Titius–Bode law

Template:Short description The Titius–Bode law (sometimes termed simply Bode's 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 (in the asteroid belt) 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 $a_n$ of each planet's orbit outward from the Sun in units such that the Earth's semi-major axis is equal to 10:$$a=4+x$$where $x=0,3,6,12,24,48,96,192,384,768\dots$ such that, with the exception of the first step, each value is twice the previous value. There is another representation of the formula:$$a=4+3\times 2^n$$where $n=-\mathrm{\infty },0,1,2,\dots .$ The resulting values can be divided by 10 to convert them into astronomical units (Template:Sc), resulting in the expression:$$a=0.4+0.3\times 2^n.$$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.Template:Efn

The "Classical" (Canonical) form of the Titius-Bode Law is $a_n=0.4+0.3\times 2^n$. This Formula also has a "Recursive" form: $a_{n+1}=2\times a_n-0.4$, where $a_0=0.55.$

Origin and history

File:Johann Daniel Titius.jpg

Johann Daniel Titius (1729–1796)

File:Johann Elert Bode.jpg

Johann Elert Bode (1747–1826)

The first mention of a series approximating Bode's law is found in a textbook by D. Gregory (1715):

A similar sentence, likely paraphrased from Gregory (1715), 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 (in later editions):Template:Efn

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.Template:Efn

Many precedents were found that predate the seventeenth century. Titius was a disciple of the German philosopher C. F. von Wolf (1679–1754), and the second part of the text that Titius inserted into Bonnet's work is in a book by von Wolf (1723), 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 (1702), 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) and Tomàs Cerdà (c. 1760) 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.

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 (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.

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. Template:Dp, 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 Template:Dp, 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:

File:Titus-Bode law.svg

Graphical plot of the eight planets, Pluto, and Ceres versus the first ten predicted distances

m	k	T–B rule distance (AU)	Planet	Semimajor axis (AU)	Deviation from prediction1
$-\mathrm{\infty }$	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	Template:Dp2	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:

File:Function f of Blagg formulation of Titius Bode Law.png

The empirical correction function  Template:Mvar  introduced in Blagg's reformulation of the Titius–Bode law.

$$\mathsf{d}\mathsf{i}\mathsf{s}\mathsf{t}\mathsf{a}\mathsf{n}\mathsf{c}\mathsf{e}=A\cdot [B+f(\alpha +n\beta )](1.7275{)}^n.$$Note in particular that in Blagg's formula, the law for the Solar System was best represented by a progression in Template:Math, rather than the original value Template:Math used by Titius, Bode, and others.

Blagg examined the satellite system of Jupiter, Saturn, and Uranus, and discovered the same progression ratio Template:Math, in each.

However, the final form of the correction function Template:Mvar 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 (the reason that points on the curve are such a close match for empirical data, for objects discovered prior to 1913, is that they are the empirical data).

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:$$\begin{array}{rl}f\left(\theta \right)& =0.4594+0.396\cos (\theta -27.4^{\circ })+0.168\cos (2(\theta -60.4^{\circ }))+0.062\cos (3(\theta -28.1^{\circ }))+\\ & \phantom{=0.4594}+0.053\cos (4(\theta -77.2^{\circ }))+0.009\cos (5(\theta -22^{\circ }))+0.012\cos (7(\theta -40.4^{\circ })).\end{array}$$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 Template:Mvar, Template:Mvar, and Template:Mvar  ambiguous; it is shown here in normalized form (i.e. this version of  Template:Mvar  is scaled to produce values ranging from Template:Math to Template:Math, inclusive):$$f\left(\theta \right)=0.249+0.860(\frac{\cos \mathrm{\Psi }}{3-\cos \left(2\mathrm{\Psi }\right)}+\frac{1}{6-4\cos (2\mathrm{\Psi }-60^{\circ })}),$$where $\mathrm{\Psi }\equiv \theta -27.5^{\circ }.$ Neither of these formulas for function Template:Mvar are used in the calculations below: The calculations here are based on a graph of function Template:Mvar which was drawn based on observed data.

Constants for Blagg's refinement of the Titius–Bode law
(as modified by )

System	Template:Mvar	Template:Mvar	Template:Mvar	Template:Mvar
Sun-orbiting bodies	0.4162	2.0250	112.4°	056.6°
Moons of Jupiter	0.4523	1.8520	113.0°	036.0°
Moons of Saturn	3.0740	0.0071	118.0°	010.0°
Moons of Uranus	2.9800	0.0805	125.7°	012.5°

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 (Jupiter IX), Lysithea (J X), Carme (J XI), Ananke (J XII), and Miranda (Uranus V).

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 Template:Mvar in the table above make them very sensitive to the exact form of the Template:Nobr

Sun-orbiting bodies Planet / minor pl. Template:Mvar obs. dist. Blagg pred. Mercury −2 0.387 0.387 Venus −1 0.723 0.723 Earth 0 1.000 1.000 Mars 1 1.524 1.524 Vesta 2 2.362 2.670 Juno 2.670 Pallas 2.774 Ceres 2.769 Jupiter 3 5.204 5.200 Saturn 4 9.583 9.550 Uranus 5 19.220 19.230 Neptune 6 30.070 30.130 (Pluto) 7 (39.48) 41.800

Moons of Jupiter Jupiter satellite Template:Mvar obs. dist. Blagg pred. Amalthea −2 0.429 0.429 — −1 — 0.708 Io 0 1.000 1.000 Europa 1 1.592 1.592 Ganymede 2 2.539 2.541 Callisto 3 4.467 4.467 — 4 — 9.260 — 5 — 15.400 Himalia 6 27.250 27.540 Elara 27.850 (Lysithea) (27.85) (Ananke) 7 (49.8)0 55.460 (Carme) (53.3)0 Pasiphae 55.700 (Sinope) (56.2)0

Moons of Saturn Saturn satellite Template:Mvar obs. dist. Blagg pred. (Janus) −3 (0.538) 0.540 Mimas −2 0.630 0.629 Enceladus −1 0.808 0.807 Tethys 0 1.000 1.000 Dione 1 1.281 1.279 Rhea 2 1.789 1.786 — 3 — 2.970 Titan 4 4.149 4.140 Hyperion 5 5.034 5.023 — 6 — 6.300 — 7 — 6.650 — 8 — 7.000 Iapetus 9 12.090 12.110 Phoebe 10 43.920 43.850

Moons of Uranus Uranus satellite Template:Mvar obs. dist. Blagg pred. (Miranda) −2 (0.678) 0.640 — −1 — 0.770 Ariel 0 1.000 1.000 Umbriel 1 1.394 1.393 Titania 2 2.293 2.286 Oberon 3 3.058 3.055

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 Template:Math rather than Template:Math. His spacing law is in the form$$R_n=(1.728{)}^n{\varrho }_n({\theta }_n),$$where ${\varrho }_n$ is an oscillatory function with period $2\pi$, representing distances ${\varrho }_n$ from an off-centered origin to points on an ellipse.

Historical inertia

Nieto, who conducted the first modern comprehensive review of the Titius–Bode Law, noted that "The psychological hold of the Law on astronomy has been such that people have always tended to regard its original form as the one on which to base theories." He was emphatic that "future theories must rid themselves of the bias of trying to explain a progression ratio of 2":

One thing which needs to be emphasized is that the historical bias towards a progression ratio of 2 must be abandoned. It ought to be clear that the first formulation of Titius (with its asymmetric first term) should be viewed as a good first guess. Certainly, it should not necessarily be viewed as the best guess to refer theories to. But in astronomy the weight of history is heavy ... Despite the fact that the number 1.73 is much better, astronomers cling to the original number 2.

Theoretical explanations

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. Since it may be a mathematical coincidence rather than a "law of nature", it is sometimes referred to as a rule instead of "law". 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".

Orbital resonance from major orbiting bodies creates regions around the Sun that are free of long-term stable orbits. Results from simulations of planetary formation support the idea that a randomly chosen, stable planetary system will likely satisfy a Titius–Bode law.

Dubrulle and Graner 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.

Natural satellite systems and exoplanetary systems

Only a limited number of systems are available upon which Bode's law can presently be tested; two solar planets have enough large moons that probably formed in a process similar to that which formed the planets: The four large satellites of Jupiter and the biggest inner satellite (i.e., Amalthea) cling to a regular, but non-Titius-Bode, spacing, with the four innermost satellites locked into orbital periods that are each twice that of the next inner satellite. Similarly, the large moons of Uranus have a regular but non-Titius-Bode spacing. However, according to Martin Harwit

The new phrasing is known as "Dermott's law".

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$$a_n=0.0142\cdot {\mathrm{e}}^{0.9975n}=0.0142\cdot 2.7115^n,$$and controversially predicts an undiscovered planet or asteroid field for $n=5$ at 2 AU. Furthermore, the orbital period and semi-major axis of the innermost planet in the 55 Cancri system have been greatly revised (from 2.817 days to 0.737 days and from 0.038 AU to 0.016 AU, respectively) since the publication of these studies.

Recent astronomical research suggests that planetary systems around some other stars may follow Titius-Bode-like laws. Template:Harvp applied a generalized Titius-Bode relation to 68 exoplanet systems that contain four or more planets. They showed that 96% of these exoplanet systems adhere to a generalized Titius-Bode relation to a similar or greater extent than the Solar System does. The locations of potentially undetected exoplanets are predicted in each system.

Subsequent research detected 5 candidate planets from the 97 planets predicted for the 68 planetary systems. The study showed that the actual number of planets could be larger. The occurrence rates of Mars- and Mercury-sized planets are unknown, so many planets could be missed due to their small size. 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.

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. TRAPPIST‑1i had a prediction based exclusively on the Titius–Bode law with an orbital period of Template:Nobr.

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, 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%) is obtained.

See also

• Dermott's law
• Phaeton (hypothetical planet)
• Logarithmic spiral
• Mysterium Cosmographicum

Further reading

• Template:Cite magazine
• – mystical and speculative; ref. diagrams missing from archive
• Template:Cite report – combination history of distance measurements and development of Titius' law, notable astronomers involved, and exposition by graphs and simple ratios of modern planetary and satellite distances