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
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 ) /
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
where ''k'=0,1,2,4,8,16,32,64,128 (sequence of powers of two
The following table compares the law's predictions with the
actual distances, where the addition of Pluto is a modern
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 :
= 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).
the number of the "Titius - Bode Law " version :
nine planets in Solar System and there are nine versions of the "
Titius - Bode Law".
There is a
connection between Titius - Bode Law and the Pascal Triangle:
||1 + 1
||1 + 2 + 1
||1 + 3 + 3 + 1
||1 + 4 + 6 + 4
5+10+ 10 + 5 + 1
15 + 6 + 1
21 + 7 + 1
28 + 8 +
We assume the
next value for the Neptune orbit:
bin( Neptune )= 32 + 64 = 96 =
25 + 26
See, also :