Fibonacci numbers and the Pascal Triangle

 

        

Titius-Bode Law

        

The Titius-Bode Law or Rule is the observation that orbits of planets in the solar system follow a simple arithmetic rule quite closely. It was discovered in 1766 by Johann Daniel Titius and "published" (without attribution) in 1772 by Johann Elert Bode, thus the name.
Johann Elert Bode was born on January 19, 1747 in Hamburg, Germany. He became a member of the Berlin Academy of Sciences and director of the Berlin Observatory. Together with Johann Heinrich Lambert, he founded the German language ephemeris, the Astronomisches Jahrbuch oder Ephemeriden [Astronomical Yearbook and Ephemeris] in 1774, later called simply Astronomisches Jahrbuch and then Berliner Astronomisches Jahrbuch, which he continued to publish until his death in 1826. In 1774, Bode started to look for nebulae and star clusters in the sky, and observed 20 of them in 1774-5. Among them are three original discoveries, M81 and M82 which he both discovered on December 31, 1774, and M53, discovered on February 3, 1775, as well as a newly cataloged asterism.

Bode merged his discoveries and other observed objects with those from other catalogs he had access, namely the existing objects and most of the asterisms and non-objects from Hevelius' catalog, the sufficiently northern objects from Lacaille's catalog, most of the 45 objects in the first 1771 edition of Messier's catalog, and some others, to a "Complete Catalog of hitherto observed Nebulous Stars and Star Clusters" of an overall 75 entries, which he published in 1777 in the "Astronomisches Jahrbuch" for 1779. Unfortunately, he added a large number of non-existing objects without verification, in particular from Hevelius, so that over 20 of his objects don't exist. In the years following, he discovered two more objects: His original discovery of M92 occurred on December 31, 1777, and he found M64 on April 4, 1779, only 12 days after Edward Pigott had first discovered it. These two discoveries were announced along with the publication of Koehler's catalog in 1779 in the Astronomisches Jahrbuch for 1782. Consequently, he continued to compile catalogs and atlasses, and in his 1782 "Vorstellung der Gestirne," publishes own independent rediscoveries of open clusters M48 (NGC 2548) and IC 4665 in Ophiuchus. On January 6, 1779, Johann Elert Bode discovered the comet of that year (C/1779 A1, 1779 Bode).

Bode was greatly interested in the new planet discovered by William Herschel in March 1781. While Herschel always referred to this planet as "Georgium Sidus" to honor King George III of England, Bode proposed the name "Uranus" which was soon adopted by the rest of the world. Bode collected virtually all observations of this planet by various astronomers, published many of them in the Astronomisches Jahrbuch, and found that Uranus had been observed before its discovery on a number of occasions, among them an observation of Tobias Mayer from 1756 and earliest by Flamsteed, in December 1690, cataloged as "star" 34 Tauri.

In 1801 Bode published his famous and popular star atlas, Uranographia, where he reproduced or introduced a number of new and strange constellaitons, including "Officina Typographica," "Apparatus Chemica," "Globus Aerostaticus," "Honores Frederici," "Felis," and "Custos Messium," all of which have not survived and vanished from modern star charts.
In 1825, after almost 40 years, Bode retired from the post of a director of the Berlin Observatory, and was succeeded by J.F. Encke. Johann Elert Bode died on November 23, 1826 in Berlin, Germany.

In 1768, Bode published his popular book, "Anleitung zur Kenntnis des gestirnten Himmels" [Instruction for the Knowledge of the Starry Heavens], which was printed in a number of editions. In this book, he stressed an empirical law on planetary distances, originally found by J.D. Titius (1729-96), now called "Bode's Law" or "Titius-Bode Law".>

The original formulation was

a = ( n + 4 ) / 10

where n=0,3,6,12,24,48 ...

The modern formulation is that the mean distance a of the planet from the Sun is, in astronomical units ( AUearth = 147.597 *106 km ):

a = 0.4 + 0.3 x k

where ''k'=0,1,2,4,8,16,32,64,128 (sequence of powers of two and 0)

The following table compares the law's predictions with the actual distances, where the addition of Pluto is a modern modification.

Planet n Titius-Bode Law Semi-Major Axis
Mercury 0.40 0.39
Venus 0 0.70 0.72
Earth 1 1.00 1.00
Mars 2 1.60 1.52
asteroid belt 3 2.80 2.8
Jupiter 4 5.20 5.20
Saturn 5 10.0 9.54
Uranus 6 19.6 19.2
Neptune - - 30.1
Pluto 7 38.8 39.4

All well and good, except that there was a big gap between Mars and Jupiter. Titius and Bode decided to skip a number, making Jupiter a particularly good fit. This law was sometimes taken to predict that a planet would be found between Mars and Jupiter. Within a few years (1781), Uranus was discovered by Sir William Herschel, and it fit right into the law. This discovery made the law respectable, and the hunt for the missing planet began. In 1801, Giuseppe Piazzi discovered the minor planet Ceres, at just the right distance. Ceres was incredibly tiny for a planet. To date, more than 9000 minor planets (asteroids) have been discovered. At first it was thought that a planet was destroyed by a collision, at that distance from the Sun. Now it is thought that the gravity of Jupiter prevented a planet from forming from the fragments there.

Neptune (discovered in Johann Galle in 1846) and Pluto (discovered by Clyde Tombaugh in 1930) do not fit the law very well. In fact, Pluto (in reality a minor planet) fits the spot where Neptune should be.

The original Rasko Jovanovic`s formulation of the " Titius-Bode Law " is now available. This formulation is that the mean distance R(k) of the planet from the Sun is :

where k = 1-Mercury, 2- Venus, 3- Earth, 4- Mars, 5- Planet V, 6- Jupiter, 7- Saturn, 8- Uranus, and 9 - Pluto.
M is 1 (Mercury, Venus and Earth), 2 (Mars, Planet V and Jupiter) and 3 ( Saturn, Uranus and Pluto).
N is the number of the "Titius - Bode Law " version :
there are nine planets in Solar System and there are nine versions of the " Titius - Bode Law".

Version N R(N) AUN*106 km
Mercury 1 1.2973 57.91
Venus 2 4.5708 108.208
Earth 3 9.694 149.597
Mars 4 21.7635 227.94
Planet V 5 46.8076 417.8796
Jupiter 6 103.8381 778.33
Saturn 7 233.8603 1429.4
Uranus 8 522.8772 2870.99
Neptune - - -
Pluto 9 1163.8905 5913.52

There is a connection between Titius - Bode Law and the Pascal Triangle:

Planet k Pascal Triangle bin(k)
Mercury 1 1 0
Venus 2 1 + 1 1
Earth 3 1 + 2 + 1 2
Mars 4 1 + 3 + 3 + 1 4
Planet V 5 1 + 4 + 6 + 4 + 1 8
Jupiter 6 1+ 5+10+ 10 + 5 + 1 16
Saturn 7 1+6 +15+20+ 15 + 6 + 1 32
Uranus 8 1+7+21+35+35+ 21 + 7 + 1 64
Neptune 9 bin(7) + bin(8) 96
Pluto 9 1+8+28+56+70+56+ 28 + 8 + 1 128

We assume the next value for the Neptune orbit:
bin( Neptune )= 32 + 64 = 96 = 25 + 26

See, also :

        

        

  2001-2003 Radoslav Jovanovic                 created:  August 2003.