Generator

The ephemeris-residuals generator

One fit, four kinds of observation, and 1.3·10⁵ between the best and the worst

One fit, four kinds of observation, and 1.3·10⁵ between the best and the worst. Post-fit residuals for a Mars ephemeris, by observation type and epoch, drawn in metres at each type's own precision. Every point is a distance, including the optical ones: a meridian-circle position is an angle, and half an arcsecond at Mars is about 500 kilometres. The four sets span 1.3·10⁵ in quality and the fit uses all of them, because they do different jobs — the century of optical data is what fixes the period, and the ranging is what fixes the position. The optical residuals are drawn with a slow undulation on top of their scatter, and that is not decoration: catalogue zone errors are correlated over decades, so the honest uncertainty of the optical set is nearer its 1.4·10⁵ m systematic than its per-point scatter, and weighting those observations by their formal errors is the classic way to produce a solution that is wrong and confident. Radar ranging arrives in 1964 and takes three orders of magnitude off in a decade; spacecraft ranging arrives with Viking in 1976 and takes another two.

Post-fit residuals for a Mars ephemeris, by observation type and epoch, drawn in metres at each type's own precision. *Every point is a distance, including the optical ones*: a meridian-circle position is an angle, and half an arcsecond at Mars is about 500 kilometres. The four sets span 1.3·10⁵ in quality and the fit uses all of them, because they do different jobs — the century of optical data is what fixes the *period*, and the ranging is what fixes the *position*. The optical residuals are drawn with a slow undulation on top of their scatter, and that is not decoration: catalogue zone errors are correlated over decades, so the honest uncertainty of the optical set is nearer its 1.4·10⁵ m systematic than its per-point scatter, and *weighting those observations by their formal errors is the classic way to produce a solution that is wrong and confident.* Radar ranging arrives in 1964 and takes three orders of magnitude off in a decade; spacecraft ranging arrives with Viking in 1976 and takes another two.

5 essays call ephemeris-residuals. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Mars to five metres and Neptune to five thousand kilometres, in the same file. Present-day heliocentric position uncertainty for each planet, in kilometres, with the range component marked separately below it. The two differ because a transponder measures a distance along the line of sight and says nothing about the two directions across it, so a planet with an orbiter is known radially some 17 times better than it is known altogether. Neptune is 10⁶ times less well determined than Mars and only 20 times further away, which is the whole point: the accuracy is a property of the observations, not of the geometry. Mars has carried a transponder almost continuously since 1976; Neptune has been visited once, in 1989, and everything else known about it is meridian-circle astrometry covering 1.07 of one orbit. The two ice giants are the only entries here whose ephemerides are still limited by nineteenth-century technology, and the only cure is a spacecraft.

The table that is a fit

A planetary ephemeris is not evaluated from Kepler's laws and is not evaluated from a theory. It is a numerical integration whose starting conditions were least-squares fitted to a century and a half of observations, and its accuracy is a property of those observations rather than of the mathematics.

After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging.

An error that is nearly all in one direction

A fitted orbit's uncertainty is not a ball. Within a few revolutions it has collapsed onto a line along the track, because an error in the size of an orbit is an error in its period and an error in period is a phase that runs away — which is why an impact probability is an integral along a curve rather than over a volume.

$Every model curve has slope −1, and four measurements agree on κ to 1.5×. Semi-major-axis drift against body diameter, for a thermal recoil in which a fraction κ = 0.085 of the absorbed sunlight comes back out along-track. The three curves are the same expression at 1, 1.6, 2.5 astronomical units, and each has a slope of exactly −1: the acceleration is the absorbed power divided by the mass, which is a cross-section over a volume, so it falls as one over the size and nothing else on this axis changes it. A kilometre-wide body drifts a few metres a year; a ten-metre one drifts hundreds. The four filled marks are the bodies whose drift has actually been measured as a fitted parameter in an orbit solution, and they do not lie on any single curve because each carries its own density, distance and obliquity. What they agree about is the number beside each: solve every measured drift for the efficiency that would produce it and the four answers are 0.084, 0.085, 0.089, 0.129 — a factor of 1.5 apart, for a quantity that could in principle have been anything from zero to a fifth. That agreement is the evidence that the mechanism is understood, and it is the only evidence there is, because the thermal conductivity that sets κ has never been measured for any of them.$ Orbits

An orbit moved by heat

A rotating body re-radiates absorbed sunlight from the hemisphere that has had time to warm, so the recoil is not aimed at the Sun. The resulting force is a few parts in ten billion of gravity, it is the only orbital force whose sign depends on which way the body spins, and it has been measured to four figures.

Drift against thermal inertia: a peak at Γ = 93, in the same place for every size. How fast an asteroid's orbit drifts under its own re-radiated heat, against the thermal inertia of its surface, at a rotation period of 4.3 hours and 1.13 astronomical units. Both axes are logarithmic, and the curves are four diameters. The non-monotonic shape is the content. A surface that conducts nothing re-radiates its heat the instant it receives it: the emission is then symmetric about the sub-solar point and the transverse push cancels exactly. A surface that conducts perfectly is isothermal, has no temperature contrast at all, and again pushes nowhere. The force lives between those two nothings, and peaks where the surface's thermal time constant is comparable to the rotation period — here at Γ = 93 in SI units, and at the same place on every curve, because the size scales the drift without moving the optimum. That separation is what makes the effect a measurement. A drift rate on its own is a single number with several unknowns in it; a drift rate together with a size from radar, a spin from a light curve and a density from a flyby leaves the thermal inertia as the only thing not measured, and solving for it says what the surface is made of. Fine dust sits near 50, bare rock in the thousands, and the values measured for the bodies spacecraft have visited — Bennu at 310, Ryugu at 225, Itokawa at 700 — straddle the peak, with the two rubble piles a factor of two or three above it and the Moon's dust well below. Being past the optimum is not a small effect but it is a gentle one: the curve falls as one over the thermal inertia on that side, so a surface three times more conductive than optimal still drifts at a third of the best rate, while one three times more insulating drifts at a third as well. The shape is symmetric in the logarithm, which is why the measurement is a good one for telling dust from pebbles and a poor one for telling pebbles from boulders. The curve is one-dimensional linear theory for a rotating half-space: it has the right limits and the right peak, and it omits the body's shape, which for an irregular asteroid changes the answer by tens of per cent.

A drift rate that says what the surface is made of

The thermal recoil that moves an asteroid's orbit depends on how long its surface holds heat, and the dependence is not monotonic — a perfect insulator and a perfect conductor both push nothing. The peak in between means a measured drift is a measurement of thermal inertia, which is a measurement of grain size.

Three passes, and the fourth is below the noise. How far the position is still wrong after each pass of the light-time iteration, for five targets, on a logarithmic scale of kilometres. The first pass uses the body's position now and is wrong by exactly the distance the body travels while its light is in transit — 6254 kilometres for Mars near opposition, which is about twenty-five arcseconds and larger than any residual in an ephemeris fit. Each further pass multiplies the error by the body's speed divided by the speed of light, so the lines are straight and their slopes are that ratio and nothing else. Three passes take every one of these below a metre, which is why the loop in an ephemeris code has a fixed trip count rather than a convergence test. The same argument run backwards is why the correction cannot be applied to the observation instead: the direction light arrived from is what the observation is, and the position it corresponds to is not known until the body's orbit is.

Where a planet is and where it is seen

An ephemeris is fitted to observations, and no observation is of a position. It is of a direction light arrived from, at a time that is not the time the light left, bent by a Sun that is nowhere near the line of sight. Three corrections stand between the two, and all three are larger than the residuals.

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