Misplaced Pages

#290709

45-422: Later work by Mary Adela Blagg and D.E. Richardson significantly revised 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". The law relates the semi-major axis  

90-508: A n   {\displaystyle ~a_{n}~} of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10: where   x = 0 , 3 , 6 , 12 , 24 , 48 , 96 , 192 , 384 , 768 …   {\displaystyle ~x=0,3,6,12,24,48,96,192,384,768\ldots ~} such that, with

135-611: A 1945 Popular Astronomy magazine article, the science writer D.E. Richardson apparently independently arrived at the same conclusion as Blagg: That the progression ratio is 1.728 rather than 2 . His spacing law is in the form:   R n = (   1.728   ) n   ϱ n ( θ n )   , {\displaystyle \ R_{n}={\bigl (}\ 1.728\ {\bigr )}^{n}\ \varrho _{n}(\theta _{n})\ ,} where ϱ n {\displaystyle \varrho _{n}}

180-425: 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". 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". Orbital resonance from major orbiting bodies creates regions around

225-536: A recluse", and rarely attended meetings. She died from heart disease on 14 April 1944 at her home in Cheadle. The crater Blagg on the Moon is named after her. In March 2023, minor planet 2000 EO 177 was also named 50753 Maryblagg in her honour. David Gregory (mathematician) David Gregory (originally spelt Gregorie ) FRS (3 June 1659 – 10 October 1708) was a Scottish mathematician and astronomer. He

270-463: A second edition of Newton's Principia . Gregory made notes of these discussions, but the second edition of 1713 was not due to Gregory. In 1695 he published Catoptricae et dioptricae sphaericae elementa which addressed chromatic aberration and the possibility of its correction with achromatic lens . In 1705 Gregory became an Honorary Fellow of the Royal College of Physicians of Edinburgh . At

315-684: A series of ten articles in the Monthly Notices , in which Turner acknowledged that a large majority of the work had been performed by Blagg. On 28 March 1906, Blagg was elected to the British Astronomical Association at the proposal of Hardcastle. After the publication of several research papers for the Royal Astronomical Society , she was elected as a fellow in January 1916, after being nominated by Professor Turner. She

360-509: A textbook by D. Gregory (1715): A similar sentence, likely paraphrased from Gregory (1715), appears in a work published by C. Wolff in 1724. In 1764, C. Bonnet wrote: In his 1766 translation of Bonnet's work, J.D. Titius added two of his own paragraphs to the statement above. The insertions were placed at the bottom of page 7 and at the top of page 8. The new paragraph is not in Bonnet's original French text, nor in translations of

405-580: Is an oscillatory function with period 2 π {\displaystyle 2\pi } , representing distances ϱ n {\displaystyle \varrho _{n}} from an off-centered origin to points on an ellipse. Nieto, who conducted the first modern comprehensive review of the Titius–Bode Law, noted that "The psychological hold of the Law on astronomy has been such that people have always tended to regard its original form as

450-441: 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. Mary Adela Blagg Mary Adela Blagg FRAS (17 May 1858 – 14 April 1944) was an English astronomer and was elected a fellow of the Royal Astronomical Society in 1916. She is noted for her work on selenography and variable stars . Blagg

495-455: The Girls' Friendly Society . By middle age, she became interested in astronomy after attending a university extension course taught by Joseph Hardcastle, John Herschel 's grandson. Her tutor suggested working in the area of selenography , particularly on the problem of developing a uniform system of lunar nomenclature . (Several major lunar maps of the period had discrepancies in terms of naming

SECTION 10

#1732880765291

540-524: 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. The Titius–Bode law predicts planets will be present at specific distances in astronomical units , which can be compared to the observed data for the planets and two dwarf planets in the Solar System: In 1913, M.A. Blagg , an Oxford astronomer, re-visited

585-512: 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.,

630-479: 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. 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

675-488: The Moon was subsequently named after him.) Together, they produced a two-volume set in 1935, titled Named Lunar Formations , that became the standard reference on the subject. During her life, Blagg performed volunteer work, including caring for Belgian refugee children during World War I . One of her favorite hobbies was chess . She was described in her obituary as being of "modest and retiring disposition, in fact very much of

720-581: The Solar System does. The locations of potentially undetected exoplanets are predicted in each system. Subsequent research detected 5 candidate planets 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 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

765-406: 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

810-601: 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 but non-Titius-Bode spacing. However, according to Martin Harwit 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

855-827: The calculations below: The calculations here are based on a graph of function   f   which was drawn based on observed data. Her paper was published in 1913, and was forgotten until 1953, when A.E. Roy came across it while researching another problem. Roy 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 , Sinope ( Jupiter IX ), Lysithea ( J X ), Carme ( J XI ), Ananke ( J XII ), and Miranda ( Uranus V ). Roy found that all six fitted very closely. This might have been an exaggeration: out of these six bodies, four were sharing positions with objects that were already known in 1913; concerning

900-425: 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. 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

945-573: The continent, including the Netherlands (where he began studying medicine at Leiden University ) and France, and did not return to Scotland until 1683. On 28 November 1683, Gregory graduated M.A. at University of Edinburgh , and in October 1683 he became Chair of Mathematics at University of Edinburgh. He was "the first to openly teach the doctrines of the Principia , in a public seminary...in those days this

SECTION 20

#1732880765291

990-428: The discovery of Uranus in 1781, which happens to fit into the series nearly exactly. 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 widely accepted at that point, until in 1846 Neptune was discovered in a location that does not conform to the law. Simultaneously, due to

1035-434: The equation and controversially predicts an undiscovered planet or asteroid field for   n = 5   {\displaystyle ~n=5~} at 2  AU . 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

1080-410: The exception of the first step, each value is twice the previous value. There is another representation of the formula: where   n = − ∞ , 0 , 1 , 2 , …   . {\displaystyle ~n=-\infty ,0,1,2,\ldots ~.} The resulting values can be divided by 10 to convert them into astronomical units ( AU ), resulting in

1125-402: The expression: 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 , the actual values are closer to 10, 19, 30, 40  AU . The first mention of a series approximating Bode's law is found in

1170-1435: The following simpler formula; however the price for the simpler form is that it produces a less accurate fit to the empirical data. Blagg gave it in an un-normalized form in her paper, which leaves the relative sizes of A , B , and f   ambiguous; it is shown here in normalized form (i.e. this version of   f   is scaled to produce values ranging from 0 to 1 , inclusive):   f (   θ   ) = 0.249 + 0.860   (   cos ⁡   Ψ     3 − cos (   2   Ψ   )   + 1   6 − 4   cos (   2   Ψ − 60 ∘ )   )   , {\displaystyle \ f{\bigl (}\ \theta \ {\bigr )}\;=\;0.249\;+\;0.860\ \left({\frac {\ \cos \ \Psi \ }{\ 3-\cos \!\left(\ 2\ \Psi \ \right)\ }}\;+\;{\frac {1}{\ 6-4\ \cos \!\left(\ 2\ \Psi -60^{\circ }\right)\ }}\right)\ ,} where   Ψ ≡ θ − 27.5 ∘   . {\displaystyle \ \Psi \equiv \theta -27.5^{\circ }~.} Neither of these formulas for function   f   are used in

1215-441: The large number of asteroids discovered in the belt , Ceres was no longer a major planet. In 1898 the astronomer and logician C.S. Peirce used Bode's law 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 very nearly at the position the law had designated for Neptune. The subsequent discovery of

1260-441: 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 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

1305-432: 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,

1350-463: The law. 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: Note in particular that in Blagg's formula, the law for the Solar System was best represented by a progression in 1.7275 , rather than

1395-461: 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

Titius–Bode law - Misplaced Pages Continue

1440-535: The orbits, seem to stem from an antique algorithm by a cossist . Many precedents were found that predate the seventeenth century. Titius was a disciple of the German philosopher C.F. von Wolf (1679–1754), and the second part of the text that Titius inserted into Bonnet's work is in a book by von Wolf (1723), suggesting that Titius learned the relation from him. Twentieth-century literature about Titius–Bode law attributes authorship to von Wolf. A prior version

1485-399: The original value 2 used by Titius, Bode, and others. Blagg examined the satellite system of Jupiter , Saturn , and Uranus , and discovered the same progression ratio 1.7275 , in each. However, the final form of the correction function   f   was not given in Blagg's 1913 paper, with Blagg noting that the empirical figures given were only for illustration. The empirical form

1530-424: The publication of these studies. Recent astronomical research suggests that planetary systems around some other stars may follow Titius-Bode-like laws. Bovaird & Lineweaver (2013) 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

1575-407: 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

1620-480: The two others, there was a ~6% overestimate for Pluto; and later, a 6% underestimate for Miranda became apparent. Bodies in parentheses were not known in 1913, when Blagg wrote her paper. Some of the calculated distances in the Saturn and Uranus systems are not very accurate. This is because the low values of constant B in the table above make them very sensitive to the exact form of the function   f  . In

1665-479: The various features.) In 1907, she was appointed by the newly formed International Association of Academies to build a collated list of all of the lunar features. She worked with Samuel Saunder on the task, and the result was published in 1913. Her work produced a long list of discrepancies that the association would need to resolve. She also performed considerable work on the subject of variable stars , in collaboration with H. H. Turner . These were published in

1710-461: The work into Italian and English. There are two parts to Titius's inserted text. The first part explains the succession of planetary distances from the Sun: In 1772, J.E. Bode , then aged twenty-five, published an astronomical compendium, in which he included the following footnote, citing Titius (in later editions): These two statements, for all their peculiar expression, and from the radii used for

1755-536: Was a daring innovation." Gregory decided to leave for England where, in 1691, he was elected Savilian Professor of Astronomy at the University of Oxford, due in large part to the influence of Isaac Newton . The same year he was elected to be a Fellow of the Royal Society . In 1692, he was elected a Fellow of Balliol College, Oxford . Gregory spent several days with Isaac Newton in 1694, discussing revisions for

1800-512: Was born in Cheadle, Staffordshire , and lived her entire life there. She was the daughter of a solicitor , John Charles Blagg, and Frances Caroline Foottit. She trained herself in mathematics by reading her brother's textbooks. In 1875, she was sent to a finishing school in Kensington , where she studied algebra and German . She later worked as a Sunday school teacher and was the branch secretary of

1845-508: Was found that her predictions had been validated by discoveries of new planetary satellites unknown at the time of publication. In 1920, she joined the Lunar Commission of the newly formed International Astronomical Union . They tasked her with continuing her work on standardizing the nomenclature . For this task, she collaborated with Karl Müller (1866–1942), a retired government official and amateur astronomer. (The crater Müller on

Titius–Bode law - Misplaced Pages Continue

1890-480: Was one of five women to be elected simultaneously, the first women to become Fellows of that society. She worked out a Fourier analysis of Bode's Law in 1913, which was detailed in Michael Martin Nieto's book "The Titius-Bode Law of Planetary Distances." Her investigation corrected a major flaw in the original law and gave it a firmer physical footing. However, her paper was forgotten until 1953, when it

1935-805: Was professor of mathematics at the University of Edinburgh , and later Savilian Professor of Astronomy at the University of Oxford , and a proponent of Isaac Newton 's Principia . The fourth of the fifteen children of David Gregorie , a doctor from Kinnairdy, Banffshire, and Jean Walker of Orchiston, David was born in Upper Kirkgate, Aberdeen. The nephew of astronomer and mathematician James Gregory , David, like his influential uncle before him, studied at Aberdeen Grammar School and Marischal College ( University of Aberdeen ), from 1671 to 1675. The Gregorys were Jacobites and left Scotland to escape religious discrimination. Young David visited several countries on

1980-2095: Was provided in the form of a graph (the reason that points on the curve are such a close match for empirical data, for objects discovered prior to 1913, is that they are the empirical data). Finding a formula that closely fit the empircal curve turned out to be difficult. Fourier analysis of the shape resulted in the following seven term approximation:   f (   θ   ) = 0.4594 + 0.396   cos (   θ − 27.4 ∘   ) + 0.168   cos (   2   (   θ − 60.4 ∘ )   ) + 0.062   cos (   3   (   θ − 28.1 ∘ )   ) + + 0.053   cos (   4   (   θ − 77.2 ∘ )   ) + 0.009   cos (   5   (   θ − 22 ∘ )   ) + 0.012   cos (   7   (   θ − 40.4 ∘ )   )   . {\displaystyle {\begin{aligned}\ f{\bigl (}\ \theta \ {\bigr )}\;=\;0.4594\;+\;\;&0.396\ \cos \!{\bigl (}\ \theta -27.4^{\circ }\ {\bigr )}\;+\;0.168\ \cos \!{\bigl (}\ 2\ (\ \theta -60.4^{\circ })\ {\bigr )}\;+\;0.062\ \cos \!{\bigl (}\ 3\ (\ \theta -28.1^{\circ })\ {\bigr )}\;+\;\\\;+\;\;&0.053\ \cos \!{\bigl (}\ 4\ (\ \theta -77.2^{\circ })\ {\bigr )}\;+\;0.009\ \cos \!{\bigl (}\ 5\ (\ \theta -22^{\circ })\ {\bigr )}\;+\;0.012\ \cos \!{\bigl (}\ 7\ (\ \theta -40.4^{\circ })\ {\bigr )}~.\end{aligned}}} After further analysis, Blagg gave

2025-564: Was written by D. Gregory (1702), in which the succession of planetary distances 4, 7, 10, 16, 52, and 100 became a geometric progression with ratio 2. This is the nearest Newtonian formula, which was also cited by Benjamin Martin (1747) and Tomàs Cerdà (c. 1760) years before Titius's expanded translation of Bonnet's book into German (1766). Over the next two centuries, subsequent authors continued to present their own modified versions, apparently unaware of prior work. Titius and Bode hoped that

#290709