The jet that pushes a comet also turns it

A comet near the Sun is a rocket with no one steering it. Sunlight warms ice on the dayside of the nucleus, the ice sublimates, and the gas streams off at half a kilometre a second, carrying momentum away. The reaction pushes the nucleus, and the push is what made Encke’s comet arrive hours early at every return for a century: a force a millionth of the Sun’s gravity, delivered mostly in the weeks around perihelion, accumulating in the timing as the square of the number of returns. That is the force’s effect on the orbit.

A push applied off-centre is also a twist. The vents on a comet’s surface are not distributed symmetrically about its spin axis — they are wherever ice happens to be exposed, in pits and on cliffs and along fractures — and the gas leaving them exerts a torque on the nucleus. The torque is small, but the nucleus is small too, and its moment of inertia falls as the fifth power of its radius. What the torque does to the spin is the subject here, and the answer turns out to decide how comets die.

A day that more than doubled #

The clearest measurement came from a small comet in 2017. 41P/Tuttle–Giacobini–Kresák is a Jupiter-family comet with a period of five and a half years and a nucleus about 1.4 kilometres across, and in early 2017 it passed within 0.15 astronomical units of the Earth, close enough for its rotation to be followed night by night through the pattern of jets in its coma.

In early March, five weeks before perihelion, the comet turned once in about twenty hours. By early May, three and a half weeks after perihelion, the period was forty-six hours or more. No other mechanism can change a body’s rotation by a factor of two in two months: there is nothing to collide with, tides from the Sun and planets are many orders of magnitude too weak, and the comet’s light and heat recoil forces act on timescales of millions of years. It was the gas.

The calculation drawn through the two measurements turns them into a number. Write the torque as the momentum flux of the gas, $\dot M v_\text{th}$, times the radius $R$, times a dimensionless efficiency $k_T$: the fraction of the momentum that acts as a lever. For a perfectly symmetric distribution of vents $k_T$ would be zero; for a single jet aimed tangentially at the equator it would be one. The spin rate $\omega$ changes as

$$\frac{\mathrm{d}\omega}{\mathrm{d}t} = \frac{k_T\, v_\text{th}\, R\, \dot M}{I}, \qquad I = 0.4\, M R^2,$$

with $I$ the moment of inertia of a uniform sphere. Taking 41P’s water production as about $3 \times 10^{27}$ molecules a second at perihelion, falling as the inverse square of the distance from the Sun, and its radius and density as 0.7 kilometres and 500 kilograms a cubic metre, the efficiency that takes the period from twenty hours to forty-six between those dates is $k_T = 0.044$. Four per cent of the momentum in the gas, acting off-centre, was enough to halve the comet’s rotation rate.

Continued, the same torque would have taken the period past a hundred hours within two months more; it weakened because the comet receded from the Sun. Had it not, the rotation would have slowed through zero and reversed. What happened to its spin afterwards has not been followed as closely; a nucleus driven by torques that do not line up with its spin is expected to end up tumbling about more than one axis, which is harder to measure than a period.

How much the twist depends on size #

41P is one comet, but the arithmetic generalises immediately, and the generalisation is where the consequences are.

Over an orbit a nucleus loses a mass $\Delta M$ to sublimation, and the spin changes by

$$\Delta\omega = \frac{2.5\, k_T\, v_\text{th}\, \Delta M}{M R}.$$

For nuclei on the same orbit with the same fraction of their surface active, the mass lost per orbit scales as the surface area, $R^2$, while the mass scales as $R^3$. So the change of spin per orbit scales as $1/R^2$: halving the radius quadruples the twist. And the number of orbits before the nucleus has lost all its volatiles, $M/\Delta M$, scales as $R$: halving the radius halves the active life.

The spin a nucleus can survive does not depend on its size at all. A body held together only by gravity — as the measured spins of asteroids show most bodies more than a few hundred metres across are — comes apart when material at its equator is no longer bound, at a period of about 4.7 hours for a density of 500 kilograms per cubic metre. Comet nuclei are fragile, porous aggregates with tensile strengths of a few pascals, and the same limit applies to them.

The figure sets the two clocks side by side. The solid line is the number of perihelion passages until the nucleus has lost its mass, scaled from 41P’s own mass loss per orbit. The dashed lines are the number of passages until the torques have spun it to break-up.

If every orbit’s torque acted in the same sense, the spin would change steadily and the number of orbits to break-up would grow as $R^2$. It does not, because the torque’s direction is set by where the active vents are, and the vents change. Fresh ice is exposed by collapsing cliffs, old vents choke on the dust they lift, and the pattern of activity on one return is not the pattern on the next. The spin changes on successive orbits are then closer to random in sense than to consistent, and a random walk takes the square of the number of steps to cover a distance that steady steps cover in proportion. With random-walk torques the orbits to break-up grow as $R^4$.

Against a clock growing as $R$, a clock growing as $R^4$ must cross it. With 41P’s own efficiency the crossing falls at 3.3 kilometres; with an efficiency of 0.01, about what the spin changes measured on other comets suggest, at 1.2 kilometres. Smaller nuclei are spun apart before they run out of ice. Larger ones run out of ice first.

The push and the twist come from the same gas #

The torque and the non-gravitational acceleration that moves a comet’s orbit are two projections of one momentum flux. The gas leaving the nucleus carries momentum at $\dot M v_\text{th}$; the component of its sum along the nucleus’s motion changes the orbit, and its moment about the centre changes the spin. A comet whose non-gravitational acceleration has been measured from its orbit, and whose spin change has been measured from its rotation, therefore has two independent estimates of how much gas it is losing and in which directions — and their ratio is a lever arm, a geometric property of where the vents are, that does not depend on the production rate at all.

That is the same structure as the thermal recoil forces of the other kinds of body. An asteroid’s orbit drifts because it re-emits sunlight from its warm afternoon side, and its spin changes because the same re-emission acts off-centre. A spacecraft’s tracking showed an anomalous acceleration that turned out to be its own waste heat radiated preferentially in one direction. In each case a force that looks external is an exhaust, and a body that exhausts asymmetrically is both pushed and turned.

The scales differ enormously. Thermal photons carry momentum $E/c$, and a kilometre-sized asteroid’s sunlight torque changes its spin over hundreds of thousands to millions of years. Gas leaving at half a kilometre a second, having cost its latent heat of sublimation to release, carries some fifty thousand times more momentum per unit of energy spent, and a comet loses a substantial fraction of the solar energy it absorbs to sublimation. The same lever arm, applied to the gas, changes a comet’s spin in weeks. A comet is an asteroid with its thermal torque turned up by four or five orders of magnitude for a few months of every orbit.

The efficiency is the uncertain number #

Every number in that comparison is measured or computed except one. The efficiency $k_T$ is known only where a comet’s spin change and its gas production have both been measured, and the list is short.

41P’s value of 0.044 is near the top. Deep Impact watched 9P/Tempel 1’s period shorten by about a quarter of an hour between successive returns; 103P/Hartley 2’s period lengthened from about 17 to about 19 hours during its 2010 approach, while a spacecraft watched it; 67P/Churyumov–Gerasimenko, orbited for two years by Rosetta, shortened its period from 12.76 to 12.40 hours across its 2015 perihelion. Each implies its own efficiency, and the values scatter from below a thousandth to a few hundredths, with a typical value near one per cent. That scatter is itself expected: $k_T$ depends on the arrangement of vents relative to the spin axis, which is a property of each nucleus’s shape and history.

Because the crossing radius moves only as the two-thirds power of $k_T$, the uncertainty compresses. A factor of forty in efficiency, from a thousandth to four hundredths, moves the crossing by a factor of twelve, from about a quarter of a kilometre to three. Wherever in that range the true typical efficiency lies, there is a radius below which comets do not survive to run out of ice, and it is of order a kilometre.

Histories of individual spins #

The comparison of clocks is a statement about averages. Individual nuclei follow random walks, and the walks end in one of two ways.

The small nuclei’s walks have steps thirty-six times larger than the large ones’, and their ice lasts a sixth as long. They lurch between slow and fast rotation from one return to the next, and all four in the figure reach the break-up rate within a few dozen passages — a few hundred years on 41P’s orbit, a small fraction of the time a comet spends as an active Jupiter-family comet. The large nuclei wander too: their spins drift down to near zero and back, and a slow-spinning nucleus is one whose rotation is easily excited into tumbling about more than one axis. But their steps are small, and in the two thousand or so passages their ice lasts none of the four reaches break-up.

A nucleus spun to break-up does not necessarily vanish at once. It sheds material from its equator, or splits into two or more pieces, the same outcomes that rubble-pile asteroids spun up by sunlight show. But for a comet the fragments are worse off than the parent. Each is smaller, so its torques are larger in proportion to its spin, and each exposes fresh ice that drives new activity; the fragments of a split comet are seen to brighten, split again and fade, sometimes over a single apparition. Comet 73P/Schwassmann–Wachmann 3 broke into several pieces in 1995 and dozens in 2006. The disintegration of C/1999 S4 (LINEAR) near its 2000 perihelion left nothing but a cloud of dust and a few small fragments. A spin pushed past break-up is one of the mechanisms proposed for both, and it is the one that grows most steeply as nuclei get smaller.

Spins that stop being simple #

A torque that is not aligned with the spin axis does more than speed or slow the rotation. It tilts the angular momentum relative to the body, and a body whose angular momentum is not along one of its principal axes does not rotate cleanly: it tumbles, with two periods instead of one, a motion called non-principal-axis rotation. Internal friction damps tumbling back to clean rotation over a time that grows steeply as the body gets smaller and its spin slower, and outgassing torques re-excite it every orbit. Small, slowly rotating comets should therefore spend much of their lives tumbling, and the walks above, which treat the spin as a single rate, are a simplification precisely for the nuclei they say are most at risk.

The prediction is checkable. The comets whose rotation has been measured in detail include several tumblers — 103P/Hartley 2 among them, a nucleus barely a kilometre across that was observed by a spacecraft rotating with two periods at once while its jets drove its spin down. A survey of nucleus rotation states that found tumbling to be commonest among the smallest nuclei would be evidence that their torques are as large, relative to their spin, as the calculation implies.

The fraction spun apart, by size #

Many such histories, at many sizes, give the fraction of nuclei that end by spinning apart rather than by losing their ice.

Every nucleus of two hundred metres is spun apart. At one kilometre, nine in ten are. At five kilometres, none are. Half are spun apart at about 1.6 kilometres, a little above the crossing of the average clocks because the random walk has a long tail: some nuclei of more than the crossing radius take a few large steps in the same direction early, and are lost.

A nucleus that ends by losing its ice has a different afterlife. It stops being a comet without coming apart — a dark, inert body on a comet’s orbit, indistinguishable at a distance from an asteroid, and some of the asteroids on Jupiter-crossing orbits with low albedos and comet-like orbits are thought to be exactly that. So the radius that divides the two outcomes also divides the two kinds of remains: below it, dust trails and meteor showers; above it, dormant nuclei.

A shortage the jets would make #

If small nuclei are destroyed faster than large ones, there should be fewer of them among active comets than the supply delivers.

The figure supplies nuclei with a power-law distribution of sizes, as they arrive from the reservoirs beyond Neptune — the scattered disc for the Jupiter-family comets, the Oort cloud for the comets arriving for the first time — and as the collisional and accretional processes that formed them would produce, and counts how many are active at any time: each size is present in proportion to its supply and to how long it stays active. With only the loss of ice to end them, the cumulative count keeps rising towards small sizes. With spin disruption as well, it flattens below about two kilometres, and at two hundred metres there are half as many as there would otherwise be.

The observed nuclei of Jupiter-family comets do show a cumulative size distribution that flattens below about a kilometre, and the flattening has usually been attributed to the difficulty of seeing a small, dark nucleus inside the bright coma that surrounds it. That difficulty is real, and the surveys that measured the distribution account for it as well as they can. The torque calculation adds a second reason that does not depend on the surveys at all: small nuclei may be missing because their own jets have destroyed them. David Jewitt made the argument in 2021, from the spin changes then measured, and the figures here reproduce its structure with one comet’s calibration.

What the model leaves out #

The nucleus is a uniform sphere with a fixed active fraction, losing mass at 41P’s rate per unit area; real nuclei are irregular, their active fractions differ by orders of magnitude, and highly active small nuclei like 41P lose mass faster per unit area than large ones, which shortens their ice clock and strengthens their torques together. The torque efficiency is a single number per figure, where a nucleus’s real efficiency changes as its shape and active areas change. The random walk assumes no memory from one orbit to the next, where real nuclei probably have some: the same large vent may stay active for several returns, which pushes the walk towards the steady case and the crossing to larger sizes. And every nucleus is on 41P’s orbit; a comet with a larger perihelion loses less mass per orbit and has weaker torques, and both clocks slow together.

What survives the simplifications is the comparison of exponents. The change of spin per orbit goes as the inverse square of the radius and the supply of ice as the radius, so whatever the efficiency and the orbit, a small enough nucleus is spun apart first.

Still open: whether the missing nuclei are missing or hidden #

The question that decides whether outgassing torques shape the comet population is observational. A survey that measured the nuclei of faint comets far from the Sun, where their comae have faded and a small nucleus is no longer hidden, would find either the small nuclei the power law predicts or the deficit the torques predict. Infrared observations from space, which see the nucleus’s own heat rather than reflected light and are less confused by dust, have begun to measure nuclei below a kilometre. The other test is the spin changes themselves: every comet whose rotation is followed across a perihelion adds one more value of $k_T$, and a dozen more would turn a range spanning a factor of forty into a distribution. The calculation predicts that the smallest nuclei whose spins are measured should be changing fastest — which 41P, among the smallest yet followed, already was.