The basic theme of this article is elementary celestial mechanics but many other areas are touched on. Specifically, this article is concerned with an oddity from the history of astronomy, Bode's Law of the solar system. The law is stated and explained for those with no previous knowledge of this curiosity and then, using Kepler's equations of planetary motion a series of new and even more unusual relationships are derived. The article concludes with one or two bold speculations.Bode's Law
Like many of the well known laws of astronomy, Bode's Law was not discovered by Bode but by another German Johann Titius in 1772. Bode brought it to the attention of his contemporaries and therefore got the credit. The law concerns the distances of the planets of the solar system from the Sun. (In Bode's day the solar system extended as far as the planet Saturn)
To derive the law is quite simple. First take a simple number series based on the number 3, where the next term of the series is twice the previous term:
0, 3, 6, 12, 24, 48, 96
Then add four to each term and divide the result by ten to give the series:
0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10
This series will be referred to as Bode's Series. The application of it arises as follows.
The distance of the Earth from the Sun is 93 million miles and astronomers refer to this distance as 1 Astronomical Unit (AU). If the distances from the Sun of the other planets of the solar system are expressed in AU the table below is produced. The last column of the table gives a comparison with Bode's Series.
As can be seen, Bode's series seems to correspond very well with the planetary distances with a gap for the 2.8 term. Bode thought this was a fundamental law of celestial mechanics and that a planet would be found corresponding to the missing term. In fact the next term in the series is 192+4 divided by 10 which equals 19.6 and when in 1781 Herschel discovered the new planet Uranus at a distance of 19.18 AU from the Sun it appeared that Bode's Law, as he now called it, really did work. Further evidence was provided when in 1801 the first of the asteroids was discovered at a distance of 2.8 AU from the Sun. Several more asteroids were found and a theory arose suggesting that a planet had once existed and had been torn apart by an unknown cataclysm.
Whatever the explanation for the asteroids the fact remained that Bode's Law, which had no scientific basis or support from physical theory, seemed to predict planetary orbits. It was used in the search for the trans-Uranian planet, the next term in the series being 38.8. However, when Neptune was discovered in 1846 it was found to be at a distance of 30 AU from the Sun.
With the prevailing hardening scientific attitudes of the time Bode's Law was dismissed as a fortuitous coincidence and fell from grace.
When Pluto was discovered in 1930 it was found to be at 39.44 AU, in agreement with Bode's predicted Neptune position. The trans-Pluto planet, if it exists, is predicted to lie at between 50 and 100 AU from the Sun. The next term in Bode's series is 77.2 AU - perhaps he will yet have his revenge.Beyond Bode
Bode's Law concerns itself with the distances of the planets from the Sun. If the periods of the planets, that is the time taken to complete one orbit of the sun, are considered more relationships between the planets emerge.
Kepler's third law of planetary motion tells us that the period, T, of a planet is related to the distance from the Sun, R, by:
T2 is proportional to R3
Using this law the terms of Bode's series can be converted into a new series related to the planetary periods. Again the Earth has a value of 1.0 which in this case means one year:
There is close agreement between Bode's periods and the actual values. This is to be expected as Kepler's third law is a scientifically proven fact. However, with respect to planetary periods Bode's Law no longer holds as a simple series relationship. No similar series can be found to fit the periods.
Up to this point the distances and periods of the planets have been referred to the Earth as the baseline. More fruitful results can be obtained if all the planetary periods are related to each other and to this end the table in Annex 1 is given. It is a list of ratios of planetary periods for all the planets of the solar system. The Earth's moon has been included not from any intrinsic interest but because its period becomes of interest as will be shown.
Many coincidences in the ratios can be extracted by inspection of the table. For example, the ratio of the periods of Mercury and Venus is the same as the ratio of the periods of Jupiter and Saturn.
The Mars/Venus relationship is 3,the same as that for Uranus/Pluto and close to that for Saturn/Uranus. Many more individual coincidences come to light on closer examination of the table and this is left as an exercise for the reader. A closer study of some of these follows in a later article.
Groups of planets may show a numerical relationship but no one 'law' emerges linking all the planetary periods. However a rather interesting set of coincidences based on the ratio of the Mars/Jupiter period the period ratios of the planets in pairs moving outward and inward from these two can be found:
|Earth-Saturn||29.45||=||6.3  ||x 4.67|
|Mercury-Neptune||684.00||=||137    ||x 4.99|
The ratios of the pairs are increasing each time by a similar factor. A further coincidence emerges with the Pluto/Moon ratio:
The final coincidence is derived from the fact that the orbits of Mars and Jupiter straddle the asteroids orbits. The Bode distance of 2.8 AU for the asteroids yields a period of 4.69 years - numerically similar to the multiplying factor that has emerged in the above.
A series thus results incorporating all the planets of the solar system and the asteroids thereby putting one over on Bode as his Law excluded Neptune. The multiplying factor varies between 4.65 and 4.99 but if the average value is taken, 4.79, then all the values lie within 4%. Thus a series for calculating the planetary periods based on this multiplying factor would give closer agreement to observation than Bode's Law did for the planetary distances.
The inclusion of the Moon would have more significance if this period referred to a planet within the orbit of Mercury, completing the set of pairs. The fact that the moon has a similar period would then be yet another coincidence. Such a planet would be 0.18 AU from the Sun and if it were small would be extremely difficult to observe. In fact the ancient astronomers postulated such a planet and named it Vulcan.
A Law can now be formerly stated. If the asteroid orbital period, taken as our average multiplying factor 4.77, is called Pa and the periods of the planets of the solar system are numbered as odd numbers starting with Jupiter(1) and moving out from the Sun, and even numbers starting with Mars(2) and moving inward then:
P3/P4 = Pa x P1/P2 where Pn is a planetary period.Conclusions
Bode's Law has been derived and shown to be a rather interesting series of coincidences that fits observations of the solar system quite well. By taking the planetary periods further coincidences have been demonstrated and another, more accurate, law has been derived. In true speculative tradition this law has been used to postulate the existence of a planet inside the orbit of Mercury and has inferred that the average period of the asteroids is of structural significance to the solar system.
The following table lists the ratio of periods between all the planets. The planetary periods used are: