A Pedagogical Discussion Concerning the Gravitational Energy Radiated by Keplerian Systems

We first discuss the use of dimensional arguments (and of the quadrupolar emission hypothesis) in the derivation of the gravitational power radiated on a circular orbit. Then, we show how to simply obtain the instantaneous power radiated on a general Keplerian orbit by approximating it locally by a circle. This allows recovering with a good precision, in the case of an ellipse, the highly non trivial dependence on the eccentricity of the average power given by general relativity. The whole approach is understandable by undergraduate students.Comment: A simpler method has been used in the calculations, which requires now only standard knowledge (the radius of curvature is defined by the normal acceleration). Two figures have been added. Concerning the dimensional analysis, the comparison with electromagnetism has been detaile

Analysis of Super-Kamiokande 5-day Measurements of the Solar Neutrino Flux

Data in 5-day bins, recently released by the Super-Kamiodande Consortium, has been analyzed by a likelihood procedure that has certain advantages over the Lomb-Scargle procedure used by the consortium. The two most prominent peaks in the power spectrum of the 10-day data were at 9.42 y-1 and 26.57 y-1, and it was clear that one was an alias of the other caused by the regularity of the binning. There were reasons to believe that the 9.42 y-1 peak was an alias of the 26.57 y-1 peak, but analysis of the 5-day data makes it clear that the reverse is the case. In addition to a strong peak near 9.42 y-1, we find peaks at 43.72 y-1and at 39.28 y-1. After comparing this analysis with a power-spectrum analysis of magnetic-field data, we suggest that these three peaks may be attributed to a harmonic of the solar rotation rate and to an r-mode oscillation with spherical harmonic indices l = 2, m = 2.Comment: Accepted for publication in Astrophysical Journa

A simple derivation of Kepler's laws without solving differential equations

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to en explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by beginners at university (or even before) can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest

Had the planet mars not existed: Kepler's equant model and its physical consequences

We examine the equant model for the motion of planets, which has been the starting point of Kepler's investigations before he modified it because of Mars observations. We show that, up to first order in eccentricity, this model implies for each orbit a velocity which satisfies Kepler's second law and Hamilton's hodograph, and a centripetal acceleration with an inverse square dependence on the distance to the sun. If this dependence is assumed to be universal, Kepler's third law follows immediately. This elementary execice in kinematics for undergraduates emphasizes the proximity of the equant model coming from Ancient Greece with our present knowledge. It adds to its historical interest a didactical relevance concerning, in particular, the discussion of the Aristotelian or Newtonian conception of motion