Deflection of the vertical — where it appears
Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.
The plumb line is not the normal
A latitude measured from the stars and a latitude that means a position on the ellipsoid are different angles, because a plumb bob hangs along gravity and gravity is not perpendicular to a mathematical surface. Ten arcseconds of difference is 309 metres of ground.
The equator is not a circle either
Eleven rungs price what pretending the Earth is a sphere costs, and every one of them replaces the sphere with a surface of revolution — a body whose equator is a circle. It is not. The two equatorial radii differ by seventy metres, the two surfaces part by thirty-five, and the auxiliary latitudes every ellipsoidal formula is written in stop existing.
A deflection is the slope of a mass
One arcsecond of deflection of the vertical is 4.85 millimetres of geoid slope in a kilometre, and that exact relation produces no deflection anywhere by itself, because a deflection is the gradient of a geoid and a geoid model is cited rather than computed. State a mass instead, and every quantity is a closed form: a salt dome five kilometres across bends the plumb line by 2.21 arcseconds.
The line a height is measured along
Five rungs have argued about the surface a height is measured *from* and every one of them took the line it is measured *along* to be straight and known. It is neither: through a stated buried mass a plumb line arrives 47 millimetres from the point below the summit, and the height it gives depends on the density of rock nobody has seen — 342 millimetres of spread at 4,000 metres and 1.37 metres at 8,000.
How far the plumb line bends
An earlier measurement dropped a plumb line down a four-kilometre column of rock beside one buried mass and found it arrived 47 millimetres from the point below the summit. That is a number about that mass. Parameterising by the deflection of the vertical instead — the quantity surveyors actually measure — gives 12.54 mm per arcsecond, exactly linear across a fortyfold range.
The geoid model stops at a degree
Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.
A mountain is not a buried sphere
The plumb line's drift was measured over a compact buried body and grows as the 0.69 power of the column's height — an exponent that is a statement about how quickly a buried sphere's field weakens with distance rather than about mountains. Spread the same mass into a crustal root and the exponent climbs to 0.84, while the proportionality to the deflection survives exactly.
A levelling loop reads the circulation, and the geoid has none
A triangle of angles reads the geoid as the flux of the deflection of the vertical through its sides. A levelling loop is the other integral round the same curve, the circulation, and at the geoid that one is identically zero, because the deflection is the slope of a height. What a levelling loop run on the ground misses by is the Jacobian of gravity and height over the loop: 31 mm round a fifty-kilometre loop in a stated mountain range, none of it from the free-air gradient that changes gravity by 1,333 mGal there, uncorrelated with what the triangle reads, and two and a half times the noise of the levelling where the triangle's reading is a fiftieth of its own.
A levelling loop weighs the root, and a triangle weighs the mountain
A mountain range is carried partly by a root of light crust beneath it, and a survey that knows the ground can ask how much. Put a root thirty kilometres down under a stated range and it moves a levelling loop's misclosure by 85 per cent as much as the mountains' own rock does, and a triangle's excess by 16 per cent: the root's gravity is a smoothed copy of the ground, and a smoothed copy crosses the contours. Read against their noise, pinning the root's share to a tenth takes about 2,700 fifty-kilometre loops, a million triangles or 550 astronomic stations — and if the rock's density is unknown too, a triangle cannot tell a compensated mountain from a lighter one.
Named alongside it
The objects these essays reach for when they reach for this one.
GeoidLevellingOrthometric heightGravity anomalyEquipotentialGeopotential numberPlumb lineSurface normalVerificationVertical datumAstronomic latitudeAuxiliary latitude