Astronomy & Celestial Mechanics
the sky, computed
Concept

Ecliptic — where it appears

The great circle the Sun appears to trace against the stars over a year — the plane of the Earth's orbit projected onto the sky. It is tilted 23.4° to the celestial equator, and the projection between the two is one of the two terms in the equation of time.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The same ellipse at 955 times the size, and a quarter of a year out of step. The aberration ellipse (solid) and the parallax ellipse (dashed) for γ Draconis at four ecliptic latitudes, drawn to one scale. Aberration is v/c towards the Earth's own direction of travel, so its ellipse has semi-major axis 20.49551″ for every star in the sky; parallax is 1/d towards the Sun, so its semi-major axis is that star's own 0.02147″ — 955 times smaller, and drawn 955 times smaller here. At the ecliptic pole both are circles; on the ecliptic both collapse to lines; in between both are ellipses of semi-minor axis sin β times the major, and the ratio is identical at every latitude because both effects project the same way. The marks are the same four dates on each: they are a quarter of a year apart between the two, because the aberration displacement follows the Earth's velocity and the parallax displacement follows its position, and velocity leads position by 90° on a circular orbit. That quarter-year is the only thing distinguishing the two phenomena on the sky, and it is why Bradley, looking for the dashed ellipse in 1728, spent months unable to interpret the solid one he had found instead.

The other twenty arcseconds

Every star in the sky traces a small ellipse over a year, of the same shape as its parallax ellipse and 90° out of step with it — and the same size for all of them, near or far. Bradley found it in 1728 while hunting for parallax, and it proved the Earth moves a century before anything's distance was known.

sky · Aberration The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

Two fictitious suns in sequence

The equation of time is usually described as one quantity with two causes. It is better read as two separate reductions, each with its own imaginary body — one that moves uniformly along the ecliptic and one that moves uniformly along the equator — and the second is not the first.

sky · Equation of time Heliocentric latitude bought three ways, for the same launch. The highest inclination to the ecliptic a spacecraft can reach, against the hyperbolic excess speed it leaves the Earth with. Launched directly into an inclined orbit at 1 AU (dashed), it pays 2v⊕ sin(i/2) — 15.4° for 8 km/s and 23.2° for 12. Launched on a transfer to Mars whose plane is tilted — the wall across the porkchop plot, where transfers near half a turn are forced steeply out of the plane — it reaches latitude i half-way, for 13.7° at 8 km/s and 21.5° at 12: the same list price, starting from the cost of getting to Mars at all. Launched to Jupiter, it can turn there: a flyby rotates its Jupiter-relative velocity, and the steepest heliocentric orbit a flyby at 6 Jupiter radii can leave it on climbs to 90° for a 12 km/s launch, reaching the 79° of Ulysses's orbit (dotted) from about 10.6 km/s — close to the launch speed that mission actually used. Inclination bought near the Earth costs the Earth's orbital speed; bought at Jupiter it costs Jupiter's, less than half as much, and the flyby pays.

The wall sells inclination at the dearest price

Transfers to Mars that go almost exactly half-way round the Sun are forced into planes tilted steeply out of the ecliptic, which is why they form a wall of expensive dates across every launch chart. A mission that wanted to leave the ecliptic could use the wall on purpose. It would be buying inclination at the Earth's orbital speed — the dearest place to buy it — and the routes that reach the Sun's poles have always gone somewhere slower to make the turn.

spaceflight · Launch windows Where on the Earth a month begins on which evening. The line west of which the new crescent can be seen with the naked eye at local sunset, for a new Moon at 0 h universal time near the spring equinox, on the ecliptic. On the first evening (blue) the crescent is seen only west of the curve, because sunset comes later in universal time the further west an observer is and the Moon is that much older; on the second evening it is seen everywhere except beyond the green line, deep in the hemisphere where the evening ecliptic lies shallow. The first-evening line bulges furthest east at about twenty degrees north, where the spring ecliptic stands steepest, and sweeps far west towards the south, where the same season is autumn. An observed calendar that begins each month at first sight begins it on the first evening west of the line and on the second evening east of it — on two different dates within one religious community spread across the map.

A month that begins when someone sees it

A purely lunar calendar that starts each month at the first sighting of the crescent has abandoned the Sun to keep the Moon, and has handed the start of the month to the weather, the season and the observer's latitude. The line on the Earth west of which the crescent can first be seen sweeps across the globe, and two cities a few time zones apart begin the same month on different evenings in a measurable share of months.

sky · Calendars Three sails against the two routes that have flown. Inclination reached against years since launch. The Venus staircase and the Jupiter step are the arithmetic of the launch-window figures — repeated Venus flybys at a v∞ of 15 km/s reaching 23°, and a single Jupiter flyby after 1.2 years reaching 90° on a heavy launch. The sails, of lightness number 0.020, 0.040, 0.080, spiral in to 0.48 AU — flat lines, 4.6, 2.3, 1.1 years — and then climb at a constant rate until they are polar. They pass the Venus route after 9.0, 4.5, 2.2 years; only the lightest catches Jupiter inside the decade, and none ever stops climbing.

A climb out of the ecliptic paid for in orbits

A solar sail can tilt its orbit without propellant by leaning against sunlight and changing which way it leans twice an orbit. Each orbit then adds the same few degrees whatever its size, so the climb runs faster the closer to the Sun the sail goes. Nothing in the orbit fixes how close that can be. The sail film's temperature does.

spaceflight · Launch windows The bands of sky from which each planet would be seen to transit. Ecliptic latitude against ecliptic longitude, with the vertical scale stretched 11 times: the band from which each of Mercury, Venus, Earth, Mars, Jupiter would be seen to cross the Sun's disc. The Earth's band lies along the ecliptic, ±0.266° wide. Mercury's is the widest, ±0.69°, and tilted by its 7° inclination; the giants' are narrow, ±0.051° for Jupiter. Each crosses the Earth's at the two nodes of its orbit, and only near those crossings can an observer see two planets transit. From nowhere on the sky can more than 3 be seen.

The stars that could catch the Earth in transit

The geometry that lets one Earth twin in two hundred be seen crossing its star works in reverse: the Earth crosses the Sun for observers in a band of sky half a degree wide, under half a per cent of the sky. About two thousand stars within a hundred parsecs sit in it. The band is fixed to the Earth's orbit and the stars are not, so the list changes on the timescale of recorded history — and from no direction can more than three of the eight planets be seen transiting at once.

exoplanets · Detection bias The Moon's reach north and south, over two cycles of its node. The greatest declination the Moon reaches in a month, north or south, from 2010 to 2045: about 23.44° plus the orbit's 5.145° tilt times the cosine of the node's longitude, which regresses once in 18.61 years. At the major standstill (2025.1, and 2006.5, 2043.7) the Moon reaches 28.6°, five degrees beyond the Sun's solstice declination (dashed); at the minor standstill, 18.3°, five degrees short of it. Near each extreme the reach changes slowly for two or three years, which is the standstill.

A Moon that rises further north than the Sun

The Sun's rising point swings north and south with the seasons between limits that do not change from one year to the next. The Moon's swings with the month, and its limits themselves open and close over 18.6 years as the plane of its orbit turns. At the widest, in 2024 and 2025, the Moon rose five degrees of declination further north than the midsummer Sun and skimmed the southern horizon of northern Scotland; at the narrowest, nine years later, it stays well inside the Sun's range.

sky · Precession The Moon is late by anything from a quarter of an hour to well over an hour. Minutes between one moonrise and the next at latitude 52° N, against the Moon's longitude along its orbit, with the node at 0°. The daily delay is 10 minutes at longitude 354° and 84 at 222°. The mean of 50 minutes everyone quotes is true only as an average. A full Moon stands opposite the Sun, so it passes the shallow point near the September equinox, and the crescent passes it in spring.

A full Moon that stops rising later

The Moon rises about fifty minutes later each day. That number is an average that hardly any night obeys. At fifty-two degrees north, in late September, the full Moon rises only ten or twenty minutes later for several nights running, so that a bright Moon stands in the evening sky for most of a week, just when harvest once needed it. Nothing about the Moon changes. It is the angle at which its path meets the horizon, and that angle goes through its lowest value at one point of the ecliptic, which the autumn's full Moon happens to occupy.

sky · Seasons

Named alongside it

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

Gravity assistInclinationObliquityOrbital inclinationPlane changeProper motionAberrationAnalemmaArgument of latitudeAstrometryAzimuthCelestial coordinates

All concepts