From astronomy to your watch.

Everything starts with four angles that grow almost linearly with time. Astronomers call them the fundamental arguments. They are the slow hands of the lunar clock: the Moon relative to the Sun, the Sun relative to its own orbit, the Moon relative to its nearest approach, and the Moon relative to the point where it crosses the ecliptic.

Each correction term is a sine or a cosine of a whole number combination of those four angles, scaled by an amplitude. Longitude takes 13 of them, ecliptic latitude takes 7, and the distance in kilometres takes 5.

The amplitudes fall away quickly, and each one has a physical name and a physical cause.

• 6.289°The equation of the centre. The orbit is an ellipse, so the Moon runs ahead of a uniform circle near perigee and lags near apogee.
• 1.274°Evection. The Sun stretches the orbit differently depending on where the long axis points, discovered by Ptolemy.
• 0.658°Variation. The Moon speeds up near new and full, and slows near the quarters.
• 0.186°The annual equation. The Earth's own orbit is elliptical, so the solar pull breathes over a year.

A sidereal day is the time the Earth takes to turn once with respect to the stars rather than the Sun. It is about four minutes shorter than a solar day, which is exactly why the same star rises four minutes earlier each night. Sidereal time is a polynomial in days since J2000, and adding your longitude turns it into local sidereal time.

Once you hold the hour angle, the rest is one rotation of a sphere. Altitude tells you how high above the horizon an object stands. Azimuth tells you which way to turn.

The size of the effect is set by one angle: how large the radius of the Earth appears from the Moon. That is the horizontal parallax, and at mean distance it comes to about 57 arcminutes.

A textbook would now divide by the cosine of the altitude, which is fine until you look straight up or stand at a pole. Moon Metrics Pro applies the correction as a single two argument arctangent instead, so there is no division to blow up and no special case to remember.

Two corrections stack. The first is the semidiameter, half the apparent width of the disc, which changes with distance. The second is refraction at the horizon, conventionally fixed at 34 arcminutes. The watch tracks a single signal that already contains both, and an event is simply the moment that signal crosses zero.

Whether that matters depends on how steeply the Moon meets the horizon, and that depends on your latitude. In the tropics the Moon drops almost vertically and the convention is worth a few minutes. In the far north it can slide along the horizon for a long time, and the same 0.83° becomes a large correction.

The first angle is the position angle of the bright limb, measured from celestial north. It depends only on where the Sun and the Moon sit relative to one another on the sphere.

The second angle is the parallactic angle, which rotates celestial north into the direction you call up. This is where the usual formula goes wrong. Written with a tangent of latitude it runs away to infinity at the poles, and because a single argument arctangent only covers half a turn it also needs a separate quadrant repair. Written as a two argument arctangent it simply keeps working.

Each signal is arranged so that zero is the thing you care about. The Moon signal already carries its semidiameter and refraction. The four solar signals are the Sun's altitude offset by the standard thresholds. A sign change between two samples means an event lies between them.

• s₀Upper limb altitude of the Moon. Zero is moonrise or moonset.
• s₁The sine of the hour angle. Zero is the transit, the highest point of the night.
• s₂Solar altitude plus 0.8333°. Zero is sunrise or sunset.
• s₃Solar altitude plus 6°. Zero is the edge of civil twilight.
• s₄Solar altitude plus 12°. Zero is the edge of nautical twilight.
• s₅Solar altitude plus 18°. Zero is the start of astronomical night.

The case that breaks naive code

Above the Arctic Circle the Moon can rise to graze the horizon and sink back within a single 15 minute step. Both endpoints are negative, the product is positive, and simple sign checking reports nothing at all. Two real events vanish.

So when the signs agree but the signal stays within one degree of zero, the scan fits a parabola through the two endpoints and the midpoint, solves for the vertex, and then brackets both halves separately. If the curve really did break the surface, both contacts are recovered.

A half lit Moon is not half as bright as a full one. It is roughly a tenth. Lunar soil scatters light back towards its source, so brightness surges sharply near opposition. Krisciunas and Schaefer captured that with a magnitude that bends with phase angle.

Light arriving near the horizon has crossed far more air than light from overhead. The Kasten and Young airmass describes that, and unlike the naive secant it stays finite all the way down to the horizon itself.