The Astronomy Book

54

set. Kepler was lucky to have chosen
Mars, since its orbit has a high
eccentricity, or stretch: 0.093 (where
0 is a circle and 1 is a parabola).
This is 14 times the eccentricity
of Venus. It took him another
12 years to show that the other
planets also had elliptical orbits.
Studying Brahe’s observations,
Kepler was also able to work out the
planets’ orbital periods. Earth goes
around the sun in one year, Mars
in 1.88 Earth years, Jupiter in 11.86,
and Saturn in 29.45. Kepler realized
that the square of the orbital period
was proportional to the cube of
the planet’s average distance from
the sun. This became his third
law and he published it in 1619
in his book Harmonices Mundi,
alongside lengthy tracts on
astrology, planetary music, and
platonic figures. The book had
taken him 20 years to produce.

Searching for meaning
Kepler was fascinated by patterns
he found in the orbits of the planets.
He noted that, once you accepted
the Copernican system for the
cosmos, the size of the orbits
of the six planets—Mercury,
Venus, Earth, Mars, Jupiter,
and Saturn—appeared in the
ratios 8 : 15 : 20 : 30 : 115 : 195.

Today, astronomers might look at

a list of planetary orbital sizes and

eccentricities and regard them as

the result of the planetary formation

process coupled with a few billion

years of change. To Kepler, however,

the numbers needed explanation.

A deeply religious man, Kepler

ELLIPTICAL ORBITS

searched for a divine purpose

within his scientific work. Since

he saw six planets, he presumed

that the number six must have a

profound significance. He produced

an ordered geometric model of the

solar system in which the sun-

centered spheres that contained

each planetary orbit were inscribed

and circumscribed by a specific

regular “platonic” solid (the five

possible solids whose faces and

internal angles are all equal). The

sphere containing the orbit of

Mercury was placed inside an

octahedron. The sphere that just

touched the points of this regular

solid contained the orbit of Venus.

This in its turn was placed inside

an icosahedron. Then followed the

orbit of Earth, a dodecahedron,

Mars, a tetrahedron, Jupiter, a cube,

and finally Saturn. The system was

beautifully ordered, but inaccurate.

According to Kepler’s second law, the line joining a planet to the sun

sweeps out equal areas in equal times. This is also known as the law of equal

areas. It is represented by the equal areas of the three shaded areas ABS,

CDS, and EFS. It takes as long to travel from A to B as from C to D and from E

to F. A planet moves most rapidly when it is nearest the sun, at perihelion; a

planet’s slowest motion occurs when it is farthest from the sun, at aphelion.

When just one body
goes around a larger body
undisturbed, the paths it can
follow are known as Kepler
orbits. These are a group of
curves called conic sections,
which include ellipses,
parabolas, and hyperbolas.
The shape of the orbit is
defined by a property called
eccentricity. An eccentricity
of 0 is a circle (A). Eccentricity
between 0 and 1 is an ellipse
(B). Eccentricity equal to 1
produces a parabola (C), and
greater than 1 a hyperbola (D).

A Focus 1 (the sun)

B

C

D

E

F

Focus 2

(empty point

in space)

Large

body

Planet near

perihelion

Planet near

aphelion

D

B

A

C

Elliptical orbit