Kepler's celestial mechanics

Vistas in Astronomy, 1979, Vol. 23, pp. 69-74. Pergamon Press Ltd.

Printed in Great Britain.

KEPLER'S CELESTIAL MECHANICS UlrichHoyer University of Milnster, Germany.

The complete success of Newton's Pr/ncip~

seams in general to have overshadowed the fact

that Johannes Kepler had already conceived a celestial mechanics.

Far from being a vague

project, kept in the background of his astronomical endeavours, his dynamical theory of planetary motions played a central part in his investigations.

Moreover, Kepler's celestial

mechanics is analogous to Newton's version and is therefore most interesting from the historical as well as from the physical or philosophical point of view.

Kepler's planetary theory assumed

its most mature form in 1620 as part of the fourth book of his

Epitome Astronomiae Copernicanae. work Mysteri~n Cosmo-

However, mechanical considerations can already be found in his early

graphicum, dating from 1596. The search for a mechanical foundation of planetary motions can justly be said to he the first of the two main motives of his lifework, the other being his harmonical investigations, for the decisive argument in favour of the heliocentric system and the reason why he so fervently defended Copernicus against Tycho was

dynamical in nature.

Kepler was convinced from an

early date that the sun is the true point of reference of planetary motions, because he felt sure that the largest body of theplanetary system must also be the source of all the forces which keep the celestial machine in motion.

Apart from the discovery of the kinematic laws

that govern this system, the dynamical interpretation of planetary motions may thus be called Kepler's decisive advance upon Copernican astronomy.

It is curious that this remarkable phase of science has left but little trace in current histories of physics.

Ernst Math, for example, passes over Kepler's celestial mechanics in

almost complete silence.

Dijksterhuis is surely more particular in examining Kepler's dynamics.

However, in his co,anent upon the

Epitome he confines himself to the lapidary statement that

Kepler "endeavoured to find a physical basis for the third law, but.., did not succeed ''I. On the other hand, Max Jammer calls the

Epitome "perhaps the first true treatise on celestial

mechanics ''2 and Max Caspar, who probably gave the most profound appreciation of Kepler's celestial mechanics, writes in his wonderful Kepler biography of 19483: "It is Kepler's merit to have replaced the formal scheme of former astronomers by a dynamical system, mathematical rule by law of nature, and geometrical description of motion by causal explanation."

After this short introductory survey let us now turn to the details of Kepler's celestial mechanics and see whether they corroborate the exposition given above.

The earliest testimony to Kepler's dynamical speculations seams to be his letter to Michael M~stlin of TUbingen, dating from 3 October 1595. from the sun (the Latin word used is "vigor")

Kepler there explains why the force emanating

cannot be constant. 69

In this case, he argues, one

70

U. Hoyer

would expect that

r/T = const,

(i)

r being the distance of the planet from the sun and T its orbital period.

Obviously,Kepler

based his considerations on Aristotelian dynamics, where

(2)

F ~ v,

F; motive force and V; velocity. tion (2).

If F=const, then, of course, eqn. (i) follows from propor-

However, Kepler knew that the periodic times increase at a higher rate than the

distances from the sun.

Therefore he made the assumption that the motive force might decrease

with the distance from the sun, analogous to the intensity of light, I, which decreases with the distance from the radiating centre. of the distance.

~

The latter intensity he assumed to vary as the inverse

So

(3)

nI t ,

and correspondingly

F ~

(4)

lit,

whence, by way of (2), he obtained

v ~

l/r,

(5)

or

r2/T

~ const.

(6)

From a letter to Raimarus Ursus, written on November 15th,

1595, we know that even at this time

Kepler suspected that eqn. (6) might not give a true account of Copernicus' harmonic law, and in his

M~dsterium cosmographicum he chose the relation

+TI

2

TI

(7)

r2

rl ,

holding for adjacent planets.

By the way, eqn. (17) is the first approximation of eqn. (6).

This equation was more in accordance with observation than the original version.

In Chapter 22 of the M ~ s t ~ u m ,

Kepl~r deals with Ptolemy's equant; i.e. he discusses the

question why the planet moves slowly when far from the sun and swiftly when it approaches it. According to Kepler the reason is again that the motive force decreases with the distance from the sun.

So everything seemed to be well matched, at least qualitatively.

Towards the end of the year 1601 Kepler found out that the velocity in the apsides of the planet's path is given by the proportion V ~ I/r,

(8)

Kepler's Celestial Mechanics

71

with small deviations occuring at points far away from the apsides.

Thus Kepler had even

achieved quantitative agreement; for his new result, derived by induction from the single planet's motion, fitted in surprsingly well with his former assumption, (5), deduced from the behaviour of different planets. of areas.

Equation (8) is Kepler's radial law, the root of his rule

As Aiton pointed out 4, the radial law was formulated correctly by Kepler in the

Epitome, whence it is obvious that Kepler fully realized that (8) does not hold for the velocity of the planet in its orbit but for its component perpendicular to the radius; i.e.

Vm

:

r(d#/dt)

:

clr

(9)

.

Since Kepler still adhered to Aristotelian mechanics, he found that according to eqn. (2) the force in the azimuthal direction must yield

F~ ~

(i0)

llr.

Kepler explained this force in the following way: the rays of force emanating from the sun rotate with the sun and whirl the planets about. the AstronomY2 Nova.

This idea he first expounded in detail in

There the rays of force, like the spokes of a wheel, move the planets

along on a circular path.

Moreover, Kepler postulated constancy of the flux of force in the

plane of rotation

E# 2~r

=

const,

(ii)

and thus was able to give an explanation of the inverse distance law.

However, in the Epitome there can be found remarkable passages, from which it is obvious that Kepler suspected the true law might be

F ~

(12)

1/r 2

For in the Paz~lipomena o~ Vitellionem of 1604 Kepler had discovered that the law of intensity of light is not in accordance with eqn. (3), but rather is given by

I ~

l/r 2

(13)

.

This meant that his former analogy of light and force was either untenable or pointed in a different direction.

Thus he asked in the Epitome5: "If light decreases as the square of

the distance, i.e. in proportion to the surfaces, why then doesn't the motive force decrease as the square of the distance, instead of as the distance itself?"

Already in his Mysterium Cosmog~phicumKepler

was aware of the fact that there must be a

second principle besides the one responsible for circular motion, namely a force generating the motion in the radial direction.

In the Aetronomia Nova, Kepler, applying Gilbert's

researches on magnetism, proceeded from the assumption that the planets are quasi-magnetic dipoles moving in a quasi-magnetic field originating from the sun - the surface of the sun being a quasi-magnetic North pole, the centre a South pole.

Under the combined effects of

the rotational and radial forces from the sun the orbit of the planet becomes an ellipse. Incidentally, Kepler found that the radial force varies according to F

~ sin ~ ,

(14)

72

U. Hoyer

which is in line with the exact result, derived from the ellipse and area law

F

¢ a (1 - ~2)

sin ¢,

(15)

where a is the major axis and ~ the numerical eccentricity of the ellipse.

On 15 May 1618, Kepler discovered the third law

(16)

T2/a 3 = const

It was published in Haz~onices Mundi Libri V in 1619.

The H~r~oniee Mundi was not the

appropriate place for a discussion of eqn. (16) from a dynamical point of view.

Kepler made

up for this in the fourth book of the Epitome, where he started from Aristotelian dynamics for finite bodies according to the assumption

fV=mv, where f can be interpreted as an average of the force per volume unit, V is the planet's volume, and m the planet's mass, which Kepler calls pondus 8eu c o p ~ m a t e r ~ e .

Kepler's

argument was then as follows: From the single planet's behaviour in its orbit, i.e. from the radial or area law, we know that

f@l

r2

_

(18)

f@2

rl

i.e. the azimuthal forces acting upon the planet at different points of its path are inversely proportional to the distances from the sun. 8a~

Kepler further assumed that this law is univeP-

valid, i.e. that it holds for different planets as well.

Confining himself to circular

motions he derived from eqns. (17) and (18)

m=Z~ g

.7 . V

(19)

2ikz 2 .

By insertion of the third law (16) he obtained from (19)

m/V

~

(20)

a-1/2

i.e. the densities of the planets are inversely proportional to the square roots of the distances from the sun.

On the assumption that

V - a

(21)

he finally obtained from (20)

m ~

a 1]2

(22)

Kepler's Celestial Mechanics

73

i.e. the planetary masses are proportional to the square roots of the solar distances.

Thus Kepler believed he had accon~odated his three laws of planetary motion to Aristotelian mechanics.

Of course, the question to raised here is: Did Kepler really succeed in achieving

his aim?

A critical examination of his arguments yields the following result: Kepler's assumption that the solar forces universally extend through space from the sun to any given planet means that we would have to displace a reference body with mass and volume I from orbit to orbit.

There-

fore we must study the action of the force per unit volume on a body of density I; or in other words: we have to divide the force per unit volume by the planet's density.

Then we obtain

eqn. (17)

m•V ffi

2~a

V = -~

(23)

Now, if

f / O ~ I/a

(~: density)

,

(24)

as is evident from the area law, then eqns. (23) and (24) yield the law (6) of 1595.

But

this is in contradiction to Kepler's third l~! Hence it follows that Kepler's celestial mechanics is not self-consistent, as was pointed out by Hans-JUrgen Treder in 19737 and independently by myself in 19768.

To sum up, therefore: In the Epitome Kepler came close to realizing that, because of the inconsistency in question, Aristotelian mechanics does not yield a sufficient basis for a physical explanation of planetary motions.

Had Kepler seen this discrepancy, he might have

raised objections to Aristotelian mechanics very similar to those put forward a dozen years later by Galieo, although from very different reasons.

I think there can be no doubt that

Kepler in fact did not realize the deficiencies of Aristotelian mechanics, contrary to what Treder seems to suppose 9.

However, I hope that the foregoing considerations make it evident

that in spite of the inconsistency in question Kepler had good reason to write to Peter CrHger in 162410:"M~stlin usually laughed at my endeavours to reduce everything, even in the case of the moon, to physical causes. of my work that I succeeded."

But indeed, this is my joy, my consolation, and the pride

74

U. Hoyer

REFERENCES i.

E. J. Dijksterhu~s, The Mechanization of the World Picture. London, Oxford, New York,

2.

Max Jammer, Co~ept8 of Force. Cambridge, 1957, p. 87.

3.

Max Caspar, Jo~nne8 Kepler. Stuttgart, 1948, p. 156.

4.

Eric J. Aiton, Infinitesimals and the Area Law.

1969, p. 323.

der S~dt 1971. p. 285-305.

Inte~tio~les Kepler-S~oei~ Well

Edited by Fritz Krafft, Karl Meyer and Bernhard Sticker,

Hildesheim, 1973. 5.

Johannes Kepler, Epitome Astronomiae Copernicanae, Ges~elte Werke Bd VII. p. 305. Ed Max Caspar, MUnchen 1953.

6.

See ref. 5., p. 306.

7.

Hans-JUrgen Treder, Kepler und die BegrHndung der Dynamik.

Die Sterne, 49. Jg, 1973,

p. 44-48. 8.

Ulrich Hoyer, Uber die Unvereinbarkeit der drei Keplerschen Gesetze mlt der Arlstotelischen Mechanik.

9.

Centre.

2__OO,1976, p. 196-209.

Hans-JHrgen Treder, Die Dynamik der Kreisbewegungen der Himmelsk~rper und des freien Falls bei Aristoteles, Copernicus, Kepler und Descartes.

Colloquia C o p e ~ i c ~

2-V,

Ossolineum 1975, p. 105-150; especially p. 107. i0.

Johannes Kepler, Geeo3r~elte Werke Bd XVIII, Briefe 1620-1630. p. 165.

Ed. Max Caspar,

MUnchen 1959.

Dedicated with admiration to Dr Martha List of the Kepler-Konlnission, Munich.