\n2 Ceres and Pluto are [[dwarf planet]]s rather than [[major planets]]."}},"i":0}}]}" id="mwAQU">^{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. In 1913, M.A. Blagg, an Oxford astronomer, re-visited the law.^{ [14] } 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 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 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:^{ [14] } After further analysis, Blagg gave 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):^{ [15] } where Neither of these formulas for function f are used in the calculations below: The calculations here are based on a graph of function f which was drawn based on observed data. Her paper was pulished in 1913, and was forgotten until 1953, when A.E. Roy came across it while researching another problem.^{ [16] } 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 the two others, there was a ~6% overestimate for Pluto; and later, a 6% underestimate for Miranda became apparent.^{ [15] } 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 a 1945 magazine article,^{ [17] } 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: where is an oscillatory function with period representing distances from an off-centered origin to points on an ellipse. Nieto, who conducted the first modern comprehensive review of the Titius–Bode Law,^{ [18] } 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] } 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".^{ [19] } 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".^{ [13] } 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.^{ [20] } Dubrulle and Graner^{ [21] }^{ [22] } 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. 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 but non-Titius-Bode spacing.^{ [23] } 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 the equation and controversially^{ [25] } predicts an undiscovered planet or asteroid field for at 2 AU.^{ [26] } 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.^{ [27] } Recent astronomical research suggests that planetary systems around some other stars may follow Titius-Bode-like laws.^{ [28] }^{ [29] } Bovaird & Lineweaver (2013) ^{ [30] } 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.^{ [30] } 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 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.^{ [31] } 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.^{ [32] } 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%)^{ [33] } is obtained.^{ [34] } In mathematics, the arithmetic–geometric mean of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some mathematical constants, in particular, computing π. Johann Elert Bode was a German astronomer known for his reformulation and popularisation of the Titius–Bode law. Bode determined the orbit of Uranus and suggested the planet's name. In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that: In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation, in contrast to rotation around a fixed axis. In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control. A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously. In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi. Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one. In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees. The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter perpendicular to its angular momentum vector. In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them. The force may be either attractive or repulsive. The problem is to find the position or speed of the two bodies over time given their masses, positions, and velocities. Using classical mechanics, the solution can be expressed as a Kepler orbit using six orbital elements. Jitendra Jatashankar Rawal is an Indian astrophysicist and scientific educator, recognized for his work in the popularisation of science. In classical mechanics, Newton's theorem of revolving orbits identifies the type of central force needed to multiply the angular speed of a particle by a factor k without affecting its radial motion. Newton applied his theorem to understanding the overall rotation of orbits that is observed for the Moon and planets. The term "radial motion" signifies the motion towards or away from the center of force, whereas the angular motion is perpendicular to the radial motion. In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k^{2}. The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion. In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways. In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial distribution of mass inside a planet or satellite. Since a moment of inertia has dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.
^{2} Ceres and Pluto are dwarf planets rather than major planets.Blagg formulation
(as modified by Nieto 1970)System A B α β Sun-orbiting bodies 0.4162 2.025 112.4° 56.6° Moons of Jupiter 0.4523 1.852 113.0° 36.0° Moons of Saturn 3.074 0.0071 118.0° 10.0° Moons of Uranus 2.98 0.0805 125.7° 12.5° Comparison of the Blagg formulation with observation
Planet / minor pl. n obs. dist. Blagg pred. Mercury −2 0.387 0.387 Venus −1 0.723 0.723 Earth 0 1.000 1.000 Mars 1 1.524 1.524 Vesta 2 2.362 2.67 Juno 2.670 Pallas 2.774 Ceres 2.769 Jupiter 3 5.204 5.200 Saturn 4 9.583 9.550 Uranus 5 19.22 19.23 Neptune 6 30.07 30.13 (Pluto) 7 (39.48) 41.8 Jupiter satellite n obs. dist. Blagg pred. Amalthea −2 0.429 0.429 −1 0.708 Io 0 1.000 1.000 Europa 1 1.592 1.592 Ganymede 2 2.539 2.541 Callisto 3 4.467 4.467 4 9.26 5 15.4 Himalia 6 27.25 27.54 Elara 27.85 (Lysithea) (27.85) (Ananke) 7 (49.8) 55.46 (Carme) (53.3) Pasiphae 55.7 (Sinope) (56.2) Saturn satellite n obs. dist. Blagg pred. (Janus) −3 (0.538) 0.54 Mimas −2 0.630 0.629 Enceladus −1 0.808 0.807 Tethys 0 1.000 1.000 Dione 1 1.281 1.279 Rhea 2 1.789 1.786 3 2.97 Titan 4 4.149 4.140 Hyperion 5 5.034 5.023 6 6.3 7 6.65 8 7.00 Iapetus 9 12.09 12.11 Phoebe 10 43.92 43.85 Uranus satellite n obs. dist. Blagg pred. (Miranda) −2 (0.678) 0.64 −1 0.77 Ariel 0 1.000 1.000 Umbriel 1 1.394 1.393 Titania 2 2.293 2.286 Oberon 3 3.058 3.055 Richardson formulation
Historical inertia
Theoretical explanations
Natural satellite systems and exoplanetary systems
See also
Footnotes
Related Research Articles
References
Further reading
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: CS1 maint: multiple names: authors list (link) — combination history of distance measurements and development of Titius' law, notable astronomers involved, and exposition by graphs and simple ratios of modern planetary and satellite distances