# 800-year-old mathematical trick could help with lunar navigation

Jul 26, 2023
Read Time:3 Minute, 29 Second

Kamilla Cziráki, a geophysics student at the Faculty of Science of Eötvös Loránd University (ELTE), has taken a new approach to researching the navigation systems that can be used on the surface of the Moon to plan future journeys. Together with Professor Gábor Timár, head of the Department of Geophysics and Space Sciences, they calculated the parameters used in the Earth’s GPS system for the Moon using the method of mathematician Fibonacci, who lived 800 years ago. Their findings have been published in the journal Acta Geodaetica et Geophysica. Acta Geodaetica et Geophysica.

Now, as humanity prepares to return to the Moon after half a century, the focus is on possible methods of lunar navigation. It seems likely that the modern successors to the lunar vehicles of the Apollo missions will now be assisted by some form of satellite navigation, similar to the GPS system on Earth. In the case of the Earth, these systems do not take into account the actual shape of our planet, the geoid, not even the surface defined by sea level, but a rotation ellipsoid that best fits the geoid. Its intersection is an ellipse that is furthest from the Earth’s centre of mass at the equator and closest to it at the poles. The radius of the Earth is just under 6400 kilometres, and the poles are about 21.5 kilometres closer to the centre than the equator.

Why is the shape of the ellipsoid that best fits the Moon interesting, and what parameters can be used to describe it? Why is it interesting that, compared to the Moon’s mean radius of 1737 kilometres, its poles are about half a kilometre closer to its centre of mass than its equator? If we want to apply the software solutions tried and tested in the GPS system to the Moon, we need to specify two numbers, the semi-major and the semi-minor axis of this ellipsoid, so that the programmes can be easily transferred from the Earth to the Moon.

The Moon rotates more slowly, with a rotation period equal to its orbital period around the Earth. This makes the Moon more spherical. It is almost a sphere, but not quite. Nevertheless, for the mapping of the Moon that has been done so far, it has been sufficient to approximate the shape of a sphere, and those who have been more interested in the shape of our celestial companion have used more complex models.

Interestingly, the approximation of the Moon’s shape with a rotating ellipsoid has never been done before.

The last time such calculations were made was in the 1960s by Soviet space scientists, using data from the side of the Moon visible from Earth.

Kamilla Cziráki, a second-year geosciences student specialising in geophysics, worked with her supervisor, Gábor Timár, head of the Department of Geophysics and Space Sciences, to calculate the parameters of the rotating ellipsoid that best fit the theoretical shape of the Moon. To do this, they used a database of an existing potential surface, called the lunar selenoid, from which they took a height sample at evenly spaced points on the surface and searched for the semi-major and semi-major axes that best fit a rotation ellipsoid. By gradually increasing the number of sampling points from 100 to 100,000, the values of the two parameters stabilised at 10000 points.

One of the main steps of the work was to investigate how to arrange N points uniformly on a spherical surface, with several possible solutions; Kamilla Cziráki and Gábor Timár chose the simplest one, the so-called Fibonacci sphere. The Fibonacci spiral can be implemented with very short and intuitive code, and the foundations of this method were laid by the 800-year-old mathematician Leonardo Fibonacci. The method has also been applied to the Earth as a verification, reconstructing a good approximation of the WGS84 ellipsoid used by GPS.

#### JOURNAL

Acta Geodaetica et Geophysica

#### DOI

10.1007/s40328-023-00415-w

#### ARTICLE TITLE

Parameters of the best fitting lunar ellipsoid based on GRAIL’s selenoid model

27-Jun-2023

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