The field a satellite is allowed to feel

A satellite’s node regresses because the Earth is not a sphere, and the rate of that regression measures the flattening to seven figures. That is the first rung of this ladder and it is a clean measurement: one number out, one number in.

The second number is harder to state, because there is no second number. Past the flattening the field is not a correction with a value; it is a spectrum with a shape, and what a spacecraft can measure of it is decided almost entirely by how high it is flying.

The field as a sum #

Outside all the mass, the gravitational potential satisfies Laplace’s equation, and every solution of Laplace’s equation that decays at infinity is a sum of spherical harmonics:

$$U(r,\theta,\lambda) = \frac{GM}{r}\left[1 + \sum_{\ell=2}^{\infty}\sum_{m=0}^{\ell}\left(\frac{R}{r}\right)^{\ell}P_{\ell m}(\sin\theta)\bigl(C_{\ell m}\cos m\lambda + S_{\ell m}\sin m\lambda\bigr)\right].$$

The degree $\ell$ counts how many times the pattern changes sign from pole to pole, the order $m$ how many times it does so around the equator. Degree 1 is absent by choice of origin — it vanishes when the coordinates are centred on the centre of mass, which is how the centre of mass is defined. Degree 2, order 0 is $-J_2$: the flattening.

This is not a fit to data with harmonics chosen for convenience. It is the complete set of solutions of the field equation in the region where there is no mass, so every possible external gravity field of every possible body is exactly one such sum, and determining a planet’s field means determining the coefficients.

Kaula’s rule #

Written out, the coefficients look like a list of unrelated numbers. Plotted, they are not.

In 1963, working with the first satellite-derived solutions, William Kaula noticed that the root-mean-square coefficient at each degree fell as

$$\sigma_\ell \approx \frac{10^{-5}}{\ell^2},$$

and every gravity model published since has stayed on that line. It has no derivation. It is an empirical statement about the Earth, and the same form with a different constant holds for the Moon, Mars and Venus.

That exception is the reason the first rung of this ladder is possible. If $J_2$ sat on the Kaula line with everything else, a satellite’s node would regress under the combined influence of a hundred comparable terms and no single number could be extracted from it. Because the flattening is a thousand times larger than its neighbours, the nodal regression is a measurement of one coefficient with the rest as a small correction.

The physical reading of the slope is worth stating. A surface load of horizontal wavelength $\lambda$ produces a gravity signal at the corresponding degree, and its amplitude at the surface is proportional to the load. Kaula’s rule then says the Earth’s density anomalies have roughly equal power per decade of wavelength — mountains, basins, subducting slabs and mantle plumes all contributing at their own scales without any of them dominating. It is a statement about geology dressed as a statement about gravity.

Upward continuation #

Now the part that decides everything about how the field is measured.

Each term in the expansion carries the factor $(R/r)^{\ell}$ in the potential, which becomes $(R/r)^{\ell+1}$ in the acceleration. At the surface that factor is one. At altitude $h$ it is $\left(\frac{R}{R+h}\right)^{\ell+1}$, and the exponent means the suppression compounds.

For the Earth and a 400-kilometre orbit, $R/r = 0.941$. At degree 2 the factor is 0.83 — the flattening is barely attenuated. At degree 100 it is $0.941^{101} = 0.0019$. At degree 200 it is $3.6\times10^{-6}$.

This has an intuitive form that is worth carrying: a gravity anomaly is invisible from a distance greater than its own width. A feature 100 kilometres across produces a field that has decayed by a factor of $e$ after 100 kilometres of altitude, and by three orders of magnitude after 700. The relation between degree and half-wavelength on the ground is $\pi R/\ell$, so degree 100 is a 200-kilometre feature, and it is unmeasurable from anything much above 200 kilometres.

That is not a technological limitation. It is a property of Laplace’s equation, and no instrument improves it.

What a low orbit found on the Moon #

The clearest demonstration that altitude is the aperture came from a body with no atmosphere at all.

In 1966 the Lunar Orbiter spacecraft were placed in low, eccentric lunar orbits to photograph landing sites. Their radio tracking showed unexplained accelerations, and in 1968 Muller and Sjogren published the map: five large positive gravity anomalies, each centred on one of the circular maria, each strong enough to shift a low orbiter’s velocity by tens of metres per second per pass. They named them mascons — mass concentrations.

Nothing about the Moon had changed. The mascons had been there for three and a half billion years, they are among the largest gravity anomalies in the solar system relative to the body’s own field, and they had been completely invisible. Earth-based tracking of lunar orbiters had been going on for years; the difference was that the earlier orbits were higher.

The consequence was immediate and practical. A lunar orbit’s inclination determines how often it passes over a mascon, and the resulting eccentricity growth can drive perilune into the surface within months. There are only four inclinations — 27°, 50°, 76° and 86° — at which a low lunar orbit is stable indefinitely. The Moon has frozen orbits and the Earth has station-keeping, and the difference is entirely in the size of the harmonics past degree 2.

Three ways to measure a spectrum #

The instruments that have been flown are worth listing because each attacks a different part of the same limitation.

Tracking a single satellite is the classical method and the one that produced Kaula’s rule. Range and range-rate from the ground, accumulated over years, are inverted for the coefficients that best reproduce the orbit. It works to degree 30 or so and no further, because the higher terms produce accelerations below what ground tracking can separate from the modelling error in everything else.

Satellite-to-satellite tracking removes the ground. GRACE, launched in 2002, flew two identical spacecraft 220 kilometres apart in the same orbit and measured the distance between them with a microwave link to a micron. When the leading spacecraft crosses a positive anomaly it accelerates and the separation grows; the rate of change of that separation is a difference of accelerations, which removes everything common to both. GRACE reached degree 160 and, more importantly, could repeat the measurement monthly — turning gravity into a time series and making it possible to weigh the water leaving an ice sheet.

Gradiometry removes the orbit as well. GOCE carried three pairs of accelerometers on orthogonal baselines half a metre long and measured the second derivative of the potential directly. A gradient is a difference of accelerations over a short baseline, so it is blind to everything the whole spacecraft experiences together — including drag, once the drag has been cancelled — and it is more sensitive to short wavelengths than the acceleration is, because differentiating multiplies each harmonic by another factor of $\ell$. That gains a degree of resolution at the cost of a factor of $\ell$ in the attenuation, which is why gradiometry only pays at very low altitude.

What the resolution limit is for #

The number that comes out of all this is a map: the geoid, the equipotential surface that mean sea level would follow, quoted to a resolution of about 80 kilometres and an accuracy of one to two centimetres.

Two centimetres of geoid over 80 kilometres is what makes it possible to measure ocean circulation. Sea surface height is measured by radar altimetry to a couple of centimetres; subtracting the geoid from it gives the dynamic topography, which is the slope the ocean currents balance against. Before the geoid was known to that accuracy the subtraction left a residual larger than the signal, and ocean circulation had to be inferred from ship measurements of density instead. The same measurement runs the other way for planets nobody has visited twice. Mars’s field, from Mars Global Surveyor and its successors, reaches degree 120 and shows the Tharsis rise as an anomaly so large that it dominates the planet’s degree-2 terms — which means Mars’s flattening is not a rotational figure at all but a load, and reading it as a rotation rate the way the Earth’s is read would give the wrong answer by a wide margin. The interpretation of $J_2$ is not the same on every body, and knowing which reading applies requires the rest of the spectrum. The hierarchy also explains an asymmetry in the literature that is otherwise puzzling. There are published gravity fields to degree two thousand for the Earth, to degree nine hundred for the Moon, to degree a hundred and twenty for Mars, and to degree two or three for most of the rest of the solar system — and the ordering is not a ranking of interest. It is a ranking of how long something has orbited each body and how low it flew, which is a statement about mission history rather than about the planets.

Where the model stops #

Three limits are worth naming.

The expansion is valid outside all the mass, and the Earth’s topography is not inside a sphere of radius $R$. Continuing the series downwards to the actual surface — which is what a geoid model at ground level requires — is a divergent operation in principle and a controlled one in practice, and it is why gravity models are quoted with an explicit reference sphere.

The coefficients are not separable from the orbit model. Every solution is derived by fitting an orbit, and an error in the drag model, the solar radiation pressure, the tidal deformation of the solid Earth or the position of the tracking station appears in the recovered coefficients. Two independent gravity models differ by more than either one’s stated uncertainty, and the difference is a fair estimate of the systematic floor.

And the field changes. Water moves, ice melts, the mantle rebounds from the last glaciation, and the solid Earth flexes twice a day under the tides. Those are not noise in the measurement of a static field; on the timescale GRACE resolved them they are the signal, and the static field is what is left after they are removed. A gravity model has a date on it, which is not something the first rung of this ladder had to say.

The rule and the attenuation are the two halves of the argument, and each is worth reading over a range different from the one used above.

The term that was set to zero #

The expansion begins at degree 2 because degree 1 was removed by choosing the origin, and the choice is worth examining because the thing it removes is measurable and is not zero.

Degree 1 describes a displacement of the body’s centre of mass from the coordinate origin. Setting it to zero is the statement that the origin is at the centre of mass — which is not a physical claim about the Earth but a definition of the frame.

The trouble is that the frame in which everything else is measured is defined by the ground. A tracking station is bolted to the crust, and the crust is not the whole Earth: water moves between the oceans and the continents, ice melts, the atmosphere shifts, and the centre of mass of the whole system moves relative to the crust by a few millimetres over a year.

So there are two origins available — the centre of mass of the Earth system, and the centre of figure of the crust — and they oscillate relative to each other. That relative motion is geocentre motion, it is a degree-one signal, and the expansion cannot see it because the expansion is written in a frame that defines it away.

Measuring it requires a technique whose reference is genuinely the centre of mass rather than the ground. Satellite laser ranging is one: a satellite orbits the centre of mass of the whole system, so tracking one from ground stations and solving for the stations’ positions returns the offset directly.

The amplitude is a few millimetres and the consequence is larger than that suggests. Sea level rise is a global average measured against a reference frame, so an error of a millimetre in where the origin is is an error of a millimetre in sea level — which is a substantial fraction of the annual rate being measured.

A term the mathematics sets to zero is a quantity somebody has to measure, and the frame the whole gravity field is expressed in depends on it.

Both are cases where the expansion’s own conventions decide what counts as a measurement, and both are easy to lose because the mathematics disposes of them tidily.

The field of a body nothing orbits #

Everything above assumes an orbiter, and most bodies in the solar system have never had one. What a single flyby measures is worth setting out, because the answer explains the shape of the published tables.

A flyby is a hyperbolic passage lasting hours, during which the spacecraft samples the field along one arc at one range of altitudes over one part of the body. The trajectory’s deflection is measured by Doppler tracking, and the deflection is fitted for the coefficients.

What one arc constrains well is the monopole — the body’s mass, which comes out to several figures from a single close pass and is often the flyby’s most durable result. What it constrains poorly is everything else, because a single arc cannot separate a degree-two term from a degree-three one: both produce a deflection, and only their variation over the surface distinguishes them, which one pass cannot sample.

So a flyby gives a mass, and a mass plus a radius from imaging gives a bulk density, and a bulk density is the single most informative number about a small body. Whether it is rock, ice or a rubble pile with half its volume empty is decided by that one figure.

Getting the quadrupole requires either several flybys at different geometries or an orbiter. That is why the moment-of-inertia factors — which need the degree-two coefficients — exist for a handful of bodies and not for the dozens whose masses are known.

There is one exception that works without any spacecraft at all. A body with a satellite of its own reveals its mass through the satellite’s orbit, measured by imaging over months, and a body with two satellites in a mutual orbit reveals rather more.

The gravitational information available about a small body is a strict hierarchy — mass from one pass, shape from images, quadrupole from many passes, spectrum from an orbiter — and almost every body in the solar system stops at the second rung.

Each of the three limits above is a place where the recovered coefficients contain something other than the planet’s static field, and none of them is reduced by a better instrument.

And the field of a body that is not spherical to begin with, since the whole expansion is an expansion about a sphere.

Where this ladder goes next #

This rung establishes that the field is a spectrum, that the spectrum has a shape, and that altitude decides how much of it is accessible.

Above it lies the inverse problem: turning a set of coefficients into a statement about the interior. Gravity alone cannot do it — a shell and a point of the same mass are indistinguishable from outside, which is the shell theorem read as a limitation — so the inversion needs a second measurement, usually topography or seismology, and the combination is what says whether an anomaly is a load on the surface or a density change at depth.

Sideways lies the same expansion on other bodies, where the low degrees carry the information the Earth’s high degrees do. A small body’s degree-2 coefficients give its moments of inertia; those give the degree of central condensation; and that says whether the body ever melted and separated a core, which is a question about the first few million years of the solar system asked entirely through the second derivative of a potential.