Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

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.

Assumes Atmospheric drag and Orbital debris.

14 min read 6 figures Small perturbations, large consequencesMeasuring the unreachable

Every year an orbit has to be paid for, and in low orbit the bill is set by a quantity nobody can compute.

The drag force on a satellite is one line: half the density times the speed squared times the drag coefficient times the area. Three of those four are known well. The density is not.

At four hundred kilometres the air is a thousand million million times thinner than at sea level, its temperature is set by extreme ultraviolet from the Sun rather than by anything below, and it responds to solar and geomagnetic activity on every timescale from hours to a decade. There is no equation for it. There are empirical models, fitted to decades of measured satellite drag, and their published accuracy is about fifteen per cent.

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.
Fig. 1 Thermospheric density against altitude for three levels of solar activity, with a model uncertainty band 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 twenty-five at four hundred kilometres between solar minimum and maximum.

That fifteen per cent propagates directly. Orbital lifetime is inversely proportional to density, so a lifetime prediction inherits it in full — and predicting the density a year ahead means predicting the Sun a year ahead, which nobody can do. An orbit that speeds up as it is slowed down describes the mechanism; this essay is about the one number in it that is fitted rather than computed.

Why the density is so variable

The thermosphere is heated from above rather than from below. Solar extreme ultraviolet is absorbed at these altitudes, and its intensity varies by a factor of two over the eleven-year cycle, by tens of per cent over a solar rotation, and abruptly during flares.

Heating raises the temperature, which raises the scale height, which raises the density at any fixed altitude far more than it raises it at the base. The lever is exponential: a twenty per cent change in the scale height at four hundred kilometres is nearly a factor of two in the density.

Superposed on that are effects with their own timescales. A diurnal bulge follows the Sun with a peak in the mid-afternoon and a factor of two between day and night. Geomagnetic storms deposit energy directly at high latitudes and raise the global density by tens of per cent within hours. And there is a semiannual variation whose cause is still argued about.

There is one further variation with no solar cause at all, and it is a useful reminder of how thin the atmosphere is up there. The thermosphere at four hundred kilometres is dominated by atomic oxygen, whose scale height is set by its own molecular weight; higher up, helium and then hydrogen take over, each with a larger scale height. So the composition changes with altitude and with temperature, and the mean molecular weight — which sets the scale height — is itself a function of the conditions. A model of the density is therefore a model of the composition as well, and the two are coupled in a way that a single exponential cannot express.

What the model actually does

An empirical thermospheric model takes a date, a position, and two or three indices of solar and geomagnetic activity, and returns a density. It contains no physics beyond the shape of a hydrostatic atmosphere; everything else is a fit to past measurements of exactly the quantity being predicted.

Those measurements come from satellite drag itself. A satellite’s orbit decays at a rate proportional to the density, so tracking a large number of objects gives a density history — and the models are fitted to that history. The circularity is complete and it is not vicious: the model is an interpolation of past drag measurements to a new date and place.

What it cannot do is extrapolate. Predicting the density next year requires predicting the Sun’s activity next year, and the state of that art is a forecast with an uncertainty comparable to the range being forecast.

A density that spans a factor of 109 at 550 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 109 at 550 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.
Fig. 2 The same relation at a higher altitude and over a narrower range of activity. Two things change together: the absolute density is far lower, so the drag is weaker and the lifetime longer, and the ratio between solar minimum and maximum is larger, because the scale height enters exponentially in the altitude. So the higher the orbit, the more its lifetime depends on which part of the solar cycle it happens to spend its time in — and above about eight hundred kilometres the lifetime exceeds a cycle and the dependence becomes an average rather than a phase.
A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.
Fig. 3 Why the density matters beyond any single object’s fate. The altitude at which drag removes debris faster than collisions create it is the boundary between a self-clearing region and an accumulating one, and that boundary is set by the atmospheric density. A factor of two in the density moves it by tens of kilometres, and a shift in the solar cycle moves it up and down every eleven years. The critical altitude is not a property of the debris population; it is a property of the Sun.

The inputs are proxies for the thing that matters

The model takes indices of solar and geomagnetic activity, and it is worth being precise about what those indices are, because neither is a measurement of the quantity doing the heating.

The solar input is conventionally the 10.7 centimetre radio flux — the Sun’s output at a wavelength of ten centimetres, measured from the ground since 1947. What heats the thermosphere is extreme ultraviolet, at wavelengths a thousand times shorter, absorbed high in the atmosphere and never reaching the ground at all. The radio flux is used because it correlates with the ultraviolet and because it has an unbroken daily record longer than the space age, which is exactly what an empirical fit needs.

The correlation is good and it is not a proportionality. The two emissions come from different heights in the solar atmosphere and respond differently to the same active region, so the ratio between them drifts over a cycle and differs between cycles. It was noticeably different in the unusually deep minimum of 2008–2009, when the thermosphere was some thirty per cent less dense than the models predicted from the radio flux — the largest single model failure on record, and one that was diagnosed as the proxy having drifted rather than the atmosphere having done something new.

Direct measurements of the solar extreme ultraviolet now exist, from instruments in orbit, and they are better predictors. They also have a twenty-year record rather than a seventy-year one, which means a model fitted to them cannot use most of the drag history it would otherwise be calibrated against. The better input and the longer calibration are not available together, and that trade is the reason the radio flux is still in operational use.

The geomagnetic input has the same character. The indices are compiled from ground magnetometer readings, they are three-hourly and quantised, and they describe the disturbance at the ground rather than the energy deposited in the upper atmosphere. The energy input during a storm arrives as particle precipitation and as Joule heating in the auroral electrojets, both concentrated at high latitudes and both varying on timescales of minutes. A three-hourly index of ground disturbance is a lossy summary of that, and the density response it is asked to predict is a factor of two within hours.

So the model’s fifteen per cent is not one error but two stacked. There is the error of the fit — how well a functional form reproduces the densities it was fitted to — and the error of the drivers, which is how well a ground-measured proxy stands in for the energy actually deposited. The second dominates during disturbed conditions, which is exactly when the prediction is most needed.

The ballistic coefficient absorbs some of it

There is a partial escape and it is worth understanding because it decides what can and cannot be predicted.

The drag acceleration is proportional to the density divided by the ballistic coefficient — the mass divided by the drag coefficient times the area. For most objects the ballistic coefficient is not known: a piece of debris has an unknown shape, an unknown mass, and a tumbling attitude that changes its cross-section.

So orbit determination fits for it. Tracking an object over several days determines the product of the density and the inverse ballistic coefficient, and since the density comes from the model, what is fitted is an effective ballistic coefficient that absorbs the model’s error.

That works beautifully for interpolation: the fitted coefficient reproduces the object’s behaviour over the next few days, because the density error is nearly the same. It fails for extrapolation, because the absorbed error was specific to the conditions during the fit, and the conditions change.

There is a corollary worth stating for anybody reading a published orbital element set. The ballistic coefficient in a catalogue entry is not a physical property of the object: it is a fitted parameter that has absorbed whatever the density model got wrong during the fitting interval, and it changes from one fit to the next by amounts far larger than the object’s physical properties could. Comparing two epochs’ values and inferring that an object has changed — tumbled, shed material, deployed something — is a common and usually wrong conclusion, and distinguishing a real change from an absorbed model error requires comparing against other objects fitted over the same interval. The solar cycle’s imprint is visible in the catalogue as a whole, moving every object’s fitted coefficient together.

What a re-entry prediction is predicting

The practical consequence is a rule of thumb that surprises people: the uncertainty on a re-entry time is about ten per cent of the remaining lifetime, whatever that lifetime is.

An object predicted to re-enter in ten days has an uncertainty of about a day. One predicted to re-enter in ten hours has an uncertainty of an hour, which at orbital speed is a quarter of the way around the world. Even at the last orbit the uncertainty on the impact point is thousands of kilometres, and that has nothing to do with the tracking — the object’s position is known to metres — and everything to do with not knowing how much air it will meet on the way down.

That ratio holds because the dominant error is multiplicative on the density and the decay is exponential in time. Improving the tracking does not help. Improving the density model helps in proportion, and the fifteen per cent is the state of the art.

A year at 400 km and 1195 years at 900. Orbital lifetime against starting altitude for a circular orbit, integrated from da/dt = −ρav/β with a ballistic coefficient of 100 kg/m² and a piecewise-exponential fit to the 1976 standard atmosphere. Only the density profile is tabulated; the decay is computed. The curve rises by a factor of 1397 between 400 and 900 kilometres — 10 months against 1,195 years — because the density falls by four decades across that span while nothing else in the expression changes much. That single ratio is why an altitude either cleans itself or does not. Below about 600 km a fragment is gone before it can find anything; above 800 it is there for centuries, and the twenty-five-year disposal rule is a statement about which side of this curve an operator is required to leave the vehicle on.
Fig. 4 The quantity being predicted: orbital lifetime against altitude, which spans from days at two hundred kilometres to centuries at a thousand. The steepness is the exponential atmosphere, and the density uncertainty translates into a lifetime uncertainty of the same fractional size at every altitude. What the plot cannot show is the phase of the solar cycle, which for the higher orbits is the difference between a decade and a century.

There is a further asymmetry worth stating because it decides which predictions are trustworthy. A conjunction prediction — will these two objects pass within a kilometre in three days — is nearly immune to the density error, because both objects’ orbits are affected almost identically and the relative geometry is what matters. A lifetime prediction is entirely exposed to it. So the operational products that require the most precision are the ones the atmosphere disturbs least, and the ones that are quoted with days of uncertainty are the ones the public notices. A collision rate that needs no collision to measure sits on the first side of that divide and re-entry warnings on the second.

What was actually measured

Three kinds of measurement bear on the problem and they disagree in an interesting way.

Drag-derived densities. The historical record is decades of satellite tracking, from which densities are inferred using assumed ballistic coefficients. It is enormous, it is the basis of every model, and it carries a systematic from the assumed drag coefficient — which for a satellite in free molecular flow depends on the gas–surface interaction and is not a simple two.

Accelerometer measurements. Several missions have flown accelerometers sensitive enough to measure the drag acceleration directly, giving densities along the orbit with high spatial resolution. These are the best data available and they cover a limited set of altitudes and epochs.

Mass spectrometers. Direct measurement of the composition and density in situ, from the earliest satellite era onward, at a small number of altitudes.

The three do not agree at the ten per cent level, and reconciling them is where the models’ remaining uncertainty comes from. The largest single systematic is the drag coefficient: a change in the assumed value shifts every drag-derived density by the same factor, and the assumed value has changed over the decades.

The population peaks at 825 km, and so does the risk. Tracked objects per 25-kilometre shell against altitude, with the collision rate on a single object computed from each bin as nσv — a gas-kinetic rate, not an orbital calculation. The cross-section is not a satellite's area but an effective one, 3698 square centimetres, fixed by requiring the whole tracked population to produce the one catastrophic collision every 8 years that is observed — most tracked objects are fragments, and two fragments meeting make nothing new. The distribution is not smooth and its shape is history: the peak of 3,010 objects near 825 km is four decades of launches into sun-synchronous orbit plus the debris of two deliberate destructions, and the second rise past 1,300 km is the Soviet-era navigation constellation. At the peak one such object waits 46,409 years between strikes — which sounds safe until it is multiplied by the 3,010 objects sharing that shell, giving one collision every 31 years among them, and by the 23,680 in the whole of low orbit, giving one every 8. The rate on one object is reassuring and the rate on the population is not, and they are the same number.
Fig. 5 Why any of this matters for anything except re-entry dates: the flux of trackable debris against altitude, which peaks where drag is too weak to clear it and where the population has accumulated. The lifetime curve of the previous figure is what shapes this one — below about six hundred kilometres the atmosphere removes objects on a useful timescale, and above it nothing does. The boundary between those two regimes is set by the density, which is the quantity nobody can compute.
Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.
Fig. 6 The long-run consequence of getting the density wrong in a particular direction: the collisional cascade, whose onset depends on whether drag removes fragments faster than collisions make them. Every model of that cascade integrates over decades, uses an assumed solar-cycle history, and inherits the density uncertainty at every step. Predictions of the debris population fifty years ahead differ between groups by factors, and the differences are dominated by the assumed atmosphere rather than by any disagreement about the collision physics.

Where the picture stops

The picture stops in three places, and the second is the one that will not improve.

The drag coefficient is not two. In free molecular flow the momentum transfer depends on whether molecules reflect specularly or diffusely and on how much energy they carry away, which depends on the surface’s chemistry and on the atomic oxygen adsorbed on it. Values between two and four are used and the choice shifts the inferred densities proportionally.

Solar activity is not predictable. The eleven-year cycle’s amplitude is not forecastable more than a few years ahead, and no individual flare or storm is forecastable at all. So a lifetime prediction more than a few months out is a prediction about the Sun, and the Sun’s is the larger uncertainty.

And the object’s own behaviour changes. A satellite that fails may deploy nothing, tumble, or shed material; a rocket body may vent residual propellant. Each changes the effective ballistic coefficient discontinuously, and the fit that was tracking it starts describing a different object.

One more limit deserves stating because it is the one that has bitten most publicly. A re-entering object does not decay smoothly to the ground: below about a hundred and twenty kilometres the drag rises so steeply that the last few orbits are qualitatively different, and the object breaks up at an altitude that depends on its construction. Predicting where the pieces land requires the break-up altitude, the fragments’ ballistic coefficients, and the winds below — none of which is known. The corridor a returning capsule has to fly is narrow for the same reason and is navigated actively rather than predicted, which is the difference between a controlled re-entry and an uncontrolled one.

There is a fifth measurement that has become important recently and is worth mentioning because it changes the data situation. Large constellations of satellites in low orbit now carry precise navigation receivers and are tracked continuously, which turns each of them into a drag sensor with a well-known ballistic coefficient — the operator knows the mass and the attitude. Thousands of such sensors, distributed in altitude and local time, are a far better sampling of the thermosphere than the historical record ever was, and assimilating them into a density model in near real time is now technically possible. The obstacle is commercial rather than scientific: the data are proprietary.

Why an empirical model is not a failure

The instinct is to treat a fitted model as a placeholder for a real one, and for this problem that instinct is probably wrong.

The thermosphere is driven by a source — the Sun — whose output is measured rather than predicted, through a coupling that involves photochemistry, conduction, and the global circulation of a rotating atmosphere. Physics-based models of it exist, they are impressive, and they do not outperform the empirical ones for operational prediction, because their inputs carry the same uncertainty and their extra physics buys resolution rather than accuracy.

That is a genuinely different situation from a mixing length or a Coulomb logarithm, where the fitted parameter stands in for a calculation nobody can do. Here the calculation can be done and the input is what is missing.

The practical distinction is worth keeping. A fitted parameter that hides missing physics improves when the physics improves. A fitted model whose limit is an unpredictable driver does not, and the effort belongs on measuring the driver in real time rather than on modelling it better — which is exactly what the operational community has concluded, and why space-weather monitoring is now an operational service rather than a research programme.

One further observation about the shape of the error. The density uncertainty is multiplicative and its effect on a position accumulates as the square of the time, because a drag error is an acceleration error and an acceleration integrates twice. That is the same t2t^2 growth as a biased error in a numerical integration, and it has the same consequence: the error is negligible over a day, noticeable over a week, and dominant over a month. Every operational cadence in the low-orbit business — how often orbits are updated, how far ahead conjunctions are screened, how often a satellite is commanded — is set by that growth rate rather than by anything about the tracking.

Where the ladder goes next

The rung above this one is the drag coefficient: what free molecular flow over a real surface actually does, why the value depends on the atomic oxygen environment, and how much of the disagreement between measurement techniques it accounts for. Further up sits the assimilation problem — how a density model is updated in real time from the observed drag on hundreds of tracked objects, which is the same manoeuvre a weather forecast makes and is now being made here.

Where this goes next

Down by 280 km, and faster by 164 m/s. A circular orbit at 400 km with a ballistic coefficient of 100 kg/m², integrated down to 120 km through a tabulated atmosphere at solar minimum and solar maximum. At solar min it takes 1.2 years; at solar max it takes 147 days — a factor of 2.9 for the same satellite in the same orbit, decided by an eleven-year cycle nobody controls. The rising curves are the orbital speed on the right-hand scale, and they are the point: the drag force is opposite the motion and takes energy out, and the body goes faster, from 7673 to 7836 m/s. There is no contradiction in it. The specific energy is −μ/2a, so removing energy shrinks a, and the circular speed √(μ/a) rises when a falls; the kinetic energy gained is exactly half the potential energy lost, and the other half is what the air took. Every point on every curve was integrated from da/dt = −(ρ/β)√(μa), and the speed at each point is √(μ/a) at that point rather than a separate model. An orbit that speeds up as it is slowed down Drag takes energy out of a satellite and the satellite goes faster. There is no paradox in it, only a sign — and the same sign makes a re-entry date a space-weather forecast rather than an orbital computation, which is why Skylab was predicted for 1983 and came down in 1979. A corridor 0.78° wide. Peak deceleration against entry flight-path angle, from sixty-one integrated entries at 11 km/s and β = 250 kg/m². The steep edge is where the load reaches 12 g, at 6.05°. The shallow edge is skip-out: below 5.27° the vehicle passes through the upper atmosphere and leaves again at 8.55 km/s, having lost too little speed to be captured. The corridor between them is 0.78° wide, which at an approach speed of 11 km/s is a targeting problem measured in kilometres of periapsis, days out. The curve is steep everywhere, which is the other half of the difficulty: half a degree of aiming error is a factor of 1.46 in the load. Lift is what widens this, and no ballistic capsule has any. A corridor a degree and a half wide The peak deceleration of an entering vehicle contains no property of the vehicle at all. Only the speed and the angle of arrival decide how hard it is slowed — the ballistic coefficient decides where, and nothing decides whether. A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend. A collision rate that needs no collision The flux through an orbital shell is a gas-kinetic calculation with no orbits in it. Production goes as the square of the population and removal goes as the first power, so a quadratic overtakes a linear once and never comes back — and which side of that a shell is on is decided by its altitude.

What this makes readable

Essays that name this one as a prerequisite.

What links here

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The objects this essay names

Each one links to every other essay that touches it.

Ballistic coefficientDrag coefficientEmpirical modelGeomagnetic stormOrbital lifetimeRe-entry predictionSolar activitySpace-weatherSystematic errorThermospheric density