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Register a new userA translation of L. Eulers An easy method for calculating the motion of celestial bodies perturbed in any manner avoiding astronomical computations
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Sylvio R Bistafa
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This is a translation from Latin of E348 Methodus facilis motus corporum coelestium utcunque perturbatos ad rationem calculi astronomici revocandi, in which Euler develops a method to alleviate the astronomical computations in a typical celestial three-body problem represented by Sun, Earth and Moon. In this work, Eulers approach consists of two parts: geometrical and mechanical. The geometrical part contains most of the analytical developments, in which Euler makes use of Cartesian and spherical trigonometry as well - the latter not always in a clear enough way. With few sketches to show the geometrical constructions envisaged by Euler - represented by several geometrical variables -, it is a hard to follow publication. The Translator, on trying to clear the way to the non-specialized reader, used the best of his abilities to add his own figures to the translation. In the latter part of the work, Euler particularizes his developments to the Moon, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body, which makes his claim of an easy method as being rather fallacious. Despite showing great analytical skills, Euler did not give indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moons orbit.
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This is a translation from Latin of E840 De motu cometarum in orbitis parabolicis, solem in foco habentibus, in which Euler addresses six problems related to comets in heliocentric parabolic orbits. Problem 1: Find the true anomaly of a heliocentric comet from the latus rectum of the orbit and the medium Earth to Sun distance. Problem 2: Find the orbit of a heliocentric comet from three given positions. Problem 3: Knowing the orbit of a comet, and the instant in time in which it dwells in the perihelion, define its longitude and latitude at any time. Problem 4: From two locations of a heliocentric comet, find the inclination of the comets orbit in relation to the ecliptic, and the positions of the nodes. Problem 5: From the time before or after the comet had reached the perihelion, and from the comets distance to the perihelion as seen from the Sun, find the same distance in another time before or after it had appeared in the perihelion. Problem 6: Find the orbit of a comet from three given heliocentric longitudes and latitudes. From these problems, several corollaries and scholia are derived.
This is an English translation of E579 in which the introductory remarks are in French, while Eulers original text is in Latin. By considering the balance of forces acting on a raising balloon on an isothermal atmosphere, namely the weight of the balloon, the buoyant force, and the aerodynamic drag force, Euler provides closed formulas for the calculation of the maximum altitude reached by the balloon, the altitude for which the velocity is maximum, the maximum velocity attained by the balloon, and the total ascending time.
An attempt is made to avoid the difficulty of the infinite reaction of the electron on itself, which occurs in quantum electrodynamics, by introducing difference equations instead of differential equations. This vision allows the difficulty of the relativistic wave equation emphasised by Klein, for example, to be essentially eliminated.
Scatterings of galactic dark matter (DM) particles with the constituents of celestial bodies could result in their accumulation within these objects. Nevertheless, the finite temperature of the medium sets a minimum mass, the evaporation mass, that DM particles must have in order to remain trapped. DM particles below this mass are very likely to scatter to speeds higher than the escape velocity, so they would be kicked out of the capturing object and escape. Here, we compute the DM evaporation mass for all spherical celestial bodies in hydrostatic equilibrium, spanning the mass range $[10^{-10} - 10^2]~M_odot$. We illustrate the critical importance of the exponential tail of the evaporation rate, which has not always been appreciated in recent literature, and obtain a robust result: for the geometric value of the scattering cross section and for interactions with nucleons, the DM evaporation mass for all spherical celestial bodies in hydrostatic equilibrium is approximately given by $E_c/T_chi sim 30$, where $E_c$ is the escape energy of DM particles at the core of the object and $T_chi$ is the DM temperature. The minimum value of the DM evaporation mass is obtained for super-Jupiters and brown dwarfs, $m_{rm evap} simeq 0.7$ GeV. For other values of the scattering cross section, the DM evaporation mass only varies by a factor of two or less within the range $10^{-41}~textrm{cm}^2 leq sigma_p leq 10^{-31}~textrm{cm}^2$, where $sigma_p$ is the spin-independent DM-nucleon scattering cross section. Its dependence on parameters such as the local galactic DM density and velocity, or the scattering and annihilation cross sections is only logarithmic.
This is an annotated translation from German of Untersuchung einer nach den Eulerschen Vorschlagen (1754) gebauten Wasserturbine [Investigation of a water turbine built according to Eulers proposals (1754)] that reports the tests results of a modern (1944) prototype of the so-called Segner-Euler turbine, which was strictly constructed according to Eulers prescription as laid down in E222 -- Theorie plus complete des machines qui sont mises en mouvement par la reaction de leau. (Memoires de lacademie des sciences de Berlin 1756, Vol. 10, pp. 227-295.), showing the feasibility of Eulers original proposal. A reproduction of the original paper is attached at the end of the translation.
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