A distant planet felt only as a stretch

In 1846 Neptune was found in the residuals of Uranus: the observed planet drifted from the positions its orbit predicted, and the drift was inverted to give the direction of an unseen body pulling on it. The method worked spectacularly once and has been tried ever since. Several planets have been predicted beyond Neptune from irregularities in the outer planets’ motions, and every one of those irregularities has turned out to be an error in the observations or in the masses assumed for the known planets.

The question has returned in a sharper form. The orbits of a handful of distant Kuiper-belt objects are clustered in orientation in a way that some dynamicists read as the signature of a planet several times the Earth’s mass, a few hundred astronomical units out. And the planetary ephemerides are now fitted to ranging data of extraordinary precision: a spacecraft orbited Saturn for thirteen years, and the round-trip time of its radio signal fixed Saturn’s distance from the Earth to a few tens of metres throughout. If a planet of that kind exists, does Saturn’s orbit already know?

The answer is less direct than the history suggests, and the reason it is less direct is the content of the problem. An ephemeris cannot see most of what a distant planet does to Saturn.

A pull that moves everything is invisible #

A body of ten Earth masses five hundred astronomical units away pulls on Saturn with an acceleration of under $10^{-12}$ metres per second squared. It pulls on the Sun with almost exactly the same acceleration, in almost exactly the same direction, because at five hundred astronomical units the Sun and Saturn are nearly the same place. An ephemeris measures where the planets are relative to the Sun — or, more exactly, relative to the Earth, which is also being pulled the same way. A uniform acceleration of the whole solar system changes nothing in it that can be measured from inside.

What survives is the difference between the pull on Saturn and the pull on the Sun: the tidal field. It is smaller than the pull itself by the ratio of Saturn’s distance from the Sun to the body’s distance, and it is a stretch rather than a push — outward along the line to the body, inward across it. Its strength is proportional to the body’s mass divided by the cube of its distance. That ratio is the only property of the body an ephemeris can measure. A body twice as far away and eight times as massive produces exactly the same tidal field and exactly the same effect on every planet, and nothing in the planets’ motions can tell the two apart.

The cube is the reason every line in the opening figure has the same slope. It also says something about what kind of measurement this is. A planet’s direct pull falls as the square of its distance and is what Neptune’s discoverers detected in Uranus, a body only ten astronomical units from Neptune at closest approach. A body hundreds of astronomical units away is felt only through its tide, and the tide falls one power of distance faster. Moving the proposed planet from four hundred to eight hundred astronomical units makes it eight times harder to see.

Most of the stretch looks like a different orbit #

The tidal field perturbs Saturn’s orbit, and the perturbation is not small in absolute terms: over thirteen years it moves Saturn by hundreds of metres. The ranging measures Saturn’s position to about twenty. The perturbation should be easy to see. It is not, because the ephemeris does not measure the perturbation directly.

An ephemeris is a fit. Saturn’s orbit is not known in advance; its six orbital elements are adjusted, along with everything else, until the integrated orbit matches the data. If a distant body has perturbed Saturn, the fit does not compare Saturn’s observed position with its unperturbed orbit, because there is no independent record of the unperturbed orbit. It compares it with the best orbit it can find — and a slightly different orbit can imitate most of what a smooth, slowly varying perturbation does over a fraction of a revolution.

The fit has six numbers to adjust, and they can be traded against the perturbation in six independent ways: a change in Saturn’s semi-major axis shifts its period and makes it run ahead or behind; a change in eccentricity bends its distance up and down once an orbit; a change in the orbital orientation shifts where those bends fall. A tidal field acting over thirteen years produces a perturbation that is mostly a combination of exactly those patterns, and the fit removes it. What remains is the part no Keplerian orbit can reproduce, and it is that remainder, not the perturbation, that the ranging has to detect.

In the figure the remainder is still well above twenty metres, so a body of ten Earth masses at that distance and in that direction would have been seen. But the margin is a factor of four, not the factor of thirty the raw perturbation suggested, and in other directions it is less.

Why Neptune could be found and this planet cannot #

The contrast with 1846 is worth making exact, because it is not a matter of better or worse data. Uranus had been observed for sixty years when the discrepancy became undeniable, and by then it had wandered about two minutes of arc from its predicted place. Neptune, at its closest, came within about ten astronomical units of Uranus — closer to Uranus than Uranus is to the Sun. At that range Neptune’s pull on Uranus was not a tide at all. It was a direct, strongly varying force, pulling Uranus ahead as it approached conjunction and holding it back afterwards, and the pattern of acceleration and delay pointed straight at the perturber. Le Verrier and Adams solved for the perturber’s position and the telescope found it within a degree.

A body at five hundred astronomical units never comes close to anything. Its direction from each planet changes by a few degrees over a century, its distance hardly at all, and its pull on the Sun and on every planet is nearly identical. The pattern that made Neptune visible — a force that rises and falls as the two planets pass — is entirely absent. What remains is a nearly static stretch of the whole planetary system, which is the hardest kind of perturbation to distinguish from a slightly different set of orbits.

The history of predicted planets since Neptune is a history of forgetting that distinction. The planet inside Mercury’s orbit, invoked to explain the excess precession of Mercury’s perihelion, was a direct perturber of exactly Neptune’s kind, and it was not there; the anomaly was relativity. The planets predicted beyond Neptune in the early twentieth century were inferred from residuals in Uranus and Neptune that came from an overestimate of Neptune’s mass; Pluto was found in the predicted region by a search that would have found it anywhere along the ecliptic, and it was far too small to have caused the residuals. The regularities in planetary spacing that once suggested where planets ought to be have no power to say what lies beyond the last one.

A range is a distance between two moving things #

The ranging that constrains Saturn’s orbit measures the distance from an antenna on the Earth to a spacecraft at Saturn, by timing a radio signal there and back. A position measured from a frequency and a round-trip time is a measurement of one direction only — along the line between the two — and the Earth is moving too. The tidal stretch of a distant body acts on the Earth’s orbit as well as on Saturn’s, with a strength smaller in proportion to the Earth’s smaller distance from the Sun, so the range carries a little of the Earth’s perturbation alongside Saturn’s.

The other two directions come from different data. Interferometric measurements of the spacecraft’s position against background quasars — an angle measured against a quasar — fix where Saturn is on the sky to a few milliarcseconds, which at Saturn’s distance is tens of kilometres: far less precise than the range, but in the directions the range cannot see. A tidal perturbation that happens to lie mostly across the line of sight is hidden from the ranging and constrained, weakly, by the angles.

So the sensitivity map is not only a property of the perturbation’s geometry and Saturn’s arc. It is also a property of where the Earth was when the ranging was done, and of which components of Saturn’s position each kind of data constrains. The calculation drawn here takes the radial component as a stand-in for the range, which is good to within the few degrees by which the line from the Earth to Saturn departs from the line from the Sun to Saturn. The real analyses carry the full geometry and find a sensitivity map of the same character, with its own particular blind spots.

A longer arc keeps more #

The fit absorbs the perturbation because the arc of data is short compared with Saturn’s orbit. Thirteen years is less than half of Saturn’s twenty-nine-year period, and a slow curve observed over less than half its length can be bent to imitate a great many smooth disturbances.

The growth is steep. Over a short arc a perturbation is close to a constant extra acceleration, and a constant acceleration is almost perfectly imitated by a slight change in the orbit’s curvature; over a longer arc the tidal perturbation develops structure on the scale of the orbit itself — it varies as Saturn moves round, because the tidal stretch depends on the angle between Saturn and the body — and that structure is what no single Keplerian orbit can match. A record spanning a whole orbit would be far more sensitive than one spanning a third.

This is why the old optical observations of the outer planets, imprecise as they are, still matter to the problem. A century of positions accurate to a tenth of an arcsecond covers several orbits of Saturn and Jupiter, and while each position is thousands of times less precise than a range, together they constrain the long-period structure that a short arc of ranging cannot. The two kinds of data are complementary in exactly the way the figure shows: precision on a short arc, coverage on a long one.

The directions a planet could hide in #

The fit’s ability to absorb the perturbation depends on how the tidal stretch is oriented relative to Saturn’s arc, and therefore on where the unseen body is.

The variation by a factor of five in the surviving signal is a factor of five in the mass that can be hidden. A planet in the most favourable direction for hiding can be five times heavier than one in the least favourable, at the same distance, and leave the same residual. Any honest limit from the ephemeris is therefore a map rather than a number: the allowed mass as a function of the body’s direction and distance. The published analyses present it that way, and they show regions of the sky where a planet of the proposed mass and distance is excluded and others where it is not.

The asymmetry has a simple origin. A tidal stretch whose axis lies across Saturn’s orbit squeezes and stretches Saturn’s path in a way that varies as Saturn moves along the arc; one whose axis lies along the arc acts nearly the same throughout and is easier to imitate. The precise pattern depends on the arc’s length and on which part of Saturn’s orbit it covered, which is a property of when the spacecraft happened to be there.

Saturn is not the only witness #

The calculation here fits Saturn alone, with the rest of the solar system held fixed. The real ephemerides fit every planet at once, and that changes the problem in both directions.

It makes detection easier, because the same tidal field acts on every planet, with a strength proportional to each planet’s distance from the Sun and a direction that depends on each planet’s position. Jupiter, Mars and the Earth all feel the stretch, and a fit that must absorb it in all of them at once with only their own orbital elements has less freedom than a fit to Saturn alone. A planet hidden from Saturn’s arc may show in Jupiter’s.

It also makes detection harder, because the ephemeris has other parameters that can absorb a tidal stretch. The masses of the asteroids, fitted alongside the planets, trade against one another and against the planets’ orbits; the mass of the Kuiper belt, represented as a ring, produces a tidal field of its own; and an error in the conversion from a position to an observation — the light time, the relativistic delay near the Sun — can mimic a small systematic shift in Saturn’s range. Each of these is a nuisance parameter, and each can take up some of the signal a distant planet would leave.

The net effect, in the full analyses, is limits tighter than a single-planet fit gives in the best directions and comparable in the worst. What survives every version of the calculation is the cube and the direction: the measurement is of mass over distance cubed, and it is strongest in some parts of the sky and weakest in others.

What the proposed planet would have done #

The shaded box in the opening figure is the range proposed for a planet beyond Neptune: five to ten Earth masses, four hundred to eight hundred astronomical units. It straddles the lines. At its near, heavy corner a planet would have been seen in Saturn’s ranging in almost any direction; at its far, light corner it would have been invisible in every direction. The ephemeris cannot rule the proposal in or out as a whole; it removes parts of it.

Those parts are not arbitrary. Because the sensitivity depends on direction, the ephemeris constrains where along its supposed orbit the planet can now be. A planet on a highly eccentric orbit spends most of its time near aphelion, far away and hard to see; near perihelion it would be closer and more detectable. Analyses combining the ranging with the planet’s proposed orbit have used this to exclude stretches of the orbit and favour others, and to direct the optical searches that would settle the question by finding the planet’s light.

That is the ephemeris’s proper role. It cannot discover a distant planet the way Uranus’s residuals discovered Neptune, because a distant planet is felt only through a weak, mostly absorbable tide. It can tell a search where not to look.

An argument that applies to any distant mass #

The same reasoning constrains anything far away. The Sun is pulled by the whole Galaxy, by the nearest stars, and by the dark matter in the solar neighbourhood, and none of that is visible in the ephemeris except through its tidal field. The Galactic tide on the planets is far below any detectable level; its effect is felt only by the Oort cloud, a hundred thousand times further out, where the tide is strong enough to change cometary orbits and send comets inward. The same stretch that is invisible in Saturn’s orbit is the dominant force on the outermost bodies of the solar system.

An ephemeris also bounds a spherical cloud of dark matter centred on the Sun, which would add a small extra attraction growing with distance and make the outer planets’ orbits precess. That signal is not tidal in the same sense — it does not cancel between the Sun and a planet — and the bounds on it are correspondingly tighter. The difference is again a matter of what moves the Sun and the planets together and what moves them apart.

Still open: whether the clustering needs a planet #

The case for the distant planet does not rest on the ephemeris; it rests on the clustered orientations of a few distant Kuiper-belt objects, and on whether that clustering is real or an artefact of where telescopes have looked. Surveys find distant objects most easily in some parts of the sky and at some times of year, and a sample drawn through that window can show clustering that the population does not have. As deeper surveys with more uniform coverage add objects, the clustering either survives or fades. If it survives, the planet’s direction becomes better defined, and the ephemeris’s map of where a planet of that mass can hide becomes a strong constraint on where to look. If it fades, the ephemeris will have done what it has done for every predicted planet since Neptune: shown that nothing in the inner solar system’s motion requires one.