# Titius Bode series

Johann Daniel Titius (1729–1796)
Johann Elert Bode (1747-1826)

The Titius-Bode series (also titius-bodesche series, Bode- Titiussche relation , bodesche rule and the like) is a numerical relation found empirically by Johann Daniel Titius and made known by Johann Elert Bode , according to which the distances of most planets from the Use a simple mathematical formula to derive the sun approximately from the number of its sequence alone.

From a mathematical point of view it is a sequence (and not a series ), but the name has become common.

## formula

Titius took the sequence of numbers 0, 3, 6, 12, 24, 48, 96, etc., in which after the 3 every number is double the previous one, and added 4 to each number. In the resulting sequence of numbers he placed the middle one Orbital radius of the earth to the number 10 and obtained with this measure the distances of all known planets from the sun.

According to the formulation of Titius and Bode, the original formula results:

${\ displaystyle R_ {n} = 4 + 3 \ cdot 2 ^ {n}}$

The exponent n , starting with Mercury, stands for the index of the sequence −∞ , 0, 1, 2, 3, 4, 5, 6 etc.
From Mercury to Saturn the associated terms of the sequence result (in short: the sequence of numbers) 4, 7, 10, 16, 28, 52, 100 ...

Only in the modern form of the formula by Johann Friedrich Wurm from 1787 is a the mean distance of a planet from the sun, which is measured by the mean distance from the earth in astronomical units :

${\ displaystyle a_ {n} = 0 {,} 4 + 0 {,} 3 \ cdot 2 ^ {n}}$

## Comparison with measured values

planet n Distance
after TB
Real
distance (AE)
deviation
Mercury −∞ 0.4 (0.39) (+ 2.56%)
Venus 0 0.7 (0.72) (- 2.78%)
earth 1 1.0 (1.00) (0.00%)
Mars 2 1.6 (1.52) (+ 5.26%)
( Ceres ) 3 2.8 (2.77) (+ 1.08%)
Jupiter 4th 5.2 (5.20) (0.00%)
Saturn 5 10.0 (9.54) (+ 4.82%)
Uranus 6th 19.6 (19.19) (+ 2.14%)
Neptune - - (30.06) (-)
( Pluto ) 7th 38.8 (39.48) (- 1.72%)
( Eris ) 8th 77.2 (67.7) (+ 14.00%)

The rule mostly agrees with the actual circumstances except for a few percent. However, there are some inconsistencies:

• For Mercury, according to the rest of the sequence, the value n would not have to be −∞ , but −1.
• The asteroid belt is located between Mars and Jupiter . The largest body in this is the Ceres , which is not a planet but a dwarf planet .
• Neptune has no place in this series. In the case of Neptune, however, the possibility is being discussed that it originally originated elsewhere in the solar system and that it migrated to its current location through interaction with the other planets or with large objects passing by the solar system (see section Origin and Migration in the article on Neptune)
• In contrast to the inner planets, Pluto itself has a strongly eccentric orbit that fluctuates between 29.7 and 49.3  AU . This difference corresponds roughly to the diameter of Saturn's orbit or the distance between Uranus and the sun, so the value of the prediction of the Titius-Bode series for the mean orbit radius of Pluto is even lower than for the other planets.
• Eris is also a dwarf planet like Ceres and Pluto, but in contrast to these does not fit into the series as well.

## history

Johannes Kepler was already looking for geometric relationships for planets and their orbits. In his book Mysterium cosmographicum ("Das Weltgeheimnis"), published in 1596, Kepler related the orbits of the then known planets Mercury to Saturn as a cross section of spherical shells with the surface of the five Platonic solids . After a few corrections, the individual surfaces of the five Platonic solids just fit into the nested spheres of the six planets, depending on their shape, as spacers. In his work Harmonice mundi (“World Harmony”), published in 1619, he developed this theory further.

Isaac Newton explained the gap between Mars and Jupiter in 1692 by divine foresight that otherwise the large planets would have severely disturbed the orbits of the little ones near the sun.

David Gregory published in his widespread astronomy textbook The Elements of Astronomy (Latin first published in 1702, English first published in 1715) a series of numbers for the average distances between known planets, according to which the mean orbit radius of the earth is composed of ten units and for the planets Mercury to Saturn gives the values ​​4, 7, 10, 15, 52 and 95. This was taken up by the philosopher Christian Wolff, without any indication of the origin, in his book, Reasonable Thoughts, first published in 1724, about the intentions of natural things .

In 1761 Johann Heinrich Lambert saw the cause of the gap between Mars and Jupiter in the large gravitational interaction between Jupiter and Saturn, which would have destabilized a planet that might have existed there earlier in its orbit.

In 1766, Johann Daniel Titius designed a formula with almost the same spacing series as David Gregory. Johann Elert Bode found it in a footnote in the widespread book Contemplation de la nature by Charles Bonnet - translated by Titius - and made it widely known in 1772 in his Guide to Knowledge of the Starry Sky . At first he did not mention Titius, but did so later.

In the formulation of Titius:

Prayer once upon the vastness of the planets respect for one another; and perceive that they are almost all distant from one another in proportion as their physical sizes increase. Prayer of the distance from the sun to a part of Saturn , Mercurius is such parts away from the sun, Venus , the earth , Mars . But see, from Mars to Jupiter there is a deviation from this so precise progression. From Mars follows a space of such parts, in which neither a principal nor a subsidiary planet is seen at the moment. And should the client have left this room empty? Never more! Let us confidently assume that this space belongs particularly to the previously undiscovered satellites of Mars; let us add that Jupiter may still have some around him who have not yet been seen with a glass. From this space, unknown to us, rises Jupiter's sphere of activity and Saturnus his, in such parts. What an admirable proportion!${\ displaystyle 100}$${\ displaystyle 4}$${\ displaystyle 4 + 3 = 7}$${\ displaystyle 4 + 6 = 10}$${\ displaystyle 4 + 12 = 16}$${\ displaystyle 4 + 24 = 28}$${\ displaystyle 4 + 48 = 52}$${\ displaystyle 4 + 96 = 100}$

The values ​​are not exactly the same as those of Wolff or Gregory (which in turn did not correspond exactly to the observational values ​​known at the time, published for example by William Whiston ), but as Titius wrote in the fourth edition of the translation, he first got them from Wolff.

The accidental discovery of Uranus in 1781 by Wilhelm Herschel , who initially thought it was a nebula or comet, confirmed this rule and made it appear as a law for all planets known at the time . Many astronomers were now looking for a planet in the gap between Mars and Jupiter, starting with Franz Xaver von Zach (from 1787), the court astronomer in Gotha. In 1788 six astronomers, including Zach and Heinrich Wilhelm Olbers , met in Lilienthal near Bremen, which provided the seeds for a Europe-wide network of observers to search for the missing planet. On the night of January 1, 1801, one of the members of this observation network, Giuseppe Piazzi , found a celestial body in Palermo that could be assigned to this distance. It was the asteroid Ceres , the first minor planet to be discovered and by far the largest of these bodies (also known as planetoids), which, together with the entire asteroid belt, closed this gap. Since August 2006, Ceres has the new status of a dwarf planet . For a long time Piazzi himself was in doubt whether it was a comet after all (with a parabola as the orbit as a first approximation). Carl Friedrich Gauß calculated the elliptical planetary orbit for Ceres in such a way that Zach was able to find it again at the end of 1801. On the one hand, this was a triumph for the young Gauss in the classical mathematical field of celestial mechanics, who was also just publishing his epoch-making number theory textbook Disquisitiones Arithmeticae , and at the same time meant that Ceres was not a comet. Incidentally, Gauss thought Titius-Bode's law was only an accidental coincidence. William Herschel found out that same year that Ceres was smaller than the known planets. In 1804 with Juno and 1807 with Vesta (by Olbers) further minor planets were found in the asteroid belt.

But when the planet Neptune was discovered in 1846, it didn't fit into the Titius-Bode series at all. The apparent failure of the law now led astronomers to be skeptical of such number games, for example in the analogy of Daniel Kirkwood (1849). Charles Sanders Peirce usually saw an example of flawed thinking in the sciences towards the end of the 19th century.

## Hegel

A widespread anecdote claims that Georg Wilhelm Friedrich Hegel claimed in his dissertation in 1801, using a geometric series that he proposed instead of the Titius-Bode series, to have proven that there was no planet between Mars and Jupiter; and this in the same year when Piazzi discovered Ceres and thus Hegel would have refuted. This was used long afterwards by astronomers and others to ridicule Hegel. However, Hegel was later taken under protection by other astronomers. In the short appendix to his dissertation, he did not claim to have shown that no planet existed in this gap (and Ceres later only turned out to be a minor planet with numerous other asteroids in this area), but only criticized the efforts of the astronomers at the time to search for a planet there on the basis of a purely speculative mathematical formula, the Titius-Bode formula. As a justification, he constructed his own series without a planet in this gap, which was based on a geometric series in Plato's dialogue Timaeus, as an example of how easily such hypotheses could be made.

Hegel started from the two sequences and to be found in Plato (i.e. geometric sequences and ), together with the sequence (which forms a certain conclusion, since it is the sum of the preceding numbers). Since 8 and 9 are close together, he replaces 8 with 16 (the next term in the first sequence of powers of two after 8) without a more detailed explanation, so that the sequence is created. The large distance between 4 and 9 is important to him, as it bridges the location of an unknown planet predicted in the Titius-Bode series. Then he replaces the sequence with (Mercury) and the remaining numbers with . From becomes and then for the remaining parts of the sequence: (for ), (for ), (for ), (for ) and (for ). Hegel himself gives the result: Mercury 1.4, Venus 2.56, Earth 4.37, Mars 6.34, Jupiter 18.75, Saturn 40.34, Uranus 81. So there is a series without a gap between Jupiter and mars. If you divide these values ​​by the distance from the earth, you get in astronomical units (AU): 0.32 (Mercury), 0.58 (Venus), 1 (Earth), 1.45 (Mars), 4.3 (Jupiter ), 9.2 (Saturn), 18.5 (Uranus). Then he also briefly investigates the relationships between the satellites of Jupiter and Saturn. ${\ displaystyle 1,2,4,8}$${\ displaystyle 1,3,9,27}$ ${\ displaystyle 2 ^ {n}}$${\ displaystyle 3 ^ {n}}$${\ displaystyle 1,2,3,4,8,9,27}$${\ displaystyle 27}$${\ displaystyle 1,2,3,4,9,16,27}$${\ displaystyle 1}$${\ displaystyle 3 ^ {\ frac {1} {3}} \ approx 1 {,} 44}$${\ displaystyle x}$${\ displaystyle x ^ {\ frac {4} {3}}}$${\ displaystyle 2}$${\ displaystyle 2 {,} 5}$${\ displaystyle 4 {,} 3}$${\ displaystyle 3}$${\ displaystyle 6 {,} 34}$${\ displaystyle 4}$${\ displaystyle 18 {,} 7}$${\ displaystyle 9}$${\ displaystyle 40 {,} 3}$${\ displaystyle 16}$${\ displaystyle 81}$${\ displaystyle 27}$

However, Hegel accepted the newly discovered minor planets in his lectures on natural philosophy and classified them under the planets.

## Interpretation and controversy

A widespread opinion is that the Titius-Bode series only fits the inner planets, already fails with the asteroid belt and has been considered an outdated number game at the latest since the discovery of the planet Neptune. So far, no physical mechanism is known that creates a certain series of distances between the planets.

The observation of the orbital times is more informative for the celestial mechanical organization of the planetary system. The orbital periods of the neighboring planets are in commensurability with one another ; that is, they have a relationship that is based on a common measure and can be expressed - partly approximately, partly fairly exactly - using fractions with small whole numbers in the numerator and denominator (see table on the right).

 Mercury 2: 5 (2: 5.11) Venus Venus 8:13 (8: 13.004) earth earth 1: 2 (1: 1.88) Mars Mars 2: 5 (2: 4.89) (Ceres) (Ceres) 2: 5 (2: 5.15) Jupiter Jupiter 2: 5 (2: 4.97) Saturn Saturn 1: 3 (1: 2.85) Uranus Uranus 1: 2 (1: 1.96) Neptune Neptune 2: 3 (2: 3.01) (Pluto)

Such resonances (Near Mean Motion Resonance, NMMR) can also be found when considering the orbital times of moons around planets. There are disruptive and stabilizing resonances depending on the ratio of the cycle times. Seen in this way, the success of the Titius-Bode series is based in general on the commensurable circulation ratios and in particular on the empirical bending of the uniform formula in order to capture all the different ratios as precisely as possible.

In newer applications, such as with the exoplanets, generalized Titius-Bode laws are used, for example by Timothy Bovaird of the form:

${\ displaystyle a_ {n} = a \ cdot C ^ {n}}$

with the semi-major axis for the n-th planet, and the parameters , whereby the semi-major axis of the first planet is adapted. As a partial explanation, it is stated that based on Kepler's third law ( ) there follows a similar dependence for the periods of revolution : ${\ displaystyle a_ {n}}$${\ displaystyle n = 0,1,2,3, ...}$${\ displaystyle a, C}$${\ displaystyle a}$${\ displaystyle n = 0}$${\ displaystyle T_ {n} \ sim a_ {n} ^ {\ frac {3} {2}}}$${\ displaystyle T_ {n}}$

${\ displaystyle T_ {n} = \ alpha \ cdot D ^ {n}}$

with and thus a ratio of the orbital times for neighboring planets: ${\ displaystyle D = C ^ {\ frac {3} {2}}}$

${\ displaystyle {\ frac {T_ {n + 1}} {T_ {n}}} = D}$

That would correspond to a system in which the values ​​of the resonance ratios correspond to a single value and a Titius-Bode law describes reality the better the less the values ​​scatter around a main value. In the solar system this is included . ${\ displaystyle {\ frac {5} {2}}}$

Simulations of the formation of planetary systems show the preference for certain resonance relationships such as and between the orbital times of neighboring planets, which are therefore particularly stable (Hills 1970). This is all the more evident, the greater the interaction between neighboring planets when they are formed. Jacques Laskar (2000) simulated a system of planetesimals and found that for radial initial surface densities of the form (with the radius) series of the Titius-Bode type resulted. This density distribution was also found in the minimum-mass solar nebula model (MMSN model) of the formation of the solar system (C. Hayashi 1981, SJ Weidenschilling 1977). ${\ displaystyle {\ frac {5} {2}}}$${\ displaystyle {\ frac {3} {2}}}$${\ displaystyle 10 ^ {5}}$${\ displaystyle \ Sigma (r) \ sim r ^ {- {\ frac {3} {2}}}}$${\ displaystyle r}$

However, statistical tests also showed that a simple formula can almost always be adapted to a hypothetical planetary system if one allows similar deviations as in the Titius-Bode sequence. These series are mostly different for each system. They result in number games that have not yet revealed any new celestial mechanical law.

Under the assumption that the Titius-Bode series is not a coincidence or just a statistical effect, hypotheses were made for the above-mentioned exceptions. Objects in the asteroid belt were thought to contain fragments of a former planet that went down in fantastic scientific literature under the name of Phaeton . Later investigations showed that the total mass of all asteroids is only about five percent of the mass of the Earth's moon and that many of the small bodies rather emerged from different, once larger asteroids. Today, the majority of the people are of the opinion that the asteroid belt arose naturally from the planetary nebula, but the formation of a larger planet was prevented by the gravitational effects of Jupiter. In the asteroid belt there are also various gaps ( Kirkwood gap ) in which the resonance relationships with Jupiter led to the destabilization. Another hypothesis is that a nearby massive object could have changed the orbits of Neptune and Pluto.

## Application of the Titius-Bode series to extrasolar planetary systems

When astronomers working with Tim Bovaird at the Australian National University in Canberra analyzed 27 extrasolar planetary systems, it was noticeable that these mostly follow the Titius-Bode formula more precisely than celestial bodies in our planetary system - almost 96%. Of 27 systems examined, 22 have the planets lined up according to the Titius-Bode rule. In three cases the Titius-Bode rule does not apply. The solar system is very extensive. In contrast, those 27 systems are much more compact. Sometimes four or five planets orbit the central star within Mercury's orbit.

Since the orbit duration and the maximum size of possible neighboring planets follow from the Titius-Bode series, the astronomers predicted the orbit of an unknown planet in the star system KOI 2722. Two months later, this exoplanet was found with the “Kepler” space telescope .

## literature

• Michael Martin Nieto: The Titius-Bode law of planetary distances: its history and theory , Oxford: Pergamon Press 1972
• Günther Wuchterl: The order of the planetary orbits , stars and space , part 1, issue 6, 2002, part 2, issue 12, 2002

## Individual evidence

1. Hoskin, Bode's law and the discovery of Ceres , Observatory Palermo. It refers to Newton's letter to the Bentley of December 2, 1692.
2. ^ Considerations on nature by Mr. Karl Bonnet , Leipzig 1774, volume 1, p. 9, footnote, digitized
3. Hoskin, Bode's law and the discovery of Ceres.
4. ^ Margaret Wertheim, Physics on the Fringe, Walker Books 2011
5. ^ Margaret Wertheim, Physics on the Fringe, 2011
6. E. Craig, M. Hoskin, Hegel and the seven planets, Journal of the History of Astronomy, Volume 23, 1992, p. XXIII, Online
7. Dieter B. Herrmann , Hegel's dissertation and the number seven of planets, stars and space. Controversies and legends about an alleged error. Stars and Space, Volume 31, 1992, pp. 688-691
8. ^ Dissertation by Hegel: De orbis planetarum, digitized version of the Bayerische Staatsbibliothek
9. See also Thomas Sören Hoffmann , Georg Wilhelm Friedrich Hegel. A Propaedeutic, Brill, 2015, pp. 103f
10. ^ Bertrand Beaumont, Hegel and the seven planets, in: Jon Stewart, The Hegel myths and legends, Northwestern University Press, 1996, pp. 285–288
11. Timothy Bovaird, Charles Lineweaver, Exoplanet predictions based on the generalized Titius – Bode relation, Monthly Notices Royal Astron. Soc., Vol. 435, 2013, pp. 1126-1139, Arxiv
12. ^ P. Goldreich, An explanation of the frequent occurrence of commensurable mean motions in the solar system, Monthly Notices Roy. Astron. Soc., Volume 1, 1965
13. ^ SF Dermott, On the origin of commensurabilities in the solar system, Monthly Notices Roy. Astron. Soc., Vol. 141, 1968, pp. 349, 363
14. JG Hills, Dynamical relaxations of planetary systems and Bode's law, Nature, Volume 225, 1970, p. 840
15. J. Laskar, On the spacing of planetary systems, Phys. Rev. Lett., Vol. 84, 2000, p. 3240
16. Hayashi, Progress of Theoretical Physics, Suppl, Volume 70, 1981, 35
17. Weidenschilling, Monthly Notices Roy. Astron. Soc., Vol. 180, 1977, p. 57
18. Timothy Bovaird, Charles Lineweaver, Exoplanet predictions based on the generalized Titius – Bode relation, Monthly Notices Royal Astron. Soc., Vol. 435, 2013, pp. 1126-1139, Arxiv
19. Guido Meyer: Planetary formula - crazy coincidence or natural law? (accessed on July 22, 2014)

## Remarks

1. Strictly speaking, in the case it is not a question of the value, but of the limit value of the sequence.${\ textstyle n \ to - \ infty}$