157-566: The asteroid belt is a torus -shaped region in the Solar System , centered on the Sun and roughly spanning the space between the orbits of the planets Jupiter and Mars . It contains a great many solid, irregularly shaped bodies called asteroids or minor planets . The identified objects are of many sizes, but much smaller than planets , and, on average, are about one million kilometers (or six hundred thousand miles) apart. This asteroid belt
314-585: 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)
471-449: 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
628-418: A donut or doughnut . If the axis of revolution does not touch 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
785-514: 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
942-593: 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
1099-752: 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
1256-521: 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
1413-631: A cloud of interstellar dust and gas collapsed under the influence of gravity to form a rotating disc of material that then conglomerated to form the Sun and planets. During the first few million years of the Solar System's history, an accretion process of sticky collisions caused the clumping of small particles, which gradually increased in size. Once the clumps reached sufficient mass, they could draw in other bodies through gravitational attraction and become planetesimals. This gravitational accretion led to
1570-583: A collection of data called an ephemeris . NASA 's Jet Propulsion Laboratory HORIZONS System provides one of several ephemeris computation services. In 1976, to establish a more precise measure for the astronomical unit, the IAU formally adopted a new definition . Although directly based on the then-best available observational measurements, the definition was recast in terms of the then-best mathematical derivations from celestial mechanics and planetary ephemerides. It stated that "the astronomical unit of length
1727-487: A common origin in the breakup of a larger body. Graphical displays of these element pairs, for members of the asteroid belt, show concentrations indicating the presence of an asteroid family. There are about 20 to 30 associations that are likely asteroid families. Additional groupings have been found that are less certain. Asteroid families can be confirmed when the members display similar spectral features. Smaller associations of asteroids are called groups or clusters. Some of
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#17327725382071884-469: A considerable improvement in parallax measurement. Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931. Direct radar measurements of the distances to Venus and Mars became available in the early 1960s. Along with improved measurements of the speed of light, these showed that Newcomb's values for the solar parallax and the constant of aberration were inconsistent with one another. The unit distance A (the value of
2041-447: A distance within the Solar System without specifying the frame of reference for the measurement is problematic. The 1976 definition of the astronomical unit was incomplete because it did not specify the frame of reference in which to apply the measurement, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and "vigorous debate" ensued until August 2012 when
2198-429: A factor in reducing the total population of this group. Torus In geometry , 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
2355-538: A few hundred micrometres . This fine material is produced, at least in part, from collisions between asteroids, and by the impact of micrometeorites upon the asteroids. Due to the Poynting–Robertson effect , the pressure of solar radiation causes this dust to slowly spiral inward toward the Sun. The combination of this fine asteroid dust, as well as ejected cometary material, produces the zodiacal light . This faint auroral glow can be viewed at night extending from
2512-516: A few metres. The asteroid material is so thinly distributed that numerous uncrewed spacecraft have traversed it without incident. Nonetheless, collisions between large asteroids occur and can produce an asteroid family , whose members have similar orbital characteristics and compositions. Individual asteroids within the belt are categorized by their spectra , with most falling into three basic groups: carbonaceous ( C-type ), silicate ( S-type ), and metal-rich ( M-type ). The asteroid belt formed from
2669-492: 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
2826-629: A low albedo . Their surface compositions are similar to carbonaceous chondrite meteorites . Chemically, their spectra match the primordial composition of the early Solar System, with hydrogen, helium, and volatiles removed. S-type ( silicate -rich) asteroids are more common toward the inner region of the belt, within 2.5 AU of the Sun. The spectra of their surfaces reveal the presence of silicates and some metal, but no significant carbonaceous compounds. This indicates that their materials have been significantly modified from their primordial composition, probably through melting and reformation. They have
2983-452: A major source of the Earth's oceans because the deuterium-hydrogen ratio is too low for classical comets to have been the principal source. Most asteroids within the asteroid belt have orbital eccentricities of less than 0.4, and an inclination of less than 30°. The orbital distribution of the asteroids reaches a maximum at an eccentricity around 0.07 and an inclination below 4°. Thus, although
3140-416: A mean radius of 10 km are expected to occur about once every 10 million years. A collision may fragment an asteroid into numerous smaller pieces (leading to the formation of a new asteroid family ). Conversely, collisions that occur at low relative speeds may also join two asteroids. After more than 4 billion years of such processes, the members of the asteroid belt now bear little resemblance to
3297-538: A mean semi-major axis of 1.9 AU) is the Hungaria family of minor planets. They are named after the main member, 434 Hungaria ; the group contains at least 52 named asteroids. The Hungaria group is separated from the main body by the 4:1 Kirkwood gap and their orbits have a high inclination. Some members belong to the Mars-crossing category of asteroids, and gravitational perturbations by Mars are likely
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#17327725382073454-483: A much larger planet that once occupied the Mars–Jupiter region, with this planet having suffered an internal explosion or a cometary impact many million years before, while Odesan astronomer K. N. Savchenko suggested that Ceres, Pallas, Juno, and Vesta were escaped moons rather than fragments of the exploded planet. The large amount of energy required to destroy a planet, combined with the belt's low combined mass, which
3611-477: A planet," in his Mysterium Cosmographicum , stating his prediction that a planet would be found there. While analyzing Tycho Brahe 's data, Kepler thought that too large a gap existed between the orbits of Mars and Jupiter to fit his own model of where planetary orbits should be found. In an anonymous footnote to his 1766 translation of Charles Bonnet 's Contemplation de la Nature , the astronomer Johann Daniel Titius of Wittenberg noted an apparent pattern in
3768-429: A population of comets had been discovered within the asteroid belt beyond the snow line, which may have provided a source of water for Earth's oceans. According to some models, outgassing of water during the Earth's formative period was insufficient to form the oceans, requiring an external source such as a cometary bombardment. The outer asteroid belt appears to include a few objects that may have arrived there during
3925-466: A ratio of solar to lunar distance of approximately 19, matching Aristarchus's figure. Although Ptolemy's procedure is theoretically workable, it is very sensitive to small changes in the data, so much so that changing a measurement by a few per cent can make the solar distance infinite. After Greek astronomy was transmitted to the medieval Islamic world, astronomers made some changes to Ptolemy's cosmological model, but did not greatly change his estimate of
4082-429: 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
4239-406: 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
4396-453: A relatively high albedo and form about 17% of the total asteroid population. M-type (metal-rich) asteroids are typically found in the middle of the main belt, and they make up much of the remainder of the total population. Their spectra resemble that of iron-nickel. Some are believed to have formed from the metallic cores of differentiated progenitor bodies that were disrupted through collision. However, some silicate compounds also can produce
4553-430: A similar appearance. For example, the large M-type asteroid 22 Kalliope does not appear to be primarily composed of metal. Within the asteroid belt, the number distribution of M-type asteroids peaks at a semimajor axis of about 2.7 AU. Whether all M-types are compositionally similar, or whether it is a label for several varieties which do not fit neatly into the main C and S classes is not yet clear. One mystery
4710-454: 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
4867-426: A solar parallax of 8.6″ . Although Huygens' estimate is remarkably close to modern values, it is often discounted by historians of astronomy because of the many unproven (and incorrect) assumptions he had to make for his method to work; the accuracy of his value seems to be based more on luck than good measurement, with his various errors cancelling each other out. Jean Richer and Giovanni Domenico Cassini measured
Asteroid belt - Misplaced Pages Continue
5024-544: 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
5181-484: A star's shift enabled the star's distance to be calculated. But all measurements are subject to some degree of error or uncertainty, and the uncertainties in the length of the astronomical unit only increased uncertainties in the stellar distances. Improvements in precision have always been a key to improving astronomical understanding. Throughout the twentieth century, measurements became increasingly precise and sophisticated, and ever more dependent on accurate observation of
5338-461: 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
5495-425: 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
5652-426: 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
5809-563: 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
5966-469: 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
6123-611: A typical asteroid has a relatively circular orbit and lies near the plane of the ecliptic , some asteroid orbits can be highly eccentric or travel well outside the ecliptic plane. Sometimes, the term "main belt" is used to refer only to the more compact "core" region where the greatest concentration of bodies is found. This lies between the strong 4:1 and 2:1 Kirkwood gaps at 2.06 and 3.27 AU, and at orbital eccentricities less than roughly 0.33, along with orbital inclinations below about 20°. As of 2006, this "core" region contained 93% of all discovered and numbered minor planets within
6280-459: 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
6437-1005: 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
Asteroid belt - Misplaced Pages Continue
6594-411: Is a compositional trend of asteroid types by increasing distance from the Sun, in the order of S, C, P, and the spectrally-featureless D-types . Carbonaceous asteroids , as their name suggests, are carbon-rich. They dominate the asteroid belt's outer regions, and are rare in the inner belt. Together they comprise over 75% of the visible asteroids. They are redder in hue than the other asteroids and have
6751-461: 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
6908-411: 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
7065-457: 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
7222-455: Is also called the main asteroid belt or main belt to distinguish it from other asteroid populations in the Solar System. The asteroid belt is the smallest and innermost known circumstellar disc in the Solar System. Classes of small Solar System bodies in other regions are the near-Earth objects , the centaurs , the Kuiper belt objects, the scattered disc objects, the sednoids , and
7379-514: Is constant for all observers, the terrestrial metre appears to change in length compared with the "planetary metre" on a periodic basis. The metre is defined to be a unit of proper length . Indeed, the International Committee for Weights and Measures (CIPM) notes that "its definition applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored". As such,
7536-466: Is formed by rotating a disk , rather than a circle, around an axis. A solid torus 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
7693-418: Is increasingly becoming the norm. A 2004 analysis of radiometric measurements in the inner Solar System suggested that the secular increase in the unit distance was much larger than can be accounted for by solar radiation, + 15 ± 4 metres per century. The measurements of the secular variations of the astronomical unit are not confirmed by other authors and are quite controversial. Furthermore, since 2010,
7850-447: 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
8007-430: Is missing. Until 2001, most basaltic bodies discovered in the asteroid belt were believed to originate from the asteroid Vesta (hence their name V-type), but the discovery of the asteroid 1459 Magnya revealed a slightly different chemical composition from the other basaltic asteroids discovered until then, suggesting a different origin. This hypothesis was reinforced by the further discovery in 2007 of two asteroids in
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#17327725382078164-434: Is only about 4% of the mass of Earth's Moon, does not support these hypotheses. Further, the significant chemical differences between the asteroids become difficult to explain if they come from the same planet. A modern hypothesis for the asteroid belt's creation relates to how, in general for the Solar System, planetary formation is thought to have occurred via a process comparable to the long-standing nebular hypothesis ;
8321-482: Is related to the Earth–Sun distance as measured in Earth radii by The smaller the solar parallax, the greater the distance between the Sun and Earth: a solar parallax of 15″ is equivalent to an Earth–Sun distance of 13,750 Earth radii. Christiaan Huygens believed that the distance was even greater: by comparing the apparent sizes of Venus and Mars , he estimated a value of about 24,000 Earth radii, equivalent to
8478-531: Is that length ( A ) for which the Gaussian gravitational constant ( k ) takes the value 0.017 202 098 95 when the units of measurement are the astronomical units of length, mass and time". Equivalently, by this definition, one au is "the radius of an unperturbed circular Newtonian orbit about the sun of a particle having infinitesimal mass, moving with an angular frequency of 0.017 202 098 95 radians per day "; or alternatively that length for which
8635-545: 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
8792-491: 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
8949-419: Is the relative rarity of V-type (Vestoid) or basaltic asteroids in the asteroid belt. Theories of asteroid formation predict that objects the size of Vesta or larger should form crusts and mantles, which would be composed mainly of basaltic rock, resulting in more than half of all asteroids being composed either of basalt or of olivine . However, observations suggest that 99% of the predicted basaltic material
9106-400: 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
9263-626: The Zhoubi Suanjing ( c. 1st century BCE ), shows how the distance to the Sun can be computed geometrically, using the different lengths of the noontime shadows observed at three places 1,000 li apart and the assumption that Earth is flat. According to Eusebius in the Praeparatio evangelica (Book XV, Chapter 53), Eratosthenes found the distance to the Sun to be "σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας" (literally "of stadia myriads 400 and 80,000″ ) but with
9420-454: 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
9577-487: The Greek asteroeides , meaning "star-like". Upon completing a series of observations of Ceres and Pallas, he concluded, Neither the appellation of planets nor that of comets can with any propriety of language be given to these two stars ... They resemble small stars so much as hardly to be distinguished from them. From this, their asteroidal appearance, if I take my name, and call them Asteroids; reserving for myself, however,
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#17327725382079734-585: The Oort cloud objects. About 60% of the main belt mass is contained in the four largest asteroids: Ceres , Vesta , Pallas , and Hygiea . The total mass of the asteroid belt is estimated to be 3% that of the Moon . Ceres, the only object in the asteroid belt large enough to be a dwarf planet , is about 950 km in diameter, whereas Vesta, Pallas, and Hygiea have mean diameters less than 600 km. The remaining mineralogically classified bodies range in size down to
9891-577: The Seven Years' War , dozens of astronomers were dispatched to observing points around the world at great expense and personal danger: several of them died in the endeavour. The various results were collated by Jérôme Lalande to give a figure for the solar parallax of 8.6″ . Karl Rudolph Powalky had made an estimate of 8.83″ in 1864. Another method involved determining the constant of aberration . Simon Newcomb gave great weight to this method when deriving his widely accepted value of 8.80″ for
10048-597: The University of Palermo , Sicily, found a tiny moving object in an orbit with exactly the radius predicted by this pattern. He dubbed it "Ceres", after the Roman goddess of the harvest and patron of Sicily. Piazzi initially believed it to be a comet, but its lack of a coma suggested it was a planet. Thus, the aforementioned pattern predicted the semimajor axes of all eight planets of the time (Mercury, Venus, Earth, Mars, Ceres, Jupiter, Saturn, and Uranus). Concurrent with
10205-480: The heliocentric gravitational constant (the product G M ☉ ) is equal to ( 0.017 202 098 95 ) au /d , when the length is used to describe the positions of objects in the Solar System. Subsequent explorations of the Solar System by space probes made it possible to obtain precise measurements of the relative positions of the inner planets and other objects by means of radar and telemetry . As with all radar measurements, these rely on measuring
10362-478: 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
10519-540: The parsec and light-year are widely used. The parsec (parallax arcsecond ) is defined in terms of the astronomical unit, being the distance of an object with a parallax of 1″ . The light-year is often used in popular works, but is not an approved non-SI unit and is rarely used by professional astronomers. When simulating a numerical model of the Solar System , the astronomical unit provides an appropriate scale that minimizes ( overflow , underflow and truncation ) errors in floating point calculations. The book On
10676-454: 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
10833-433: 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
10990-459: The "planetary second" (conventionally measured in TDB). This is because the distance between Earth and the Sun is not fixed (it varies between 0.983 289 8912 and 1.016 710 3335 au ) and, when Earth is closer to the Sun ( perihelion ), the Sun's gravitational field is stronger and Earth is moving faster along its orbital path. As the metre is defined in terms of the second and the speed of light
11147-425: The 16th century. Johannes Kepler was the first to realize that Ptolemy's estimate must be significantly too low (according to Kepler, at least by a factor of three) in his Rudolphine Tables (1627). Kepler's laws of planetary motion allowed astronomers to calculate the relative distances of the planets from the Sun, and rekindled interest in measuring the absolute value for Earth (which could then be applied to
11304-595: The 1850 translation (by Elise Otté ) of Alexander von Humboldt's Cosmos : "[...] and the regular appearance, about the 13th of November and the 11th of August, of shooting stars, which probably form part of a belt of asteroids intersecting the Earth's orbit and moving with planetary velocity". Another early appearance occurred in Robert James Mann 's A Guide to the Knowledge of the Heavens : "The orbits of
11461-496: The 2009 estimate. With the definitions used before 2012, the astronomical unit was dependent on the heliocentric gravitational constant , that is the product of the gravitational constant , G , and the solar mass , M ☉ . Neither G nor M ☉ can be measured to high accuracy separately, but the value of their product is known very precisely from observing the relative positions of planets ( Kepler's third law expressed in terms of Newtonian gravitation). Only
11618-406: 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
11775-588: The 50,000 meteorites found on Earth to date, 99.8 percent are believed to have originated in the asteroid belt. In 1918, the Japanese astronomer Kiyotsugu Hirayama noticed that the orbits of some of the asteroids had similar parameters, forming families or groups. Approximately one-third of the asteroids in the asteroid belt are members of an asteroid family. These share similar orbital elements , such as semi-major axis , eccentricity , and orbital inclination as well as similar spectral features, which indicate
11932-530: The Earth–Sun distance. For example, in his introduction to Ptolemaic astronomy, al-Farghānī gave a mean solar distance of 1,170 Earth radii, whereas in his zij , al-Battānī used a mean solar distance of 1,108 Earth radii. Subsequent astronomers, such as al-Bīrūnī , used similar values. Later in Europe, Copernicus and Tycho Brahe also used comparable figures ( 1,142 and 1,150 Earth radii), and so Ptolemy's approximate Earth–Sun distance survived through
12089-451: The Greek stadium of 185 to 190 metres, the former translation comes to 754,800 km to 775,200 km , which is far too low, whereas the second translation comes to 148.7 to 152.8 billion metres (accurate within 2%). In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as 1,210 times Earth's radius . To determine this value, Ptolemy started by measuring
12246-409: The IAU adopted the current definition of 1 astronomical unit = 149,597,870,700 metres . The astronomical unit is typically used for stellar system scale distances, such as the size of a protostellar disk or the heliocentric distance of an asteroid, whereas other units are used for other distances in astronomy . The astronomical unit is too small to be convenient for interstellar distances, where
12403-540: The Late Heavy Bombardment was likely affected by the passages of large Centaurs and trans-Neptunian objects (TNOs). Centaurs and TNOs that reach the inner Solar System can modify the orbits of main belt asteroids, though only if their mass is of the order of 10 M ☉ for single encounters or, one order less in case of multiple close encounters. However, Centaurs and TNOs are unlikely to have significantly dispersed young asteroid families in
12560-496: The Moon and concluded that the apparent diameter of the Sun was equal to the apparent diameter of the Moon at the Moon's greatest distance, and from records of lunar eclipses, he estimated this apparent diameter, as well as the apparent diameter of the shadow cone of Earth traversed by the Moon during a lunar eclipse. Given these data, the distance of the Sun from Earth can be trigonometrically computed to be 1,210 Earth radii. This gives
12717-401: The Moon's parallax, finding what amounted to a horizontal lunar parallax of 1° 26′, which was much too large. He then derived a maximum lunar distance of 64 + 1 / 6 Earth radii. Because of cancelling errors in his parallax figure, his theory of the Moon's orbit, and other factors, this figure was approximately correct. He then measured the apparent sizes of the Sun and
12874-479: The Sizes and Distances of the Sun and Moon , which is ascribed to Aristarchus , says the distance to the Sun is 18 to 20 times the distance to the Moon , whereas the true ratio is about 389.174 . The latter estimate was based on the angle between the half-moon and the Sun, which he estimated as 87° (the true value being close to 89.853° ). Depending on the distance that van Helden assumes Aristarchus used for
13031-432: The Solar System, the asteroids melted to some degree, allowing elements within them to be differentiated by mass. Some of the progenitor bodies may even have undergone periods of explosive volcanism and formed magma oceans. Because of the relatively small size of the bodies, though, the period of melting was necessarily brief compared to the much larger planets, and had generally ended about 4.5 billion years ago, in
13188-443: The Solar System. The JPL Small-Body Database lists over 1 million known main-belt asteroids. The semimajor axis of an asteroid is used to describe the dimensions of its orbit around the Sun, and its value determines the minor planet's orbital period . In 1866, Daniel Kirkwood announced the discovery of gaps in the distances of these bodies' orbits from the Sun. They were located in positions where their period of revolution about
13345-459: The Sun was an integer fraction of Jupiter's orbital period. Kirkwood proposed that the gravitational perturbations of the planet led to the removal of asteroids from these orbits. When the mean orbital period of an asteroid is an integer fraction of the orbital period of Jupiter, a mean-motion resonance with the gas giant is created that is sufficient to perturb an asteroid to new orbital elements . Primordial asteroids entered these gaps because of
13502-549: The Sun. This has led to calls to abandon the astronomical unit as a unit of measurement. As the speed of light has an exact defined value in SI units and the Gaussian gravitational constant k is fixed in the astronomical system of units , measuring the light time per unit distance is exactly equivalent to measuring the product G × M ☉ in SI units. Hence, it is possible to construct ephemerides entirely in SI units, which
13659-577: The additional note that in the Greek text the grammatical agreement is between myriads (not stadia ) on the one hand and both 400 and 80,000 on the other: all three are accusative plural, while σταδιων is genitive plural ("of stadia") . All three words (or all four including stadia ) are inflected . This has been translated either as 4 080 000 stadia (1903 translation by Edwin Hamilton Gifford ), or as 804,000,000 stadia (edition of Édouard des Places , dated 1974–1991). Using
13816-401: The asteroid belt, dynamically exciting the region's population and increasing their velocities relative to each other. In regions where the average velocity of the collisions was too high, the shattering of planetesimals tended to dominate over accretion, preventing the formation of a planet. Instead, they continued to orbit the Sun as before, occasionally colliding. During the early history of
13973-687: The asteroids are placed in a wide belt of space, extending between the extremes of [...]". The American astronomer Benjamin Peirce seems to have adopted that terminology and to have been one of its promoters. Over 100 asteroids had been located by mid-1868, and in 1891, the introduction of astrophotography by Max Wolf accelerated the rate of discovery. A total of 1,000 asteroids had been found by 1921, 10,000 by 1981, and 100,000 by 2000. Modern asteroid survey systems now use automated means to locate new minor planets in ever-increasing numbers. On 22 January 2014, European Space Agency (ESA) scientists reported
14130-447: The astronomer Karl Ludwig Hencke detected a fifth object ( 5 Astraea ) and, shortly thereafter, new objects were found at an accelerating rate. Counting them among the planets became increasingly cumbersome. Eventually, they were dropped from the planet list (as first suggested by Alexander von Humboldt in the early 1850s) and Herschel's coinage, "asteroids", gradually came into common use. The discovery of Neptune in 1846 led to
14287-454: The astronomical unit by John Flamsteed , which accomplished it alone by measuring the martian diurnal parallax . Another colleague, Ole Rømer , discovered the finite speed of light in 1676: the speed was so great that it was usually quoted as the time required for light to travel from the Sun to the Earth, or "light time per unit distance", a convention that is still followed by astronomers today. A better method for observing Venus transits
14444-468: The astronomical unit in metres) can be expressed in terms of other astronomical constants: where G is the Newtonian constant of gravitation , M ☉ is the solar mass, k is the numerical value of Gaussian gravitational constant and D is the time period of one day. The Sun is constantly losing mass by radiating away energy, so the orbits of the planets are steadily expanding outward from
14601-531: The astronomical unit was conceived as the average Earth-Sun distance (the average of Earth's aphelion and perihelion ), before its modern redefinition in 2012. The astronomical unit is used primarily for measuring distances within the Solar System or around other stars. It is also a fundamental component in the definition of another unit of astronomical length, the parsec . One au is equivalent to 499 light-seconds to within 10 parts per million . A variety of unit symbols and abbreviations have been in use for
14758-666: The astronomical unit. In a 1976 resolution, the International Astronomical Union (IAU) had used the symbol A to denote a length equal to the astronomical unit. In the astronomical literature, the symbol AU is common. In 2006, the International Bureau of Weights and Measures (BIPM) had recommended ua as the symbol for the unit, from the French "unité astronomique". In the non-normative Annex C to ISO 80000-3 :2006 (later withdrawn),
14915-405: The axis of revolution passes through the center of 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
15072-566: The belt's total mass, with 39% accounted for by Ceres alone. The present day belt consists primarily of three categories of asteroids: C-type carbonaceous asteroids, S-type silicate asteroids, and a hybrid group of X-type asteroids. The hybrid group have featureless spectra, but they can be divided into three groups based on reflectivity, yielding the M-type metallic, P-type primitive, and E-type enstatite asteroids. Additional types have been found that do not fit within these primary classes. There
15229-553: The best IAU 2009 estimate was A = c 0 τ A = 149,597,870,700 ± 3 m , based on a comparison of Jet Propulsion Laboratory and IAA–RAS ephemerides. In 2006, the BIPM reported a value of the astronomical unit as 1.495 978 706 91 (6) × 10 m . In the 2014 revision of the SI ;Brochure, the BIPM recognised the IAU's 2012 redefinition of the astronomical unit as 149,597,870,700 m . This estimate
15386-638: The case of Ceres with the Gefion family .) The Vesta family is believed to have formed as the result of a crater-forming impact on Vesta. Likewise, the HED meteorites may also have originated from Vesta as a result of this collision. Three prominent bands of dust have been found within the asteroid belt. These have similar orbital inclinations as the Eos, Koronis, and Themis asteroid families, and so are possibly associated with those groupings. The main belt evolution after
15543-429: The celestial police, discovered a second object in the same region, Pallas. Unlike the other known planets, Ceres and Pallas remained points of light even under the highest telescope magnifications instead of resolving into discs. Apart from their rapid movement, they appeared indistinguishable from stars . Accordingly, in 1802, William Herschel suggested they be placed into a separate category, named "asteroids", after
15700-517: The change was an improved method of measuring the speed of light.) The speed of light could then be expressed exactly as c 0 = 299,792,458 m/s , a standard also adopted by the IERS numerical standards. From this definition and the 2009 IAU standard, the time for light to traverse an astronomical unit is found to be τ A = 499.004 783 8061 ± 0.000 000 01 s , which is slightly more than 8 minutes 19 seconds. By multiplication,
15857-447: The curve at about 5 km and 100 km , where more asteroids than expected from such a curve are found. Most asteroids larger than approximately 120 km in diameter are primordial, having survived from the accretion epoch, whereas most smaller asteroids are products of fragmentation of primordial asteroids. The primordial population of the main belt was probably 200 times what it is today. The absolute magnitudes of most of
16014-649: The detection, for the first definitive time, of water vapor on Ceres, the largest object in the asteroid belt. The detection was made by using the far-infrared abilities of the Herschel Space Observatory . The finding was unexpected because comets , not asteroids, are typically considered to "sprout jets and plumes". According to one of the scientists, "The lines are becoming more and more blurred between comets and asteroids". In 1802, shortly after discovering Pallas, Olbers suggested to Herschel and Carl Gauss that Ceres and Pallas were fragments of
16171-418: The direction of the Sun along the plane of the ecliptic . Asteroid particles that produce visible zodiacal light average about 40 μm in radius. The typical lifetimes of main-belt zodiacal cloud particles are about 700,000 years. Thus, to maintain the bands of dust, new particles must be steadily produced within the asteroid belt. It was once thought that collisions of asteroids form a major component of
16328-452: The discovery of Ceres, an informal group of 24 astronomers dubbed the " celestial police " was formed under the invitation of Franz Xaver von Zach with the express purpose of finding additional planets; they focused their search for them in the region between Mars and Jupiter where the Titius–Bode law predicted there should be a planet. About 15 months later, Heinrich Olbers , a member of
16485-456: The discrediting of the Titius–Bode law in the eyes of scientists because its orbit was nowhere near the predicted position. To date, no scientific explanation for the law has been given, and astronomers' consensus regards it as a coincidence. The expression "asteroid belt" came into use in the early 1850s, although pinpointing who coined the term is difficult. The first English use seems to be in
16642-438: The distance to the Moon, his calculated distance to the Sun would fall between 380 and 1,520 Earth radii. Hipparchus gave an estimate of the distance of Earth from the Sun, quoted by Pappus as equal to 490 Earth radii. According to the conjectural reconstructions of Noel Swerdlow and G. J. Toomer , this was derived from his assumption of a "least perceptible" solar parallax of 7 ′ . A Chinese mathematical treatise,
16799-408: The early history of the Solar System. The Hungaria asteroids lie closer to the Sun than the 4:1 resonance, but are protected from disruption by their high inclination. When the asteroid belt was first formed, the temperatures at a distance of 2.7 AU from the Sun formed a " snow line " below the freezing point of water. Planetesimals formed beyond this radius were able to accumulate ice. In 2006,
16956-418: The effects described by Einstein 's theory of relativity and upon the mathematical tools it used. Improving measurements were continually checked and cross-checked by means of improved understanding of the laws of celestial mechanics , which govern the motions of objects in space. The expected positions and distances of objects at an established time are calculated (in au) from these laws, and assembled into
17113-430: The field of topology , 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
17270-495: The first 100 million years of the Solar System's history. Some fragments eventually found their way into the inner Solar System, leading to meteorite impacts with the inner planets. Asteroid orbits continue to be appreciably perturbed whenever their period of revolution about the Sun forms an orbital resonance with Jupiter. At these orbital distances, a Kirkwood gap occurs as they are swept into other orbits. In 1596, Johannes Kepler wrote, "Between Mars and Jupiter, I place
17427-403: The first few tens of millions of years), surface melting from impacts, space weathering from radiation, and bombardment by micrometeorites . Although some scientists refer to the asteroids as residual planetesimals, other scientists consider them distinct. The current asteroid belt is believed to contain only a small fraction of the mass of the primordial belt. Computer simulations suggest that
17584-482: The first tens of millions of years of formation. In August 2007, a study of zircon crystals in an Antarctic meteorite believed to have originated from Vesta suggested that it, and by extension the rest of the asteroid belt, had formed rather quickly, within 10 million years of the Solar System's origin. The asteroids are not pristine samples of the primordial Solar System. They have undergone considerable evolution since their formation, including internal heating (in
17741-526: The formation of the planets. Planetesimals within the region that would become the asteroid belt were strongly perturbed by Jupiter's gravity. Orbital resonances occurred where the orbital period of an object in the belt formed an integer fraction of the orbital period of Jupiter, perturbing the object into a different orbit; the region lying between the orbits of Mars and Jupiter contains many such orbital resonances. As Jupiter migrated inward following its formation, these resonances would have swept across
17898-406: 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
18055-455: The known asteroids are between 11 and 19, with the median at about 16. On average the distance between the asteroids is about 965,600 km (600,000 miles), although this varies among asteroid families and smaller undetected asteroids might be even closer. The total mass of the asteroid belt is estimated to be 2.39 × 10 kg, which is 3% of the mass of the Moon. The four largest objects, Ceres, Vesta, Pallas, and Hygiea, contain an estimated 62% of
18212-414: The last few hundred years, the list includes (457175) 2008 GO 98 also known as 362P. Contrary to popular imagery, the asteroid belt is mostly empty. The asteroids are spread over such a large volume that reaching an asteroid without aiming carefully would be improbable. Nonetheless, hundreds of thousands of asteroids are currently known, and the total number ranges in the millions or more, depending on
18369-477: The latter is taken to be 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
18526-472: The layout of the planets, now known as the Titius-Bode Law . If one began a numerical sequence at 0, then included 3, 6, 12, 24, 48, etc., doubling each time, and added four to each number and divided by 10, this produced a remarkably close approximation to the radii of the orbits of the known planets as measured in astronomical units , provided one allowed for a "missing planet" (equivalent to 24 in
18683-661: The liberty of changing that name, if another, more expressive of their nature, should occur. By 1807, further investigation revealed two new objects in the region: Juno and Vesta . The burning of Lilienthal in the Napoleonic wars , where the main body of work had been done, brought this first period of discovery to a close. Despite Herschel's coinage, for several decades it remained common practice to refer to these objects as planets and to prefix their names with numbers representing their sequence of discovery: 1 Ceres, 2 Pallas, 3 Juno, 4 Vesta. In 1845, though,
18840-414: The lower size cutoff. Over 200 asteroids are known to be larger than 100 km, and a survey in the infrared wavelengths has shown that the asteroid belt has between 700,000 and 1.7 million asteroids with a diameter of 1 km or more. The number of asteroids in the main belt steadily increases with decreasing size. Although the size distribution generally follows a power law , there are 'bumps' in
18997-432: The main belt, although they can have perturbed some old asteroid families. Current main belt asteroids that originated as Centaurs or trans-Neptunian objects may lie in the outer belt with short lifetime of less than 4 million years, most likely orbiting between 2.8 and 3.2 AU at larger eccentricities than typical of main belt asteroids. Skirting the inner edge of the belt (ranging between 1.78 and 2.0 AU, with
19154-497: The measurement of the time itself must be translated to a standard scale that accounts for relativistic time dilation . Comparison of the ephemeris positions with time measurements expressed in Barycentric Dynamical Time (TDB) leads to a value for the speed of light in astronomical units per day (of 86,400 s ). By 2009, the IAU had updated its standard measures to reflect improvements, and calculated
19311-453: The metre (exactly 149,597,870,700 m ). The new definition recognizes as a consequence that the astronomical unit has reduced importance, limited in use to a convenience in some applications. This definition makes the speed of light, defined as exactly 299,792,458 m/s , equal to exactly 299,792,458 × 86,400 ÷ 149,597,870,700 or about 173.144 632 674 240 au/d, some 60 parts per trillion less than
19468-562: The migration of Jupiter's orbit. Subsequently, asteroids primarily migrate into these gap orbits due to the Yarkovsky effect , but may also enter because of perturbations or collisions. After entering, an asteroid is gradually nudged into a different, random orbit with a larger or smaller semimajor axis. The high population of the asteroid belt makes for an active environment, where collisions between asteroids occur frequently (on deep time scales). Impact events between main-belt bodies with
19625-461: The most prominent families in the asteroid belt (in order of increasing semi-major axes) are the Flora , Eunomia , Koronis , Eos , and Themis families. The Flora family, one of the largest with more than 800 known members, may have formed from a collision less than 1 billion years ago. The largest asteroid to be a true member of a family is 4 Vesta. (This is in contrast to an interloper, in
19782-430: 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
19939-409: The original asteroid belt may have contained mass equivalent to the Earth's. Primarily because of gravitational perturbations, most of the material was ejected from the belt within about 1 million years of formation, leaving behind less than 0.1% of the original mass. Since its formation, the size distribution of the asteroid belt has remained relatively stable; no significant increase or decrease in
20096-430: The original population. Evidence suggests that most main belt asteroids between 200 m and 10 km in diameter are rubble piles formed by collisions. These bodies consist of a multitude of irregular objects that are mostly bound together by self-gravity, resulting in significant amounts of internal porosity . Along with the asteroid bodies, the asteroid belt also contains bands of dust with particle radii of up to
20253-399: The other planets). The invention of the telescope allowed far more accurate measurements of angles than is possible with the naked eye. Flemish astronomer Godefroy Wendelin repeated Aristarchus’ measurements in 1635, and found that Ptolemy's value was too low by a factor of at least eleven. A somewhat more accurate estimate can be obtained by observing the transit of Venus . By measuring
20410-487: The outer belt, 7472 Kumakiri and (10537) 1991 RY 16 , with a differing basaltic composition that could not have originated from Vesta. These two are the only V-type asteroids discovered in the outer belt to date. The temperature of the asteroid belt varies with the distance from the Sun. For dust particles within the belt, typical temperatures range from 200 K (−73 °C) at 2.2 AU down to 165 K (−108 °C) at 3.2 AU. However, due to rotation,
20567-547: The parallax of Mars between Paris and Cayenne in French Guiana when Mars was at its closest to Earth in 1672. They arrived at a figure for the solar parallax of 9.5″ , equivalent to an Earth–Sun distance of about 22,000 Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague Jean Picard in 1669 as 3,269,000 toises . This same year saw another estimate for
20724-500: 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. Astronomical units The astronomical unit (symbol: au or AU ) is a unit of length defined to be exactly equal to 149,597,870,700 m . Historically,
20881-417: The points of its extremes defined the exact shape mathematically, and made possible calculations for the entire orbit as well as predictions based on observation. In addition, it mapped out exactly the largest straight-line distance that Earth traverses over the course of a year, defining times and places for observing the largest parallax (apparent shifts of position) in nearby stars. Knowing Earth's shift and
21038-406: The primordial solar nebula as a group of planetesimals , the smaller precursors of the protoplanets . However, between Mars and Jupiter gravitational perturbations from Jupiter disrupted their accretion into a planet, imparting excess kinetic energy which shattered colliding planetesimals and most of the incipient protoplanets. As a result, 99.9% of the asteroid belt's original mass was lost in
21195-500: The product is required to calculate planetary positions for an ephemeris, so ephemerides are calculated in astronomical units and not in SI units. The calculation of ephemerides also requires a consideration of the effects of general relativity . In particular, time intervals measured on Earth's surface ( Terrestrial Time , TT) are not constant when compared with the motions of the planets: the terrestrial second (TT) appears to be longer near January and shorter near July when compared with
21352-431: 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
21509-497: The sequence) between the orbits of Mars (12) and Jupiter (48). In his footnote, Titius declared, "But should the Lord Architect have left that space empty? Not at all." When William Herschel discovered Uranus in 1781, the planet's orbit closely matched the law, leading some astronomers to conclude that a planet had to be between the orbits of Mars and Jupiter. On January 1, 1801, Giuseppe Piazzi , chairman of astronomy at
21666-435: The solar parallax (and for the constant of aberration and the Gaussian gravitational constant) were incorporated into the first international system of astronomical constants in 1896, which remained in place for the calculation of ephemerides until 1964. The name "astronomical unit" appears first to have been used in 1903. The discovery of the near-Earth asteroid 433 Eros and its passage near Earth in 1900–1901 allowed
21823-424: The solar parallax (close to the modern value of 8.794 143 ″ ), although Newcomb also used data from the transits of Venus. Newcomb also collaborated with A. A. Michelson to measure the speed of light with Earth-based equipment; combined with the constant of aberration (which is related to the light time per unit distance), this gave the first direct measurement of the Earth–Sun distance in metres. Newcomb's value for
21980-555: The speed of light at 173.144 632 6847 (69) au/d (TDB). In 1983, the CIPM modified the International System of Units (SI) to make the metre defined as the distance travelled in a vacuum by light in 1 / 299,792,458 s. This replaced the previous definition, valid between 1960 and 1983, which was that the metre equalled a certain number of wavelengths of a certain emission line of krypton-86. (The reason for
22137-439: 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 ,
22294-449: 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
22451-1057: 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
22608-469: The surface temperature of an asteroid can vary considerably as the sides are alternately exposed to solar radiation then to the stellar background. Several otherwise unremarkable bodies in the outer belt show cometary activity. Because their orbits cannot be explained through the capture of classical comets, many of the outer asteroids are thought to be icy, with the ice occasionally exposed to sublimation through small impacts. Main-belt comets may have been
22765-629: The symbol of the astronomical unit was also ua. In 2012, the IAU, noting "that various symbols are presently in use for the astronomical unit", recommended the use of the symbol "au". The scientific journals published by the American Astronomical Society and the Royal Astronomical Society subsequently adopted this symbol. In the 2014 revision and 2019 edition of the SI Brochure, the BIPM used
22922-439: 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
23079-426: 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
23236-436: The time taken for photons to be reflected from an object. Because all photons move at the speed of light in vacuum, a fundamental constant of the universe, the distance of an object from the probe is calculated as the product of the speed of light and the measured time. However, for precision the calculations require adjustment for things such as the motions of the probe and object while the photons are transiting. In addition,
23393-484: 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
23550-534: 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
23707-460: The transit in two different locations, one can accurately calculate the parallax of Venus and from the relative distance of Earth and Venus from the Sun, the solar parallax α (which cannot be measured directly due to the brightness of the Sun ). Jeremiah Horrocks had attempted to produce an estimate based on his observation of the 1639 transit (published in 1662), giving a solar parallax of 15 ″ , similar to Wendelin's figure. The solar parallax
23864-420: 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
24021-452: The typical dimensions of the main-belt asteroids has occurred. The 4:1 orbital resonance with Jupiter, at a radius 2.06 astronomical units (AUs), can be considered the inner boundary of the asteroid belt. Perturbations by Jupiter send bodies straying there into unstable orbits. Most bodies formed within the radius of this gap were swept up by Mars (which has an aphelion at 1.67 AU) or ejected by its gravitational perturbations in
24178-456: The unit symbol "au". ISO 80000-3:2019, which replaces ISO 80000-3:2006, does not mention the astronomical unit. Earth's orbit around the Sun is an ellipse . The semi-major axis of this elliptic orbit is defined to be half of the straight line segment that joins the perihelion and aphelion . The centre of the Sun lies on this straight line segment, but not at its midpoint. Because ellipses are well-understood shapes, measuring
24335-410: The zodiacal light. However, computer simulations by Nesvorný and colleagues attributed 85 percent of the zodiacal-light dust to fragmentations of Jupiter-family comets, rather than to comets and collisions between asteroids in the asteroid belt. At most 10 percent of the dust is attributed to the asteroid belt. Some of the debris from collisions can form meteoroids that enter the Earth's atmosphere. Of
24492-473: Was devised by James Gregory and published in his Optica Promata (1663). It was strongly advocated by Edmond Halley and was applied to the transits of Venus observed in 1761 and 1769, and then again in 1874 and 1882. Transits of Venus occur in pairs, but less than one pair every century, and observing the transits in 1761 and 1769 was an unprecedented international scientific operation including observations by James Cook and Charles Green from Tahiti. Despite
24649-417: Was still derived from observation and measurements subject to error, and based on techniques that did not yet standardize all relativistic effects, and thus were not constant for all observers. In 2012, finding that the equalization of relativity alone would make the definition overly complex, the IAU simply used the 2009 estimate to redefine the astronomical unit as a conventional unit of length directly tied to
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