### THE TITIUS-BODE FORMULA

In 1772 an astronomer named Titius living in Wittenberg wrote to a colleague named Bode who was in Berlin. Titius had discovered a very interesting relationship between the average distances of the planets from the Sun.

Starting with the series 0, 3, 6, 12, 24,.. etc and then adding four and dividing by 10, the resultant series was very close to the actual distances of the planets from the Sun when measured in astronomical units. (An astronomical unit or AU is the mean distance of the Earth from the Sun.)

The table below compares the actual distances to three significant figures with the distances given by the series formula:

```     Planet    Actual      Formula

Mercury     0.387        0.4
Venus       0.723        0.7
Earth       1.00         1.0
Mars        1.52         1.6
???                      2.8
Jupiter     5.20         5.2
Saturn      9.55        10.0
Uranus     19.2         19.6
Neptune    30.1         38.8
```

Note that neither Uranus nor Neptune were known at the time. Uranus was not discovered until 1781, and Neptune not until 1846. However, there was an anomaly, with the formula predicting that a planet should exist at 2.8 AU.

Bode adopted this formula with so much enthusiasm that it came to be known as Bode's Law. It was not until a few decades ago that historical research restored the originator to equal status, and the formula is now referred to in most books as the Titius-Bode law or rule.

Bode urged that a search be made for a planet at the 2.8 AU distance. The formula was given the status of a "law" when William Herschal discovered Uranus in 1781, and this was reinforced in 1801 when Giuseppe Piazzi of Sicily discovered the first asteroid Ceres, which happened to have a mean solar distance of 2.8 AU.

Unfortunately, this happy situation did not last. The error between the formula prediction for Neptune was quite large. And for Pluto it is enormous. In fact, those who still hold out some hope that the formula really expresses some underlying physics of solar system formation must have been very happy at the recent International Astronomical Union (IAU) decision to exclude Pluto from the family of planets and relegate it to a minor body, or dwarf planet.

We can express the Titius-Bode rule by the following mathematical formula:

D(n) = ( 3 x 2n + 4 ) / 10 AU

where n = -infinity, 0, 1, 2, 3, 4, .....
We unfortunately have to use a value of negative infinity for Mercury because the initial series of Titius is not a true geometric series because of the zero in the initial place.

It is also possible to write the Titius series in terms of kilometres rather than astronomical units. We start off with the series 0, 45, 90, 180, 360, 720, ... and then add 60 to each term. This gives us a planetary distance in millions of kilometres, as seen below:

```       Planet      Distance      Formula
Name      million km    million km

Mercury       58            60
Venus        108           105
Earth        150           150
Mars         228           240
Asteroids    390           420
Jupiter      778           780
Saturn      1429          1500
Uranus      2875          2940
Neptune     4505          5820
```

Despite many years of investigation, no explanation has been found for any underlying reason that planets should or do follow this formula, and most scientists now regard the Titius-Bode formula as just an interesting near coincidence with reality.