1.
Standard gravitational parameter
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In celestial mechanics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the body. μ = G M For several objects in the Solar System, the SI units of the standard gravitational parameter are m3 s−2. However, units of km3 s−2 are frequently used in the scientific literature, the central body in an orbital system can be defined as the one whose mass is much larger than the mass of the orbiting body, or M ≫ m. This approximation is standard for planets orbiting the Sun or most moons, conversely, measurements of the smaller bodys orbit only provide information on the product, μ, not G and M separately. This can be generalized for elliptic orbits, μ =4 π2 a 3 / T2 where a is the semi-major axis, for parabolic trajectories rv2 is constant and equal to 2μ. For elliptic and hyperbolic orbits μ = 2a| ε |, where ε is the orbital energy. The value for the Earth is called the gravitational constant. However, the M can be out only by dividing the MG by G. The uncertainty of those measures is 1 to 7000, so M will have the same uncertainty, the value for the Sun is called the heliocentric gravitational constant or geopotential of the Sun and equals 1. 32712440018×1020 m3 s−2. Note that the mass is also denoted by μ. Astronomical system of units Planetary mass
2.
Mass
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In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
3.
Compton wavelength
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The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons, the Compton wavelength of a particle is equivalent to the wavelength of a photon whose energy is the same as the mass of the particle. The standard Compton wavelength, λ, of a particle is given by λ = h m c, where h is the Planck constant, m is the particles mass, the significance of this formula is shown in the derivation of the Compton shift formula. The CODATA2014 value for the Compton wavelength of the electron is 6988242631023670000♠2. 4263102367×10−12 m, other particles have different Compton wavelengths. When the Compton wavelength is divided by 2π, one obtains the “reduced” Compton wavelength ƛ, i. e. the Compton wavelength for 1 radian instead of 2π radians, where ħ is the “reduced” Planck constant. The reduced Compton wavelength is a representation for mass on the quantum scale. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle and it appears in the Dirac equation, − i γ μ ∂ μ ψ + ψ =0. The reduced Compton wavelength also appears in Schrödingers equation, although its presence is obscured in traditional representations of the equation. Dividing through by ℏ c, and rewriting in terms of the structure constant, one obtains. The reduced Compton wavelength is a representation for mass on the quantum scale. Equations that pertain to inertial mass like Klein-Gordon and Schrödingers, use the reduced Compton wavelength, the non-reduced Compton wavelength is a natural representation for mass that has been converted into energy. Equations that pertain to the conversion of mass energy, or to the wavelengths of photons interacting with mass. A particle of mass m has a rest energy of E = mc2, the non-reduced Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by E = h f = h c λ = m c 2, the Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. This limitation depends on the m of the particle. To see how, note that we can measure the position of a particle by bouncing light off it –, light with a short wavelength consists of photons of high energy. If the energy of these photons exceeds mc2, when one hits the particle position is being measured the collision may yield enough energy to create a new particle of the same type. This renders moot the question of the particles location
4.
Einstein's field equation
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First published by Einstein in 1915 as a tensor equation, the EFE equate local spacetime curvature with the local energy and momentum within that spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor, the inertial trajectories of particles and radiation in the resulting geometry are then calculated using the geodesic equation. As well as obeying local energy–momentum conservation, the EFE reduce to Newtons law of gravitation where the field is weak. Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry, special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation and these equations are used to study phenomena such as gravitational waves. The EFE is an equation relating a set of symmetric 4 ×4 tensors. Each tensor has 10 independent components, although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations, the vacuum field equations define Einstein manifolds. Despite the simple appearance of the equations they are quite complicated. In fact, when written out, the EFE are a system of 10 coupled, nonlinear. The EFE can then be written as G μ ν + Λ g μ ν =8 π G c 4 T μ ν. Using geometrized units where G = c =1, this can be rewritten as G μ ν + Λ g μ ν =8 π T μ ν. The expression on the left represents the curvature of spacetime as determined by the metric, the EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime. These equations, together with the equation, which dictates how freely-falling matter moves through space-time. The above form of the EFE is the established by Misner, Thorne. The sign of the term would change in both these versions, if the metric sign convention is used rather than the MTW metric sign convention adopted here. Taking the trace with respect to the metric of both sides of the EFE one gets R − D2 R + D Λ =8 π G c 4 T where D is the spacetime dimension
5.
Radius
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Remote Authentication Dial-In User Service is a networking protocol that provides centralized Authentication, Authorization, and Accounting management for users who connect and use a network service. RADIUS was developed by Livingston Enterprises, Inc. in 1991 as an access server authentication and accounting protocol and these networks may incorporate modems, DSL, access points, VPNs, network ports, web servers, etc. RADIUS is a protocol that runs in the application layer. Network access servers, the gateways that control access to a network, RADIUS is often the back-end of choice for 802. 1X authentication as well. The RADIUS server is usually a background process running on a UNIX or Microsoft Windows server, RADIUS is a AAA protocol which manages network access in the following two-step process, also known as a AAA transaction. AAA stands for authentication, authorization and accounting, Authentication and authorization characteristics in RADIUS are described in RFC2865 while accounting is described by RFC2866. The user or machine sends a request to a Network Access Server to gain access to a network resource using access credentials. The credentials are passed to the NAS device via the link-layer protocol - for example, in turn, the NAS sends a RADIUS Access Request message to the RADIUS server, requesting authorization to grant access via the RADIUS protocol. This request includes access credentials, typically in the form of username, the RADIUS server checks that the information is correct using authentication schemes such as PAP, CHAP or EAP. Historically, RADIUS servers checked the users information against a locally stored flat file database, modern RADIUS servers can do this, or can refer to external sources — commonly SQL, Kerberos, LDAP, or Active Directory servers — to verify the users credentials. The RADIUS server then returns one of three responses to the NAS, 1) Access Reject, 2) Access Challenge, or 3) Access Accept, Access Reject The user is unconditionally denied access to all requested network resources. Reasons may include failure to provide proof of identification or an unknown or inactive user account, Access Challenge Requests additional information from the user such as a secondary password, PIN, token, or card. Access Accept The user is granted access, once the user is authenticated, the RADIUS server will often check that the user is authorized to use the network service requested. A given user may be allowed to use a wireless network. Again, this information may be stored locally on the RADIUS server, each of these three RADIUS responses may include a Reply-Message attribute which may give a reason for the rejection, the prompt for the challenge, or a welcome message for the accept. The text in the attribute can be passed on to the user in a web page. Authorization attributes are conveyed to the NAS stipulating terms of access to be granted and this might be with a customizable login prompt, where the user is expected to enter their username and password. Alternatively, the user use a link framing protocol such as the Point-to-Point Protocol
6.
Event horizon
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In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In laymans terms, it is defined as the shell of points of no return, i. e. the points at which the gravitational pull becomes so great as to make escape impossible, an event horizon is most commonly associated with black holes. Light emitted from inside the event horizon can never reach the outside observer, likewise, any object approaching the horizon from the observers side appears to slow down and never quite pass through the horizon, with its image becoming more and more redshifted as time elapses. This means that the wavelength is getting longer as the object away from the observer. The traveling object, however, experiences no strange effects and does, in fact, more specific types of horizon include the related but distinct absolute and apparent horizons found around a black hole. Often, this is described as the boundary within which the black holes escape velocity is greater than the speed of light, the surface at the Schwarzschild radius acts as an event horizon in a non-rotating body that fits inside this radius. The Schwarzschild radius of an object is proportional to its mass, theoretically, any amount of matter will become a black hole if compressed into a space that fits within its corresponding Schwarzschild radius. For the mass of the Sun this radius is approximately 3 kilometers, in practice, however, neither the Earth nor the Sun has the necessary mass and therefore the necessary gravitational force, to overcome electron and neutron degeneracy pressure. The minimal mass required for a star to be able to collapse beyond these pressures is the Tolman-Oppenheimer-Volkoff limit, black hole event horizons are widely misunderstood. As with any mass in the Universe, matter must come within its scope for the possibility to exist of capture or consolidation with any other mass. Equally common is the idea that matter can be observed “falling into” a black hole, astronomers can only detect accretion disks around black holes, where material moves with such speed that friction creates high-energy radiation which can be detected. Furthermore, a distant observer will never see something cross the horizon. Instead, while approaching the hole, the object will seem to go ever more slowly, in cosmology, the event horizon of the observable universe is the largest comoving distance from which light emitted now can ever reach the observer in the future. This differs from the concept of particle horizon, which represents the largest comoving distance from which light emitted in the past could have reached the observer at a given time. For events beyond that distance, light has not had time to reach our location, how the particle horizon changes with time depends on the nature of the expansion of the Universe. If the expansion has certain characteristics, there are parts of the Universe that will never be observable, the boundary past which events cannot ever be observed is an event horizon, and it represents the maximum extent of the particle horizon. The criterion for determining whether a particle horizon for the Universe exists is as follows, define a comoving distance d p by d p = ∫0 t 0 c a d t. In this equation, a is the factor, c is the speed of light
7.
Germany
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Germany, officially the Federal Republic of Germany, is a federal parliamentary republic in central-western Europe. It includes 16 constituent states, covers an area of 357,021 square kilometres, with about 82 million inhabitants, Germany is the most populous member state of the European Union. After the United States, it is the second most popular destination in the world. Germanys capital and largest metropolis is Berlin, while its largest conurbation is the Ruhr, other major cities include Hamburg, Munich, Cologne, Frankfurt, Stuttgart, Düsseldorf and Leipzig. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity, a region named Germania was documented before 100 AD. During the Migration Period the Germanic tribes expanded southward, beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation, in 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the German Revolution of 1918–1919, the Empire was replaced by the parliamentary Weimar Republic, the establishment of the national socialist dictatorship in 1933 led to World War II and the Holocaust. After a period of Allied occupation, two German states were founded, the Federal Republic of Germany and the German Democratic Republic, in 1990, the country was reunified. In the 21st century, Germany is a power and has the worlds fourth-largest economy by nominal GDP. As a global leader in industrial and technological sectors, it is both the worlds third-largest exporter and importer of goods. Germany is a country with a very high standard of living sustained by a skilled. It upholds a social security and universal health system, environmental protection. Germany was a member of the European Economic Community in 1957. It is part of the Schengen Area, and became a co-founder of the Eurozone in 1999, Germany is a member of the United Nations, NATO, the G8, the G20, and the OECD. The national military expenditure is the 9th highest in the world, the English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine. This in turn descends from Proto-Germanic *þiudiskaz popular, derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- people, the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a mine in Schöningen where three 380, 000-year-old wooden javelins were unearthed
8.
Karl Schwarzschild
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Karl Schwarzschild was a German physicist and astronomer. He was also the father of astrophysicist Martin Schwarzschild, Schwarzschild accomplished this while serving in the German army during World War I. He died the year from the autoimmune disease pemphigus, which he developed while at the Russian front. Various forms of the disease particularly affect people of Ashkenazi Jewish origin, asteroid 837 Schwarzschilda is named in his honour, as is the large crater Schwarzschild, on the far side of the moon. Schwarzschild was born in Frankfurt am Main to Jewish parents and his father was active in the business community of the city, and the family had ancestors in the city dating back to the sixteenth century. Karl attended a Jewish primary school until 11 years of age and he was something of a child prodigy, having two papers on binary orbits published before he was sixteen. He studied at Straßburg and Munich, obtaining his doctorate in 1896 for a work on Henri Poincarés theories, from 1897, he worked as assistant at the Kuffner observatory in Vienna. From 1901 until 1909 he was a professor at the institute at Göttingen. Schwarzschild became the director of the observatory in Göttingen and he married Else Posenbach, the daughter of a professor of surgery at Göttingen, in 1909, and later that year moved to Potsdam, where he took up the post of director of the Astrophysical Observatory. This was then the most prestigious post available for an astronomer in Germany and he and Else had three children, Agathe, Martin, and Alfred. From 1912, Schwarzschild was a member of the Prussian Academy of Sciences, at the outbreak of World War I in 1914 he joined the German army despite being over 40 years old. He served on both the western and eastern fronts, rising to the rank of lieutenant in the artillery, while serving on the front in Russia in 1915, he began to suffer from a rare and painful skin disease called pemphigus. Nevertheless, he managed to three outstanding papers, two on relativity theory and one on quantum theory. Schwarzschilds struggle with pemphigus may have led to his death. Thousands of dissertations, articles, and books have since been devoted to the study of Schwarzschilds solutions to the Einstein field equations, while at Vienna in 1897, Schwarzschild developed a formula, now known as the Schwarzschild law, to calculate the optical density of photographic material. It involved an exponent now known as the Schwarzschild exponent, which is the p in the formula and this formula was important for enabling more accurate photographic measurements of the intensities of faint astronomical sources. Two points on two world lines contribute to the Lagrangian only if they are a zero Minkowskian distance, hence the term δ, the idea was further developed by Tetrode and Fokker in the 1920s and Wheeler and Feynman in the 1940s and constitutes an alternative/equivalent formulation of electrodynamics. Einsteins approximate solution was given in his famous 1915 article on the advance of the perihelion of Mercury, there, Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass
9.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
10.
Gravitational constant
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Its measured value is 6. 67408×10−11 m3⋅kg−1⋅s−2. The constant of proportionality, G, is the gravitational constant, colloquially, the gravitational constant is also called Big G, for disambiguation with small g, which is the local gravitational field of Earth. The two quantities are related by g = GME/r2 E. In the general theory of relativity, the Einstein field equations, R μ ν −12 R g μ ν =8 π G c 4 T μ ν, the scaled gravitational constant κ = 8π/c4G ≈2. 071×10−43 s2·m−1·kg−1 is also known as Einsteins constant. The gravitational constant is a constant that is difficult to measure with high accuracy. This is because the force is extremely weak compared with other fundamental forces. In SI units, the 2014 CODATA-recommended value of the constant is. In cgs, G can be written as G ≈6. 674×10−8 cm3·g−1·s−2, in other words, in Planck units, G has the numerical value of 1. In astrophysics, it is convenient to measure distances in parsecs, velocities in kilometers per second, in these units, the gravitational constant is, G ≈4.302 ×10 −3 p c M ⊙ −12. In orbital mechanics, the period P of an object in orbit around a spherical object obeys G M =3 π V P2 where V is the volume inside the radius of the orbit. It follows that P2 =3 π G V M ≈10.896 h r 2 g c m −3 V M. This way of expressing G shows the relationship between the density of a planet and the period of a satellite orbiting just above its surface. Cavendish measured G implicitly, using a torsion balance invented by the geologist Rev. John Michell and he used a horizontal torsion beam with lead balls whose inertia he could tell by timing the beams oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused, cavendishs aim was not actually to measure the gravitational constant, but rather to measure Earths density relative to water, through the precise knowledge of the gravitational interaction. In modern units, the density that Cavendish calculated implied a value for G of 6. 754×10−11 m3·kg−1·s−2, the accuracy of the measured value of G has increased only modestly since the original Cavendish experiment. G is quite difficult to measure, because gravity is weaker than other fundamental forces. Published values of G have varied rather broadly, and some recent measurements of precision are, in fact. This led to the 2010 CODATA value by NIST having 20% increased uncertainty than in 2006, for the 2014 update, CODATA reduced the uncertainty to less than half the 2010 value
11.
Schwarzschild metric
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The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. The solution is named after Karl Schwarzschild, who first published the solution in 1916, according to Birkhoffs theorem, the Schwarzschild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a hole that has no electric charge or angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, the Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole, the analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle. In practice, the ratio rs/r is almost always extremely small, for example, the Schwarzschild radius rs of the Earth is roughly 8.9 mm, while the Sun, which is 3. 3×105 times as massive has a Schwarzschild radius of approximately 3.0 km. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion, the ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars. The Schwarzschild metric is a solution of Einsteins field equations in empty space and that is, for a spherical body of radius R the solution is valid for r > R. It was the first exact solution of the Einstein field equations other than the flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I, johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation. In the early years of general relativity there was a lot of confusion about the nature of the found in the Schwarzschild. In Schwarzschilds original paper, he put what we now call the event horizon at the origin of his coordinate system, in this paper he also introduced what is now known as the Schwarzschild radial coordinate, as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius, a more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r =0 and r = rs. Although there was consensus that the singularity at r =0 was a genuine physical singularity. They, however, did not recognize that their solutions were just coordinate transforms, later, in 1932, Georges Lemaître gave a different coordinate transformation to the same effect and was the first to recognize that this implied that the singularity at r = rs was not physical. A similar result was later rediscovered by George Szekeres, and independently Martin Kruskal, the new coordinates nowadays known as Kruskal-Szekeres coordinates were much simpler than Synges but both provided a single set of coordinates that covered the entire spacetime. This led to identification of the r = rs singularity in the Schwarzschild metric as an event horizon. The Schwarzschild solution appears to have singularities at r =0 and r = rs, since the Schwarzschild metric is only expected to be valid for radii larger than the radius R of the gravitating body, there is no problem as long as R > rs
12.
Mathematical singularity
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See Singularity theory for general discussion of the geometric theory, which only covers some aspects. For example, the function f =1 x on the line has a singularity at x =0. The function g = |x| also has a singularity at x =0, similarly, the graph defined by y2 = x also has a singularity at, this time because it has a corner at that point. The algebraic set defined by in the system has a singularity at because it does not admit a tangent there. In real analysis singularities are either discontinuities or discontinuities of the derivative, there are four kinds of discontinuities, type I, which has two sub-types, and type II, which also can be divided into two subtypes, but normally is not. To describe these two limits are used. There are some functions for which these limits do not exist at all, for example, the function g = sin does not tend towards anything as x approaches c =0. The limits in this case are not infinite, but rather undefined, borrowing from complex analysis, this is sometimes called an essential singularity. A point of continuity, is a value of c for which f = f = f, all the values must be finite. Two subtypes occur, A jump discontinuity occurs when f ≠ f, regardless of whether f exists, and regardless of what value it might have if it does exist. A removable discontinuity occurs when f = f, but either the value of f does not match the limits, a type II discontinuity occurs when either f or f does not exist. An essential singularity is a term borrowed from complex analysis and this is the case when either one or the other limits f or f does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if legal answers are extended to include ± ∞, in real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, a coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example is the apparent singularity at the 90 degree latitude in spherical coordinates, an object moving due north on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole. This discontinuity, however, is apparent, it is an artifact of the coordinate system chosen. A different coordinate system would eliminate the apparent discontinuity, e. g. by replacing latitude/longitude with n-vector, in complex analysis there are several classes of singularities, described below. Suppose U is a subset of the complex numbers C, and the point a is an element of U
13.
Black hole
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A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it. The theory of relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon, although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed. In many ways a black hole acts like a black body. Moreover, quantum theory in curved spacetime predicts that event horizons emit Hawking radiation. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. Black holes were considered a mathematical curiosity, it was during the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality, black holes of stellar mass are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings, by absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses may form. There is general consensus that supermassive black holes exist in the centers of most galaxies, despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter that falls onto a black hole can form an accretion disk heated by friction. If there are other stars orbiting a black hole, their orbits can be used to determine the black holes mass, such observations can be used to exclude possible alternatives such as neutron stars.3 million solar masses. On 15 June 2016, a detection of a gravitational wave event from colliding black holes was announced. The idea of a body so massive that light could not escape was briefly proposed by astronomical pioneer John Michell in a letter published in 1783-4. Michell correctly noted that such supermassive but non-radiating bodies might be detectable through their effects on nearby visible bodies. In 1915, Albert Einstein developed his theory of general relativity, only a few months later, Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the solution for the point mass. This solution had a peculiar behaviour at what is now called the Schwarzschild radius, the nature of this surface was not quite understood at the time
14.
Sun
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The Sun is the star at the center of the Solar System. It is a perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process. It is by far the most important source of energy for life on Earth. Its diameter is about 109 times that of Earth, and its mass is about 330,000 times that of Earth, accounting for about 99. 86% of the total mass of the Solar System. About three quarters of the Suns mass consists of hydrogen, the rest is mostly helium, with smaller quantities of heavier elements, including oxygen, carbon, neon. The Sun is a G-type main-sequence star based on its spectral class and it formed approximately 4.6 billion years ago from the gravitational collapse of matter within a region of a large molecular cloud. Most of this matter gathered in the center, whereas the rest flattened into a disk that became the Solar System. The central mass became so hot and dense that it eventually initiated nuclear fusion in its core and it is thought that almost all stars form by this process. The Sun is roughly middle-aged, it has not changed dramatically for more than four billion years and it is calculated that the Sun will become sufficiently large enough to engulf the current orbits of Mercury, Venus, and probably Earth. The enormous effect of the Sun on Earth has been recognized since prehistoric times, the synodic rotation of Earth and its orbit around the Sun are the basis of the solar calendar, which is the predominant calendar in use today. The English proper name Sun developed from Old English sunne and may be related to south, all Germanic terms for the Sun stem from Proto-Germanic *sunnōn. The English weekday name Sunday stems from Old English and is ultimately a result of a Germanic interpretation of Latin dies solis, the Latin name for the Sun, Sol, is not common in general English language use, the adjectival form is the related word solar. The term sol is used by planetary astronomers to refer to the duration of a solar day on another planet. A mean Earth solar day is approximately 24 hours, whereas a mean Martian sol is 24 hours,39 minutes, and 35.244 seconds. From at least the 4th Dynasty of Ancient Egypt, the Sun was worshipped as the god Ra, portrayed as a falcon-headed divinity surmounted by the solar disk, and surrounded by a serpent. In the New Empire period, the Sun became identified with the dung beetle, in the form of the Sun disc Aten, the Sun had a brief resurgence during the Amarna Period when it again became the preeminent, if not only, divinity for the Pharaoh Akhenaton. The Sun is viewed as a goddess in Germanic paganism, Sól/Sunna, in ancient Roman culture, Sunday was the day of the Sun god. It was adopted as the Sabbath day by Christians who did not have a Jewish background, the symbol of light was a pagan device adopted by Christians, and perhaps the most important one that did not come from Jewish traditions
15.
Earth
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Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
16.
Observable universe
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There are at least two trillion galaxies in the observable universe, containing more stars than all the grains of sand on planet Earth. Assuming the universe is isotropic, the distance to the edge of the universe is roughly the same in every direction. That is, the universe is a spherical volume centered on the observer. Every location in the Universe has its own universe, which may or may not overlap with the one centered on Earth. The word observable used in this sense does not depend on modern technology actually permits detection of radiation from an object in this region. It simply indicates that it is possible in principle for light or other signals from the object to reach an observer on Earth, in practice, we can see light only from as far back as the time of photon decoupling in the recombination epoch. That is when particles were first able to emit photons that were not quickly re-absorbed by other particles, before then, the Universe was filled with a plasma that was opaque to photons. The detection of gravitational waves indicates there is now a possibility of detecting signals from before the recombination epoch. The surface of last scattering is the collection of points in space at the distance that photons from the time of photon decoupling just reach us today. These are the photons we detect today as cosmic microwave background radiation, however, with future technology, it may be possible to observe the still older relic neutrino background, or even more distant events via gravitational waves. It is estimated that the diameter of the universe is about 28.5 gigaparsecs. The total mass of matter in the universe can be calculated using the critical density. Some parts of the Universe are too far away for the light emitted since the Big Bang to have had time to reach Earth. In the future, light from distant galaxies will have had time to travel. This fact can be used to define a type of cosmic event horizon whose distance from the Earth changes over time, both popular and professional research articles in cosmology often use the term universe to mean observable universe. It is plausible that the galaxies within our observable universe represent only a fraction of the galaxies in the Universe. If the Universe is finite but unbounded, it is possible that the Universe is smaller than the observable universe. In this case, what we take to be very distant galaxies may actually be duplicate images of nearby galaxies and it is difficult to test this hypothesis experimentally because different images of a galaxy would show different eras in its history, and consequently might appear quite different
17.
Milky Way
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The Milky Way is the galaxy that contains our Solar System. The descriptive milky is derived from the appearance from Earth of the galaxy – a band of light seen in the night sky formed from stars that cannot be distinguished by the naked eye. The term Milky Way is a translation of the Latin via lactea, from Earth, the Milky Way appears as a band because its disk-shaped structure is viewed from within. Galileo Galilei first resolved the band of light into individual stars with his telescope in 1610, until the early 1920s, most astronomers thought that the Milky Way contained all the stars in the Universe. Following the 1920 Great Debate between the astronomers Harlow Shapley and Heber Curtis, observations by Edwin Hubble showed that the Milky Way is just one of many galaxies, the Milky Way is a barred spiral galaxy with a diameter between 100,000 light-years and 180,000 light-years. The Milky Way is estimated to contain 100–400 billion stars, there are probably at least 100 billion planets in the Milky Way. The Solar System is located within the disk, about 26,000 light-years from the Galactic Center, on the edge of one of the spiral-shaped concentrations of gas. The stars in the inner ≈10,000 light-years form a bulge, the very center is marked by an intense radio source, named Sagittarius A*, which is likely to be a supermassive black hole. Stars and gases at a range of distances from the Galactic Center orbit at approximately 220 kilometers per second. The constant rotation speed contradicts the laws of Keplerian dynamics and suggests much of the mass of the Milky Way does not emit or absorb electromagnetic radiation. This mass has been termed dark matter, the rotational period is about 240 million years at the position of the Sun. The Milky Way as a whole is moving at a velocity of approximately 600 km per second with respect to frames of reference. The oldest stars in the Milky Way are nearly as old as the Universe itself, the Milky Way has several satellite galaxies and is part of the Local Group of galaxies, which is a component of the Virgo Supercluster, which is itself a component of the Laniakea Supercluster. The Milky Way can be seen as a band of white light some 30 degrees wide arcing across the sky. Dark regions within the band, such as the Great Rift, the area of the sky obscured by the Milky Way is called the Zone of Avoidance. The Milky Way has a low surface brightness. Its visibility can be reduced by background light such as light pollution or stray light from the Moon. The sky needs to be darker than about 20.2 magnitude per square arcsecond in order for the Milky Way to be seen and it should be visible when the limiting magnitude is approximately +5.1 or better and shows a great deal of detail at +6.1
18.
Sagittarius A*
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Sagittarius A* is a bright and very compact astronomical radio source at the center of the Milky Way, near the border of the constellations Sagittarius and Scorpius. Astronomers have been unable to observe Sgr A* in the spectrum because of the effect of 25 magnitudes of extinction by dust. Several teams of researchers have attempted to image Sagittarius A* in the spectrum using very-long-baseline interferometry. The current highest-resolution measurement, made at a wavelength of 1.3 mm, at a distance of 26,000 light-years, this yields a diameter of 44 million kilometers. For comparison, Earth is 150 million kilometers from the Sun, the proper motion of Sgr A* is approximately −2.70 mas per year for the right ascension and −5.6 mas per year for the declination. There are plans to image Sagittarius A* in much greater detail than before using interferometry combining images from widely spaced observatories at different places on Earth and it is hoped the measurements will test Einsteins theory of relativity more rigorously than has previously been done. Discrepancies may be found between the theory and actual observation or, if there are no discrepancies relativity may be confirmed. Karl Jansky was the first person to determine that a signal was coming from a location at the center of the Milky Way. Sgr A* was discovered on February 13 and 15,1974, by astronomers Bruce Balick, the name Sgr A* was coined by Brown in a 1982 paper because the radio source was exciting, and excited states of atoms are denoted with asterisks. The observations of S2 used near-infra red interferometry because of reduced interstellar extinction in this band, siO masers were used to align NIR images with radio observations, as they can be observed in both NIR and radio bands. The rapid motion of S2 easily stood out against slower-moving stars along the line-of-sight so these could be subtracted from the images, the VLBI radio observations of Sagittarius A* could also be aligned centrally with the images so S2 could be seen to orbit Sagittarius A*. From examining the Keplerian orbit of S2, they determined the mass of Sagittarius A* to be 2.6 ±0.2 million solar masses, confined in a volume with a radius no more than 17 light-hours. Later observations of the star S14 showed the mass of the object to be about 4.1 million solar masses within a volume with no larger than 6.25 light-hours or about 6.7 billion kilometres. They also determined the distance from Earth to the Galactic Center, in November 2004 a team of astronomers reported the discovery of a potential intermediate-mass black hole, referred to as GCIRS 13E, orbiting three light-years from Sagittarius A*. This black hole of 1,300 solar masses is within a cluster of seven stars and this observation may add support to the idea that supermassive black holes grow by absorbing nearby smaller black holes and stars. After monitoring stellar orbits around Sagittarius A* for 16 years, Gillessen et al. estimate the mass at 4.31 ±0.38 million solar masses. The result was announced in 2008 and published in The Astrophysical Journal in 2009, reinhard Genzel, team leader of the research, said the study has delivered what is now considered to be the best empirical evidence that super-massive black holes do really exist. The stellar orbits in the Galactic Center show that the mass concentration of four million solar masses must be a black hole
19.
Andromeda Galaxy
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The Andromeda Galaxy, also known as Messier 31, M31, or NGC224, is a spiral galaxy approximately 780 kiloparsecs from Earth. It is the nearest major galaxy to the Milky Way and was referred to as the Great Andromeda Nebula in older texts. It received its name from the area of the sky in which it appears, the constellation of Andromeda, which was named after the mythological princess Andromeda. Andromeda is approximately 220,000 light years across, and it is the largest galaxy of the Local Group, which contains the Milky Way, the Triangulum Galaxy. The mass of the Andromeda Galaxy is estimated to be 1. 5×1012 solar masses, the Milky Way and Andromeda galaxies are expected to collide in 4.5 billion years, eventually merging to form a giant elliptical galaxy or perhaps a large disc galaxy. Around the year 964, the Persian astronomer Abd al-Rahman al-Sufi described the Andromeda Galaxy, star charts of that period labeled it as the Little Cloud. In 1612, the German astronomer Simon Marius gave a description of the Andromeda Galaxy based on telescopic observations. In 1764, Charles Messier catalogued Andromeda as object M31 and incorrectly credited Marius as the discoverer despite it being visible to the naked eye, in 1785, the astronomer William Herschel noted a faint reddish hue in the core region of Andromeda. In 1850, William Parsons, 3rd Earl of Rosse, saw, in 1864, William Huggins noted that the spectrum of Andromeda differs from a gaseous nebula. The spectra of Andromeda displays a continuum of frequencies, superimposed with dark lines that help identify the chemical composition of an object. Andromedas spectrum is similar to the spectra of individual stars. In 1885, a supernova was seen in Andromeda, the first, at the time Andromeda was considered to be a nearby object, so the cause was thought to be a much less luminous and unrelated event called a nova, and was named accordingly, Nova 1885. In 1887, Isaac Roberts took the first photographs of Andromeda, Roberts actually mistook Andromeda and similar spiral nebulae as solar systems being formed. In 1912, Vesto Slipher used spectroscopy to measure the velocity of Andromeda with respect to our solar system—the largest velocity yet measured. In 1917, Heber Curtis observed a nova within Andromeda, searching the photographic record,11 more novae were discovered. Curtis noticed that these novae were, on average,10 magnitudes fainter than those that occurred elsewhere in the sky, as a result, he was able to come up with a distance estimate of 500,000 light-years. He became a proponent of the island universes hypothesis, which held that spiral nebulae were actually independent galaxies. In 1920, the Great Debate between Harlow Shapley and Curtis took place, concerning the nature of the Milky Way, spiral nebulae, in 1922 Ernst Öpik presented a method to estimate the distance of Andromeda using the measured velocities of its stars
20.
NGC 4889
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NGC4889 is a class-4 supergiant elliptical galaxy. It was discovered in 1785 by the British astronomer Frederick William Herschel I, the brightest galaxy within the northern Coma Cluster, it is located at a distance of 94 million parsecs from Earth. At the core of the galaxy is a black hole that heats the intracluster medium through the action of friction from infalling gases. The X-ray emission from the galaxy extends out to several light years of the cluster. As with other elliptical galaxies, only a fraction of the mass of NGC4889 is in the form of stars. They have a flattened, unequal distribution that bulges within its edge, between the stars is a dense interstellar medium full of heavy elements emitted by evolved stars. The diffuse stellar halo extends out to one light years in diameter. Orbiting the galaxy is a large population of globular clusters. NGC4889 is also a source of soft X-ray, ultraviolet. NGC4889 was not included by the astronomer Charles Messier in his famous Messier catalogue despite being a bright object quite close to some Messier objects. In 1864, Herschels son, John Frederick William Herschel, published the General Catalogue of Nebulae and Clusters of Stars. He included the objects catalogued by his father, including the one later to be called NGC4889, plus others he found that were missed by his father.0 projects. It was then decided that the object to be called by its designation, NGC4889. In December 1995, Patrick Caldwell Moore compiled the Caldwell catalogue, the list also includes NGC4889, which is given the designation Caldwell 35. NGC4889 is located along the high region of Coma Berenices. It can be traced by following the line from Beta Comae Berenices to Gamma Comae Berenices. With an apparent magnitude of 11.4, it can be seen by telescopes with 12 inch aperture, but its visibility is greatly affected by light pollution due to glare of the light from Beta Comae Berenices. However, under very dark, moonless skies, it can be seen by small telescopes as a faint smudge, NGC4889 is far enough that its distance can be measured using redshift
21.
Human
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Modern humans are the only extant members of Hominina tribe, a branch of the tribe Hominini belonging to the family of great apes. Several of these hominins used fire, occupied much of Eurasia and they began to exhibit evidence of behavioral modernity around 50,000 years ago. In several waves of migration, anatomically modern humans ventured out of Africa, the spread of humans and their large and increasing population has had a profound impact on large areas of the environment and millions of native species worldwide. Humans are uniquely adept at utilizing systems of communication for self-expression and the exchange of ideas. Humans create complex structures composed of many cooperating and competing groups, from families. Social interactions between humans have established a wide variety of values, social norms, and rituals. These human societies subsequently expanded in size, establishing various forms of government, religion, today the global human population is estimated by the United Nations to be near 7.5 billion. In common usage, the word generally refers to the only extant species of the genus Homo—anatomically and behaviorally modern Homo sapiens. In scientific terms, the meanings of hominid and hominin have changed during the recent decades with advances in the discovery, there is also a distinction between anatomically modern humans and Archaic Homo sapiens, the earliest fossil members of the species. The English adjective human is a Middle English loanword from Old French humain, ultimately from Latin hūmānus, the words use as a noun dates to the 16th century. The native English term man can refer to the species generally, the species binomial Homo sapiens was coined by Carl Linnaeus in his 18th century work Systema Naturae. The generic name Homo is a learned 18th century derivation from Latin homō man, the species-name sapiens means wise or sapient. Note that the Latin word homo refers to humans of either gender, the genus Homo evolved and diverged from other hominins in Africa, after the human clade split from the chimpanzee lineage of the hominids branch of the primates. The closest living relatives of humans are chimpanzees and gorillas, with the sequencing of both the human and chimpanzee genome, current estimates of similarity between human and chimpanzee DNA sequences range between 95% and 99%. The gibbons and orangutans were the first groups to split from the leading to the humans. The splitting date between human and chimpanzee lineages is placed around 4–8 million years ago during the late Miocene epoch, during this split, chromosome 2 was formed from two other chromosomes, leaving humans with only 23 pairs of chromosomes, compared to 24 for the other apes. There is little evidence for the divergence of the gorilla, chimpanzee. Each of these species has been argued to be an ancestor of later hominins
22.
Deriving the Schwarzschild solution
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The Schwarzschild solution describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. Of the solutions to the Einstein field equations, it is considered by some to be one of the simplest and most useful, as a result of this, some textbooks omit the rigorous derivation of this metric, provided below. Working in a chart with coordinates labelled 1 to 4 respectively. The solution is assumed to be symmetric, static and vacuum. For the purposes of article, these assumptions may be stated as follows, A spherically symmetric spacetime is one that is invariant under rotations. A static spacetime is one in which all components are independent of the time coordinate t. A vacuum solution is one that satisfies the equation T a b =0, from the Einstein field equations, this implies that R a b =0 since contracting R a b − R2 g a b =0 yields R =0. The first simplification to be made is to diagonalise the metric, under the coordinate transformation, →, all metric components should remain the same. On each hypersurface of constant t, constant θ and constant ϕ, g 11 should only depend on r, note that if A or B is equal to zero at some point, the metric would be singular at that point. Using the metric above, we find the Christoffel symbols, where the indices are =, the sign ′ denotes a total derivative of a function. Subtracting the first and third equations produces, A ′ B + A B ′ =0 ⇒ A B = K where K is a real constant. The geodesics of the metric must, in some limit, agree with the solutions of Newtonian motion, the metric singularity is not a physical one, as can be shown by using a suitable coordinate transformation. The Schwarzschild metric can also be derived using the physics for a circular orbit. Start with the metric with coefficients that are unknown coefficients of r, − c 2 =2 = A2 + r 22 + B2, now apply the Euler-Lagrange equation to the arc length integral J = ∫ τ1 τ2 −2 d τ. Since d s / d τ is constant, the integrand can be replaced with 2, keplers third law of motion is T2 r 3 =4 π2 G. In a circular orbit, the period T equals 2 π /, so B ′ = −2 G M / r 2 and integrating this yields B =2 G M / r + C, where C is an unknown constant of integration. C can be determined by setting M =0, in case the space-time is flat. So C = − c 2 and B =2 G M / r − c 2 = c 2 = c 2, when the point mass is temporarily stationary, r ˙ =0 and ϕ ˙ =0
23.
Rotating black hole
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A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry, there are four known, exact, black hole solutions to the Einstein field equations, which describe gravity in general relativity. Two of those rotate, the Kerr and Kerr–Newman black holes and these numbers represent the conserved attributes of an object which can be determined from a distance by examining its electromagnetic and gravitational fields. All other variations in the hole will either escape to infinity or be swallowed up by the black hole. This is because anything happening inside the black hole horizon cannot affect events outside of it, as most stars rotate it is expected that most black holes in nature are rotating black holes. In late 2006, astronomers reported estimates of the rates of black holes in the The Astrophysical Journal. A black hole in the Milky Way, GRS 1915+105, may rotate 1,150 times per second, the formation of a rotating black hole by a collapsar is thought to be observed as the emission of gamma ray bursts. A rotating black hole can produce large amounts of energy at the expense of its rotational energy and this happens through the Penrose process in the black holes ergosphere, an area just outside its event horizon. A rotating black hole is a solution of Einsteins field equation, there are two known exact solutions, the Kerr metric and the Kerr–Newman metric, which are believed to be representative of all rotating black hole solutions, in the exterior region. Kerr black holes as wormholes BKL singularity – solution representing interior geometry of black holes formed by gravitational collapse, melia, Fulvio, The Galactic Supermassive Black Hole, Princeton U Press,2007 Macvey, John W
24.
Supermassive black hole
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In the case of the Milky Way, the SMBH corresponds with the location of Sagittarius A*. Supermassive black holes have properties that distinguish them from lower-mass classifications, first, the average density of a supermassive black hole can be less than the density of water in the case of some supermassive black holes. This is because the Schwarzschild radius is proportional to mass. In addition, the forces in the vicinity of the event horizon are significantly weaker for massive black holes. Unlike with stellar mass black holes, one would not experience significant tidal force until very deep into the black hole, donald Lynden-Bell and Martin Rees hypothesized in 1971 that the center of the Milky Way galaxy would contain a supermassive black hole. Sagittarius A* was discovered and named on February 13 and 15,1974, by astronomers Bruce Balick and they discovered a radio source that emits synchrotron radiation, it was found to be dense and immobile because of its gravitation. This was, therefore, the first indication that a black hole exists in the center of the Milky Way. The origin of black holes remains an open field of research. Astrophysicists agree that once a black hole is in place in the center of a galaxy, it can grow by accretion of matter, there are, however, several hypotheses for the formation mechanisms and initial masses of the progenitors, or seeds, of supermassive black holes. The most obvious hypothesis is that the seeds are black holes of tens or perhaps hundreds of masses that are left behind by the explosions of massive stars. Yet another model involves a dense stellar cluster undergoing core-collapse as the heat capacity of the system drives the velocity dispersion in the core to relativistic speeds. Finally, primordial black holes may have been produced directly from external pressure in the first moments after the Big Bang, formation of black holes from the deaths of the first stars has been extensively studied and corroborated by observations. The other models for black hole formation listed above are theoretical, the difficulty in forming a supermassive black hole resides in the need for enough matter to be in a small enough volume. This matter needs to have very little angular momentum in order for this to happen, normally, the process of accretion involves transporting a large initial endowment of angular momentum outwards, and this appears to be the limiting factor in black hole growth. This is a component of the theory of accretion disks. Gas accretion is the most efficient and also the most conspicuous way in black holes grow. The majority of the growth of supermassive black holes is thought to occur through episodes of rapid gas accretion. Observations reveal that quasars were more frequent when the Universe was younger, indicating that supermassive black holes formed
25.
Stellar black hole
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A stellar black hole is a black hole formed by the gravitational collapse of a massive star. They have masses ranging from about 5 to several tens of solar masses, the process is observed as a hypernova explosion or as a gamma ray burst. These black holes are also referred to as collapsars, by the no-hair theorem, a black hole can only have three fundamental properties, mass, electric charge and angular momentum. It is believed that black holes formed in all have spin. The spin of a black hole is due to the conservation of angular momentum of the star that produced it. The gravitational collapse of a star is a process that can produce a black hole. It is inevitable at the end of the life of a star, if the collapsing star has a mass exceeding the TOV limit, the crush will continue until zero volume is achieved and a black hole is formed around that point in space. The maximum mass that a star can possess is not fully understood. In 1939, it was estimated at 0.7 solar masses, in 1996, a different estimate put this upper mass in a range from 1.5 to 3 solar masses. In the theory of relativity, a black hole could exist of any mass. The lower the mass, the higher the density of matter has to be in order to form a black hole, there are no known processes that can produce black holes with mass less than a few times the mass of the Sun. If black holes that small exist, they are most likely black holes. Until 2016, the largest known black hole was 15. 65±1.45 solar masses. In September 2015, a hole of 62±4 solar masses was discovered in gravitational waves as it formed in a merger event of two smaller black holes. As of April 2008, XTE J1650-500 was reported by NASA and others to be the black hole currently known to science, with a mass 3.8 solar masses. However, this claim was subsequently retracted, the more likely mass is 5–10 solar masses. There is observational evidence for two types of black holes, which are much more massive than stellar black holes. They are intermediate-mass black holes and supermassive black holes in the centre of the Milky Way, stellar black holes in close binary systems are observable when matter is transferred from a companion star to the black hole
26.
Galactic Center
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The Galactic Center is the rotational center of the Milky Way. The estimates for its range from 7.6 to 8.7 kiloparsecs from Earth in the direction of the constellations Sagittarius, Ophiuchus. There is strong evidence consistent with the existence of a black hole at the Galactic Center of the Milky Way. Because of interstellar dust along the line of sight, the Galactic Center cannot be studied at visible, the available information about the Galactic Center comes from observations at gamma ray, hard X-ray, infrared, sub-millimetre and radio wavelengths. In the early 1940s Walter Baade at Mount Wilson Observatory took advantage of wartime conditions in nearby Los Angeles to conduct a search for the center with the 100 inch Hooker Telescope. This gap has been known as Baades Window ever since, by 1954 they had built an 80 feet fixed dish antenna and used it to make a detailed study of an extended, extremely powerful belt of radio emission that was detected in Sagittarius. In 1958 the International Astronomical Union decided to adopt the position of Sagittarius A as the true zero co-ordinate point for the system of latitude and longitude. In the equatorial system the location is, RA 17h 45m 40. 04s. The exact distance between the Solar System and the Galactic Center is not certain, although estimates since 2000 have remained within the range 7. 2–8.8 kpc. The nature of the Milky Ways bar, which extends across the Galactic Center, is also debated, with estimates for its half-length. Certain authors advocate that the Milky Way features two bars, one nestled within the other. The bar is delineated by red-clump stars, however, RR Lyr variables do not trace a prominent Galactic bar. The bar may be surrounded by a called the 5-kpc ring that contains a large fraction of the molecular hydrogen present in the Milky Way. Viewed from the Andromeda Galaxy, it would be the brightest feature of the Milky Way, accretion of gas onto the black hole, probably involving a disk around it, would release energy to power the radio source, itself much larger than the black hole. The latter is too small to see with present instruments, a study in 2008 which linked radio telescopes in Hawaii, Arizona and California measured the diameter of Sagittarius A* to be 44 million kilometers. For comparison, the radius of Earths orbit around the Sun is about 150 million kilometers, thus the diameter of the radio source is slightly less than the distance from Mercury to the Sun.3 million solar masses. On 5 January 2015, NASA reported observing an X-ray flare 400 times brighter than usual, the central cubic parsec around Sagittarius A* contains around 10 million stars. Although most of them are old red giant stars, the Galactic Center is also rich in massive stars, more than 100 OB and Wolf–Rayet stars have been identified there so far
27.
Neutron star
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A neutron star is the collapsed core of a large star. Neutron stars are the smallest and densest stars known to exist, though neutron stars typically have a radius on the order of 10 km, they can have masses of about twice that of the Sun. They result from the explosion of a massive star, combined with gravitational collapse. They are supported against further collapse by neutron degeneracy pressure, a described by the Pauli exclusion principle. If the remnant has too great a density, something which occurs in excess of a limit of the size of neutron stars at 2–3 solar masses. Neutron stars that can be observed are very hot and typically have a temperature around 6×105 K. They are so dense that a normal-sized matchbox containing neutron-star material would have a mass of approximately 3 billion tonnes and their magnetic fields are between 108 and 1015 times as strong as that of the Earth. The gravitational field at the stars surface is about 2×1011 times that of the Earth. As the stars core collapses, its rotation rate increases as a result of conservation of angular momentum, some neutron stars emit beams of electromagnetic radiation that make them detectable as pulsars. Indeed, the discovery of pulsars in 1967 was the first observational suggestion that stars exist. The radiation from pulsars is thought to be emitted from regions near their magnetic poles. The fastest-spinning neutron star known is PSR J1748-2446ad, rotating at a rate of 716 times a second or 43,000 revolutions per minute, giving a linear speed at the surface on the order of 0.24 c. There are thought to be around 100 million neutron stars in the Milky Way, however, most are old and cold, and neutron stars can only be easily detected in certain instances, such as if they are a pulsar or part of a binary system. Soft gamma repeaters are conjectured to be a type of neutron star with strong magnetic fields, known as magnetars, or alternatively. Additionally, such accretion can recycle old pulsars and potentially cause them to mass and spin-up to very fast rotation rates. The merger of binary stars may be the source of short-duration gamma-ray bursts and are likely strong sources of gravitational waves. Any main-sequence star with a mass of above 8 times the mass of the sun has the potential to produce a neutron star. As the star evolves away from the sequence, subsequent nuclear burning produces an iron-rich core
28.
Mount Everest
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Mount Everest, also known in Nepal as Sagarmāthā and in China as Chomolungma/珠穆朗玛峰, is Earths highest mountain. Its peak is 8,848 metres above sea level, Mount Everest is in the Mahalangur Range. The international border between China and Nepal runs across Everests summit point and its massif includes neighbouring peaks Lhotse,8,516 m, Nuptse,7,855 m, and Changtse,7,580 m. In 1856, the Great Trigonometrical Survey of India established the first published height of Everest, then known as Peak XV, at 8,840 m. The current official height of 8,848 m as recognised by China and Nepal was established by a 1955 Indian survey, in 2005, China remeasured the height of the mountain and got a result of 8844.43 m. An argument regarding the height between China and Nepal lasted five years from 2005 to 2010, China argued it should be measured by its rock height which is 8,844 m but Nepal said it should be measured by its snow height 8,848 m. In 2010, an agreement was reached by both sides that the height of Everest is 8,848 m and Nepal recognises Chinas claim that the rock height of Everest is 8,844 m. In 1865, Everest was given its official English name by the Royal Geographical Society upon a recommendation by Andrew Waugh, the British Surveyor General of India. As there appeared to be several different local names, Waugh chose to name the mountain after his predecessor in the post, Sir George Everest, Mount Everest attracts many climbers, some of them highly experienced mountaineers. There are two main climbing routes, one approaching the summit from the southeast in Nepal and the other from the north in Tibet, as of 2016 there are well over 200 corpses on the mountain, with some of them even serving as landmarks. The first recorded efforts to reach Everests summit were made by British mountaineers, with Nepal not allowing foreigners into the country at the time, the British made several attempts on the north ridge route from the Tibetan side. Tragedy struck on the descent from the North Col when seven porters were killed in an avalanche. They had been spotted high on the mountain that day but disappeared in the clouds, never to be seen again, Tenzing Norgay and Edmund Hillary made the first official ascent of Everest in 1953 using the southeast ridge route. Tenzing had reached 8,595 m the previous year as a member of the 1952 Swiss expedition, the Chinese mountaineering team of Wang Fuzhou, Gonpo, and Qu Yinhua made the first reported ascent of the peak from the north ridge on 25 May 1960. In 1802, the British began the Great Trigonometric Survey of India to fix the locations, heights, starting in southern India, the survey teams moved northward using giant theodolites, each weighing 500 kg and requiring 12 men to carry, to measure heights as accurately as possible. They reached the Himalayan foothills by the 1830s, but Nepal was unwilling to allow the British to enter the country due to suspicions of political aggression, several requests by the surveyors to enter Nepal were turned down. The British were forced to continue their observations from Terai, a region south of Nepal which is parallel to the Himalayas, conditions in Terai were difficult because of torrential rains and malaria. Three survey officers died from malaria while two others had to retire because of failing health, nonetheless, in 1847, the British continued the survey and began detailed observations of the Himalayan peaks from observation stations up to 240 km distant
29.
Big Bang
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The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. If the known laws of physics are extrapolated to the highest density regime, detailed measurements of the expansion rate of the universe place this moment at approximately 13.8 billion years ago, which is thus considered the age of the universe. After the initial expansion, the universe cooled sufficiently to allow the formation of subatomic particles, giant clouds of these primordial elements later coalesced through gravity in halos of dark matter, eventually forming the stars and galaxies visible today. Since Georges Lemaître first noted in 1927 that a universe could be traced back in time to an originating single point. More recently, measurements of the redshifts of supernovae indicate that the expansion of the universe is accelerating, the known physical laws of nature can be used to calculate the characteristics of the universe in detail back in time to an initial state of extreme density and temperature. American astronomer Edwin Hubble observed that the distances to faraway galaxies were strongly correlated with their redshifts, assuming the Copernican principle, the only remaining interpretation is that all observable regions of the universe are receding from all others. Since we know that the distance between galaxies increases today, it must mean that in the past galaxies were closer together, the continuous expansion of the universe implies that the universe was denser and hotter in the past. Large particle accelerators can replicate the conditions that prevailed after the early moments of the universe, resulting in confirmation, however, these accelerators can only probe so far into high energy regimes. Consequently, the state of the universe in the earliest instants of the Big Bang expansion is still poorly understood, the first subatomic particles to be formed included protons, neutrons, and electrons. Though simple atomic nuclei formed within the first three minutes after the Big Bang, thousands of years passed before the first electrically neutral atoms formed, the majority of atoms produced by the Big Bang were hydrogen, along with helium and traces of lithium. Giant clouds of primordial elements later coalesced through gravity to form stars and galaxies. The framework for the Big Bang model relies on Albert Einsteins theory of relativity and on simplifying assumptions such as homogeneity. The governing equations were formulated by Alexander Friedmann, and similar solutions were worked on by Willem de Sitter, extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. This singularity indicates that general relativity is not a description of the laws of physics in this regime. How closely models based on general relativity alone can be used to extrapolate toward the singularity is debated—certainly no closer than the end of the Planck epoch. This primordial singularity is itself called the Big Bang, but the term can also refer to a more generic early hot. The agreement of independent measurements of this age supports the model that describes in detail the characteristics of the universe. The earliest phases of the Big Bang are subject to much speculation, in the most common models the universe was filled homogeneously and isotropically with a very high energy density and huge temperatures and pressures and was very rapidly expanding and cooling
30.
Primordial black hole
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A primordial black hole is a hypothetical type of black hole that is formed in the early Universe due to the gravitational collapse of important density fluctuations. Several mechanisms have been proposed to produce the inhomogeneities at the origin of black hole formation, like in the context of cosmic inflation, reheating. Depending on the model, primordial black holes could have initial masses ranging from 10−8 kg to more than thousands of solar masses. However primordial black holes with a lower than 1011 kg would have evaporated in a time much shorter than the age of the Universe. A noticeable exception is the case of Planck relics that could eventually be stable, the abundance of primordial black holes could be as important as the one of Dark Matter, to which they are a plausible candidate. Primordial black holes are also candidates for being the seeds of the Supermassive Black Holes at the center of massive galaxies. Primordial black holes belong to the class of MAssive Compact Halo Objects and they are naturally a good dark matter candidate, they are collision-less and stable, they have non-relativistic velocities, and they form very early in the history of the Universe. The unexpected high mass of the black holes detected by LIGO has strongly revived the interest for black holes with masses in the range of 1-100 solar masses. Primordial black holes could have formed in the very early Universe, the essential ingredient for a primordial black hole to form is a fluctuation in the density of the Universe, inducing its gravitational collapse. One typically requires density contrasts δ ρ / ρ ∼0.1 to form a black hole, there are several mechanisms able to produce such inhomogeneities, for instance in the context of cosmic inflation, reheating or cosmological phase transitions. Stephen Hawking theorized in 1974 that large numbers of such smaller primordial black holes might exist in the Milky Way in our galaxys halo region, all black holes are theorized to emit Hawking radiation at a rate inversely proportional to their mass. A regular black hole cannot lose all of its mass within the current age of the universe, however, since primordial black holes are not formed by stellar core collapse, they may be of any size. A black hole with a mass of about 1011 kg would have a lifetime about equal to the age of the universe. If such low-mass black holes were created in sufficient number in the Big Bang, nASAs Fermi Gamma-ray Space Telescope satellite, launched in June 2008, was designed in part to search for such evaporating primordial black holes. Fermi data set up the limit that less than one percent of Dark Matter could be made of black holes with masses up to 1013 kg. Evaporating primordial black holes would have also an impact on the Big Bang Nucleosythesis, lensing of gamma-ray bursts, Compact objects can induce a change in the luminosity of Gamma-ray bursts when passing close to their line-of-sight, through the Gravitational lensing effect. The Fermi Gamma-Ray Burst Monitor experiment found that black holes cannot contribute importantly to the dark matter within the mass range 5 x 1014 –1017 kg. The observation of stars in globular clusters can thus be used to set a limit on primordial black holes abundances
31.
Gravitational time dilation
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Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The weaker the gravitational potential, the time passes. Albert Einstein originally predicted this effect in his theory of relativity and this has been demonstrated by noting that atomic clocks at differing altitudes will eventually show different times. The effects detected in such Earth-bound experiments are small, with differences being measured in nanoseconds. Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source, Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of time at different positions as described by a metric tensor of spacetime. The existence of time dilation was first confirmed directly by the Pound–Rebka experiment in 1959. Clocks that are far from massive bodies run more quickly, for example, considered over the total lifetime of the earth, a clock set at the peak of Mount Everest would be about 39 hours ahead of a clock set at sea level. This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects. According to general relativity, inertial mass and gravitational mass are the same, let us consider a family of observers along a straight vertical line, each of whom experiences a distinct constant g-force directed along this line. Let g be the dependence of g-force on height, a coordinate along the aforementioned line. For simplicity, in a Rindlers family of observers in a flat space-time, the dependence would be g = c 2 / with constant H, which yields T d = e ln − ln H = H + h H. On the other hand, when g is constant and g h is much smaller than c 2. See Ehrenfest paradox for application of the formula to a rotating reference frame in flat space-time. In comparison, a clock on the surface of the sun will accumulate around 66.4 fewer seconds in one year, in the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than 32 r s. T0 = t f 1 −32 ⋅ r s r, according to the general theory of relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame. An exception is the center of a distribution of matter. Additionally, all phenomena in similar circumstances undergo time dilation equally according to the equivalence principle used in the general theory of relativity
32.
Speed of light
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The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299792458 metres per second, it is exact because the unit of length, the metre, is defined from this constant, according to special relativity, c is the maximum speed at which all matter and hence information in the universe can travel. It is the speed at which all particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the reference frame of the observer. In the theory of relativity, c interrelates space and time, the speed at which light propagates through transparent materials, such as glass or air, is less than c, similarly, the speed of radio waves in wire cables is slower than c. The ratio between c and the speed v at which light travels in a material is called the index n of the material. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft, the light seen from stars left them many years ago, allowing the study of the history of the universe by looking at distant objects. The finite speed of light limits the theoretical maximum speed of computers. The speed of light can be used time of flight measurements to measure large distances to high precision. Ole Rømer first demonstrated in 1676 that light travels at a speed by studying the apparent motion of Jupiters moon Io. In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave, in 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame is a constant and is independent of the motion of the light source. He explored the consequences of that postulate by deriving the theory of relativity and in doing so showed that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of increasingly precise measurements, in 1975 the speed of light was known to be 299792458 m/s with a measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units as the distance travelled by light in vacuum in 1/299792458 of a second, as a result, the numerical value of c in metres per second is now fixed exactly by the definition of the metre. The speed of light in vacuum is usually denoted by a lowercase c, historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant later shown to equal √2 times the speed of light in vacuum, in 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, sometimes c is used for the speed of waves in any material medium, and c0 for the speed of light in vacuum. This article uses c exclusively for the speed of light in vacuum, since 1983, the metre has been defined in the International System of Units as the distance light travels in vacuum in 1⁄299792458 of a second
33.
Circular orbit
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A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis. Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions, here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion. In this case, not only the distance, but also the speed, angular speed, potential, there is no periapsis or apoapsis. This orbit has no radial version, transverse acceleration causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have a = v 2 r = ω2 r where, v is velocity of orbiting body. The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of a is measured in meters per second per second, then the values for v will be in meters per second, r in meters. μ = G M is the standard gravitational parameter, the orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to, r = h 2 μ where, h = r v is specific angular momentum of the orbiting body. Maneuvering into a circular orbit, e. g. It is also a matter of maneuvering into the orbit, for the sake of convenience, the derivation will be written in units in which c = G =1. The four-velocity of a body on an orbit is given by. The dot above a variable denotes derivation with respect to proper time τ
34.
Elliptic orbit
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In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1, this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0, in a wider sense it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1, in a gravitational two-body problem with negative energy both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit, examples of elliptic orbits include, Hohmann transfer orbit, Molniya orbit and tundra orbit. A is the length of the semi-major axis, the velocity equation for a hyperbolic trajectory has either +1 a, or it is the same with the convention that in that case a is negative. Conclusions, For a given semi-major axis the orbital energy is independent of the eccentricity. ν is the true anomaly. The angular momentum is related to the cross product of position and velocity. Here ϕ is defined as the angle which differs by 90 degrees from this and this set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit, the two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit, another set of six parameters that are commonly used are the orbital elements. In the Solar System, planets, asteroids, most comets, the following chart of the perihelion and aphelion of the planets, dwarf planets and Halleys Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity, note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halleys Comet and Eris. A radial trajectory can be a line segment, which is a degenerate ellipse with semi-minor axis =0. Although the eccentricity is 1, this is not a parabolic orbit, most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed and it is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, the velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity. The radial elliptic trajectory is the solution of a problem with at some instant zero speed
35.
Semi-major axis
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In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
36.
Photon sphere
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A photon sphere is a spherical region of space where gravity is strong enough that photons are forced to travel in orbits. This equation entails that photon spheres can only exist in the surrounding a extremely compact object. As photons approach the event horizon of a hole, those with the appropriate energy avoid being pulled into the black hole by traveling in a nearly tangential direction known as an exit cone. A photon on the boundary of this cone does not possess the energy to escape the gravity well of the black hole, instead, it orbits the black hole. These orbits are stable in the long term. The photon sphere is located farther from the center of a hole than the event horizon. For non-rotating black holes, the sphere is a sphere of radius 3/2 rs. There are no free fall orbits that exist within or cross the photon sphere. Any free fall orbit that crosses it from the outside spirals into the black hole, any orbit that crosses it from the inside escapes to infinity. Another property of the sphere is centrifugal force reversal. Outside the photon sphere, the faster one orbits the greater the force one feels. Centrifugal force falls to zero at the sphere, including non-freefall orbits at any speed, i. e. you weigh the same no matter how fast you orbit. Inside the photon sphere the faster you orbit the greater your felt weight or inward force and this has serious ramifications for the fluid dynamics of inward fluid flow. A rotating black hole has two photon spheres, as a black hole rotates, it drags space with it. The photon sphere that is closer to the hole is moving in the same direction as the rotation. The greater the velocity of the rotation of a black hole. Since the black hole has an axis of rotation, this holds true if approaching the black hole in the direction of the equator. If approaching at a different angle, such as one from the poles of the hole to the equator
37.
Chandrasekhar limit
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The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The limit was first indicated in papers published by Wilhelm Anderson, stoner, and was named after Subrahmanyan Chandrasekhar, the Indian astrophysicist who independently discovered and improved upon the accuracy of the calculation in 1930, at the age of 19, in India. White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, the Chandrasekhar limit is the mass above which electron degeneracy pressure in the stars core is insufficient to balance the stars own gravitational self-attraction. Those with masses under the limit remain stable as white dwarfs, the currently accepted value of the limit is about 1.4 M ⊙. Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle, since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels, compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons will increase upon compression, so pressure must be exerted on the gas to compress it. With sufficient compression, electrons are forced into nuclei in the process of electron capture, relieving the pressure. In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P = K1 ρ53, where P is the pressure, ρ is the mass density, and K1 is a constant. As the mass of a white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The velocities of the approach the speed of light. In the strongly relativistic limit, the equation of state takes the form P = K2 ρ43 and this will yield a polytrope of index 3, which will have a total mass, Mlimit say, depending only on K2. For a fully relativistic treatment, the equation of state used will interpolate between the equations P = K1 ρ53 for small ρ and P = K2 ρ43 for large ρ. When this is done, the model radius still decreases with mass, the curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively, μe has been set equal to 2. Radius is measured in standard solar radii or kilometers, and mass in standard solar masses, calculated values for the limit will vary depending on the nuclear composition of the mass. Chandrasekhar, eq. eq. eq. mH is the mass of the hydrogen atom, ω30 ≈2.018236 is a constant connected with the solution to the Lane-Emden equation. As ℏ c / G is the Planck mass, the limit is of the order of M Pl 3 m H2, lieb and Yau have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation
38.
John Michell
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John Michell was an English clergyman and natural philosopher who provided pioneering insights in a wide range of scientific fields, including astronomy, geology, optics, and gravitation. He also invented an apparatus to measure the mass of the Earth and he has been called both the father of seismology and the father of magnetometry. According to one source, a few specifics of Michells work really do sound like they are ripped from the pages of a twentieth century astronomy textbook. John Michell was born in 1724 in Eakring, in Nottinghamshire, the son of Gilbert Michell, a priest, Gilbert was the son of William Michell and Mary Taylor of Kenwyn, Cornwall, Obedience was the daughter of Ralph and Hannah Gerrard of London. He was educated at Queens College, Cambridge, and later became a Fellow of Queens and he obtained his M. A. degree in 1752 and B. D. degree in 1761. He was nominated Rector of St Botolphs, Cambridge, on 28th March 1760, from 1762 to 1764, he held the Woodwardian Chair of Geology till he was obliged to relinquish it on his marriage. There is no surviving portrait of Michell, he is said to have been a little short Man, of a black Complexion, Michell proceeded to take up clerical positions in Compton and then Havant, both in Hampshire. During this period he unsuccessfully sought positions at Cambridge and as astronomer royal, in 1767, he was appointed rector of St. Michaels Church of Thornhill, near Leeds, Yorkshire, England, a post he held for the rest of his life. He did most of his important scientific work in Thornhill, where he died on 21 April 1793, after local pressure, a blue plaque went up on the church wall to commemorate him. At one point, Michell attempted to measure the pressure of light by focusing sunlight onto one side of a compass needle. The experiment was not a success, the needle melted, until the late 20th century Michell was considered important primarily because of his work on geology. His most important geological essay, written after the 1755 Lisbon earthquake, was entitled Conjectures concerning the Cause, in this paper he introduced the idea that earthquakes spread out as waves through the Earth and that they involve the offsets in geological strata now known as faults. He was able to both the epicentre and the focus of the Lisbon earthquake, and may also have been the first to suggest that a tsunami is caused by a submarine earthquake. Michell’s essay not only provided insights on earthquakes but also, more broadly and he recognized that the Earth is composed of regular and uniform strata, some of which have been interrupted by upheavals. The most important part of Michells Earthquake paper, in the view of one commentator, is the account which it contains of what is now known as the crust of the Earth. K. In 1760, as a result of work, he was elected a member of the Royal Society. A1788 letter to Cavendish indicated that Michell continued to be interested in several decades after his paper on earthquakes. Michell studied magnetism and discovered the law, the fact that the magnetic force exerted by each pole of a magnet decreases in proportion to the square of the distance between them
39.
Schwarzschild black hole
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The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. The solution is named after Karl Schwarzschild, who first published the solution in 1916, according to Birkhoffs theorem, the Schwarzschild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a hole that has no electric charge or angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, the Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole, the analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle. In practice, the ratio rs/r is almost always extremely small, for example, the Schwarzschild radius rs of the Earth is roughly 8.9 mm, while the Sun, which is 3. 3×105 times as massive has a Schwarzschild radius of approximately 3.0 km. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion, the ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars. The Schwarzschild metric is a solution of Einsteins field equations in empty space and that is, for a spherical body of radius R the solution is valid for r > R. It was the first exact solution of the Einstein field equations other than the flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I, johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation. In the early years of general relativity there was a lot of confusion about the nature of the found in the Schwarzschild. In Schwarzschilds original paper, he put what we now call the event horizon at the origin of his coordinate system, in this paper he also introduced what is now known as the Schwarzschild radial coordinate, as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius, a more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r =0 and r = rs. Although there was consensus that the singularity at r =0 was a genuine physical singularity. They, however, did not recognize that their solutions were just coordinate transforms, later, in 1932, Georges Lemaître gave a different coordinate transformation to the same effect and was the first to recognize that this implied that the singularity at r = rs was not physical. A similar result was later rediscovered by George Szekeres, and independently Martin Kruskal, the new coordinates nowadays known as Kruskal-Szekeres coordinates were much simpler than Synges but both provided a single set of coordinates that covered the entire spacetime. This led to identification of the r = rs singularity in the Schwarzschild metric as an event horizon. The Schwarzschild solution appears to have singularities at r =0 and r = rs, since the Schwarzschild metric is only expected to be valid for radii larger than the radius R of the gravitating body, there is no problem as long as R > rs
40.
Micro black hole
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Micro black holes, also called quantum mechanical black holes or mini black holes, are hypothetical tiny black holes, for which quantum mechanical effects play an important role. They might be observed by astrophysicists in the future, through the particles they are expected to emit by Hawking radiation. Some hypotheses involving additional space dimensions predict that micro black holes could be formed at energies as low as the TeV range, popular concerns have then been raised over end-of-the-world scenarios. However, such black holes would instantly evaporate, either totally or leaving only a very weakly interacting residue. Beside the theoretical arguments, the cosmic rays hitting the Earth do not produce any damage, in principle, a black hole can have any mass equal to or above the Planck mass. To make a hole, one must concentrate mass or energy sufficiently that the escape velocity from the region in which it is concentrated exceeds the speed of light. This condition gives the Schwarzschild radius, R =2 G M / c 2, where G is the constant, c is the speed of light. On the other hand, the Compton wavelength, λ = h / M c, for sufficiently small M, the reduced Compton wavelength exceeds half the Schwarzschild radius, and no black hole description exists. This smallest mass for a hole is thus approximately the Planck mass. Some extensions of present physics posit the existence of extra dimensions of space, in higher-dimensional spacetime, the strength of gravity increases more rapidly with decreasing distance than in three dimensions. With certain special configurations of the dimensions, this effect can lower the Planck scale to the TeV range. Examples of such extensions include large extra dimensions, special cases of the Randall–Sundrum model, in such scenarios, black hole production could possibly be an important and observable effect at the large hadron collider. It would also be a natural phenomenon induced by the cosmic rays. All this assumes that the theory of general relativity remains valid at small distances. If it does not, then other, presently unknown, effects will limit the size of a black hole. Elementary particles are equipped with a quantum-mechanical, intrinsic angular momentum, the correct conservation law for the total angular momentum of matter in curved spacetime requires that spacetime is equipped with torsion. The simplest and most natural theory of gravity with torsion is the Einstein-Cartan theory, torsion modifies the Dirac equation in the presence of the gravitational field and causes fermion particles to be spatially extended. The spatial extension of fermions limits the mass of a black hole to be on the order of 1016 kg
41.
Visible universe
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There are at least two trillion galaxies in the observable universe, containing more stars than all the grains of sand on planet Earth. Assuming the universe is isotropic, the distance to the edge of the universe is roughly the same in every direction. That is, the universe is a spherical volume centered on the observer. Every location in the Universe has its own universe, which may or may not overlap with the one centered on Earth. The word observable used in this sense does not depend on modern technology actually permits detection of radiation from an object in this region. It simply indicates that it is possible in principle for light or other signals from the object to reach an observer on Earth, in practice, we can see light only from as far back as the time of photon decoupling in the recombination epoch. That is when particles were first able to emit photons that were not quickly re-absorbed by other particles, before then, the Universe was filled with a plasma that was opaque to photons. The detection of gravitational waves indicates there is now a possibility of detecting signals from before the recombination epoch. The surface of last scattering is the collection of points in space at the distance that photons from the time of photon decoupling just reach us today. These are the photons we detect today as cosmic microwave background radiation, however, with future technology, it may be possible to observe the still older relic neutrino background, or even more distant events via gravitational waves. It is estimated that the diameter of the universe is about 28.5 gigaparsecs. The total mass of matter in the universe can be calculated using the critical density. Some parts of the Universe are too far away for the light emitted since the Big Bang to have had time to reach Earth. In the future, light from distant galaxies will have had time to travel. This fact can be used to define a type of cosmic event horizon whose distance from the Earth changes over time, both popular and professional research articles in cosmology often use the term universe to mean observable universe. It is plausible that the galaxies within our observable universe represent only a fraction of the galaxies in the Universe. If the Universe is finite but unbounded, it is possible that the Universe is smaller than the observable universe. In this case, what we take to be very distant galaxies may actually be duplicate images of nearby galaxies and it is difficult to test this hypothesis experimentally because different images of a galaxy would show different eras in its history, and consequently might appear quite different