Incoprea research · Earth and planetary measurement · 3 November 2026

Earth and planetary measurement: two hundred leads, four computations and one proof

1. How to read this page

Four sections of fifty leads. Each lead has a number, a single falsifiable sentence, and a disposition: test means the claim can be checked with code or a public table that already exists and would change what I do next; note means the claim is a design rule, a reporting convention, or blocked on data nobody has assembled. There is no third tag because there is no third state of knowledge on this page: everything here is either checked in §7 or not checked at all.

The page runs from the inside out. §2 is about what a field measured at a surface can say about depth. §3 is about what an age is an age of. §4 is about what a catalogue contains and what a derived column contains. §5 is about what a number in a data product is a number of. §6 is one short proof about §2, and it is the only thing on the page that is proved rather than computed, conjectured or stipulated. §7 is four computations on stipulated models.

What this page is for: the questions that decide whether a depth, an age, a count or a reflectance means what the sentence containing it says. What it is not: a review, a benchmark, or evidence about any real body.

2. Section U · Geophysical inference: what a surface measurement cannot distinguish at depth (50 leads, M1–M50)

A potential field measured on a surface is the boundary value of a function that obeys Laplace's equation in the empty space outside its sources. That is simultaneously a strong and a weak statement. It is strong because the field outside is determined by the field on the boundary, so no interior detail can hide from a complete exterior measurement. It is weak because the map from sources to field is not injective, and because continuing the field inward is unstable at every wavelength. §6 proves the cleanest case of the first problem; §7.1 measures how much of it survives when the symmetry that produces it is broken. The seismic leads in this section are the same problem in a different basis: a travel time is a functional of a velocity field, and the functionals are neither complete nor orthogonal.

The distinction that matters in this section is between a measurement that constrains a moment of the source and one that constrains its distribution. Total mass, mean density, moment of inertia and the first few gravitational harmonics are moments; the depth to a interface, the thickness of a layer and the shape of a density profile are distributions. A moment is a finite number of numbers and a distribution is a function, and no finite set of moments determines a function unless the function is restricted to a finite-dimensional family, which is a prior and not a measurement. The leads that look like they contradict this are the ones where an independent constraint — a seismic velocity, a petrological model, a measured gravity gradient at two altitudes — supplies the missing dimensions.

3. Section V · Geochronology and isotope systematics: closure, inheritance, and the meaning of an age (50 leads, M51–M100)

An age is a model output. It is a ratio of isotopes divided by a decay constant, corrected for what was present at the start and multiplied by an assumption about when the system stopped exchanging with its surroundings. Three of those inputs are routinely invisible in the quoted uncertainty: the initial ratio, which is a correlation between samples; the closure temperature, which is a thermal model; and the decay constant, which is a constant only in the sense that its uncertainty is not reduced by more analyses. §7.2 measures the second of the three, because it is the one that looks most like a measurement and is least like one.

The unifying failure in this section is a quantity that is well defined only inside a model being reported as if it were independent of the model. A closure temperature is the temperature at which a diffusion model says retention became effective; it depends on the cooling rate, the diffusion domain size, the geometry and the activation energy, and a number quoted without those is a number for a family of situations rather than for a mineral. An isochron age is well defined only if the samples share an initial ratio; an inheritance correction is well defined only if the inherited component can be identified. The leads are written so that each can be falsified by a computation or a table rather than argued about.

4. Section W · Planetary catalogues: completeness, selection, and derived quantities (50 leads, M101–M150)

A catalogue is a measurement of what was detectable, not of what exists. Every derived quantity in a planetary table is the composition of a detection function, a conversion from the observable to the physical quantity, and a model that supplies whatever the observation does not constrain. Completeness is a function of the thing being counted, so a catalogue is not complete or incomplete in general; selection operates on every variable that entered the detection, which is usually more variables than the analyst has in mind. §7.3 measures one link in that chain, and finds the bias in the least obvious place.

Two habits would remove most of the leads in this section. The first is to report, beside every count, the function that converts a count into a rate, and to test that function by injecting synthetic objects into the same pipeline. The second is to report every derived quantity as a function of the assumptions that were needed to derive it, which for a diameter means the albedo, for a mass means the density, and for an age means the production function. Neither habit is expensive, and both are rarer than the leads below imply.

5. Section X · Remote sensing conventions: reflectance, emissivity and unit traps (50 leads, M151–M200)

Radiance is a physical quantity with units. Reflectance, emissivity, albedo and brightness temperature are conventions applied to radiance, and each convention carries a factor: pi for a Lambertian surface, 1.76 between the two Planck peaks, 10 between per-micrometre and per-wavenumber radiance, 100 between a fraction and a percent. The factors are exact, so they are not errors of measurement; they are errors of transcription, and they survive because a table that reports the result usually does not report the convention. §7.4 computes four of them.

The section is ordered by how far the convention sits from the measurement. The Planck convention is the deepest: the same spectrum has two different peaks depending on which axis the distribution is written on, and neither is wrong. The factor of pi is next, and it is the one that most often turns a dark surface into a bright one. The unit conversions are the shallowest and the most common. None of these is a subtle physical effect; all of them are invisible in a number without its definition.

6. One proof: a surface gravity measurement does not determine a depth profile

This is the only proved statement on the page. It is analytic, it involves no numbers, and the computation in §7.1 is a numerical check of its first table rather than a part of the argument.

Statement. Let a body be spherically symmetric and static, with total mass M and outer radius R, in Newtonian gravity. Then the gravitational field at and outside the surface is g(r) = −GM/r² r̂, for every radial density profile ρ(r) whose integral over the body is M. Consequently no measurement of g at any point outside the body, at any precision, distinguishes two such profiles: the radial density distribution is not identifiable from an exterior gravity measurement, and the set of profiles consistent with a measured surface gravity is infinite-dimensional.

Assumptions. (i) Newtonian gravity, with no modified dynamics and no fifth force. (ii) Exact spherical symmetry: no lateral density variation, no topography, no ellipticity, no rotation, no tides, no atmosphere. (iii) A static configuration, so that no mass redistribution occurs during the measurement. (iv) M and R are known independently; if they are not, a single value of g determines only the ratio M/R², and even the mean density is unconstrained. Assumption (ii) is false for every real body, and the proof is therefore a statement about an idealisation — but it is the idealisation that every spherical inversion silently adopts.

Proof. By Gauss's law for gravity, the outward flux of g through any closed surface S is −4πGM_enc, where M_enc is the mass enclosed by S. Take S to be a sphere of radius r ≥ R concentric with the body. Spherical symmetry forces g to be radial and of constant magnitude on S, so the flux is −4πr²g(r), and therefore g(r) = −GM_enc(r)/r². For r ≥ R the enclosed mass is the total mass M, whatever its radial distribution inside r < R. Hence g(r) = −GM/r² for all r ≥ R, and two profiles ρ₁ ≠ ρ₂ with the same integral give identical fields everywhere outside the body. ∎

Corollary 1 (counting). A single scalar cannot determine a function. The measurement supplies one number per observation height; the model has as many degrees of freedom as the discretisation allows, and the residual freedom is the entire space of rearrangements of mass inside R. Adding observation heights adds equations but not independence: the kernel (1 − z/√(z²+R²)) tends to a constant as R grows, which is why the infinite-sheet limit has a one-dimensional range and a null space of every other dimension.

Corollary 2 (harmonicity). The same argument applies to any harmonic field outside its source, because a gravitational potential satisfies Laplace's equation in a source-free region and is fixed by its boundary values. The "equivalent source" ambiguity is therefore not a numerical inconvenience or a consequence of bad data; it is a property of the field. Every inversion that reports a depth has supplied the missing information from somewhere, and the only question is where.

What the proof does not say. It does not say that gravity is uninformative. It says that the information is in the departures from the idealisation: the quadrupole and higher harmonics, the lateral variations, the vertical gradient, and the joint use of fields with different kernels. §7.1 measures how much is recovered when the symmetry is broken by a finite lateral extent and the observation is a vertical profile at six heights. The answer, at 1% noise, is one or two of twenty-four layers — so breaking the symmetry converts an infinite ambiguity into a large finite one, which is progress and not a solution.

7. Tests run

Four computations, all on stipulated models or synthetic samples, all in incoprea_nov3/code/earth_measure_tests.py, with results in incoprea_nov3/results/earth_measure_tests.json. numpy 2.5.4, numpy.random.default_rng(20261103) for every stochastic cell. No observational data set was read: every physical parameter below is a stipulated value chosen to put the model in a plausible range, and is labelled as such where it is used. A table number that is not in the JSON does not exist; the builder reads the JSON rather than retyping it.

7.1 What a depth profile looks like from outside

Model. A stack of 24 uniform horizontal discs, each 1250 m thick, spanning 0 to 30 km depth, with density ρ_i and total mass column 8.1×10⁷ kg/m² (2700 kg/m³ over 30 km). The vertical gravity on the axis above a disc of radius R at depth z, measured at height h, is 2πGσ(1 − (z+h)/√((z+h)²+R²)), which tends to 2πGσ as R grows and is therefore independent of z in the infinite-sheet limit. The first table is the spherical case, evaluated by quadrature of G dM/R² over shells rather than by the closed form, as a numerical check on §6.

U1 spherical bodies, identical mass and radius. Four radial profiles, each normalised to the same mass and outer radius. The last column is the relative difference from the uniform case.

Reading. The surface gravity is 9.819973426 m/s² in all four rows and the largest relative difference is 5.4×10⁻¹⁶, which is round-off. The rows differ in central density by a factor of 33 (2637 to 86146 kg/m³). This is §6 evaluated numerically, and it is in the table only so that the rest of the section has a baseline: if the machinery cannot reproduce an identity it can be trusted with, it cannot be trusted with the case where the identity fails.

U2 five depth profiles, one mass column (8.1e7 kg/m2). Five depth profiles with the same mass column, at three lateral extents. 1 mGal = 10⁻⁵ m/s²; 1 Eötvös = 10⁻⁹ s⁻².

Reading. The sheet column is 3396.8 mGal in all five rows, to the last digit: the infinite-sheet value depends on the mass column and on nothing else, which is §6 in a different geometry. At 30 km lateral extent the same five mass columns give 1989.6, 2868.7, 1700.6, 2151.2 and 1020.3 mGal — a factor of 2.8 between the extremes, in an order that no simple rule predicts. The two-layer profile and the uniform profile have the same mean depth (15.0 km) and differ by 8.1%, so the mean depth is not the parameter that surface gravity measures; the exponential profile is the shallowest and gives the highest value, and the basal profile is the deepest and gives the lowest, which is the one ordering that does follow from the kernel.

The gradient column is the observable that costs money and buys information: at R = 30 km the surface gradient runs from −413 to −1062 E across the same five columns, a factor of 2.6, while the potential itself differs by less than the sheet value. A gradiometer does not break the degeneracy; it moves it.

U3 kernel singular values, 24 layers over 30 km, 6 observation heights. Singular values of the 6×24 kernel matrix at five lateral extents, the number of singular values above 1% of the largest, and the effect of a perturbation along the weakest singular direction.

Reading. In the sheet limit the condition number is 3.8×10¹⁶ and only one singular value is above 1% of the largest: the six observation heights are six copies of the same equation. At 30 km extent the condition number falls to 3.4×10⁶ and two directions survive at 1% noise. The last two columns are the same statement in the units an interpreter cares about: a profile perturbation of 43.5% in L2 norm changes the predicted gravity by 1.3×10⁻⁵% at R = 30 km, and by 1.8×10⁻¹⁵% in the sheet limit. A 1% gravity error is 7.5×10⁴ times larger than the signal of that perturbation.

U4 two indistinguishable models at R = 30 km. Two models at R = 30 km: a two-layer truth and the truth displaced along the weakest kernel direction by the largest step that keeps every layer above 10% of its density.

Reading. The second model differs from the first by 91.7% in L2 density, has a mean depth that moves by 0.2 km, and produces gravity that differs by 6.4×10⁻⁷ relative — four orders of magnitude below the 1% noise level assumed in the table above. This is the quantitative form of the claim in §6: what is not identifiable is not the mean depth, which is pinned, but everything about how the mass is arranged around it.

Expectation contradicted. I expected the finite lateral extent to break the degeneracy usefully — that a 30 km disc would make a depth profile recoverable in practice. It does not. It removes one degree of freedom out of twenty-three, and the model that is 92% wrong in density is 0.2 km right in mean depth, which is exactly the model an inversion would return and no diagnostic in a standard report would flag.

7.2 What a closure temperature does to an age

Model. A cooling history T(t) = 10 + 590(1 − exp(−t/100 Myr)) °C, with t measured backwards from the present, so the body cools from 600 °C at 200 Ma to 10 °C now and the cooling rate at the present is 5.9 K/Myr. This is a stipulated curve, not a reconstruction of any rock. The closure temperature comes from iterating T_c = E/(R ln(A R T_c²/(E · |dT/dt|))) with E = 150 kJ/mol and A = D₀/a² as stated; those are illustrative values chosen to put T_c between 70 and 200 °C, and they are not diffusion parameters measured on any mineral.

V1 Dodson closure temperature (E = 150 kJ/mol, stipulated). Closure temperature against the diffusion coefficient and the cooling rate, three values each.

Reading. Across two decades of A and two of cooling rate, T_c runs from 77.7 to 199.1 °C: a factor of 2.6 on the Celsius scale and 1.35 in kelvin, from parameters that are all "the same mineral". The dependence on cooling rate alone is 16–26 °C per decade, which is larger than the difference between many pairs of thermochronometers.

V2 apparent age against assumed closure temperature. Apparent age for a range of assumed closure temperatures on the stipulated cooling curve, with the local cooling rate and the sensitivity.

Reading. The age runs from 16.55 Myr at T_c = 100 °C to 96.40 Myr at 375 °C. The last column is the derivative d(age)/dT_c = τ/(T₀ − T_c), which grows from 0.200 to 0.444 Myr per °C: the sensitivity of an age to its own closure temperature is itself a function of the closure temperature. An error bar on an age that does not say which T_c it belongs to is an error bar on nothing.

V3 age error produced by a closure-temperature error. Age error produced by a closure-temperature error of 20 or 40 °C, at four assumed closure temperatures.

Reading. At T_c = 120 °C a 40 °C error moves the age by −38.8% or +42.2%; at 300 °C the same error moves it by −18.5% or +21.2%. The asymmetry is the shape of the cooling curve, not a mistake: the age is exponential in 1/(T₀ − T_c), so an error toward lower T_c costs less than the same error toward higher. A ±40 °C closure uncertainty is routine, and at these ages it is a ±40% age.

V4 diffusion simulation against the closure-temperature approximation. A full spherical diffusion simulation of the same cooling history, against the single-closure-temperature approximation, and the geometric multiplier that reconciles them.

Reading. The diffusion simulation is a sphere of unit radius with zero concentration at the surface and uniform production, stepped implicitly on 61 radial nodes; 400 and 1200 time steps give 15.09 and 15.16 Myr, a 0.52% difference, so the time step is not controlling the answer. The closure-temperature construction with the bare diffusion coefficient returns 21.33 Myr, which is 40.7% older. Multiplying D₀/a² by 54.0 — the geometric constant of the sphere model, found here by bisection and not assumed — brings the closure temperature down from 123.3 to 93.0 °C and the age to 15.16 Myr, a residual of 0.00%.

Expectation contradicted. I expected the diffusion simulation to disagree with the closure-temperature approximation by something like 10%, and to disagree in a way that pointed at the approximation's age. It agrees to 0.3% — but only once a factor of 54 is inserted, and the factor of 54 is a property of the assumed geometry. The same mineral, the same activation energy, the same cooling history and the same physics therefore produce apparent ages of 15.2 and 21.3 Myr depending on whether the geometry constant is inside the closure temperature or not. The construct is exact; the number that goes into it is a model choice, and 30 °C of closure temperature is 6.2 Myr of age.

7.3 What a detection threshold does to a size-frequency distribution

Model. Diameters drawn from a Pareto distribution with cumulative form N(>D) ∝ D^−b between 0.2 km and infinity, 20000 objects per trial, 150 trials per cell. The catalogue is what survives a detection rule, and the fit is the maximum-likelihood Pareto exponent computed from the surviving diameters above an assumed threshold of 1.0 km — the fit an analyst would run after deciding that the catalogue is complete above 1 km. Two detection rules are compared: a hard cut on the measured diameter with lognormal measurement error, and a soft roll-off p(detect) = 1/(1+(D₅₀/D)⁴) applied to the true diameter.

W1 selection on the measured diameter, threshold at 1.0 km. Selection on the measured diameter with a hard threshold at 1.0 km: fitted slope, its 16–84% range, and the cumulative count extrapolated to 0.5 km relative to the truth.

Reading. The fitted slope is unbiased to within 0.4% at every measurement error up to 0.2 dex, which is larger than the precision of most real diameter catalogues. The number of objects above the threshold, however, rises from 836 to 875 (+4.7%) and the extrapolated cumulative count at 0.5 km rises by 8% at b = 2 and 13% at b = 2.5. The slope survives the threshold; the normalisation does not, because the objects that scatter into the sample from below are more numerous than those that scatter out.

This is the Eddington effect with the sign and the size that the arithmetic gives it, and it is worth stating plainly because it is usually stated the other way round: for a cumulative power law the bias from symmetric measurement error at a hard threshold is a normalisation bias, not a slope bias.

W2 soft detection roll-off, b = 2.0, no measurement noise. Soft detection roll-off applied to the true diameter, no measurement noise. The fit still assumes a sharp threshold at 1.0 km.

Reading. Here the slope moves. At D₅₀ = 0.75 km, below the adopted threshold, the fitted exponent is 1.877 against a true 2.0 (−6.2%) and the extrapolated count at 0.5 km is 0.84 of the truth. At D₅₀ = 1.5 km the exponent is 1.345 (−32.8%) and the count is 0.32 of the truth. The exponent is biased shallow, which means the small-body population is systematically under-counted, which means any surface age derived from the small craters is systematically under-estimated by the same factor. The failure is not that a threshold exists; it is that the threshold is soft and the fit assumes it is hard.

W3 differential histogram slope versus the maximum-likelihood fit. The same data fitted two ways: maximum likelihood above a threshold, and an ordinary least-squares fit to a differential histogram over the full diameter range.

Reading. The histogram slope is biased low by 8.8–12.0% at every true exponent tested, with no detection effect in the model at all: it is the bias of fitting a line to binned counts of a steep power law, and it is present in the best case. The maximum-likelihood fit recovers the exponent to 0.001–0.019. Any comparison between two catalogues that uses binned slopes will therefore agree with itself and disagree with the truth by about a tenth of the exponent.

Expectation contradicted. I expected the dangerous case to be the hard threshold with measurement noise, because that is the case the Eddington argument is usually told about. It is the safe case: the slope is unbiased to 0.4% at 0.2 dex of noise. The damage is in the soft roll-off and in the binning, and both of those are invisible in the error bar that the fit reports.

7.4 What a convention does to a number

Model. Planck's law with the SI values h = 6.62607015×10⁻³⁴ J s, c = 2.99792458×10⁸ m/s and k = 1.380649×10⁻²³ J/K, evaluated directly. The peaks are found by Newton iteration on the two transcendental equations; nothing is taken from a table.

X1 peak of the Planck function by convention, and the ratio. The Planck peak in the two abscissa conventions, and their ratio, at three temperatures.

Reading. At 300 K the peak is at 9.659 µm if the ordinate is radiance per unit wavelength and 16.998 µm if it is radiance per unit frequency. The ratio is 1.760 at every temperature, as it must be: the two peaks are the roots of 5(1−e^−x) = x and 3(1−e^−x) = x, which the code finds at x = 4.965114 and 2.821439. "The thermal peak of a 300 K body is at 10 µm" and "it is at 17 µm" are both true statements about different distributions of the same photons.

X2 brightness temperature error for a 300 K surface, emissivity assumed 1. Brightness temperature error, in kelvin, from retrieving with emissivity set to one on a 300 K surface of the stated emissivity.

Reading. At 10 µm an emissivity of 0.95 returns a brightness temperature 3.15 K too low, and 0.80 returns one 13.24 K too low. The error grows with wavelength: −2.54 K at 8 µm against −3.73 K at 12 µm for the same emissivity. A 3 K error is small against a 300 K surface and large against the 5 K contrasts that most thermal maps are used to detect.

X3 the same radiance in per-micron and per-wavenumber units, 300 K. One radiance expressed in the two spectral units that both appear in the literature.

Reading. The conversion factor is λ²/10⁴ with λ in micrometres, so it runs from 1.6×10⁻³ at 4 µm to 4.0×10⁻² at 20 µm — a factor of 25 across the thermal infrared, and 0.01 at 10 µm, which is the factor-of-a-hundred trap in a different dress. A radiance of 9.924×10⁶ in per-micrometre units is 9.924×10⁴ in per-wavenumber units, and both describe the same 300 K surface.

X4 rectangular band integral against the band-centre value, 10 um, 300 K. A rectangular band integral against the band-centre value times the width, at 10 µm.

Reading. The monochromatic approximation is good to 0.05% at 0.5 µm width, 0.75% at 2 µm and 4.97% at 5 µm, always in the same direction, because the Planck function is concave at its peak. A five-micrometre-wide "10 µm" band is a different measurement from a monochromatic 10 µm radiance by 5%, which is larger than most of the emissivity effects above.

X5 Lambertian radiance with and without the factor of pi. Lambertian radiance with and without the factor of pi, for four reflectances at 30 degrees incidence and 1000 W/m².

Reading. The ratio is 3.142 in every row, which is the least interesting column. The last column is the one that matters: a surface of reflectance 0.12 analysed with the π-less formula is reported as 0.377, and a surface of 0.30 is reported as 0.942. The error is not a scaling of the answer; it changes which side of every classification threshold the answer falls on, and it does so in the direction of making dark surfaces look bright, which is the direction that makes a detection look like a discovery.

8. Counts

Two hundred leads, of which 162 are marked test and 38 note. The tags are a priority claim, not a quality claim: "test" means a check is cheap and would change what I do next, "note" means it would not, yet.

9. Limits

• Every computation is synthetic. Four stipulated models: a disc stack, a spherical diffusion domain, a Pareto size distribution and a black body. No catalogue, spectrum, gravity map or age was read, so nothing here is evidence about any real data set, including the ones that motivated the page.
• The proof is about an idealisation. Assumption (ii) of §6 — exact spherical symmetry — is false for every real body, and assumption (iv) is false whenever mass and radius are not independently known. The proof bounds one route of inference, not all of them, and §7.1 shows how much survives when the symmetry is broken, which is not much but is not nothing.
• The gravity kernel is a stack of uniform discs. No topography, no ellipticity, no tides, no isostatic compensation, no noise correlated between observation heights. The 1–2 degrees of freedom in §7.1 belong to that kernel at those six heights; a satellite gravity mission with thousands of observations at one altitude has a different, and not obviously better, null space.
• One diffusion geometry and one activation energy. A sphere, a single domain, uniform production, no recoil, no alteration, no multi-domain diffusion, and an argon production rate treated as constant over 200 Myr. The 54.0 found in §7.2 is the sphere constant for this production model; a different production profile gives a different number, and the point of the table is that the constant exists, not that its value transfers.
• The cooling history is a formula. 10 + 590(1 − e^{−t/100 Myr}) °C is chosen so that every number in §7.2 is an exact consequence of it. No rock cooled this way, and the sensitivities would differ for an isothermal hold or a two-stage history.
• The size distribution is exactly Pareto above 0.2 km. Real crater and asteroid distributions are curved, and a curved distribution changes both the sign and the size of the histogram bias in §7.3. The detection models are logistic and lognormal because those are the simplest functions with the right behaviour, not because any instrument has that response.
• Grey bodies only. §7.4 assumes emissivity independent of wavelength within each table. A real surface with a spectral feature has a brightness temperature that depends on the band, and no single-band retrieval can separate the two.
• No literature search. Several leads are surely known results under other names, and the constructs used here are named after their authors rather than cited. Nothing on the page depends on a citation, and nothing should be read as one.
• The leads are untested claims. Some are probably false, and a few are certainly false as stated — that is what a lead is. The tags say which ones I would check first, not which ones I believe.

If one line survives: a surface measurement is a boundary condition on a field that is harmonic outside its source, an age is a thermal model wearing a ratio of isotopes, a catalogue is a detection function wearing a count, and a reflectance is a convention wearing a radiance — in all four cases the number is exact and the thing it is supposed to be a number of is supplied from somewhere else.

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