On the convergence of elliptic motion

The convergence of Lagrange series is studied on a part of the elliptical orbit for values of eccentricity exceeding the Laplace limit. The regions in the vicinity of the two apses of the orbit are identified in which the Lagrange series converge absolutely and uniformly for the values of the eccentricity greater than the Laplace limit. The obtained results are of practical interest for astronomy when studying motions of stellar bodies in orbits with high eccentricity. In particular, these series may be used to calculate the orbits of comets or asteroids with high eccentricity as they pass through the neighborhood of perihelion or to calculate the orbits of artificial satellites with high eccentricity “hanging” in the vicinity of apogee. In stellar dynamics, these series may be used in cases of close binary stars, many of which move in orbits with an eccentricity greater than the Laplace limit.