The team applied a method based on the mathematical principle established 800 years ago by renowned mathematician Fibonacci, to calculate parameters for a GPS-like system on the Moon. Their research has recently been featured in the esteemed journal Acta Geodaetica et Geophysica.
As humans plan to set foot on the Moon once again after half a century, the prime focus is the potential methods of lunar navigation. It's anticipated that the modern lunar vehicles, unlike their Apollo mission predecessors, will be equipped with some sort of satellite navigation akin to Earth's GPS system.
Earth's GPS systems don't take into account the geoid, the actual shape of our planet, nor the surface determined by sea level. Instead, they are based on a rotation ellipsoid which aligns most closely with the geoid. The intersecting ellipse is furthest from the Earth's center of mass at the equator and nearest at the poles. With an approximate radius of 6400 kilometers, the Earth's poles are around 21.5 kilometers closer to its center than the equator.
To transpose the established GPS solutions onto the Moon, we first need to ascertain the parameters that best describe the shape of the Moon's corresponding ellipsoid. This is particularly relevant as, unlike Earth, the Moon's poles are approximately half a kilometer closer to its center of mass than its equator, with a mean radius of 1737 kilometers. These parameters, the semi-major and semi-minor axes of the ellipsoid, are integral to transitioning the programs from Earth to the Moon.
Due to its slower rotation, which is equal to its orbital period around Earth, the Moon's shape is more spherical. It's not a perfect sphere, but previous mappings of the Moon have deemed it sufficient to approximate its shape as such, with more comprehensive models used for more detailed research.
The last attempt to calculate these parameters occurred in the 1960s by Soviet space scientists, utilizing data from the Earth-visible side of the Moon. However, Kamilla Cziraki and Professor Gabor Timar have revisited this task, using a database of an existing potential surface known as the lunar selenoid.
From this, they extracted height samples at regularly spaced points on the surface and sought the best fitting semi-major and semi-minor axes of a rotation ellipsoid. As they increased the sampling points from 100 to 100,000, the parameters' values stabilized at 10,000 points.
One of the central aspects of their study was exploring how to uniformly distribute N points on a spherical surface. Their method of choice was the Fibonacci sphere, known for its simplicity and efficiency. This method, founded on the work of 800-year-old mathematician Leonardo Fibonacci, involves a Fibonacci spiral and is conducive to a short, intuitive code. This method has also been tested on Earth, yielding a good approximation of the WGS84 ellipsoid used by GPS.
Research Report:Parameters of the best fitting lunar ellipsoid based on GRAIL's selenoid model
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