The Eccentricity of the Sun: Kepler's Novel Method of Calculation

Abstract

We construct in this paper the following argument:(1) Johannes Kepler (1571-1630) was critical of the traditional method for calculating the solar eccentricity; it was not sufficiently accurate for his needs.(2) He therefore developed his own method which was based on the variation of the apparent diameter of the Sun.(3) The results of the application of his novel method gave him confidence that the magnitude of the solar eccentricity, which Tycho Brahe (1546-1601) assumed, had to be halved.(4) This was the first substantial change in the calculated magnitude of the solar eccentricity since the time of Ptolemy.Kepler's new value of the eccentricity of the Sun is an essential piece in the revolutionary scheme of his New astronomy (1609).1In July 1600, in a letter to Herwart von Hohenburg, Kepler argued that in order to confirm his polyhedral hypothesis he needed accurate values of eccentricities and positional data.2 Kepler's insistence on accurate observational records was one of the reasons for his meeting with Tycho at Benatky Castle in April 1600. However, Tycho' s calculations of the planetary eccentricities and especially that of the Sun's circle were not sufficiently accurate for Kepler's purpose. Moreover, he realized that Tycho, like Ptolemy and Copernicus, had considered the mean motion of the Sun around the centre of its eccentric circle and not the true motion.3 Already in his Mysterium cosmographicum (1596) Kepler expressed his concerns about the disagreements between astronomical predictions resulted from the equations of the solid-sphere theories and astronomical observations. He remarked,For though the approach to the whole history of the heavenly motions is slippery, and requires lengthy and difficult observations, yet it is particularly apparent that in establishing the eccentricities and the positions of the apogee, the solar (or terrestrial) eccentricity should be the most accurately known of all.4Kepler believed that the main obstacle, which had prevented him from attaining knowledge of the exact planetary motions [exactam motuum cognitionem], was due to errors in the calculations of the eccentricities.5 He therefore decided to reconsider the value of the solar eccentricity.Kepler published his novel calculations of the eccentricity of the Sun in chapters 10 and 1 1 of his Astronomiae pars optica (1604). He made the following revealing statement which we quote at length:Ptolemy was deceived when he raised the planetary epicycles too high on the one side and pushed them down on the other, because the slowness of the one place, and the speed of the other, seemed to require as much. But the error was immediately obvious from the apparent magnitude, for the epicycles increased less at perigee than accords with such a close approach, for which reason another cause for slowing down was seized upon, which, as I have just said, Ptolemy ascribed to the circle of the equant. In the Sun, no epicycle was needed, and as a consequence this error has remained to this day. It was, however, first discovered by me, through an exact observation of the visible diameter [of the Sun], as I shall say below [in the Astronomiae pars optica], and then by Tycho's most precise observations taken of the star Mars, as I shall make plain at the proper time and place [appeared in 1609 as New astronomy ... treated by means of commentaries on the motions of the star Mars]. By both arguments it is established that the Sun recedes from us by only half of the eccentricity that was attributed to it by Albategnius and Tycho, and thus that an equant circle governs the motion in the Sun too.6Three issues are worthy of emphasis. In the first place, Kepler noted that in the Ptolemaic and, following it, in the Copernican, system the Sun's circle does not require an epicycle. Kepler explained the reason for this claim: he discovered that the apparent solar diameter required by the standard solar theory was greater than the apparent diameter of the Sun, measured at perigee, which he obtained in his observations of the Sun's disc through the year (to be discussed further by Kepler in the optical treatise of 1604, chaps. …