The Madrid Codices I and II are two collections of Leonardo da Vinci's manuscripts found in a collection in the National Library of Madrid at the end of the 1960s. In particular, the Madrid Codex I consists of 382 pages of notes accompanied by something like 1600 between sketches and drawings and addresses a problem for which Leonardo is in some way particularly known as an engineer and designer: gears.
Leonardo's starting point is the study of friction. This is a force that opposes motion, but thanks to its opposition it is possible for us to walk without slipping or losing balance. As we all know today, however, the intensity of the frictional force depends on the surfaces that are in contact with each other, on how smooth or rough they are, which is independent of the area in contact and which can be reduced by using, for example, a lubricant or cylinders. All this, however, was already known to Leonardo, as it is possible to observe from the reading of the Madrid Code I. Furthermore, it is always Leonardo who introduced the concept of friction coefficient, defining it as the ratio between the force required to slide two surfaces horizontally on top of each other and the pressure between the two surfaces. Leonardo also estimated the value of this friction coefficient in 1/4, consistent with the materials best known to the florentine and with which he could carry out experiments (wood on wood, bronze on steel, etc.)(1).
At this point Leonardo is ready to develop a series of gears capable of carrying mechanical energy and producing motion, minimizing friction with the use of spheres and cylinders, as can be seen from his numerous drawings. In particular, however, it is Leonardo's mechanical use of two particular geometric shapes that is striking, because it anticipates their actual adoption by centuries: the epicloidal teeth and the globoidal gear.
The architecture of Pi Mensae
Few days before the formal acceptance of this paper, an independent study about the architecture of the π Men planetary system was published(1). The results of that work, based on public data and not including the ESPRESSO observations, confirm the high mutual inclination of the orbital planes of π Men b and c. Our results are in agreement with those of Xuan & Wyatt and are characterized by a better formal precision.(2)Pi Mensae, or π Men, is a yellow dwarf star in the constellation of Mensa. We know that it has a little planetary system, constituted by two planets (or, if you prefer, we discover only two planets orbiting around Pi Mensae): Pi Mensae b, one of the most massive planets ever discovered, about 14.1 the mass of Jupiter, and Pi Mensae c, a super-Earth, about 4.5 the mass of our planet.
In 2020, an analysis with Gaia DR2 and Hipparcos astrometry showed that planets b and c are located on orbits mutually inclined by 49°-131°, which causes planet c to not transit most of the time, and acquire large misalignments with its host star's spin axis(1).
This result was discovered also by an italian team(2), but published just few days after on arXiv, using ESPRESSO (Echelle Spectrograph for Rocky Exoplanet- and Stable Spectroscopic Observations), a sèectrograph designed and developed in Italy by researchers of Brera's Astronomical Observatory at the Merate's laboratories. The instrument, mounted on the Very Large Telescope, has in a certain sense been tested with the planetary system of Pi Mensae, therefore, even coming seconds for a few, the result can be considered a success for the young ESPRESSO.
Xuan, J. W., & Wyatt, M. C. (2020). Evidence for a high mutual inclination between the cold Jupiter and transiting super Earth orbiting π Men. Monthly Notices of the Royal Astronomical Society, 497(2), 2096-2118. doi:10.1093/mnras/staa2033 (arXiv) ↩︎ ↩︎
Damasso, M., Sozzetti, A., Lovis, C., Barros, S. C. C., Sousa, S. G., Demangeon, O. D. S., ... & Rebolo, R. (2020). A precise architecture characterization of the Men planetary system. A&A, Forthcoming article doi:10.1051/0004-6361/202038416 (arXiv) ↩︎ ↩︎