Titius-Bode law
Overview
 
The Titius–Bode law is a hypothesis that the bodies in some orbital systems, including the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

's, orbit at semi-major axes
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

 in a function of planetary sequence. The hypothesis correctly predicted the orbits of Ceres and Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

, but failed as a predictor of Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

's orbit.
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:
a = 4 + n


where n = 0, 3, 6, 12, 24, 48 …, each value of n > 3 twice the previous value.
Discussions
Encyclopedia
The Titius–Bode law is a hypothesis that the bodies in some orbital systems, including the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

's, orbit at semi-major axes
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

 in a function of planetary sequence. The hypothesis correctly predicted the orbits of Ceres and Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

, but failed as a predictor of Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

's orbit.

Formulation

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:
a = 4 + n


where n = 0, 3, 6, 12, 24, 48 …, each value of n > 3 twice the previous value. The resulting values can be divided by 10 to convert them into astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

s (AU), which would result in the expression
a = 0.4 + 0.3 • 2m

for m = -∞, 0, 1, 2 …

For the outer planets, each planet is predicted to be roughly twice as far from the Sun as the previous object.

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 six." A similar sentence, likely paraphrased from Gregory, appears in a work published by Christian Wolff
Christian Wolff (philosopher)
Christian Wolff was a German philosopher.He was the most eminent German philosopher between Leibniz and Kant...

 in 1724.

In 1764, Charles Bonnet
Charles Bonnet
Charles Bonnet , Swiss naturalist and philosophical writer, was born at Geneva, of a French family driven into Switzerland by the religious persecution in the 16th century.-Life and work:Bonnet's life was uneventful...

 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." To this, in his 1766 translation of Bonnet's work, Johann Daniel Titius
Johann Daniel Titius
Johann Daniel Titius was a German astronomer and a professor at Wittenberg.Titius was born in Konitz , Royal Prussia, and died in Wittenberg...

 added the following unattributed addition, removed to a footnote in later editions:

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
Johann Elert Bode
Johann Elert Bode was a German astronomer known for his reformulation and popularization of the Titius-Bode law. Bode determined the orbit of Uranus and suggested the planet's name.-Biography:...

, aged only twenty-five, completed the second edition of his astronomical compendium Anleitung zur Kenntniss des gestirnten Himmels, into which he added the following footnote, initially unsourced, but credited to Titius in later versions:

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.


When originally published, the law was approximately satisfied by all the known planets — Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

 through Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.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
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

 in 1781 which happens to fit neatly into the series. Based on this discovery, Bode urged a search for a fifth planet. , the largest object in the asteroid belt
Asteroid belt
The asteroid belt is the region of the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets...

, was found at Bode's predicted position in 1801. Bode's law was then widely accepted until Neptune was discovered
Discovery of Neptune
Neptune was mathematically predicted before it was directly observed. With a prediction by Urbain Le Verrier, telescopic observations confirming the existence of a major planet were made on the night of September 23, 1846, and into the early morning of the 24th, at the Berlin Observatory, by...

 in 1846 and found not to satisfy Bode's law. Simultaneously, the large number of known asteroids in the belt resulted in Ceres no longer being considered a planet. (It is now understood that no planet could have formed in the belt, due to the gravitational influence of Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

.) Bode's law was discussed as an example of fallacious reasoning by the astronomer and logician Charles Sanders Peirce in 1898.

The discovery of Pluto
Pluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

 in 1930 confounded the issue still further. While 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
Kuiper belt
The Kuiper belt , sometimes called the Edgeworth–Kuiper belt, is a region of the Solar System beyond the planets extending from the orbit of Neptune to approximately 50 AU from the Sun. It is similar to the asteroid belt, although it is far larger—20 times as wide and 20 to 200 times as massive...

, and in particular of the object , which is larger than Pluto yet does not fit Bode's law, have further discredited the formula.

Data

Here are the distances of planets in our solar system, calculated from the rule and compared with the real ones:

Planet k T-B rule distance (AU) Real distance (AU) % error (using real distance as the accepted value)
Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

0 0.4 0.39 2.56 %
Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...

1 0.7 0.72 2.78 %
Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

2 1.0 1.00 0.00 %
Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...

4 1.6 1.52 5.26 %
1 8 2.8 2.77 1.08 %
Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

16 5.2 5.20 0.00 %
Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

32 10.0 9.54 4.82 %
Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

64 19.6 19.2 2.08 %
Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

128 38.8 30.06 29.08 %
Pluto
Pluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

1
256 77.22 39.44 95.75 %



1 Ceres was considered a planet from 1801 until the 1860s. Pluto was considered a planet from 1930 to 2006. Both are now classified as dwarf planet
Dwarf planet
A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...

s.

2 While the difference between the T-B rule distance and real distance seems very large here, if Neptune is 'skipped,' the T-B rule's distance of 38.8 is quite close to Pluto's real distance with an error of only 1.62%.

Theoretical explanations

There is no solid theoretical explanation of the Titius–Bode law, but if there is one it is possibly a combination of orbital resonance
Orbital resonance
In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of...

 and shortage of degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

: any stable planetary system has a high probability of satisfying a Titius–Bode-type relationship. Because of this, it has been called a "rule" rather than a "law". However, astrophysicist
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...

 Alan Boss
Alan Boss
Alan P. Boss is a United States astrophysicist and NASA scientist.-Life and career:Educated at the University of South Florida and the University of California, Santa Barbara, Boss is a prominent scientist in stellar and planetary system formation and the study of extrasolar planets who has made...

 states that it is just a coincidence, and the planetary science
Planetary science
Planetary science is the scientific study of planets , moons, and planetary systems, in particular those of the Solar System and the processes that form them. It studies objects ranging in size from micrometeoroids to gas giants, aiming to determine their composition, dynamics, formation,...

 journal Icarus
Icarus (journal)
Icarus is a premier scientific journal dedicated to the field of planetary science. It is published under the auspices of the American Astronomical Society's Division for Planetary Sciences . The longtime publisher was Academic Press, which is now part of Elsevier...

no longer accepts papers attempting to provide improved versions of the law.

Orbital resonance from major orbiting bodies creates regions around the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

 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 have shown 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 being a feature of many phenomena considered to play a role in planetary formation, such as turbulence.

Lunar systems and other planetary systems

There is a decidedly limited number of systems on which Bode's law can presently be tested. Two of the solar planets have a number of big moons that appear possibly to have been created by a process similar to that which created the planets themselves. The four big satellites of Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

 and the biggest inner satellite, Amalthea
Amalthea (moon)
Amalthea is the third moon of Jupiter in order of distance from the planet. It was discovered on September 9, 1892, by Edward Emerson Barnard and named after Amalthea, a nymph in Greek mythology. It is also known as '....

, cling to a regular, but non-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, but non-Bode, spacing. However, according to Martin Harwit
Martin Harwit
Martin Harwit is a Czech-American astronomer, author, and was director of the National Air and Space Museum in Washington, D.C. from 1987 to 1995...

, "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 to other solar systems. An attempt with 55 Cancri
55 Cancri
55 Cancri , also cataloged Rho1 Cancri or abbreviated 55 Cnc, is a binary star approximately 41 light-years away from Earth in the constellation of Cancer...

 suggested the equation a = 0.0142 e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

0.9975 n, and predicts for n = 5 an undiscovered planet or asteroid field at 2 AU. This is controversial. 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.

Recent astronomical research suggests that planetary systems around some other stars may fit Titius-Bode-like laws. Kepler-11
Kepler-11
Kepler-11 is a sun-like star slightly larger than the Sun in the constellation Cygnus, located some 2,000 light years from Earth. It is located within the field of vision of the Kepler spacecraft, the satellite that NASA's Kepler Mission uses to detect planets that may be transiting their stars...

looks to be an exception, as its planets don't look to follow any sort of Bode's Law.
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