Astronomy & Celestial Mechanics
the sky, computed
Concept

Spherical harmonics — where it appears

The family of functions on a sphere that play the part sines and cosines play on a line. Each is labelled by a degree that sets the angular scale and an order that sets the orientation, and any field on a sphere can be expanded in them, from a planet's gravity to the microwave background.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

255 km sees degree 154; 35786 km sees degree 6. The same field spectrum, multiplied by the upward continuation factor (R/r)ˡ⁺¹ for orbits at 255 km, 450 km, 800 km, 35786 km. A harmonic of degree ℓ has ℓ bumps around the planet, so it falls off with height like a wave of that wavelength — fast, and faster the finer it is. The horizontal rule is a measurement floor; where each curve crosses it is the highest degree that orbit can feel at all, and the answer is 154 at 255 km, 101 at 450 km, 65 at 800 km, 6 at 35786 km, or 130 km, 198 km, 308 km, 3340 km of horizontal resolution on the ground. Nothing in that arithmetic is an instrument. It is why GOCE flew at 255 kilometres with drag compensation rather than at a comfortable altitude with a better gradiometer, why the Moon's mascons were not seen until something orbited low over them, and why a geostationary satellite's ephemeris needs a field with four terms in it.

The field a satellite is allowed to feel

Past the flattening, a planet's gravity is a sum of harmonics whose sizes follow a rule with no physics in it. How much of that sum a spacecraft can measure is decided by its altitude and by nothing else — which is why one mission flew at 255 kilometres and had to push itself along.

gravitation · Oblateness Gravity per kilometre of topography, by wavelength. The free-air gravity anomaly per kilometre of topography — the admittance — on Earth, against the wavelength of the topography, for elastic plates 5, 25, 100 km thick, with crust of density 2800 kg/m³ over mantle of 3300 and compensation at 35 km. Short wavelengths are uncompensated and give the full attraction of a slab of crust, 117 mGal per km. Long wavelengths are compensated and give nearly nothing, because the mass of the mountain is cancelled by the root beneath it. Where the curve falls between the two is set by the plate's thickness, so a measured admittance spectrum is a measurement of how strong the lithosphere was when the load was applied.

A mountain's gravity says what holds it up

A mountain can stand because the rock beneath it is strong, or because it floats on a deep root of light crust like an iceberg. The two leave different gravity behind them, and the difference depends on the mountain's size — so the ratio of gravity to topography, measured wavelength by wavelength, is a measurement of how stiff a planet's outer shell was when the mountain was built.

gravitation · Moment of inertia

Named alongside it

The objects these essays reach for when they reach for this one.

AdmittanceDegree varianceDrag-free controlElastic thicknessFlexureFree air anomalyGeoidGeopotentialGravity anomalyGravity gradiometryIsostasyKaula's rule

All concepts