Researchers tap into 800-year-old mathematical theory to design GPS on the Moon

According to a recent study, the mathematical technique devised by the Italian mathematician Leonardo Fibonacci shows potential for the development of lunar GPS.
Mrigakshi Dixit
Representational picture.
Representational picture.

Vitaly Kusaylo/iStock 

GPS proves invaluable whenever we find ourselves disoriented in any location on Earth. This system, known as the Global Positioning System, utilizes a network of satellites to accurately determine our exact position in real-time.

As humanity gears up to invest more time on the Moon, certain experts are contemplating the potential use of a comparable technology to aid future explorers in navigating the lunar surface.

ELTE (Eötvös Loránd University) has been conducting research on navigation methodologies that could be implemented in forthcoming GPS systems on the Moon.

As per their official statement, they are considering a potential approach for developing a precise lunar GPS system, which draws upon an 800-year-old mathematical technique created by the Italian mathematician Leonardo Fibonacci.

Shape is a crucial factor for a precise GPS system

The establishment of a highly accurate global GPS system is the culmination of numerous interrelated mechanisms working in tandem.

Among the essential factors in creating a precise GPS system for any celestial body or planet is its specific shape.

For instance, the Earth's 3D geometric representation adopts an ellipsoid shape, and this ellipsoid coordinate system is utilized by GPS technology on Earth for both horizontal and vertical datums. These datums act as reference points for specific locations, such as when providing directions from a particular point.

While the prevailing theory suggests that the Moon's shape is nearly spherical, the authors of this new study aimed to estimate the Moon's shape by applying the concept of a rotating ellipsoid, as used for the Earth.

“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 researchers said in the press release.

The Fibonacci sphere theory arranged points uniformly

The scientists utilized existing surface data of the lunar selenoid to derive the parameters of the rotating ellipsoid that best fit the hypothesized shape of the moon. 

They searched the database for "semi-major and semi-minor axes" that best matched a rotating ellipsoid. Furthermore, the data supplied sample height to uniformly distributed points on the surface.

“By gradually increasing the number of sampling points from 100 to 100,000, the values of the two parameters stabilized at 10,000 points,” noted the official release. 

The researchers then used the Fibonacci sphere theory to arrange points uniformly across a spherical surface of the Moon. 

This method has also been applied to the WGS84 ellipsoid standard system used by GPS, according to the statement.

By employing this approach, scientists believe that it could potentially pave the way for the development of precise software for the Moon's GPS system in the future.

The findings have been published in the journal Acta Geodaetica et Geophysica.

Study Abstract:

Since the Moon is less flattened than the Earth, most lunar GIS applications use a spherical datum. However, with the renaissance of lunar missions, it seems worthwhile to define an ellipsoid of revolution that better fits the selenoid. The main long-term benefit of this might be to make the lunar adaptation of methods already implemented in terrestrial GNSS and gravimetry easier and somewhat more accurate. In our work, we used the GRGM 1200A Lunar Geoid, a 660th degree and order potential surface, developed in the frame of the GRAIL project. Samples were taken from the potential surface along a mesh that represents equal area pieces of the surface, using a Fibonacci sphere. We tried Fibonacci spheres with several numbers of points and also separately examined the effect of rotating the network for a given number of points on the estimated parameters. We estimated the best-fitting rotation ellipsoid’s semi-major axis and flatness data by minimizing the selenoid undulation values at the network points, which were obtained for a = 1,737,576.6 m and f = 0.000305. This parameter pair is already obtained for a 10,000 point grid, while the case of reducing the points of the mesh to 3000 does not cause a deviation in the axis data of more than 10 cm. As expected, the absolute value of the selenoid undulations have decreased compared to the values taken with respect to the spherical basal surface, but significant extreme values still remained as well.

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