Satnav for the Moon could benefit from Fibonacci’s expertise

Middle Ages maths to the rescue

Future satellite navigation systems intended for Earth's Moon may be aided by a model of it developed with methods that go back to mathematician Fibonacci, who lived 800 years ago.

With increasing interest in missions to return people to the Moon and even establish some sort of permanent outpost on Earth’s satellite, it seems that modern successors to the lunar vehicles of the Apollo missions may be assisted by some form of navigation system, similar to the GPS system on Earth.

US space agency NASA successfully sent an uncrewed Orion spacecraft into lunar orbit and back earlier this year as part of the Artemis I mission, and plans to repeat the trick with a human crew for Artemis II in 2024. The space agency’s long-term plans are to build a sustainable presence on the lunar surface, including at least one moon base.

Both NASA and the European Space Agency (ESA) have already conceived of potential GPS-like satellite constellations around the Moon, named LunaNet and Moonlight, respectively, in order to provide accurate position, navigation, and timing (PNT) services for lunar activity.

However, it appears that the Earthly GPS systems do not take into account the actual shape of the planet itself, using an approximation based on a rotational ellipsoid that is the best fit for its true shape, which is wider at the equator than the distance between the poles.

In the case of the moon, it rotates more slowly, with a rotational period equal to its orbital period around the Earth. This leads to the Moon being more spherical than the Earth, but still not truly spherical.

For the mapping of the Moon done so far, it has been sufficient to approximate the shape of a sphere, but the last time such calculations were made was in the 1960s. With greater lunar activity planned for the future, a more accurate representation is called for.

Scientists at the Faculty of Science of Eötvös Loránd University (ELTE) at Budapest in Hungary have sought to address this by calculating the parameters of the rotating ellipsoid that best fit the theoretical shape of the Moon.

The scientists, Kamilla Cziráki, a second-year geosciences student specialising in geophysics, and Gábor Timár, head of the Department of Geophysics and Space Sciences, made use of lunar surface data derived from the NASA GRAIL mapping mission, and took height samples at evenly spaced points on the surface, then used these to search for the axes of a best-fit rotational ellipsoid.

This is where Fibonacci comes in. One of the simplest solutions to equally distribute a given number of points on a spherical surface is the Fibonacci Sphere, related to the Fibonacci sequence, which the medieval Italian mathematician is credited with introducing to Europe. He also helped Europe kick the habit of using Roman numerals in favor of Indo-Arabic digits, paving the way for major advances in European mathematics.

An article describing the physicists' work has been published in the academic journal Acta Geodaetica et Geophysica.

The abstract states that in the case of LunaNet, which NASA plans to create for geographic information system (GIS) applications, the reference surface is currently planned to be the Lunar Reference Frame Standard, a sphere with a radius of 1,737.4km. However, in the future, it is conceivable that a rotational ellipsoid will be needed to replace it.

The scientists also performed the same calculations for the Earth, with the aim of showing that this method can give a good estimate of the ideally fitting ellipsoid. Their results differed from the existing model closely, with a deviation of only 60cm (24in). ®

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