Turn time into sky orientation

1. Count turns of Earth

Think of a clock hand that can make many full turns. To describe where it points now, we keep the fraction of its total turns and discard the completed circles. At engine day offset u = 0, the rotation expression starts at 0.779057273264 turns. Multiplication by 360 gives 280.46061837504°.

The engine separates the everyday one-turn-per-day part from the extra rotation. At u = 0.25 days, the everyday part is a quarter turn, while the extra part contributes 0.00273781191135448 × 0.25 turns. Adding the reference fraction gives 1.0297417262418386 turns; discarding one full turn and multiplying by 360 gives 10.7070214470619°.

f = rem(u, 1); θ = 360 × rem(0.779057273264 + 0.00273781191135448u + f, 1)

The symbol rem here means the nonnegative remainder: rem(−0.25, 1) is 0.75. The values u and f are numerical day offsets. The f term carries an implicit factor of one turn per day, so f = 0.25 days contributes 0.25 turns. The coefficient 0.00273781191135448 supplies the extra turns per day. θ, the Earth rotation angle, is in degrees. Keeping the one-day fraction separate also avoids carrying a large integer number of ordinary daily turns into the final wrapping operation.

const rem = (x: number, modulus: number): number => { const r = x % modulus; return r < 0 ? r + modulus : r; };
const earthRotationDegrees = (ut: number): number =>
  360 * rem(0.7790572732640 + 0.00273781191135448 * ut + rem(ut, 1), 1);

2. Orient the equinox from Greenwich

A turning Earth supplies one orientation, but chart coordinates also refer to an equinox: the crossing of the equator and ecliptic used as longitude zero. That reference changes with the moving planes from the earlier lessons. Sidereal time expresses the equinox’s orientation relative to a meridian, a north–south reference through the observer.

The engine adds a slowly changing polynomial and a small wobble correction to θ. The polynomial uses T = tt / 36525, centuries from the reference epoch in the engine’s approximate tt model. Its result P is in arcseconds of angle, not seconds on a clock. The wobble helper e_tilt supplies ee, the equation of equinoxes, in seconds of sidereal time. One such second represents 15 arcseconds, so multiplying ee by 15 makes the units agree.

P(T) = 0.014506 + 4612.156534T + 1.3915817T² − 0.00000044T³ − 0.000029956T⁴ − 0.0000000368T⁵

st = P(T) + 15ee, in arcseconds; G = wrap(θ + st/3600), in degrees

Greenwich apparent sidereal hours = G/15; gast_degrees = hours × 15

For a deliberately isolated arithmetic example, supply θ = 280°, T = 0 and ee = 0.2 seconds of sidereal time. Then P = 0.014506 arcseconds and 15ee = 3 arcseconds. Their sum is 3.014506 arcseconds, or 0.0008373627778°. The result G is 280.0008373627778°, equivalent to 18.6667224908519 sidereal hours. These inputs exercise unit conversion; they are not a self-consistent engine epoch.

A full circle takes 24 sidereal hours, so 360/24 = 15° per hour. These hours label an orientation; they are not a UTC wall-clock reading. “Greenwich” identifies the starting meridian. “Apparent” means the reference includes the wobble of the equinox; “mean” omits that correction. Sidereal time here is also distinct from choosing a sidereal zodiac, which would change the longitude zero used for chart positions.

const wrap = (x: number): number => { const r = x % 360; return r < 0 ? r + 360 : r; };
const greenwichDegrees = (theta: number, ttDays: number, eeSeconds: number): number => {
  const t = ttDays / 36525;
  const p = 0.014506 + (((((-0.0000000368 * t - 0.000029956) * t - 0.00000044) * t + 1.3915817) * t + 4612.156534) * t);
  return wrap(theta + (p + 15 * eeSeconds) / 3600);
};

3. Move the orientation to the observer

Two observers at different longitudes face different parts of the sky at the same instant. Starting at Greenwich, moving 20° east advances the meridian direction by 20°. If G is 350°, adding 20° gives 370°, which wraps to 10°. Moving west instead contributes a negative longitude.

α = wrap(G + L), where L is east-positive geographic longitude in degrees

The engine calls α the right ascension of the midheaven (RAMC)The meridian direction on the equatorClose This is the equatorial angle of the local meridian, rather than the ecliptic longitude returned as the Midheaven. The letters abbreviate Right Ascension of the Midheaven. United States Naval Observatory: observer longitude and local hour angle. Right ascension is an angle along the equator from the equinox. Here it tells us which equatorial direction lies on the local meridian; the next lesson converts that orientation into an ecliptic midheaven longitude. RAMC and midheaven longitude generally differ because those two circles are tilted relative to one another.

For a second invented example, G = 15° and L = −30° give α = wrap(−15°) = 345°. Latitude does not enter this addition. It will affect the horizon angle in the next lesson. Keeping longitude and latitude separate prevents an easy mistake: both describe location, but they enter different parts of the geometry.

const localMeridianDegrees = (greenwichDegrees: number, eastLongitude: number): number => {
  const r = (greenwichDegrees + eastLongitude) % 360;
  return r < 0 ? r + 360 : r;
};