Astronomy & Celestial Mechanics
the sky, computed
Concept

Thermospheric density — where it appears

The density of the air a few hundred kilometres up, which decides every low orbit's future and cannot be computed. It varies by a factor of twenty-five over the solar cycle and by two within a day, and every model of it is a fit to past satellite drag.

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

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 The drag coefficient of a sphere runs from 2.03 to 2.79, and 2.2 is a convention. The free-molecular drag coefficient of a sphere against the accommodation coefficient — the fraction of striking molecules that thermalise with the surface and leave in a cosine distribution rather than bouncing — at speed ratios 2, 4, 8, with the surface at 0.3 times the flow's temperature. Specular reflection gives 2.469 at the lowest speed ratio drawn and 2.001 in the hypersonic limit, where every molecule delivers exactly twice its own momentum. Accommodation adds the re-emitted flux, which leaves at the wall temperature in a direction the flow did not choose, and it adds most where the speed ratio is smallest — which is high up, where the light species dominate. The conventional 2.2 lies outside this family at both ends: at s = 8 a sphere reaches only 2.112 even at full accommodation, and at s = 2 it is already 2.469 with none. That is not a defect of the arithmetic — 2.2 is a fitted average for satellite shapes, whose flat panels have a higher coefficient than a sphere of the same projected area, and the sphere is drawn because it is the one geometry with a closed form. What survives the shape is the dependence: a satellite's drag coefficient is an assumption about its surface chemistry and its attitude, and every density inferred from drag carries it in inverse proportion.

A coefficient that belongs to the surface, not the satellite

Every density ever inferred from satellite drag was divided by a drag coefficient, and that coefficient is not a property of the spacecraft. It is a property of what happens when an oxygen atom at eight kilometres a second strikes a surface it has already coated — and the conventional 2.2 is a convention.

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 corridor 0.2 km wide, and a density known to a factor of 2. The apoapsis a vehicle is left on after a single atmospheric pass, against the periapsis altitude it aimed at, for ballistic coefficients of 60, 130, 300 kg/m² arriving at 3 km/s. The energy removed is the density at periapsis times an effective path length of √(2πrₚH), divided by the ballistic coefficient — so it falls exponentially with altitude and the curve is steep. Hitting a 1000 km apoapsis to ±10 per cent requires a periapsis inside 0.2 kilometres. Getting the atmosphere wrong by a factor of 2 moves the aim point by 7.6 kilometres, which is 30.7 times the corridor's own width — so a ballistic vehicle aiming at a planet whose density is known to a factor of two misses by more than the tolerance allows, and the manoeuvre has to be flown rather than aimed.

A manoeuvre that has never been flown once

Arrive on a hyperbola, dip once through the atmosphere, leave on a bound orbit having spent no propellant. The saving is a kilometre a second or more, the physics is the same as an entry corridor, and nobody has done it — because the corridor is a tenth of a kilometre wide and the density is known to a factor of two.

spaceflight · Atmospheric drag 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.

Ballistic coefficientSpace-weatherSystematic errorConjunction assessmentDrag coefficientEmpirical modelOrbital decayRe-entry predictionAccommodation coefficientAerobrakingAerocaptureAlong-track error

All concepts