The Earth's shape, read off a satellite's node

A sphere pulls exactly like a point, and the whole of two-body celestial mechanics rests on that. It is the reason an orbit closes, the reason the elements are constants, and the reason a planet can be treated as a number rather than as an object.

The Earth is not a sphere. It is an oblate spheroid, flattened by one part in 298 — a departure from the sphere whose exterior field is a point mass exactly, and the departure is not aesthetic: it is the largest single perturbation acting on almost every satellite ever flown, larger than the Moon, larger than the Sun, larger than the air above about 600 km. What it does is not to change the size or shape of an orbit at all. It turns the orbit’s plane.

A thousandth of the field, written as a series #

Any static gravitational field outside the body producing it can be expanded in spherical harmonics, and for a body with an axis of symmetry the expansion collapses to a sum over Legendre polynomials in the sine of the latitude:

$$U = -\frac{\mu}{r}\left[1 - \sum_{n\ge2} J_n \left(\frac{R}{r}\right)^n P_n(\sin\varphi)\right].$$

The leading term is the monopole, and it is the whole of the shell theorem: any spherically symmetric body has $J_n = 0$ for every $n$ and pulls like a point. The first correction that survives is $n = 2$, because $n = 1$ can always be removed by putting the origin at the centre of mass.

$J_2 = 1.08263\times10^{-3}$. Every coefficient after it is at least a thousand times smaller: $J_3$ is $-2.5\times10^{-6}$, $J_4$ is $-1.6\times10^{-6}$. So the Earth’s external field is a monopole, plus a term one part in a thousand, plus a residue three orders of magnitude below that. One correction dominates, which is what makes the subject tractable.

Why the plane turns and the ellipse does not #

A perturbation acting on an orbit can do three things: change its size, change its shape, or change its orientation. Averaged over one revolution, $J_2$ does only the third.

The reason is that the perturbing potential depends on latitude and not on longitude — the body is symmetric about its spin axis — so the perturbing force does no net work around a complete orbit. No work means no change in energy, and energy is the semi-major axis. Similarly the component of angular momentum along the spin axis is conserved, which fixes the combination that would otherwise change the eccentricity secularly. What is left is the direction of the angular momentum vector, and that direction is the orbital plane. Elements that were supposed to be constants are constants in five of six cases here, and the exceptions are exactly the two angles that orient the ellipse.

The transverse component of the perturbation, drawn in the first figure, is the mechanism. It pulls a satellite crossing the mid-latitudes back towards the equatorial plane. On a body that is already turning, a torque does not restore — it precesses, in the same way that a torque on a spinning top makes it circle rather than fall. The line of nodes therefore turns steadily, at

$$\dot\Omega = -\frac{3}{2}\,J_2\left(\frac{R}{p}\right)^2 n\cos i,$$

and the argument of periapsis turns too, at $\dot\omega = \tfrac34 J_2 (R/p)^2 n(5\cos^2 i - 1)$.

The two inclinations that stand still #

Two special values fall straight out of those formulae, and both are used.

Ninety degrees. $\cos i$ vanishes, so a polar orbit’s node does not move at all. Its plane stays fixed against the stars while the Earth turns underneath, which is exactly what a mapping satellite wants: successive passes cover successive strips of longitude, and the geometry of each pass is the same.

63.4349 degrees. $5\cos^2 i = 1$, so the apsides stand still: the ellipse keeps its orientation within its plane. That is the critical inclination, and it is the reason the Molniya orbit exists. A highly eccentric orbit with apogee over the northern hemisphere gives long dwell times at high latitude, but only if apogee stays in the north — and at any other inclination the argument of periapsis walks, carrying apogee into the southern hemisphere within a few years. At 63.4° it does not walk.

The perturbation used on purpose #

The node rate is negative for a prograde orbit — the plane regresses, westward — and positive beyond 90°. Somewhere in the retrograde range it passes through $+0.9856$ degrees a day, which is the rate at which the Sun appears to move along the ecliptic.

An orbit turning its plane at exactly that rate keeps a fixed angle to the Sun for ever. Every pass over a given latitude happens at the same local solar time, year after year, at no propellant cost whatever — which is worth comparing against what a plane change costs when it has to be bought, where a 28.5° rotation in low orbit is 3.8 km/s. That is a sun-synchronous orbit, and essentially every Earth-observation satellite ever flown is in one: consistent illumination is what makes images from different dates comparable, and a shadow that changes length from one pass to the next is a change in the data that has nothing to do with the ground.

That ceiling is worth dwelling on, because it is a limit of an unusual kind. It is not that a sun-synchronous orbit above 5,974 km is expensive, or hard to reach, or unstable. It is that the Earth’s oblateness cannot supply the required rate at that distance, at any inclination — the perturbation being exploited has run out. A design constraint whose boundary is set by a coefficient of the gravity field is not a constraint anybody chose.

What was actually measured #

$J_2$ was not measured on the ground. It cannot be: a gravimeter on the surface measures the local field, and separating the oblateness of the potential from the topography, the density anomalies and the centrifugal term of the Earth’s own rotation is a problem in geodesy that had been argued over since Newton.

What settled it was Vanguard 1, launched on 17 March 1958 into a 654 × 3,969 km orbit at 34.25° inclination. Its node regressed, and the regression was measured by timing the satellite’s passes over tracking stations, over months. The rate came out at a few degrees a day; the formula above turns a rate, an inclination and a semi-major axis into $J_2$; and the value that emerged — $1.0827\times10^{-3}$ — was better than the best ground-based estimate by an order of magnitude within a year of launch.

The observation is a set of pass times. Not a shape, not a mass distribution, not a picture of a bulge: a sequence of moments at which a radio beacon rose above a horizon, differenced against where a spherical Earth would have put them. Everything about the Earth’s figure that follows is an inference from that difference through the potential expansion, and the inference is only as good as the expansion is complete.

Vanguard 1 gave a second result the same way, and it is the better story. Its node data, extended over two more years, showed a term the $J_2$ model could not fit — an oscillation with the period of the orbit’s own rotation rather than a steady drift. That is the signature of $J_3$, the odd harmonic, which is antisymmetric about the equator: it says the southern hemisphere’s sea level is slightly further from the centre than the northern. The Earth is very slightly pear-shaped, by about 17 metres, and it was found in 1959 in the residuals of a satellite’s node.

Where the model stops #

One term is not the field. The modern gravity models carry harmonics to degree 2,190 — millions of coefficients — determined by GRACE and GOCE, which measure the field by the separation of two spacecraft and by an onboard gradiometer respectively. $J_2$ is the largest by three orders of magnitude and the rest are not zero: precise orbit determination for altimetry satellites needs hundreds of them, and the residuals after that are what oceanography reads.

$J_2$ itself is not constant. It has been decreasing since satellite measurements began, at a few parts in $10^{11}$ a year, as the mantle rebounds from the ice sheets removed at the end of the last glaciation. Since about 1998 the trend has reversed, and the reversal is attributed to present-day ice mass loss redistributing water towards the equator. A coefficient in a gravity model is a time series about the Earth’s interior and its climate.

Averaging over a revolution throws away the short-period terms. The formulae above are secular rates; the actual node oscillates about the mean trend with the period of the orbit, at an amplitude of a fraction of a degree. For most purposes that is noise. For a satellite whose ground track has to repeat exactly, it is the thing being controlled.

And the whole framework assumes the perturbation is small. It is, for the Earth. It is not for a rapidly rotating body: Jupiter’s $J_2$ is $1.47\times10^{-2}$, fourteen times the Earth’s, and Saturn’s is $1.63\times10^{-2}$. For the inner satellites of those planets the first-order rates above are a first term in a series that needs several more, and the orbits of the ring particles at Saturn are shaped by exactly this, alongside the resonances with the moons.

What the picture cannot show #

The flattening. Every drawing of an oblate Earth exaggerates, this one by twenty-eight times, and no honest drawing can do otherwise: at any size a page can carry, 21 km on 6,378 is thinner than the line used to draw it.

The time. The node rates here are degrees per day, and the interesting behaviour is over years. A figure showing five degrees a day cannot also show that this is a complete turn every seventy days and forty turns over a satellite’s life.

The three dimensions. The plane’s rotation is a rotation of a vector about an axis, and every figure here is a plot of a rate against an angle. The thing that is actually happening — an orbit’s normal sweeping out a cone in space — has no representative on these axes.

The first term of a series that became a discipline #

The question of whether the Earth is flattened at the poles or elongated was an argument between the Newtonians and the Cassinis, settled in the 1730s by two expeditions — one to Lapland and one to Peru — sent to measure the length of a degree of latitude at the extremes. The Lapland degree came out longer, which means the curvature is gentler there, which means the Earth is flattened. It took eight years and two continents to establish a fact that a satellite’s node delivered in three decimal places in eighteen months.

The surprising part is which direction the information now runs. Geodesy began as the business of measuring the Earth’s shape in order to make maps. The satellite era inverted it: the shape is now inferred from orbits, and the orbits are then computed from the shape, so that a spacecraft’s position depends on a gravity model which was derived from other spacecraft’s positions — the same circularity, and the same escape from it, as a distance ladder whose every rung is calibrated on the one below. It is a bootstrap, and it closes — the residuals of each generation of satellites determine the model the next generation is tracked against. The Earth’s figure and the orbits above it are one self-consistent set of measurements, and neither is prior to the other.

Two satellites and a ruler between them #

The modern determination of the field does not use a node at all. It uses the distance between two spacecraft.

Fly two identical satellites in the same low orbit, a couple of hundred kilometres apart, and connect them by a microwave link that measures their separation continuously. As the leading one passes over a region of excess mass it is accelerated first, so the separation shortens; a couple of hundred kilometres later the trailing one feels the same pull and the separation returns.

The separation is therefore a direct readout of the along-track gradient of the field, sampled along the ground track, and it is measured to about a micron — a precision that would be pointless for anything except this.

What that buys is not a better static field, though it gives one. It is the time variation. Repeating the mapping every month gives a sequence of gravity fields, and differencing them gives the mass that has moved between one month and the next.

The results are hydrological rather than geophysical. Groundwater depletion in aquifers appears as a steady loss of mass over a region; the seasonal cycle of a monsoon appears as an annual oscillation; and the mass loss of the Greenland and Antarctic ice sheets appears as a secular trend that is now the primary measurement of that quantity.

The technique has one severe limitation, which is resolution. The field’s short-wavelength components attenuate with altitude, so a satellite at four hundred kilometres cannot resolve features much smaller than a few hundred kilometres across. Everything is a smoothed map, and separating a signal from an adjacent one requires a model of what the adjacent one should look like.

A geodetic mission became a climate instrument, and the quantity it reports — gigatonnes of ice per year — is a mass difference inferred from a distance measured in microns.

The two longitudes a satellite drifts towards #

Everything above concerns the zonal harmonics, which depend on latitude alone. The Earth’s field also varies with longitude, and for one class of orbit that variation is not a small correction.

A geostationary satellite has an orbital period equal to the Earth’s rotation, so it sits over a fixed longitude and feels the same part of the longitude-dependent field continuously rather than averaging over it. A perturbation that would ordinarily average away instead accumulates.

The dominant term is the second-degree, second-order harmonic — the ellipticity of the equator, which amounts to about seventy metres between the long and short axes. Its effect on a satellite fixed above a point is a small tangential acceleration, pushing the satellite along the equator towards the nearer end of the long axis.

The result is four equilibrium longitudes: two stable, at the ends of the equatorial minor axis, and two unstable, at the ends of the major one. A satellite released anywhere else drifts toward the nearer stable point and, having no damping, oscillates about it with a period of a couple of years.

Holding a station therefore costs propellant. The east–west station-keeping budget for a geostationary satellite is a couple of metres per second a year, and it is spent fighting a seventy-metre departure from a circle in a body forty thousand kilometres away.

The same effect decides where uncontrolled objects end up. Defunct satellites and upper stages left in the geostationary belt librate about the two stable longitudes, so the debris population is not uniform around the belt — it is concentrated in two regions, and that concentration is a direct observation of a harmonic coefficient.

The regression rate depends on altitude as steeply as it does on inclination, and both dependences are worth reading at a second value.

Where the ladder goes next #

Later rungs on this anchor: the tesseral harmonics, which depend on longitude and produce the 24-hour and 12-hour resonances that make a geostationary satellite drift towards one of two stable longitudes. Frozen orbits, where $J_2$ and $J_3$ balance so that the eccentricity and argument of periapsis both stay put. The geoid as an equipotential surface, and why “sea level” is a gravitational object rather than a geometric one. GRACE, and mass change measured as the separation of two spacecraft to a micron. The $J_2$ of other bodies — Jupiter’s, Saturn’s, and what the value implies about the density profile inside. And the relativistic terms that sit alongside these: geodetic precession and frame dragging, which turn an orbital plane for a reason with no bulge in it at all.

The satellite that produced all this is still in orbit. Vanguard 1 stopped transmitting in 1964 and is the oldest human object in space, in an orbit whose node has now turned some thousands of times, at a rate the satellite itself was the first thing to measure.