Earth ellipsoid

An Earth ellipsoid, also called an Earth spheroid, is a mathematical figure used to approximate the shape and size of our planet.

Because the Earth is not a perfect, smooth sphere, scientists need a reliable geometric model to perform calculations. These models are essential for the fields of geodesy, astronomy, and the geosciences. Geodesy is the science of measuring the Earth's shape and its gravitational field. By using an ellipsoid, researchers can create a stable reference frame. This allows them to define precise coordinates like latitude, longitude, and elevation. Without these mathematical models, modern navigation and mapping would be much less accurate.

The Earth is specifically an oblate ellipsoid of revolution. This shape is created by the centrifugal force caused by the Earth's rotation. This force causes the planet to bulge at its center. As a result, the Earth is wider at the equator than it is from pole to pole. The major axis, or equatorial axis, is longer than the minor axis, which connects the geographical poles. The difference between these two axes is slightly more than 21 kilometers. This difference represents about 0.335% of the Earth's size.

To define this shape, scientists use two main parameters. They use the semi-major axis, which is the equatorial radius. They also use the semi-minor axis, which is the distance from the center to a pole. Another way to describe this is through flattening. Flattening is the amount of bulging at the equator relative to the poles. Scientists often express this as the inverse flattening.

There are two distinct types of ellipsoids used in science: mean and reference. A mean Earth ellipsoid describes the global average of the Earth's surface curvature. It aims for a theoretical connection between geographic latitude and the curvature of the geoid. The geoid is the irregular, true shape of the Earth based on gravity. An ideal mean ellipsoid would have the same volume as the geoid. However, for specific regions, a reference ellipsoid is often a better choice. A reference ellipsoid is a surface that matches the local curvature of the regional geoid. If the mathematical surface does not match the local area, measurements can suffer from small distortions. This is why different models are used for different parts of the world.

History shows how our understanding of this shape has evolved through discovery. In 1669, Jean Picard found a reliable value for the Earth's radius. Later, in 1687, Isaac Newton published his work in the Principia. He proved that a rotating fluid body in equilibrium takes the form of an oblate ellipsoid. This explained why the Earth bulges. In 1743, Alexis Clairaut proposed that studying variations in gravity could help determine the Earth's shape. By the late 18th century, scientists worked to combine gravity measurements with physical arc measurements. In 1799, the Weights and Measures Commission adopted a specific flattening based on the work of Pierre-Simon Laplace.

Many different ellipsoids have been used for national and international surveys. The Bessel ellipsoid was developed in 1841. The international Hayford ellipsoid was created in 1910 and adopted in 1924. Even though modern axes have changed, some older ellipsoids are still used for legal reasons. For example, the coordinates of millions of boundary stones are fixed to specific models. If the reference surface changes, those coordinates would change too. In the modern era, the WGS84 ellipsoid is the standard for GPS positioning and satellite navigation. The WGS84 has a flattening of approximately 1/298.257. This model is incredibly precise and is used for global mapping.

Different celestial bodies show how much their rotation affects their shape. The Earth is more elliptical than our Moon, which has a flattening of less than 1/825. However, Jupiter is much more oblate, with a flattening of about 1/15. Some moons, like Saturn's moon Telesto, are even more extreme. Telesto is a triaxial ellipsoid with a flattening between 1/3 and 1/2. This means its polar diameter is only about half or two-thirds of its equatorial diameter. These variations show that the faster or more massive an object is, the more its shape might deviate from a sphere.

Today, scientists determine the Earth's shape using advanced technology. In the past, they used meridian arcs, which involved measuring long curves on the ground. Modern geodesy relies on satellite geodesy and satellite gravimetry. These methods use satellites to measure the Earth's gravity and its center of mass with extreme accuracy. This precision is vital for astronautics and international networks. The International Geoscientific Union (IUGG) regularly updates the axes of the Earth ellipsoid to match the best available data. This ensures that our mathematical models stay as close to reality as possible.