A comet that arrives a day early

In 1819 Johann Encke worked out that four comets recorded over thirty-three years were the same object on a three-and-a-third-year orbit, predicted its return, and was right. He also noticed something that should not have been there: each apparition arrived a little earlier than gravity alone allowed, and the discrepancy was the same each time — about two and a half hours per revolution, accumulating.

Nothing in the solar system was supposed to do that. Every other object’s arrival time is a computation from the masses of the planets, and by 1819 that computation was very good indeed. A residual that repeats rather than scattering is not an error; it is a force that has been left out.

What the force is #

A comet’s nucleus is a few kilometres of ice and dust. Sunlight heats the surface; ice sublimates rather than melting, because the pressure is far too low for a liquid phase; the vapour leaves at several hundred metres a second, carrying dust with it; and the nucleus recoils.

The recoil is a rocket, and like any rocket its net effect depends on the direction of the exhaust averaged over time. If sublimation were symmetric about the sub-solar point the average thrust would be exactly radial — directly away from the Sun — and a purely radial force does nothing to the period at leading order. It merely reduces the effective solar gravity by a constant factor.

What produces the observed effect is the asymmetry. The nucleus rotates, so the warmest part of the surface is not the sub-solar point but a region rotated away from it by thermal lag, exactly as the afternoon side of an asteroid is its warmest. The average thrust therefore has a component along the direction of motion, and a transverse thrust changes the orbital energy.

The sign is set by the rotation. A prograde-rotating nucleus is pushed forwards and spirals outwards; a retrograde one is pushed backwards and spirals inwards, shortening its period. Encke’s comet arrives early, so its period is shortening, so its nucleus rotates retrograde.

That is a statement about the spin of an object nobody had resolved, deduced from arrival times. It was confirmed directly a century and a half later, and it is one of the cleanest examples in this collection of a physical property extracted from a residual: nothing was imaged, nothing was resolved, and a rotation direction came out of a table of dates.

The same logic runs through the thermal drift of an asteroid, which is the identical argument at a thousandth of the magnitude — a recoil from re-radiated heat rather than from vapour, with the sign again set by the spin, and detected only because it too is secular.

The mechanism was not available in 1819 and was not proposed until 1950, when Whipple’s “dirty snowball” model of the nucleus supplied it. In the intervening century and a quarter the effect was real, repeatedly measured, entirely unexplained, and generally attributed to a resisting medium — a fluid filling interplanetary space, which would slow a comet, shrink its orbit and shorten its period. That account had the sign right and everything else wrong, and it failed on the comets that arrive late, which a drag cannot produce.

Marsden’s parameterisation, and why it has three components #

The standard form used in every modern orbit fit writes the acceleration as

$$\mathbf{a} \;=\; \left[A_1\hat{\mathbf{r}} + A_2\hat{\mathbf{t}} + A_3\hat{\mathbf{n}}\right] g(r),$$

with $g$ the empirical function of heliocentric distance drawn above, normalised to one at one astronomical unit. Three constants and one shape.

The shape is chosen to match the sublimation rate of water ice, which is why it turns over near two and a half to three astronomical units — the same threshold that puts the snow line where it is. A comet whose activity is driven by carbon monoxide instead, as some distant ones are, needs a different shape entirely, and fits that force water’s shape onto such an object return meaningless parameters. The asymmetry of the sublimation about perihelion is a fourth parameter in modern fits, written as a time offset in $g(r)$. A nucleus with thermal inertia is more active after perihelion than before, at the same distance, and that lag changes the net impulse — so the offset is a thermal measurement in the same way the lag angle is.

Of the three, $A_2$ is the one that shows up in arrival times, because it is the one that changes the period. $A_1$ is typically an order of magnitude larger in absolute terms and much harder to detect, since it mimics a slightly weaker Sun. $A_3$ is usually small and is detectable only in objects observed over many returns.

The reason it is harder to detect is worth spelling out, because it is a degeneracy rather than a matter of precision. A purely radial acceleration proportional to $1/r^2$ is indistinguishable from a reduction in the Sun’s mass, as far as one comet is concerned: both give a slightly wider orbit with a slightly longer period and no secular drift at all. Only the departure of $g(r)$ from an inverse square breaks the degeneracy, and $g(r)$ is close to an inverse square over most of the range where the comet is observed.

So $A_1$ is measured essentially from the part of the orbit where sublimation is turning off, which is the part where the comet is faintest and the astrometry worst. For most comets it is quoted with an uncertainty comparable with its value, and for many it is fixed at a nominal value rather than fitted — one of the places in this subject where a parameter appears in a published solution having been assumed rather than determined.

The degeneracy also runs upwards, into the ephemeris the comet is being fitted against. A radial acceleration common to many small bodies would be absorbed, in a global solution, by a small adjustment to the solar mass parameter or to the asteroid masses that the same solution estimates. Cometary radial terms are therefore fitted one object at a time and are never allowed to inform the underlying ephemeris, on the reasoning that a systematic shared by a class of objects is exactly what a global fit would misattribute. That is a deliberate loss of information, taken because the alternative failure is silent.

An awkward consequence follows for any comet observed at only one apparition. With a single arc there is no secular signal to separate the terms, so the fit returns a transverse parameter correlated with the radial one and with the epoch of perihelion, and the reported uncertainty on the arrival time of the next return is dominated by that correlation rather than by the astrometry. The predictions that fail are almost never the ones made from bad observations.

Why the error grows as the square of the time #

A constant transverse acceleration produces a constant change in the semi-major axis per revolution, and a constant change in the semi-major axis is a constant change in the period. That much is a linear effect.

The timing is not linear, because the period errors accumulate. If each revolution is short by $\delta P$, then after $n$ revolutions the comet has arrived early by $\delta P(1+2+\cdots+n) = \delta P\, n(n+1)/2$, which grows as $n^2$.

This is the same arithmetic as the growth of a position error in any orbit determination, and it is worth seeing the two together. The practical consequence is that the effect is undetectable in one apparition and unmissable in ten. A two-and-a-half-hour period change is about one part in ten thousand of a three-year orbit, which no eighteenth-century observation could have caught in a single return. Over thirty returns it is three months.

How it is measured now #

The modern procedure is to fit $A_1$, $A_2$ and sometimes $A_3$ as free parameters alongside the six orbital elements, using all available astrometry.

The parameters are small: $A_2$ for Encke is a few times $10^{-10}$ astronomical units per day squared, which is roughly $10^{-9}$ of the solar gravity at one astronomical unit. It is measured because it is secular — it acts in the same direction every revolution — and secular effects are detectable at amplitudes far below what a periodic effect of the same size would require. That is the general principle behind most of the small measurements in this collection: a perturbation that averages to zero over an orbit is a nuisance and one that does not is an instrument.

The complications that make it a difficult parameter #

The model is a rocket with a fixed thrust profile and a fixed direction, and a real comet is neither.

The activity is not steady. Outgassing comes from discrete active regions that rotate in and out of sunlight, so the thrust is modulated on the rotation period, and averaging over a rotation gives a net that depends on where the vents are. Comets are also observed to have outbursts, in which the production rate jumps by an order of magnitude for days.

The rotation state changes. The same torques that produce the net thrust also spin the nucleus up or down and shift its axis, so $A_2$ is not a constant over decades. Several comets have shown their non-gravitational parameters changing between apparitions, and at least one has been observed to change its rotation period measurably during a single perihelion passage. And the nucleus can split. A fragmenting comet’s non-gravitational parameters change discontinuously, because both the mass and the active area do, and after a split the fragments have different parameters and separate slowly.

The comet as its own worst instrument #

There is an irony in the arrangement worth stating, because it recurs whenever a body is measured by its own activity.

The quantity that makes a comet visible at all — its outgassing — is the quantity that makes it impossible to predict. A dormant object, dark and inert, has an orbit that can be propagated for centuries to within kilometres. An active one is bright, easy to find, spectacular, and cannot be located a decade ahead to better than hours of arrival time. The observability and the predictability are in direct competition, and both come from the same ice.

The competition has a second edge. Astrometry of a comet is astrometry of a coma, and the brighter the coma the worse the position: a faint, barely active comet is measured against the nucleus, and a bright one is measured against a cloud whose photometric centre wanders. So the observations get worse exactly as the object gets more interesting, and the two effects — a larger force and noisier data — pull the fit apart together.

That is why the best-determined non-gravitational parameters in the catalogue belong to comets of moderate activity observed over many returns, rather than to the famous ones. Encke is the archetype for exactly that reason: short period, many apparitions, and an activity level that has never been dramatic.

What it is good for #

The parameters are a nuisance in an orbit fit and a measurement in their own right.

$A_2$ divided by $A_1$ is a measurement of the thermal lag, which is a measurement of the rotation period and the surface conductivity. The absolute size of the parameters, combined with an independently measured production rate of water, gives the mass of the nucleus — because the acceleration is the thrust divided by the mass, and the thrust follows from how much gas is leaving and how fast. That is one of very few routes to a cometary mass, and for the comets visited by spacecraft it agrees with the mass derived from the spacecraft’s own trajectory.

The force also puts a floor under how long a comet lasts. A body losing a metre of surface per perihelion passage has a few thousand returns in it at most, which for a short-period comet is a fraction of a per cent of the age of the solar system — so every active comet now visible arrived recently, and the population must be resupplied. That inference is made from the mass-loss rate rather than from any observation of a supply, and it is the reason the outer solar system was expected to contain reservoirs long before either was found.

And the same force, in a very different regime, is what makes the deflection of a hazardous object a real engineering problem rather than a calculation: an object that outgasses cannot be predicted decades ahead to the precision that a keyhole passage requires, so knowing whether a threatening body is active is part of knowing whether it is threatening.

The same drift law read at another thermal conductivity, and the obliquity dependence at a stronger seasonal term.

A rocket that is running out #

Encke’s comet has been watched for two centuries, which is long enough to see its non-gravitational parameter change rather than merely to measure it.

The period shortening that Encke identified was about two and a half hours per revolution. It is now well under an hour. The decline is not smooth and it is not a measurement artefact: it is present in fits made with different methods, over different arcs, by different groups, and it amounts to a fall of roughly a factor of four in $A_2$ across the observed record.

The natural reading is that the comet is running out of active surface. A nucleus does not sublimate uniformly; the volatiles retreat beneath an insulating crust of the dust too heavy to be carried away, and activity survives only where that crust has failed. As the active fraction shrinks, so does the thrust, and the object approaches a state in which it is indistinguishable from an asteroid except by its orbit.

That has a consequence for the population argument at the end of this essay. If a comet’s activity declines steeply before its ice is exhausted, then the lifetime estimated from the mass-loss rate is a lower bound on how long the body survives, and the number of dormant former comets in near-Earth space is correspondingly larger than an accounting based on active objects would suggest. Several near-Earth asteroids are on orbits that are difficult to reach without a cometary history, and at least one is associated with a meteor stream.

Encke supplies its own example of that. It is the parent of the Taurid stream, and the stream is massive — considerably more material than the comet could have shed at anything like its present rate over the interval since the stream’s orbits were established. Either the comet was very much more active in the recent past, or the stream and the comet are both fragments of a larger body that broke up some tens of thousands of years ago. Both readings say the same thing about the measurement in this essay: a non-gravitational parameter is a snapshot of an object that is changing, and extrapolating it backwards is the one thing it is not entitled to be used for.

Where the picture stops #

The shape function is empirical and its parameters were fitted to a handful of comets. It is used universally, including for objects whose composition and activity are known to be different, because nothing better is available that does not require modelling the nucleus.

And the fit is degenerate with the astrometry’s systematic errors. A comet’s astrometric position is measured on a fuzzy coma whose brightness centre is offset from the nucleus by an amount that varies with the illumination and the activity, and that offset is systematic across an apparition. It is not always separable from a genuine non-gravitational term, and the separation is one of the harder judgements in cometary orbit work.

And the thermal-inertia dependence over a range of sizes and a slower spin.

Where this ladder goes next #

Later rungs on this anchor: the rotational torques that accompany the same outgassing, and the spin-up that eventually disrupts a nucleus; the mass of a nucleus derived from its own acceleration, and the density that implies; the transition from an active comet to a dormant one, and the objects that are classified as asteroids and are not; the interstellar objects and what their accelerations do and do not require; and the deflection problem, where an unmodelled rocket is the dominant uncertainty in a prediction that matters.