Kepler’s derivation of the bisection of the earth’s orbit in Astronomia Nova

Abstract

Chapter 22 of Astronomia Nova is focused on the calculation of the Earth’s eccentricity. This is carried out by observing the effect of the Earth’s motion on the apparent position of Mars. Kepler’s method to derive the exact eccentricity, however, requires as data a set of longitudes of Mars while that planet and the Earth are in a very particular and restricted number of possible configurations. This paper explains how Kepler understood and tackled the Earth problem in theoretical terms, and also how he drew information from Tycho’s observational registers in a methodical way in order to obtain the necessary data to calculate the desired parameter, i.e. the eccentricity of the Earth’s orbit. In doing so, I will analyze not only Astronomia Nova’s relevant passages, but also Kepler’s preliminary annotations, as published in the Gesammelte Werke.

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Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 Kepler 2015, 226. As Kepler was already aware, though, the incorporation of a bisected equant to the Earth’s orbit would only imply an improvement of about 30′ in the predictions. This is much less than the 1;45° Tycho denounced in his letter. The rest is due to other problems in Tycho’s model, which are beyond the scope of this paper.

2 Given that the mean Sun is merely a theoretical longitude, it is strange to denote a point as ‘the centre of the mean Sun’s eccentric’. Nevertheless, this way of presenting the transformation is, I think, the clearest way of expressing what Kepler depicts in his diagram.

3 The difference would be greater if the planet were on the two points where the observer’s line of sight is tangent to the epicycle. To calculate those positions, though, involves some extra steps, and the improvement is minimal.

4 A rounded value. It should also be pointed out that this value corresponds to a model referenced to the mean Sun, and not the true Sun. This fits well with the rest of Kepler’s results.

5 There are small discrepancies between the Astronomia Nova and the preparatory study due to some rounding in one or the other. In this case, in the study he gives 42;30° and 222;30° for the true eccentric anomaly, because he had previously rounded the value of the solar apogee to 95°.

6 Kepler’s table has four columns: the first and third are the ones I copy here. To avoid an overcomplicated exposition, and because in the end the relevant dates were not taken from them, the second and fourth columns are ommited.

7 Times are calculated from noon.

8 Magini predicts 150;53° for 18 May at Venice’s noon, and 151;21° for 19 May (Magini 1582, 68). A linear interpolation gives 151;5,15° for that time, in Venice. Kepler is here just truncating, because the time difference between Venice and Uraniborg is less than 2 min, which is not enough for those 15" of difference in longitude.

9 Donahue’s translation reads April 18, as does the 1609 edition of the Astronomia Nova. However, the 33 days of difference noted afterwards, and Tycho’s observational log, show that it was the 15th (Brahe 1923, 399).

10 For 15 April at noon, Magini gives 137;52°, and 138;10° for the next day (Magini, 1582, 68). An interpolation gives 137;59,30° for the time Kepler is using.

11 For 30 May at noon, Magini gives 156;42°, and 157;12° for 31 May (Magini 1582, 68). This means 156;48,15° for 30 May, 5h.

12 Calculated from (Meeus 1998).

13 In his Progymnasmata, Tycho had given separate refraction tables for the sun (Brahe 1915, 64) and the fixed stars (Brahe 1915, 287). The first one gives, for 16°, a value of 7′, and the second one 0;2;30°. At 7a.m. Mars was almost culminating, therefore almost all the difference in altitude due to refraction translates to a difference in declination. So it seems that Kepler is using the table for the sun, or even applying the method of averaging the values from both tables, a practice Tycho had carried out for several years (Thoren 1990, 235).

14 He explicitly says that because the period elapsed between dates is short, he will not take into account the motion of the apogee.

15 In Astronomia Nova he gives $231;{34}^{\circ }-193;{28}^{\circ }=38;5,{30}^{\circ }$, which does not add up. Hence I corrected it to 231;33,30°.

16 He had previously rounded the value to 78°.

Additional information

Funding

This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas.