A shower that is a reading of an orbit

Six numbers fix an orbit: the size and shape of the ellipse, the tilt of its plane, the direction in which that plane crosses the reference plane, the direction of perihelion within the plane, and a time. Five of them are geometry and one is a clock, and they are usually met as the output of a calculation — a fit to a comet’s positions, a catalogue entry for a satellite. A meteor shower presents them the other way round. It is a natural measurement of a stream’s orbit, made by the Earth itself, read in the date of the shower, the speed of the meteors and the point in the sky from which they appear to come.

The meteoroids are grains of dust and grit shed by a comet, or in a few cases an asteroid, as it passes the Sun. Each grain leaves the parent at a few metres a second, which is nothing beside the parent’s own orbital speed, so it follows nearly the same orbit, a little ahead or a little behind. Over many returns the grains spread along the whole orbit into a stream, and when the Earth crosses that stream, the grains strike the atmosphere at tens of kilometres a second and burn up as meteors. The shower is the crossing. Everything about it is the geometry of two orbits that meet.

A meeting only at a node #

Two orbits in different planes can meet only on the line where the planes intersect. For the Earth’s orbit and a stream’s, that line is the line of nodes, and the stream crosses the Earth’s orbital plane at two points along it: the ascending node, where it passes northward through the plane, and the descending node, where it passes southward. Whether a shower happens depends on whether one of those two points lies at the Earth’s distance from the Sun.

The node’s distance is set by the orbit’s shape and by where the perihelion lies relative to the node — the argument of perihelion, ω. Swift–Tuttle’s orbit is long and thin, reaching from just inside the Earth’s orbit to beyond Pluto’s, and for most values of ω both its nodes would be far from 1 AU: one near perihelion, inside the Earth’s orbit, and one out among the giant planets. Only in four narrow windows of a few degrees each does a node fall at the Earth’s distance. Swift–Tuttle’s present argument of perihelion puts its descending node at 1.01 astronomical units, within a few hundredths of the Earth’s orbit, and the Perseids are the result.

That is the first thing a shower says about its orbit: one of its elements sits in a window a few degrees wide, and the shower exists because it does. The planets’ pull slowly turns every orbit’s perihelion and node — which direction of push moves which element is the standard result of perturbation theory — so the window is crossed eventually, and showers begin and end on timescales of thousands of years as their streams’ nodes drift through the Earth’s distance.

A speed that is an inclination #

The date at which the Earth reaches the node is the node’s longitude, the element Ω, read directly off the calendar as the Sun’s position. The speed of the meteors is a different element.

At the node the meteoroid and the Earth are at the same place, and the meteoroid’s velocity differs from the Earth’s in direction by an angle set mostly by the orbit’s inclination. A stream in the plane of the Earth’s orbit, going the same way round the Sun, overtakes the Earth from behind with a small relative speed — twelve kilometres a second for a nearly parabolic orbit, to which the Earth’s own gravity adds enough to bring the meteors in at seventeen. A stream on a retrograde orbit, going the other way round the Sun, meets the Earth head-on and the speeds add, to seventy-two. The curve between the two extremes is the law of cosines applied to two velocity vectors, and every shower lies near it.

The four showers in the figure, computed from the orbital elements of their parent bodies alone, come out at the speeds the meteors are measured to have. The Draconids, from a comet on a low-inclination prograde orbit, are among the slowest meteors seen; the Leonids, from a comet on a nearly reversed orbit, are the fastest. The Geminids sit above the curve because their parent’s perihelion is far inside the Earth’s orbit, so they arrive with a large radial velocity that the curve, drawn for perihelion at 1 AU, leaves out. A meteor’s speed, in other words, is a reading of its orbit’s inclination, corrected for its eccentricity and perihelion distance, and a meteor camera network that triangulates the speed of a single fireball measures its heliocentric orbit to a few per cent from that alone.

A direction that is a velocity #

The meteors of a shower all seem to come from one point in the sky, the radiant — the Perseids from Perseus, the Leonids from Leo — and the reason is perspective. All the meteoroids in a stream move along nearly parallel paths, and parallel lines seen in projection converge on a vanishing point, as railway tracks do. The radiant is the direction from which the stream’s grains approach the Earth, and so it is the direction of their velocity relative to the Earth. It combines the meteoroids’ heliocentric velocity with the Earth’s own, which is why the radiants of retrograde showers lie near the direction the Earth is heading in and appear high in the sky before dawn, when the observer is on the Earth’s leading side.

The radiant and the speed together give the meteoroids’ full geocentric velocity vector; subtract the Earth’s velocity and the result is the heliocentric velocity at the node. A position (the node, at the Earth) and a velocity together fix an orbit — the same reconstruction that an incoming asymptote and its speed give for a body that arrives once — and so a shower’s date, speed and radiant are sufficient to determine every element of its stream’s orbit except the time of perihelion. That is how the Perseids were tied to comet Swift–Tuttle in 1866, by Schiaparelli, who computed the stream’s orbit from the shower and found it matched the comet’s. It was the first proof that meteors are cometary dust.

Two showers from one comet, and neither node at the Earth #

Halley’s comet shows what a stream is, as distinct from its parent’s present orbit.

Neither of Halley’s nodes lies at the Earth’s distance today: one is at 1.78 astronomical units and the other at 0.85. A shower from its present orbit should not exist. Yet the Earth meets Halley’s dust twice a year, in early May as the Eta Aquariids and in October as the Orionids. The resolution is that the stream is older than the orbit. Halley has shed dust for thousands of returns, and over that time its argument of perihelion and its node have been turned by the planets through angles of tens of degrees. Dust shed long ago still follows the orbits it was released on, which had their own nodes at their own distances. The stream is a fan of orbits spanning the parent’s history, wide enough that part of it crosses the Earth’s distance at both nodes even though the comet’s current orbit crosses at neither.

The Geminids show the same thing from a different parent.

Phaethon’s descending node is at 0.89 astronomical units, a tenth of an astronomical unit inside the Earth’s orbit, and the window that would put it at 1 AU lies a few degrees of argument of perihelion away. The Geminids are nonetheless the richest annual shower. Their meteoroids were released on orbits slightly different from Phaethon’s present one, and the planets have since spread them; the Earth crosses the part of the stream that lies where Phaethon’s orbit did not. A shower measures its stream’s elements, not its parent’s, and the difference between the two is a measure of the stream’s age.

A duration that is a width #

How long a shower lasts is a separate measurement, of how wide the stream is where the Earth crosses it.

The Earth moves through space at thirty kilometres a second, an astronomical unit in fifty-eight days, so a shower lasting two days is a stream a few hundredths of an astronomical unit across, and one lasting five weeks is a stream half an astronomical unit wide. The Perseids have both: a core a couple of days wide, made of dust released in the last few thousand years, and a broad low background, made of older dust spread further. The Leonids show the other extreme. Their annual shower is weak, but on occasion the Earth passes through a single trail of dust released at one particular return of the comet, only a few hundred thousand kilometres wide, and the result is a storm of thousands of meteors an hour lasting an hour or two. The storm of 1966 produced tens of meteors a second for a few minutes.

Those trails are thin enough to be followed individually, and since the late 1990s the dates and even the hours of Leonid storms have been predicted in advance by integrating the dust from each past return of the comet forward under the planets’ gravity, a calculation that depends on knowing each trail’s elements to a precision of thousandths of an astronomical unit. The trails’ positions are moved also by the push of sunlight on the grains, and by the gas jets that push a comet and turn it when the dust is released, and the predictions that succeeded were the ones that modelled both.

A shower that moves through the calendar #

There is one change a shower makes that its orbit does not. The Perseids’ peak falls on the twelfth or thirteenth of August now. Chinese records of the first centuries AD place the shower in July.

The calendar is tied to the equinox, so that the seasons stay put, and the equinox moves westward round the ecliptic by about fifty arcseconds a year as the Earth’s axis precesses. A point fixed among the stars — the node of a stream that is not itself moving — is reached about twenty minutes later each year by the calendar, a day and a half later each century, nearly four weeks later in two thousand years. The drift of the Perseids from July to August over two millennia is almost entirely this. The stream’s node, fixed in space to within the slow turning the planets produce, has barely moved; the calendar has moved under it.

That separation is the point of reading the shower as an orbit. A date in the calendar mixes the orbit’s node with the observer’s reference frame. The same date expressed as the Sun’s longitude in a fixed frame — the convention meteor astronomers use — is the node itself, and changes in it measure the stream’s real evolution.

When the window closes #

Because a shower depends on a node lying within a narrow window of distance, showers are temporary. As the planets turn a stream’s orbit, its node moves through the Earth’s distance and out again, and the shower brightens, peaks and fades over centuries to tens of thousands of years. Comet Biela split into two pieces in the 1840s, disintegrated, and left behind the Andromedids, which produced great storms in 1872 and 1885 and have since dwindled to almost nothing, because the stream’s node has moved away from the Earth’s orbit. Other streams have no known parent at all: the parent is gone, and the stream remains, slowly being turned out of reach.

The same turning sets which showers will exist. A stream whose node is approaching the Earth’s distance will produce a new shower in some future century, and integrations of the orbits of known comets and asteroids predict several. The prediction is a statement about one element — where the node will lie — and how fast the planets move it.

A stream with no node #

The argument has a limiting case that shows what the node is. A stream whose orbit lay exactly in the plane of the Earth’s orbit would have no line of nodes at all: its plane and the Earth’s would coincide, and the ascending node would be undefined, as it is in the elements that stop existing at zero inclination. Such a stream would meet the Earth’s orbit not at two points but along an arc, wherever its distance from the Sun equalled the Earth’s, and the shower would last as long as the Earth took to traverse that arc — potentially weeks or months — with meteors arriving at the low speed of a prograde orbit. The low-inclination streams near the ecliptic behave roughly like this: the complex of weak showers associated with comet Encke, whose orbit is inclined by twelve degrees and whose nodes swing slowly, produces broad, slow activity in October and November, and in June on the daytime side, where it was found by radar because the meteors arrive from near the Sun.

The opposite limit, a polar orbit at ninety degrees, meets the Earth at right angles to its motion: the meteors arrive at the speed that combines the two velocities in quadrature, about fifty kilometres a second, from a radiant near the ecliptic’s pole. Between the two limits every inclination is represented among the known showers, and each sits where its speed says it should. The order in which three rotations put an orbit in space — node, inclination, perihelion — is the order in which a shower reveals them: the date, then the speed, then, with the radiant, the rest.

What survives to be counted #

A shower also says something about which parents can produce one. A comet on its first passage from the Oort cloud — one of the first-time visitors that make the famous spike in the distribution of original orbits — sheds dust onto an orbit that has never been travelled before and will not be travelled again for millions of years. Its dust trail is a single thin strand, and the Earth would have to cross the comet’s orbit within a few years of the comet’s passage to meet it. Most known showers come from comets with periods of decades to centuries, which have laid down dust over many returns along nearly the same orbit, and from the occasional asteroid that sheds dust as it is heated near perihelion. The showers are a census of the small bodies that come back.

They are also a hazard measured. The same streams that produce meteors are crossed by every satellite in Earth orbit, and the meteoroid flux during a storm can exceed the background flux by orders of magnitude for an hour; satellite operators turn solar panels edge-on to the radiant during predicted Leonid storms, as they plan around the collision rates that space debris imposes. The prediction of a storm to the hour is the prediction of a trail’s elements to the thousandth of an astronomical unit, which is the most exacting use of the six numbers anywhere in the solar system’s small bodies.

One orbit standing for a fan of them #

The figures treat each stream as a single orbit, the Earth’s orbit as a circle and the Earth’s plane as the ecliptic, and the speeds are computed at the node of the parent’s present orbit rather than of the stream. Real streams are fans of orbits with spreads in every element; the Earth’s eccentricity moves its distance from the Sun by three per cent through the year, which matters for which part of a stream it meets; and the meteoroids themselves are pushed by sunlight in proportion to their size, so a stream is sorted, with the smallest grains on somewhat different orbits from the largest. The calendar figure ignores the change from the Julian to the Gregorian calendar, which shifts the historical dates by a further few days. The shower is a measurement of a stream’s elements, not an exact one, and the precision with which a stream is known is set by how many meteors have had their orbits triangulated.

Still open: what the streams with no parents came from #

Camera networks now measure the orbits of tens of thousands of meteors a year, each triangulated from two or more stations, and the catalogues contain hundreds of streams, most of them weak and most without a known parent. Some will be the dust of comets that have died; some, like the Geminids, of asteroids that shed dust in ways not yet understood. The question the elements can answer is which: a stream whose orbit can be integrated back to a common origin with a known asteroid or a dormant comet nucleus gains a parent, and the number that do measures how often small bodies near the Earth are, or were, active comets. The answer has a practical edge, because the parents of meteor streams are, by construction, bodies whose orbits come close to the Earth’s.