Titius–Bode law

The Titius–Bode law is a formulaic prediction of spacing between planets in any given planetary system. The formula suggests that, extending outward, each planet should be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres and Uranus, but failed as a predictor of Neptune's orbit. It is named after Johann Daniel Titius and Johann Elert Bode.
Later work by Mary Adela Blagg and D. E. Richardson significantly revised the original formula, and made predictions that were subsequently validated by new discoveries and observations. It is these re-formulations that offer "the best phenomenological representations of distances with which to investigate the theoretical significance of Titius–Bode type Laws".

Original formulation

The law relates the semi-major axis of each planet's orbit outward from the Sun in units such that the Earth's semi-major axis is equal to 10:where such that, with the exception of the first step, each value is twice the previous value. There is another representation of the formula:where The resulting values can be divided by 10 to convert them into astronomical units, resulting in the expression:For the far outer planets, beyond Saturn, each planet is predicted to be roughly twice as far from the Sun as the previous object. Whereas the Titius–Bode law predicts Saturn, Uranus, Neptune, and Pluto at about 10, 20, 39, and 77 AU, the actual values are closer to 10, 19, 30, 40 AU.
The "Classical" form of the Titius-Bode Law is. This Formula also has a "Recursive" form:, where

Origin and history

The first mention of a series approximating Bode's law is found in a textbook by D. Gregory :
A similar sentence, likely paraphrased from Gregory, appears in a work published by C. Wolff in 1724.
In 1764, C. Bonnet wrote:
In his 1766 translation of Bonnet's work, J. D. Titius added two of his own paragraphs to the statement above. The insertions were placed at the bottom of page 7 and at the top of page 8. The new paragraph is not in Bonnet's original French text, nor in translations of the work into Italian and English.
There are two parts to Titius's inserted text. The first part explains the succession of planetary distances from the Sun:
In 1772, J. E. Bode, then aged twenty-five, published an astronomical compendium, in which he included the following footnote, citing Titius :
These two statements, for all their peculiar expression, and from the radii used for the orbits, seem to stem from an antique algorithm by a cossist.
Many precedents were found that predate the seventeenth century. Titius was a disciple of the German philosopher C. F. von Wolf, and the second part of the text that Titius inserted into Bonnet's work is in a book by von Wolf, suggesting that Titius learned the relation from him. Twentieth-century literature about Titius–Bode law attributes authorship to von Wolf. A prior version was written by D. Gregory, in which the succession of planetary distances 4, 7, 10, 16, 52, and 100 became a geometric progression with ratio 2. This is the nearest Newtonian formula, which was also cited by Benjamin Martin and Tomàs Cerdà years before Titius's expanded translation of Bonnet's book into German. Over the next two centuries, subsequent authors continued to present their own modified versions, apparently unaware of prior work.
Titius and Bode hoped that the law would lead to the discovery of new planets, and indeed the discovery of Uranus and Ceres – both of whose distances fit well with the law – contributed to the law's fame. Neptune's distance was very discrepant, however, and indeed Pluto – no longer considered a planet – is at a mean distance that roughly corresponds to that the Titius–Bode law predicted for the next planet out from Uranus.
When originally published, the law was approximately satisfied by all the planets then known – i.e., Mercury through Saturn – with a gap between the fourth and fifth planets. Vikarius Wurm proposed a modified version of the Titius–Bode Law that accounted for the then-known satellites of Jupiter and Saturn, and better predicted the distance for Mercury.
The Titius–Bode law was regarded as interesting, but of no great importance until the discovery of Uranus in 1781, which happens to fit into the series nearly exactly. Based on this discovery, Bode urged his contemporaries to search for a fifth planet., the largest object in the asteroid belt, was found at Bode's predicted position in 1801.
Bode's law was widely accepted at that point, until in 1846 Neptune was discovered in a location that does not conform to the law. Simultaneously, due to the large number of asteroids discovered in the belt, Ceres was no longer considered a major planet. In 1898 the astronomer and logician C. S. Peirce used Bode's law as an example of fallacious reasoning.
The discovery of Pluto in 1930 confounded the issue still further: Although nowhere near its predicted position according to Bode's law, it was very nearly at the position the law had designated for Neptune. The subsequent discovery of the Kuiper belt – and in particular the object, which is more massive than Pluto, yet does not fit Bode's law – further discredited the formula.

Data

The Titius–Bode law predicts planets will be present at specific distances in astronomical units, which can be compared to the observed data for the planets and two dwarf planets in the Solar System:
Image:Titus-Bode law.svg|thumb|right|upright=1.6|Graphical plot of the eight planets, Pluto, and Ceres versus the first ten predicted distances

m	k	T–B rule distance	Planet	Semimajor axis	Deviation from prediction¹
	0	0.4	Mercury	0.39	−3.23%
0	1	0.7	Venus	0.72	+3.33%
1	2	1.0	Earth	1.00	0.00%
2	4	1.6	Mars	1.52	−4.77%
3	8	2.8	²	2.77	−1.16%
4	16	5.2	Jupiter	5.20	+0.05%
5	32	10.0	Saturn	9.58	−4.42%
6	64	19.6	Uranus	19.22	−1.95%
–	–	–	Neptune	30.07	–
7	128	38.8	Pluto²	39.48	+1.02%

Blagg formulation

In 1913, M. A. Blagg, an Oxford astronomer, revisited the law. She analyzed the orbits of the planetary system and those of the satellite systems of the outer gas giants, Jupiter, Saturn and Uranus. She examined the log of the distances, trying to find the best "average" difference. Her analysis resulted in a different formula:
Note in particular that in Blagg's formula, the law for the Solar System was best represented by a progression in, rather than the original value used by Titius, Bode, and others.
Blagg examined the satellite system of Jupiter, Saturn, and Uranus, and discovered the same progression ratio, in each.
However, the final form of the correction function was not given in Blagg's 1913 paper, with Blagg noting that the empirical figures given were only for illustration. The empirical form was provided in the form of a graph.
Finding a formula that closely fit the empircal curve turned out to be difficult. Fourier analysis of the shape resulted in the following seven-term approximation:After further analysis, Blagg gave the following simpler formula; however the price for the simpler form is that it produces a less accurate fit to the empirical data. Blagg gave it in an un-normalized form in her paper, which leaves the relative sizes of,, and ambiguous; it is shown here in normalized form :where Neither of these formulas for function are used in the calculations below: The calculations here are based on a graph of function which was drawn based on observed data.

System
Sun-orbiting bodies	0.4162	2.025	112.4°	56.6°
Moons of Jupiter	0.4523	1.852	113.0°	36.0°
Moons of Saturn	3.074	0.0071	118.0°	10.0°
Moons of Uranus	2.98	0.0805	125.7°	12.5°

Her paper was published in 1913, and was forgotten until 1953, when A. E. Roy came across it while researching another problem. Roy noted that Blagg herself had suggested that her formula could give approximate mean distances of other bodies still undiscovered in 1913. Since then, six bodies in three systems examined by Blagg had been discovered: Pluto, Sinope, Lysithea, Carme, Ananke, and Miranda.
Roy found that all six fitted very closely. This might have been an exaggeration: out of these six bodies, four were sharing positions with objects that were already known in 1913; concerning the two others, there was a ~6% overestimate for Pluto; and later, a 6% underestimate for Miranda became apparent.

Comparison of the Blagg formulation with observation

Bodies in parentheses were not known in 1913, when Blagg wrote her paper. Some of the calculated distances in the Saturn and Uranus systems are not very accurate. This is because the low values of constant in the table above make them very sensitive to the exact form of the

Richardson formulation

In a 1945 Popular Astronomy magazine article, the science writer D. E. Richardson apparently independently arrived at the same conclusion as Blagg: That the progression ratio is rather than. His spacing law is in the formwhere is an oscillatory function with period, representing distances from an off-centered origin to points on an ellipse.