A translation of L. Euler's "On the motion of comets in parabolic orbits, having the Sun in the focus"

TL;DR

This paper provides a translation of Euler's work on the mathematical analysis of cometary orbits in parabolic trajectories with the Sun at the focus, addressing key problems in orbital determination.

Contribution

It offers a detailed translation and explanation of Euler's original solutions to six fundamental problems in comet orbit calculations.

Findings

Formulas for true anomaly from orbit parameters

Methods to derive orbit from three positions

Techniques to determine orbit inclination and nodes

Abstract

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 comet's 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 comet's 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.