Definition
Topocentric means centred on the observer's place (from the Greek topos, place). Topocentric coordinates give the direction of a body as seen from a point on the Earth's surface, while geocentric coordinates give it as seen from the Earth's centre. Ephemerides are computed geocentrically, because the centre of the Earth is the same for everybody; the topocentric correction then adapts them to each observer.
The difference is the diurnal parallax. The observer is up to 6,378 km from the Earth's centre, so a nearby body appears shifted, always away from the zenith. For the stars the shift is utterly negligible; for the Moon it reaches almost a degree.
Size of the effect
| Body | Distance | Maximum topocentric shift |
|---|---|---|
| Moon | 356,000 – 407,000 km | 54′ to 61′ (on the horizon) |
| Venus at inferior conjunction | about 40 million km | about 30″ |
| Sun | 1 AU | 8.8″ |
| Mars at opposition | 56 – 100 million km | 13″ to 23″ |
| Nearest star | 4.2 light-years | about 0.00003″ |
For the planets the correction matters only for precise work, such as timing transits or occultations. For the Moon it is essential: without it, rising times would be wrong by several minutes, and occultation and eclipse predictions would be meaningless.
The observer's position
The Earth is flattened (flattening 1/298.257), so the observer's geocentric latitude φ′, measured at the Earth's centre, differs from the geographic latitude φ by up to 11.5′ at 45°. Meeus computes two quantities that place the observer relative to the centre, in units of the equatorial radius, from φ and the height H above sea level in metres:
tan u = 0.99664719 · tan φρ sin φ′ = 0.99664719 · sin u + (H / 6378140) · sin φρ cos φ′ = cos u + (H / 6378140) · cos φAt Paris (φ = 48.86°, H = 60 m), ρ cos φ′ = 0.6592 and ρ sin φ′ = 0.7495: the observer is 0.9981 equatorial radii from the centre.
The correction
With the geocentric right ascension α, declination δ, hour angle H and the equatorial horizontal parallax π of the body (sin π = 6378.14 km / distance), Meeus's rigorous formulas give the topocentric α′ and δ′:
tan Δα = −ρ cos φ′ · sin π · sin H / (cos δ − ρ cos φ′ · sin π · cos H)α′ = α + Δαtan δ′ = (sin δ − ρ sin φ′ · sin π) · cos Δα / (cos δ − ρ cos φ′ · sin π · cos H)The topocentric coordinates are then converted to altitude and azimuth as usual, and refraction is applied last. The order matters: parallax is geometric, refraction is atmospheric.
The Moon from the surface
- The Moon appears lower than its geocentric position by up to about 1°, when it is on the horizon.
- It appears larger when high in the sky: the observer is then up to one Earth radius closer to it, and its semidiameter grows by up to about 1.8 %. This is the opposite of the popular Moon illusion, which makes it look larger near the horizon.
- Two observers a few thousand kilometres apart see the Moon against different stars, which is why a lunar occultation is visible only along a band of the Earth, and its timing changes from city to city.
- In a solar eclipse, the Moon's shadow sweeps across the Earth; the local circumstances, partial, total or annular, and their times are entirely topocentric.
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Frequently asked questions
What is the difference between geocentric and topocentric?
Geocentric coordinates are measured from the centre of the Earth, topocentric coordinates from the observer's place on its surface. They differ by the diurnal parallax, which is significant only for the Moon and, at a fine level, for the nearest planets and the Sun.
When do I need topocentric coordinates?
Whenever the Moon is involved at better than a degree, and for any eclipse, occultation or transit timing. For drawing the planets and stars in the sky, geocentric positions are enough.
Does parallax change the Moon's apparent size?
Yes. When the Moon is near the zenith, the observer is almost one Earth radius closer to it than the Earth's centre is, and its disc appears up to about 1.8 % larger than when it is on the horizon.
