How far the plumb line bends

The previous rung took the ladder’s own argument seriously for the first time. Five essays before it argue about the surface a height is measured from — the ellipsoid, the geoid, a level surface, a quasigeoid — and every one of them takes the line the height is measured along to be straight. It is not: it is the plumb line, it follows the gravity field, and the field is bent by whatever mass is nearby.

Dropped down a four-kilometre column beside one buried sphere, the line arrived 47 millimetres from the point vertically below the summit, and was 0.3 micrometres longer than the straight line beside it.

That essay recorded the 47 mm as a shortfall. It is a number about one buried sphere at one offset, whose deflection of the vertical at the surface is under three arcseconds, and real deflections in mountainous country run to tens.

Parameterising by the thing that is measured #

The fix is to stop parameterising by the mass. A buried sphere has a radius, a depth and a density contrast, none of which anybody knows for a real mountain, and all three of which feed into an answer through a formula the reader has to take on trust.

The deflection of the vertical is different. It is the angle between the plumb line and the ellipsoid normal at the surface, it is what an astronomical observation minus a geodetic position gives directly, and it is published for real stations. Parameterising the family by it means every figure below has an input a reader can look up for their own country.

So the body is solved for rather than assumed. Given a required deflection and a depth, the mass follows from the peak-deflection formula, and the density contrast follows from the radius. Holding the radius at half the depth keeps the point-mass field valid — the field is exact only outside the body — and leaves the contrast as the output.

That output is itself informative, and it is the essay’s first limit rather than its first finding.

Rock is about 2,670 kg/m³, and the largest contrasts anybody measures between one rock and another are a few hundred. A compact buried body producing ten arcseconds of deflection would have to be denser than rock; one producing forty would have to be denser than any material on the planet. A large deflection is not produced by a buried body at all. It is produced by topography and by the crustal root under it — a mass distribution spread over tens of kilometres, not concentrated in one.

Why the linearity rescues the number anyway #

The implausible densities would matter if the answer depended on the shape of the mass. It does not, to the precision that matters here, and that is what makes the measurement transferable.

The drift is proportional to the deflection with a fitted exponent of 0.9999 across a fortyfold range, and the residual departure from exact proportionality is two parts in ten thousand. The mechanism is straightforward: the plumb line’s local tilt is the gravity vector’s local tilt, and to first order the whole field scales with the disturbing mass. Doubling the mass doubles the deflection everywhere and doubles the drift.

So 12.54 mm per arcsecond at a 4,000 m column is a scaling law rather than a case study, and it can be applied to a real deflection from a real station even though the model that produced it is a body no geologist would accept. What would break the transfer is a mass distribution whose shape differs enough to change the field’s variation with depth — which a broad crustal root does, and which is stated below as the limit it is.

Doubling the mountain does not double the drift #

The second finding is the one that inverts an expectation, and it is about the height rather than the deflection.

The instinct is that a plumb line accumulates its bending along its length, so a taller line should drift more than proportionally. The measurement says the opposite: 4.3 mm at 1,000 m, 7.7 at 2,000, 12.5 at 4,000 and 18.0 at 8,000, for one arcsecond — a doubling of height buying about 1.6 times the drift each time.

The reason is where the mass is. A disturbing body at eight kilometres’ depth produces a field whose horizontal component falls off with distance, and the top of an eight-kilometre column is far enough away that the tilt there is much smaller than the tilt at the base. The integral is dominated by the bottom few kilometres, and adding height on top adds very little.

The consequence for a reader is a warning about quoting: a drift quoted without the height it was measured over is not transferable, and it is not transferable in the direction opposite to the obvious guess. Scaling 47 mm at 4,000 m up to 8,000 m by a factor of two gives 94 mm and the answer is 67.

What the drift is worth to a survey #

Twelve and a half millimetres per arcsecond is a rate, and turning it into a consequence needs one more number: what deflections are.

Over most of a stable continental interior the deflection of the vertical is one to three arcseconds, so a plumb line down a hundred-metre shaft drifts a fraction of a millimetre and nobody has ever needed to care. In alpine terrain twenty to forty arcseconds is ordinary, and there the rate says a four-kilometre line drifts a quarter to half a metre.

Nobody drops a four-kilometre plumb line. What they do is level up a mountain, which is the same integral taken in steps — and the accumulated horizontal displacement of the vertical reference between the valley station and the summit station is exactly what the orthometric correction exists to account for. The correction is applied as a length adjustment along the route; this essay is the same quantity seen sideways.

The reading that follows is short. The straight line in every textbook picture of an orthometric height is wrong by a quantity that is proportional to the local deflection and to about the two-thirds power of the height, and in alpine country that quantity is a matter of centimetres to decimetres. It does not invalidate anything the ladder has said, because every height system that matters is defined through the potential rather than through the line. It does mean the picture should not be drawn straight.

What was computed, and how #

The plumb line is integrated downwards in five hundred steps. At each step the horizontal pull of the disturbing mass and the vertical pull of normal gravity plus the mass’s own vertical component are computed, the ratio is the local tilt, and the line moves sideways by that tilt times the step.

The drift reported is the relative one: the tilt at the top of the column is subtracted, so what is measured is how much the line’s direction changes between the summit and the base rather than how far it is from the ellipsoid normal. That is the quantity that matters for levelling, because a survey referenced to the summit station carries the summit’s own deflection in its datum and cares about the change — the same distinction the ladder’s base rung draws between a height and a height difference.

Normal gravity comes from the ladder’s own normal field, which is checked against published constants elsewhere in the collection, and gravity inside the rock column uses the free-air gradient with the Bouguer slab removed and restored — the same Poincaré–Prey reduction the orthometric-height rung is built on.

Two numbers, and which of them to keep #

The rung produces two results and they have different lives ahead of them.

Twelve and a half millimetres per arcsecond, at a four-kilometre line. This one transfers. It is exactly linear in the deflection, the linearity is a consequence of the field scaling with the mass rather than of the model’s shape, and the deflection is a published quantity for real stations. Anyone with a deflection can use it.

An exponent of 0.69 in the height. This one does not transfer, and the essay is careful to say so rather than presenting the pair as equally solid. The exponent is a statement about how quickly the disturbing field weakens with distance from the mass, and a compact buried sphere weakens far faster than a broad crustal root does. For a real mountain the exponent would be larger — how much larger is not established here.

Keeping the two apart is the point of separating them. A result whose parameter enters linearly is robust to the model being wrong about that parameter; a result whose parameter enters through the geometry is not, and quoting both with the same confidence would misrepresent the second.

Where the model stops #

A compact body, and real deflections do not come from one. This is the limit the density table above makes explicit and it is the honest boundary of the measurement. A broad crustal root produces a field that varies much more slowly with depth than a small buried sphere’s, so the integral would be less bottom-heavy and the height exponent would be larger — closer to 1 than to 0.69. The linearity in the deflection survives any shape, because it is a statement about scaling the mass; the exponent in the height is a statement about the shape, and it is this shape’s.

One offset. The line is dropped at the horizontal distance where the deflection peaks, which is the worst case for a body at that depth. A line directly above the mass has no horizontal pull at all and no drift; a line far to one side has a small pull that barely varies with depth and so, again, little relative drift.

And the terrain is not there. A real mountain has the column’s own mass beside the line, not just an anomaly beneath it, and the topographic deflection is usually the larger term. Including it would need a terrain model, which is a dataset, and this collection does not buy datasets — so what is measured is the anomaly’s contribution alone, and it is a lower bound on the real thing.

Where the drift is already accounted for #

Nothing in modern geodesy is broken by any of this, and saying where the effect already lives is the fair way to close the measurement.

A geopotential number is a potential difference between a point and the geoid. It is path-independent, so the plumb line’s curvature does not enter it at all, and it is the quantity a modern height system stores for exactly that reason. A dynamic height is a geopotential number divided by a constant, and it inherits the independence.

An orthometric height is a length along the curved line, so the curvature is in it — and it is absorbed by the mean gravity along that line, which is what Helmert’s formula estimates and what the density assumption is about. The drift measured here is the horizontal partner of the same integral.

A levelling network carries the effect as the orthometric correction, applied per section, which is a standard step in reducing a levelling run through mountainous country. It is not an approximation anybody has forgotten; it is a line item.

So the finding is not a defect anywhere. It is a picture correction: every diagram of a height system draws the line from the summit to the geoid straight, and in alpine country the real line arrives a quarter of a metre away.

Whether an instrument would ever see it #

A quantity that sits below the noise of the only measurement that could detect it is a curiosity rather than a correction, so it is worth asking which side of that line this one falls on.

Precise levelling accumulates random error as the square root of the distance run, at something like a millimetre per kilometre for first-order work. A line reaching the top of a four-kilometre column does not climb vertically: at a working gradient of one in ten it runs about forty kilometres, and the accumulated random error over that run is around six millimetres.

Set that beside the drift. At a deflection of a few arcseconds the relative drift over the column is a few tens of millimetres — the essay’s own figure is forty-seven — which is roughly eight times the random error of the survey that would be used to measure it. So the quantity is not hidden by the noise of the instrument. It is comfortably above it, by nearly an order of magnitude, and a levelling network run through mountainous country without an orthometric correction would show it as a systematic misclosure rather than as scatter.

That is the reason the correction exists and is applied as a matter of course. It is also the reason the effect is invisible to anybody who has not run such a network: a systematic error that is eight times the random one is easy to see in a closed loop and impossible to see in a single line, because a single line has nothing to close against.

The check cuts the other way too. If the drift had come out at a millimetre, the honest report would have been that the straight line in the textbook picture is straight enough, and the rung would have ended there.

The generalisation #

A measurement’s transferability depends on which of its inputs it is linear in, and finding that out is worth more than making the model realistic. The model here is unphysical at the deflections that matter, and the result transfers anyway, because the quantity it is unphysical about enters linearly.

That is a general and slightly counter-intuitive point about modelling. Effort spent making a model’s parameters realistic is wasted if the answer is linear in them, and effort spent checking which parameters the answer is linear in says where realism is required. Here it is required in the mass’s spatial extent — where the answer is not linear, as the height exponent shows — and it is not required in its magnitude.

The same reasoning is why a scaling law is a better deliverable than a number. Forty-seven millimetres is an anecdote about a model. Twelve and a half millimetres per arcsecond, with a stated height and a stated exponent for other heights, is something a reader can apply to their own country’s published deflections without believing anything about buried spheres.

Who found it, and when #

The deflection of the vertical has been measurable since the eighteenth century and was measured famously early: Pierre Bouguer’s 1749 account of the Peru expedition reports the attraction of Chimborazo pulling a plumb line off the vertical, and Nevil Maskelyne’s 1774 Schiehallion experiment used exactly that deflection to weigh the Earth. Both were measuring the surface deflection, which is the input to this essay rather than its output.

That the plumb line is curved, and that its curvature is what makes an orthometric height depend on a path, is the standing problem of vertical datums and is why the geopotential number — a potential difference, which is path-independent — is the quantity a modern height system stores. Helmert’s 1890 orthometric height is the classical approximation to the mean gravity along the curved line, and this collection has measured what its density assumption costs.

The specific drift between the top and the bottom of a column does not seem to be a quantity anybody publishes, which is reasonable: for levelling it is absorbed by the orthometric correction, and for a plumb bob nobody is trying to hit the point below the summit. It is worth computing because it makes the abstraction concrete, and because it is the quantity that says how wrong the picture in every textbook — a straight line from the summit to the geoid — actually is.

Where the ladder goes next #

This rung gives the deflection its scaling law. The identify ladder, the youngest on the site, has a question the same shape as the one this one just answered: it names a projection from a set of control points by the residual it leaves, and a residual has more than one explanation.