The Titius–Bode law (sometimes termed just Bode's law) is a formulaic prediction of spacing between planets in any given solar 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the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit and was eventually superseded as a theory of solar system formation. It is named after Johann Daniel Titius and Johann Elert Bode.
The law relates the semi-major axis of each planet 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 (AU), resulting in the expression:
for For the outer planets, each planet is predicted to be roughly twice as far from the Sun as the previous object.
Origin and history
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."
In 1764, Charles Bonnet said in his Contemplation de la Nature: "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."
To the above statement, 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 (and further delineated in Bode’s memoir by a reference to Titius, with clear recognition of his priority):
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.[a] Many precedents were found that predate 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 found in a von Wolf work dated 1723, Vernünftige Gedanken von den Wirkungen der Natur. Twentieth-century literature about Titius–Bode law attributes authorship to von Wolf; if so, Titius could have learned it from him. Another older reference was written by James Gregory in 1702, in his Astronomiae physicae et geometricae elementa, in which 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 was cited by 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 — 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. 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 his contemporaries to 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 conform to 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.
The discovery of Pluto in 1930 confounded the issue still further. Although nowhere near its predicted position according to Bode's law, it was roughly at the position the law had delineated for Neptune. 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.
A potentially earlier explanation
The Jesuit Tomàs Cerdà (1715–1791) taught 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). From the original manuscript preserved in the Royal Academy of History in Madrid, Lluís Gasiot revised Tratado de Astronomía from Cerdá — published in 1999, based on Astronomiae physicae from James Gregory (1702) and Philosophia Britannica from Benjamin Martin (1747). Cerdàs's Tratado denotes the planetary distances obtained from the periodic times applying Kepler's third law, with an accuracy of 10−3. Quantifying distance from the Earth as 10 and rounding to whole numbers, the geometric progression [(Dn × 10) − 4]/[(Dn−1 × 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 in 1.82; 1.84; 1.86; 1.88 and 1.90, in which rn = 2 − 0.02(12 − n), the ratio between Keplerian succession and Titius–Bode Law, would be a casual numerical coincidence. The ratio 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).
|m||k||T–B rule distance (AU)||Planet||Semimajor axis (AU)||Deviation from prediction1|
|Planet Nine (hypothetical)||ca. 400–800|
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 that the semi-major 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, Pluto, Eris, and possibly Sedna are dwarf planets, which are considered planets by many planetary scientists.
3 No obvious known bodies have been found close to these distances.
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." On the one hand, 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." On the other hand, a growing amount of data from exoplanetary systems points to a generalized fulfillment of this rule in other planetary systems.
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.
Lunar systems and other planetary 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. The large moons of Uranus have a regular, non-Titius–Bode spacing. 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." 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 predicts for n = 5 an undiscovered planet or asteroid field at 2 AU. Furthermore, the orbital period and semi-major 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.
Recent astronomical research suggests that planetary systems around some other stars may follow Titius–Bode-like laws. Bovaird and Lineweaver 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 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 currently 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 27.53 ± 0.83 days.
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.
- 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 modern mathematicians use . 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 a reputation as the only person capable of solving a particular problem.
- "Dawn: Where Should the Planets Be? The Law of Proportionalities". Archived from the original on 7 March 2016. Retrieved 16 March 2018.
- Hoskin, Michael (26 June 1992). "Bodes' law and the discovery of Ceres". Observatorio Astronomico di Palermo "Giuseppe S. Vaiana". Retrieved 5 July 2007.
- Peirce, Charles Sanders; Ketner, Kenneth Laine (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.
- Boss, Alan (October 2006). "Ask Astro". Astronomy. 30 (10): 70.
- Dr. Ramon Parés. Distancias planetarias y ley de Titius–Bode (Historical essay). www.ramonpares.com
- Carroll, Bradley W.; Ostlie, Dale A. (2007). An Introduction to Modern Astrophysics. Pearson Addison-Wesley. pp. 716–717. ISBN 978-0-8053-0402-2.
- Wayne Hayes; Scott Tremaine (October 1998). "Fitting Selected Random Planetary Systems to Titius–Bode Laws" (PDF). Icarus. 135 (2): 549. arXiv:astro-ph/9710116. Bibcode:1998Icar..135..549H. CiteSeerX 10.1.1.27.8254. doi:10.1006/icar.1998.5999.
- 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.
- 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.
- Cohen, Howard L. "The Titius–Bode Relation Revisited". Archived from the original on 28 September 2007. Retrieved 24 February 2008.
- Harwit, Martin. Astrophysical Concepts (Springer 1998), pages 27–29.
- Ivan Kotliarov (21 June 2008). "The Titius–Bode Law Revisited But Not Revived". arXiv:0806.3532 [physics.space-ph].
- 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.
- 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 (1): 937–953. arXiv:1005.4050. Bibcode:2010ApJ...722..937D. doi:10.1088/0004-637X/722/1/937.
- "The HARPS search for southern extra-solar planets" (PDF). 23 August 2010. Retrieved 24 August 2010. Section 8.2: "Extrasolar Titius–Bode-like laws?"
- Lara, P. (2012). "On the structural law of exoplanetary systems". doi:10.1063/1.4756667. Cite journal requires
- 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:1304.3341. Bibcode:2013MNRAS.435.1126B. doi:10.1093/mnras/stt1357.
- Huang, Chelsea X.; Bakos, Gáspár Á. (9 May 2014). "Testing the Titius–Bode law predictions for Kepler multi-planet systems". Monthly Notices of the Royal Astronomical Society. 442 (1): 674–681. arXiv:1405.2259. Bibcode:2014MNRAS.442..674H. doi:10.1093/mnras/stu906.
- Kipping, David (2018). "Predicting the Orbit of TRAPPIST-1i". Research Notes of the American Astronomical Society. 2 (3): 136. arXiv:1807.10835. Bibcode:2018RNAAS...2..136K. doi:10.3847/2515-5172/aad6e8.
- Lara, Patricia; Cordero-Tercero, Guadalupe; Allen, Christine (2020). "The reliability of the Titius-Bode relation and its implications for the search for exoplanets". arXiv:2003.05121 [astro-ph.EP].
- F. J. Ballesteros; A. Fernandez-Soto; V. J. Martinez (2019). "Title: Diving into Exoplanets: Are Water Seas the Most Common?". Astrobiology. 19 (5): 642–654. doi:10.1089/ast.2017.1720. hdl:10261/213115. PMID 30789285.
- The ghostly hand that spaced the planets New Scientist 9 April 1994, p13
- Plants and Planets: The Law of Titius–Bode explained by H.J.R. Perdijk
- Distancias planetarias y ley de Titius-Bode (Spanish) by Dr. Ramon Parés