Earth Shape, Geodetic Coordinates, and WGS84 #

The celestial sphere is centered on the observer — but where exactly is the observer? This article defines the Earth’s shape as it matters for chart calculation, specifies the coordinate system used to locate the observer on that shape, and quantifies how much positional errors in the observer’s location actually affect a chart.

The Ellipsoidal Earth #

The Earth is not a sphere. It is closer to an oblate spheroid (an ellipsoid of revolution) — a sphere that has been slightly flattened at the poles and bulges at the equator. This flattening arises from the Earth’s rotation: centrifugal force reduces the effective gravity at the equator, causing the equatorial radius to exceed the polar radius by about 21 km.

An ellipsoid of revolution is defined by two parameters:

• The semi-major axis $a$ (equatorial radius)
• The flattening $f = (a - b)/a$, where $b$ is the semi-minor axis (polar radius)

Equivalently, one can specify the eccentricity $e$:

$$ e^2 = 2f - f^2 = \frac{a^2 - b^2}{a^2} $$

The true Earth surface departs from this ellipsoid by up to about ±100 meters (the geoid, discussed below), but for astronomical purposes the ellipsoid is the standard model.

WGS84 #

The World Geodetic System 1984 (WGS84) is the reference ellipsoid used by GPS and by virtually all modern mapping, geodesy, and astronomical software. Its defining parameters are:

The semi-major axis is exact by definition. The flattening is defined to the full precision shown. All other geometric quantities are derived from these two.

For chart calculation, the WGS84 ellipsoid enters through two mechanisms:

1. It defines the relationship between the observer’s geodetic latitude (what GPS reports) and the geocentric latitude (what astronomical formulas require).
2. It determines the observer’s geocentric distance, which affects topocentric corrections (parallax for the Moon and chart angles).

Geodetic vs. Geocentric Latitude #

The distinction between geodetic and geocentric latitude is a persistent source of confusion and bugs.

Geodetic latitude ($\phi$) is the angle between the normal to the ellipsoid surface at the observer’s location and the equatorial plane. This is the latitude reported by GPS receivers, shown on maps, and listed in atlases and birth-record databases. It is the standard input to astrological software.

Geocentric latitude ($\phi’$) is the angle between the line from the Earth’s center to the observer and the equatorial plane. Because the Earth is oblate, the surface normal does not pass through the center (except at the equator and poles), so $\phi’ \neq \phi$ in general.

The difference is:

$$ \phi’ = \phi - \Delta\phi $$

where $\Delta\phi$ is always positive (geocentric latitude is closer to the equator). The maximum difference occurs at latitude $\pm 45°$ and is:

$$ \Delta\phi_{\max} \approx \frac{f \sin 2\phi_{\max}}{2} \approx 0.19° \approx 11.5’ $$

The exact relationship is:

$$ \tan\phi’ = (1 - e^2) \tan\phi = \frac{b^2}{a^2} \tan\phi $$

or equivalently:

$$ \tan\phi’ = (1 - f)^2 \tan\phi $$

This is exact for a point on the ellipsoid surface. Most astronomical formulas from Meeus and the Explanatory Supplement use geocentric latitude (or the equivalent $\rho\sin\phi’$ and $\rho\cos\phi’$ quantities defined below). Software must convert geodetic input to geocentric before applying these formulas.

The Quantities $\rho\sin\phi’$ and $\rho\cos\phi’$ #

Rather than separately computing the geocentric latitude $\phi’$ and the geocentric distance $\rho$ (both measured from the Earth’s center), it is more practical to compute two combined quantities that appear directly in parallax formulas:

$$ \rho\sin\phi’ = \frac{(1 - e^2) \sin\phi}{\sqrt{1 - e^2 \sin^2\phi}} + \frac{h}{a} \sin\phi $$

$$ \rho\cos\phi’ = \frac{\cos\phi}{\sqrt{1 - e^2 \sin^2\phi}} + \frac{h}{a} \cos\phi $$

Here $h$ is the observer’s height above the ellipsoid in meters, $a$ is the semi-major axis, and $\rho$ is the geocentric distance expressed as a fraction of the equatorial radius $a$ (dimensionless). The quantity $1/\sqrt{1 - e^2\sin^2\phi}$ is the radius of curvature in the prime vertical, often denoted $N/a$ or $C$.

For an observer at sea level ($h = 0$) at geodetic latitude $\phi = 45°$:

$$ \rho\sin\phi’ = \frac{(1 - 0.00669438)\sin 45°}{\sqrt{1 - 0.00669438 \sin^2 45°}} = \frac{0.99331 \times 0.70711}{\sqrt{1 - 0.003347}} = \frac{0.70238}{0.99833} = 0.70355 $$

$$ \rho\cos\phi’ = \frac{\cos 45°}{\sqrt{1 - 0.00669438 \sin^2 45°}} = \frac{0.70711}{0.99833} = 0.70829 $$

From these, the geocentric latitude and distance can be recovered:

$$ \phi’ = \text{atan2}(\rho\sin\phi’, \rho\cos\phi’) = \text{atan2}(0.70355, 0.70829) = 44.81° = 44°48’36’’ $$

$$ \rho = \sqrt{(\rho\sin\phi’)^2 + (\rho\cos\phi’)^2} = \sqrt{0.70355^2 + 0.70829^2} = 0.99775 $$

The geocentric latitude at $45°$ geodetic is about $44°48’36’‘$ — a difference of $11’24’'$ or about $0.19°$. The geocentric distance is about $0.998,a$, slightly less than the equatorial radius.

The Geoid #

The geoid is the equipotential surface of the Earth’s gravity field that best approximates mean sea level. It is not an ellipsoid — it has bumps and dips caused by the uneven distribution of mass within the Earth. The geoid departs from the WGS84 ellipsoid by up to about ±100 meters.

For chart calculation, the geoid is irrelevant. The observer’s height above the ellipsoid enters the $\rho\sin\phi’$ and $\rho\cos\phi’$ formulas, but the terms involving $h/a$ are negligible for any reasonable altitude. Even at 10 km altitude (the summit of the atmosphere), $h/a \approx 1.6 \times 10^{-3}$, which contributes less than $0.01’'$ to topocentric parallax. The distinction between height above the geoid (orthometric height) and height above the ellipsoid (geodetic height) is entirely negligible.

Chart Sensitivity to Location Errors #

How much does a chart change if the birth location is wrong? This question matters practically because historical birth locations are often imprecise, and it determines how much effort is warranted in location research.

Effect on Chart Angles #

The Ascendant moves approximately 1° every 4 minutes of sidereal time, which corresponds roughly to 1° of geographic longitude (at mid-latitudes; the exact relationship depends on latitude and the ecliptic geometry at the moment). The MC moves at approximately the same rate.

Longitude error: A 1° error in geographic longitude shifts the local sidereal time by about 4 minutes (since the Earth rotates 360° in 24 sidereal hours, or 1° in 4 minutes). This shifts the MC by roughly 1° and the Ascendant by roughly 1°, though the Ascendant shift varies with latitude and the oblique ascension rate of the rising sign.

Latitude error: A 1° error in geographic latitude changes the Ascendant by a variable amount depending on the latitude and the ecliptic longitude on the horizon. Near the equator, the effect is small (a fraction of a degree). At high latitudes, the effect is larger and can exceed 1° per degree of latitude error. The MC is essentially independent of latitude (it depends only on sidereal time, which depends only on longitude and time).

Effect on Planetary Positions #

Planetary ecliptic longitudes are nearly independent of the observer’s location on Earth. The geocentric longitude of a planet is the same regardless of whether the observer is in New York or Tokyo. Topocentric corrections (parallax) shift the apparent position, but for all bodies except the Moon, these corrections are far below 1’. For the Moon, the topocentric correction can reach about 1° — significant enough to change the Moon’s sign in rare cases. See Viewpoints for details.

Practical Sensitivity #

The following table estimates the effect of a 1° error in geographic longitude and latitude on various chart elements, for a mid-latitude location ($\phi \approx 45°$):

The conclusion: for chart angles and house cusps, location must be accurate to within a few arcminutes of longitude and latitude to keep the error below 1° of arc. For planetary longitudes, location accuracy is essentially irrelevant. This is why birth-time accuracy (which determines sidereal time) matters far more than location accuracy for most chart elements — but location still matters for the Ascendant and houses.

Implementation Notes #

1. Always use WGS84 unless you have a specific reason not to. Older reference ellipsoids (Clarke 1866, GRS 80) differ from WGS84 by amounts too small to matter for chart calculation, but consistency is free and prevents confusion.

2. Convert geodetic to geocentric early. Compute $\rho\sin\phi’$ and $\rho\cos\phi’$ once from the input latitude and store them. Every subsequent formula that needs the observer’s position will use these quantities.

3. Altitude is negligible for chart calculation unless you are computing topocentric positions for the Moon with sub-arcminute precision. For all practical astrological purposes, $h = 0$ is safe.

4. GPS coordinates are geodetic. When a user enters coordinates from a GPS device or a mapping service, those are geodetic (WGS84) coordinates. No further conversion of the latitude system is needed — but the geodetic-to-geocentric conversion for astronomical formulas is still required.

5. Southern hemisphere: geodetic and geocentric latitudes are both negative in the southern hemisphere. The formulas work correctly with negative $\phi$; no special-casing is needed.

References #

• Meeus, J. (1998). Astronomical Algorithms, 2nd ed. Willmann-Bell. Chapter 11.
• Explanatory Supplement to the Astronomical Almanac, 3rd ed. (2013). University Science Books. Chapter 4.
• NIMA (2000). Department of Defense World Geodetic System 1984. Technical Report TR8350.2, 3rd ed.
• Torge, W., & Müller, J. (2012). Geodesy, 4th ed. De Gruyter. Chapters 4–5.