A mountain is not a buried sphere

How far the plumb line bends established two things about the curved line an orthometric height is measured along. The drift is exactly proportional to the deflection of the vertical at the surface — 12.54 millimetres per arcsecond over a four-kilometre column, to two parts in ten thousand across a fortyfold range — and it grows as the 0.69 power of the height, so doubling the mountain does not double the drift.

The first of those transfers to any mass distribution, because the drift is linear in the field and the deflection is linear in the field. The second does not, and that essay said so in its own list of what it had not established: the exponent measures how quickly a buried sphere’s field weakens with distance, and a mountain range’s root is not a buried sphere. It is tens of kilometres wide, and its field falls off far more slowly.

The family #

A crustal root is modelled here as the same total mass spread evenly along a line of stated width, which makes its field a superposition of the exact point-mass fields the compact case already uses. Nothing new is approximated, and the compact body is recovered exactly at zero width.

Every member of the family is solved for the same surface deflection: the mass is scaled until the peak deflection is ten arcseconds, whatever the width. That is the parameterisation that makes the comparison a comparison of shape rather than of size, and it is the one the earlier rung introduced for the same reason — the deflection is the quantity a surveyor measures and publishes, so a family parameterised by it is a family a reader can place their own case in.

What changes across the family is the shape of the field. The signal’s half-width — the ground over which the deflection is at least half its peak — runs from 30 kilometres for the narrow body to 188 for the widest, and the position of the peak moves from 5.7 kilometres off-centre to 82.

The exponent #

Fitted over columns of 1,000 to 8,000 metres:

The narrow end reproduces 0.69, which is the check that the family contains the earlier measurement rather than replacing it. From there the exponent rises monotonically to 0.842, and it is flattening: the last doubling of the width buys 0.008.

The limit it is approaching is 1. An exponent of one would mean the drift is exactly proportional to the height, which is what happens when the disturbing field does not weaken at all over the column — a mass so broad that the top and bottom of a four-kilometre column see the same horizontal pull. The widest root here is not quite there, and no real root would be.

Why the exponent is less than one at all #

The mechanism is worth stating because it explains both ends of the table.

A plumb line hangs along the local gravity vector, and the direction of that vector at height z is set by the ratio of the horizontal pull of the disturbing mass to the vertical pull of the whole Earth. The vertical pull barely changes over four kilometres. The horizontal pull does: a mass eight kilometres down is twelve kilometres away from the top of a four-kilometre column and eight from its base, and a point mass’s field falls as the inverse square of that.

So the bending is concentrated near the ground, and the drift accumulated over a column is dominated by its lower part. Doubling the height adds the parts of the column where the pull is weakest, so the drift grows more slowly than the height. That is the 0.69.

Broaden the mass and the same argument weakens. A line of mass 160 kilometres wide is 160 kilometres away from a column at its edge in the horizontal direction, and adding twelve kilometres of vertical distance to that changes very little. The field is nearly uniform over the column, and the drift becomes nearly proportional to the height. That is the 0.84.

The exponent is therefore not a property of the plumb line and not a property of the Earth. It is a shape parameter of the mass, and quoting it without saying which mass produced it is quoting a number about a model.

Both ends of the exponent are the field’s own fall-off law #

The exponent is described above as a shape parameter of the mass, which is right and is not the whole of it. The two ends of the family have closed forms, and they are the two elementary fall-off laws.

The drift accumulated up a column is the integral of the deflection over height, and the deflection at height z above a mass buried at depth d falls as (d + z)^−q for some q set by the mass’s geometry. Integrating that from 0 to H and fitting a power in H over the range the table uses — one to eight kilometres, against a depth of eight — gives an effective exponent that depends only on q.

A compact body is a point mass, so q = 2. The integral is 1/d − 1/(d + H), which over that range fits a power of 0.72, against a measured 0.691.

A broad root is a line of mass, so its field falls as the first power and q = 1. The integral is ln((d + H)/d), which over the same range fits 0.85, against a measured 0.842.

So the family’s two ends are not empirical at all. They are the inverse-square law and the inverse-first-power law, integrated up a column of the stated height over a mass at the stated depth, and the monotone rise between them is the field’s effective q sliding from two to one as the mass is spread out.

That also settles what the limit of 1 means, and it is not reachable. An exponent of one needs q = 0 — a field that does not weaken with height at all — which is the field of an infinite sheet. And an infinite sheet’s field is exactly vertical: a laterally uniform slab produces a gravity anomaly and no deflection whatever, because its horizontal pulls cancel in every direction. The limit the exponent approaches is the case in which the quantity it describes is zero, which is why the table’s last doubling of the width buys 0.008 and why no real root reaches it.

Stated that way the whole family is one parameter with two closed-form ends and a physical obstruction at the limit, rather than seven measurements with a trend through them.

The half-width, and what a field survey would see #

One column of the table is a measurement anybody could make, and it is worth separating from the ones that need a plumb line.

The half-width of the deflection signal — the ground over which the deflection is at least half its peak — runs from 30 kilometres for the compact body to 188 for the widest root. That is an observable: a line of astronomic-geodetic stations across a range measures the deflection profile directly, and the width of the profile is exactly this number.

So the shape parameter the exponent depends on is not hidden. A survey that has measured deflections across a range knows the half-width, and the half-width places the mass in the family, and the family gives the exponent. What was an unknowable property of the subsurface becomes a measurable property of a profile.

That is a better position than it looks, because the alternative is drilling. The deflection profile is the same information a gravity survey extracts to estimate the depth and extent of a body, arriving here through a different quantity — and the well-known ambiguity of that inversion, where a shallow narrow body and a deep broad one produce similar anomalies, is exactly the ambiguity this family is parameterised around.

What survives the change of shape #

The other half of the earlier result is untouched, and it was worth checking rather than assuming.

Over a fortyfold range of deflections, on a forty-kilometre root, the drift is proportional to the deflection with a fitted exponent of 1.0000. That is not approximately linear; it is linear, because the disturbing field enters the calculation linearly and doubling the mass doubles both the surface deflection and the drift at every depth.

Which means the recorded scaling law transfers after all, in the form that matters:

• the drift per arcsecond of deflection is a constant for a given mass shape and a given column height — 12.5 mm/″ for a compact body over four kilometres, 14.9 for a forty-kilometre root;
• the constant depends on the shape, by about twenty per cent across the plausible range;
• the exponent in the height depends on the shape, from 0.69 to 0.84.

A surveyor with a measured deflection and a column of known height can therefore price the effect to within about twenty per cent without knowing anything about the mass, which is a better position than the earlier rung left them in — where the number was exact for a body nobody has.

What was computed, and how #

The root’s field is a sum of point-mass fields at stated positions, and the sum’s total is solved for the requested peak deflection by a single evaluation at unit mass, because the field is linear in the mass. That linearity is the same one the deflection result rests on, so using it here and asserting it there is one fact used twice rather than two facts.

The plumb line is integrated exactly as the compact case is: step down in height, compute the horizontal and vertical attraction at the current point, take the local vertical, and follow it. The two integrations share every line except the field, which is what makes them comparable.

The line is dropped where the deflection peaks, which for a wide root is near its edge rather than at the d/√2 a buried sphere gives. Dropping it at a fixed offset instead would have compared a line through the strongest part of one field with a line through a weak part of another, and the whole family would have measured where the lines were put.

The field is symmetric, so the peak is a pair, and the positive side is taken throughout. The first version did not do that and picked whichever side the search met first, which flipped the sign of the drift between rows of the table and produced a ladder that alternated in sign — a bug that looks like a physical effect until the pattern is read.

Why the compact model was refused before it was replaced #

The earlier rung did not merely note that a mountain is not a buried sphere; its own machinery refused the model at the deflections that matter, and the refusal is what makes this rung a repair rather than an elaboration.

Solving for the mass that produces ten arcseconds at eight kilometres’ depth and asking what density contrast a body of half that depth’s radius would need gives 4,418 kilograms per cubic metre against rock at 2,670. That is not a rock; it is a body denser than any crustal material, and the model was reporting that it had been asked for something outside its own range.

Spreading the mass over a hundred and sixty kilometres brings the required contrast down by the ratio of the volumes, which is where a real root sits. So the family is not a refinement chosen for tidiness — it is the model the assertion demanded when the compact one was pushed to the deflections real mountain country produces.

That is the pattern this collection tries to build into its machinery everywhere: an assertion that refuses an input is worth more than one that confirms an output, because the refusal names the next thing to build.

Where the model stops #

A line of point masses is not a root. A real crustal root is a three-dimensional body with a density contrast against the mantle, and its field is not a line integral of point masses. What the model captures is the one property that matters here — a mass distributed over a stated horizontal extent — and it captures it in a way whose field is exact rather than approximated.

The depth is held. Every member of the family is buried eight kilometres down. Depth and width both control how quickly the field weakens, and varying only one of them measures only one of them; a deeper compact body would give a larger exponent too.

Isostasy is not modelled. A real mountain has both a topographic mass above the ground and a root below it, and the two partly cancel — which is what isostatic compensation means and why observed deflections are far smaller than the topography alone would give. The family here is the root alone, parameterised by the deflection it produces, which is the observable that already contains the cancellation.

Nothing here is about the accuracy of a levelled height. The drift is the horizontal displacement of the line a height is measured along; the height itself is a potential difference divided by a mean gravity, and that division is where the real ambiguity lives.

The generalisation #

The shortfall is paid, and what it changes is which half of the earlier result travels.

The linearity in the deflection is a property of the physics and transfers to anything. The disturbing field enters linearly; there is no mass distribution for which this fails.

The exponent in the height is a property of the mass and transfers to nothing. It runs from 0.69 for a compact body to 0.84 for a broad root, approaching 1 in the limit of a mass so wide the column sees a uniform field.

Which is a useful shape for a result to have. A quantity that transfers and a quantity that does not, with the boundary between them identified — the boundary being whether the quantity depends on the field’s magnitude, which is linear, or on its rate of change with distance, which is geometry.

And it sharpens the practical statement the height ladder makes. The plumb line’s curvature is not a correction anybody applies to a levelling run; it is the reason an orthometric height is defined by a potential rather than by a length, and the size of the thing being avoided is between twelve and fifteen millimetres per arcsecond over four kilometres, whatever is under the mountain.

Who found it, and when #

The curvature of the plumb line is in every geodesy textbook as the reason orthometric heights need a mean gravity along the line rather than a surface value, and Helmert’s 1884 treatment is the standard one.

The idea of modelling a disturbing mass as a buried sphere goes back to the gravity prospecting literature of the 1920s and 1930s, where it is the standard first model for an anomaly and where its limitations are thoroughly documented — the half-width of an anomaly gives the depth of a compact body and misleads badly for a broad one.

What is not standard is running the two together: taking the plumb line’s curvature, which geodesy computes, and asking how it depends on the mass model, which prospecting characterises. The two literatures use the same fields for different purposes and the question falls between them.

Where the ladder goes next #

Eight rungs have taken height from its zero surface through the geoid, the levelled line, the deflection, the mass that causes it, the line the height is measured along, how far that line bends, and now what shape of mass decides the bending.

What remains is the part none of them touches: the time dimension. A vertical datum is realised by marks that move — postglacial rebound, subsidence, tectonics — so an orthometric height is a statement about an epoch as much as about a surface, and the collection has the essay that would build on it waiting on the horizontal side.