Titius–Bode law
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".
The new paragraph is not in Bonnet's original French text, nor in translations of the work into Italian and English.
Bode, then aged twenty-five, published an astronomical compendium,[5] in which he included the following footnote, citing Titius (in later editions):[b][6] 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.
[citation needed] A prior version was written by D. Gregory (1702),[8] 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 (1747)[9] and Tomàs Cerdà (c. 1760)[10] years before Titius's expanded translation of Bonnet's book into German (1766).
Over the next two centuries, subsequent authors continued to present their own modified versions, apparently unaware of prior work.
Vikarius (Johann Friedrich) Wurm (1787) 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.
[11] 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.
Ceres, the largest object in the asteroid belt, was found at Bode's predicted position in 1801.
Simultaneously, due to the large number of asteroids discovered in the belt, Ceres was no longer a major planet.
The subsequent discovery of the Kuiper belt – and in particular the object Eris, which is more massive than Pluto, yet does not fit Bode's law – further discredited the formula.
Blagg examined the satellite system of Jupiter, Saturn, and Uranus, and discovered the same progression ratio 1.7275, in each.
Finding a formula that closely fit the empircal curve turned out to be difficult.
[16] 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 (Jupiter IX), Lysithea (J X), Carme (J XI), Ananke (J XII), and Miranda (Uranus V).
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.
This is because the low values of constant B in the table above make them very sensitive to the exact form of the function f .
[1]No solid theoretical explanation underlies the Titius–Bode law – but it is possible that, given a combination of orbital resonance and shortage of degrees of freedom, any stable planetary system has a high probability of satisfying a Titius–Bode-type relationship.
[19] Astrophysicist Alan Boss states that it is just a coincidence, and the planetary science journal Icarus no longer accepts papers attempting to provide improved versions of the "law".
[20] Dubrulle and Graner[21][22] showed that power-law distance rules can be a consequence of collapsing-cloud models of planetary systems possessing two symmetries: rotational invariance (i.e., the cloud and its contents are axially symmetric) and scale invariance (i.e., the cloud and its contents look the same on all scales).
The latter is a feature of many phenomena considered to play a role in planetary formation, such as turbulence.
Of the recent discoveries of extrasolar planetary systems, few have enough known planets to test whether similar rules apply.
An attempt with 55 Cancri suggested the equation and controversially[25] predicts an undiscovered planet or asteroid field for
[27] Recent astronomical research suggests that planetary systems around some other stars may follow Titius-Bode-like laws.
[28][29] Bovaird & Lineweaver (2013)[30] applied a generalized Titius-Bode relation to 68 exoplanet systems that contain four or more planets.
Other possible reasons that may account for apparent discrepancies include planets that do not transit the star or circumstances in which the predicted space is occupied by circumstellar disks.
Despite these types of allowances, the number of planets found with Titius–Bode law predictions was lower than expected.
[31] In a 2018 paper, the idea of a hypothetical eighth planet around TRAPPIST-1 named "TRAPPIST‑1i", was proposed by using the Titius–Bode law.
[32] Finally, raw statistics from exoplanetary orbits strongly point to a general fulfillment of Titius-Bode-like laws (with exponential increase of semi-major axes as a function of planetary index) in all the exoplanetary systems; when making a blind histogram of orbital semi-major axes for all the known exoplanets for which this magnitude is known,[33] and comparing it with what should be expected if planets distribute according to Titius-Bode-like laws, a significant degree of agreement (i.e., 78%)[34] is obtained.
