Winter model in the quasi-continuum limit

We study Winter or [Formula: see text]-shell model at finite volume (length), describing a small resonating cavity weakly-coupled to a large one. For generic values of the coupling, a resonance of the usual model corresponds, in the finite-volume case, to a compression of the spectral lines; for specific values of the coupling, a resonance corresponds instead to a degenerate or a quasi-degenerate doublet. A secular term of the form [Formula: see text] occurs in the perturbative expansion of the momenta (or of the energies) of the particle at third order in [Formula: see text], where [Formula: see text] is the coupling among the cavities and [Formula: see text] is the ratio of the length of the large cavity over the length of the small one. These secular terms, which tend to spoil the convergence of the perturbative series in the large volume case, [Formula: see text], are resummed to all orders in [Formula: see text] by means of standard multiscale methods. The resulting improved perturbative expansions provide a rather complete analytic description of resonance dynamics at finite volume.

The Motion of Halley's Comet from 837 to 1910

Numerical experiments have been made in an attempt to remove the residuals of P/Halley and link the seven apparitions from 1456 to 1910. All efforts to link more than two apparitions using Newtonian equations have invariably failed. However, by the addition of a secular term to the equations of motion, the four apparitions from 1910 back to 1682 can be linked by a numerical integration which represents the observations to contemporary accuracy. When this integration is continued, the apparitions of 1607, 1531, and 1456 show residuals of less than one day in the time of perihelion passage. Prior to 1456 the residuals begin to run off but, with the exception of 1222 and 1066, the apparitions back to 837 show residuals no greater than four days in the time of perihelion passage. The residuals of 20 days in 1222 and 7 days in 1066 appear anomalous but can be made reasonable if the Chinese records are adopted in preference to the European records.