68
5 Nearly Half a Million Stars
series, the ‘higher order sines’. You can represent any mathematical function
(such as the sine of an angle or the logarithm of a number) as a sum of regular
terms that become smaller and smaller, so that you can calculate a reasonable
approximation with the first ones. This series expansion has purely mathematical significance. The Kapteyn brothers published two long articles in French
about it, with dozens of pages full of formulas. Properties of mathematical
series was a specialty of Willem Kapteyn, who became an authority in it. He
published a lot about it during his career, mostly in French. So William must
have done most of the work. For those interested in mathematics, I briefly
summarize what higher order sines and cosines are in Appendix A.3.
5.2 Kepler’s Equation
However, in his first years in Groningen, Kapteyn also did original and significant astronomical research. I start here with his work on Kepler’s equation,
which he published in 1883. For that publication he chose a fairly new magazine, Copernicus, which was published in Ireland. It did not exist for very
long; after less than three years it was terminated. It presented itself as an international journal and published papers in French and German in addition
to English. Kapteyn was used to German as being the language in which astronomical literature was published, at least on the European continent, where the
German Astronomische Gesellschaft set the tone; many publications went to
the Astronomische Nachrichten published by this Gesellschaft. Kapteyn’s paper,
and the next one on absolute declinations, both were in German.
Kepler described around 1600 how the orbits of all objects in the Solar System (except for the disturbances by the planets among themselves of course)
could be described. These are elliptical, with the Sun in one of the foci and the
planets move faster when closer to the Sun; there also is a relationship between
the size of the orbit (or mean distance from the Sun) and the period in it.
Even before the end of the seventeenth century Newton showed that Kepler’s
laws were a direct consequence of his theory for gravitation. If one knew the
characteristics of the orbit and at a certain moment the position of the planet,
asteroid or comet in it, it was possible using Kepler’s laws to calculate for any
given moment where that object would be in its orbit. However, there was a
serious problem. One had to solve a certain equation (named after Kepler),
which was very labor-intensive. Methods had been devised, but it remained
still very time-consuming. In Appendix A.4 I give a bit of an explanation for
readers who want to know a little more about this (see the online version of
the appendix for a bit more algebra). It was a serious problem because obser-
5 Nearly Half a Million Stars
series, the ‘higher order sines’. You can represent any mathematical function
(such as the sine of an angle or the logarithm of a number) as a sum of regular
terms that become smaller and smaller, so that you can calculate a reasonable
approximation with the first ones. This series expansion has purely mathematical significance. The Kapteyn brothers published two long articles in French
about it, with dozens of pages full of formulas. Properties of mathematical
series was a specialty of Willem Kapteyn, who became an authority in it. He
published a lot about it during his career, mostly in French. So William must
have done most of the work. For those interested in mathematics, I briefly
summarize what higher order sines and cosines are in Appendix A.3.
5.2 Kepler’s Equation
However, in his first years in Groningen, Kapteyn also did original and significant astronomical research. I start here with his work on Kepler’s equation,
which he published in 1883. For that publication he chose a fairly new magazine, Copernicus, which was published in Ireland. It did not exist for very
long; after less than three years it was terminated. It presented itself as an international journal and published papers in French and German in addition
to English. Kapteyn was used to German as being the language in which astronomical literature was published, at least on the European continent, where the
German Astronomische Gesellschaft set the tone; many publications went to
the Astronomische Nachrichten published by this Gesellschaft. Kapteyn’s paper,
and the next one on absolute declinations, both were in German.
Kepler described around 1600 how the orbits of all objects in the Solar System (except for the disturbances by the planets among themselves of course)
could be described. These are elliptical, with the Sun in one of the foci and the
planets move faster when closer to the Sun; there also is a relationship between
the size of the orbit (or mean distance from the Sun) and the period in it.
Even before the end of the seventeenth century Newton showed that Kepler’s
laws were a direct consequence of his theory for gravitation. If one knew the
characteristics of the orbit and at a certain moment the position of the planet,
asteroid or comet in it, it was possible using Kepler’s laws to calculate for any
given moment where that object would be in its orbit. However, there was a
serious problem. One had to solve a certain equation (named after Kepler),
which was very labor-intensive. Methods had been devised, but it remained
still very time-consuming. In Appendix A.4 I give a bit of an explanation for
readers who want to know a little more about this (see the online version of
the appendix for a bit more algebra). It was a serious problem because obser-