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133-585: Although the closest scattered-disc objects approach the Sun at about 30–35 AU, their orbits can extend well beyond 100 AU. This makes scattered disc objects among the coldest and most distant objects in the Solar System. The innermost portion of the scattered disc overlaps with a torus -shaped region of orbiting objects traditionally called the Kuiper belt , but its outer limits reach much farther away from

266-585: A Riemannian manifold , as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when L = Z 2 {\displaystyle \mathbb {Z} ^{2}} : R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} , which can also be described as the Cartesian plane under the identifications ( x , y ) ~ ( x + 1, y ) ~ ( x , y + 1) . This particular flat torus (and any uniformly scaled version of it)

399-477: A Tisserand parameter (relative to Neptune) greater than 3 and have a time-averaged eccentricity greater than 0.2. An alternative classification, introduced by B. J. Gladman , B. G. Marsden and C. Van Laerhoven in 2007, uses 10-million-year orbit integration instead of the Tisserand parameter. An object qualifies as an SDO if its orbit is not resonant, has a semi-major axis no greater than 2000 AU, and, during

532-449: A closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If

665-514: A fiber bundle over S (the Hopf bundle ). The surface described above, given the relative topology from R 3 {\displaystyle \mathbb {R} ^{3}} , is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R 3 {\displaystyle \mathbb {R} ^{3}} from

798-593: A maximal torus ; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G . Toroidal groups are examples of protori , which (like tori) are compact connected abelian groups, which are not required to be manifolds . Automorphisms of T are easily constructed from automorphisms of the lattice Z n {\displaystyle \mathbb {Z} ^{n}} , which are classified by invertible integral matrices of size n with an integral inverse; these are just

931-460: A perturbed orbit oscillate synchronously (increasing eccentricity while decreasing inclination and vice versa). This resonance applies only to bodies on highly inclined orbits; as a consequence, such orbits tend to be unstable, since the growing eccentricity would result in small pericenters , typically leading to a collision or (for large moons) destruction by tidal forces . In an example of another type of resonance involving orbital eccentricity,

1064-752: A product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem , giving: A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r , V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 . {\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}} These formulas are

1197-409: A torus ( pl. : tori or toruses ) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut . If the axis of revolution does not touch

1330-521: A 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical. These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads . A flat torus

1463-429: A 1:2 resonance a few hundred million years ago; the moons have drifted away from each other since then because Proteus is outside a synchronous orbit and Larissa is within one. Passage through the resonance is thought to have excited both moons' eccentricities to a degree that has not since been entirely damped out. In the case of Pluto 's satellites, it has been proposed that the present near resonances are relics of

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1596-415: A 2:3 resonance early in the Solar System's history. This would have led to orbital eccentricity and tidal heating that may have warmed Tethys' interior enough to form a subsurface ocean. Subsequent freezing of the ocean after the moons escaped from the resonance may have generated the extensional stresses that created the enormous graben system of Ithaca Chasma on Tethys. The satellite system of Uranus

1729-475: A clear distinction between the Kuiper belt and the scattered disc, separating those objects in stable orbits (the Kuiper belt) from those in scattered orbits (the scattered disc and the centaurs). However, the difference between the Kuiper belt and the scattered disc is not clear-cut, and many astronomers see the scattered disc not as a separate population but as an outward region of the Kuiper belt. Another term used

1862-500: A few percent larger than a mean-motion resonance ratio than a few percent smaller (particularly in the case of first order resonances, in which the integers in the ratio differ by one). This was predicted to be true in cases where tidal interactions with the star are significant. A number of near- integer -ratio relationships between the orbital frequencies of the planets or major moons are sometimes pointed out (see list below). However, these have no dynamical significance because there

1995-492: A flat torus into 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} was found. It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals , yielding

2128-408: A member of the scattered disc, because, with a perihelion distance of 76 AU, it is too remote to be affected by the gravitational attraction of the outer planets. Under this definition, an object with a perihelion greater than 40 AU could be classified as outside the scattered disc. Sedna is not the only such object: (148209) 2000 CR 105 (discovered before Sedna) and 474640 Alicanto have

2261-498: A passing star. A scheme introduced by a 2005 report from the Deep Ecliptic Survey by J. L. Elliott et al. distinguishes between two categories: scattered-near (i.e. typical SDOs) and scattered-extended (i.e. detached objects). Scattered-near objects are those whose orbits are non-resonant, non-planetary-orbit-crossing and have a Tisserand parameter (relative to Neptune) less than 3. Scattered-extended objects have

2394-481: A perihelion too far away from Neptune to be influenced by it. This led to a discussion among astronomers about a new minor planet set, called the extended scattered disc ( E-SDO ). 2000 CR 105 may also be an inner Oort-cloud object or (more likely) a transitional object between the scattered disc and the inner Oort cloud. More recently, these objects have been referred to as "detached" , or distant detached objects ( DDO ). There are no clear boundaries between

2527-605: A period equal to the entire age of the Solar System; a second posits that the scattering took place relatively quickly, during Neptune's early migration epoch. Models for a continuous formation throughout the age of the Solar System illustrate that at weak resonances within the Kuiper belt (such as 5:7 or 8:1), or at the boundaries of stronger resonances, objects can develop weak orbital instabilities over millions of years. The 4:7 resonance in particular has large instability. KBOs can also be shifted into unstable orbits by close passage of massive objects, or through collisions. Over time,

2660-569: A process by which satellites gain orbital energy at the expense of the primary's rotational energy, affecting inner moons disproportionately. In the Uranian system, however, due to the planet's lesser degree of oblateness , and the larger relative size of its satellites, escape from a mean-motion resonance is much easier. Lower oblateness of the primary alters its gravitational field in such a way that different possible resonances are spaced more closely together. A larger relative satellite size increases

2793-584: A ratio of 18:22:33 (or, in terms of the near resonances with Charon's period, 3+3/11:4:6; see below ); the respective ratio of orbits is 11:9:6. Based on the ratios of synodic periods , there are 5 conjunctions of Styx and Hydra and 3 conjunctions of Nix and Hydra for every 2 conjunctions of Styx and Nix. As with the Galilean satellite resonance, triple conjunctions are forbidden. Φ {\displaystyle \Phi } librates about 180° with an amplitude of at least 10°. The dwarf planet Pluto

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2926-427: A ratio of small integers . Most commonly, this relationship is found between a pair of objects (binary resonance). The physical principle behind orbital resonance is similar in concept to pushing a child on a swing , whereby the orbit and the swing both have a natural frequency , and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance

3059-429: A rectangle together, choosing the other two sides instead will cause the same reversal of orientation. The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian ). The 2-torus is a twofold branched cover of the 2-sphere, with four ramification points . Every conformal structure on the 2-torus can be represented as such

3192-406: A regular torus. For example, in the following map: If R and P in the above flat torus parametrization form a unit vector ( R , P ) = (cos( η ), sin( η )) then u , v , and 0 < η < π /2 parameterize the unit 3-sphere as Hopf coordinates . In particular, for certain very specific choices of a square flat torus in the 3-sphere S , where η = π /4 above, the torus will partition

3325-467: A resonance governed the motions of Jupiter's moons Io , Europa , and Ganymede . It is now also often applied to other 3-body resonances with the same ratios, such as that between the extrasolar planets Gliese 876 c, b, and e. Three-body resonances involving other simple integer ratios have been termed "Laplace-like" or "Laplace-type". A Lindblad resonance drives spiral density waves both in galaxies (where stars are subject to forcing by

3458-444: A secular resonance will change the eccentricity and inclination of the small body. Several prominent examples of secular resonance involve Saturn. There is a near-resonance between the precession of Saturn's rotational axis and that of Neptune's orbital axis (both of which have periods of about 1.87 million years), which has been identified as the likely source of Saturn's large axial tilt (26.7°). Initially, Saturn probably had

3591-424: A similar 2:3 resonance with Neptune, called a plutino , is the probable dwarf planet Orcus . Orcus has an orbit similar in inclination and eccentricity to Pluto's. However, the two are constrained by their mutual resonance with Neptune to always be in opposite phases of their orbits; Orcus is thus sometimes described as the "anti-Pluto". Neptune's innermost moon, Naiad , is in a 73:69 fourth-order resonance with

3724-507: A simple integer ratio of each other. It does not depend only on the existence of such a ratio, and more precisely the ratio of periods is not exactly a rational number, even averaged over a long period. For example, in the case of Pluto and Neptune (see below), the true equation says that the average rate of change of 3 α P − 2 α N − ϖ P {\displaystyle 3\alpha _{P}-2\alpha _{N}-\varpi _{P}}

3857-454: A so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics. In

3990-544: A sphere — by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with three points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, M* may be constructed by glueing together two congruent geodesic triangles in

4123-455: A tilt closer to that of Jupiter (3.1°). The gradual depletion of the Kuiper belt would have decreased the precession rate of Neptune's orbit; eventually, the frequencies matched, and Saturn's axial precession was captured into a spin-orbit resonance, leading to an increase in Saturn's obliquity. (The angular momentum of Neptune's orbit is 10 times that of Saturn's rotation rate, and thus dominates

Scattered disc - Misplaced Pages Continue

4256-461: A torus is a closed surface defined as the product of two circles : S  ×  S . This can be viewed as lying in C and is a subset of the 3-sphere S of radius √2. This topological torus is also often called the Clifford torus . In fact, S is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S as

4389-405: A torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one. An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Klein bottle ). Torus

4522-425: A torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of

4655-426: A torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R , the distance from the center of the coordinate system, and θ and φ , angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in

4788-563: A torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric . It can not be analytically embedded ( smooth of class C , 2 ≤ k ≤ ∞ ) into Euclidean 3-space. Mapping it into 3 -space requires one to stretch it, in which case it looks like

4921-469: A two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points . In fact, the conformal type of the torus is determined by the cross-ratio of the four points. The torus has a generalization to higher dimensions, the n-dimensional torus , often called the n -torus or hypertorus for short. (This is the more typical meaning of

5054-422: A white or greyish appearance. One explanation is the exposure of whiter subsurface layers by impacts; another is that the scattered objects' greater distance from the Sun creates a composition gradient, analogous to the composition gradient of the terrestrial and gas giant planets. Michael E. Brown, discoverer of the scattered object Eris, suggests that its paler colour could be because, at its current distance from

5187-459: Is R n {\displaystyle \mathbb {R} ^{n}} modulo the action of the integer lattice Z n {\displaystyle \mathbb {Z} ^{n}} (with the action being taken as vector addition). Equivalently, the n -torus is obtained from the n -dimensional hypercube by gluing the opposite faces together. An n -torus in this sense is an example of an n- dimensional compact manifold . It

5320-403: Is "scattered Kuiper-belt object" (or SKBO) for bodies of the scattered disc. Morbidelli and Brown propose that the difference between objects in the Kuiper belt and scattered-disc objects is that the latter bodies "are transported in semi-major axis by close and distant encounters with Neptune," but the former experienced no such close encounters. This delineation is inadequate (as they note) over

5453-1005: Is a Latin word for "a round, swelling, elevation, protuberance". A torus of revolution in 3-space can be parametrized as: x ( θ , φ ) = ( R + r cos ⁡ θ ) cos ⁡ φ y ( θ , φ ) = ( R + r cos ⁡ θ ) sin ⁡ φ z ( θ , φ ) = r sin ⁡ θ {\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \\\end{aligned}}} using angular coordinates θ , φ ∈ [ 0 , 2 π ) , {\displaystyle \theta ,\varphi \in [0,2\pi ),} representing rotation around

Scattered disc - Misplaced Pages Continue

5586-461: Is a member of the Lie group SO(4). It is known that there exists no C (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of

5719-408: Is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings , non-inflatable lifebuoys , ring doughnuts , and bagels . In topology , a ring torus is homeomorphic to the Cartesian product of two circles : S 1 × S 1 {\displaystyle S^{1}\times S^{1}} , and the latter is taken to be

5852-411: Is a torus with the metric inherited from its representation as the quotient , R 2 {\displaystyle \mathbb {R} ^{2}} / L , where L is a discrete subgroup of R 2 {\displaystyle \mathbb {R} ^{2}} isomorphic to Z 2 {\displaystyle \mathbb {Z} ^{2}} . This gives the quotient the structure of

5985-451: Is actually far more likely. For instance: Most bodies that are in resonance orbit in the same direction; however, the retrograde asteroid 514107 Kaʻepaokaʻawela appears to be in a stable (for a period of at least a million years) 1:−1 resonance with Jupiter. In addition, a few retrograde damocloids have been found that are temporarily captured in mean-motion resonance with Jupiter or Saturn . Such orbital interactions are weaker than

6118-457: Is also an example of a compact abelian Lie group . This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Toroidal groups play an important part in the theory of compact Lie groups . This is due in part to the fact that in any compact Lie group G one can always find

6251-402: Is an oscillator that receives periodic kicks via a weak coupling to some driving motor. The analog here would be that a more massive body provides a periodic gravitational kick to a smaller body, as it passes by. The mode-locking regions are named Arnold tongues . In general, an orbital resonance may A mean-motion orbital resonance occurs when two bodies have periods of revolution that are

6384-518: Is common, this resonance stabilizes the orbits by maximizing separation at conjunction, but it is unusual for the role played by orbital inclination in facilitating this avoidance in a case where eccentricities are minimal. While most extrasolar planetary systems discovered have not been found to have planets in mean-motion resonances, chains of up to five resonant planets and up to seven at least near resonant planets have been uncovered. Simulations have shown that during planetary system formation ,

6517-434: Is exactly zero, where α P {\displaystyle \alpha _{P}} is the longitude of Pluto, α N {\displaystyle \alpha _{N}} is the longitude of Neptune, and ϖ P {\displaystyle \varpi _{P}} is the longitude of Pluto's perihelion . Since the rate of motion of the latter is about 0.97 × 10 degrees per year,

6650-440: Is following an orbit trapped in a web of resonances with Neptune . The resonances include: One consequence of these resonances is that a separation of at least 30 AU is maintained when Pluto crosses Neptune's orbit. The minimum separation between the two bodies overall is 17 AU, while the minimum separation between Pluto and Uranus is just 11 AU (see Pluto's orbit for detailed explanation and graphs). The next largest body in

6783-447: Is known as the "square" flat torus. This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into

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6916-406: Is no appropriate precession of perihelion or other libration to make the resonance perfect (see the detailed discussion in the section above ). Such near resonances are dynamically insignificant even if the mismatch is quite small because (unlike a true resonance), after each cycle the relative position of the bodies shifts. When averaged over astronomically short timescales, their relative position

7049-401: Is notably different from those of Jupiter and Saturn in that it lacks precise resonances among the larger moons, while the majority of the larger moons of Jupiter (3 of the 4 largest) and of Saturn (6 of the 8 largest) are in mean-motion resonances. In all three satellite systems, moons were likely captured into mean-motion resonances in the past as their orbits shifted due to tidal dissipation ,

7182-441: Is one of the defining characteristics of scattered objects, as it allows Neptune to exert its gravitational influence. The classical objects ( cubewanos ) are very different from the scattered objects: more than 30% of all cubewanos are on low-inclination, near-circular orbits whose eccentricities peak at 0.25. Classical objects possess eccentricities ranging from 0.2 to 0.8. Though the inclinations of scattered objects are similar to

7315-408: Is random, just like bodies that are nowhere near resonance. For example, consider the orbits of Earth and Venus, which arrive at almost the same configuration after 8 Earth orbits and 13 Venus orbits. The actual ratio is 0.61518624, which is only 0.032% away from exactly 8:13. The mismatch after 8 years is only 1.5° of Venus' orbital movement. Still, this is enough that Venus and Earth find themselves in

7448-476: Is still poorly understood: no model of the formation of the Kuiper belt and the scattered disc has yet been proposed that explains all their observed properties. According to contemporary models, the scattered disc formed when Kuiper belt objects (KBOs) were "scattered" into eccentric and inclined orbits by gravitational interaction with Neptune and the other outer planets . The amount of time for this process to occur remains uncertain. One hypothesis estimates

7581-517: Is that there was another moon around Saturn whose orbit destabilized about 100 million years ago, perturbing Saturn. The perihelion secular resonance between asteroids and Saturn ( ν 6 = g − g 6 ) helps shape the asteroid belt (the subscript "6" identifies Saturn as the sixth planet from the Sun). Asteroids which approach it have their eccentricity slowly increased until they become Mars-crossers , at which point they are usually ejected from

7714-545: Is the n -fold product of the circle, the n -torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, T n = ( S 1 ) n {\displaystyle \mathbb {T} ^{n}=(\mathbb {S} ^{1})^{n}} . The configuration space of unordered , not necessarily distinct points is accordingly the orbifold T n / S n {\displaystyle \mathbb {T} ^{n}/\mathbb {S} _{n}} , which

7847-491: Is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates). For n = 2, the quotient is the Möbius strip , the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle , with a twist ; equivalently, as a triangular prism whose top and bottom faces are connected with

7980-400: Is the standard 2-torus, T 2 {\displaystyle \mathbb {T} ^{2}} . And similar to the 2-torus, the n -torus, T n {\displaystyle \mathbb {T} ^{n}} can be described as a quotient of R n {\displaystyle \mathbb {R} ^{n}} under integral shifts in any coordinate. That is, the n -torus

8113-468: Is thought to be in a 7:12 resonance with Neptune, and Gonggong is thought to be in a 3:10 resonance with Neptune. The simple integer ratios between periods hide more complex relations: As illustration of the latter, consider the well-known 2:1 resonance of Io-Europa. If the orbiting periods were in this relation, the mean motions n {\displaystyle n\,\!} (inverse of periods, often expressed in degrees per day) would satisfy

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8246-488: Is used in the current definition of a planet . A binary resonance ratio in this article should be interpreted as the ratio of number of orbits completed in the same time interval, rather than as the ratio of orbital periods , which would be the inverse ratio. Thus, the 2:3 ratio above means that Pluto completes two orbits in the time it takes Neptune to complete three. In the case of resonance relationships among three or more bodies, either type of ratio may be used (whereby

8379-454: The Euler characteristic of the n -torus is 0 for all n . The cohomology ring H ( T n {\displaystyle \mathbb {T} ^{n}} ,  Z ) can be identified with the exterior algebra over the Z - module Z n {\displaystyle \mathbb {Z} ^{n}} whose generators are the duals of the n nontrivial cycles. As the n -torus

8512-483: The Galilean moons (see below). Before Newton, there was also consideration of ratios and proportions in orbital motions, in what was called "the music of the spheres", or musica universalis . The article on resonant interactions describes resonance in the general modern setting. A primary result from the study of dynamical systems is the discovery and description of a highly simplified model of mode-locking; this

8645-574: The Gliese 876 system, in contrast, is associated with one triple conjunction per orbit of the outermost planet, ignoring libration.) The graph illustrates the positions of the moons after 1, 2, and 3 Io periods. Φ L {\displaystyle \Phi _{L}} librates about 180° with an amplitude of 0.03°. Another "Laplace-like" resonance involves the moons Styx , Nix , and Hydra of Pluto: This reflects orbital periods for Styx, Nix, and Hydra, respectively, that are close to

8778-537: The Observatoire de la Côte d'Azur in Nice suggested the formation of a 1:2 resonance between Jupiter and Saturn due to interactions with planetesimals that caused them to migrate inward and outward, respectively. In the model, this created a gravitational push that propelled both Uranus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would have doubled Neptune's distance from

8911-410: The asteroid belt by a close pass to Mars . This resonance forms the inner and "side" boundaries of the asteroid belt around 2 AU , and at inclinations of about 20°. Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between Mercury and Jupiter ( g 1 = g 5 ) has the potential to greatly increase Mercury's eccentricity and possibly destabilize

9044-532: The centaurs , a class of icy planetoids that orbit between Jupiter and Neptune, may simply be SDOs thrown into the inner reaches of the Solar System by Neptune, making them "cis-Neptunian" rather than trans-Neptunian scattered objects. Some objects, like (29981) 1999 TD 10 , blur the distinction and the Minor Planet Center (MPC), which officially catalogues all trans-Neptunian objects , now lists centaurs and SDOs together. The MPC, however, makes

9177-478: The hyperbolic plane along their (identical) boundaries, where each triangle has angles of π/2, π/3, and 0. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the Gauss-Bonnet theorem shows that the area of each triangle can be calculated as π - (π/2 + π/3 + 0) = π/6, so it follows that the compactified moduli space M* has area equal to π/3. The other two cusps occur at

9310-402: The resonant Kuiper-belt objects , those which Neptune has locked into a precise orbital ratio such as 2:3 (the object goes around twice for every three Neptune orbits) and 1:2 (the object goes around once for every two Neptune orbits). These ratios, called orbital resonances , allow KBOs to persist in regions which Neptune's gravitational influence would otherwise have cleared out over the age of

9443-454: The square root gives a quartic equation , ( x 2 + y 2 + z 2 + R 2 − r 2 ) 2 = 4 R 2 ( x 2 + y 2 ) . {\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).} The three classes of standard tori correspond to

9576-433: The " moduli space " of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space M may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle = π and the other has total angle = 2π/3. M may be turned into a compact space M* — topologically equivalent to

9709-400: The 2:3 resonance between Neptune and Pluto . Unstable resonances with Saturn 's inner moons give rise to gaps in the rings of Saturn . The special case of 1:1 resonance between bodies with similar orbital radii causes large planetary system bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood , an effect that

9842-406: The 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary . One example is the torus T defined by Other tori in S having this partitioning property include the square tori of the form Q ⋅ T , where Q is a rotation of 4-dimensional space R 4 {\displaystyle \mathbb {R} ^{4}} , or in other words Q

9975-609: The CCD captured more light than film (about 90% versus 10% of incoming light) and the blinking could now be done at an adjustable computer screen, the surveys allowed for higher throughput. A flood of new discoveries was the result: over a thousand trans-Neptunian objects were detected between 1992 and 2006. The first scattered-disc object (SDO) to be recognised as such was 1996 TL 66 , originally identified in 1996 by astronomers based at Mauna Kea in Hawaii. Three more were identified by

10108-630: The Oort cloud but to have been drawn into the inner Solar System by the gravity of the giant planets, whereas the JFCs are thought to have originated in the scattered disc. The centaurs are thought to be a dynamically intermediate stage between the scattered disc and the Jupiter family. There are many differences between SDOs and JFCs, even though many of the Jupiter-family comets may have originated in

10241-456: The Oort cloud have been made. Some researchers further suggest a transitional space between the scattered disc and the inner Oort cloud, populated with " detached objects ". The Kuiper belt is a relatively thick torus (or "doughnut") of space, extending from about 30 to 50 AU comprising two main populations of Kuiper belt objects (KBOs): the classical Kuiper-belt objects (or "cubewanos"), which lie in orbits untouched by Neptune, and

10374-453: The Solar System, because these objects would move between two exposures—this involved time-consuming steps like exposing and developing photographic plates or films , and people then using a blink comparator to manually detect prospective objects. During the 1980s, the use of CCD -based cameras in telescopes made it possible to directly produce electronic images that could then be readily digitized and transferred to digital images . Because

10507-549: The Solar System, since the objects are never close enough to Neptune to be scattered by its gravity. Those in 2:3 resonances are known as " plutinos ", because Pluto is the largest member of their group, whereas those in 1:2 resonances are known as " twotinos ". In contrast to the Kuiper belt, the scattered-disc population can be disturbed by Neptune. Scattered-disc objects come within gravitational range of Neptune at their closest approaches (~30 AU) but their farthest distances reach many times that. Ongoing research suggests that

10640-434: The Sun and farther above and below the ecliptic than the Kuiper belt proper. Because of its unstable nature, astronomers now consider the scattered disc to be the place of origin for most periodic comets in the Solar System, with the centaurs , a population of icy bodies between Jupiter and Neptune, being the intermediate stage in an object's migration from the disc to the inner Solar System. Eventually, perturbations from

10773-428: The Sun, its atmosphere of methane is frozen over its entire surface, creating an inches-thick layer of bright white ice. Pluto, conversely, being closer to the Sun, would be warm enough that methane would freeze only onto cooler, high- albedo regions, leaving low-albedo tholin -covered regions bare of ice. The Kuiper belt was initially thought to be the source of the Solar System's ecliptic comets . However, studies of

10906-589: The Sun. The resultant expulsion of objects from the proto-Kuiper belt as Neptune moved outwards could explain the Late Heavy Bombardment 600 million years after the Solar System's formation and the origin of Jupiter's Trojan asteroids . An outward migration of Neptune could also explain the current occupancy of some of its resonances (particularly the 2:5 resonance) within the Kuiper belt. While Saturn's mid-sized moons Dione and Tethys are not close to an exact resonance now, they may have been in

11039-486: The age of the Solar System, since bodies "trapped in resonances" could "pass from a scattering phase to a non-scattering phase (and vice versa) numerous times." That is, trans-Neptunian objects could travel back and forth between the Kuiper belt and the scattered disc over time. Therefore, they chose instead to define the regions, rather than the objects, defining the scattered disc as "the region of orbital space that can be visited by bodies that have encountered Neptune" within

11172-449: The appearance of resonant chains of planetary embryos is favored by the presence of the primordial gas disc . Once that gas dissipates, 90–95% of those chains must then become unstable to match the low frequency of resonant chains observed. Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the transit method are reported to have an example of this (with period ratios in

11305-400: The circle, the surface has a ring shape and is called a torus of revolution , also known as a ring torus . If the axis of revolution is tangent to the circle, the surface is a horn torus . If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus ). If the axis of revolution passes through the center of

11438-437: The circle, the surface is a degenerate torus, a double-covered sphere . If the revolved curve is not a circle, the surface is called a toroid , as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings , inner tubes and ringette rings . A torus should not be confused with a solid torus , which is formed by rotating a disk , rather than a circle, around an axis. A solid torus

11571-456: The corresponding interactions between bodies orbiting in the same direction. The trans-Neptunian object 2011 KT 19 has an orbital inclination of 110 ° with respect to the planets' orbital plane and is currently in a 7:9 polar resonance with Neptune. A Laplace resonance is a three-body resonance with a 1:2:4 orbital period ratio (equivalent to a 4:2:1 ratio of orbits). The term arose because Pierre-Simon Laplace discovered that such

11704-476: The definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space , but another way to do this is the Cartesian product of the embedding of S 1 {\displaystyle S^{1}} in the plane with itself. This produces a geometric object called the Clifford torus , a surface in 4-space . In the field of topology ,

11837-519: The early evolution of the Solar System, perhaps through exchanges of angular momentum with scattered objects. Once the orbits of Jupiter and Saturn shifted to a 2:1 resonance (two Jupiter orbits for each orbit of Saturn), their combined gravitational pull disrupted the orbits of Uranus and Neptune, sending Neptune into the temporary "chaos" of the proto-Kuiper belt. As Neptune traveled outward, it scattered many trans-Neptunian objects into higher and more eccentric orbits. This model states that 90% or more of

11970-488: The eccentricities of Ganymede and Callisto vary with a common period of 181 years, although with opposite phases. There are only a few known mean-motion resonances (MMR) in the Solar System involving planets, dwarf planets or larger satellites (a much greater number involve asteroids , planetary rings , moonlets and smaller Kuiper belt objects, including many possible dwarf planets ). Additionally, Haumea

12103-468: The entire trans-Neptunian population would show a similar red surface colour, as they were thought to have originated in the same region and subjected to the same physical processes. Specifically, SDOs were expected to have large amounts of surface methane, chemically altered into tholins by sunlight from the Sun. This would absorb blue light, creating a reddish hue. Most classical objects display this colour, but scattered objects do not; instead, they present

12236-474: The following Substituting the data (from Misplaced Pages) one will get −0.7395° day , a value substantially different from zero. Actually, the resonance is perfect, but it involves also the precession of perijove (the point closest to Jupiter), ω ˙ {\displaystyle {\dot {\omega }}} . The correct equation (part of the Laplace equations) is: In other words,

12369-447: The giant planets send such objects towards the Sun, transforming them into periodic comets. Many objects of the proposed Oort cloud are also thought to have originated in the scattered disc. Detached objects are not sharply distinct from scattered disc objects, and some such as Sedna have sometimes been considered to be included in this group. Traditionally, devices like a blink comparator were used in astronomy to detect objects in

12502-411: The inner Solar System several billion years from now. The Titan Ringlet within Saturn's C Ring represents another type of resonance in which the rate of apsidal precession of one orbit exactly matches the speed of revolution of another. The outer end of this eccentric ringlet always points towards Saturn's major moon Titan . A Kozai resonance occurs when the inclination and eccentricity of

12635-406: The integral matrices with determinant ±1. Making them act on R n {\displaystyle \mathbb {R} ^{n}} in the usual way, one has the typical toral automorphism on the quotient. The fundamental group of an n -torus is a free abelian group of rank n . The k -th homology group of an n -torus is a free abelian group of rank n choose k . It follows that

12768-419: The integration, its semi-major axis shows an excursion of 1.5 AU or more. Gladman et al. suggest the term scattering disk object to emphasize this present mobility. If the object is not an SDO as per the above definition, but the eccentricity of its orbit is greater than 0.240, it is classified as a detached TNO . (Objects with smaller eccentricity are considered classical.) In this scheme, the disc extends from

12901-475: The interaction.) However, it seems that the resonance no longer exists. Detailed analysis of data from the Cassini spacecraft gives a value of the moment of inertia of Saturn that is just outside the range for the resonance to exist, meaning that the spin axis does not stay in phase with Neptune's orbital inclination in the long term, as it apparently did in the past. One theory for why the resonance came to an end

13034-436: The list – meaning the relationship that seems most likely to have not just be by random chance – is that between Io and Metis, followed by those between Rosalind and Cordelia, Pallas and Ceres, Jupiter and Pallas, Callisto and Ganymede, and Hydra and Charon, respectively. A past resonance between Jupiter and Saturn may have played a dramatic role in early Solar System history. A 2004 computer model by Alessandro Morbidelli of

13167-434: The mean motion of Io is indeed double of that of Europa taking into account the precession of the perijove. An observer sitting on the (drifting) perijove will see the moons coming into conjunction in the same place (elongation). The other pairs listed above satisfy the same type of equation with the exception of Mimas-Tethys resonance. In this case, the resonance satisfies the equation The point of conjunctions librates around

13300-470: The mechanism of this escape is believed to explain why its orbital inclination is more than 10 times those of the other regular Uranian moons (see Uranus' natural satellites ). Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and Thebe are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively. Neptune's regular moons Proteus and Larissa are thought to have passed through

13433-404: The midpoint between the nodes of the two moons. The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the orbital phase of the moons: where λ {\displaystyle \lambda } are mean longitudes of the moons (the second equals sign ignores libration). This relation makes a triple conjunction impossible. (A Laplace resonance in

13566-464: The more extreme KBOs, very few scattered objects have orbits as close to the ecliptic as much of the KBO population. Although motions in the scattered disc are random, they do tend to follow similar directions, which means that SDOs can become trapped in temporary resonances with Neptune. Examples of possible resonant orbits within the scattered disc include 1:3, 2:7, 3:11, 5:22 and 4:79. The scattered disc

13699-439: The mutual gravitational influence of the bodies (i.e., their ability to alter or constrain each other's orbits). In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of Jupiter 's moons Ganymede , Europa and Io , and

13832-420: The next outward moon, Thalassa . As it orbits Neptune, the more inclined Naiad successively passes Thalassa twice from above and then twice from below, in a cycle that repeats every ~21.5 Earth days. The two moons are about 3540 km apart when they pass each other. Although their orbital radii differ by only 1850 km, Naiad swings ~2800 km above or below Thalassa's orbital plane at closest approach. As

13965-430: The north pole of S . The torus can also be described as a quotient of the Cartesian plane under the identifications or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA B . The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: Intuitively speaking, this means that

14098-502: The numbers of objects in the Kuiper belt and the scattered disc are hypothesized to be roughly equal, observational bias due to their greater distance means that far fewer SDOs have been observed to date. Known trans-Neptunian objects are often divided into two subpopulations: the Kuiper belt and the scattered disc. A third reservoir of trans-Neptunian objects, the Oort cloud , has been hypothesized, although no confirmed direct observations of

14231-562: The objects in the scattered disc may have been "promoted into these eccentric orbits by Neptune's resonances during the migration epoch...[therefore] the scattered disc might not be so scattered." Scattered objects, like other trans-Neptunian objects, have low densities and are composed largely of frozen volatiles such as water and methane . Spectral analysis of selected Kuiper belt and scattered objects has revealed signatures of similar compounds. Both Pluto and Eris, for instance, show signatures for methane. Astronomers originally supposed that

14364-476: The opposite relative orientation to the original every 120 such cycles, which is 960 years. Therefore, on timescales of thousands of years or more (still tiny by astronomical standards), their relative position is effectively random. The presence of a near resonance may reflect that a perfect resonance existed in the past, or that the system is evolving towards one in the future. Some orbital frequency coincidences include: The least probable orbital correlation in

14497-423: The orbit of Neptune to 2000 AU, the region referred to as the inner Oort cloud. The scattered disc is a very dynamic environment. Because they are still capable of being perturbed by Neptune, SDOs' orbits are always in danger of disruption; either of being sent outward to the Oort cloud or inward into the centaur population and ultimately the Jupiter family of comets. For this reason Gladman et al. prefer to refer to

14630-553: The points corresponding in M* to a) the square torus (total angle = π) and b) the hexagonal torus (total angle = 2π/3). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation. Orbital resonance In celestial mechanics , orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by

14763-442: The radius of a Hill sphere , and the Kuiper belt as its "complement ... in the a > 30 AU region"; the region of the Solar System populated by objects with semi-major axes greater than 30 AU. The Minor Planet Center classifies the trans-Neptunian object 90377 Sedna as a scattered-disc object. Its discoverer Michael E. Brown has suggested instead that it should be considered an inner Oort-cloud object rather than

14896-453: The range 1.83–2.18), as well as one sixth of planetary systems characterized by Doppler spectroscopy (with in this case a narrower period ratio range). Due to incomplete knowledge of the systems, the actual proportions are likely to be higher. Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a commensurability . It is much more common for pairs of planets to have orbital period ratios

15029-479: The ratio of periods is actually 1.503 in the long term. Depending on the details, mean-motion orbital resonance can either stabilize or destabilize the orbit. Stabilization may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance: Orbital resonances can also destabilize one of the orbits. This process can be exploited to find energy-efficient ways of deorbiting spacecraft. For small bodies, destabilization

15162-414: The region as the scattering disc, rather than scattered. Unlike Kuiper-belt objects (KBOs), the orbits of scattered-disc objects can be inclined as much as 40° from the ecliptic . SDOs are typically characterized by orbits with medium and high eccentricities with a semi-major axis greater than 50 AU, but their perihelia bring them within influence of Neptune. Having a perihelion of roughly 30 AU

15295-553: The region since 1992 have shown that the orbits within the Kuiper belt are relatively stable, and that ecliptic comets originate from the scattered disc, where orbits are generally less stable. Comets can loosely be divided into two categories: short-period and long-period—the latter being thought to originate in the Oort cloud. The two major categories of short-period comets are Jupiter-family comets (JFCs) and Halley-type comets . Halley-type comets, which are named after their prototype, Halley's Comet , are thought to have originated in

15428-416: The relatively high eccentricities of the orbits of Uranus' inner satellites, and the anomalously high orbital inclination of Miranda. High past orbital eccentricities associated with the (1:3) Umbriel-Miranda and (1:4) Titania-Ariel resonances may have led to tidal heating of the interiors of Miranda and Ariel, respectively. Miranda probably escaped from its resonance with Umbriel via a secondary resonance, and

15561-431: The same as for a cylinder of length 2π R and radius r , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. Expressing the surface area and the volume by the distance p of an outermost point on

15694-540: The same survey in 1999: 1999 CV 118 , 1999 CY 118 , and 1999 CF 119 . The first object presently classified as an SDO to be discovered was 1995 TL 8 , found in 1995 by Spacewatch . As of 2011, over 200 SDOs have been identified, including Gǃkúnǁʼhòmdímà (discovered by Schwamb, Brown, and Rabinowitz), Gonggong (Schwamb, Brown, and Rabinowitz) 2002 TC 302 ( NEAT ), Eris (Brown, Trujillo, and Rabinowitz), Sedna (Brown, Trujillo, and Rabinowitz), and 474640 Alicanto ( Deep Ecliptic Survey ). Although

15827-500: The scattered and detached regions. Gomes et al. define SDOs as having "highly eccentric orbits, perihelia beyond Neptune, and semi-major axes beyond the 1:2 resonance." By this definition, all distant detached objects are SDOs. Since detached objects' orbits cannot be produced by Neptune scattering, alternative scattering mechanisms have been put forward, including a passing star or a distant, planet-sized object . Alternatively, it has been suggested that these objects have been captured from

15960-466: The scattered disc would gradually form from these isolated events. Computer simulations have also suggested a more rapid and earlier formation for the scattered disc. Modern theories indicate that neither Uranus nor Neptune could have formed in situ beyond Saturn, as too little primordial matter existed at that range to produce objects of such high mass. Instead, these planets, and Saturn, may have formed closer to Jupiter, but were flung outwards during

16093-773: The scattered disc. Although the centaurs share a reddish or neutral coloration with many SDOs, their nuclei are bluer, indicating a fundamental chemical or physical difference. One hypothesis is that comet nuclei are resurfaced as they approach the Sun by subsurface materials which subsequently bury the older material. Solar System   → Local Interstellar Cloud   → Local Bubble   → Gould Belt   → Orion Arm   → Milky Way   → Milky Way subgroup   → Local Group → Local Sheet → Virgo Supercluster → Laniakea Supercluster   → Local Hole   → Observable universe   → Universe Each arrow ( → ) may be read as "within" or "part of". Torus In geometry ,

16226-465: The smallest whole-integer ratio sequences are not necessarily reversals of each other), and the type of ratio will be specified. Since the discovery of Newton's law of universal gravitation in the 17th century, the stability of the Solar System has preoccupied many mathematicians, starting with Pierre-Simon Laplace . The stable orbits that arise in a two-body approximation ignore the influence of other bodies. The effect of these added interactions on

16359-491: The spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r , and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices. Topologically ,

16492-484: The spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons ). A secular resonance occurs when the precession of two orbits is synchronised (usually a precession of the perihelion or ascending node ). A small body in secular resonance with a much larger one (e.g. a planet ) will precess at the same rate as the large body. Over long times (a million years, or so)

16625-405: The stability of the Solar System is very small, but at first it was not known whether they might add up over longer periods to significantly change the orbital parameters and lead to a completely different configuration, or whether some other stabilising effects might maintain the configuration of the orbits of the planets. It was Laplace who found the first answers explaining the linked orbits of

16758-567: The strength of their interactions. Both factors lead to more chaotic orbital behavior at or near mean-motion resonances. Escape from a resonance may be associated with capture into a secondary resonance, and/or tidal evolution-driven increases in orbital eccentricity or inclination . Mean-motion resonances that probably once existed in the Uranus System include (3:5) Ariel-Miranda, (1:3) Umbriel-Miranda, (3:5) Umbriel-Ariel, and (1:4) Titania-Ariel. Evidence for such past resonances includes

16891-449: The study of Riemann surfaces , one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature . In the case of a torus, the constant curvature must be zero. Then one defines

17024-1057: The surface of the torus to the center, and the distance q of an innermost point to the center (so that R = ⁠ p + q / 2 ⁠ and r = ⁠ p − q / 2 ⁠ ), yields A = 4 π 2 ( p + q 2 ) ( p − q 2 ) = π 2 ( p + q ) ( p − q ) , V = 2 π 2 ( p + q 2 ) ( p − q 2 ) 2 = 1 4 π 2 ( p + q ) ( p − q ) 2 . {\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}} As

17157-439: The term " n -torus", the other referring to n holes or of genus n . ) Just as the ordinary torus is topologically the product space of two circles, the n -dimensional torus is topologically equivalent to the product of n circles. That is: The standard 1-torus is just the circle: T 1 = S 1 {\displaystyle \mathbb {T} ^{1}=\mathbb {S} ^{1}} . The torus discussed above

17290-426: The three possible aspect ratios between R and r : When R ≥ r , the interior ( x 2 + y 2 − R ) 2 + z 2 < r 2 {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}} of this torus is diffeomorphic (and, hence, homeomorphic) to

17423-484: The torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem , which was proven in the 1950s, an isometric C embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit C (continuously differentiable) isometric embedding of

17556-534: The torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z {\displaystyle z} - axis is ( x 2 + y 2 − R ) 2 + z 2 = r 2 . {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.} Algebraically eliminating

17689-420: The tube and rotation around the torus' axis of revolution, respectively, where the major radius R {\displaystyle R} is the distance from the center of the tube to the center of the torus and the minor radius r {\displaystyle r} is the radius of the tube. The ratio R / r {\displaystyle R/r} is called the aspect ratio of

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