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Titius–Bode law: Difference between revisions

Revision as of 09:42, 19 April 2021

Later work by Blagg and Richardson significantly corrected 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".^[1]

Formulation

The law relates the semi-major axis ${\displaystyle ~a_{n}~}$ 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 ${\displaystyle ~x=0,3,6,12,24,48,96,192,384,768\ldots ~}$ such that, with the exception of the first step, each value is twice the previous value. There is another representation of the formula:

${\displaystyle ~a=4+3\times 2^{n}~}$

where ${\displaystyle ~n=-\infty ,0,1,2,\ldots ~.}$ The resulting values can be divided by 10 to convert them into astronomical units (AU), resulting in the expression:

${\displaystyle a=0.4+0.3\times 2^{n}~.}$

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, but the actual values are closer to 10, 19, 30, 39 AU.^[a]

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."^[2]

A similar sentence, likely paraphrased from Gregory,^[2] appears in a work published by Christian Wolff in 1724.

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."^[2]

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):^[3]

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.^[b] Many precedents were found that predate the seventeenth century.^{[citation needed]} 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 David 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. (Over the next two centuries, subsequent authors would continue to present their own modifications, often unaware of previous attempts.^[1])

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^[4] 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. 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.^[5]

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.^[6]

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).^[7] 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 David 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 [(D_n × 10) − 4]/[(D_n−1 × 10) − 4] = 2, from n = 2 to n = 8, can be expressed. And using the circular uniform fictitious movement to Kepler's Anomaly, R_n values corresponding to each planet's ratios may be obtained as r_n = (R_n − R₁)/(R_n−1 − R₁), resulting in 1.82; 1.84; 1.86; 1.88 and 1.90, in which r_n = 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).

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:

m	k	T–B rule distance (AU)	Planet	Semimajor axis (AU)	Deviation from prediction1
${\displaystyle -\infty }$	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	Ceres2	2.77	−1.16%
4	16	5.2	Jupiter	5.20	+0.05%
5	32	10.0	Saturn	9.55	−4.45%
6	64	19.6	Uranus	19.22	−1.95%
–	–	–	Neptune	30.11	–
7	128	38.8	Pluto2	39.54	+1.02%
8	256	77.2	Eris2	67.78	−12.9%
9	512	154.0	– 3	–	–
10	1024	307.6	– 3	–	–
11	2048	614.8	Sedna2	506.24	−17.66%
			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.

³ No obvious known bodies have been found close to these distances.

⁴ Note that the exceptional difference in aphelion and perihelion gives rise to a large semi-major axis, which is not reflected by the object's current distance from the Earth.

Blagg Formulation

In 1913, Mary Blagg, an Oxford astronomer, re-visited the law.^[8] 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:

where:

Note that in her formulation, the Law for the solar system was best represented by a progression in 1.7275, not 2.

Blagg examined the satellite systems of Jupiter, Saturn, and Uranus, and discovered the same progression ratio (1.7275) in each.

Constants of the Blagg formulation of the Titius-Bode Law

System	A	B	${\displaystyle \alpha }$	${\displaystyle \beta }$
Planets	0.4162	2.025	112.4°	56.6°
Jupiter	0.4523	1.852	113.0°	36.0°
Saturn	3.074	0.0071	118.0°	10.0°
Uranus	2.98	0.0805	125.7°	12.5°

Her paper appeared in the Monthly Notices of the Royal Astronomical Society for 1913, and was forgotten until 1953, when A. E. Roy at Glasgow University Observatory came across it while researching another problem.^[9] He 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, Jupiter IX, X, XI, XII, and Uranus V.

Roy found that all six fitted very closely. In addition, another of Blagg's predictions was confirmed: that some bodies were clustered at particular distances.

Her formula also predicted that if a transplutonian planet existed, it would be at ~68 AU from the Sun.

Richardson Formulation

In 1945 D. E. Richardson^[10] independently arrived at the same conclusion as Blagg, that the progression ratio was not 2, but 1.728:

where ${\displaystyle \varrho _{n}}$ is an oscillatory function of ${\displaystyle 2\pi }$, represented by distances ${\displaystyle \varrho _{n}}$ from an off-centered origin to angularly varying points on a "distribution ellipse".

Historical Inertia

Nieto, who conducted the first modern comprehensive review of the Titius-Bode Law^[11] 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.^[1]

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."^[12] 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."^[6] On the other hand, a growing amount of data from exoplanetary systems points to a generalized fulfillment of this rule in other planetary systems ^{[citation needed]}.

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.^[13]

Dubrulle and Graner^[14][15] 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, non-Titius–Bode spacing.^[16] 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."^[17]

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 ${\displaystyle ~a_{n}=0.0142\cdot \mathrm {e} ^{\left(\,0.9975\,n\,\right)}=0.0142\cdot {\bigl (}\,2.7115\,{\bigr )}^{n}~,}$ and controversially^[18] predicts for ${\displaystyle ~n=5~}$ an undiscovered planet or asteroid field at 2 AU.^[19] 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.^[20]

Recent astronomical research suggests that planetary systems around some other stars may follow Titius–Bode-like laws.^[21][22] Bovaird and Lineweaver^[23] 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.^[24]

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.^[25]

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%)^[26] is obtained.^[27]

See also

• Dermott's law
• Phaeton (hypothetical planet)
• Logarithmic spiral
• Lennard-Jones potential

Footnotes

1. ^ The spacing seems to transition from the complicated Titius-series to simple equal-spacing starting at Saturn, with Neptune being the first major planet that does not fit the Titius-Bode rule.

^ 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 ${\displaystyle x}$. 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.^{[citation needed]}

References

Further reading

• 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