Cosmic Harmony and Titius and Bode's Law

Experience proves that very often the outcome of human activity does not fit the expectations. Very rarely, however, does the wrong result turn into something positive. Nevertheless, this is what happened in the case of a Danish astronomer Tycho Brahe (1546 - 1601). This most eminent critic of Copernicus' heliocentric theory (and author of a rivaling system) unwillingly contributed to its victory. Namely, his very precise observations of the position of planets - peformed with a precision of 1' at a time when telescopes were still unknown - enabled his student, Johannes Kepler (1571 - 1630), to formulate the well-known laws of the motion of planets which remain a permanent contribution to modern astronomy.

Besides these unquestionable merits Kepler has on his account some, let's say, more controversial achievement. This refers to his attempts at finding a law which would properly describe the distances between the planets and the sun, known by that time and first correctly determined by Copernicus. Kepler, filled with Pythagorean mysticism and seeking "cosmic harmony" everywhere, arrived at the following result, published in his treatise "Mysterium cosmographicum" (1596). If we circumscribe a regular octahedron about a sphere of the dimensions of the orbit of Mercury, the sphere circumscribed about the octahedron will be of the dimensions of the orbit of Venus. If now we circumscribe a regular icosahedron about this second sphere, the sphere circumscribed about the icosahedron will be of the dimensions of the orbit of Earth.

Proceeding likewise, we successively construct a regular dodecahedron, tetrahedron and a cube and the spheres separating the polyhedrons turn out to have the dimensions of the orbits of Mars and Jupiter, correspondingly, whereas the last sphere, circumscribed about the cube, has the dimension of the orbit of Saturn, the sixth - the last planet known to Kepler. Thus five regular polyhedrons alternate with six planetary spheres producing an astounding harmony indeed. It is easily seen that this "empirical model" of Kepler would break down upon a discovery of another planet - after all, there are only five regular polyhedrons. And this is exactly what occurred when F. W. Herschel discovered Uranus in 1781. Fortunately, nobody interpreted this fact as the collapse of a serious theory, for Kepler's construction had never been taken too seriously. Besides, as early as 1766 J. D. Titius produced a different method for determining the radii r(n) of the planetary orbits, namely

(in astronomical units). The value of n corresponding to Mercury is (-infinity, while to other planets known to Kepler successively correspond the numbers 1, 2, 3, 5, 6. The first corroboration of the formula broadly diffused by J. E. Bode by the end of the 18th century (and therefore often called Titius - Bode's law or rule) was brought by the discovery of Uranus. The dimensions of the orbit of this new planet turned out to be in full accordance with Titius - Bode's rule for n=7. The next success came in 1801, when G. Piazzi discovered the first greatest planetoid, called Ceres, at a distance from the sun corresponding to n=4, thus filling an existing gap. Later still it turned out that the orbit of Neptune does not fit the general rule, but the orbit of Pluto corresponds to n=8. Incidentally, this fact gave way to speculations as to whether Pluto could have previously been a satellite of Neptune until a disruption of the system threw Neptune out of its previous orbit. Finally, in very recent times analogous formulas have been found - as a result of amateurish mental entertainment - for the distances of the satellites of Jupiter, Saturn and Neptune from their mother planets. All this formulas are of the form

- very similar to Titius - Bode's law.

A natural question inevitably suggests itself at this point. Is it pure coincidence that all these formulas fit reality or are they rather a reflection of some yet unknown laws of nature? Kepler's polyhedrons are just curiosity, but Titius - Bode's law is correct even for objects discovered long after its formulation! On the one hand, we are aware of the fact that there is always some formula which fits a few observations, particularly so if the observations are somehow selected as in the case of the satellites of Jupiter and Saturn, but on the other, the simplicity of the formulas is striking. In short - at present astronomy is simply not capable of interpreting these facts. Titius - Bode's law exists and this cannot be denied, but neither can we give its explanation now.

Tomasz KWAST


Mean distances of planets from the sun (in astronomical units, 1 astronomical unit = 149600000 km).

Planet n distance from the Sun
Mercury infinity 0.387
Venus 1 0.723
Earth 2 1
Mars 3 1.524
Ceres 4 2.767
Jupiter 5 5.204
Saturn 6 9.575
Uranus 7 19.30
Neptune - 30.21
Pluto 8 39.91
Radii of the orbits of some satellites of Jupiter (in thousands of km)

Amalthea 181
Thebe 222
Io 422
Europe 671
Ganimedes 1070
Callisto 1880
Radii of the orbits of some satellites of Saturn (in thousands of km)

Mimas 186
Enceladus 238
Tethys 295
Dione 377
Rhea 527

Radii of the orbits of the satellites of Uranus (in thousands of km)

Miranda 130
Ariel 191
Umbriel 266
Titania 436
Oberon 583

The reader will easily find an exponential formula for the radii of the satellite orbits.