# Titius–Bode law

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The Titius–Bode law (sometimes termed just Bode's law) is a hypothesis that the bodies in some orbital systems, including the Sun's, orbit at semi-major axes in a function of planetary sequence. The formula suggests that, extending outward, each planet would 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orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit and has eventually been superseded as a theory of solar system formation. It is named for Johann Daniel Titius and Johann Elert Bode.

## Formulation

The law relates the semi-major axis ${\displaystyle a}$ of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10:

${\displaystyle a=4+x}$

where ${\displaystyle x=0,3,6,12,24,48...}$ with the exception of the first step, each value is twice the previous value. There is another representation of the formula: ${\displaystyle a=2^{n}\times 3+4}$ where ${\displaystyle n=-\infty ,0,1,2...}$. The resulting values can be divided by 10 to convert them into astronomical units (AU), resulting in the expression

${\displaystyle a=0.4+0.3\times 2^{m}}$

for ${\displaystyle m=-\infty ,0,1,2...}$ For the outer planets, each planet is predicted to be roughly twice as far from the Sun as the previous object.

## Paradoxical origin and subsequent course

Johann Daniel Titius
Johann Elert Bode

The first mention of a series approximating Bode's law is found in David Gregory's The Elements of Astronomy, published in 1715. In it, he says, "...supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety five."[1] A similar sentence, likely paraphrased from Gregory,[1] appears in a work published by Christian Wolff in 1724.

In 1764, Charles Bonnet said in his Contemplation de la Nature that, "We know seventeen planets that enter into the composition of our solar system [that is, major planets and their satellites]; but we are not sure that there are no more."[1] To this, in his 1766 translation of Bonnet's work, Johann Daniel Titius added two of his own paragraphs, at the bottom of page 7 and at the beginning of page 8. The new interpolated paragraph is not found in Bonnet's original text, nor in translations of the work into Italian and English.

There are two parts to Titius's intercalated text. The first part explains the succession of planetary distances from the Sun:

Take notice of the distances of the planets from one another, and recognize that almost all are separated from one another in a proportion which matches their bodily magnitudes. Divide the distance from the Sun to Saturn into 100 parts; then Mercury is separated by four such parts from the Sun, Venus by 4+3=7 such parts, the Earth by 4+6=10, Mars by 4+12=16. But notice that from Mars to Jupiter there comes a deviation from this so exact progression. From Mars there follows a space of 4+24=28 such parts, but so far no planet was sighted there. But should the Lord Architect have left that space empty? Not at all. Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope. Next to this for us still unexplored space there rises Jupiter's sphere of influence at 4+48=52 parts; and that of Saturn at 4+96=100 parts.

In 1772, Johann Elert Bode, aged twenty-five, completed the second edition of his astronomical compendium Anleitung zur Kenntniss des gestirnten Himmels ("Manual for Knowing the Starry Sky"), into which he added the following footnote, initially unsourced, but credited to Titius in later versions (in Bode's memoir can be found a reference to Titius with clear recognition of his priority):[2]

This latter point seems in particular to follow from the astonishing relation which the known six planets observe in their distances from the Sun. Let the distance from the Sun to Saturn be taken as 100, then Mercury is separated by 4 such parts from the Sun. Venus is 4+3=7. The Earth 4+6=10. Mars 4+12=16. Now comes a gap in this so orderly progression. After Mars there follows a space of 4+24=28 parts, in which no planet has yet been seen. Can one believe that the Founder of the universe had left this space empty? Certainly not. From here we come to the distance of Jupiter by 4+48=52 parts, and finally to that of Saturn by 4+96=100 parts.

These two statements, for all their particular typology and the radii of the orbits, seem to stem from an antique cossist.[3] Many precedents were found from before the seventeenth century. Titius was a disciple of the German philosopher Christian Freiherr von Wolf (1679-1754). The second part of the inserted text in Bonnet's work is founded in a von Wolf work dated 1723, Vernünftige Gedanken von den Wirkungen der Natur. Twentieth century literature about Titius–Bode law assigns the German philosopher authorship; if so, Titius could have learned from him. Another older reference was written by James Gregory in 1702, in his Astronomiae physicae et geometricae elementa, where the succession of planetary distances 4, 7, 10, 16, 52 and 100 became a geometric progression of ratio 2. This is the nearest Newtonian formula, which is also contained in Benjamin Martin and Tomàs Cerdà years before the German publication of Bonnet's book.

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—now discredited as a planet—is at a mean distance that roughly corresponds to that the Titus-Bode law predicted for the next planet out from Uranus.

When originally published, the law was approximately satisfied by all the known planets—Mercury through Saturn—with a gap between the fourth and fifth planets. It was regarded as interesting, but of no great importance until the discovery of Uranus in 1781, which happens to fit into the series. Based on this discovery, Bode urged a search for a fifth planet. Ceres, the largest object in the asteroid belt, was found at Bode's predicted position in 1801. Bode's law was then widely accepted until Neptune was discovered in 1846 and found not to satisfy the law. Simultaneously, the large number of asteroids discovered in the belt removed Ceres from the list of planets. Bode's law was discussed by the astronomer and logician Charles Sanders Peirce in 1898 as an example of fallacious reasoning.[4]

The discovery of Pluto in 1930 confounded the issue still further. Although nowhere near its position as predicted by Bode's law, it was roughly at the position the law had predicted for Neptune. However, the subsequent discovery of the Kuiper belt, and in particular of the object Eris, which is more massive than Pluto yet does not fit Bode's law, further discredited the formula.[5]

## A potentially earlier explanation

The Jesuit Tomàs Cerdà (1715-1791) gave a famous astronomy course in Barcelona in 1760, at the Royal Chair of Mathematics of the College of Sant Jaume de Cordelles (Imperial and Royal Seminary of Nobles of Cordellas).[6] From the original manuscript preserved in the Royal Academy of History in Madrid, Lluís Gasiot remade Tratado de Astronomía from Cerdá, published in 1999, based on Astronomiae physicae from James Gregory (1702) and Philosophia Britannica from Benjamin Martin (1747). In the Cerdàs's Tratado appears the planetary distances obtained from the periodic times applying Kepler's third law, with an accuracy of 10−3. Taking as reference the distance from Earth as 10 and rounding to whole, the geometric progression [(Dn x 10) - 4] / [(Dn-1 x 10) - 4] = 2, from n=2 to n=8 can be expressed. And using the circular uniform fictitious movement to Kepler's Anomaly, Rn values corresponding to each planet's ratios may be obtained as rn = (Rn - R1) / (Rn-1 - R1) resulting 1.82; 1.84; 1.86; 1.88 and 1.90, which rn = 2 - 0.02 (12 - n), the ratio between Keplerian succession and Titius-Bode Law, which would be a casual numerical coincidence. The reason is close to 2, but increases harmonically from 1.82.

The planet's average speed from n=1 to n=8 decreases moving away the Sun and differs from uniform descent in n=2 to recover from n=7 (orbital resonance).

## 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 several planets and dwarf planets in the Solar System:

Graphical plot of the traditional nine planets (including Pluto) and Ceres versus the first ten predicted distances.
k T–B rule distance (AU) Planet Semimajor axis (AU) Deviation from prediction1
0 0.4 Mercury 0.39 −3.23%
1 0.7 Venus 0.72 +3.33%
2 1.0 Earth 1.00 0.00%
4 1.6 Mars 1.52 −4.77%
8 2.8 Ceres2 2.77 −1.16%
16 5.2 Jupiter 5.20 +0.05%
32 10.0 Saturn 9.55 −4.45%
64 19.6 Uranus 19.22 −1.95%
128 38.8 Neptune 30.11 −22.40%
Orcus3 39.17 +0.96%
Pluto3 39.54 +1.02%
Haumea3 43.22 +11.39%
Quaoar3 43.40 +11.87%
256 77.2 2007 OR103 66.85 −14.07%
Eris3 67.78 −12.9%
512 154.00 2000 CR1053 230.12 +49.4%
1024 307.6 2010 GB1743 ~351.0 +14%
2048 614.8 Sedna3 506.2 −17.66%
Planet Nine
(hypothetical)
~665[7] +8%

1 For large k, each Titius–Bode rule distance is approximately twice the preceding value. Hence, an arbitrary planet may be found within −25% to +50% of one of the predicted positions. For small k the predicted distances do not fully double, so the range of potential deviation is smaller. Note the semimajor axis is proportional to the 2/3 power of the orbital period. For example, planets in a 2:3 orbital resonance (such as plutinos relative to Neptune) will vary in distance by (2/3)2/3 = -23.69% and +31.04% relative to one another.

2 Ceres, now classified as a dwarf planet, was considered a small planet from 1801 until the 1860s. It orbits near the middle of the asteroid belt, most of whose objects fall in three bands between -25% and +18% of the Titius-Bode rule distance (2.06 to 2.5 AU, 2.5 to 2.82 AU, 2.82 to 3.28 AU), which are separated by Kirkwood gaps representing 3:1, 5:2, and 2:1 resonances with Jupiter.

3 Pluto was considered a planet from 1930 to 2006, when it was reclassified as a dwarf planet. Aside from the hypothetical Planet Nine, none of the small objects beyond Neptune currently qualifies as a planet.

## 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".[8] However, 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.[5]

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.[9]

Dubrulle and Graner[10][11] showed that power-law distance rules can be a consequence of collapsing-cloud models of planetary systems possessing two symmetries: rotational invariance (the cloud and its contents are axially symmetric) and scale invariance (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.

### Lunar systems and other planetary systems

Only a limited number of systems are available on which Bode's law can presently be tested. Two solar planets have enough large moons that probably have formed in a process similar to that which formed the planets. The four big satellites of Jupiter and the biggest inner satellite, Amalthea, cling to a regular, but non-Titius–Bode, spacing, with the four innermost locked into orbital periods that are each twice that of the next inner satellite. The big moons of Uranus have a regular, non-Titius–Bode spacing.[12] However, according to Martin Harwit, "a slight new phrasing of this law permits us to include not only planetary orbits around the Sun, but also the orbits of moons around their parent planets."[13] 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 = 0.0142 e 0.9975 n, and controversially[14] predicts for n = 5 an undiscovered planet or asteroid field at 2 AU.[15] Furthermore, the orbital period and semimajor axis of the innermost planet in the 55 Cancri system have been significantly 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.[16]

Recent astronomical research suggests that planetary systems around some other stars may follow Titius–Bode-like laws.[17][18] Bovaird and Lineweaver[19] 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 five planet candidates from predicted 97 planets from the 68 planetary systems. The study showed that the actual number of planets could be larger. The occurrence rate of Mars- and Mercury-sized planets are currently unknown, so many planets could be missed due to their small size. Other reasons includes planets that do not transit the star or that the predicted space is occupied by circumstellar disks. Despite this, the number of planets found with Titius–Bode law predictions were lower than expected.[20]

## References

1. ^ a b c
2. ^ Hoskin, Michael (1992-06-26). "Bodes' Law and the Discovery of Ceres". Observatorio Astronomico di Palermo "Giuseppe S. Vaiana". Retrieved 2007-07-05.
3. ^ The cossists were experts in calculations of all kinds and were employed by merchants and businessmen to solve complex accounting problems. Their name derives from the Italian word cosa, meaning "thing", because they used symbols to represent an unknown quantity, similar to the way mathematicians use x today. All professional problem-solvers of this era invented their own clever methods for performing calculations and would do their utmost to keep these methods secret in order to maintain their reputation as the only person capable of solving a particular problem.
4. ^ Laine, Charles Sanders Peirce; Ketner, Kenneth (1992). Reasoning and the logic of things: the Cambridge conferences lectures of 1898. Harvard University Press. pp. 194–196. ISBN 978-0-674-74966-5. HUP catalog page.
5. ^ a b Alan Boss (October 2006). "Ask Astro". Astronomy. 30 (10): 70.
6. ^ Dr. Ramon Parés. Distancias planetarias y ley de Titius-Bode (Historical essay). www.ramonpares.com
7. ^ Renu Malhotra; Kathryn Volk; Xianyu Wang (2016-03-09). "Corralling a distant planet with extreme resonant Kuiper belt objects". The Astrophysical Journal. 824 (2): L22. arXiv:. doi:10.3847/2041-8205/824/2/L22.
8. ^ Carroll, Bradley W.; Ostlie, Dale A. (2007). An Introduction to Modern Astrophysics. Pearson Addison-Wesley. pp. 716–717. ISBN 978-0-8053-0402-2.
9. ^ Wayne Hayes; Scott Tremaine (October 1998). "Fitting Selected Random Planetary Systems to Titius–Bode Laws" (PDF). Icarus. 135 (2): 549. Bibcode:1998Icar..135..549H. doi:10.1006/icar.1998.5999.
10. ^ F. Graner; B. Dubrulle (1994). "Titius-Bode laws in the solar system. Part I: Scale invariance explains everything". Astronomy and Astrophysics. 282: 262–268. Bibcode:1994A&A...282..262G.
11. ^ B. Dubrulle; F. Graner (1994). "Titius–Bode laws in the solar system. Part II: Build your own law from disk models". Astronomy and Astrophysics. 282: 269–276. Bibcode:1994A&A...282..269D.
12. ^ Cohen, Howard L. "The Titius-Bode Relation Revisited". Archived from the original on 28 September 2007. Retrieved 2008-02-24.
13. ^ Harwit, Martin. Astrophysical Concepts (Springer 1998), pages 27–29.
14. ^ Ivan Kotliarov (21 June 2008). "The Titius-Bode Law Revisited But Not Revived". arXiv: [physics.space-ph].
15. ^ Arcadio Poveda & Patricia Lara (2008). "The Exo-Planetary System of 55 Cancri and the Titus-Bode Law" (PDF). Revista Mexicana de Astronomía y Astrofísica (44): 243–246.
16. ^ Rebekah I. Dawson; Daniel C. Fabrycky (2010). "Title: Radial velocity planets de-aliased. A new, short period for Super-Earth 55 Cnc e". Astrophysical Journal. 722: 937–953. arXiv:. Bibcode:2010ApJ...722..937D. doi:10.1088/0004-637X/722/1/937.
17. ^ "The HARPS search for southern extra-solar planets" (PDF). 2010-08-23. Retrieved 2010-08-24. Section 8.2: "Extrasolar Titius-Bode-like laws?"
18. ^ P. Lara, A. Poveda, and C. Allen. On the structural law of exoplanetary systems. AIP Conf. Proc. 1479, 2356 (2012); doi:10.1063/1.4756667
19. ^ Timothy Bovaird; Charles H. Lineweaver (2013). "Title: Exoplanet predictions based on the generalized Titius–Bode relation". Monthly Notices of the Royal Astronomical Society. 435 (2): 1126. arXiv:. Bibcode:2013MNRAS.tmp.2080B. doi:10.1093/mnras/stt1357.
20. ^ Huang, Chelsea X.; Bakos, Gáspár Á. (2014-05-09). "Testing the Titius-Bode law predictions for Kepler multi-planet systems". Monthly Notices of the Royal Astronomical Society. 442: 674. arXiv:. doi:10.1093/mnras/stu906.