Lunar Parallax: Geocentric and Topocentric Moon Positions

What is lunar parallax?

Before comparing numbers, ask where each observer is. A shared time alone does not make an Earth-centred position and a surface-observer position equivalent.

Try a small viewpoint experiment

Hold a finger in front of a distant edge, such as a door frame. Look with one eye and then the other while keeping your head and finger still. The finger appears to move against the background because the sightline begins at a different place. Moving the finger farther away reduces this shift.

This illustrates a change of viewpoint, not movement of the object. The lunar comparison replaces the two eyes with Earth's centre and an observing site. The baseline and distances are much larger, but the need to specify both ends of a sightline remains.

Do not use a pair of city photographs' raw altitude difference as a direct parallax measurement. Their local horizons can point in different directions, and the photographs may use different times, rotations or camera scales. A careful comparison first brings the directions into a common reference frame.

A right triangle explains horizontal parallax

Use an ideal spherical Earth with a chosen radius R = 6,378 km. Let d be the distance from Earth's centre to the Moon's centre. Ignore the atmosphere, surface height, Earth's flattening and the Moon's angular radius. This model is for understanding geometry, not calculating tonight's moonrise.

Place the lunar centre exactly on the observer's geometric horizon. The line from the observer to the Moon is perpendicular to the Earth-radius line. In that right triangle, the angle between the Earth-centred and observer-centred sightlines obeys sin π = R/d, so π = arcsin(R/d). Here π is a label for horizontal parallax, not the number 3.14159.

The middle row is about 0.95°, or 57 arcminutes. The US Naval Observatory's lunar-rise definitions describe horizontal parallax varying roughly from 54 to 61 arcminutes. Our chosen distances illustrate the scale and the direction of change; they are not the minimum, maximum or current distances for a specified month.

Reducing d while keeping R fixed increases the angle. Dividing by a surface-to-Moon distance instead of the stated centre-to-centre distance would change the triangle and invalidate this formula.

The shift gets smaller towards the zenith

Horizontal parallax names the horizon case. It is not a constant angle to subtract at every height. In the same spherical model, let hₜ be the Moon centre's geometric altitude at the site and let p be the actual parallax at that altitude. Then sin p = (R/d) cos hₜ.

For comparison, measure the geocentric direction above a plane through Earth's centre parallel to that site's horizon. Call that angle hɢ. In this model, hɢ = hₜ + p. The surface observer's geometric direction is lower than the corresponding geocentric direction until the zenith, where the two sightlines coincide.

The first row does not set both altitudes to zero. The Moon centre is on the site's horizon, while the centre-based sightline is above the parallel plane by the horizontal-parallax angle. Keeping those two planes distinct prevents a common diagram error.

Parallax, refraction and the lunar edge answer different questions

Parallax changes the origin of the sightline. Atmospheric refraction changes the path of light. The angular radius determines the offset between the Moon's centre and its upper edge. They are separate ingredients in a rise/set calculation.

USNO defines the standard rise/set event using the upper limb and a refraction convention, while accounting for parallax when working from geocentric coordinates. A centre placed on an airless geometric horizon, as in our model, is therefore not that complete observational event.

NASA JPL's Horizons documentation also distinguishes airless and refracted elevation and requires a surface location for topocentric elevation. Read those settings before comparing an almanac value with a telescope or photograph.

A practical way to compare two Moon positions

1. Match the instant, including the time zone and date.
2. Identify the origin: Earth's centre or a site with latitude, longitude and height.
3. Match the coordinate type and reference frame. Ecliptic longitude and local altitude are not interchangeable.
4. Check whether refraction is included and whether the number refers to the centre or an edge.
5. Compare unrounded values only after the conventions match.

Skyfield's position documentation makes the observer's origin explicit and requires a geographic position for a local horizon calculation. This is a useful reading principle even when the report was produced by different software.

A geocentric phase or tithi calculation and a local horizon event can legitimately use different conventions. Changing a phase's clock display to your time zone does not by itself turn its coordinate into a topocentric one.

Why an angle is not a fixed number of moonrise minutes

Do not convert a roughly one-degree parallax into one universal correction to moonrise. The rate at which the Moon crosses your horizon depends on the geometry and its motion. A fixed angular difference does not imply the same time difference for every site and date.

Use local moonrise and moonset times for the timing question. Use the model above to understand why the observer's location belongs in that calculation. Its rounded sphere and selected distances are teaching inputs, not an alternative ephemeris.

Finally, distinguish parallax from libration. A changed sightline and a changed view of lunar surface features are related geometrically, but the terms do not name the same complete effect.

Frequently asked questions