Live sky

1Clocks: from your watch to the stars

Astronomy counts time in Julian days, a running day count since 4713 BC, so any two moments subtract cleanly. The orbits run on Terrestrial Time, an even clock that ignores the Earth's slightly wobbly spin; it is ahead of ordinary time by ΔT, about a minute.

The stars come back to the same place every sidereal day, 3 minutes 56 seconds short of 24 hours, because the Earth also moves along its orbit. Local sidereal time says which right ascension is on your meridian right now: it is what turns the fixed star map into tonight's sky.

JD = t_ms / 86 400 000 + 2 440 587.5Unix milliseconds to Julian dayJD now = …

TT = UT + ΔTTerrestrial Time, which the orbits useΔT = …

GMST = 280.46062° + 360.98564737° · dsidereal time at Greenwich, d days since JD 2 451 545d = …, GMST = …

LST = GMST + λ + Δψ cos εadd your longitude λ (east positive) and the equation of the equinoxes… … … → LST = …

2Stars: from the map to your sky

A star's right ascension α and declination δ are its longitude and latitude on the sky, fixed for centuries. The hour angle H = LST − α says how far it has turned past your meridian; one spherical triangle between the pole, the zenith and the star gives its height h and bearing A.

The sky disc puts the zenith in the middle and the horizon at the edge, with north up and east on the left, as you see it lying on your back. Brighter stars (smaller magnitude m) are drawn bigger, and their colour comes from the B − V colour index.

Right now the highest bright star over … is ….

sin h = sin δ sin φ + cos δ cos φ cos Hheight h, at latitude φα = …, δ = …, H = … → h = …

tan A = −cos δ sin H / (sin δ cos φ − cos δ sin φ cos H)bearing A from north through eastA = …

r = R (90° − h) / 90°, x = −r sin A, y = −r cos Awhere it lands on the disc of radius R

size = max(0.6, 2.7 − 0.42 m) pxstar size from magnitudesize = …

3Planets: Kepler's ellipses

Each planet follows an ellipse with the Sun at one focus, described by six orbital elements: size a, shape e, tilt i, node N, perihelion w and the mean anomaly M, which grows evenly with time. The elements drift slowly, and the page uses Paul Schlyter's set.

The hard part is Kepler's equation: it gives M from the eccentric anomaly E, but we need it the other way round, and there is no formula for that. Newton's method guesses E = M and corrects itself, usually landing in two or three steps. Then the point on the ellipse is turned through the node, tilt and perihelion into the ecliptic, and the Earth's position is subtracted to see it from here.

Try it: slide the shape and the mean anomaly, or load a planet as it is today. A very stretched orbit near perihelion takes Newton longest.

M = E − e sin EKepler's equation, 1609E = … after … Newton steps

E_k+1 = E_k − (E_k − e sin E_k − M) / (1 − e cos E_k)Newton's method, from E₀ = M

tan v = √(1 − e²) sin E / (cos E − e), r = a (1 − e cos E)true anomaly v (the real angle from the Sun) and distance rv = …, r = …

x = r (cos N cos(v + w) − sin N sin(v + w) cos i)one of three rotations into the ecliptic; then subtract the Earth

4Jupiter and Saturn pull on each other

Pure ellipses ignore the planets' pulls on each other, and for the two giants that matters. Jupiter goes round about five times while Saturn goes round twice, so the same alignments repeat and their tugs build up instead of cancelling: the great inequality. Over about 938 years it pushes Saturn up to 0.81° ahead then behind, and Jupiter 0.33° the other way.

A dozen sine terms in their mean anomalies put it right. Without them the 2020 great conjunction came out fifteen hours late; with them it lands within about three hours, and the one in 1623 on the right day with the right 5′ gap.

Δλ_S = 0.812° sin(2M_J − 5M_S − 67.6°) − 0.229° cos(2M_J − 4M_S − 2°) + …Saturn's main termsM_J = …, M_S = …, 2M_J − 5M_S − 67.6° = …

Δλ_J = −0.332° sin(2M_J − 5M_S − 67.6°) − 0.056° sin(2M_J − 2M_S + 21°) + …Jupiter's, the same rhythm the other waynow: Jupiter …, Saturn …

2 · 0.08309°/day − 5 · 0.03344°/day ≈ −0.00105°/dayso one cycle takes 360° ÷ 0.00105°/day ≈ 938 years

5The Moon

The Moon is close and pulled hard by the Sun, so its ellipse is bent the most. The biggest corrections have names from the 1600s: the evection (up to 1.27°), the variation (0.66°) and the annual equation (0.19°), each a sine of the angles between the Sun, the Moon and the Moon's orbit.

Its phase needs only its elongation ψ, the angle between the Moon and the Sun as seen from Earth: half of the Moon is always lit, and ψ says how much of that half faces us.

Δλ = −1.274° sin(M − 2D) + 0.658° sin 2D − 0.186° sin M_☉ + …M: Moon's mean anomaly, D: its mean elongation, M☉: the Sun's

k = (1 − cos ψ) / 2the fraction of the disc litψ = … → k = …

6The Sun, to the arcsecond

The Sun's place sets the seasons, so it gets the most care. The Earth's orbit comes from VSOP87, a sum of about 190 cosine terms that captures the Moon and planets tugging on it. That gives the geometric longitude: where the Sun really is.

Two small effects move where it appears. Nutation is a nod of the Earth's axis, mostly from the Moon's orbit turning every 18.6 years: it shifts the equinox by up to 17″ (Δψ) and tilts the equator by up to 9″ (Δε). Aberration, from the Earth's 30 km/s speed, leans the sunlight back by 20.5″, like rain on a moving car. Together they are worth up to a quarter of an hour, so the equinoxes and solstices now land within a minute of the published times instead of up to 17 minutes early.

Next: …

L = Σ_k τ^k Σ_i A_i cos(B_i + C_i τ), λ_geo = L + 180°VSOP87: τ in millennia from J2000 (TT)τ = …, λ_geo = …, R = …

Δψ ≈ −17.20″ sin Ω − 1.32″ sin 2L_☉ − 0.23″ sin 2L_☾ + 0.21″ sin 2Ωnutation in longitude, Ω the Moon's nodeΔψ = …, Δε = …

ε = 23.43929° − 46.815″ T + … + Δεthe true tilt of the Earth's axisε = …

λ_app = λ_geo + Δψ − 20.4898″ / Rwhere the Sun appearsaberration … → λ_app = …

7Twilight and what you can see

Once the Sun is below the horizon the sky darkens in steps named by its depth: civil twilight to 6°, nautical to 12°, astronomical to 18°, then full night. The page colours the sky by the Sun's height and hides stars fainter than the sky allows, so the map shows what you could actually see.

day > 0° > civil > −6° > nautical > −12° > astronomical > −18° > nightthe Sun's height h☉h☉ = …, faintest star shown: …

8Meteor showers

A shower happens when the Earth crosses a comet's dust trail, at the same point of its orbit every year. That point is a solar longitude λ☉, the Sun's angle along the ecliptic, in the fixed J2000 frame the tables use. The calendar date drifts about six hours a year and jumps back after a leap day, so the page solves for when the Sun gets there.

Here it uses the Sun's geometric longitude: the tables describe where the Earth is, which neither aberration nor nutation changes. How many you see depends on how high the radiant is.

Showing: …, peak at ….

λ_J2000(t) = λ_geo(t) − 1.396971° · Tundo precession since 2000, T in centuriestarget λ_☉ = …

t ← t + (λ_shower − λ_J2000(t)) / 0.98565°/dayNewton steps on the Sun's mean motion

rate ≈ ZHR · sin hZHR: the hourly rate with the radiant overheadZHR …, radiant h = … → …

9The solar system view

The view from above uses the same heliocentric positions. Real distances run from 0.39 AU for Mercury to 30 AU for Neptune, so at true scale the inner planets crowd the Sun; the squashed view uses the square root of the distance so all eight fit. Each planet's year follows from Kepler's third law, and light takes 8 minutes 19 seconds to cross each AU.

P = a^3/2 yearsKepler's third law, a in AUJupiter: a^3/2 = …

r_screen = R √(a / 30.07 AU)the squashed scaleJupiter sits … of the way out, not 17%

t_light = d · 499.005 s/AUlight timeJupiter: … from the Sun, … from us, light takes …

10Names that never overlap, and the quiz

Names are placed greedily by importance: a selected shower's radiant first, then the planets, then the stars from brightest down, then the constellations. Each tries six spots round its point and takes the first that doesn't touch a name already placed or leave the sky disc; if none fit it stays hidden until you tap.

The quiz draws one constellation and its neighbours with a stereographic projection about its centre, which keeps shapes true, so the patterns look as they do in the sky.

[x₀, x₁] ∩ [x₀′, x₁′] ≠ ∅ and [y₀, y₁] ∩ [y₀′, y₁′] ≠ ∅two name boxes overlapnow: … names shown, … hidden

k = 2 / (1 + sin δ₀ sin δ + cos δ₀ cos δ cos Δα)stereographic scale about the centre (α₀, δ₀)

x = −k cos δ sin Δα, y = k (cos δ₀ sin δ − sin δ₀ cos δ cos Δα)east on the left, as seen from the ground