the field rather than the line of sight, so the ones that reach a given planet left a longitude about 61° to the west of it. A magnetically well-connected flare on the western limb delivers a particle storm and a larger one at disc centre does not, and the difference is this geometry rather than anything about the flare.

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.

sky · Stellar winds 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 Assimilation buys a factor of 3.6 at 2 hours and 1.03 at 14 days. The along-track position error of a low-orbit object against how far ahead the prediction reaches, on logarithmic axes. The upper curve uses a climatological density model, whose error stays at 15 per cent however long it is run — the limitation is the functional form and the proxies driving it rather than a shortage of data. The others assimilate the observed drag on objects already in orbit, which replaces that with an observation error of 3 per cent and then lets the thermosphere forget, with memories of 0.5, 1.5, 4 days. Every curve rises as the square of the time, because an error in a drag acceleration integrates twice into a position. The advantage is a factor of 3.6 at 2 hours and 1.03 at 14 days, so assimilation changes what a conjunction screening can do and changes nothing about a re-entry date — and the dashed line is the kilometre at which a close approach becomes a manoeuvre decision.

A weather forecast made out of orbits

A density model fitted to fifty years of satellite drag is a climatology, and its error does not shrink with more data. Updating it from the drag observed on objects in orbit right now is the manoeuvre a weather forecast makes — and it buys a factor of several for a day and nothing at all for a fortnight.

spaceflight · Atmospheric drag 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 a change of attitude moves a satellite before the encounter. The along-track shift a 5-kg satellite achieves by switching from 0.01 to 0.1 m² of frontal area when warned of a conjunction, against the warning time, at 400, 500, 600 km, between solar minimum (lower edge) and maximum (upper edge), logarithmic. The shift grows as the square of the time. In a day it is 26.1 to 156 km at 400 km; 3.9 to 39 km at 500 km; 0.6 to 10 km at 600 km. The dashed line is the 556 m needed to bring a 300-m position uncertainty's collision risk below one in ten thousand.

A dodge made of air

A small satellite with no thruster can still move along its orbit by turning broadside to the thin air, and the question is whether that is enough to get out of the way of debris. Below six hundred kilometres the answer, measured in distance, is easily yes: a day's warning buys kilometres when a few hundred metres would do. Measured in predictability it is no. The density of the air is forecast to about thirty per cent, so the dodge goes somewhere nobody, including the satellite's own operator, can say in advance to the precision the dodge was meant to achieve.

spaceflight · Rendezvous

Named alongside it

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

Thermospheric densityBallistic coefficientConjunction assessmentEmpirical modelGeomagnetic stormRe-entry predictionSolar windSystematic errorAlfven radiusAlong-track errorAtmospheric dragCharge exchange

All concepts