An orbit that speeds up as it is slowed down

A satellite in low orbit is being braked. The residual atmosphere at 400 km is thin beyond any laboratory vacuum — about $10^{-12}$ kg per cubic metre, some fifteen orders of magnitude below sea level — but the vehicle is moving through it at 7.7 kilometres a second, and the drag force is real. Nothing else in the six numbers describing the orbit changes secularly; only the size does.

The satellite gets faster.

Not eventually, not after some intermediate process: continuously, from the moment the drag begins, at a rate proportional to the drag itself. The harder it is braked the more it accelerates, and this is not a curiosity of the model but a direct consequence of the same relation that governs every other orbit on this site.

The sign is in the vis-viva relation #

The specific energy of an orbit is $-\mu/2a$, and everything follows from the minus sign in front.

Drag removes energy, so $\varepsilon$ becomes more negative, so $a$ decreases. On a circular orbit the speed is $\sqrt{\mu/a}$, which rises as $a$ falls. Braking a satellite makes its orbit smaller, and a smaller orbit is a faster one.

The bookkeeping closes exactly. Lowering a circular orbit from $a$ to $a - da$ changes the potential energy by $+\mu\,da/a^2$ — the vehicle has fallen, so the potential energy has decreased by that amount — and changes the kinetic energy by $+\mu\,da/2a^2$. The kinetic energy gained is exactly half the potential energy lost. The other half is what the air took.

So a satellite in decay is a machine that converts potential energy into kinetic energy and heat in a fixed two-to-one ratio, and it does so regardless of how strong the drag is, because the ratio is a property of the orbit rather than of the force.

The same statement, run the other way #

The essay next to this one is the same sentence with the sign reversed. A low-thrust spiral applies thrust along the velocity, adds energy, raises the orbit — and the vehicle ends up slower than it started, having spent 4.7 km/s of velocity change to lose 3 km/s of speed.

Drag is that manoeuvre run backwards, performed by the atmosphere without permission, and at a similar magnitude. A satellite at 400 km loses about 100 metres of altitude a day at solar minimum, which is an energy loss the same size as a small ion engine’s output. The two processes are described by the same differential equation with opposite signs on the tangential acceleration:

$$\frac{da}{dt} = \frac{2a^2}{\mu}\,a_T v,$$

with $a_T$ positive for thrust and negative for drag.

Two numbers decide a lifetime #

The rate of decay is

$$\frac{da}{dt} = -\frac{\rho}{\beta}\sqrt{\mu a}, \qquad \beta = \frac{m}{C_D A},$$

where $\beta$ is the ballistic coefficient: mass divided by drag coefficient times cross-sectional area, in kilograms per square metre. Dense and compact means large $\beta$ and slow decay; light and spread out means small $\beta$ and rapid decay.

Everything about a satellite’s design enters through that one number, and everything about the environment enters through $\rho$.

The parallel curves are the useful result. The lifetime factorises: one function of altitude, times $\beta$. A designer who wants a satellite to deorbit within twenty-five years — which is now the international guideline — a rule about where a satellite may be left rather than about how it flies — has two independent levers, and the altitude lever is enormously the stronger of the two. Moving from 700 km to 600 costs almost nothing in mission capability and changes the lifetime by more than any conceivable change in the vehicle’s shape.

What was actually measured #

The atmosphere at 400 km cannot be sampled by anything that stays there. What is measured is the drag on satellites, and the density is inferred from it.

This is a genuine inversion and it is worth being careful about. A tracked object’s orbit is determined from radar and optical observations; the semi-major axis is differenced over time; the decay rate is combined with an assumed $\beta$ to give $\rho$. So every thermospheric density model — Jacchia, MSIS, and their successors — is fitted to the observed decay of hundreds of objects, and its output is then used to predict the decay of others. It has the shape of every calibration chain in this subject: the model is fitted to the observable it is later used to predict.

The ballistic coefficient is the weak link. A satellite’s mass is known, its area is known when it is not tumbling, and $C_D$ is not measurable in flight: in free-molecular flow it depends on how gas molecules reflect from the surface, which depends on what the surface is made of and what has accumulated on it. Values between 2.0 and 2.4 are used, and the choice is a 20 per cent systematic in every inferred density.

The other observable is the solar cycle, and it is measured independently. The extreme-ultraviolet flux that heats the thermosphere is tracked by the F10.7 index — the solar radio flux at 10.7 cm, measured daily since 1947 from Penticton — which correlates well with the EUV and, unlike it, penetrates the atmosphere to a ground station. Density models take F10.7 and the geomagnetic index as inputs.

The swing over a cycle is a factor of about four at 400 km and about seven at 800, computed from the tabulated model densities rather than quoted. That is smaller than the “two orders of magnitude” the subject is often described with, and the discrepancy is worth stating: the very large ratios apply above 800 km and to the exospheric temperature, not to the density at the altitudes most satellites use.

The prediction that failed by four years #

Skylab was left in a 433 km orbit in February 1974 with no propulsion. NASA’s analysis at the time gave a re-entry around 1983, which was comfortably beyond the point at which the Space Shuttle was expected to be available to raise it.

Solar cycle 21, which began in 1976, was one of the strongest on record. The thermosphere expanded, the density at Skylab’s altitude rose several-fold, and the decay accelerated. By 1978 the projected date had moved to 1979; the Shuttle’s first flight slipped to 1981; and Skylab re-entered on 11 July 1979, scattering debris across Western Australia.

The failure was not in the orbital mechanics. The equation of motion was known exactly, the ballistic coefficient was known reasonably well, and the integration was straightforward. What was wrong was a forecast of solar activity five years ahead — a quantity that is still not predictable, because the solar dynamo is not understood well enough to forecast a cycle’s amplitude before it begins.

That is the sense in which a re-entry date is a space-weather forecast. Everything else in the calculation is deterministic; the input is not.

What the thermosphere is heated by #

The factor of four over a solar cycle is not a change in how much air there is. It is a change in how far up the air reaches, and the distinction explains why the swing grows with altitude.

Above about 100 km, solar extreme-ultraviolet radiation is absorbed by molecular oxygen and by atomic oxygen, and there is almost nothing to radiate the heat away — the density is too low for efficient infrared cooling, and conduction downward is slow. So the temperature rises steeply with height and then levels off at an exospheric value which is between about 700 and 1,500 kelvin depending on how active the Sun is.

The density at a given altitude then follows from hydrostatic balance, exactly as it does inside a star: the pressure falls exponentially with a scale height proportional to the temperature. A hotter thermosphere has a larger scale height, so the exponential falls off more slowly, so there is more air at any given altitude.

The amplification is the important part. The density at height $h$ goes roughly as $\exp(-h/H)$ with $H$ proportional to the temperature, so a fractional change in temperature produces a fractional change in density of about $(h/H)$ times as large. At 400 km with a scale height of 60 km that multiplier is nearly seven, which turns a 40 per cent swing in temperature into a factor of several in density. At 800 km it is larger still, which is why the observed swing there is a factor of seven rather than four.

The satellite is not flying through more air; it is flying through air that has been lifted to where it is. That is why a density model takes a temperature as its state variable and computes the density from it, and why the input driving the whole thing is a measurement of the Sun rather than of the atmosphere.

The drag term that is not a ballistic coefficient #

There is a practical trap in this subject that catches everybody once, and it concerns the number that looks like it should be the ballistic coefficient and is not.

The orbital elements of tracked objects are distributed as two-line element sets, and each carries a drag term conventionally written $B^*$. It is used by one specific propagator, and it is a fitted parameter: whatever value makes that propagator reproduce the recent observations best. It therefore absorbs not only the true drag but every other unmodelled effect, every deficiency in the propagator’s own gravity model, and any manoeuvre the object performed during the fit interval.

The consequences are what a fitted catch-all always produces. The term is sometimes negative, which no physical drag can be. It changes when the solar activity changes, even though the object has not. It differs between two element sets generated for the same object on the same day from different tracking data. And it cannot be converted into a physical ballistic coefficient except by assuming the density model the fit implicitly used.

That does not make it useless — it makes the propagator work, which is what it is for — but it makes it the wrong number to take out of the file and put into the physics. A lifetime estimate built on it inherits every approximation in the propagator, and the honest procedure is to fit the object’s observed decay directly against a density model of one’s own choosing.

A parameter named after a physical quantity is not thereby a measurement of it, and the distinction between a fitted coefficient and a measured one is the same distinction this collection keeps meeting under other names.

Measuring the drag rather than inferring it #

The inversion described above — density inferred from decay, decay then predicted from density — was broken by putting an accelerometer on the satellite.

A precision accelerometer in free fall measures the non-gravitational acceleration directly: it cannot feel gravity, which acts on the instrument and its proof mass alike, so what it registers is drag, radiation pressure, and the vehicle’s own thermal emission. Several geodesy missions have flown them, primarily because separating gravity from everything else is exactly what a gravity-mapping mission needs, and the drag signal is a by-product.

What that provides is a density measurement along a track, sampled at seconds rather than averaged over an orbit, and obtained without assuming a drag coefficient — although a cross-section and a reflection model are still needed to convert force to density.

The results changed the models. Density variations associated with geomagnetic storms turned out to be faster and larger than the empirical models allowed, and there are persistent structures — enhancements over the poles during disturbed conditions — that an orbit-averaged decay measurement cannot see at all, because it averages over exactly the scale on which they vary.

Replacing an inference with a measurement usually reveals structure the inference had been averaging over, and here the structure is most of what makes a re-entry prediction uncertain in the last few days.

Where the model stops #

The atmosphere is not static. The tabulated densities used here are averages. The real thermosphere has a diurnal bulge on the sunlit side, of amplitude a factor of two; it responds to geomagnetic storms within hours, by up to a factor of several; and it has semi-annual variations of tens of per cent whose cause is still argued about. A lifetime computed against a mean density is a mean lifetime.

Circular orbits are the easy case. An elliptical orbit’s drag is concentrated at perigee, where the density is highest and the speed is greatest, so the effect is to circularise: apogee comes down and perigee stays put, until the orbit is nearly circular and then the whole thing comes down together. The formulae above apply only to the last part of that history.

Attitude is not constant. A tumbling satellite presents a varying cross-section and therefore a varying $\beta$; a stabilised one with large solar arrays presents a $\beta$ that depends on where the arrays are pointed. The International Space Station’s drag varies by more than a factor of two with array orientation, and its operators exploit that deliberately to manage the reboost schedule.

And below about 150 km the model changes kind. The flow stops being free-molecular, aerodynamic and thermal effects dominate, the vehicle begins to break up, and the trajectory becomes a re-entry problem rather than an orbital one. The integrations here stop at 120 km for that reason and the last part of every decay is outside them. The same instruments also settled a question the decay measurements could not even pose, which is how quickly the density responds. The answer is minutes rather than hours, and a model whose fastest input is a daily index cannot represent that however well it is fitted.

What the picture cannot show #

The exponential. The density falls by a factor of $e$ every 50 to 80 kilometres, so a plot of altitude against time is a plot of a quantity whose governing coefficient changes by orders of magnitude down the vertical axis. The curves look gentle for most of their length and then fall vertically, and that final plunge — the last 50 km in a few orbits — is compressed to a point.

The variability. Two curves are drawn for solar minimum and maximum, and the real quantity is a stochastic process with storms, a diurnal bulge and an unpredictable cycle amplitude. An honest figure would be a band rather than a line, and the band would be wider than the difference between the two lines shown.

The population. One satellite is drawn. The actual subject is 30,000 tracked objects and an estimated million fragments above a centimetre, whose collective decay is the only thing removing them — and whose collision probability depends on the density of the population, which depends on the decay. That feedback is the whole of the debris problem and it is not on any axis here.

The lifetime is a two-parameter answer and the inspiral is a one-parameter one, and both are worth reading at a second value.

Where the ladder goes next #

Later rungs on this anchor: free-molecular aerodynamics and where the drag coefficient of 2.2 comes from. Aerobraking, in which the same effect is used deliberately to circularise an interplanetary arrival — Magellan at Venus and every Mars orbiter since — for a saving of hundreds of kilograms. Aerocapture, which is aerobraking in a single pass and has never been flown. The 25-year rule and the design compliance it forces. Drag makeup and the propellant budget of a station. Differential drag as a control authority, used to spread a satellite constellation without any thruster at all. And the debris environment, where the decay rate computed here is the only sink in a system whose source is growing.

The first artificial satellite decayed in three months. Sputnik 1 was in a 215 × 939 km orbit with a perigee deep in the thermosphere, and its orbital decay was tracked worldwide and used, within weeks, to make the first measurement of upper-atmosphere density above 200 km. The very first satellite’s most durable scientific result was a measurement of the thing that destroyed it.