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121-398: Interplanetary transfer orbits are solutions to the gravitational three-body problem , which, for the general case, does not have analytical solutions, and is addressed by numerical analysis approximations. However, a small number of exact solutions exist, most notably the five orbits referred to as " Lagrange points ", which are orbital solutions for circular orbits in the case when one body

242-456: A protoplanetary disk and powered mainly by the conversion of gravitational energy. The period of gravitational contraction lasts about 10 million years for a star like the sun, up to 100 million years for a red dwarf. Early stars of less than 2  M ☉ are called T Tauri stars , while those with greater mass are Herbig Ae/Be stars . These newly formed stars emit jets of gas along their axis of rotation, which may reduce

363-467: A stellar wind of particles that causes a continual outflow of gas into space. For most stars, the mass lost is negligible. The Sun loses 10   M ☉ every year, or about 0.01% of its total mass over its entire lifespan. However, very massive stars can lose 10 to 10   M ☉ each year, significantly affecting their evolution. Stars that begin with more than 50  M ☉ can lose over half their total mass while on

484-484: A brief period of carbon fusion before the core becomes degenerate. During the AGB phase, stars undergo thermal pulses due to instabilities in the core of the star. In these thermal pulses, the luminosity of the star varies and matter is ejected from the star's atmosphere, ultimately forming a planetary nebula. As much as 50 to 70% of a star's mass can be ejected in this mass loss process. Because energy transport in an AGB star

605-491: A burst of electron capture and inverse beta decay . The shockwave formed by this sudden collapse causes the rest of the star to explode in a supernova. Supernovae become so bright that they may briefly outshine the star's entire home galaxy. When they occur within the Milky Way, supernovae have historically been observed by naked-eye observers as "new stars" where none seemingly existed before. A supernova explosion blows away

726-512: A certain large mass, e.g. a star , and a smaller, orbiting mass, e.g. a planet ) are not stable equilibrium points. If a spacecraft placed at the Earth–Moon L 1 point is given even a slight nudge away from the equilibrium point, the spacecraft's trajectory will diverge away from the L 1 point. The entire system is in motion, so the spacecraft will not actually hit the Moon, but will travel in

847-410: A continuous image due to the effect of refraction from sublunary material, citing his observation of the conjunction of Jupiter and Mars on 500 AH (1106/1107 AD) as evidence. Early European astronomers such as Tycho Brahe identified new stars in the night sky (later termed novae ), suggesting that the heavens were not immutable. In 1584, Giordano Bruno suggested that the stars were like

968-440: A difference between " fixed stars ", whose position on the celestial sphere does not change, and "wandering stars" ( planets ), which move noticeably relative to the fixed stars over days or weeks. Many ancient astronomers believed that the stars were permanently affixed to a heavenly sphere and that they were immutable. By convention, astronomers grouped prominent stars into asterisms and constellations and used them to track

1089-509: A much larger gravitationally bound structure, such as a star cluster or a galaxy. The word "star" ultimately derives from the Proto-Indo-European root "h₂stḗr" also meaning star, but further analyzable as h₂eh₁s- ("to burn", also the source of the word "ash") + -tēr (agentive suffix). Compare Latin stella , Greek aster , German Stern . Some scholars believe the word is a borrowing from Akkadian " istar " ( Venus ). "Star"

1210-539: A net release of energy. Some massive stars, particularly luminous blue variables , are very unstable to the extent that they violently shed their mass into space in events supernova impostors , becoming significantly brighter in the process. Eta Carinae is known for having underwent a supernova impostor event, the Great Eruption, in the 19th century. As a star's core shrinks, the intensity of radiation from that surface increases, creating such radiation pressure on

1331-509: A relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole . However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity. Star A star

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1452-421: A search for "periodic free-fall orbits" three-body problem, limited to the equal-mass case, and found 12,409 distinct solutions. Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. There have been attempts of creating computer programs that numerically solve the three-body problem (and by extension,

1573-463: A series of star maps and applied Greek letters as designations to the stars in each constellation. Later a numbering system based on the star's right ascension was invented and added to John Flamsteed 's star catalogue in his book "Historia coelestis Britannica" (the 1712 edition), whereby this numbering system came to be called Flamsteed designation or Flamsteed numbering . The internationally recognized authority for naming celestial bodies

1694-549: A set of nominal solar values (defined as SI constants, without uncertainties) which can be used for quoting stellar parameters: The solar mass M ☉ was not explicitly defined by the IAU due to the large relative uncertainty ( 10 ) of the Newtonian constant of gravitation G . Since the product of the Newtonian constant of gravitation and solar mass together ( G M ☉ ) has been determined to much greater precision,

1815-420: A set of initial conditions of measure zero. But there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps: This finishes the proof of Sundman's theorem. The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution

1936-499: A star begins with gravitational instability within a molecular cloud, caused by regions of higher density—often triggered by compression of clouds by radiation from massive stars, expanding bubbles in the interstellar medium, the collision of different molecular clouds, or the collision of galaxies (as in a starburst galaxy ). When a region reaches a sufficient density of matter to satisfy the criteria for Jeans instability , it begins to collapse under its own gravitational force. As

2057-434: A star of more than 9 solar masses expands to form first a blue supergiant and then a red supergiant . Particularly massive stars (exceeding 40 solar masses, like Alnilam , the central blue supergiant of Orion's Belt ) do not become red supergiants due to high mass loss. These may instead evolve to a Wolf–Rayet star , characterised by spectra dominated by emission lines of elements heavier than hydrogen, which have reached

2178-407: A white dwarf is no longer a plasma. Eventually, white dwarfs fade into black dwarfs over a very long period of time. In massive stars, fusion continues until the iron core has grown so large (more than 1.4  M ☉ ) that it can no longer support its own mass. This core will suddenly collapse as its electrons are driven into its protons, forming neutrons, neutrinos , and gamma rays in

2299-493: A winding path, off into space. There is, however, a semi-stable orbit around each of these points, called a halo orbit . The orbits for two of the points, L 4 and L 5 , are stable, but the halo orbits for L 1 through L 3 are stable only on the order of months . In addition to orbits around Lagrange points, the rich dynamics that arise from the gravitational pull of more than one mass yield interesting trajectories, also known as low energy transfers . For example,

2420-495: Is a luminous spheroid of plasma held together by self-gravity . The nearest star to Earth is the Sun . Many other stars are visible to the naked eye at night ; their immense distances from Earth make them appear as fixed points of light. The most prominent stars have been categorised into constellations and asterisms , and many of the brightest stars have proper names . Astronomers have assembled star catalogues that identify

2541-514: Is a special case of the n -body problem . Historically, the first specific three-body problem to receive extended study was the one involving the Earth , the Moon , and the Sun . In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles. The mathematical statement of the three-body problem can be given in terms of

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2662-400: Is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom. Within the point vortex model , the motion of vortices in a two-dimensional ideal fluid is described by equations of motion that contain only first-order time derivatives. I.e. in contrast to Newtonian mechanics, it

2783-420: Is cognate (shares the same root) with the following words: asterisk , asteroid , astral , constellation , Esther . Historically, stars have been important to civilizations throughout the world. They have been part of religious practices, divination rituals, mythology , used for celestial navigation and orientation, to mark the passage of seasons, and to define calendars. Early astronomers recognized

2904-516: Is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10 terms. In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form

3025-465: Is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem. Mathematically, the problem is stated as follows. Let m 1 , 2 {\displaystyle m_{1,2}} be the masses of

3146-513: Is possible especially for such a system like that of our Earth , the Moon , and the Sun. Guided by major Renaissance astronomers Nicolaus Copernicus , Tycho Brahe and Johannes Kepler he introduced later generations to the beginning of the gravitational three-body problem. In Proposition 66 of Book 1 of the Principia , and its 22 Corollaries, Newton took the first steps in the definition and study of

3267-409: Is primarily by convection , this ejected material is enriched with the fusion products dredged up from the core. Therefore, the planetary nebula is enriched with elements like carbon and oxygen. Ultimately, the planetary nebula disperses, enriching the general interstellar medium. Therefore, future generations of stars are made of the "star stuff" from past stars. During their helium-burning phase,

3388-588: Is significantly more massive. The key to discovering the Interplanetary Transport Network was the investigation of the nature of the winding paths near the Earth-Sun and Earth-Moon Lagrange points. They were first investigated by Henri Poincaré in the 1890s. He noticed that the paths leading to and from any of those points would almost always settle, for a time, on an orbit about that point. There are in fact an infinite number of paths taking one to

3509-433: Is simply the total energy of the system, gravitational plus kinetic. In the restricted three-body problem formulation, in the description of Barrow-Green, two... bodies revolve around their centre of mass in circular orbits under the influence of their mutual gravitational attraction, and... form a two body system... [whose] motion is known. A third body (generally known as a planetoid), assumed massless with respect to

3630-536: Is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies. A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom , in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction . Like the gravitational three-body problem, the helium atom cannot be solved exactly. In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides

3751-1185: Is the Hamiltonian : H = − G m 1 m 2 | r 1 − r 2 | − G m 2 m 3 | r 3 − r 2 | − G m 3 m 1 | r 3 − r 1 | + p 1 2 2 m 1 + p 2 2 2 m 2 + p 3 2 2 m 3 . {\displaystyle {\mathcal {H}}=-{\frac {Gm_{1}m_{2}}{|\mathbf {r_{1}} -\mathbf {r_{2}} |}}-{\frac {Gm_{2}m_{3}}{|\mathbf {r_{3}} -\mathbf {r_{2}} |}}-{\frac {Gm_{3}m_{1}}{|\mathbf {r_{3}} -\mathbf {r_{1}} |}}+{\frac {\mathbf {p_{1}} ^{2}}{2m_{1}}}+{\frac {\mathbf {p_{2}} ^{2}}{2m_{2}}}+{\frac {\mathbf {p_{3}} ^{2}}{2m_{3}}}.} In this case, H {\displaystyle {\mathcal {H}}}

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3872-555: Is the International Astronomical Union (IAU). The International Astronomical Union maintains the Working Group on Star Names (WGSN) which catalogs and standardizes proper names for stars. A number of private companies sell names of stars which are not recognized by the IAU, professional astronomers, or the amateur astronomy community. The British Library calls this an unregulated commercial enterprise , and

3993-485: Is the Orion Nebula . Most stars form in groups of dozens to hundreds of thousands of stars. Massive stars in these groups may powerfully illuminate those clouds, ionizing the hydrogen, and creating H II regions . Such feedback effects, from star formation, may ultimately disrupt the cloud and prevent further star formation. All stars spend the majority of their existence as main sequence stars , fueled primarily by

4114-776: Is the gravitational constant . As astronomer Juhan Frank describes, "These three second-order vector differential equations are equivalent to 18 first order scalar differential equations." As June Barrow-Green notes with regard to an alternative presentation, if P i {\displaystyle P_{i}} represent three particles with masses m i {\displaystyle m_{i}} , distances P i {\displaystyle P_{i}} P j {\displaystyle P_{j}} = r i j {\displaystyle r_{ij}} , and coordinates q i j {\displaystyle q_{ij}} (i,j = 1,2,3) in an inertial coordinate system ...

4235-503: Is the velocity and not the acceleration that is determined by their relative positions. As a consequence, the three-vortex problem is still integrable , while at least four vortices are required to obtain chaotic behavior. One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics. The gravitational three-body problem has also been studied using general relativity . Physically,

4356-533: The Genesis mission , the first to return solar wind samples to Earth. The network is also relevant to understanding Solar System dynamics; Comet Shoemaker–Levy 9 followed such a trajectory on its collision path with Jupiter. The ITN is based around a series of orbital paths predicted by chaos theory and the restricted three-body problem leading to and from the orbits around the Lagrange points – points in space where

4477-491: The Earth and the Moon . Beginning in 1997, Martin Lo , Shane D. Ross , and others wrote a series of papers identifying the mathematical basis that applied the technique to the Genesis solar wind sample return , and to lunar and Jovian missions. They referred to it as an Interplanetary Superhighway (IPS). As it turns out, it is very easy to transit from a path leading to the point to one leading back out. This makes sense, since

4598-685: The M87 and M100 galaxies of the Virgo Cluster , as well as luminous stars in some other relatively nearby galaxies. With the aid of gravitational lensing , a single star (named Icarus ) has been observed at 9 billion light-years away. The concept of a constellation was known to exist during the Babylonian period. Ancient sky watchers imagined that prominent arrangements of stars formed patterns, and they associated these with particular aspects of nature or their myths. Twelve of these formations lay along

4719-513: The New York City Department of Consumer and Worker Protection issued a violation against one such star-naming company for engaging in a deceptive trade practice. Although stellar parameters can be expressed in SI units or Gaussian units , it is often most convenient to express mass , luminosity , and radii in solar units, based on the characteristics of the Sun. In 2015, the IAU defined

4840-456: The angular momentum of the collapsing star and result in small patches of nebulosity known as Herbig–Haro objects . These jets, in combination with radiation from nearby massive stars, may help to drive away the surrounding cloud from which the star was formed. Early in their development, T Tauri stars follow the Hayashi track —they contract and decrease in luminosity while remaining at roughly

4961-612: The central configurations for the three-body problem. These solutions are valid for any mass ratios, and the masses move on Keplerian ellipses . These four families are the only known solutions for which there are explicit analytic formulae. In the special case of the circular restricted three-body problem , these solutions, viewed in a frame rotating with the primaries, become points called Lagrangian points and labeled L 1 , L 2 , L 3 , L 4 , and L 5 , with L 4 and L 5 being symmetric instances of Lagrange's solution. In work summarized in 1892–1899, Henri Poincaré established

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5082-399: The gravity between various bodies balances with the centrifugal force of an object there. For any two bodies in which one body orbits around the other, such as a star/planet or planet/moon system, there are five such points, denoted L 1 through L 5 . For instance, the Earth–Moon L 1 point lies on a line between the two, where gravitational forces between them exactly balance with

5203-632: The interstellar medium . These elements are then recycled into new stars. Astronomers can determine stellar properties—including mass, age, metallicity (chemical composition), variability , distance , and motion through space —by carrying out observations of a star's apparent brightness , spectrum , and changes in its position in the sky over time. Stars can form orbital systems with other astronomical objects, as in planetary systems and star systems with two or more stars. When two such stars orbit closely, their gravitational interaction can significantly impact their evolution. Stars can form part of

5324-529: The n-body problem ) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as special relativity . In addition, using the theory of random walks , an approximate probability of different outcomes may be computed. The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Philosophiæ Naturalis Principia Mathematica , in which Newton attempted to figure out if any long term stability

5445-449: The photographic magnitude . The development of the photoelectric photometer allowed precise measurements of magnitude at multiple wavelength intervals. In 1921 Albert A. Michelson made the first measurements of a stellar diameter using an interferometer on the Hooker telescope at Mount Wilson Observatory . Important theoretical work on the physical structure of stars occurred during

5566-517: The radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the three-body problems. As is briefly discussed below, the only singularities in the three-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant). Collisions of any number are somewhat improbable, since it has been shown that they correspond to

5687-555: The thermonuclear fusion of hydrogen into helium in its core. This process releases energy that traverses the star's interior and radiates into outer space . At the end of a star's lifetime as a fusor , its core becomes a stellar remnant : a white dwarf , a neutron star , or—if it is sufficiently massive—a black hole . Stellar nucleosynthesis in stars or their remnants creates almost all naturally occurring chemical elements heavier than lithium . Stellar mass loss or supernova explosions return chemically enriched material to

5808-443: The two-body problem , the three-body problem has no general closed-form solution , meaning there is no equation that always solves it. When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions . Because there are no solvable equations for most three-body systems, the only way to predict the motions of the bodies is to estimate them using numerical methods . The three-body problem

5929-575: The 11th century, the Persian polymath scholar Abu Rayhan Biruni described the Milky Way galaxy as a multitude of fragments having the properties of nebulous stars, and gave the latitudes of various stars during a lunar eclipse in 1019. According to Josep Puig, the Andalusian astronomer Ibn Bajjah proposed that the Milky Way was made up of many stars that almost touched one another and appeared to be

6050-472: The 2015 IAU nominal constants will remain the same SI values as they remain useful measures for quoting stellar parameters. Large lengths, such as the radius of a giant star or the semi-major axis of a binary star system, are often expressed in terms of the astronomical unit —approximately equal to the mean distance between the Earth and the Sun (150 million km or approximately 93 million miles). In 2012,

6171-407: The Earth and the Sun. The physical problem was first addressed by Amerigo Vespucci and subsequently by Galileo Galilei , as well as Simon Stevin , but they did not realize what they contributed. Though Galileo determined that the speed of fall of all bodies changes uniformly and in the same way, he did not apply it to planetary motions. Whereas in 1499, Vespucci used knowledge of the position of

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6292-521: The Earth's Moon or between the Galilean moons of Jupiter , within a few months or years. For trips from Earth to other planets, they are not useful for crewed or uncrewed probes, as the trip would take many generations. Nevertheless, they have already been used to transfer spacecraft to the Earth–Sun L 1 point, a useful point for studying the Sun that was employed in a number of recent missions, including

6413-413: The IAU defined the astronomical constant to be an exact length in meters: 149,597,870,700 m. Stars condense from regions of space of higher matter density, yet those regions are less dense than within a vacuum chamber . These regions—known as molecular clouds —consist mostly of hydrogen, with about 23 to 28 percent helium and a few percent heavier elements. One example of such a star-forming region

6534-413: The IAU defined the nominal solar mass parameter to be: The nominal solar mass parameter can be combined with the most recent (2014) CODATA estimate of the Newtonian constant of gravitation G to derive the solar mass to be approximately 1.9885 × 10  kg . Although the exact values for the luminosity, radius, mass parameter, and mass may vary slightly in the future due to observational uncertainties,

6655-655: The ITN to travel from lunar orbit to the Earth-Sun L 2 point, then on to fly by the asteroid 4179 Toutatis . The asteroid 39P/Oterma 's path from outside Jupiter's orbit, to inside, and back to outside is said to follow these low energy paths. Three-body problem In physics , specifically classical mechanics , the three-body problem is to take the initial positions and velocities (or momenta ) of three point masses that orbit each other in space and calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation . Unlike

6776-526: The Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem. In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem. In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem. This

6897-620: The Moon around Earth. Jean le Rond d'Alembert and Alexis Clairaut , who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747. It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" ( French : Problème des trois Corps ) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that

7018-460: The Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea , solved in practice by John Harrison 's invention of the marine chronometer . However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of

7139-2569: The Newtonian equations of motion for vector positions r i = ( x i , y i , z i ) {\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})} of three gravitationally interacting bodies with masses m i {\displaystyle m_{i}} : r ¨ 1 = − G m 2 r 1 − r 2 | r 1 − r 2 | 3 − G m 3 r 1 − r 3 | r 1 − r 3 | 3 , r ¨ 2 = − G m 3 r 2 − r 3 | r 2 − r 3 | 3 − G m 1 r 2 − r 1 | r 2 − r 1 | 3 , r ¨ 3 = − G m 1 r 3 − r 1 | r 3 − r 1 | 3 − G m 2 r 3 − r 2 | r 3 − r 2 | 3 . {\displaystyle {\begin{aligned}{\ddot {\mathbf {r} }}_{\mathbf {1} }&=-Gm_{2}{\frac {\mathbf {r_{1}} -\mathbf {r_{2}} }{|\mathbf {r_{1}} -\mathbf {r_{2}} |^{3}}}-Gm_{3}{\frac {\mathbf {r_{1}} -\mathbf {r_{3}} }{|\mathbf {r_{1}} -\mathbf {r_{3}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {2} }&=-Gm_{3}{\frac {\mathbf {r_{2}} -\mathbf {r_{3}} }{|\mathbf {r_{2}} -\mathbf {r_{3}} |^{3}}}-Gm_{1}{\frac {\mathbf {r_{2}} -\mathbf {r_{1}} }{|\mathbf {r_{2}} -\mathbf {r_{1}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {3} }&=-Gm_{1}{\frac {\mathbf {r_{3}} -\mathbf {r_{1}} }{|\mathbf {r_{3}} -\mathbf {r_{1}} |^{3}}}-Gm_{2}{\frac {\mathbf {r_{3}} -\mathbf {r_{2}} }{|\mathbf {r_{3}} -\mathbf {r_{2}} |^{3}}}.\end{aligned}}} where G {\displaystyle G}

7260-491: The Solar System, Isaac Newton suggested that the stars were equally distributed in every direction, an idea prompted by the theologian Richard Bentley . The Italian astronomer Geminiano Montanari recorded observing variations in luminosity of the star Algol in 1667. Edmond Halley published the first measurements of the proper motion of a pair of nearby "fixed" stars, demonstrating that they had changed positions since

7381-439: The Sun enters the helium burning phase, it will expand to a maximum radius of roughly 1 astronomical unit (150 million kilometres), 250 times its present size, and lose 30% of its current mass. As the hydrogen-burning shell produces more helium, the core increases in mass and temperature. In a red giant of up to 2.25  M ☉ , the mass of the helium core becomes degenerate prior to helium fusion . Finally, when

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7502-449: The Sun, and may have other planets , possibly even Earth-like, in orbit around them, an idea that had been suggested earlier by the ancient Greek philosophers , Democritus and Epicurus , and by medieval Islamic cosmologists such as Fakhr al-Din al-Razi . By the following century, the idea of the stars being the same as the Sun was reaching a consensus among astronomers. To explain why these stars exerted no net gravitational pull on

7623-495: The band of the ecliptic and these became the basis of astrology . Many of the more prominent individual stars were given names, particularly with Arabic or Latin designations. As well as certain constellations and the Sun itself, individual stars have their own myths . To the Ancient Greeks , some "stars", known as planets (Greek πλανήτης (planētēs), meaning "wanderer"), represented various important deities, from which

7744-428: The beginning of the 21st century. George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury . At the beginning of the 20th century, Karl Sundman approached the problem mathematically and systematically by providing a functional theoretical proof to the problem valid for all values of time. It was the first time scientists theoretically solved

7865-399: The centrifugal force of an object placed in orbit there. These five points have particularly low delta-v requirements, and appear to be the lowest-energy transfers possible, even lower than the common Hohmann transfer orbit that has dominated orbital navigation since the start of space travel. Although the forces balance at these points, the first three points (the ones on the line between

7986-496: The chemical composition of the stellar atmosphere to be determined. With the exception of rare events such as supernovae and supernova impostors , individual stars have primarily been observed in the Local Group , and especially in the visible part of the Milky Way (as demonstrated by the detailed star catalogues available for the Milky Way galaxy) and its satellites. Individual stars such as Cepheid variables have been observed in

8107-452: The circular orbits. (That is, it is useful to consider the effective potential . ) With respect to a rotating reference frame , the two co-orbiting bodies are stationary, and the third can be stationary as well at the Lagrangian points , or move around them, for instance on a horseshoe orbit . The restricted three-body problem is easier to analyze theoretically than the full problem. It

8228-445: The clean numerical simulation (CNS), with the use of a national supercomputer, to successfully gain 695 families of periodic solutions of the three-body system with equal mass. In 2019, Breen et al. announced a fast neural network solver for the three-body problem, trained using a numerical integrator. In September 2023, several possible solutions have been found to the problem according to reports. The term "three-body problem"

8349-408: The cloud collapses, individual conglomerations of dense dust and gas form " Bok globules ". As a globule collapses and the density increases, the gravitational energy converts into heat and the temperature rises. When the protostellar cloud has approximately reached the stable condition of hydrostatic equilibrium , a protostar forms at the core. These pre-main-sequence stars are often surrounded by

8470-612: The cloud into multiple stars distributes some of that angular momentum. The primordial binaries transfer some angular momentum by gravitational interactions during close encounters with other stars in young stellar clusters. These interactions tend to split apart more widely separated (soft) binaries while causing hard binaries to become more tightly bound. This produces the separation of binaries into their two observed populations distributions. Stars spend about 90% of their lifetimes fusing hydrogen into helium in high-temperature-and-pressure reactions in their cores. Such stars are said to be on

8591-400: The core. The blown-off outer layers of dying stars include heavy elements, which may be recycled during the formation of new stars. These heavy elements allow the formation of rocky planets. The outflow from supernovae and the stellar wind of large stars play an important part in shaping the interstellar medium. Binary stars ' evolution may significantly differ from that of single stars of

8712-417: The direction of the Milky Way core . His son John Herschel repeated this study in the southern hemisphere and found a corresponding increase in the same direction. In addition to his other accomplishments, William Herschel is noted for his discovery that some stars do not merely lie along the same line of sight, but are physical companions that form binary star systems. The science of stellar spectroscopy

8833-1311: The distance between the two massive bodies, as well as the gravitational constant, are both equal to 1 {\displaystyle 1} . Then, the motion of the planetoid is given by: d 2 x d t 2 = − m 1 x − x 1 r 1 3 − m 2 x − x 2 r 2 3 , d 2 y d t 2 = − m 1 y − y 1 r 1 3 − m 2 y − y 2 r 2 3 , {\displaystyle {\begin{aligned}{\frac {d^{2}x}{dt^{2}}}=-m_{1}{\frac {x-x_{1}}{r_{1}^{3}}}-m_{2}{\frac {x-x_{2}}{r_{2}^{3}}},\\{\frac {d^{2}y}{dt^{2}}}=-m_{1}{\frac {y-y_{1}}{r_{1}^{3}}}-m_{2}{\frac {y-y_{2}}{r_{2}^{3}}},\end{aligned}}} where r i = ( x − x i ) 2 + ( y − y i ) 2 {\displaystyle r_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}} . In this form

8954-405: The end of the star's life, fusion continues along a series of onion-layer shells within a massive star. Each shell fuses a different element, with the outermost shell fusing hydrogen; the next shell fusing helium, and so forth. The final stage occurs when a massive star begins producing iron. Since iron nuclei are more tightly bound than any heavier nuclei, any fusion beyond iron does not produce

9075-404: The equations of motion carry an explicit time dependence through the coordinates x i ( t ) , y i ( t ) {\displaystyle x_{i}(t),y_{i}(t)} ; however, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis. There is no general closed-form solution to

9196-465: The existence of an infinite number of periodic solutions to the restricted three-body problem, together with techniques for continuing these solutions into the general three-body problem. In 1893, Meissel stated what is now called the Pythagorean three-body problem: three masses in the ratio 3:4:5 are placed at rest at the vertices of a 3:4:5 right triangle , with the heaviest body at the right angle and

9317-517: The first decades of the twentieth century. In 1913, the Hertzsprung-Russell diagram was developed, propelling the astrophysical study of stars. Successful models were developed to explain the interiors of stars and stellar evolution. Cecilia Payne-Gaposchkin first proposed that stars were made primarily of hydrogen and helium in her 1925 PhD thesis. The spectra of stars were further understood through advances in quantum physics . This allowed

9438-412: The form of a Puiseux series , specifically a power series in terms of powers of t . This series converges for all real t , except for initial conditions corresponding to zero angular momentum . In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero. An important issue in proving this result is the fact that

9559-518: The gravity environment of the Sun–Earth–Moon system allows spacecraft to travel great distances on very little fuel, albeit on an often circuitous route. Launched in 1978, the ISEE-3 spacecraft was sent on a mission to orbit around one of the Lagrange points. The spacecraft was able to maneuver around the Earth's neighborhood using little fuel by taking advantage of the unique gravity environment. After

9680-460: The inverse-square force that do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force. This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it

9801-514: The known stars and provide standardized stellar designations . The observable universe contains an estimated 10 to 10 stars. Only about 4,000 of these stars are visible to the naked eye—all within the Milky Way galaxy . A star's life begins with the gravitational collapse of a gaseous nebula of material largely comprising hydrogen , helium, and trace heavier elements. Its total mass mainly determines its evolution and eventual fate. A star shines for most of its active life due to

9922-401: The lightest at the smaller acute angle. Burrau further investigated this problem in 1913. In 1967 Victor Szebehely and C. Frederick Peters established eventual escape of the lightest body for this problem using numerical integration, while at the same time finding a nearby periodic solution. In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of

10043-437: The main sequence and are called dwarf stars. Starting at zero-age main sequence, the proportion of helium in a star's core will steadily increase, the rate of nuclear fusion at the core will slowly increase, as will the star's temperature and luminosity. The Sun, for example, is estimated to have increased in luminosity by about 40% since it reached the main sequence 4.6 billion ( 4.6 × 10 ) years ago. Every star generates

10164-677: The main sequence. The time a star spends on the main sequence depends primarily on the amount of fuel it has and the rate at which it fuses it. The Sun is expected to live 10 billion ( 10 ) years. Massive stars consume their fuel very rapidly and are short-lived. Low mass stars consume their fuel very slowly. Stars less massive than 0.25  M ☉ , called red dwarfs , are able to fuse nearly all of their mass while stars of about 1  M ☉ can only fuse about 10% of their mass. The combination of their slow fuel-consumption and relatively large usable fuel supply allows low mass stars to last about one trillion ( 10 × 10 ) years;

10285-412: The main sequence. Besides mass, the elements heavier than helium can play a significant role in the evolution of stars. Astronomers label all elements heavier than helium "metals", and call the chemical concentration of these elements in a star, its metallicity . A star's metallicity can influence the time the star takes to burn its fuel, and controls the formation of its magnetic fields, which affects

10406-420: The mass and orbital parameters, which makes it possible for such orbits to be observed in the physical universe. But it has been argued that this is unlikely since the domain of stability is small. For instance, the probability of a binary–binary scattering event resulting in a figure-8 orbit has been estimated to be a small fraction of a percent. In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at

10527-456: The most extreme of 0.08  M ☉ will last for about 12 trillion years. Red dwarfs become hotter and more luminous as they accumulate helium. When they eventually run out of hydrogen, they contract into a white dwarf and decline in temperature. Since the lifespan of such stars is greater than the current age of the universe (13.8 billion years), no stars under about 0.85  M ☉ are expected to have moved off

10648-445: The motions of the planets and the inferred position of the Sun. The motion of the Sun against the background stars (and the horizon) was used to create calendars , which could be used to regulate agricultural practices. The Gregorian calendar , currently used nearly everywhere in the world, is a solar calendar based on the angle of the Earth's rotational axis relative to its local star, the Sun. The oldest accurately dated star chart

10769-421: The name was first used in 1747. From the end of the 19th century to early 20th century, the approach to solve the three-body problem with the usage of short-range attractive two-body forces was developed by scientists, which offered P.F. Bedaque, H.-W. Hammer and U. van Kolck an idea to renormalize the short-range three-body problem, providing scientists a rare example of a renormalization group limit cycle at

10890-479: The names of the planets Mercury , Venus , Mars , Jupiter and Saturn were taken. ( Uranus and Neptune were Greek and Roman gods , but neither planet was known in Antiquity because of their low brightness. Their names were assigned by later astronomers.) Circa 1600, the names of the constellations were used to name the stars in the corresponding regions of the sky. The German astronomer Johann Bayer created

11011-403: The nuclear fusion of hydrogen into helium within their cores. However, stars of different masses have markedly different properties at various stages of their development. The ultimate fate of more massive stars differs from that of less massive stars, as do their luminosities and the impact they have on their environment. Accordingly, astronomers often group stars by their mass: The formation of

11132-400: The orbit is unstable, which implies one will eventually end up on one of the outbound paths after spending no energy at all. Edward Belbruno coined the term " weak stability boundary " or "fuzzy boundary" for this effect. With careful calculation, one can pick which outbound path one wants. This turns out to be useful, as many of these paths lead to some interesting points in space, such as

11253-413: The other two, moves in the plane defined by the two revolving bodies and, while being gravitationally influenced by them, exerts no influence of its own. Per Barrow-Green, "[t]he problem is then to ascertain the motion of the third body." That is to say, this two-body motion is taken to consist of circular orbits around the center of mass , and the planetoid is assumed to move in the plane defined by

11374-417: The outer convective envelope collapses and the star then moves to the horizontal branch. After a star has fused the helium of its core, it begins fusing helium along a shell surrounding the hot carbon core. The star then follows an evolutionary path called the asymptotic giant branch (AGB) that parallels the other described red-giant phase, but with a higher luminosity. The more massive AGB stars may undergo

11495-404: The outer shell of gas that it will push those layers away, forming a planetary nebula. If what remains after the outer atmosphere has been shed is less than roughly 1.4  M ☉ , it shrinks to a relatively tiny object about the size of Earth, known as a white dwarf . White dwarfs lack the mass for further gravitational compression to take place. The electron-degenerate matter inside

11616-411: The point and away from it, and all of which require nearly zero change in energy to reach. When plotted, they form a tube with the orbit about the Lagrange point at one end. The derivation of these paths traces back to mathematicians Charles C. Conley and Richard P. McGehee in 1968. Hiten , Japan's first lunar probe, was moved into lunar orbit using similar insight into the nature of paths between

11737-640: The positions of the stars. They built the first large observatory research institutes, mainly to produce Zij star catalogues. Among these, the Book of Fixed Stars (964) was written by the Persian astronomer Abd al-Rahman al-Sufi , who observed a number of stars, star clusters (including the Omicron Velorum and Brocchi's Clusters ) and galaxies (including the Andromeda Galaxy ). According to A. Zahoor, in

11858-475: The primary mission was completed, ISEE-3 went on to accomplish other goals, including a flight through the geomagnetic tail and a comet flyby. The mission was subsequently renamed the International Cometary Explorer (ICE). The first low energy transfer using what would later be called the ITN was the rescue of Japan 's Hiten lunar mission in 1991. Another example of the use of the ITN

11979-1038: The problem is described by nine second-order differential equations. The problem can also be stated equivalently in the Hamiltonian formalism , in which case it is described by a set of 18 first-order differential equations, one for each component of the positions r i {\displaystyle \mathbf {r_{i}} } and momenta p i {\displaystyle \mathbf {p_{i}} } : d r i d t = ∂ H ∂ p i , d p i d t = − ∂ H ∂ r i , {\displaystyle {\frac {d\mathbf {r_{i}} }{dt}}={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p_{i}} }},\qquad {\frac {d\mathbf {p_{i}} }{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {r_{i}} }},} where H {\displaystyle {\mathcal {H}}}

12100-403: The problem of deriving an orbit of binary stars from telescope observations was made by Felix Savary in 1827. The twentieth century saw increasingly rapid advances in the scientific study of stars. The photograph became a valuable astronomical tool. Karl Schwarzschild discovered that the color of a star and, hence, its temperature, could be determined by comparing the visual magnitude against

12221-403: The problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory , the motion of the Moon under the gravitational influence of Earth and the Sun. Later, this problem was also applied to other planets' interactions with

12342-497: The proper motion of the star Sirius and inferred a hidden companion. Edward Pickering discovered the first spectroscopic binary in 1899 when he observed the periodic splitting of the spectral lines of the star Mizar in a 104-day period. Detailed observations of many binary star systems were collected by astronomers such as Friedrich Georg Wilhelm von Struve and S. W. Burnham , allowing the masses of stars to be determined from computation of orbital elements . The first solution to

12463-771: The same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In this family, the three objects all have the same mass and can exhibit both retrograde and direct forms. In some of Broucke's solutions, two of the bodies follow the same path. In 1993, physicist Cris Moore at the Santa Fe Institute found a zero angular momentum solution with three equal masses moving around a figure-eight shape. In 2000, mathematicians Alain Chenciner and Richard Montgomery proved its formal existence. The solution has been shown numerically to be stable for small perturbations of

12584-461: The same mass. For example, when any star expands to become a red giant, it may overflow its Roche lobe , the surrounding region where material is gravitationally bound to it; if stars in a binary system are close enough, some of that material may overflow to the other star, yielding phenomena including contact binaries , common-envelope binaries, cataclysmic variables , blue stragglers , and type Ia supernovae . Mass transfer leads to cases such as

12705-451: The same temperature. Less massive T Tauri stars follow this track to the main sequence, while more massive stars turn onto the Henyey track . Most stars are observed to be members of binary star systems, and the properties of those binaries are the result of the conditions in which they formed. A gas cloud must lose its angular momentum in order to collapse and form a star. The fragmentation of

12826-500: The star's outer layers, leaving a remnant such as the Crab Nebula. The core is compressed into a neutron star , which sometimes manifests itself as a pulsar or X-ray burster . In the case of the largest stars, the remnant is a black hole greater than 4  M ☉ . In a neutron star the matter is in a state known as neutron-degenerate matter , with a more exotic form of degenerate matter, QCD matter , possibly present in

12947-400: The strength of its stellar wind. Older, population II stars have substantially less metallicity than the younger, population I stars due to the composition of the molecular clouds from which they formed. Over time, such clouds become increasingly enriched in heavier elements as older stars die and shed portions of their atmospheres . As stars of at least 0.4  M ☉ exhaust

13068-485: The supply of hydrogen at their core, they start to fuse hydrogen in a shell surrounding the helium core. The outer layers of the star expand and cool greatly as they transition into a red giant . In some cases, they will fuse heavier elements at the core or in shells around the core. As the stars expand, they throw part of their mass, enriched with those heavier elements, into the interstellar environment, to be recycled later as new stars. In about 5 billion years, when

13189-468: The surface due to strong convection and intense mass loss, or from stripping of the outer layers. When helium is exhausted at the core of a massive star, the core contracts and the temperature and pressure rises enough to fuse carbon (see Carbon-burning process ). This process continues, with the successive stages being fueled by neon (see neon-burning process ), oxygen (see oxygen-burning process ), and silicon (see silicon-burning process ). Near

13310-455: The temperature increases sufficiently, core helium fusion begins explosively in what is called a helium flash , and the star rapidly shrinks in radius, increases its surface temperature, and moves to the horizontal branch of the HR diagram. For more massive stars, helium core fusion starts before the core becomes degenerate, and the star spends some time in the red clump , slowly burning helium, before

13431-452: The three-body problem. In other words, it does not have a general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases. However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in

13552-509: The three-body problem. However, because there was not a qualitative enough solution of this system, and it was too slow for scientists to practically apply it, this solution still left some issues unresolved. In the 1970s, implication to three-body from two-body forces had been discovered by V. Efimov , which was named the Efimov effect . In 2017, Shijun Liao and Xiaoming Li applied a new strategy of numerical simulation for chaotic systems called

13673-400: The time of the ancient Greek astronomers Ptolemy and Hipparchus. William Herschel was the first astronomer to attempt to determine the distribution of stars in the sky. During the 1780s, he established a series of gauges in 600 directions and counted the stars observed along each line of sight. From this, he deduced that the number of stars steadily increased toward one side of the sky, in

13794-416: The two massive bodies, with (planar) coordinates ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} , and let ( x , y ) {\displaystyle (x,y)} be the coordinates of the planetoid. For simplicity, choose units such that

13915-571: Was NASA 's 2001–2003 Genesis mission , which orbited the Sun–Earth L 1 point for over two years collecting material, before being redirected to the L 2 Lagrange point, and finally redirected from there back to Earth. The 2003–2006 SMART-1 of the European Space Agency used another low energy transfer from the ITN. In a more recent example, the Chinese spacecraft Chang'e 2 used

14036-435: Was developed by Annie J. Cannon during the early 1900s. The first direct measurement of the distance to a star ( 61 Cygni at 11.4 light-years ) was made in 1838 by Friedrich Bessel using the parallax technique. Parallax measurements demonstrated the vast separation of the stars in the heavens. Observation of double stars gained increasing importance during the 19th century. In 1834, Friedrich Bessel observed changes in

14157-511: Was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses. In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem. The free-fall formulation starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forward and backward along an open "track". In 2023, Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published

14278-419: Was pioneered by Joseph von Fraunhofer and Angelo Secchi . By comparing the spectra of stars such as Sirius to the Sun, they found differences in the strength and number of their absorption lines —the dark lines in stellar spectra caused by the atmosphere's absorption of specific frequencies. In 1865, Secchi began classifying stars into spectral types . The modern version of the stellar classification scheme

14399-586: Was the SN 1006 supernova, which was observed in 1006 and written about by the Egyptian astronomer Ali ibn Ridwan and several Chinese astronomers. The SN 1054 supernova, which gave birth to the Crab Nebula , was also observed by Chinese and Islamic astronomers. Medieval Islamic astronomers gave Arabic names to many stars that are still used today and they invented numerous astronomical instruments that could compute

14520-603: Was the result of ancient Egyptian astronomy in 1534 BC. The earliest known star catalogues were compiled by the ancient Babylonian astronomers of Mesopotamia in the late 2nd millennium BC, during the Kassite Period ( c.  1531 BC  – c.  1155 BC ). The first star catalogue in Greek astronomy was created by Aristillus in approximately 300 BC, with the help of Timocharis . The star catalog of Hipparchus (2nd century BC) included 1,020 stars, and

14641-480: Was used to assemble Ptolemy 's star catalogue. Hipparchus is known for the discovery of the first recorded nova (new star). Many of the constellations and star names in use today derive from Greek astronomy. Despite the apparent immutability of the heavens, Chinese astronomers were aware that new stars could appear. In 185 AD, they were the first to observe and write about a supernova , now known as SN 185 . The brightest stellar event in recorded history

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