A spike only first-time visitors can be in

Oort found the cloud that bears his name as a histogram of a reciprocal. Catalogues listed hundreds of comets with eccentricities above one, which on paper were unbound; Oort worked instead with $1/a$, the reciprocal of the semi-major axis, which is proportional to the orbital energy and passes smoothly through zero from bound to unbound. Computed for each comet before it entered the planetary region — its original $1/a$ — the distribution was a narrow spike just on the bound side of zero, a few hundred-thousandths of an inverse astronomical unit wide. Comets were arriving from a reservoir tens of thousands of astronomical units away, on orbits that barely held them to the Sun.

That was 1950, and the spike is still the cloud’s main evidence. But the spike says more than that the cloud exists. Its narrowness says that the comets in it are seen for the first time; its position says how the Galaxy delivers them; and what is missing from the rest of the distribution says that comets do not last.

Ten times the spike’s width, in one passage #

A comet falling in from the Oort cloud passes through the planetary region once per orbit, and there the giant planets — Jupiter above all — tug on it. The tug is brief and its effect on the orbit’s shape is modest, but its effect on the orbital energy is large compared with the energy the comet has. A comet with a semi-major axis of 25,000 AU has $1/a = 4\times10^{-5}$ AU⁻¹, and the energy change from a single passage past Jupiter’s distance amounts, typically, to a change in $1/a$ of about $5\times10^{-4}$ — more than ten times the comet’s own binding energy.

The size of that kick can be estimated in one line. A comet crossing Jupiter’s orbit feels Jupiter’s pull for about as long as it takes to cross a region the size of that orbit, and the energy it gains or loses per unit mass is of order the planet’s mass fraction times the energy scale of an orbit at that distance: $\Delta(1/a) \sim (m_J/M_\odot)/a_J$, which is $10^{-3}$ divided by 5.2 AU, or about $2\times10^{-4}$ AU⁻¹, with a factor of a few from geometry. The comet’s own binding is set by the size of the cloud, a thousand times larger than Jupiter’s orbit, and Jupiter’s thousandth of the Sun’s mass is not small enough to compensate. The planets are small compared with the Sun but the comet’s binding is smaller still, and that single comparison is why the spike cannot survive a passage.

The consequence is drawn above. Starting from the spike, a comet’s $1/a$ after one passage is spread over a range ten times wider than the spike, symmetrically about where it started. Almost half end with negative $1/a$: the planets have given them more energy than they had, and they leave the Solar System on hyperbolic orbits, never to return. The other half end more tightly bound, with semi-major axes of a few thousand AU or less and periods of tens to hundreds of thousands of years. Only a few per cent end within the spike’s own width.

A comet in the spike is therefore on its first passage through the planetary region, almost by definition. Nothing the planets do can put a comet back into the spike once it has been through them, because their kicks are ten times larger than the target. The comets of the spike are called “new” comets for exactly that reason: their original orbits have never been perturbed by the planets, and they carry, in their $1/a$, the imprint of the cloud and nothing else.

Original and future #

For every well-observed long-period comet the same calculation is made in both directions. Its orbit is fitted to astrometry near perihelion, then integrated backwards until it is far outside the planets, giving the original $1/a$, and forwards until it is far outside again, giving the future $1/a$. The published elements near perihelion are neither: they are osculating elements, the conic the comet would follow if the planets were switched off at that instant, and it is these that produce eccentricities above one on paper.

The pair of numbers carries an asymmetry that settles a question the eccentricities cannot. Negative future values are common: nearly half of all new comets leave on hyperbolic orbits, ejected by the planets. Negative original values are absent: no comet of the thousands with good orbits has arrived on a convincingly hyperbolic orbit, once non-gravitational forces from outgassing are allowed for — the jets that make a comet arrive a day early also bias the fitted energy, and a comet’s original $1/a$ is only as good as the model of its outgassing. That asymmetry is the Solar System’s own record that it keeps what it has and throws out a steady stream of comets, and it is why the two interstellar objects found since 2017, with original $1/a$ thousands of times more negative than the spike’s width, were recognisable at once.

A random walk with a wall at zero #

The half of the new comets that stay bound return, and on every return they receive another kick of the same typical size, in a direction unrelated to the last. Their $1/a$ performs a random walk, with steps of $5\times10^{-4}$ AU⁻¹, and with an absorbing wall at zero: any step that takes the walk below zero ejects the comet for good.

A symmetric random walk starting next to an absorbing wall has a well-known survival law: the fraction still alive after $n$ steps falls as $1/\sqrt{n}$. Most walks end at the first step or soon after; a few wander far from the wall and last a long time. The simulated comets follow that law. Forty-seven per cent make only one passage; a quarter are still long-period comets after eight; four per cent after two hundred. The mean number of passages is about 47, but the mean is carried by the rare survivors that drift deep into the bound region, towards the short periods where they stop counting as long-period comets at all. About one in a hundred of the comets arriving from the cloud end that way — which is a small number and not a negligible one, since it is one supply route for the comets of shorter period.

Too few come back #

The walk makes a prediction that fails, and the failure is one of the oldest results in the subject.

If every comet from the cloud went on being visible for as long as its orbit kept bringing it back, most of the long-period comets seen at any time would be returning comets, deep in the random walk, and the new ones in the spike would be a small minority — each new comet contributes one apparition to the spike and, on average, dozens outside it.

Computed from the same walk, comets that never faded would make first passages only about two per cent of all apparitions. The observed fraction of long-period comets with original $1/a$ in the spike is about a third. The planets alone cannot remove comets fast enough to make the returning population as small as it is. Oort saw this in his 1950 paper and concluded that comets must fade: lose their volatile ices, disintegrate, or become dormant and too faint to be discovered, within a few passages. The figure puts a number on it. For the share of new comets to be a third, a comet must stop being seen after about eight passages on average.

That is the fading problem, and it is not a problem with the model but a measurement of comets. Comets are known to break up — some have been watched splitting into fragments near perihelion — and to exhaust their ice in outer layers while retaining it deeper down. What the spike’s prominence says is that this happens fast: a comet from the Oort cloud is, in a statistical sense, a thing that survives only a handful of close approaches to the Sun. How the fading depends on a comet’s size and perihelion distance, and whether large comets fade at all, is still argued about, and the models that fit the distribution of $1/a$ best require the fading to be severe for the first few passages and gentler afterwards.

What delivers comets past the giants #

The last question the spike answers is why it is where it is. The cloud extends from a few thousand to perhaps a hundred thousand astronomical units, and the comets in it have perihelia far outside the planets. To be seen, a comet’s perihelion must be brought within a few AU of the Sun, and something must change its orbit to do that. For a comet whose perihelion is gradually reduced, there is an obstacle. When its perihelion first comes within ten or fifteen AU, it passes Saturn and Jupiter, and their kicks — as large for such a comet as for the ones drawn above — randomise its energy and eject or capture it before its perihelion can decrease further. The giants form a barrier, and a comet that creeps inward is almost always stopped by it before it is visible.

What gets a comet past the barrier is the Galaxy. The Sun and its cloud sit in the disc of the Milky Way, and the disc’s own mass exerts a tidal force that pulls towards the midplane: stronger the further a body is from the Sun. On a comet at tens of thousands of AU that tide exerts a torque, and averaged over an orbit it changes the orbit’s angular momentum — and therefore its perihelion — steadily, in a cycle that is the same mathematics as the Kozai–Lidov cycle that turns an inclination into an eccentricity, with the Galactic disc in the role of the distant perturber.

The change in angular momentum per orbit from the tide is largest when the orbit is highly eccentric and suitably oriented, and at its largest it is $5\pi G\rho_0 a^2 P$, where $\rho_0$ is the local density of the disc — about a tenth of a solar mass per cubic parsec — and $P$ the orbital period. Since $P$ goes as $a^{3/2}$, the change goes as $a^{7/2}$: steeply. At 10,000 AU it barely moves a perihelion. At 20,000 it moves it by several AU per orbit. Beyond about 24,000 AU it can take a perihelion from outside Saturn to inside Jupiter in one orbit, before the giants have had a chance to act on it. Those comets arrive, visible, on their first pass through the inner Solar System, having jumped the barrier in a single step.

A semi-major axis of 24,000 AU is $1/a \approx 4\times10^{-5}$ AU⁻¹. That is the position of the Oort spike. The spike sits where it does not because the cloud has most of its comets there — it probably has more further in — but because that is the smallest distance from which the Galaxy’s tide can deliver a comet past the giant planets in one orbit. Comets from further in exist and creep inward, and are stopped at the barrier; comets from further out are delivered too, but they are fewer. The spike’s position is, in effect, a measurement of the density of the Galactic disc at the Sun’s position, made with a histogram of comet orbits.

The same astronomer, eighteen years earlier #

There is a coincidence in the history that is worth drawing out, because it turns the spike into a second measurement of something Oort had measured first. In 1932 he estimated the density of matter in the Galactic disc near the Sun from the vertical motions of stars: stars oscillate up and down through the disc’s midplane, and the disc’s gravity, which sets the frequency of the oscillation, is set by the density. The result, a tenth of a solar mass or so per cubic parsec, is called the Oort limit, and whether it exceeds the mass in visible stars and gas has been argued ever since as one of the local tests for dark matter in the disc.

The spike’s position depends on exactly that density. The tide that delivers comets past the giant planets is the vertical pull of the same disc, and the semi-major axis at which it can jump the barrier in one orbit scales as the density to the power $-2/7$ — weakly, but directly. Oort’s comets and Oort’s stars measure the same quantity, one through a histogram of orbital energies and the other through the velocities of stars moving through the plane of a galaxy seen from inside it. They agree, to the precision the comets allow, which is modest; the comets are a check rather than a competitor, but a check made with objects bound to the Sun rather than to the Galaxy.

Which comets are which #

The random walk also explains a division that comet catalogues make on other grounds. Long-period comets are defined by their periods, above two hundred years, but they are also distinguished by how they move relative to Jupiter, through the one combination of elements that a close encounter with Jupiter nearly preserves. Comets with small values of that parameter cannot have been captured from the Jupiter-family population and must have come from the cloud; those with values near three are Jupiter’s own, recycled from the scattered disc beyond Neptune. The walk in $1/a$ occasionally carries a long-period comet into the Jupiter-family range, but only occasionally — the one per cent captured in the figure, with inclinations still randomly distributed — which is why comets of intermediate period, the Halley type, have inclinations spread over the whole sphere while the Jupiter family’s are low. The two populations are two reservoirs, and the number an encounter preserves says which one a comet came from even after the planets have rearranged its orbit.

What the drawing leaves out #

The kicks here are drawn from a Gaussian of fixed width. Real planetary perturbations have a much heavier tail — a comet that passes close to Jupiter receives a kick hundreds of times larger than typical — and their size depends on the comet’s perihelion distance and inclination: comets on retrograde orbits move past the planets faster and are kicked less. The heavy tail changes the ejection fraction at the first passage only a little, and it matters most for the rare comets captured into short-period orbits. The fading threshold of eight passages depends on the width of the kicks and on where the long-period class is cut off, and is a demonstration of the size of the effect rather than a fit.

The tide has been reduced to its largest possible effect per orbit, which a comet only experiences at the right orientation, and to its vertical component; the radial component of the Galactic tide and the random encounters with passing stars both add to it. Passing stars matter most in bursts, when one comes within a few tens of thousands of AU and perturbs a large part of the cloud at once — a comet shower, in which the rate of new comets rises for a few million years. The present rate appears to be quiet, but the evidence for that is itself statistical.

Still open: how big the cloud is behind the spike #

The spike measures the part of the cloud that the tide can deliver past the giants: its outer half, beyond twenty thousand AU. The inner cloud, from a few thousand to twenty thousand AU, is invisible in the spike because its comets are stopped at the barrier — except during showers, when a passing star kicks them past it. How many comets it contains is not known to within a factor of ten, and it matters: the total mass of the cloud, and therefore how many planetesimals the giant planets threw out while they formed, depends on it. The few long-period comets whose original $1/a$ lies well outside the spike, at a few times $10^{-4}$, are candidates for having come from the inner cloud in a recent shower, or for being returning comets that survived their fading; separating the two requires orbits good enough to compute original energies to better than the width of the spike, a precision set by the unmodelled outgassing as much as by the astrometry. The spike was found by computing a reciprocal carefully, and what lies behind it will be found the same way.