MEETING VENUS C. Sterken, P. P. Aspaas (Eds.) The Journal of Astronomical Data 19, 1, 2013 From Keplerian Orbits to Precise Planetary Predictions: the Transits of the 1630s Steinar Thorvaldsen Department of Education, University of Tromsø, 9037 Tromsø, Norway Abstract. The first transits of Mercury and Venus ever observed were important for quite different reasons than were the transit of Venus observed in the eighteenth century. Good data of planetary orbits are necessary for the prediction of plane- tary transits. Under the assumption of the central position of the Sun, Johannes Kepler published the theory of elliptical orbital motion of the planets in 1609; this new astronomy made it possible to compute noticeably improved ephemerides for the planets. In 1627 Kepler published the Tabulae Rudolphinae, and thanks to these tables he was able to publish a pamphlet announcing the rare phenomenon of Mercury and Venus transiting the Sun. Although the 1631 transit of Mercury was only observed by three astronomers in France and in Switzerland, and the 1639 transit of Venus was only predicted and observed by two self-taught astronomers in the English countryside, their observation would hardly been possible without the revolutionary theories and calculations of Kepler. The Tabulae Rudolphinae count among Kepler’s outstanding astronomical works, and during the seventeenth cen- tury they gradually found entrance into the astronomical praxis of calculation among mathematical astronomers and calendar makers who rated them more and more as the most trustworthy astronomical foundation. 1. Introduction The worldwide fame of Johannes Kepler (1571–1630) is based above all on his con- tribution to celestial mechanics and astronomy. Aided by the accurate observations of Tycho Brahe, he published a mature system of the world based on elliptical astro- nomy, celestial physics and mathematical harmonies: Epitome astronomiae coper- nicanae (Kepler 1618–21/1995). His groundbreaking and innovative work reached beyond the three laws on which his fame mainly rests today. Kepler also practiced what he preached and made astronomical predictions by his laws. Starting in 1617, he published planetary ephemerides that showed a marked increase in accuracy as compared to other tables that were in use at the time. After Kepler had published the Tabulae Rudolphinae in 1627 (Kepler 1627/1969), more and more ephemerides were calculated on the basis of this work in the following decades. Because of the high reliability of Kepler’s tables, the Copernican doctrine gained more esteem (Gingerich 1968/1993). Although this was first restricted to the mathematical foundations of the system, as well as to Kepler’s laws, it was more and more accepted that the Copernican doctrine was not just a new and useful mathematical hypothesis, but a reflection of reality. Kepler, like many of the leading natural philosophers before him, had some explicit metaphysical guiding principles for his work. He firmly believed that God had created a harmonious well-ordered universe, and he saw his work as a God-given mission to understand its nature, principles and design (Kozhamthadam 1994; Methuen 1998; Martens 2000). The driving goal for Kepler was to carefully read the “book of Nature” and discover the orderly structure God had created in the universe. This 97 98 Thorvaldsen order of nature was supposed to be expressed in mathematical language and its quantitative calculations. Tycho Brahe had estimated the distance between the Sun and the Earth at 8 mil- lion kilometer. In Epitome Kepler estimated the solar distance to be 3469 Earth radii, actually the largest of his several estimates. This meant that the distance to the Sun was 24 million kilometers, and the solar parallax 1 arcminute. The solar parallax is the angle subtended by the equatorial radius of the Earth at its mean distance of the Sun. Kepler based his estimate on Tycho’s attempted measurements of the parallax of Mars and on his own calculations of eclipses, which had convinced him that a solar parallax of more than 10 gave unsatisfactory results (van Helden 1985, 1989). We know now that this solar distance was roughly seven times too small,1 and Kepler’s understanding of absolute distances within the solar system re- mained considerably faulty although he mainly worked with relative distances based on the Earth–Sun unit, the so called astronomical unit. On the basis of Tycho Brahe’s observations, Kepler early in his career had calcu- lated that Mercury would pass in front of the Sun at the end of May, 1607. He used a pinhole camera (camera obscura) to observe the event, and on May 28 a black spot was visible on the Sun’s image between clouds. He concluded that he had seen a transit of Mercury, and published his observation in the report Phaenomenon singulare seu Mercurius in Sole (Kepler 1609/1941). Later on, when the sunspots had been discovered by the newly invented telescope, he realized that he had seen a sunspot, and he admitted his error in print. 2. Mathematical solution Kepler was operating in a long-standing tradition of numerical calculation in astron- omy. His new planetary theory was much more complicated than anything that had been done before. In the earlier theories, the orbit of the Earth and the planets were represented by two large circles, the deferent and the epicycle. Kepler’s first law states that the planetary orbits are elliptical, and to compute areas inside the ellipse is not easy. For the astronomers time was represented by an angle, and in the Keplerian model there was no simple or exact way to find explicitly the position angle corresponding to a given time, and consequently there was no direct way to calculate the planetary position exactly from its date, as it had always been possible before Kepler. However, Kepler made his non-circular astronomy operational and computable by applying a set of innovative numerical methods (Thorvaldsen 2010). In fact it was the use of an auxiliary circle that aided Kepler at this point. One of the fundamental properties of the ellipse is the ratio-property, that for any Q and P on a line perpendicular to CD, see Fig. 1: QR/PR =constant (1) Kepler’s law states that time is proportional to the ellipse area ADP . This area is composed of the triangle ARP and the segment RDP . By Eq. 1 the triangle is proportional to the triangle ARQ and the segment is proportional to the segment RDQ. Consequently, time is proportional to the circle area ADQ which is β + e sin β (2) 1The currently accepted value of the solar parallax is 800. 794. The mathematical procedure behind computing the parallax based on transits of the Sun can be found in Teets (2003). From Keplerian Orbits to Precise Planetary Predictions 99 Figure 1. A is the position of the Sun and B is the center of the auxiliary circle. The planet is at P , and Q is its projection on the circle. Kepler’s problem is to determine the angle β (the eccentric anomaly). Here e is the constant AB and β the angle QBD called the eccentric anomaly. Analytically, Kepler expressed the problem by the equation M = β + e sin β where M is the quantity used to designate time (the Mean anomaly). Kepler measured M as a fraction of the area of the full circle, i.e. an area of 360◦ (216000, or 129600000). The Mean anomaly is proportional to the time elapsed, and itisasa function of time that we wish to find the position of the planet P . So the important problem is to find β when M and e are given. This transcendental equation is called Kepler’s equation and turned out to be a famous problem as it is fundamental in celestial dynamics (Colwell 1993). Kepler could only solve it by approximations (Swerdlow 2000; Thorvaldsen 2010). The problem was stated at the very end of Astronomia nova (Kepler 1609/1992, 600). In Epitome, book 5, part II, chapter 4 (Kepler 1618-21/1995, pp. 158–59) he gives a numerical example with three iterations as shown in Table 1. The eccentricity e is determined from the area of the triangle ABF , which he says he earlier had calculated to be 1191000. Table 1. Kepler’s numerical solution of his equation in Epitome, book 5, part II, chapter 4, compared to a simple computer iteration based on βn = M −e sin βn−1 in Microsoft Excel. Eccentricity e = 0.05774, and the Mean anomaly M = 50◦0901000 (= 0.87533 radians). The convergence is quite rapid in this case. Kepler in Epitome Calculated degrees radians radians iteration# 44◦ 250 0.77522 0.77522 β ◦ 0 46 44 0.81565 0.83492 β1 ◦ 0 00 47 44 6 0.83313 0.83253 β2 ◦ 0 00 47 42 17 0.83260 0.83262 β3 100 Thorvaldsen Kepler’s method can be described as a simple numerical “fixed point method”, that is, an iterative algorithm that generates a convergent sequence whose limit is the solution of the proposed problem: βn = M − e sin βn−1. He called his method to solve equations the regula positionum (rule of suppo- sition), probably an allusion to the classical method of regula falsi (rule of false position). But he made no closer examination of the speed of convergence or errors involved. Later, in his Tabulae Rudolphinae, Kepler solved the equation for a grid of uniformly spaced angles, that determined a set of non-uniformly spaced times. By an interpolation scheme the desired values of time could be determined, and hence the tedious iterations were avoided for the users of his tables. During the years 1614–1630 John Napier’s (1550–1617) discovery of logarithms rapidly spread within mathematics, and was exploited by Kepler to speed up the as- tronomical calculations and ease the pain of the process.