The stars that could catch the Earth in transit

Every argument about which planets a survey finds is an argument about geometry, and the plainest of them is the transit probability. A planet crosses its star’s disc, as seen from the Earth, only if its orbit is tilted to the line of sight by less than about the ratio of the star’s radius to the orbit’s radius. For an Earth twin round a Sun twin that ratio is one part in two hundred and fifteen, so one such planet in two hundred and fifteen is seen to transit and the rest are invisible to the method. Every occurrence rate drawn from transits divides by that factor, and every catalogue of transiting planets is a sample drawn from a different sky than any other method’s.

The same geometry runs the other way, and it is worth running because the answer is concrete. The Earth crosses the Sun’s disc for an observer who lies within the same angle of the plane of the Earth’s orbit. That is a band of sky about the ecliptic, a little over half a degree wide, and any star inside it has a view in which the Earth transits once a year. Which stars those are, how many, and for how long they stay in it, are questions about the solar system’s own detectability — the same arithmetic as a survey’s selection effect, with the solar system as the target.

A band half a degree wide #

The band’s half-width is the angle the Sun’s radius subtends at the Earth’s distance: 0.266 degrees, about the angular radius of the Sun as seen from here, which is no coincidence. An observer exactly in the ecliptic plane sees the Earth cross the centre of the Sun’s disc; an observer 0.266 degrees above it sees the Earth graze the Sun’s limb; further away, no transit. The band covers 0.47 per cent of the sky — exactly one part in two hundred and fifteen, the transit probability of an Earth twin, because the two numbers are the same geometry seen from opposite ends.

Every planet has a band of its own, centred on its own orbital plane and with a half-width equal to the Sun’s radius divided by its own orbital radius. Mercury, closest, has the widest band, ±0.69 degrees, and its seven-degree inclination tilts it across the ecliptic in a wave that reaches seven degrees north and south. Venus’s band is about half as wide and tilted by three and a half degrees. Mars’s and Jupiter’s are narrower and less tilted. All the bands are great circles — the planets’ orbital planes all pass through the Sun — so each crosses the Earth’s at two points, the nodes of that planet’s orbit relative to the ecliptic.

Those crossings are the only places from which two planets can be seen to transit. Away from them an observer sees at most one, and over almost all the sky, none.

A system that looks single from outside #

The giants’ bands make the point more strongly, because they are so narrow.

A distant astronomer looking for Jupiter by transit would have to sit in a band a tenth of a degree wide, and would then see a transit once every twelve years, lasting about thirty hours. For Neptune the band is two hundredths of a degree and the transits come once every hundred and sixty-five years. A survey of the solar system from outside, by the method that has found most known planets, would almost certainly see one planet at most, and most likely none.

Counting the overlaps over the whole sky makes it exact.

From 2.3 per cent of the sky the solar system shows one transiting planet; from a tenth as much it shows two; from about three hundredths of a per cent it shows three; from nowhere does it show more. The solar system, with eight planets in orbits inclined by a few degrees to each other, is what an outside survey would catalogue as a single-planet system, and once in a long while as a pair. That is the direct demonstration of the point that the number of planets a star has is not a measurement that transits can make: the observed multiplicity is the true one filtered through mutual inclinations that a few degrees are enough to make fatal.

It also says something specific about what kind of system the solar system would look like. The two-planet views are dominated by pairs including Mercury, whose wide band crosses everything, and the three-planet patches sit where three bands happen to cross near one longitude. An observer in one of them would see three planets with periods spanning a factor of tens and very different transit depths, and would have no way of knowing the other five were there.

How many stars sit in the band #

The band’s share of the sky converts directly into a count of stars, if the stars near the Sun are spread evenly in space — which, within a few hundred parsecs and above the thin disc’s own thickness, is a fair approximation.

With about a tenth of a star per cubic parsec, red dwarfs and all, the band holds about two stars within ten parsecs, fifty within thirty, and two thousand within a hundred. Catalogues built from the Gaia positions give counts of the same order — about a thousand main-sequence stars within a hundred parsecs in the band at present, the difference being mostly in which faint dwarfs are counted. The closest are a few parsecs away, close enough that a telescope of the kind now being built could, in principle, detect an Earth-sized planet transiting a Sun-like star at that distance from the depth of its transit and the precision of the light it removes.

That is the sense in which the band has been discussed — as the list of places from which the Earth itself could be found by the method that finds most planets here. It is a short list. The number of stars within a hundred parsecs is around four hundred thousand; the band holds half a per cent of them.

A list that does not stay the same #

The band is fixed to the Earth’s orbital plane, and the stars are not fixed to anything. Every star near the Sun moves across the sky at a rate set by its velocity across the line of sight and its distance, and a star moving perpendicular to the ecliptic crosses the band in a time that depends on both.

A typical star near the Sun moves at a few tens of kilometres a second relative to it. At ten parsecs such a star crosses the band in about three thousand years — less time than has passed since the pyramids were built. At a hundred parsecs it takes thirty thousand. The ecliptic itself moves too, as the planets’ perturbations rock the Earth’s orbital plane by about forty-seven arcseconds a century — a motion distinct from the precession of the Earth’s axis, which moves the equator and not the orbit —, but that is far slower than the stars’ own drift for anything within a few hundred parsecs.

So the list of stars that could see the Earth transit is a list with a date on it. Several nearby stars with known planets illustrate the turnover. Ross 128, eleven light-years away, was in the band for a couple of thousand years and left it about three thousand years ago; Teegarden’s Star enters it in a few decades; the seven-planet system round TRAPPIST-1 enters it in under two thousand years. None of those dates is remarkable in itself. What they show is that the band is a moving window rather than a fixed address.

How many stars have had the view #

Integrate the drift over a span of time and the number of distinct stars that have been in the band at some point grows linearly with that span, fastest for the nearest stars.

Within a hundred parsecs about two thousand stars are in the band now, and in the five thousand years of written history about twenty per cent more have passed through it. Within thirty parsecs the turnover is much faster: the fifty or so in the band now become more than a hundred over ten thousand years, because the near stars sweep across the sky quickly. The stars with the best view — close enough that an Earth’s transit would be detectable with technology much like the present one — are precisely the stars that come and go most rapidly.

The point is not that anyone is watching. It is that a transit zone is a property of an orbit and a moment, and that catalogues of planets found by transit are selected by the same geometry: the planets catalogued from here are those whose systems happen to lie edge-on to the Sun now, and over tens of thousands of years a different set would be.

What the symmetry says about the survey #

The equality between the band’s share of the sky and the transit probability is worth a second look, because it is a cleaner statement of the transit selection effect than the usual one. The usual statement is that one Earth twin in two hundred and fifteen transits. The reversed statement is that, of all the directions from which the Earth could be viewed, two hundred and fifteen out of every two hundred and sixteen see no transit. A survey of Earth twins round a thousand nearby Sun-like stars, with orbits oriented at random, would find about five. If it found none, it would not be evidence that Earth twins are rare until the survey’s size was several times larger than the inverse of the probability.

The planets’ mutual inclinations make the reverse calculation sharper than the single-planet one. A survey that finds one transiting planet round a star learns that the star’s system has at least that planet, and very little about how many more it has, because a tilt of a degree or two removes the others from view, exactly as it would from most of the patches of sky where the solar system shows one. That is why systems with several transiting planets are informative out of proportion to their number: they exist only where the orbits are aligned to well under a degree, so they measure how flat planetary systems can be, and the solar system, by this test, is not among the flattest.

There is a second asymmetry the reversed view exposes. A transit survey’s detection threshold depends on the depth of the transit and on how many transits fall within the observing span. For the solar system seen from outside, the Earth’s transit is 84 parts per million deep and comes once a year, Jupiter’s is one per cent deep and comes once in twelve years. A survey running four years would see the Earth four times and Jupiter once or not at all — every method prefers what it can repeat — so the planet a ten-year survey from outside would most likely catalogue is not the largest but the one that transits most often within its window.

What a transit of the Earth would tell its observer #

Suppose an observer in the band, ten parsecs away, watched the Earth cross the Sun. What could be learned is a useful reminder of what any transit gives.

The transit lasts about thirteen hours for an observer on the band’s centre line and less towards its edges, and removes 84 parts per million of the Sun’s light — the ratio of the Earth’s disc to the Sun’s. The depth gives the planet’s radius relative to the star’s, and nothing else; the depth is not the area once the darkened limb and any unresolved companion star are allowed for. The duration and the period together give the Sun’s mean density, because a transit’s length is set by how fast the planet moves across a disc of known size, and Kepler’s third law ties that speed to the star’s mass. A second transiting planet, if the observer sat in one of the rare double patches, would have to agree on that density — every planet of a star must agree on it — and the agreement is what would confirm that both orbit the same star rather than one orbiting a faint companion.

The Moon would be visible too, in principle. Its disc is a seventh of the Earth’s in area, so it removes about seven parts per million, a small separate dip that leads or trails the Earth’s by up to a few hours depending on the Moon’s phase. More measurable than the Moon’s own shadow is its pull: the Earth circles the Earth–Moon barycentre at four and a half thousand kilometres, and the Earth’s transit therefore arrives up to about two and a half minutes early or late from one year to the next, in a pattern that repeats with the Moon’s synodic month folded against the year. That is the signature that a transit running late carries for a companion, applied to the only planet whose moon is known — and whether such a moon could stay bound at all is a question about the region a planet may keep a moon in, which the Earth’s Moon sits well inside.

The observer would not get the Earth’s mass from the transit. For that it would need the Sun’s wobble — nine centimetres a second from the Earth, swamped by the twelve metres a second from Jupiter — and that is where an outside view of the solar system becomes as hard as the inside view of any other system.

Why the band was proposed as a place to listen #

The band’s best-known use is not as a selection effect but as a search strategy. If there were observers anywhere who found planets the way astronomers here find them, the stars from which the Earth transits are the ones where the Earth would have been noticed, and its atmosphere — oxygen, ozone, water and the seasonal swing of carbon dioxide — read in transmission during transit. Searches for artificial radio signals have therefore given extra weight to stars in the band, on the argument that a civilisation that had found the Earth would have a reason to point at it.

The argument is worth stating because the arithmetic in this essay limits it. A transit of the Earth is a shallow, rare event — 84 parts per million, once a year, for thirteen hours — and detecting its atmosphere through it, from ten parsecs, would demand a telescope far larger than any now planned here. And because the band turns over on the timescale of history, the stars in it now are not the stars that were in it when the Earth’s atmosphere first carried the signatures of life, two and a half billion years ago, nor the ones in it when its radio leakage began a century ago. A radio signal sent by a star in the band in response to a transit would reach the Earth only after a light travel time of decades to centuries, by which time the nearest of those stars may already be leaving it. The selection effect that the band describes is real; whether it means anything for listening depends on a timeline the geometry cannot supply.

Grazing transits, uneven skies and fast halo stars #

The bands here are computed with the planets as points, a uniform solar disc and a sharp edge to the zone. A grazing transit at the band’s edge removes very little light and would not be detected, so the effective band for detection is narrower than the geometric one by an amount that depends on the observer’s precision, and somewhat narrower again because the Sun’s limb is darker than its centre. The star counts assume a uniform density, which ignores the disc’s vertical structure and the fact that the ecliptic is tilted by sixty degrees to the Galactic plane, so the band runs through denser and sparser parts of the sky. And the turnover uses a single mean speed for all stars, where the real distribution has a long tail of fast-moving halo stars that cross the band in centuries. None of these changes the size of the numbers by more than a factor of order unity, which is the precision at which the band’s contents are interesting.

Still open: how flat other systems are, seen the way the solar system would be #

The solar system would show at most three of its eight planets from any direction, and its planets are inclined to one another by a few degrees. Whether that is typical is not known. The systems with many transiting planets found so far are flatter, but they were found because they are flat, which is the same selection the reversed view describes. The measurement that would settle it is a census of planets round nearby stars that does not depend on alignment — from the stars’ wobble, measured both along the line of sight and across the sky by astrometry precise enough to reach Earth-mass planets — compared with the transiting subset of the same sample. If the full census finds most systems as tilted as the solar system, then transiting multiples are rare exceptions and the solar system is ordinary. If it finds most systems flatter, then the solar system’s few degrees of disorder are a fact about its own history, and a distant survey that saw only one of its planets would be catching it at its least typical.