B00B7H7M2E EBOK (12 page)

(b) Kepler’s arrangement of them in relation to the spheres of the planets. Saturn’s sphere is outside the cube. Jupiter’s sphere is between the cube and the tetrahedron. Mars’s sphere is between the tetrahedron and the dodecahedron. Earth’s sphere is between the dodecahedron and the icosahedron. Venus’s sphere is between the icosahedron and the octahedron. Mercury’s sphere is within the octahedron.

However, Kepler was not completely out in the cold, for the opportunity had arisen to join Tycho Brahe at Prague. Kepler had sent the older man a copy of
Mysterium
, and Tycho had recognized a serious talent. Kepler was understandably apprehensive about how well the two of them would get along, for Tycho had a reputation as a proud, imperious eccentric. His nose had been partly sliced off in a duel, and he had restored the missing bit himself with gold, silver and wax. But Tycho was also unarguably the greatest astronomer of his generation . . . and Kepler needed a job. So Kepler, now 30 years old, moved to Prague in 1601. Whatever the difficulties working with Tycho, he didn’t have to cope with them for long, for within two years Tycho died. Kepler rose to the title of as Imperial Mathematician.

This new position was a considerable improvement over teaching at Graz, but there was a negative side. Although he had an impressive title, Kepler’s salary was much lower than Tycho’s had been and often it wasn’t paid. Kepler was obliged to waste a great deal of time trying to collect what was due him. The job as Imperial Mathematician did, however, have other compensations, for, over the objections of Tycho’s relatives and fortunately for the future of astronomy, it was Kepler who fell heir to Tycho’s magnificent set of astronomical observations, the best the world had ever known. Tycho had found that circular orbits were difficult to reconcile with the actual paths of the planets, and he had undertaken an exhaustive series of observations that he hoped would throw more light on the problem. No man was better suited than Kepler to put this precious inheritance to optimum use.

Kepler brought to this work both a firm belief in Copernican astronomy and an unwillingness to accept that the skilled and meticulous Tycho’s data could be faulty. The two must somehow fit, even though Tycho himself had rejected Copernican
astronomy
in favour of the scheme that had the Sun orbiting the Earth and all the other planets orbiting the Sun. Kepler also brought to his task a concept of his own that the Sun moves the planets by a ‘whirling force’ centred in itself. Orbits centred elsewhere than on the Sun would not do.

Tycho Brahe had paid particular attention to the planet Mars, and it was in trying to make sense of these observations that Kepler finally found himself obliged to abandon circular orbits, removing the stumbling block that had hobbled Copernican astronomy since its inception. He realized that by using elliptical orbits he could explain Tycho’s observations. Kepler signalled his overwhelming joy and astonishment at his insight by falling to his knees and exclaiming, ‘My God, I am thinking Thy thoughts after Thee.’ Those who describe Kepler as dry and passionless and his life as uniformly drab and sad simply fail to appreciate the sorts of things that moved him.

Kepler’s discovery and Tycho’s earlier observations (without a telescope) that made it possible were achievements that still inspire awe in modern astronomers. The ellipse in which the Earth orbits is so nearly circular that any attempt to make it obviously an ellipse in a scale drawing ends up being a distortion. It is only slightly more obvious that the orbit of Mars is an ellipse. Yet with this seemingly trivial geometric alteration, the Copernican system fell into place. Kepler, aware of the power of his discovery and also of the reaction it would inevitably receive, commented wryly that with this introduction of elliptical orbits, he had ‘laid an enormous egg’. He had indeed, and even many who admired his work found this egg difficult to digest. Galileo, for one, never accepted it.

Saying an ellipse is an ‘egg shape’ or an ‘oval’ isn’t precise enough. It’s best to think of an ellipse in one of two ways. One is as a slice out of a cone. Not just any oval or egg shape can be produced by slicing a cone, and so not all ovals and egg shapes are ellipses. A circle, however,
is
an ellipse. It’s a slice directly across the cone.
See Figure 3.2.

Figure 3.2

A second way of thinking about an ellipse is a little more difficult to describe, but it gets the same result: imagine a piece of thick cardboard lying on the table before you. Stick two drawing pins into it, a distance apart. Then take a piece of string that is longer than the distance between the two pins and attach its ends to the pins – something like the string shown in
See Figure 3.3.

Figure 3.3

Next take a pencil and pull the string taut with its point (
Figure 3.4
), still keeping the string flat on the surface of the cardboard.

Figure 3.4

If you move the pencil, allowing it to slide along the string while the string remains taut, the point of the pencil will draw an ellipse. Changing the length of the string or moving the drawing pins closer together or further apart changes the shape of the ellipse, just as changing the tilt of the slices in the cone does. The closer together the two drawing pins are, the nearer the ellipse comes to being a circle. If they are in precisely the same location, it
is
a circle.

Each drawing pin represents a
focus
of the ellipse, so an ellipse has two foci. If a planet’s orbit is near to circular, that means that the ellipse’s foci must be very close together. By contrast, comets, which also orbit the Sun in elliptical orbits, have foci that are very far apart, producing an extremely elongated ellipse.

The first of Kepler’s two laws of planetary motion that appeared in his book titled
Astronomia Nova (New Astronomy)
states that a planet moves in an elliptical orbit, and the Sun is located at one of the two foci of that ellipse (
see Figure 3.5).

Kepler’s second law has to do with the variation in the speed of a planet as it travels in its orbit. The law states that an imaginary straight line (called the ‘radius vector’) joining the centre of the planet to the centre of the Sun ‘sweeps out’ equal areas in equal intervals of time.

Picture an elliptical orbit, the Sun (one of its foci), and a planet travelling along in the orbit (
see Figure 3.6
). Picture the
planet
at
A
and draw an imaginary straight line from
A
to the centre of the Sun. Set a stopwatch and time the planet as it orbits to
B
(which for purposes of this demonstration could be located anywhere on the part of the orbit nearer the Sun). Think of the imaginary line moving with it like the hand of a clock, shading the area it sweeps across. At
B
, stop the shading and the timing. You have now created a shaded area that we say is ‘swept out’ as the planet travels from
A
to
B
– looking like the left side of
Figure 3.6.
You know how long it took the planet to travel that distance and sweep out that area. Wait until the planet has travelled somewhat further. Decide on a point
C
somewhere on the side of the ellipse further from the Sun.
When
the planet reaches that point, again draw a line from the planet to the centre of the Sun and start the shading and the stopwatch. The planet travels along and so does the sweeping line. When the planet has travelled for
the same amount of time it took to go from
A
to
B
, stop the shading and call that final point
D
.

Figure 3.5

Figure 3.6

The resulting drawing resembles
Figure 3.6.
What does it mean? Notice that on the left side of the picture, from
A
to
B
, the planet travelled a much greater distance, in the same interval of time, than it did from
C
to
D
over on the right side. Clearly it couldn’t have maintained a constant speed. It had to travel much faster from
A
to
B
, to cover all that distance, than it did to cover the smaller distance from
C
to
D
in the same interval of time. The two shaded ‘swept out’ portions, however, are equal in
area
. Kepler’s law thus predicts that a planet will always travel at a faster rate the nearer its path comes to the Sun. Kepler thought the variation in speed occurred because the distance from the Sun affected how much or how little the planet felt the Sun’s ‘whirling force’.