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88-622: [REDACTED] Look up sdo in Wiktionary, the free dictionary. SDO may stand for: Astronomy [ edit ] Scattered disc object in the Solar System Solar Dynamics Observatory , a NASA mission to study the Sun Subdwarf O star (sdO) Computing, software, electronics [ edit ] Service Data Object protocol under

176-585: 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)

264-479: 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

352-449: 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

440-514: 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

528-593: 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

616-752: 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

704-409: 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

792-521: 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

880-571: A Dutch association football club Air Santo Domingo (ICAO airline code SDO ) Other uses [ edit ] Sub Divisional Officer , a government officer in charge of a subdivision in a district in India. Selective door operation on trains Social dominance orientation , a personality trait Bukar–Sadong language (ISO 639 language code sdo ) Sistema Direzionale Orientale  [ it ] (Italian: Eastern Directional District ), Via della Pietra Sanguigna, Rome, Italy;

968-433: A cancelled-planned business district around Quintiliani subway station See also [ edit ] [REDACTED] Search for "sdo" , "sd-o" , "s-do" , or "s-d-o" on Misplaced Pages. All pages with titles beginning with SDO All pages with titles containing SDO Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

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1056-476: 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

1144-492: 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

1232-410: 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

1320-501: 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

1408-484: 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

1496-668: 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,

1584-429: 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

1672-406: 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

1760-454: 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

1848-544: 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

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1936-461: 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

2024-405: 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

2112-425: 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

2200-426: 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

2288-563: 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

2376-469: 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

2464-473: 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

2552-459: 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

2640-405: 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

2728-1005: 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

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2816-461: 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

2904-408: 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

2992-411: 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

3080-457: 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

3168-447: 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

3256-443: 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

3344-478: 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

3432-545: 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

3520-491: 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

3608-400: 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

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3696-454: 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

3784-535: 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

3872-478: 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

3960-402: 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

4048-454: 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

4136-433: 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

4224-406: 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

4312-519: The CANopen protocol Service Data Objects , allowing uniform data access Intel Secure Device Onboard for IoT Spatial Data Option , late Oracle Spatial and Graph Silent Death Online (videogame), from Mythic Entertainment Groups, companies, organizations [ edit ] Standards developing organization San Diego Opera , California, USA Secret Double Octopus , Israeli software company SDO Bussum , Bussum, Netherlands;

4400-612: 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

4488-633: 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

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4576-459: 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

4664-453: 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

4752-552: 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

4840-436: 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

4928-429: 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

5016-487: 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

5104-400: 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

5192-437: 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

5280-468: 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

5368-476: 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 ,

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5456-447: The disc extends from 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

5544-523: 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

5632-470: 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

5720-447: 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

5808-406: 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

5896-400: 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,

5984-466: 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

6072-430: 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

6160-504: 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

6248-565: 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

6336-443: 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

6424-417: 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

6512-555: 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

6600-431: 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

6688-547: 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

6776-505: 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

6864-467: 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

6952-774: 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 ,

7040-491: 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 ,

7128-449: 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

7216-1057: 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

7304-439: 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

7392-426: 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

7480-495: The title SDO . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=SDO&oldid=1254816713 " Category : Disambiguation pages Hidden categories: Articles containing Italian-language text Short description is different from Wikidata All article disambiguation pages All disambiguation pages Scattered disc Although

7568-484: 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

7656-534: 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

7744-420: 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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