the observatory

1 · the sky as coordinates

to point at anything in the sky you need an origin. the everyday one is the horizontal (alt-azimuth) system, centred on you: altitude is the angle above the horizon (0° at the horizon, 90° at the zenith, straight up), and azimuth is the compass bearing around the horizon (0° = north, 90° = east, 180° = south, 270° = west). two numbers, and you can point your arm at any star.

the catch: alt-az depends on where and when you stand. the same star is at a different altitude in tokyo than in new york at the same instant. so astronomers also keep an equatorial system fixed to the stars — right ascension and declination, which are simply the sky's own latitude and longitude painted onto an imaginary sphere around us. declination (δ) is how far north or south of the celestial equator a star sits; right ascension is its east–west address. those two numbers don't change as the earth turns — so a catalogue can list them once and anyone, anywhere, can look the star up forever.

the bridge between the two systems is one equation. given your latitude (φ), a star's declination (δ), and its hour angle (H — how far the sky has turned the star past your due-south line, at 15° per hour), its altitude a right now is:

sin a = sin φ · sin δ + cos φ · cos δ · cos H

a = altitude (height above your horizon) · φ = your latitude · δ = the star's declination · H = its hour angle (15° for every hour off your meridian). the companion formula cos A = (sin δ − sin φ · sin a) / (cos φ · cos a) gives the azimuth A.

that's it — that single line is what the site evaluates, thousands of times, to place the sun, the moon, and every meteor radiant in your dome. plug in φ = 90° (the north pole) and the equation collapses to sin a = sin δ: a star's height never changes, the whole sky just spins flat around you. the math contains the geography.

fun fact the sin in that equation carries a 1,500-year travel story. indian astronomers (āryabhaṭa, c. 500) tabulated it and called the half-chord jyā. arabic scholars borrowed the sound as jiba, which got written without vowels as jb — and a later latin translator misread it as jaib, "a bay or fold," and rendered it sinus. every "sine" in every textbook is a 900-year-old typo for a sanskrit word. the idea crossed three civilisations to reach that one line of code.

2 · the sun, and why it's never on time

earth's axis is tilted 23.44° from the plane of its orbit (the obliquity of the ecliptic, written ε). that single tilt is the reason for seasons and the reason the sun's noon height swings through the year. the sun's declination — its angle north or south of the celestial equator — rides from +23.44° at the june solstice to −23.44° in december, and you can compute it for any day of the year from the tilt alone:

δ ≈ −23.44° · cos( 360°⁄365 · (N + 10) )

δ = the sun's declination that day · N = the day number (jan 1 = 1) · 23.44° = the axial tilt · the +10 nudges the start to the december solstice. feed this straight back into §1's altitude equation and you have the sun's height at any moment — which is exactly the chain the site runs.

here's the surprising part: the sun is almost never where a clock says it should be. a sundial and a wristwatch can disagree by up to ~16 minutes. the gap is the equation of time, and it has two causes stacked together:

• earth's orbit is an ellipse — by kepler's second law (1609) we move faster near the sun (perihelion, early january) and slower far from it, so the sun appears to drift ahead of, then behind, clock time.
• the axial tilt — the sun moves along the tilted ecliptic, not the equator, so even at constant speed its shadow-casting component speeds up and slows down through the year.

stack the two and you get a single correction in minutes — one you can actually evaluate:

E ≈ 9.87 sin 2B − 7.53 cos B − 1.5 sin B  (minutes),  B = 360°⁄365 · (N − 81)

E = how far ahead (+) or behind (−) the real sun runs the clock · N = day of year · the first term is the tilt effect, the other two the ellipse.

plot the sun at clock-noon every day for a year and that correction traces a lopsided figure-eight in the sky: the analemma. that shape is the equation of time made visible — and it's exactly the curve a serious sundial engraves on its face.

how this idea was built up. hipparchus already noticed the seasons aren't equal length (a hint the sun's motion isn't uniform). ptolemy patched it with an off-centre circle. then around 900, al-battānī in raqqa re-measured the sun for decades and caught something ptolemy couldn't have: the point of the sun's nearest approach had moved since ptolemy's day — proof the orbit's geometry drifts, not just the stars. he pinned the length of the year to within ~2 minutes. seven centuries later kepler cites al-battānī by name while replacing the patched circles with a clean ellipse. the equation-of-time wobble you just computed is the modern, tidied descendant of that exact chain of corrections.

3 · shadows & the gnomon

a gnomon is just a stick in the ground — the oldest astronomical instrument there is. its shadow points in the direction opposite the sun's azimuth, and its length follows one clean rule: as the sun climbs, the shadow shrinks. precisely,

L = h · cot a = h ⁄ tan a

L = shadow length · h = the stick's height · a = the sun's altitude (from §1). high sun → short shadow; as a → 0 at sunset, tan a → 0 and the shadow races to infinity.

that one proportion is enough to do something astonishing — measure the whole earth. around 240 bc, eratosthenes of cyrene, chief librarian at alexandria, knew that at noon on the summer solstice the sun stood straight down a well in syene (modern aswan) — zero shadow, sun directly overhead. on the same day in alexandria, a gnomon cast a shadow leaning 7.2° from vertical. since 7.2° is one-fiftieth of a circle, the two cities had to be 1/50th of earth's circumference apart:

C = d · 360° ⁄ θ = d · 360° ⁄ 7.2° = 50 d

C = earth's circumference · d = the alexandria-to-syene distance · θ = 7.2°, the shadow's lean. multiply out and you land within a few percent of the true ~40,000 km. with a stick.

the same question, a second way. twelve centuries later al-bīrūnī (c. 1020) re-measured the earth without needing two cities at all. he climbed a mountain of known height h, measured the tiny angle θ by which the horizon dips below level, and turned it into a radius:

R = h · cos θ ⁄ (1 − cos θ)

R = earth's radius · h = the mountain's height · θ = the horizon's dip angle. one mountain, one angle, a result within ~1% of today's value. eratosthenes' question, a sharper instrument — the lineage is the method, refined.

on this site, the page's drop-shadows are a stylised gnomon: they encode the sun's real east–west lean for the active location (the --sun-dx / --sun-dy vector), so the whole layout leans away from the real sun. it's a brutalist sundial you scroll through.

4 · the sundial, forward and reverse

run the gnomon idea backwards and you get a sundial: from the shadow's direction, read the time. the sun's position is set by its hour angle — just the clock turned into degrees:

H = 15° · (t − 12)

H = hour angle · t = local solar time in hours · so 1 pm is H = +15°, 3 pm is +45°. the earth turns 15° an hour; that's where the 15 comes from.

but a flat horizontal dial has a twist: because the gnomon must point at the celestial pole (so its shadow-edge tracks the sun cleanly all day), it has to be tilted up at an angle equal to your latitude — a london dial and a cairo dial are cut differently. and the hour lines aren't evenly spaced; their angle from the noon line is:

tan θ = sin φ · tan H

θ = the hour line's angle from noon on the dial face · φ = your latitude · H = the hour angle above. set φ = 90° and the lines spread evenly like a clock; nearer the equator they bunch up. the latitude is literally engraved into the spacing.

shadow clocks go back to egypt and babylon (~1500 bc), and the greeks added the angled gnomon. but the dial got its mathematical maturity in the islamic world: astronomers of the 9th–14th centuries needed exact daylight divisions for the five prayer times, and ibn al-shāṭir (1304–1375), timekeeper at the great mosque of damascus, cut a marble sundial whose gnomon pointed at the pole and whose lines marked equal clock-hours — essentially the modern dial, three centuries before europe standardised it. the prayer-time problem drove the geometry forward.

this site runs the sundial in reverse. it already knows the real sun's position for the active location, so it computes the compass bearing you'd have to aim a screen for its on-screen shadows to fall the same way the real sun is throwing them outside. line your phone up to that bearing and the digital shadows and the physical ones agree — the screen becomes a tiny, honest gnomon. (the optional live compass uses the phone's magnetometer to guide you onto that bearing.)

the full sign conventions and the worked regression examples for this live in the repo's _docs/OBSERVATORY-MATH.md — this lesson is the why; that doc is the proof.

5 · twilight & the bent horizon

sunset isn't an on/off switch — the sky dims through three defined stages, set by how far the sun has sunk below the horizon:

• civil twilight — sun 0° to −6°. you can still read outside; the brightest stars and planets appear.
• nautical twilight — −6° to −12°. the horizon at sea is still faintly visible and bright stars are out — the window sailors used to fix their position with a sextant. hence the name.
• astronomical twilight — −12° to −18°. the last glow leaves the sky; past −18° it is fully dark and the faintest galaxies are reachable.

the site grades the paper colour by exactly this number — the sun's altitude — and the optional circadian theme flips light/dark on the same threshold: dark when the real sun is actually down where the active perspective is.

the catch the physics adds. the moment you call "sunrise" isn't when the sun geometrically reaches the horizon — the air bends its light. the atmosphere acts like a weak lens, lifting the sun's image by about 34 arc-minutes (just over half a degree) right at the horizon — slightly more than the sun's own width. so when you watch the sun's lower edge "touch" the sea, the real sun is already entirely below it. the site uses the standard correction astronomers use, sunrise when the sun's centre is at:

a = −0.83°  (≈ −34′ refraction − 16′ for the sun's half-width)

a = the sun's true altitude at the instant its visible edge meets the horizon. that small negative number is why almanac sunrise times "shouldn't" line up — and do.

fun fact understanding that the air bends light is itself a thousand-year-old result. ibn al-haytham (alhazen, c. 1021, cairo) studied atmospheric refraction and even used twilight's end to estimate the height of the atmosphere — and, more importantly, insisted you settle such questions by controlled experiment rather than by quoting aristotle. that habit, not any single measurement, is the thing we now call the scientific method.

6 · the moon & the tilted crescent

the moon makes no light of its own; we see the half the sun lights, from an angle. its phase is set by the phase angle (i) — the sun–moon–earth angle — and the lit fraction you see is a clean function of it:

k = (1 + cos i) ⁄ 2

k = the illuminated fraction (0 = new, 1 = full) · i = the phase angle. at i = 0° the sun is behind us, k = 1, full moon; at i = 180°, k = 0, new. that's the exact number the little moon glyph draws.

the cycle from new moon to new moon takes 29.53 days — the synodic month — and there's a lovely reason it's longer than the 27.32 days the moon actually takes to circle the earth (the sidereal month). while the moon orbits, the earth has moved along its own orbit, so the moon has to catch a little extra to line back up with the sun. it's a beat between two cycles:

1⁄S = 1⁄T_moon − 1⁄T_year  →  S = 29.53 days

S = synodic month (new-to-new) · T_moon = 27.32 d (one true orbit) · T_year = 365.25 d. the same "two clocks drifting" math sets a sundial against a wristwatch, and a calendar against the seasons.

two subtler things the site gets right. first, libration: the moon keeps one face toward us, but it nods and rocks slightly, so over a month we actually glimpse about 59% of its surface. second — the one most people never notice — the crescent tilts. the line between light and dark (the terminator) and the "horns" of the crescent point at different angles depending on your latitude and where the moon sits in your sky. that lean is the parallactic angle (q), and it has an equation just like everything else:

tan q = sin H ⁄ ( tan φ · cos δ − sin δ · cos H )

q = the parallactic angle (how far the crescent is rotated) · H = the moon's hour angle · φ = your latitude · δ = the moon's declination. near the equator this drives the crescent flat onto its back like a bowl — "the wet moon." the site tilts the glyph to exactly this q for the active location.

how we learned to predict it. the moon is the hardest of the easy objects — its orbit wobbles. babylon found the first handle ~2,500 years ago: the saros cycle, that eclipses repeat every 18 years, 11 days, 8 hours. that's pure pattern-spotting, no theory — and it still works. hipparchus turned it into geometry; al-battānī refined the eclipse timings; ibn al-shāṭir (the same damascus timekeeper from §4) built a lunar model so good that, two centuries later, copernicus's moon model turned out to be mathematically identical to his. the strawberry-moon name the site shows on a june full moon is the folk layer on top of all that machinery.

7 · meteor showers & the radiant

a "shooting star" is a grain of comet dust — often no bigger than a sand grain — hitting the atmosphere at 11–72 km/s and burning up about 80–120 km overhead. those two speed limits aren't arbitrary: 11 km/s is the slowest anything can fall onto earth (the escape velocity, just run in reverse), and 72 km/s is a head-on collision — earth's own ~30 km/s orbital speed plus a meteoroid's ~42 km/s falling from the outer solar system. nothing in a shower goes faster than that sum.

the reason a speck makes a visible streak is pure kinetic energy, which grows with the square of speed:

E = ½ m v²

E = energy dumped into the air · m = the grain's mass · v = its speed. double the speed, quadruple the flash — which is why the fast november leonids (71 km/s) blaze and the slow ones merely drift.

a meteor shower happens when earth, on its yearly orbit, ploughs through the debris stream a comet left along its path. same stream, same week, every year — which is why the perseids are always mid-august. the particles travel on parallel tracks, but perspective makes them appear to diverge from a single point — the radiant — exactly like railway tracks vanishing to a point. each shower is named for the constellation its radiant sits in (perseids → perseus, geminids → gemini).

the standard measure of strength is the ZHR, the zenithal hourly rate: how many you'd see per hour under perfect dark skies with the radiant straight overhead. you almost never get that ideal, so the rate you'll actually see is the ZHR knocked down by how low the radiant sits and how bright your sky is:

rate ≈ ZHR · sin h_R ⁄ r^{(6.5 − m)}

h_R = the radiant's altitude (so a radiant on the horizon, sin 0° = 0, gives you almost nothing) · m = your sky's faintest visible star · r = the shower's "population index" (how the count grows toward fainter meteors). the sin h_R term is why the site only spawns a busy meteor field when the real radiant is genuinely up — and why every "/hr" readout here quotes ZHR · sin h_R computed for the active perspective's coordinates (flip me ⇄ you in settings and watch the number move), never the raw ZHR. it's also why the almanacs quote the perseids at "zhr 100" but tell you to expect 30–50: the radiant spends most of the night well below the zenith.

8 · the algorithms & the sources

none of this is guessed at runtime — it's the same vetted formulae observatories use, just evaluated in your browser. the equations above are the shape of it; the production versions carry dozens more periodic correction terms for the moon and planets, but the bones are exactly what you've read. the canonical references, if you want to go all the way down:

• jean meeus — astronomical algorithms (1991; 2nd ed. 1998). the bible. every sun/moon/phase/position formula in this lesson traces back here.
• peter duffett-smith — practical astronomy with your calculator (1979 →). the gentle on-ramp; do it by hand once and it sticks.
• suncalc, by vladimir agafonkin — the tiny, exact javascript library this site uses for sun & moon position, rise/set, and phase.
• the international meteor organization (IMO) — the working data for shower dates, radiants, and ZHR.
• the american meteor society (AMS) — the annual shower calendar the site's table is cross-checked against, and the source of the honest framing §7 now runs on: zhr is the ideal, the observed rate is what you quote.
• ptolemy's almagest (~150 ad) and, for the modern rigour, the explanatory supplement to the astronomical almanac — the deep end.
• and the site's own _docs/OBSERVATORY-MATH.md — the auditable chain from these formulae to the pixels, with sign conventions and worked examples.

9 · the long bridge — who worked it out

you've met most of these people already, scattered through the sections where their work actually lives — that was the point. but the chain has one gap most tellings skip straight over: after ptolemy (~150 ad), greek astronomy in europe went largely quiet for some 700 years. it didn't die — it moved.

from the 8th century, the house of wisdom in baghdad ran a deliberate translation movement: greek, persian, and indian science rendered into arabic, then tested, corrected, and extended. for seven centuries — baghdad, córdoba, cairo, isfahan, damascus, samarkand — scholars didn't merely preserve the inheritance. they caught ptolemy's errors, fused greek geometry with indian trigonometry (that sanskrit sine from §1), built better instruments, and pushed. when europe finally turned back to the sky, it climbed up on their shoulders — and often didn't say so. so here's the whole run in order, each line built on the one above it — the section where you met each idea is tagged in §:

1. ~1500 bcegypt & babylon — shadow clocks; the babylonians log eclipse cycles (the saros, §6) and hand us the 360° circle. the first instruments are sticks.
2. ~6th c bcthales & anaximander — the gnomon reaches greece; the sky becomes something you measure, not just watch.
3. ~500 bcpythagoras & the greeks argue the earth is a sphere — from the round shadow it throws on the moon during an eclipse.
4. ~270 bcaristarchus of samos proposes a sun-centred cosmos — and tries to measure the sun's distance. ~1,800 years early.
5. ~240 bceratosthenes measures earth's circumference from two shadows (§3).
6. ~130 bchipparchus — the first star catalogue, the magnitude scale, the discovery of precession.
7. ~500 adāryabhaṭa (india) tabulates the sine (jyā) and treats earth's spin as real — the trig that §1 still runs.
8. ~150 adptolemy writes the almagest — wrong at the centre, right in the trig; 1,400 years of authority.
9. ~830al-khwārizmī & the house of wisdom — algebra, "algorithm," and the first great islamic star tables.
10. ~900al-battānī nails the solar year to ~2 minutes and catches the sun's apogee drifting (§2).
11. 964al-ṣūfī records the andromeda galaxy and re-measures the stars (§1).
12. ~1021ibn al-haytham — optics, atmospheric refraction (§5), and the experimental method itself.
13. ~1030al-bīrūnī measures earth's radius from a mountain (§3) and debates whether it spins.
14. ~1260al-ṭūsī builds the marāgha observatory; the "ṭūsī couple" is born.
15. ~1350ibn al-shāṭir — lunar & sundial models (§4, §6) mathematically identical to copernicus, two centuries early.
16. 1437ulugh beg measures ~1,000 stars from samarkand — the last word before the telescope.
17. 1543copernicus puts the sun back at the centre — leaning, quietly, on al-battānī and al-ṭūsī.
18. 1572–77tycho brahe — a new star and a comet prove the "unchanging" heavens change; he logs the planets to unheard-of precision.
19. 1609–19kepler — elliptical orbits and the three laws (§2); galileo turns the telescope upward (§6).
20. 1687newton — one inverse-square law of gravity underneath all of kepler.
21. 1833the leonid storm — meteors become a science overnight (§7).
22. 1838bessel measures the first stellar parallax — the first true distance to a star.
23. 1866schiaparelli ties meteor streams to comets (§7).
24. 1991meeus publishes astronomical algorithms — the formulae go portable (§8).
25. todayall of it, recomputed in a browser tab, for wherever you happen to be.

you speak arabic every time you use this site's sky. azimuth (al-sumūt, "the directions"), zenith, nadir, almanac, and star names like aldebaran, betelgeuse, rigel, vega, altair, deneb, and algol ("the ghoul") all came through the islamic golden age. the words are the fossil record of who kept the lights on — the same names §1 looks up to place a star in your dome.

that's lesson 1. every line on the live card at the top is one of the equations down here, evaluated for your sky this second — the moon's tilt from §6, the page's shadows from §3, the meteor field from §7, the whole dome re-cast for a new latitude when you flip me ⇄ you (§1). the heavens aren't decoration on this site — they're arithmetic with a three-thousand-year memory. — delilah, for zayd. now you know the lore. ← back to the archive · ↑ top