Ancient Mathematical Technique Revitalized for Lunar Navigation

Researchers have introduced an innovative technique for navigation on the Moon that employs the Fibonacci spiral to determine the best settings for a Moon-based GPS equivalent. This avant-garde method improves the navigational capabilities of lunar vehicles by providing more precise mapping tailored to the Moon’s unique ellipsoidal geometry, which is defined by specific semi-major and semi-minor axes.

Navigational technologies apt for lunar terrain exploration are essential for planning forthcoming missions.

Kamilla Cziráki, a geophysics undergraduate from the Faculty of Science at Eötvös Loránd University (ELTE), is leading groundbreaking research to develop navigational technologies suitable for exploration on the lunar surface. Collaborating with Professor Gábor Timár, who heads the Department of Geophysics and Space Sciences, they have incorporated the mathematical methodologies developed by the medieval mathematician Fibonacci to modify the Earth’s GPS settings to be compatible with the Moon’s requirements.

The research outcomes have been formally published in the scholarly journal Acta Geodaetica et Geophysica.

As global efforts intensify to make a lunar return after a 50-year hiatus, attention is being paid to potential methods for Moon-based navigation. It is becoming increasingly probable that future lunar vehicles, much like the successors to the Apollo missions, will benefit from some form of satellite navigation analogous to Earth’s GPS system.

For Earth, existing navigation systems do not account for the planet’s true shape, known as the geoid. Rather, they employ a rotational ellipsoid that closely approximates the geoid. This ellipsoid’s intersection with the Earth creates an ellipse, which is furthest from Earth’s center of mass at the equator and nearest at the poles. Earth’s radius measures just below 6400 kilometers, with its poles being approximately 21.5 kilometers nearer to the center than the equator.

Why is the Moon’s best-fitting ellipsoidal shape significant, and what variables define it? It is intriguing to note that compared to the Moon’s average radius of 1737 kilometers, its poles are nearly half a kilometer closer to its center of mass than its equator. If we aim to implement software solutions proven effective in Earth’s GPS system to lunar settings, we must identify the semi-major and semi-minor axes of this ellipsoid for seamless adaptation.

The Moon has a slower rotation speed, equal to its orbital period around Earth. This characteristic renders the Moon almost spherical in shape. Existing lunar mapping efforts have sufficed with approximations based on spherical shapes, while more complex models have been employed by those interested in more detailed studies of our celestial neighbor.

Remarkably, this is the first instance where the Moon’s shape has been approximated using a rotating ellipsoid.

The most recent previous calculations of this nature date back to the 1960s and were conducted by Soviet space researchers, utilizing data from the portion of the Moon visible from Earth.

Kamilla Cziráki, a sophomore in geosciences with a focus on geophysics, collaborated with her advisor, Gábor Timár, to determine the parameters of the rotating ellipsoid that most accurately represents the theoretical shape of the Moon. They utilized a pre-existing potential surface database, known as the lunar selenoid, and sampled elevations at equidistant points on the surface to find the semi-major and semi-minor axes that best fit a rotation ellipsoid. By incrementally increasing the number of sample points from 100 to 100,000, they found that the values for these parameters stabilized at 10,000 points.

A pivotal aspect of their work involved investigating methods to uniformly distribute N points over a spherical surface. Several approaches were considered, and they eventually selected the simplest one: the Fibonacci sphere. The Fibonacci spiral’s implementation is supported by concise and intuitive code, rooted in methods developed by the medieval mathematician Leonardo Fibonacci. This technique has also been validated on Earth by approximating the WGS84 ellipsoid used by GPS systems.

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

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More about Fibonacci spiral in lunar navigation

• Published Research in Acta Geodaetica et Geophysica
• Introduction to Fibonacci Spiral
• Eötvös Loránd University (ELTE) Faculty of Science
• Understanding Ellipsoids and Geoids
• Overview of GPS Technology
• History of Soviet Space Research
• GRAIL’s Selenoid Model
• WGS84 Ellipsoid Explanation