Astronomy & Celestial Mechanics
the sky, computed
Concept

Precession of the equinoxes — where it appears

The slow westward drift of the equinox along the ecliptic, completing a circuit in about 25,800 years as the Earth's axis precesses. It is why the pole star changes and why a tropical year differs from a sidereal one.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Mercury's perihelion, term by term. The observed advance of Mercury's perihelion is 5600 arcseconds per century against the equinox. Almost all of it is the equinox: the coordinate frame itself turns, and removing it leaves 574. Subtracting the perturbations of the other planets — computed by Le Verrier in 1859 and refined many times since — leaves 42.98 arcseconds a century that nothing in Newtonian gravitation accounts for. The bars are logarithmic in nothing; they are the real proportions, which is why the residual is barely visible beside the frame term.

Forty-three arcseconds, after everything else

Mercury's perihelion moves through 5,600 arcseconds a century. Nearly all of that is the coordinate system turning, and almost all of the rest is the other planets. What was left over was 43 — under one per cent of the raw number, and the most consequential residual in the history of the subject.

gravitation · Relativistic orbits June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is.

An average that precession cannot move

Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent.

orbits · Orbital averages The same tilt, and ice ages of two different lengths. Top: the obliquity anomaly in degrees from the five largest terms of the standard series, over 2 million years of model time; it is dominated by the 41,000-year cycle and has no power at all near 100,000 years. Bottom: ice volume in a threshold model driven by it — the ice grows steadily, and it collapses whenever volume plus a multiple of the obliquity anomaly crosses a threshold, marked by dots. For the first 1 million years the threshold is low and the ice collapses at nearly every obliquity maximum, on average every 43 thousand years. At the vertical line the threshold is raised, and nothing else changes: the ice now survives one or two obliquity maxima and collapses at the second or third, every 108 thousand years on average. The forcing is identical on both sides; the ice sheet counts it differently.

An ice sheet that skips a beat of the tilt

The last eight hundred thousand years of ice ages come roughly every hundred thousand years, and the Earth's orbit has no forcing of any strength at that period. A model in which an ice sheet collapses only when it is both large and warmed by a high-obliquity summer reproduces the rhythm with nothing but the forty-one-thousand-year tilt — the ice sheet grows through one or two maxima and falls on the next — and a slow rise in how large the ice must grow turns the older forty-one-thousand-year world into the newer one without any change in the orbit.

orbits · Orbital averages The rhythms in the rock, set by the Moon's distance. The periods of the climatic precession cycles — the precession constant plus the perihelion frequency of each of the planets that dominate the Earth's eccentricity — and of the obliquity cycle, the precession constant plus the node frequency s₃, against the Earth–Moon distance. Today they are 23.8, 22.4, 19.0 thousand years for precession and 41 for obliquity. At 341,000 km they were 14.7, 14.2, 12.8 and 20.0. The planetary frequencies are held at today's values, which the chaos of the inner solar system makes uncertain beyond a few tens of millions of years; the precession constant is the term that moves.

A rhythm in the rock that measures the Moon

The Earth's axis precesses because the Sun and Moon pull on its equatorial bulge, and two-thirds of the pull is the Moon's. When the Moon was nearer and the day shorter, the axis turned faster, and the climate cycles that precession drives were shorter. Sediments laid down 1.4 billion years ago record those cycles — and with nothing in the rock but layer thicknesses, the ratio of two rhythms gives the Moon's distance at the time.

sky · Precession How fast a meteor arrives is its orbit's inclination. The speed at which a meteoroid on a near-parabolic orbit with perihelion at 1 AU meets the Earth, against the orbit's inclination to the ecliptic: relative to the Earth far away (dashed) and at the top of the atmosphere after the Earth's gravity has added its pull (solid). A prograde orbit in the ecliptic overtakes the Earth at 12 km/s and enters at 17; a retrograde one meets it head-on at 72 and enters at 73. The points are four showers computed from their parents' elements at the node nearest 1 AU: Draconids 24 km/s at 31.9°, Geminids 38 km/s at 22.3°, Perseids 61 km/s at 113.5°, Leonids 72 km/s at 162.5°. A meteor's speed is a reading of the angle between its orbit and the Earth's.

A shower that is a reading of an orbit

Every August the Earth runs through the dust of comet Swift–Tuttle, and the meteors arrive at sixty kilometres a second from a point in Perseus. Each of those facts is an orbital element in disguise. The date is the longitude of the stream's node, the speed is its inclination, the radiant is the direction it comes from, and the shower exists at all only because one element happens to put a node within a hair of the Earth's orbit.

orbits · Orbital elements A star that walks through a calendar with no leap day. The day of the 365-day Egyptian civil year on which Sirius first rose in the dawn, seen from latitude 30°, against year, from 3000 BC to AD 501, counting from a coincidence of the rising with the civil New Year in AD 139. The civil year was a quarter of a day short of the star's, so the rising arrived one civil day later every four years and went once round the whole calendar in about 1458 years — the Sothic cycle. The star's own year was 365.2504 days, within a few minutes of the Julian year of 365¼.

A year kept by a star seen for one morning

For some seventy days each year Sirius is lost in the Sun's glare, and then one morning it rises just before dawn, for a few minutes, low in the east. The Egyptians kept a calendar of exactly 365 days, and against it that first morning slipped a day every four years, going round the whole calendar in about fourteen and a half centuries. The star's own year turned out to be the Julian year to within minutes — not because it had to be, but because two effects of precession cancel for Sirius seen from Egypt.

sky · Calendars

Named alongside it

The objects these essays reach for when they reach for this one.

EccentricityMilankovitch cyclesObliquityInsolationAngular momentumApsidal precessionArgument of perihelionAscending nodeBinary pulsarCalendarCaloric half-yearComet

All concepts