Sky & Telescope - USA (2019-08)

Kepler Anniversary

62 AUGUST 2019 • SKY & TELESCOPE

Cosmographicum. Several chapters were primarily mysticism,

replete with attempts to associate human affairs with the

planets, but the fi fth and fi nal 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 quantifi able physi-

cal 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:

P^2 = Ka^3

where P is the orbital period, K is the constant of proportional-

ity, and a is the length of the semi-major axis. The law depends

only on the relative, not specifi c, distances of solar system

bodies from each other. The constant is the same for all bodies

within a specifi c system, but differs from system to system.

While Kepler’s stunning intellectual achievement ranks

with the greatest of all time, his fi ndings were empirical and

limited to the observed motion of Mars as seen from Earth.

At about the time Kepler published his fi rst two laws, Galileo

Galilei (1564–1642) was on the verge of the fi rst astronomi-

cal breakthroughs using the newly invented telescope. Most

celebrated was his 1610 discovery of Jupiter’s four largest sat-

ellites, 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 discov-

ery of Jupiter’s satellites was signifi cant, but as with Kepler’s

situation, it was confi ned to a special case.

In 1621 Kepler published the third volume of Epitome

Astronomiae Copernicanae (Epitome of Copernican Astron-

omy), a lengthy exposition of his system. Although inter-

spersed 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 fl avor 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 verifi ed, as Kepler had strongly suspected,

that they apply to all the solar system planets and by impli-

cation 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.

Mathematical Representation

of Kepler’s Third Law

Planet Sidereal

Period

P (yr)

Semi-

major Axis

a (a.u.)

P^2 ∝ a^3

Mercury 0.24 .39 0.06 0.06

Venus 0.62 .72 0.38 0.37

Ear th 1.00 1.00 1.00 1.00

Mars 1.88 1.52 3.53 3.51

Jupiter 11.86 5.20 140.66 140.61

Saturn 29.45 9.54 867.30 868.2 5

Uranus 84.02 19.19 7,059.36 7,066.83

Neptune 164.79 30.07 27,155.74 27,189.44

tSEARCHING FOR A

CAUSE Johannes Kepler

was as much a mystic as

a mathematician. Several

chapters of his Myste-

rium Cosmographicum

(1596) are dedicated to

astrology and numerol-

ogy, and the effects of

other planets on Earth.

This painting, an 18th-

century copy of a now-

lost portrait from 1610,

shows Kepler holding

a drafter’s divider in his

right hand.

pTHE SOLAR SYSTEM SORTED Kepler demonstrated that all

planets move in elliptical orbits, with the Sun occupying one focus

(F^1 ) of the ellipse. (The other focus is unoccupied.) As a planet ap-

proaches 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.

a

b

c

F^1 F^2

Sun

Planet

KE

PL

ER

:^ P

UB

LIC

DO

MA

IN;

SO

LA

R^ S

YS

TE

M^

DIA

GR

AM

:^ G

RE

GG

DI

ND

ER

MA

N^ /

S&

T