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Kepler and the resonant structure of the solar system
ICARUS
11, 111--113 (1969)
Kepler and the Resonant Structure of the Solar System OWEN
GINGERICH
Smithsonian Astrophysical Observatory and Harvard University, Cambridge, Massachusetts Received May 8, 1969 Molchanov's ordering the solar system by a table of resonance relations recalls an earlier attempt by Johannes Kepler (1571-1630). A. M. Molchanov (1968) is by no means the first to seek for the "resonant structure of the solar system" ; his research hears an interesting parallel to a thread t h a t runs through the entire work of Johannes Kepler. Both authors have addressed themselves to a deeply fundamental question : W h y are the planets arranged in their present pattern? Even in the 17th century this was an unorthodox query, and Kepler's vision of the "intelligible b u t inaudible" harmonies sung by the planets has generally been dismissed as sheer mysticism or numerology. Nevertheless, when a serious consideration of this question is reopened on the pages of Icarus, it is worthwhile to review briefly the verve and imagination t h a t characterized Kepler's inquiry. As a young teacher in Graz (Austria), fresh from the seminary in Ttibingen, Kepler posed for himself three unusual questions: W h y are there only six planets? W h y are they arranged this way? W h y do t h e y move with these periods? The first part of the answer, he tells us, came as a flash during a class lecture (Kepler, 1596, Praefatio). There are six planets because there are five, and only five, regular solids: God the Great Geometer has provided an archetype for the solar system by arranging these figures in a concentric fashion separated and surrounded by six spheres, suitably spaced for the planetary circuits (Fig. 1). Astonishingly, the scheme works with fair accuracy when space is allowed for the eccentricities of the planetary paths. The figures are given in Table I as tabulated in Kepler's Mysterium Cosmographicum of
1596 (Chap. 14). Kepler was obliged to compromise the elegance of his system by adopting the second value for Mercury, which is the radius of a sphere inscribed in the square formed by the edges of the octahedron, rather than in the octahedron itself. With this concession, everything fits within 5% except Jupiter, at which "no one will wonder, considering such a great distance." This marvelous remark reflects the state of mind of a convinced theoretician, and the fact t h a t better values of the orbital parameters improve the agreement is perhaps irrelevant. The Mysterium Cosmographicum arrived from the printer early in 1597, and within a few months K e p l e r sent a copy to the great Tycho Brahe, specifically hoping to obtain in return better observational data. TABLE I RATIOS OF THE PLANET~Y ORBITS (ASSUMING THE I N N E R M O S T PART 03' THE OUTER ORBIT TO BE 1 0 0 0 )
11I
Planets
Regular solid
Computed by Kepler
According to
Copernicus
Saturn Cube
577
635
Tetrahedron
333
333
Dodecahedron
795
757
Icosahedron
795
794
Octahedron
577 or 707
723
Jupiter Mars Earth Venus
Mercury
112
OWEN GINGERICH
? 2 ,.."
. .."" ! " ,... ."
/.'"
.
.
$¢
FIG. 1. The nesting of regular solids and planetary spheres in Kepler's Mysterium Gosmographicum.
In Kepler's view, it was divine fate that brought them together in Prague and that set him onto an analysis of Mars. This endeavor, documented in the Astronomia Nova (1609), led to the discovery of Mars' elliptical orbit and the law of areas. Kepler's successful attack on Mars, clearly founded on the finest observational data of his day, might have turned his interest from the Platonic pursuit of nested solids. The Astronomia Nova is his most enduring contribution to astronomy, but to the end of his life he ranked the nest of spheres and polyhedra among his highest achievements. The research on the Martian orbit was in many respects merely an arduous digression from his more pro-
found search for the archetypal harmonies of the universe. Armed with more accurate parameters, Kepler again plunged into his quest for the ultimate order of the planetary system, this time following up some initial ideas on the third question, W h y do the planets move with these particular periods? Kepler first sought in vain for harmonic ratios between the planetary periods. Commenting on this failure, he wrote in his Harmonices M u n d i (1619, Bk. 5, Chap. 4) that apparently God the Creator did not wish to introduce harmonic proportions between the periodic times. But, he continued, "since God has established nothing without geometric beauty which
K E P L E R A N D SOLAR SYSTEM S T R U C T U R E
was not bound beforehand by some law of necessity, we easily conclude t h a t the periodic times got their appropriate lengths because of a prior archetype." Success finally crowned Kepler's search when he dismissed the times of the total journeys around the Sun and concentrated on arcs near the aphelia and perihelia. The movements, when halved a suitable number of times, reduced to the ratios of the musician's major scale! 1 "Sing out, Urania," cried Kepler, "while I ascend through the harmonic scale of celestial motions to the heights where the true archetype of the fabric of the universe is hidden" (1619, Bk. 5, Chap. 7). Of course, Kepler noted, a planet could not be at aphelion and perihelion at the 1 Another parallel between Kepler and the c u r r e n t c o n t r o v e r s y w i t h M o l c h a n o v is reflected i n a r e c e n t discussion b y W a l k e r (1967): " B u t it is e v i d e n t t h a t , g i v e n a wide e n o u g h m a r g i n o f error, o n e c o u l d find m u s i c a l r a t i o s in a n y old s e t of n u m b e r s . W a s K e p l e r , as A t h a n a s i u s K i r c h e r s u g g e s t e d in his Musurgia (1650), p l a y i n g a g a m e w i t h s u c h lax rules t h a t h e was b o u n d t o win? Or d i d h e in f a c t d i s c o v e r a p a t t e r n , a r e g u l a r i t y w h i c h really does exist? I t h i n k t h e a n s w e r is t h a t h e was n o t p l a y i n g t o o e a s y a g a m e a n d t h a t h e h a d e v e r y r e a s o n to suppose that he had made a genuine discovery-for t h e following reason. H e h a s b e e n t r y i n g , e v e r since t h e Mysterium Cosmographicum (1596), t o find t h e s e r a t i o s in t h e h e a v e n s , in t h e d i s t a n c e s b e t w e e n t h e o r b i t s o f t h e p l a n e t s , a n d t h e n in t h e i r o r b i t a l speeds, a n d h e d i d n o t find t h e m . I t is clear, t h e r e f o r e , t h a t h e w a s n o t willing t o s t r e t c h his m a r g i n o f e r r o r so t h a t h e c o u l d find t h e m w h e r e v e r h e looked for t h e m . "
1 13
same time; hence the silent harmonies did not sound simultaneously, but only from time to time as the planets wheeled in their generally dissonant courses around the Sun. Swept on by the grandeur of his vision, he exclaimed: " I t should no longer seem strange t h a t man, the ape of his Creator, has finally discovered how to sing polyphonically, an art unknown to the ancients. With this symphony of voices man can play through the eternity of time in less than an hour and can taste in small measure through human arts the delight of God the Supreme Artist by calling forth t h a t very sweet sensation of pleasure in the music which imitates God" (1619, Bk. 5, Chap. 7). Few modern scientists expose their motivations so candidly, but the sweetness of celestial harmonies can still be savored. I:~EI~ERENCES KEPLER, J . (1596). " P r o d r o m u s . . . M y s t e r i u m Cosmographicum," Tubingae. KEPLER, J. (1609). "Astronomia Nova A~T~o~oT~roa , seu P h y s i c a Coelestis, t r a d i t a c o m m e n t a r i i s de M o t i b u s Stellae M a r t i s " [Pragae]. KEPLER, J . (1619). " H a r m o n i c e s M u n d i L i b r i V , " Lincii A u s t r i a e . KIRCHER, A. (1650). " M u s u r g i a U n i v e r s a l i s . " Romae. MOLCHANOV, A. M. (1968). T h e r e s o n a n t s t r u c t u r e of t h e solar s y s t e m . T h e law of p l a n e t a r y d i s t a n c e s . Icarus 8, 203-215. WALKER, D. P. (1967). K e p l e r ' s Celestial Music.
J.
Warburg and Courtauld Institutes 39,
228-250.
11, 111--113 (1969)
Kepler and the Resonant Structure of the Solar System OWEN
GINGERICH
Smithsonian Astrophysical Observatory and Harvard University, Cambridge, Massachusetts Received May 8, 1969 Molchanov's ordering the solar system by a table of resonance relations recalls an earlier attempt by Johannes Kepler (1571-1630). A. M. Molchanov (1968) is by no means the first to seek for the "resonant structure of the solar system" ; his research hears an interesting parallel to a thread t h a t runs through the entire work of Johannes Kepler. Both authors have addressed themselves to a deeply fundamental question : W h y are the planets arranged in their present pattern? Even in the 17th century this was an unorthodox query, and Kepler's vision of the "intelligible b u t inaudible" harmonies sung by the planets has generally been dismissed as sheer mysticism or numerology. Nevertheless, when a serious consideration of this question is reopened on the pages of Icarus, it is worthwhile to review briefly the verve and imagination t h a t characterized Kepler's inquiry. As a young teacher in Graz (Austria), fresh from the seminary in Ttibingen, Kepler posed for himself three unusual questions: W h y are there only six planets? W h y are they arranged this way? W h y do t h e y move with these periods? The first part of the answer, he tells us, came as a flash during a class lecture (Kepler, 1596, Praefatio). There are six planets because there are five, and only five, regular solids: God the Great Geometer has provided an archetype for the solar system by arranging these figures in a concentric fashion separated and surrounded by six spheres, suitably spaced for the planetary circuits (Fig. 1). Astonishingly, the scheme works with fair accuracy when space is allowed for the eccentricities of the planetary paths. The figures are given in Table I as tabulated in Kepler's Mysterium Cosmographicum of
1596 (Chap. 14). Kepler was obliged to compromise the elegance of his system by adopting the second value for Mercury, which is the radius of a sphere inscribed in the square formed by the edges of the octahedron, rather than in the octahedron itself. With this concession, everything fits within 5% except Jupiter, at which "no one will wonder, considering such a great distance." This marvelous remark reflects the state of mind of a convinced theoretician, and the fact t h a t better values of the orbital parameters improve the agreement is perhaps irrelevant. The Mysterium Cosmographicum arrived from the printer early in 1597, and within a few months K e p l e r sent a copy to the great Tycho Brahe, specifically hoping to obtain in return better observational data. TABLE I RATIOS OF THE PLANET~Y ORBITS (ASSUMING THE I N N E R M O S T PART 03' THE OUTER ORBIT TO BE 1 0 0 0 )
11I
Planets
Regular solid
Computed by Kepler
According to
Copernicus
Saturn Cube
577
635
Tetrahedron
333
333
Dodecahedron
795
757
Icosahedron
795
794
Octahedron
577 or 707
723
Jupiter Mars Earth Venus
Mercury
112
OWEN GINGERICH
? 2 ,.."
. .."" ! " ,... ."
/.'"
.
.
$¢
FIG. 1. The nesting of regular solids and planetary spheres in Kepler's Mysterium Gosmographicum.
In Kepler's view, it was divine fate that brought them together in Prague and that set him onto an analysis of Mars. This endeavor, documented in the Astronomia Nova (1609), led to the discovery of Mars' elliptical orbit and the law of areas. Kepler's successful attack on Mars, clearly founded on the finest observational data of his day, might have turned his interest from the Platonic pursuit of nested solids. The Astronomia Nova is his most enduring contribution to astronomy, but to the end of his life he ranked the nest of spheres and polyhedra among his highest achievements. The research on the Martian orbit was in many respects merely an arduous digression from his more pro-
found search for the archetypal harmonies of the universe. Armed with more accurate parameters, Kepler again plunged into his quest for the ultimate order of the planetary system, this time following up some initial ideas on the third question, W h y do the planets move with these particular periods? Kepler first sought in vain for harmonic ratios between the planetary periods. Commenting on this failure, he wrote in his Harmonices M u n d i (1619, Bk. 5, Chap. 4) that apparently God the Creator did not wish to introduce harmonic proportions between the periodic times. But, he continued, "since God has established nothing without geometric beauty which
K E P L E R A N D SOLAR SYSTEM S T R U C T U R E
was not bound beforehand by some law of necessity, we easily conclude t h a t the periodic times got their appropriate lengths because of a prior archetype." Success finally crowned Kepler's search when he dismissed the times of the total journeys around the Sun and concentrated on arcs near the aphelia and perihelia. The movements, when halved a suitable number of times, reduced to the ratios of the musician's major scale! 1 "Sing out, Urania," cried Kepler, "while I ascend through the harmonic scale of celestial motions to the heights where the true archetype of the fabric of the universe is hidden" (1619, Bk. 5, Chap. 7). Of course, Kepler noted, a planet could not be at aphelion and perihelion at the 1 Another parallel between Kepler and the c u r r e n t c o n t r o v e r s y w i t h M o l c h a n o v is reflected i n a r e c e n t discussion b y W a l k e r (1967): " B u t it is e v i d e n t t h a t , g i v e n a wide e n o u g h m a r g i n o f error, o n e c o u l d find m u s i c a l r a t i o s in a n y old s e t of n u m b e r s . W a s K e p l e r , as A t h a n a s i u s K i r c h e r s u g g e s t e d in his Musurgia (1650), p l a y i n g a g a m e w i t h s u c h lax rules t h a t h e was b o u n d t o win? Or d i d h e in f a c t d i s c o v e r a p a t t e r n , a r e g u l a r i t y w h i c h really does exist? I t h i n k t h e a n s w e r is t h a t h e was n o t p l a y i n g t o o e a s y a g a m e a n d t h a t h e h a d e v e r y r e a s o n to suppose that he had made a genuine discovery-for t h e following reason. H e h a s b e e n t r y i n g , e v e r since t h e Mysterium Cosmographicum (1596), t o find t h e s e r a t i o s in t h e h e a v e n s , in t h e d i s t a n c e s b e t w e e n t h e o r b i t s o f t h e p l a n e t s , a n d t h e n in t h e i r o r b i t a l speeds, a n d h e d i d n o t find t h e m . I t is clear, t h e r e f o r e , t h a t h e w a s n o t willing t o s t r e t c h his m a r g i n o f e r r o r so t h a t h e c o u l d find t h e m w h e r e v e r h e looked for t h e m . "
1 13
same time; hence the silent harmonies did not sound simultaneously, but only from time to time as the planets wheeled in their generally dissonant courses around the Sun. Swept on by the grandeur of his vision, he exclaimed: " I t should no longer seem strange t h a t man, the ape of his Creator, has finally discovered how to sing polyphonically, an art unknown to the ancients. With this symphony of voices man can play through the eternity of time in less than an hour and can taste in small measure through human arts the delight of God the Supreme Artist by calling forth t h a t very sweet sensation of pleasure in the music which imitates God" (1619, Bk. 5, Chap. 7). Few modern scientists expose their motivations so candidly, but the sweetness of celestial harmonies can still be savored. I:~EI~ERENCES KEPLER, J . (1596). " P r o d r o m u s . . . M y s t e r i u m Cosmographicum," Tubingae. KEPLER, J. (1609). "Astronomia Nova A~T~o~oT~roa , seu P h y s i c a Coelestis, t r a d i t a c o m m e n t a r i i s de M o t i b u s Stellae M a r t i s " [Pragae]. KEPLER, J . (1619). " H a r m o n i c e s M u n d i L i b r i V , " Lincii A u s t r i a e . KIRCHER, A. (1650). " M u s u r g i a U n i v e r s a l i s . " Romae. MOLCHANOV, A. M. (1968). T h e r e s o n a n t s t r u c t u r e of t h e solar s y s t e m . T h e law of p l a n e t a r y d i s t a n c e s . Icarus 8, 203-215. WALKER, D. P. (1967). K e p l e r ' s Celestial Music.
J.
Warburg and Courtauld Institutes 39,
228-250.