# 800 year old math trick could help lunar navigation.

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A navigation system for exploring the lunar surface to plan future expeditions.
Eötvös Lorand University (ELTE) geophysics student Kamilla Cziráki is pioneering a new approach to the study of navigation systems suitable for expeditions to the lunar surface. In collaboration with Professor Gábor Timárs, Head of the Department of Geophysics and Space Science, they used the 800-year-old mathematician Fibonacci’s method to adjust the parameters of the Earth’s GPS system to the Moon.

Their results were published in the journal Acta Geodaetica et Geophysicala. Now that humans are preparing to return to the moon after half a century, the main focus is on possible methods of lunar navigation. Today’s successors to the Apollo mission rovers are now likely to be aided by some form of satellite navigation similar to the GPS system on Earth. In the case of the Earth, these systems do not consider the actual shape of the Earth, the geoid, or even the surface defined by sea level, but instead consider the ellipsoid that best fits the geoid. Its intersection is an ellipse with the equator farthest from the Earth’s center of mass and the poles closest to the Earth’s center of mass. The Earth’s radius is less than 6,400 kilometers, and the poles are 21.5 kilometers closer to the center than the equator.

Why is the optimal ellipsoidal shape of the Moon interesting and by what parameters can it be characterized? Why is it interesting that, compared to the average radius of the Moon, which is 1737 kilometers, its poles are half a kilometer closer to the center of mass than to the equator? If we want to use the tried and tested software solutions in the lunar GPS systems, we need to specify two numbers, the semi-major axis and the semi-minor axis of the ellipsoid, so that the program can be easily transferred from Earth. up to a month. The Moon rotates slowly with a rotation period equal to its orbit around the Earth. This makes the Moon more spherical. It’s almost a ball, but not quite. Until now, however, a rough spherical shape has sufficed for mapping the Moon, and those more interested in the shape of our celestial moon have used more complex models. Interestingly, the ellipsoid has never before been used to approximate the shape of the Moon.
The last time such calculations were made was in the sixties of the last century, when Soviet space scientists used data from the side of the Moon visible from Earth. Kamilla Cziráki, a second-year geophysics student, worked with her advisor Gábor Timár, head of the Department of Geophysics and Space Science, to calculate the parameters of the spheroid that best fit the theoretical shape of the Moon.

To do this, they used an existing library of potential surfaces (called the Moon Belt), extracted the heights from the library at evenly spaced points on the surface, and searched for the semi-major axis and the semi-major axis that best fit the ellipsoid of the axis of rotation. Gradually increasing the number of sampling points from 100 to 100,000, the values ​​of both parameters stabilize at 10,000 points.

One of the main steps in this work was to investigate how to arrange N points evenly on the sphere, and there were several possible solutions; Kamilla Cziráki and Gábor Timárs chose the simplest, 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. This method has also been used on Earth as a validation that reconstructs a good approximation of the WGS84 ellipsoid used by GPS.

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