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# sun rich solar

Most of this matter gathered in the center, whereas the rest flattened into an orbiting disk that became the Solar System. He reasoned that it was not the chariot of Helios, but instead a giant flaming ball of metal even larger than the land of the Peloponnesus and that the Moon reflected the light of the Sun. The Sun is about halfway through its main-sequence stage, during which nuclear fusion reactions in its core fuse hydrogen into helium. The International Solar Alliance (ISA) and the University of Birmingham are working together to help farmers in ‘sun-rich’ countries make the most of chilled food distribution systems powered by solar and solar-hybrid solutions. [92] Discoveries included the first observations of coronal mass ejections, then called "coronal transients", and of coronal holes, now known to be intimately associated with the solar wind. Two identical spacecraft were launched into orbits that cause them to (respectively) pull further ahead of and fall gradually behind Earth. Get a quote today [92] Besides its direct solar observation, SOHO has enabled the discovery of a large number of comets, mostly tiny sungrazing comets that incinerate as they pass the Sun. Dr. Oliver Manuel, a professor of nuclear chemistry, believes that iron, not hydrogen, is the sun's most abundant element. At this rate, the Sun has so far converted around 100 times the mass of Earth into energy, about 0.03% of the total mass of the Sun. The change in opacity is due to the decreasing amount of H− ions, which absorb visible light easily. They based their claim on studies of meteorites and moon samples which showed traces of strange xenon. Corliss and Bozman (1962 cited in Biemont 1978) and Warner (1967 cited in Biemont 1978), Signer and Suess 1963; Manuel 1967; Marti 1969; Kuroda and Manuel 1970; Srinivasan and Manuel 1971, all cited in Manuel and Hwaung 1983, Kuroda and Manuel 1970 cited in Manuel and Hwaung 1983:7, The Arabian Science, C. Ronan, pp. Pioneer 9 operated for a particularly long time, transmitting data until May 1983. [29][30] The Sun is a Population I, or heavy-element-rich,[b] star. 2 Go solar for little to \$0 down, lock in low energy rates. {\displaystyle 2\pi /\kappa } Is it possible that Saudi Arabia is more forward-thinking than the U.S. when it comes to the end of … Data from NASA's Galileo probe of Jupiter's helium-rich atmosphere in 1996 reveals traces of strange xenon gases -- solid evidence against the conventional model of the solar system's creation, Manuel says. π [121][122] Studies of ancient meteorites reveal traces of stable daughter nuclei of short-lived isotopes, such as iron-60, that form only in exploding, short-lived stars. The proportions of metals (heavier elements) is unchanged. X [93] The interplanetary magnetic field is much stronger than the dipole component of the solar magnetic field. Ibn Yunus observed more than 10,000 entries for the Sun's position for many years using a large astrolabe.[171]. Despite its typical whiteness, most[note 1] people mentally picture the Sun as yellow; the reasons for this are the subject of debate. Sun First Solar provides the most efficient, customized energy systems in the San Francisco Bay Area. [71][72], The core is the only region in the Sun that produces an appreciable amount of thermal energy through fusion; 99% of the power is generated within 24% of the Sun's radius, and by 30% of the radius, fusion has stopped nearly entirely. In a frame of reference defined by the stars, the rotational period is approximately 25.6 days at the equator and 33.5 days at the poles. The Sun formed about 4.6 billion years ago from the collapse of part of a giant molecular cloud that consisted mostly of hydrogen and helium and that probably gave birth to many other stars. Its mass is about 330,000 times that of Earth, and accounts for about 99.86% of the total mass of the Solar System. At solar-cycle minimum, the toroidal field is, correspondingly, at minimum strength, sunspots are relatively rare, and the poloidal field is at its maximum strength. [169], Arabic astronomical contributions include Al-Battani's discovery that the direction of the Sun's apogee (the place in the Sun's orbit against the fixed stars where it seems to be moving slowest) is changing. Many ancient monuments were constructed with solar phenomena in mind; for example, stone megaliths accurately mark the summer or winter solstice (some of the most prominent megaliths are located in Nabta Playa, Egypt; Mnajdra, Malta and at Stonehenge, England); Newgrange, a prehistoric human-built mount in Ireland, was designed to detect the winter solstice; the pyramid of El Castillo at Chichén Itzá in Mexico is designed to cast shadows in the shape of serpents climbing the pyramid at the vernal and autumnal equinoxes. = [149] It takes the Solar System about 225–250 million years to complete one orbit through the Milky Way (a galactic year),[150] so it is thought to have completed 20–25 orbits during the lifetime of the Sun. [172] In 1672 Giovanni Cassini and Jean Richer determined the distance to Mars and were thereby able to calculate the distance to the Sun. One billion is 10. The Sun is a G2V star, with G2 indicating its surface temperature of approximately 5,778 K (5,505 °C, 9,941 °F), and V that it, like most stars, is a main-sequence star. Eratosthenes estimated the distance between Earth and the Sun in the 3rd century BC as "of stadia myriads 400 and 80000", the translation of which is ambiguous, implying either 4,080,000 stadia (755,000 km) or 804,000,000 stadia (148 to 153 million kilometers or 0.99 to 1.02 AU); the latter value is correct to within a few percent. Manuel claims that hydrogen fusion creates some of the sun's heat, as hydrogen -- the lightest of all elements -- moves to the sun's surface. [91], Theoretical models of the Sun's development suggest that 3.8 to 2.5 billion years ago, during the Archean eon, the Sun was only about 75% as bright as it is today. This energy, which can take between 10,000 and 170,000 years to escape the core, is the source of the Sun's light and heat. The number of substellar objects in that volume are expected to be comparable to the number of stars. [151] At this speed, it takes around 1,190 years for the Solar System to travel a distance of 1 light-year, or 7 days to travel 1 AU. II. At this point in the dynamo cycle, buoyant upwelling within the convective zone forces emergence of toroidal magnetic field through the photosphere, giving rise to pairs of sunspots, roughly aligned east–west and having footprints with opposite magnetic polarities. (see Oort constants), corresponding to 235 million years, and this is the time that the point takes to go once around the galaxy. Fischer and her team determined that 20 percent of metal-rich stars have planets orbiting them. [132] According to a 2008 model, Earth's orbit is shrinking due to tidal forces (and, eventually, drag from the lower chromosphere), so that it will be engulfed by the Sun near the tip of the red giant branch phase, 3.8 and 1 million years after Mercury and Venus have respectively had the same fate. , [93], It is thought that the energy necessary to heat the corona is provided by turbulent motion in the convection zone below the photosphere, and two main mechanisms have been proposed to explain coronal heating. [24], In English, the Greek and Latin words occur in poetry as personifications of the Sun, Helios /ˈhiːliəs/ and Sol /ˈsɒl/,[2][1] while in science fiction "Sol" may be used as a name for the Sun to distinguish it from other stars. Each second, more than four million tonnes of matter are converted into energy within the Sun's core, producing neutrinos and solar radiation. Robust PV Degradation Methodology and … [27], The English weekday name Sunday stems from Old English Sunnandæg "sun's day", a Germanic interpretation of the Latin phrase diēs sōlis, itself a translation of the Greek ἡμέρα ἡλίου hēmera hēliou "day of the sun". A shock wave from a nearby supernova would have triggered the formation of the Sun by compressing the matter within the molecular cloud and causing certain regions to collapse under their own gravity. This outermost layer of the Sun is defined to begin at the distance where the flow of the solar wind becomes superalfvénic—that is, where the flow becomes faster than the speed of Alfvén waves,[94] at approximately 20 solar radii (0.1 AU). Re-emission happens in a random direction and usually at a slightly lower energy. The energy of this sunlight supports almost all life[c] on Earth by photosynthesis,[36] and drives Earth's climate and weather. [46] The atmosphere in particular filters out over 70% of solar ultraviolet, especially at the shorter wavelengths. As a result, the outward-flowing solar wind stretches the interplanetary magnetic field outward, forcing it into a roughly radial structure. It is composed of four distinct parts: the chromosphere, the transition region, the corona and the heliosphere. ( This motion of the Sun is mainly due to Jupiter, Saturn, Uranus, and Neptune. The solar wind travels outward continuously through the heliosphere,[95][96] forming the solar magnetic field into a spiral shape,[93] until it impacts the heliopause more than 50 AU from the Sun. The photosphere has a particle density of ~1023 m−3 (about 0.37% of the particle number per volume of Earth's atmosphere at sea level). The enormous effect of the Sun on Earth has been recognized since prehistoric times, and the Sun has been regarded by some cultures as a solar deity. This differential rotation is caused by convective motion due to heat transport and the Coriolis force due to the Sun's rotation. In December 2019, a new type of solar magnetic explosion was observed, known as forced magnetic reconnection. 0 [153], The Sun is moved by the gravitational pull of the planets. [55], The Sun's original chemical composition was inherited from the interstellar medium out of which it formed. J-Box iP 65 rated This spacecraft was designed to observe gamma rays, X-rays and UV radiation from solar flares during a time of high solar activity and solar luminosity. Sun Rich Asia・ Billion Trend International is a Trading Company and Manufacturer for a full range of Christmas Products such as Decorative Ribbon, Shatterproof Plastic … [189], One of the most important solar missions to date has been the Solar and Heliospheric Observatory, jointly built by the European Space Agency and NASA and launched on 2 December 1995. For example, in the 17th century, the solar cycle appeared to have stopped entirely for several decades; few sunspots were observed during a period known as the Maunder minimum. Director General ISA, H.E Mr. Upendra Tripathy with Toby Peters, Professor in Cold Economy, University of Birmingham. [23][24] This is ultimately related to the word for "sun" in other branches of the Indo-European language family, though in most cases a nominative stem with an l is found, rather than the genitive stem in n, as for example in Latin sōl, Greek ἥλιος hēlios, Welsh haul and Russian солнце solntse (pronounced sontse), as well as (with *l > r) Sanskrit स्वर svár and Persian خور xvar. [67] It has a density of up to 150 g/cm3[68][69] (about 150 times the density of water) and a temperature of close to 15.7 million kelvin (K). {\displaystyle 2\Omega /\kappa \approx 1.50.} [69] This temperature gradient is less than the value of the adiabatic lapse rate and hence cannot drive convection, which explains why the transfer of energy through this zone is by radiation instead of thermal convection. [d][43], The solar constant is the amount of power that the Sun deposits per unit area that is directly exposed to sunlight. Ernest Rutherford suggested that the Sun's output could be maintained by an internal source of heat, and suggested radioactive decay as the source. ( [128][129] As a red giant, the Sun will grow so large that it will engulf Mercury, Venus, and probably Earth. The solar panel on its … The Sun has eight known planets. While Manuel's theory is seen as highly controversial by many in the scientific community, other researchers have confirmed that distant solar systems orbit stars that are rich in iron and other metals. [214], From at least the Fourth 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. [178] However, it would be Albert Einstein who would provide the essential clue to the source of the Sun's energy output with his mass–energy equivalence relation E = mc2. Solar flares and coronal-mass ejections tend to occur at sunspot groups. Helium and strange xenon are also seen together in Jupiter. Scientists Use Ulysses To Complete 3D Observation Of Solar Storms. The content herein, unless otherwise known to be public domain, are Copyright 1995-2016 - Space Media Network. The enormous effect of the Sun on Earth has been recognized since prehistoric times. 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. [129] For the Sun, four thermal pulses are predicted before it completely loses its outer envelope and starts to make a planetary nebula. [81] Presently, it is hypothesized (see Solar dynamo) that a magnetic dynamo within this layer generates the Sun's magnetic field. Worship of the Sun was central to civilizations such as the ancient Egyptians, the Inca of South America and the Aztecs of what is now Mexico. / ( [83] In the upper part of the chromosphere helium becomes partially ionized. The two methods generally agree well. [84][85] [188], Launched in 1991, Japan's Yohkoh (Sunbeam) satellite observed solar flares at X-ray wavelengths. have to take the same amount of time to go around the galaxy as the sun and thus remain in the same general vicinity as the Sun. Models vary depending on the rate and timing of mass loss. [106], An 11-year sunspot cycle is half of a 22-year Babcock–Leighton dynamo cycle, which corresponds to an oscillatory exchange of energy between toroidal and poloidal solar magnetic fields. 1.50. All numbers in this article are short scale. As a result, an orderly motion of the mass develops into thermal cells that carry the majority of the heat outward to the Sun's photosphere above. [82], The visible surface of the Sun, the photosphere, is the layer below which the Sun becomes opaque to visible light. [216][217], In Proto-Indo-European religion, the Sun was personified as the goddess *Seh2ul. [218][219] Derivatives of this goddess in Indo-European languages include the Old Norse Sól, Sanskrit Surya, Gaulish Sulis, Lithuanian Saulė, and Slavic Solntse. 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# 47 Ursae Majoris

Observation data Characteristics Epoch J2000.0      Equinox J2000.0 The red dot shows the location of 47 Ursae Majoris in Ursa Major. Constellation Ursa Major Right ascension 10h 59m 27.97s[1] Declination +40° 25′ 48.9″[1] Apparent magnitude (V) +5.03 Spectral type G1V U−B color index 0.13 B−V color index 0.61 Radial velocity (Rv) +12.6 km/s Proper motion (μ) RA: –317.01 ± 0.22[1] mas/yr Dec.: 54.64 ± 0.20[1] mas/yr Parallax (π) 71.11 ± 0.25[1] mas Distance 45.9 ± 0.2 ly (14.06 ± 0.05 pc) Absolute magnitude (MV) 4.41[2] Mass 1.08[3] M☉ Radius 1.172 ± 0.111 [4] R☉ Luminosity 1.48[note 1] L☉ Surface gravity (log g) 4.377[3] cgs Temperature 5887 ± 3.8 [5] K Metallicity 110% solar[3] Rotational velocity (v sin i) 2.80[3] km/s Age 6.03 × 109[6] years BD+41°2147, FK5 1282, GC 15087, GCTP 2556.00, Gliese 407, HD 95128, HIP 53721, HR 4277, LTT 12934, SAO 43557 SIMBAD data Exoplanet Archive data ARICNS data Extrasolar Planets Encyclopaedia data

47 Ursae Majoris (often abbreviated 47 UMa) is a solar analog, yellow dwarf star approximately 46 light-years[1] away from Earth in the constellation of Ursa Major. As of 2011, it has been confirmed that three Jupiter-like extrasolar planets orbit the star. Because of this, 47 Ursae Majoris was listed as one of top 100 target stars for NASA's former Terrestrial Planet Finder mission.[7]

## Distance and visibility

47 Ursae Majoris is located fairly close to the Solar System: according to astrometric measurements made by the Hipparcos astrometry satellite, the star exhibits a parallax of 71.11 milliarcseconds, corresponding to a distance of 45.913 light-years.[1] With an apparent magnitude of +5.03, it is visible to the naked eye under good conditions.

## Stellar components

47 Ursae Majoris has a similar mass to that of our Sun. It is slightly more metal-rich than the Sun, having around 110% of the solar abundance of iron. With a spectral type of G1V, it is also slightly hotter than the Sun, at around 5,882 K.[3] 47 Ursae Majoris has an absolute magnitude of +4.29, implying it has a visual luminosity around 60% greater than the Sun.

Like the Sun, 47 Ursae Majoris is on the main sequence, converting hydrogen to helium in its core by nuclear fusion. Based on its chromospheric activity, the star may be around 6,000 million years old, though evolutionary models suggest an older age of around 8,700 million years.[6] Other studies have yielded estimates of 4,400 and 7,000 million years for the star.[8]

## Planetary system

In 1996 an extrasolar planet was announced in orbit around 47 Ursae Majoris by Geoffrey Marcy and R. Paul Butler. The discovery was made by observing the change in the star's radial velocity as the planet's gravity pulled it around. The measurements were made by observing the Doppler shift of the star's spectrum.[9] The planet, designated 47 Ursae Majoris b, was the first long-period extrasolar planet discovered. Unlike the majority of known long-period extrasolar planets, 47 Ursae Majoris b has a low-eccentricity orbit. The planet is at least 2.53 times the mass of Jupiter and takes 1,078 days or 2.95 years to orbit its star. If it were to be located in the Solar System, it would lie between the orbits of Mars and Jupiter.[10]

Orbits of the planets in the 47 Ursae Majoris system. The orbit of 47UMa d is currently quite uncertain; both it and that of 47 UMa c may be circular.

In 2001, preliminary astrometric measurements made by the Hipparcos probe suggest the orbit of 47 Ursae Majoris b is inclined at an angle of 63.1° to the plane of the sky. If these measurements are confirmed, this implies the planet's true mass is around 2.9 times that of Jupiter.[11] However, subsequent analysis suggests that the Hipparcos measurements are not precise enough to accurately determine the orbits of substellar companions, and the inclination and true mass remain unknown.[12]

A second planet, designated 47 Ursae Majoris c, was announced in 2002 by Debra Fischer, Geoffrey Marcy, and R. Paul Butler. The discovery was made using the same radial velocity method used to detect the first planet. According to Fischer et al., the planet takes around 2,391 days or 6.55 years to complete an orbit. This configuration is similar to the configuration of Jupiter and Saturn in the Solar System, with the orbital ratio (close to 5:2), and mass ratio roughly similar.[13]

Subsequent measurements failed to confirm the existence of the second planet, and it was noted that the dataset used to determine its existence left the planet's parameters "almost unconstrained".[14] Analysis of a longer dataset spanning over 6,900 days suggests that while a second planet in the system is likely, periods near 2,500 days have a high false alarm probability, and the best fit model gives an orbital period of 7,586 days at a distance of 7.73 AU from the star. Nevertheless, the parameters of the second planet are still highly uncertain.[15] On the other hand, the Catalog of Nearby Exoplanets gives a period of 2,190 days, which would put the planets close to a 2:1 ratio of orbital periods, though the reference for these parameters is uncertain: the original Fischer et al. paper is cited as a reference in spite of the fact that it gives different parameters,[13][16] though this solution has been adopted by the Extrasolar Planets Encyclopaedia.[17]

In 2010, the discovery of a third planet, designated 47 Ursae Majoris d, was made by using the Bayesian Kepler Periodogram. Using this model of this planetary system found out it is 100,000 times more likely to have three planets than two planets. This discovery was announced by Debra Fischer and P.C. Gregory. This 1.64 MJ planet has an orbital period of 14,002 days or 38.33 years and a semimajor axis of 11.6 AU with a moderate eccentricity of 0.16.[10] It would be the longest-period planet discovered by radial velocity method, although longer-period planets only priorly been discovered by direct imaging and pulsar timing.

Simulations suggest that the inner part of the habitable zone of 47 Ursae Majoris could host a terrestrial planet in a stable orbit, though the outer regions of the habitable zone would be disrupted by the gravitational influence of the planet 47 Ursae Majoris b.[18] However, the presence of a giant planet within 2.5 AU of the star may have disrupted planet formation in the inner system, and reduced the amount of water delivered to inner planets during accretion.[19] This may mean any terrestrial planets orbiting in the habitable zone of 47 Ursae Majoris are likely to be small and dry. As of 2008, there have been two METI messages sent to 47 Ursae Majoris. Both were transmitted from Eurasia's largest radar — 70-meter (230-foot) Eupatoria Planetary Radar. The first message, the Teen Age Message, was sent on September 3, 2001, and it will arrive at 47 Ursae Majoris in July 2047. The second message, Cosmic Call 2, was sent on July 6, 2003, and it will arrive at 47 Ursae Majoris in May 2049.[20]

The 47 Ursae Majoris planetary system[10]
Companion
(in order from star)
Mass Semimajor axis
(AU)
Orbital period
(days)
b >2.53+0.07
−0.06
MJ
2.10 ± 0.02 1078 ± 2 0.032 ± 0.014
c >0.540+0.066
−0.073
MJ
3.6 ± 0.1 2391+100
−70
0.098+0.047
−0.096
d >1.64+0.29
−0.48
MJ
11.6+2.1
−2.9
14002+4018
−5095
0.16+0.09
−0.16

## Notes

1. ^ From $\begin{smallmatrix}L=4 \pi R^2 \sigma T_{\rm eff}^4 \end{smallmatrix}$, where $\begin{smallmatrix}L \end{smallmatrix}$ is the luminosity, $\begin{smallmatrix}R \end{smallmatrix}$ is the radius, $\begin{smallmatrix}T_{\rm eff}\end{smallmatrix}$ is the effective surface temperature and $\begin{smallmatrix}\sigma \end{smallmatrix}$ is the Stefan–Boltzmann constant

## References

1. F. van Leeuwen (2007). "HIP 53721". Hipparcos, the New Reduction. Retrieved 2009-12-08.
2. ^ Elgarøy, Øystein; Engvold, Oddbjørn; Lund, Niels (March 1999), "The Wilson-Bappu effect of the MgII K line - dependence on stellar temperature, activity and metallicity", Astronomy and Astrophysics 343: 222–228, Bibcode:1999A&A...343..222E
3. "Stars Table". Catalog of Nearby Exoplanets. Archived from the original on 17 October 2008. Retrieved 2008-10-04.
4. ^ G. T. van Belle, K. von Braun (2009). "Directly Determined Linear Radii and Effective Temperatures of Exoplanet Host Stars". Astrophysical Journal 694 (2): 1085–1098. arXiv:0901.1206. Bibcode:2009ApJ...694.1085V. doi:10.1088/0004-637X/694/2/1085.
5. ^ V. V. Kovtyukh; Soubiran, C. et al. (2003). "High precision effective temperatures for 181 F-K dwarfs from line-depth ratios". Astronomy and Astrophysics 411 (3): 559–564. arXiv:astro-ph/0308429. Bibcode:2003A&A...411..559K. doi:10.1051/0004-6361:20031378.
6. ^ a b C. Saffe; Gómez, M. et al. (2005). "On the Ages of Exoplanet Host Stars". Astronomy and Astrophysics 443 (2): 609–626. arXiv:astro-ph/0510092. Bibcode:2005A&A...443..609S. doi:10.1051/0004-6361:20053452.
7. ^ "#72 HIP 53721". TPF-C Top 100. Archived from the original on 19 August 2006. Retrieved 22 July 2006.
8. ^ E. E. Mamajek, L. A. Hillenbrand (2008). "Improved Age Estimation for Solar-Type Dwarfs Using Activity-Rotation Diagnostics". Astrophysical Journal 687 (2): 1264–1293. arXiv:0807.1686. Bibcode:2008ApJ...687.1264M. doi:10.1086/591785.
9. ^ R. P. Butler; Marcy, Geoffrey W. (1996). "A Planet Orbiting 47 Ursae Majoris". Astrophysical Journal Letters 464 (2): L153–L156. Bibcode:1996ApJ...464L.153B. doi:10.1086/310102.
10. ^ a b c P. C. Gregory, D. A. Fischer (2010). "A Bayesian periodogram finds evidence for three planets in 47 Ursae Majoris". Monthly Notices of the Royal Astronomical Society 403 (2): 731–747. arXiv:1003.5549. Bibcode:2010MNRAS.403..731G. doi:10.1111/j.1365-2966.2009.16233.x.
11. ^ I. Han, D. C. Black, G. Gatewood (2001). "Preliminary Astrometric Masses for Proposed Extrasolar Planetary Companions". Astrophysical Journal Letters 548 (1): L57–L60. Bibcode:2001ApJ...548L..57H. doi:10.1086/318927.
12. ^ D. Pourbaix, F. Arenou (2001). "Screening the Hipparcos-based astrometric orbits of sub-stellar objects". Astronomy and Astrophysics 372 (3): 935–944. arXiv:astro-ph/0104412. Bibcode:2001A&A...372..935P. doi:10.1051/0004-6361:20010597.
13. ^ a b D. A. Fischer; Marcy, Geoffrey W. et al. (2002). "A Second Planet Orbiting 47 Ursae Majoris". Astrophysical Journal 564 (2): 1028–1034. Bibcode:2002ApJ...564.1028F. doi:10.1086/324336.
14. ^ D. Naef; Mayor, M. et al. (2004). "The ELODIE survey for northern extra-solar planets. III. Three planetary candidates detected with ELODIE". Astronomy and Astrophysics 414: 351–359. arXiv:astro-ph/0310261. Bibcode:2004A&A...414..351N. doi:10.1051/0004-6361:20034091.
15. ^ R. A. Wittenmyer, M. Endl, W. D.Cochran (2007). "Long-Period Objects in the Extrasolar Planetary Systems 47 Ursae Majoris and 14 Herculis". Astrophysical Journal 654 (1): 625–632. arXiv:astro-ph/0609117. Bibcode:2007ApJ...654..625W. doi:10.1086/509110.
16. ^ "Planets Table". Catalog of Nearby Exoplanets. Archived from the original on 21 September 2008. Retrieved 2008-10-04.
17. ^ Jean Schneider (2011). "Notes for Planet 47 Uma c". Extrasolar Planets Encyclopaedia. Retrieved 3 October 2011.
18. ^ B. Jones; Underwood, David R. et al. (2005). "Prospects for Habitable "Earths" in Known Exoplanetary Systems". Astrophysical Journal 622 (2): 1091–1101. arXiv:astro-ph/0503178. Bibcode:2005ApJ...622.1091J. doi:10.1086/428108.
19. ^ S. Raymond (2006). "The Search for other Earths: limits on the giant planet orbits that allow habitable terrestrial planets to form". Astrophysical Journal Letters 643 (2): L131–134. arXiv:astro-ph/0605136. Bibcode:2006ApJ...643L.131R. doi:10.1086/505596.
20. ^ А. Л. Зайцев (7 June 2004). "Передача и поиски разумных сигналов во Вселенной". Пленарный доклад на Всероссийской астрономической конференции ВАК-2004 "Горизонты Вселенной", Москва, МГУ. (Russian)

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# Black Hole.... Not so Black

Discussion in 'Astronomy, Exobiology, & Cosmology' started by RajeshTrivedi, Oct 1, 2014.

Messages:
21,803

Messages:
21,803

Messages:
21,803

Apologies, as I somehow missed your post among the rest of my notifications.
Actually, an Interesting point you have raised.
What you say appears logical with regard to stellar mass and SMBH.
All I can add is that once matter/energy reaches that quantum stage, it is highly degenerate anyway, and we still only have the one GR theory applying to all.
What I may do, is a "tashja"who is our expert professional opinion getter,

and E-Mail a few of my own and one in particular that did frequent another forum I was on.

KilljoyKlown likes this.

7. ### KilljoyKlownWhateverValued Senior Member

Messages:
6,493
I have great admiration for the GR theory, but don't think it is the end all. Highly degenerate is an apt description that many still take to mean singularity and I just don't believe in them anymore than I believe in God. I think the degenerate process continues as the mass increases. We get to observe some of that process on this side of the EH (white dwarfs & neutron stars). Just because the mass contracts below the EH I don't believe it should be viewed as fully compressed no matter how massive it gets. Since we can't know what the mass on the other side of an EH is like, a newly formed solar BH mass could be just a single millimeter below the EH and we'd never know it.

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21,803
Agreed, GR is not the be all or end all, still its the best we have got.
And using it, and the associated Schwarzchild radius, once that is reached, further collapse is compulsory.

Agreed again, I don't believe too many physicists do accept the classical BH point Singularity.
Since GR gives up the ghost, so to speak, at the quantum/Planck level, this is where I see the Singularity beginning, and according to some definitions of singularity, in that they do begin when all accepted models break down.

Here in your finality, you seem to be invoking a "Michell/Newtonian Dark Star.....or some may call it a Frozen Star.

9. ### KilljoyKlownWhateverValued Senior Member

Messages:
6,493
I hadn't heard about this before, but I have wondered if there are any more original thoughts or ideas that haven't been thought of before. I don't really have a firm belief about the possible forms matter might have on the other side of the EH, but I do like to keep an open mind about it. If current theories are not being challenged all the time we would never make any advancements would we.

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21,803

wiki
http://en.wikipedia.org/wiki/John_Michell
John Michell (25 December 1724 – 29 April 1793) was anEnglish clergyman and natural philosopher who provided pioneering insights in a wide range of scientific fields, including astronomy, geology, optics, and gravitation. Considered "one of the greatest unsung scientists of all time",[1] he was the first person known to propose the existence of black holes in publication, the first to suggest that earthquakes travel in waves, the first to explain how to manufacture artificial magnets, and the first to apply statistics to the study of the cosmos, recognizing that double stars were a product of mutual gravitation. He also invented an apparatus to measure the mass of the Earth. He has been called both the father of seismology and the father of magnetometry.

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11. ### rcscwcRegistered Senior Member

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721
Surely, mass cannot vanish into a poof. Mass is energy and Laws of Thermodynamics do hold.

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21,803

The point is that the BH is mainly just critically curved spacetime.
The mass is in a degenerate state at the Singularity.
The following explains it far better than I can.
http://curious.astro.cornell.edu/question.php?number=219
extract:
There is a rough analogy between a black hole and an atom. In both cases, the mass is concentrated in a tiny region at the center, but the "size" of the object is much bigger. You can use the Schwarzschild radius to calculate the "density" of the black hole - i.e., the mass divided by the volume enclosed within the Schwarzschild radius. This is roughly equal to (1.8x1016 g/cm3) x (Msun / M)2, where M is defined as above. From the point of view of an outside observer, this might as well be the actual black hole density, since the distribution of matter within the Schwarzschild radius has no effect on the outside.

Messages:
21,803

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(–Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

extract:
Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution.

14. ### SchmelzerValued Senior Member

Messages:
4,177
First, the argument if "gravity gravitates" is void. What is considered to be the "source", which "gravitates", is quite arbitrary - as arbitrary as the decision what to write on the left side or the right side of the equations. Write it this way:

$g^{ab}\partial_a \partial_b g^{mn} + T_g^{mn}(g^{ab}, g^{ab}_{,k}) + T_m^{mn}(...) = 0$

and you can, as well, consider the non-linear expression containing only first derivatives as a source term too - don't forget that the energy-momentum tensor depends on the metric too, so, adding this term means only a little bit more dependence on the metric.

It is definitely not "excuse" - the Sun radiates as gravitational waves something in the order of a few kilowatts. So, there was never a justified expectation that one can observe gravitational waves from known sources.

But what about unknown sources? What about gravitational waves created new the Big Bang? They would travel almost undistorted through the universe, and they could be much more powerful than what the known sources give.

Of course, the question how much is justified to spend for fundamental science is a subtle one. But spending a few billions on gravitational radiation detection is clearly better for humankind than to spend the same money for some more tanks and nuclear rockets. And, once it is accepted to spend some billions for fundamental physics at all, then the part of these billions spend on gravity waves is unproblematic. Because it is quite clear that, after some time, one will be able to observe gravitational waves from known sources, and this will be a great increase in the observational power of astronomers. And don't forget that negative results are also important scientific results: They can falsify some theories, say, about the Big Bang, which would lead to extremely powerful sources of gravitational waves which we would observe already today.

15. ### Q-reeusValued Senior Member

Messages:
3,165
According to standard GR, LHS and RHS of EFE's equates uniquely as curvature = SET (stress-energy tensor) - RHS exclusive of 'gravitational energy-momentum density' as possible source term):
https://en.wikipedia.org/wiki/Einstein_field_equations
https://en.wikipedia.org/wiki/Einstein_field_equations#Vacuum_field_equations (explicitly, Ricci curvature tensor is zero if SET is zero.)
http://en.wikipedia.org/wiki/Stress–energy_tensor (defined first para).

Sure one can mathematically swap terms around, but then what is being equated? It's no longer curvature = non-gravitational source density. Not in any full sense. Given the Ricci tensor is zero whenever SET is zero, any vacuum 'gravitational field' is guaranteed thereby to be source free i.e. cannot be a source of it's own gravity.
That must be so regardless of determination in an inertial or accelerated frame, hence independent of any 'g' field felt in a given frame. Hence independent of any computed 'local gravitational field energy density'. In good analogy to an EM wave having frame-dependent energy density but zero charge density in any frame.
Only by means of subterfuge is this artfully and seemingly overcome.

Such is the actual situation in GR, though not as practiced by many. In say Yilmaz gravity, the claim is made gravity does indeed self-gravitate in a fully self-consistent way. Certainly the Yilmaz exponential metric equivalent to Schwarzschild metric in GR, does have the property of a non-zero vacuum Ricci curvature as required for such claim.
You are I presume referring to so-called HFGW's owing to thermal collisions mainly in the core region. Interestingly, it was shortly after posting links to just such worked calculations, in a thread elsewhere, I came to reverse my opinion from arguing in favor of such GW's (not just HFGW's), to arguing against. Once seen, it's laughably obvious the so-called quadrupole mode TT-gauge GW's with '+' and 'x' polarizations, are an impossibility. I have yet to figure if any other mode is allowable (GR says no) but won't elaborate here.
You are surely aware, after all the initial fanfare, BICEP 2 is dead and buried - 'bitten the dust' as some like to pun. A great personal relief for me, and obvious bitter disappointment to many others. Hope lives on that BICEP 3 (or maybe 4 or 5 or...) will recover some scraps or hints of primordial GW's, but even then, others have worked out alternate possible non-GW sources for 'B' modes. It might be very hard to tease one possibility from another. So far, no need. AS for LIGO, VIRGO etc., well hopes are kept alive. If there are eventually any detections such 'direct' routes, I will bet any money you like it would not be of the quadrupole mode TT-gauge variety. Binary pulsar results must have another explanation.


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# Which is the best motor type for this application?

## Which is the best motor type for this application?

(OP)
Hello,
I need to build a test instrumentation for a customer and I would like to ask your feedback regarding the best motor type to be used.
The system will be made like this:

The goal is to actuate a torque profile in a way that the force acted on the rope by the device is opposed.
The requirements are:
1- The control needs to be with torque set point (I need to create a torque profile)
2- The torque required ON THE MOTOR is from 1Nm to 4 Nm and I can not decrease it adapting the mechanical transmission
3- The motor will actuate this torque very often with 0 speed
4- The power supply will be 230 VAC 1-phase and, if required, I can use a Vac-Vdc converter
5- The cost of the equiment must be as low as possible

If I didn’t have any cost costraints my choice would be AC brushless motor + servo drive and by this solution I could satisfy all the requirements but, unfortunately, it is absolutely too expensive for the customer.
So I am searching for cheaper solution. I searched for DC motors and AC motors but I don’t have a lot of experience with this kind of motors and I ask your help.

Please write me if something is not clear and thanks in advance!

Thanks!:)

### RE: Which is the best motor type for this application?

Why not use an available electric winch or hoist?

Walt

### RE: Which is the best motor type for this application?

(OP)

I searched about winch and hoist but it seems that is not possible to define torque profile (not fixed torque) or to change (by software algorythm) the desired torque while the test is in progress..do you know if it is possible with these devices?

Thanks

### RE: Which is the best motor type for this application?

Our VFD Gurus may be taking the day off. Wait until tomorrow.

Bill
--------------------
"Why not the best?"
Jimmy Carter

### RE: Which is the best motor type for this application?

Since force seems to be your most important criteria and force control in electric motors requires expensive feedback loops involving shaft encoders and drives, perhaps pneumatics are a better choice. A simple I-P converter and a pneumatic motor can give you variable force control in an open loop system with no risk of overheating when stalled.

### RE: Which is the best motor type for this application?

Where are the VFD guys? I'll offer my ideas in the meantime. Torque is roughly proportional to current and it is possible to control current in motors. Certainly servo motors can do it and I believe VFDs and 3 phase AC motors can do it. How accurate are your needs for torque (or force) control?

### RE: Which is the best motor type for this application?

The VFD guys have noted that the desired price is -free- and there are no solutions close to that.

### RE: Which is the best motor type for this application?

Perhaps a shunt wound DC motor would work. The output torque of such a motor is linearly related to the current consumed by the armature. To control the torque, you control the armature current. Probably be accurate to better than 5%.

### RE: Which is the best motor type for this application?

Using torque control of a motor usually results in very poor control of force on the other side of a gearbox. This is due to unknown friction in the gearbox that varies with load and speed. Thus you need a load cell to measure the actual force and feed the information back to a controller. This gets pretty expensive.

Cheaper ways to control force are weights, levers, springs, pneumatics or hydraulics.

### RE: Which is the best motor type for this application?

Plain induction motor driving through a magnetic clutch. For fine control use a dancer arm.

Bill
--------------------
"Why not the best?"
Jimmy Carter

### RE: Which is the best motor type for this application?

Is heating of the clutch an issue or not really at that low of a power level?

### RE: Which is the best motor type for this application?

Something to look at would be vector control drives. I believe they do well controlling torque at low (and zero) speeds.

### RE: Which is the best motor type for this application?

Well, as a "VFD guy", the problem I see here is that at 4Nm maximum torque at the motor, that comes to about 0.6kW on a 4 pole 50Hz AC induction motor. Can it be done? Sure, but a motor even that small, with a separately powered blower (because it may not actually "spin" to cool itself), plus an encoder, plus a VFD capable of true Flux Vector Control and intelligent enough to make decisions will likely cost, in US$, close to$2000. It didn't sound as though this is a tenable solution. A small servo would likely cost less actually.

"You measure the size of the accomplishment by the obstacles you had to overcome to reach your goals" -- Booker T. Washington

### RE: Which is the best motor type for this application?

Hi Tugboat. I haven't seen a variable speed magnetic clutch for decades now. They were used for speed control in some applications. I don't remember heating issues. The clutch that I was responsible for ran fairly fast. Lots of air movement for cooling and less heat generated than when running at slower speeds.

Bill
--------------------
"Why not the best?"
Jimmy Carter

### RE: Which is the best motor type for this application?

(OP)
Hello guys,
thanks to everyone for the reply.

Customer would like to avoid the usage of pneumatic air and now the possible solutions are:
1- VFD with flux control: I agree with jraef and I think that this solution is feasible but not extremely cheap
2- Servo Motor: from my point of view (except for the price) this would be the best solution and I could really use torque profiles with closed loop (with the torque on the motor shaft).
3- DC Motor: I never used one but probably this solution would be the cheapest.

In all these solutions the problem said by Compositepro remains and the way to decrease it would be to find a way to decrease the gear ratio or to find a smart mechanical transmission.

I have two more questions: speaking about the DC motor torque control. In your experience, the performance of the torque control are similar to the servo ones? And, moreover, considering that the motor will often actuate the torque at zero speed, will it damage the motor on the long run?

Thanks!

### RE: Which is the best motor type for this application?

#### Quote:

will often actuate the torque at zero speed, will it damage the motor on the long run

Depends. Most servo motors can run motionless continuously without frying. When you reach certain power levels and sizes that breaks down and they start limiting you to duty cycles.

Keith Cress
kcress - http://www.flaminsystems.com

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# Sudoku grid guaranteed to be solvable?

I want to generate random sudoku grids.

My approach is to fill the three 3x3 groups on a diagonal of the grid, each with the numbers 1-9 randomly shuffled. That looks e.g. like the grid below:

+-------+-------+-------+
| 5 6 4 | · · · | · · · |
| 1 7 2 | · · · | · · · |
| 9 8 3 | · · · | · · · |
+-------+-------+-------+
| · · · | 3 2 5 | · · · |
| · · · | 7 9 6 | · · · |
| · · · | 8 1 4 | · · · |
+-------+-------+-------+
| · · · | · · · | 1 5 9 |
| · · · | · · · | 3 2 7 |
| · · · | · · · | 6 4 8 |
+-------+-------+-------+

Then I let my sudoku solver process it until it finds a solution to fill all remaining gaps. The example above resulted in the filled grid below:

+-------+-------+-------+
| 5 6 4 | 9 3 7 | 2 8 1 |
| 1 7 2 | 5 4 8 | 9 6 3 |
| 9 8 3 | 1 6 2 | 4 7 5 |
+-------+-------+-------+
| 7 4 9 | 3 2 5 | 8 1 6 |
| 2 1 8 | 7 9 6 | 5 3 4 |
| 6 3 5 | 8 1 4 | 7 9 2 |
+-------+-------+-------+
| 8 2 6 | 4 7 3 | 1 5 9 |
| 4 5 1 | 6 8 9 | 3 2 7 |
| 3 9 7 | 2 5 1 | 6 4 8 |
+-------+-------+-------+

My question is if this approach is mathematically safe, i.e. can you prove me that when I fill my grid using the described approach (randomly assigning the first 27 numbers to fields in the groups on a diagonal line), there will be always at least one possible way to complete the grid from there on?

Or are there chances that the randomly placed numbers can interfere with each other and result in an impossible grid?

• And, also nice to know, are there cases with more than one solution? – Paul Jan 21 '16 at 18:42

WLOG the top block is in the canonical order, because we can just relabel, as in 5 -> 1, 6 -> 2, … in your example. I'm not going to make that replacement through the rest of your grid, but will just pretend you started off like that.

+-------+-------+-------+
| 1 2 3 | · · · | · · · |
| 4 5 6 | · · · | · · · |
| 7 8 9 | · · · | · · · |
+-------+-------+-------+
| · · · | 3 2 5 | · · · |
| · · · | 7 9 6 | · · · |
| · · · | 8 1 4 | · · · |
+-------+-------+-------+
| · · · | · · · | 1 5 9 |
| · · · | · · · | 3 2 7 |
| · · · | · · · | 6 4 8 |
+-------+-------+-------+

Making minor-row swaps within one of the three major rows, or making minor-column swaps within one of the three major columns, doesn't change solveability of the sudoku. Therefore we may WLOG that the top-left entry of each of the three grids is 1, by rotating:

+-------+-------+-------+
| 1 2 3 | · · · | · · · |
| 4 5 6 | · · · | · · · |
| 7 8 9 | · · · | · · · |
+-------+-------+-------+
| · · · | 1 4 8 | · · · |
| · · · | 9 6 7 | · · · |
| · · · | 2 5 3 | · · · |
+-------+-------+-------+
| · · · | · · · | 1 5 9 |
| · · · | · · · | 3 2 7 |
| · · · | · · · | 6 4 8 |
+-------+-------+-------+

Similarly we may WLOG that the 5 of the middle block appears in one of two places:

+-------+
| 1 *   |
| ! *   |
|       |
+-------+

because if it's in any of the others, we can row/column swap it into one of those. The exception is if it's in the ! position, which is actually equivalent to the top *. This is because we may transpose the entire grid, and relabel the top-left block again, without affecting the middle block's 1 or 5 except in moving them to the correct positions.

Likewise (but without the option of reflecting this time, because that could mess up the middle block) we may WLOG that the bottom-right block's 5 is in one of three positions:

+-------+
| 1 *   |
| * *   |
|       |
+-------+

There are now $2 \times 7! \times 3 \times 7! = 152409600$ remaining cases, which are left as an exercise to the reader.

• Your explanations seems nice to me, but I didn't get how you get the numbers in your last paragraph? – Byte Commander Jan 21 '16 at 15:34
• There are 2 possible places for the 5 in the central block, and then 7! possible placements for the remaining numbers in that block. Then there are 3 possible places for the 5 in the bottom-right block, and 7! possible places for the remaining numbers in that block. – Patrick Stevens Jan 21 '16 at 16:41

I worked this out on paper by drawing a sudoku grid and putting down all the numbers for each individual digit place, column and row it could and couldn't be. As long as the programme you're using is 100% reliable, there'll will be at least one right solution to each 27 numbers in the diagonal. Sorry I can't give you a formula for proof, this sort of stuff I'm better at visualizing it.

• That was my guess as well and running my program for a few hundred times without errors supports it, but I was looking forward to some proof... – Byte Commander Jan 21 '16 at 14:57
• Sorry! Did give it my best shot! As a sudoku enthusiast, this is such a cool question that's been put up on this site. – Sophie Meaden Jan 21 '16 at 14:58


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# Math Help - Find counterexamples to the following statements.

1. ## Find counterexamples to the following statements.

I'm having trouble getting started with the following question:

Find counterexamples to the following statements.

$A \subseteq B \Rightarrow A^c \subseteq B^c$

$(A \not\subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$

$(A \subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$

How should I make stuff up?

2. ## Re: Find counterexamples to the following statements.

Originally Posted by maclunian
I'm having trouble getting started with the following question:

Find counterexamples to the following statements.

$A \subseteq B \Rightarrow A^c \subseteq B^c$

$(A \not\subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$

$(A \subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$
Let $\mathcal{U}=\{0,1,2,3,4,5,6,7,8,9\}$
If $A=\{2,3,5,6\}$ then $A^c=\{0,1,4,7,8,9\}$.
Now you do some of your own work.

3. ## Re: Find counterexamples to the following statements.

What would be an example to the first one though? How would I notate it?

4. ## Re: Find counterexamples to the following statements.

suppose B = {1,2,3,5,6}. is it true that A⊆B? what can you say about $B^c$?

5. ## Re: Find counterexamples to the following statements.

Originally Posted by Deveno
suppose B = {1,2,3,5,6}. is it true that A⊆B?
Well that would depend on what A = ?

If A = {2,3,5,6} and B = {1,2,3,5,6} then it is true that A⊆B, right?

6. ## Re: Find counterexamples to the following statements.

assume the same A as Plato gave you.

7. ## Re: Find counterexamples to the following statements.

then it is true that A⊆B, right?

8. ## Re: Find counterexamples to the following statements.

So here's my answer for the first one:

$A \subseteq B \Rightarrow A^c \not\subseteq B^c$

because $\{2,3,5,6\} \subseteq \{1,2,3,5,6\} \Rightarrow \{0,1,4,7,8,9\} \not\subseteq \{0,4,7,8,9\}$

Is this correct?

9. ## Re: Find counterexamples to the following statements.

Originally Posted by maclunian
So here's my answer for the first one:

$A \subseteq B \Rightarrow A^c \not\subseteq B^c$

because $\{2,3,5,6\} \subseteq \{1,2,3,5,6\} \Rightarrow \{0,1,4,7,8,9\} \not\subseteq \{0,4,7,8,9\}$

Is this correct?
You tell us if it is correct or not.

10. ## Re: Find counterexamples to the following statements.

Originally Posted by maclunian
So here's my answer for the first one:

$A \subseteq B \Rightarrow A^c \not\subseteq B^c$
whether this statement is correct or not, it is not what you were asked to show. you were asked to find a counter-example, that is, sets A and B for which:

$A\subseteq B \not\implies A^c \subseteq B^c$

so you should have a statement like:

$A \subseteq B$ but $A^c \not \subseteq B^c$,

for some specific A and B (one example, not a general case which the counter-example "proves").


text:
# Equation of a Circle when centre of the Circle lies on y-axis

Expressing a circle in terms of a mathematical form expression when the centre of the circle lies on the y-axis is defined as equation of a circle when centre of the circle lies on $y$-axis.

Geometrically, centre of a circle can be lied on the vertical axis without passing through the origin of the Cartesian coordinate system. The circle can be written in a mathematical expression to understand this case in mathematics. It can be developed as per the relation of the circle with the Cartesian coordinate system.

## Derivation

Imagine a circle which have radius of $r$ units and assume the centre of the circle lies on the vertical axis and also assume it does not pass through the origin. Assume, the centre of the circle is called as point $P$ and it is located at a distance of $b$ units from the origin of the Cartesian coordinate system. The centre of the circle is lied on the y-axis. So, the horizontal distance from centre of the circle to origin is zero. Therefore, the centre of the circle is located at $P\left(0,b\right)$.

Consider a point on the circle and assume it is called point $R$. Actually, it is a point which represents every one of the points on the circle. Assume it is located at $x$ units and $y$ units distance in horizontal and vertical axis from the origin of the Cartesian coordinate system. Therefore, the point $R$ is located at $R\left(x,y\right)$ in the Cartesian coordinate system.

Now, draw a line from point $P$ and it should be parallel to the horizontal axis, and draw another line from point $R$ and it should be parallel to the vertical axis and they both perpendicularly intersect at a point which is assumed to call point $Q$. Thus, a right angled triangle $\Delta RPQ$ is formed inside the circle.

In this right angled triangle, $RQ,PQ$ and $PR$ are opposite side, adjacent side and hypotenuse respectively. According to the Pythagorean Theorem, the sides of the right angled triangle can be expressed in a mathematical form as follows.

${PR}^{2}={PQ}^{2}+{RQ}^{2}$

The lengths of the right angled triangle can be evaluated from the right angled triangle.

Length of the opposite side is

Length of the adjacent side is $PQ=x$

Length of the hypotenuse is $PR=r$

Substitute these values in the mathematical relation of the three sides.

This algebraic equation is the required circle equation when the circle’s centre lies on the vertical axis. It can expand further as given below.

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Existence results for some fourth-order nonlinear elliptic problems. (English) Zbl 0981.35016
From the introduction: Let $$\Omega$$ be a bounded open set in $$\mathbb{R}^n$$. We are concerned with the fourth-order semilinear elliptic boundary value problem \begin{aligned} &\Delta^2 u+ c\Delta u= f(x,u)\quad\text{in }\Omega,\\ & u|_{\partial\Omega}=\Delta u|_{\partial\Omega}= 0,\end{aligned}\tag{1} where $$\Delta^2$$ denotes the biharmonic operator and $$c\in \mathbb{R}$$.
We also consider the fourth-order quasilinear elliptic boundary value problem \begin{aligned} &\Delta(g_1((\Delta u)^2)\Delta u)+ c\text{ div}(g_2(|\nabla u|^2)\nabla u)= f(x,u)\quad\text{in }\Omega,\\ & u|_{\partial\Omega}=\Delta u|_{\partial\Omega} =0.\end{aligned}\tag{2} It is the purpose of this paper to use variational methods for the fourth-order semilinear problem and the fourth-order quasilinear problem. We show the existence of solutions of problems (1) and (2) for a more general nonlinearity $$f$$ under weak assumptions.

##### MSC:
 35J60 Nonlinear elliptic equations 35J35 Variational methods for higher-order elliptic equations 35J30 Higher-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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##### References:
 [1] Drabek, P.; Kufner, A.; Nicolosi, F., On the solvability of degenerated quasilinear elliptic equations of higher order, J. differential equations, 109, 325-347, (1994) · Zbl 0847.35054 [2] Grigis, A.; Rothchila, L.P., A criterion for analytic hypoellipticity of a class of differential operators with polynormal coefficients, Ann. math., 108, 443-460, (1983) · Zbl 0541.35017 [3] Lazer, A.C.; Mckenna, P.J., Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM rev., 32, 537-578, (1990) · Zbl 0725.73057 [4] Lazer, A.C.; Mckenna, P.J., Global bifurcation and a theorem of tarantello, J. math. anal. appl., 181, 648-655, (1994) · Zbl 0797.34021 [5] Mckenna, P.J.; Walter, W., Nonlinear oscillations in a suspension bridge, Arch. rational mech. anal., 98, 167-177, (1987) · Zbl 0676.35003 [6] Mckenna, P.J.; Walter, W., Travelling waves in a suspension bridge, SIAM J. appl. math., 50, 703-715, (1990) · Zbl 0699.73038 [7] Micheletti, A.M.; Pistoia, A., Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear anal., 31, 895-908, (1998) · Zbl 0898.35032 [8] Tarantello, G., A note on a semilinear elliptic problem, Differential integral equations, 5, 561-566, (1992) · Zbl 0786.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.