800-Year-Old Math Trick Could Help With Navigating the Moon

Lunar Navigation Mathematics

Scientists have developed a new method for lunar navigation, utilizing the Fibonacci spiral to calculate the optimal parameters for a GPS-like system on the Moon. The innovative approach helps modernize lunar vehicle navigation, offering more accurate mapping based on the Moon’s unique ellipsoidal shape, which is characterized by distinctive semi-major and semi-minor axes.

Navigation systems suitable for lunar surface exploration to chart future expeditions.

Kamilla Cziráki, a geophysics student from the Faculty of Science at Eötvös Loránd University (ELTE), is pioneering a novel approach to study navigation systems suitable for lunar surface expeditions. In collaboration with Professor Gábor Timár, the head of the Department of Geophysics and Space Sciences, they’ve applied the methodology of the 800-year-old mathematician, Fibonacci, to adapt the parameters of Earth’s GPS system for the Moon.

Their findings have been published in the journal 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 of 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 center of mass at the equator and closest to it at the poles. The radius of the Earth is just under 6400 kilometers, and the poles are about 21.5 kilometers closer to the center 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 kilometers, its poles are about half a kilometer closer to its center 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 programs 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 specializing 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 stabilized 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.

Reference: “Parameters of the best fitting lunar ellipsoid based on GRAIL’s selenoid model” by Kamilla Cziráki and Gábor Timár, 27 June 2023, Acta Geodaetica et Geophysica.
DOI: 10.1007/s40328-023-00415-w

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