Planets and their Orbits
Kamal Lodaya, The Institute of Mathematical Sciences, Chennai

The great 16th century Danish astronomer Tyge Brahe (known as Tycho) had a huge amount of data on the observations which he had made, of positions of planets and stars in the sky over many years. He was suspicious of his assistant Johannes Kepler; he feared that Kepler would steal his data.
Tycho did not believe in Copernicus's model of the solar system, in which all planets went around the Sun. Kepler thought Copernicus's model was correct. Tycho had his own system, where all of them did go around the Sun, except that the Sun went around the Earth! He wanted to prove this by analyzing all his data. Today we know that Tycho's model is nothing but an Earth-based view of Copernicus's model, but Tycho did not know that.
To keep Kepler occupied, Tycho assigned to him the task of understanding the orbit of the planet Mars, which seemed particularly troublesome. Ironically this was what would make Kepler famous! Today school students read about Kepler's laws whereas Tycho is forgotten.
Mars gave Kepler many headaches (he suffered from severe migraines), but after Tycho's death in 1601 Kepler got to see all his earlier observations. Using them he was able to show by a very clever triangulation with the position of Mars in the night sky, that in January, the Earth is at {perihelion}, closest to the Sun, and in July at {aphelion}, farthest from the Sun.
(Here is a trick question: If the earth is nearest the Sun in January, why do we have winter? Shouldn't we be having summer?)
Orbits are elliptical
Even after making all the corrections for the position of the Earth, Kepler found that he could not account for the orbit of Mars. But then he realized that the orbit of Mars was not circular either, it was an ellipse with the Sun at one focus.
Kepler soon discovered his famous law: the Earth's orbit around the Sun, and of all the planets, are ellipses. Today we know that in January the Earth is 14.7 crore kilometres from the Sun, in July, 15.2 crore kilometres. So the orbit is almost circular. The difference can only be observed with precise observations like Tycho's. Mars is a little less circle-like, 20.7 crore kilometres from the Sun at perihelion, and 24.9 crore kilometres from the Sun at aphelion. This was why it gave so much trouble to early astronomers trying to calculate its orbit when they thought it was a circle.
Mercury is the most elliptical of the planets, it is 4.6 crore kilometres from the Sun at perihelion and 7 crore kilometres at aphelion. Whereas Neptune is almost circular: 446 crore km at perihelion and 454 crore km at aphelion, remaining around 25 times the mean distance of the Earth (astronomers say 25 astronomical units, or 25 AU). Pluto, now demoted to a dwarf planet, has a very elliptical orbit. Its perihelion is 444 crore km, so inside Neptune's orbit, whereas the aphelion at 738 crore km is far out.
The figure shows the variation of the distance from the Sun of different planets. The thicker the line, the more the variation and the more elliptical the orbit.
All these numbers we know today. In Kepler's time distances between the planets and the Sun were not known precisely. To verify Kepler's laws, astronomers such as Isaac Newton used comets, whose orbits are the flattest ellipses one can find. For example, today we know that Halley's comet is 90 crore km from the Sun at perihelion (inside Venus's orbit) and 630 crore km at aphelion (outside Neptune's)! The same figure also shows the great variation in the distance of Halley's comet from the Sun.