Sky & Telescope - August 2019 - 62

Kepler Anniversary

Planet

Cosmographicum. Several chapters were primarily mysticism,
replete with attempts to associate human affairs with the
planets, but the ﬁfth and ﬁnal chapter centered on Kepler's
Third Law, which states that the square of a planet's period
of revolution around the Sun, its sidereal year, is proportional to
the cube of its semimajor axis. (The semimajor axis is half the
length of the major axis.)
Here Kepler linked the distances of the planets with the
Sun's force upon them, again invoking a quantiﬁable physical cause for the observed behavior. The relationship can be
expressed with the following simple equation, which includes
a constant of proportionality valid for all solar system planets:
P2 = Ka3
where P is the orbital period, K is the constant of proportionality, and a is the length of the semi-major axis. The law depends
only on the relative, not speciﬁc, distances of solar system
bodies from each other. The constant is the same for all bodies

Mathematical Representation
of Kepler's Third Law
Planet

∝

Sidereal
Period
P (yr)

Semimajor Axis
a (a.u.)

Mercury

0.24

.39

0.06

Venus

0.62

.72

0.38

0.37

Earth

1.00

Mars

1.88

1.52

3.53

3.51

Jupiter

11.86

5.20

140.66

140.61

Saturn

29.45

9.54

867.30

868.25

Uranus

84.02

19.19

7,059.36

7,066.83

Neptune

164.79

30.07

27,155.74

27,189.44

AUGU S T 2 019 * SK Y & TELESCOPE

Sun
F1

p THE SOLAR SYSTEM SORTED Kepler demonstrated that all
planets move in elliptical orbits, with the Sun occupying one focus
(F1) of the ellipse. (The other focus is unoccupied.) As a planet approaches the Sun, it moves more quickly through space, traveling
through a longer section of the orbit than it would in the same amount
of time when more distant from the Sun. A line drawn from the Sun to
a planet sweeps out equal areas (a, b, and c) in equal times.

within a speciﬁc system, but differs from system to system.
While Kepler's stunning intellectual achievement ranks
with the greatest of all time, his ﬁndings were empirical and
limited to the observed motion of Mars as seen from Earth.
At about the time Kepler published his ﬁrst two laws, Galileo
Galilei (1564-1642) was on the verge of the ﬁrst astronomical breakthroughs using the newly invented telescope. Most
celebrated was his 1610 discovery of Jupiter's four largest satellites, and, with the planet itself serving as a surrogate Sun,
Galileo envisioned a miniature solar system analogue, which
we now know dutifully obeys Kepler's laws. Galileo's discovery of Jupiter's satellites was signiﬁcant, but as with Kepler's
situation, it was conﬁned to a special case.
In 1621 Kepler published the third volume of Epitome
Astronomiae Copernicanae (Epitome of Copernican Astronomy), a lengthy exposition of his system. Although interspersed with much of the occult, it recapitulates his work and
extends the three laws to the known planets, our Moon, and
Jupiter's moons, but doesn't range beyond the solar system,
remaining silent with regard to the realm of the stars.
In the 1660s Isaac Newton (1642-1727), armed with
calculus, of which he was a co-inventor, developed laws
of motion and gravitation, which encompassed the entire
universe, more or less subsuming Kepler and Galileo. Kepler's
laws retained a ﬂavor of mutual independence, but with
the new mathematics, Newton was able to derive enhanced
versions of and unite the three. In his 1687 book, Principia
Mathematica, he veriﬁed, as Kepler had strongly suspected,
that they apply to all the solar system planets and by implication to everything everywhere. Newton modestly told the
world that he stood on the shoulders of giants, and Johannes
Kepler was certainly one of the giants.

¢ MIKE WITKOSKI has been an avid sky observer for decades
and has contributed articles to several publications over the
years. He is an advocate for dark skies and volunteers at
public events under the stars at Muddy Run Observatory in
southeastern Pennsylvania.

K EPLER: PUBLIC DO M AIN; SOL A R SYSTEM DIAG R A M: G REGG DINDER M A N / S&T

t SEARCHING FOR A
CAUSE Johannes Kepler
was as much a mystic as
a mathematician. Several
chapters of his Mysterium Cosmographicum
(1596) are dedicated to
astrology and numerology, and the effects of
other planets on Earth.
This painting, an 18thcentury copy of a nowlost portrait from 1610,
shows Kepler holding
a drafter's divider in his
right hand.

Sky & Telescope - August 2019

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