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The Titius–Bode law

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Solar System diagram showing planetary spacing in whole numbers, when the Sun-Neptune distance is normalized to 100. The numbers listed are distinct from the Bode sequence, but can give an appreciation for the harmonic resonances that are generated by the gravitational "pumping" action of the gas giants.
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 axesin a function of planetary sequence. The hypothesis correctly predicted the orbits of Ceres and Uranus, but failed as a predictor of Neptune's orbit. It is named forJohann Daniel Titius and Johann Elert Bode.

Formulation[edit]

The law relates the semi-major axis a 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... with the exception of the first step, each value is twice the previous value. There is another representation of the formula: a=1.5 \times 2^{(n-1)}+4 where n=-\infty,2,3,4.... The resulting values can be divided by 10 to convert them into astronomical units(AU), resulting in the expression
a=0.4+0.3 \times 2^m
for 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.

History[edit]

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 six."[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 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, aged only 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:[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.
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 neatly 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 Bode's law. Simultaneously, the large number of known asteroids in the belt resulted in Ceres no longer being considered a planet at that time. Bode's law was discussed as an example of fallacious reasoning by the astronomer and logician Charles Sanders Peirce in 1898.[3]
The discovery of Pluto 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, and in particular of the object Eris, which is larger than Pluto yet does not fit Bode's law, have further discredited the formula.[4]

Data[edit]

Here are the distances of planets in the Solar System, calculated from the rule and compared with the real ones:
Graphical plot using data from table to the left
PlanetkT-B rule distance (AU)Real distance (AU)% error (using real distance as the accepted value)
Mercury00.40.392.56%
Venus10.70.722.78%
Earth21.01.000.00%
Mars41.61.525.26%
Ceres182.82.771.08%
Jupiter165.25.200.00%
Saturn3210.09.544.82%
Uranus6419.619.22.08%
Neptune12838.830.0629.08%
Pluto225677.2239.4495.75%
1 Ceres was considered a small planet from 1801 until the 1860s. Pluto was considered a planet from 1930 to 2006. Both are now classified as dwarf planets.
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[edit]

There is no solid theoretical explanation of the Titius–Bode law, but if there is one it is possibly a combination of orbital resonance and shortage ofdegrees of freedom: any stable planetary system has a high probability of satisfying a Titius–Bode-type relationship. Since it may simply be a mathematical coincidence rather than a "law of nature", it is sometimes referred to as a rule instead of "law".[5] However, astrophysicist Alan Bossstates that it is just a coincidence, and the planetary science journal Icarus no longer accepts papers attempting to provide improved versions of the law.[4]
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[6][7] 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[edit]

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 and the biggest inner satellite, Amalthea, 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.[8] 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."[9] 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 planetary systems. An attempt with 55 Cancri suggested the equation a = 0.0142 e 0.9975 n, and predicts for n = 5 an undiscovered planet or asteroid field at 2 AU.[10] This is controversial.[11] 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.[12]
Recent astronomical research suggests that planetary systems around some other stars may fit Titius–Bode-like laws.[13][14] Bovaird and Lineweaver[15] applied a generalized Titius-Bode relation to 68 exoplanet systems which 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 managed to detect 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 were accounted to planet not transiting the star or the predicted space being occupied by circumstellar disks. Despite this, the number of planets found with Titius-Bode law predictions were still lower than expected.[16]

Plants and Planets

The Law of Titius-Bode explained

H.J.R. Perdijk

table-5.jpg

Plants and Planets

The Law of Titius-Bode explained

H.J.R. Perdijk

 
INTRODUCTION
This book developed unintentionally while I was trying to find out what would be the result if fertile soil was made unfertile without radical interventions.
I had figured it as follows: first I would carefully work with a rotary cultivator which, I supposed, would give rise to ruderal plants mainly. I also assumed some trees would start to grow as meadows are often fringed with trees. These trees would be cut time and again and the mishmash of ruderal plants would gradually disappear as the soil was calming down. Then flowering plants would have the chance of growing and the goal would be reached in, say, 10 years. One book even mentioned that it would be possible to have quite a nice flower meadow in 5 years. But none of this happened..
The way I had pictured it did not correspond at all with reality and this reinforced my interest in the behaviour of plants. It will be essential for a follow-up investigation to mow with a scythe and locally with a sickle. Mechanized methods yield less plant species on account of which figure 3 still contains a few traces of "noise" instead of waves. The system of plant communities was used to make order out of the chaos that occurred each year. Completely unexpectedly and surprisingly the investigation revealed an influence of the sun wind on plant species, next a connection with the revolution of the earth and finally an indication of the origin of the planetary system. The Law of Titius-Bode comes to the fore. This all relies on the correctness of figure 3 , which can only be achieved when working with the utmost accuracy.
The above experiment should be repeated as the result was found more or less accidentally. It is expected that the sun wind will exert more influences on other forms of life on earth. A continuation of this inquiry is therefore deemed important, especially since the lifetime of people usally is 84 years.
The greater part of this report is available in Dutch (Planten en Planeten) by Profiel at Bedum ( click here )
The developments occurring in the meadow are explained more elaborately in the book: "Plantengemeenschappen en zonnevlekken"(Plant communities and sun spots), which was published by Profiel at Bedum, the Netherlands, in 1994.
H.J.R. Perdijk, August 1999

PLANTS AND PLANETS
June 4, 1998 ; Completed August 18, 2003
H.J.R. Perdijk
The attempt was to make a highly fertile meadow more unfertile and to follow the annual changes in the vegetation occurring in the process. It proved difficult to do this well on such a large surface (1800 square metres). The various species of plants are subdivided into three groups with the help of the categorization used to describe plant communities: ruderal species, meadow and wood species. The behaviour of these groups gives a good insight into the recovery of the soil after rotary cultivation. However, it was also found that the three groups had varying numbers of varieties. This seems to be occasioned indirectly by the sun wind who modulates cosmic radiation. Finally a connection is demonstrated between the revolution of the earth and the three groups: ruderal, meadow and wood species. This in turn yields a work model for further investigations into the planetary system.
The soil
The soil (1800 square metres) of the meadow consists of loamy sand ( figure 1 ). The meadow is located at the rear of a former field as a result of which the soil bulges. This neglected, wet meadow was superficially milled in 1982 with a hand cultivator. The differences in altitude were intensified by creating a bank of the fertile soil adjacent to it. This soil was cut off superficially leaving a thinner layer of fertile soil. A "marsh" was also dug. In addition, a "mountain" was formed using the excavated soil. The mountain was then covered with fertile soil from the gullies. Drainage is such that groundwater - and rainwater too- will never get too high.
Due to the bulging character of the meadow there is not much difference in altitude. However, there is a substantial difference in altitude between mountain and marsh bottom. Currently, the entire meadow has a highly fertile milled soil as upper layer varying in altitude and thickness.
Groundwater
In wintertime, the groundwater is at a high level but it is 80 cms lower in summer. During the summer, the meadow is kept moist to fairly wet through capillary action. During the winter the groundwater levels out fertility.
Lime
To protect the soil from acid rain it was scattered with lime each year (150 à 200 kgs Ca+Mg). Lime also generates calcium phosphate causing the soil to become impoverished. As the meadow was cut twice yearly with scythe and sickle thereby removing the plants, the soil had impoverished even more. The easily dissociating phosphates in the soil reduce the dissociation of the ill dissolving calcium phosphate. This is why calcium is of hardly any contribution to the available food for plants. This situation continues until the well dissolvable phosphates become exhausted.
Conditions constant
The food remains constant for many years while the weather conditions are being neutralized through the differences in altitude and the density of the fertile layer. In actual fact nothing about the conditions is changed. This justifies the expectation that after the soil has set, the same plant species will be found each year although on varying places depending on the weather conditions.
The remarkable thing is that it is sometimes hard to trace a large number of plant species whereas in other years they are found quite easily. After 1984, 80% of the species annually returns in varying quantities (masses), the remaining 20% vary in type. This accounts for the fluctuation in figure 3 as was found later on.

Diagrams
In connection with this two diagrams are significant, the figure 2 and 3. In figure 2 an estimate is given of the massive presence or otherwise of some plants. This could only be done through estimation, counting was impossible in view of the large surface.
As already known, pioneer vegetation is the first to appear after working with a rotary cultivator. The following year ruderal perennials appear, next wood formation tends to occur. This was also found in our case, yet we had expected a more gradual course than is shown in figure 2 . This fluctuation (in a better way, using counts) is described in the book Wilde Planten  (1)  ("Wild Plants") in the chapter on succession. The only thing that can be said, however, is that a repeating pattern was found in which other plants appear over and over again.
The method of making estimates of frequently occurring plants proved unsuitable to keep up with the developments in vegetation over a long time. The minutes state that in spring 1986 the meadow was first full with flowers but also that an increasing number of alders was found in the course of the same year. It was the first time that the meadow was remarkably in bloom and that trees were growing. The notes say: the meadow gives the impression that plants will split up into groups, groups that will develop themselves in different ways. This, however, is not so clear from figure 2 , yet it induced us to look into the groups of plant species.

Plant communities
We had to look for a different method that would include all plant species in the meadow and would be of long-term use. This method was found in the classification used to describe plant communities (seeWesthoff and Den Held  (2) ). A plant community is a group of plants that remarkably often are found close to each other. However, they usually do not belong to the same family. It is rather something of a commune. As for these communities, further distinctions are made such as Formation, class, order, union, association and sometimes sub-association. But these do not apply in our case. What we needed was an overview of the entire meadow, a distinction in large-scale groups. We will refer to these as ruderal, meadow and wood (see table 1 ), sometimes indicated by the abbrevations : r, m, and w.
To form large groups like these we will depart from the class a plant belongs to. When the plants have been grouped in classes, the groups ruderal, meadow and wood are found through counting. This, however, is quite a complex procedure so we will continue first with an illustration of it.
The flora of Heukels-van Ooststroom (3) (elaborately dealt with in "Plantengemeenschappen in Nederland") first gives a description of a plant and then continues with the class it belongs to together with any subdivisions. For the purpose of our inquiry, however, we only need the first numbers, the classes.
The large stinging-nettle is recorded as: P17; 17Acl; 38Aa, this means the plant is found in classes 17 and 38. It belongs to class 17 but the class printed in italics, class 38, indicates that the plant plays a special role in class 38. For the purpose of our study we will consider the connection with class 38 as a "weak connection". However, this connection will become a strong one when it appears a large number of plants has a weak connection with class 38. This is frequently found. With the help of a few plants the arithmetic method of figure 3 is clarified (see also table 1):

Examples of  ‘species found’classes
RuderalMeadowWood
couch grass121621
tansy1217
large stinging-nettle1738
raspberry34
cow parsley253538
knapweed25
number of classes21212112
sum 12=100%534
as a percentage41.625.033.4
 

In figure 3 ), these percentages could yield points that would practically correspond to 1984 or 1985. In actual fact the computation over 1984 yielded 38 species and over 1985 even 62. This number usually fluctuates between 52 and 68.
As can be deduced from this example with 6 plant species, the amount of data increases from 6 to 12 when introducing classes. In practice the number of data usually increases by a factor of 1.5.
The example also indicates that it is not always necessary to find a large number of plants in the meadow. Sometimes that's just all there is. A limited number of plant species particularly occurs after working with a rotary cultivator. In 1982, only 9 plant species were found. Still, these 9 species showed that due to their weak connection with the wood (classes printed in italics) that a wood might eventually develop (figure 3).
The method proved highly suitable to observe the soil returning to a state of balance after working with the rotary cultivator: the dotted lines in figure 3 . As can be seen, this takes quite some time even after careful working with the rotary cultivator.
If after some time 60 plant species are found and classes are introduced, the plants will be strongly interrelated in the groups ruderal, wood and meadow. Groups of entwined species are at issue here. All class characteristics of plants are registered and the groups are balanced against each other. In the end the weighing will determine the place of the points in figure 3 . The group which is represented most by the classes, will be located at the highest point in this figure.
It becomes plausible from table 2 that when finding a large number of plant species, the groups of ruderal, wood and meadow will yield the same number of classes, each 33.3%. This may deviate in case of domination of one of the three groups. As the measurements show, this is usually the case. One wonders where the energy causing these fluctuations, originates.
Connections between the groups
Ruderal plants and meadow plants too are quite strongly related with the wood beause of the relatively large numbers of weak connections between them. However, there is only a slight connection between ruderal and meadow plants as they require conflicting types of soil. Ruderal plants grow steadily in places where the soil has been disturbed whereas meadow plants need soil that has settled down. If peace and quiet in the meadow are disturbed by a mole passing through, yarrow is likely to appear on the molehill (13 Bal; 20Bc4; 34A). Actually, this plant has no special connection to whatsoever. Common grass, too, (18 Aa; 20Ba4) has a weak connection to ruderal and meadow plants. Disturbance by animals is the likely cause for the occurrence of such species. Ruderal plants and meadow plants do not get along, however, they are both the precursors of the wood.

The hexagon
Above, a few plant species came up, but when we start talking in terms of classes, as in figure 3 , we lose track of them. With figure 4 , however, we can subdivide plants into groups that can be mentioned separately so we can follow what is happening to the plants themselves. The list of plants that were found in 9 years has also been classified that way.

Comparision between figures 2 and 3
In figure 3 wavy lines connect the corresponding points to each other with the axis drawn in stripes. Curves above the axis refer to dominating groups. Some of these curves are specified in thicker lines. If we compare these with the peaks in figure 2 , they appear to correspond with the abundantly growing plants mentioned there. This leads us to the following conclusion:
If a certain group (ruderal, wood or meadow) is represented by relatively many species, some of those species will be found in large numbers.
It also becomes clear from this, however, that the seemingly ongoing line in figure 2 cannot be considered a whole. It consists of line fragments that correspond to the plant groups (ruderal, wood or meadow) that alternately come to the fore.

Recovery of the disturbance
The curved axes in figure 3 indicate the recovery of the soil after working with the rotary cultivator in 1982. Up to 1985, the problems caused by the disturbance are noticeable but then practically disappear. For after 1985, the 33.3-axis is almost reached and each year some 60 plant species are found. Yet it will take at least 10 years for the recovery to be complete.
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References[edit]

  1. ^ Jump up to:a b c "Dawn: A Journey to the Beginning of the Solar System". Space Physics Center: UCLA. 2005. Retrieved 2007-11-03.
  2. Jump up^ Hoskin, Michael (1992-06-26). "Bodes' Law and the Discovery of Ceres". Observatorio Astronomico di Palermo "Giuseppe S. Vaiana". Retrieved 2007-07-05.
  3. Jump up^ Pages 194-196 in
  4. ^ Jump up to:a b Alan Boss (October 2006). "Ask Astro". Astronomy 30 (10): 70.
  5. Jump up^ Carroll, Bradley W.; Ostlie, Dale A. (2007). An Introduction to Modern Astrophysics (Second ed.). Addison-Wesley. pp. 716–717. ISBN 0-8053-0402-9.
  6. Jump up^ 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.
  7. Jump up^ B. Dubrulle, F. Graner (1994). "Titius–Bode laws in the solar system. Part II: Build your own law from disk models". Astronomy and Astrophysics282: 269–276. Bibcode:1994A&A...282..269D.
  8. Jump up^ Cohen, Howard L. "The Titius-Bode Relation Revisited". Retrieved 2008-02-24.[dead link]
  9. Jump up^ Harwit, Martin. Astrophysical Concepts (Springer 1998), pages 27-29.
  10. Jump up^ Arcadio Poveda and 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.
  11. Jump up^ Ivan Kotliarov (21 June 2008). "The Titius-Bode Law Revisited But Not Revived". arXiv:0806.3532 [physics.space-ph].
  12. Jump up^ 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:1005.4050.Bibcode:2010ApJ...722..937D. doi:10.1088/0004-637X/722/1/937.
  13. Jump up^ "The HARPS search for southern extra-solar planets". 2010-08-23. Retrieved 2010-08-24. Section 8.2: "Extrasolar Titius-Bode-like laws?"
  14. Jump up^ 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
  15. Jump up^ Timothy Bovaird, Charles H. Lineweaver (2013). "Title: Exoplanet predictions based on the generalized Titius–Bode relation". Monthly Notices of the Royal Astronomical Society. arXiv:1304.3341.Bibcode:2013MNRAS.tmp.2080B. doi:10.1093/mnras/stt1357.
  16. Jump up^ http://arxiv.org/abs/1405.2259

See also[edit]

Further reading[edit]

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