The Majipoor series is a series of novels and stories by American writer Robert Silverberg , set on the planet Majipoor. The setting is a mixture of science fiction and fantasy elements.
129-398: Books in the series: The works in the series are as follows, in internal chronological order. Majipoor is a planet much larger than Earth, though far less dense, so that its surface gravity is almost the same as on Earth. It has been settled by humans, Ghayrogs, Skandars, Vroons, Liimen, Hjorts and other alien races for many thousands of years. The aboriginal inhabitants of the planet,
258-415: A = ∂ ∂ t + Ω ∂ ∂ φ {\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}} , the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, where Ω {\displaystyle \Omega }
387-929: A = ( 1 , 0 , 0 , 0 ) {\displaystyle k^{a}=(1,0,0,0)} . Performing a coordinate change to the advanced Eddington–Finklestein coordinates v = t + r + 2 M ln | r − 2 M | {\textstyle v=t+r+2M\ln |r-2M|} causes the metric to take the form d s 2 = − ( 1 − 2 M r ) d v 2 + ( d v d r + d r d v ) + r 2 ( d θ 2 + sin 2 θ d φ 2 ) . {\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+\left(dv\,dr+\,dr\,dv\right)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).} Under
516-517: A ′ = g a ′ v = ( − 1 + 2 M r , 1 , 0 , 0 ) . {\textstyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).} Considering the b = v {\displaystyle v} entry for k a ∇ a k b = κ k b {\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}} gives
645-410: A := J / M {\displaystyle a:=J/M} . Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing. Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a timelike Killing vector (field) . Several definitions have been proposed over
774-730: A carbon white dwarf of 0.59 M ☉ with a hydrogen atmosphere. After initially taking approximately 1.5 billion years to cool to a surface temperature of 7140 K, cooling approximately 500 more kelvins to 6590 K takes around 0.3 billion years, but the next two steps of around 500 kelvins (to 6030 K and 5550 K) take first 0.4 and then 1.1 billion years. Most observed white dwarfs have relatively high surface temperatures, between 8000 K and 40 000 K . A white dwarf, though, spends more of its lifetime at cooler temperatures than at hotter temperatures, so we should expect that there are more cool white dwarfs than hot white dwarfs. Once we adjust for
903-418: A main-sequence star of low or intermediate mass ends, such a star will expand to a red giant and fuse helium to carbon and oxygen in its core by the triple-alpha process . If a red giant has insufficient mass to generate the core temperatures required to fuse carbon (around 10 K ), an inert mass of carbon and oxygen will build up at its center. After such a star sheds its outer layers and forms
1032-419: A planet or star , will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy ). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached. For most celestial objects,
1161-468: A planetary nebula , it will leave behind a core, which is the remnant white dwarf. Usually, white dwarfs are composed of carbon and oxygen ( CO white dwarf ). If the mass of the progenitor is between 7 and 9 solar masses ( M ☉ ), the core temperature will be sufficient to fuse carbon but not neon , in which case an oxygen–neon– magnesium ( ONeMg or ONe ) white dwarf may form. Stars of very low mass will be unable to fuse helium; hence,
1290-506: A band of lowest-available energy states, the Fermi sea . This state of the electrons, called degenerate , meant that a white dwarf could cool to zero temperature and still possess high energy. Compression of a white dwarf will increase the number of electrons in a given volume. Applying the Pauli exclusion principle, this will increase the kinetic energy of the electrons, thereby increasing
1419-514: A companion star or other source, its radiation comes from its stored heat, which is not replenished. White dwarfs have an extremely small surface area to radiate this heat from, so they cool gradually, remaining hot for a long time. As a white dwarf cools, its surface temperature decreases, the radiation that it emits reddens, and its luminosity decreases. Since the white dwarf has no energy sink other than radiation, it follows that its cooling slows with time. The rate of cooling has been estimated for
SECTION 10
#17327808810091548-611: A consequence of a physical law he had proposed, which stated that an uncharged, rotating body should generate a magnetic field proportional to its angular momentum . This putative law, sometimes called the Blackett effect , was never generally accepted, and by the 1950s even Blackett felt it had been refuted. In the 1960s, it was proposed that white dwarfs might have magnetic fields due to conservation of total surface magnetic flux that existed in its progenitor star phase. A surface magnetic field of c. 100 gauss (0.01 T) in
1677-447: A density of between 10 and 10 g/cm . White dwarfs are composed of one of the densest forms of matter known, surpassed only by other compact stars such as neutron stars and the hypothetical quark stars . White dwarfs were found to be extremely dense soon after their discovery. If a star is in a binary system, as is the case for Sirius B or 40 Eridani B, it is possible to estimate its mass from observations of
1806-460: A density of over 25 000 times that of the Sun , which was so high that he called it "impossible". As Arthur Eddington put it later, in 1927: We learn about the stars by receiving and interpreting the messages which their light brings to us. The message of the companion of Sirius when it was decoded ran: "I am composed of material 3000 times denser than anything you have ever come across;
1935-498: A general change of coordinates the Killing vector transforms as k v = A t v k t {\displaystyle k^{v}=A_{t}^{v}k^{t}} giving the vectors k a ′ = δ v a ′ = ( 1 , 0 , 0 , 0 ) {\displaystyle k^{a'}=\delta _{v}^{a'}=(1,0,0,0)} and k
2064-441: A given district, so most races can be found in some numbers virtually anywhere. On the village level, there are monocultures, however. Some areas also have a small number of other intelligent beings visiting from off-world. There are also indigenous sea dragons, discovered to be highly intelligent. The characters of any given book often travel vast distances across the face of Majipoor. The planet itself has three main continents,
2193-413: A helium white dwarf may form by mass loss in an interacting binary star system. Because the material in a white dwarf no longer undergoes fusion reactions, it lacks a heat source to support it against gravitational collapse . Instead, it is supported only by electron degeneracy pressure , causing it to be extremely dense. The physics of degeneracy yields a maximum mass for a non-rotating white dwarf,
2322-415: A high color temperature , will lessen and redden with time. Over a very long time, a white dwarf will cool enough that its material will begin to crystallize, starting with the core. The star's low temperature means it will no longer emit significant heat or light, and it will become a cold black dwarf . Because the length of time it takes for a white dwarf to reach this state is calculated to be longer than
2451-408: A hydrogen or mixed hydrogen-helium atmosphere. This makes old white dwarfs with this kind of atmosphere bluer than the main cooling sequence. Hence these white dwarfs are called IR-faint white dwarfs . White dwarfs with hydrogen-poor atmospheres, such as WD J2147–4035, are less affected by CIA and therefore have a yellow to orange color. White dwarf core material is a completely ionized plasma –
2580-404: A limiting mass that no white dwarf can exceed without collapsing to a neutron star is another consequence of being supported by electron degeneracy pressure. Such limiting masses were calculated for cases of an idealized, constant density star in 1929 by Wilhelm Anderson and in 1930 by Edmund C. Stoner . This value was corrected by considering hydrostatic equilibrium for the density profile, and
2709-477: A luminosity from over 100 times that of the Sun to under 1 ⁄ 10 000 that of the Sun. Hot white dwarfs, with surface temperatures in excess of 30 000 K , have been observed to be sources of soft (i.e., lower-energy) X-rays . This enables the composition and structure of their atmospheres to be studied by soft X-ray and extreme ultraviolet observations . White dwarfs also radiate neutrinos through
SECTION 20
#17327808810092838-489: A major island, several smaller islands of note, and one hemisphere which is the Great Sea (entirely water). Surface gravity The surface gravity , g , of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which
2967-405: A match for the possible quantum states available to that electron, hence radiative heat transfer within a white dwarf is low; it does, however, have a high thermal conductivity . As a result, the interior of the white dwarf maintains an almost uniform temperature as it cools down, starting at approximately 10 K shortly after the formation of the white dwarf and reaching less than 10 K for
3096-416: A mixture of nuclei and electrons – that is initially in a fluid state. It was theoretically predicted in the 1960s that at a late stage of cooling, it should crystallize into a solid state, starting at its center. The crystal structure is thought to be a body-centered cubic lattice. In 1995 it was suggested that asteroseismological observations of pulsating white dwarfs yielded a potential test of
3225-455: A multiple of the Earth 's standard surface gravity , which is equal to In astrophysics , the surface gravity may be expressed as log g , which is obtained by first expressing the gravity in cgs units , where the unit of acceleration and surface gravity is centimeters per second squared (cm/s ), and then taking the base-10 logarithm of the cgs value of the surface gravity. Therefore,
3354-630: A multiple of the Earth's, m is its mass, expressed as a multiple of the Earth 's mass ( 5.976 × 10 kg ) and r its radius, expressed as a multiple of the Earth's (mean) radius (6,371 km). For instance, Mars has a mass of 6.4185 × 10 kg = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. The surface gravity of Mars is therefore approximately 0.107 0.532 2 = 0.38 {\displaystyle {\frac {0.107}{0.532^{2}}}=0.38} times that of Earth. Without using
3483-441: A non-radiating black dwarf in approximate thermal equilibrium with its surroundings and with the cosmic background radiation . No black dwarfs are thought to exist yet. At very low temperatures (<4000 K) white dwarfs with hydrogen in their atmosphere will be affected by collision induced absoption (CIA) of hydrogen molecules colliding with helium atoms. This affects the optical red and infrared brightness of white dwarfs with
3612-531: A position on the Hertzsprung–Russell diagram between the asymptotic giant branch and the white dwarf region. They may be called pre-white dwarfs . These variables all exhibit small (1%–30%) variations in light output, arising from a superposition of vibrational modes with periods of hundreds to thousands of seconds. Observation of these variations gives asteroseismological evidence about the interiors of white dwarfs. White dwarfs are thought to represent
3741-474: A problem because the acceleration of a test body at the event horizon of a black hole turns out to be infinite in relativity. Because of this, a renormalized value is used that corresponds to the Newtonian value in the non-relativistic limit. The value used is generally the local proper acceleration (which diverges at the event horizon) multiplied by the gravitational time dilation factor (which goes to zero at
3870-527: A remnant white dwarf composed chiefly of oxygen , neon, and magnesium , provided that its core does not collapse, and provided that fusion does not proceed so violently as to blow apart the star in a supernova . Although a few white dwarfs have been identified that may be of this type, most evidence for the existence of such comes from the novae called ONeMg or neon novae. The spectra of these novae exhibit abundances of neon, magnesium, and other intermediate-mass elements that appear to be only explicable by
3999-598: A shell that then ignites the core. The dominant mechanism by which type Ia supernovae are produced remains unclear. Despite this uncertainty in how type Ia supernovae are produced, type Ia supernovae have very uniform properties and are useful standard candles over intergalactic distances. Some calibrations are required to compensate for the gradual change in properties or different frequencies of abnormal luminosity supernovae at high redshift, and for small variations in brightness identified by light curve shape or spectrum. White dwarfs have low luminosity and therefore occupy
Majipoor series - Misplaced Pages Continue
4128-465: A spectrum by a symbol that consists of an initial D, a letter describing the primary feature of the spectrum followed by an optional sequence of letters describing secondary features of the spectrum (as shown in the adjacent table), and a temperature index number, computed by dividing 50 400 K by the effective temperature . For example, a white dwarf with only He I lines in its spectrum and an effective temperature of 15 000 K could be given
4257-499: A star will have a carbon–oxygen core that does not undergo fusion reactions, surrounded by an inner helium-burning shell and an outer hydrogen-burning shell. On the Hertzsprung–Russell diagram, it will be found on the asymptotic giant branch. It will then expel most of its outer material, creating a planetary nebula , until only the carbon–oxygen core is left. This process is responsible for the carbon–oxygen white dwarfs that form
4386-424: A star, leading to the commonly quoted value of 1.4 M ☉ . (Near the beginning of the 20th century, there was reason to believe that stars were composed chiefly of heavy elements, so, in his 1931 paper, Chandrasekhar set the average molecular weight per electron, μ e , equal to 2.5, giving a limit of 0.91 M ☉ .) Together with William Alfred Fowler , Chandrasekhar received
4515-458: A strip at the bottom of the Hertzsprung–Russell diagram , a graph of stellar luminosity versus color or temperature. They should not be confused with low-luminosity objects at the low-mass end of the main sequence, such as the hydrogen-fusing red dwarfs , whose cores are supported in part by thermal pressure, or the even lower-temperature brown dwarfs . The relationship between the mass and radius of low-mass white dwarfs can be estimated using
4644-515: A ton of my material would be a little nugget that you could put in a matchbox." What reply can one make to such a message? The reply which most of us made in 1914 was — "Shut up. Don't talk nonsense." As Eddington pointed out in 1924, densities of this order implied that, according to the theory of general relativity , the light from Sirius B should be gravitationally redshifted . This was confirmed when Adams measured this redshift in 1925. Such densities are possible because white dwarf material
4773-439: A white dwarf, which is For a more accurate computation of the mass-radius relationship and limiting mass of a white dwarf, one must compute the equation of state that describes the relationship between density and pressure in the white dwarf material. If the density and pressure are both set equal to functions of the radius from the center of the star, the system of equations consisting of the hydrostatic equation together with
4902-488: A wide color range, from the whitish-blue color of an O-, B- or A-type main sequence star to the yellow-orange of a late K- or early M-type star. White dwarf effective surface temperatures extend from over 150 000 K to barely under 4000 K. In accordance with the Stefan–Boltzmann law , luminosity increases with increasing surface temperature (proportional to T ); this surface temperature range corresponds to
5031-561: Is κ = 1 4 M {\displaystyle \kappa ={\frac {1}{4M}}} ( κ = c 4 / 4 G M {\displaystyle \kappa ={c^{4}}/{4GM}} in SI units). The surface gravity for the uncharged, rotating black hole is, simply κ = g − k , {\displaystyle \kappa =g-k,} where g = 1 4 M {\textstyle g={\frac {1}{4M}}}
5160-430: Is DA have hydrogen-dominated atmospheres. They make up the majority, approximately 80%, of all observed white dwarfs. The next class in number is of DBs, approximately 16%. The hot, above 15 000 K , DQ class (roughly 0.1%) have carbon-dominated atmospheres. Those classified as DB, DC, DO, DZ, and cool DQ have helium-dominated atmospheres. Assuming that carbon and metals are not present, which spectral classification
5289-520: Is able to reveal the presence of a magnetic field of 1 megagauss or more. Thus the basic identification process also sometimes results in discovery of magnetic fields. It has been estimated that at least 10% of white dwarfs have fields in excess of 1 million gauss (100 T). The magnetic fields in a white dwarf may allow for the existence of a new type of chemical bond , perpendicular paramagnetic bonding , in addition to ionic and covalent bonds , though detecting molecules bonded in this way
Majipoor series - Misplaced Pages Continue
5418-408: Is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's. These proportionalities may be expressed by the formula: g ∝ m r 2 {\displaystyle g\propto {\frac {m}{r^{2}}}} where g is the surface gravity of an object, expressed as
5547-412: Is expected to be difficult. The highly magnetized white dwarf in the binary system AR Scorpii was identified in 2016 as the first pulsar in which the compact object is a white dwarf instead of a neutron star. A second white dwarf pulsar was discovered in 2023. Early calculations suggested that there might be white dwarfs whose luminosity varied with a period of around 10 seconds, but searches in
5676-487: Is just these exceptions that lead to an advance in our knowledge", and so the white dwarfs entered the realm of study! The spectral type of 40 Eridani B was officially described in 1914 by Walter Adams . The white dwarf companion of Sirius, Sirius B, was next to be discovered. During the nineteenth century, positional measurements of some stars became precise enough to measure small changes in their location. Friedrich Bessel used position measurements to determine that
5805-438: Is kept from cooling very quickly only by its outer layers' opacity to radiation. The first attempt to classify white dwarf spectra appears to have been by G. P. Kuiper in 1941, and various classification schemes have been proposed and used since then. The system currently in use was introduced by Edward M. Sion , Jesse L. Greenstein and their coauthors in 1983 and has been subsequently revised several times. It classifies
5934-412: Is measured in standard solar radii and mass in standard solar masses. These computations all assume that the white dwarf is non-rotating. If the white dwarf is rotating, the equation of hydrostatic equilibrium must be modified to take into account the centrifugal pseudo-force arising from working in a rotating frame . For a uniformly rotating white dwarf, the limiting mass increases only slightly. If
6063-456: Is more than 10 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s , about a third of the speed of light . For black holes, the surface gravity must be calculated relativistically. In the Newtonian theory of gravity , the gravitational force exerted by an object is proportional to its mass: an object with twice
6192-419: Is no real property of mass. The existence of numberless visible stars can prove nothing against the existence of numberless invisible ones. Bessel roughly estimated the period of the companion of Sirius to be about half a century; C.A.F. Peters computed an orbit for it in 1851. It was not until 31 January 1862 that Alvan Graham Clark observed a previously unseen star close to Sirius, later identified as
6321-455: Is not composed of atoms joined by chemical bonds , but rather consists of a plasma of unbound nuclei and electrons . There is therefore no obstacle to placing nuclei closer than normally allowed by electron orbitals limited by normal matter. Eddington wondered what would happen when this plasma cooled and the energy to keep the atoms ionized was no longer sufficient. This paradox was resolved by R. H. Fowler in 1926 by an application of
6450-420: Is not constant, but increases as the planet grows in size, as they are not incompressible bodies. That is why the experimental relationship between surface gravity and mass does not grow as 1/3 but as 1/2: g = M 1 / 2 {\displaystyle g=M^{1/2}} here with g in times Earth's surface gravity and M in times Earth's mass. In fact, the exoplanets found fulfilling
6579-431: Is seen depends on the effective temperature. Between approximately 100 000 K to 45 000 K , the spectrum will be classified DO, dominated by singly ionized helium. From 30 000 K to 12 000 K , the spectrum will be DB, showing neutral helium lines, and below about 12 000 K , the spectrum will be featureless and classified DC. Molecular hydrogen ( H 2 ) has been detected in spectra of
SECTION 50
#17327808810096708-525: Is smaller at the equator than at the poles. This effect was exploited by Hal Clement in his SF novel Mission of Gravity , dealing with a massive, fast-spinning planet where gravity was much higher at the poles than at the equator. To the extent that an object's internal distribution of mass differs from a symmetric model, the measured surface gravity may be used to deduce things about the object's internal structure. This fact has been put to practical use since 1915–1916, when Roland Eötvös 's torsion balance
6837-458: Is that the Universe's age is finite; there has not been enough time for white dwarfs to cool below this temperature. The white dwarf luminosity function can therefore be used to find the time when stars started to form in a region; an estimate for the age of our galactic disk found in this way is 8 billion years. A white dwarf will eventually, in many trillions of years, cool and become
6966-571: Is the Schwarzschild surface gravity, and k := M Ω + 2 {\displaystyle k:=M\Omega _{+}^{2}} is the spring constant of the rotating black hole. Ω + {\displaystyle \Omega _{+}} is the angular velocity at the event horizon. This expression gives a simple Hawking temperature of 2 π T = g − k {\displaystyle 2\pi T=g-k} . The surface gravity for
7095-643: Is the angular velocity. Since k a {\displaystyle k^{a}} is a Killing vector k a ∇ a k b = κ k b {\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}} implies − k a ∇ b k a = κ k b {\displaystyle -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}} . In ( t , r , θ , φ ) {\displaystyle (t,r,\theta ,\varphi )} coordinates k
7224-395: Is the electric charge, J {\displaystyle J} is the angular momentum, define r ± := M ± M 2 − Q 2 − J 2 / M 2 {\textstyle r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}} to be the locations of the two horizons and
7353-440: Is thought that, over a lifespan that considerably exceeds the age of the universe ( c. 13.8 billion years), such a star will eventually burn all its hydrogen, for a while becoming a blue dwarf , and end its evolution as a helium white dwarf composed chiefly of helium-4 nuclei. Due to the very long time this process takes, it is not thought to be the origin of the observed helium white dwarfs. Rather, they are thought to be
7482-403: Is thought to have a surface field of approximately 300 million gauss (30 kT). Since 1970, magnetic fields have been discovered in well over 200 white dwarfs, ranging from 2 × 10 to 10 gauss (0.2 T to 100 kT). The large number of presently known magnetic white dwarfs is due to the fact that most white dwarfs are identified by low-resolution spectroscopy, which
7611-405: Is usually at least 1000 times more abundant than all other elements. As explained by Schatzman in the 1940s, the high surface gravity is thought to cause this purity by gravitationally separating the atmosphere so that heavy elements are below and the lighter above. This atmosphere, the only part of the white dwarf visible to us, is thought to be the top of an envelope that is a residue of
7740-460: Is very dense : its mass is comparable to the Sun 's, while its volume is comparable to Earth 's. No nuclear fusion takes place in a white dwarf. Instead, the light it radiates comes from the residual heat stored in it. The nearest known white dwarf is Sirius B , at 8.6 light years, the smaller component of the Sirius binary star . There are currently thought to be eight white dwarfs among
7869-454: Is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere. Surface gravity is measured in units of acceleration, which, in the SI system, are meters per second squared . It may also be expressed as
SECTION 60
#17327808810097998-580: The Chandrasekhar limit — approximately 1.44 times M ☉ — beyond which electron degeneracy pressure cannot support it. A carbon–oxygen white dwarf that approaches this mass limit, typically by mass transfer from a companion star, may explode as a Type Ia supernova via a process known as carbon detonation ; SN 1006 is a likely example. A white dwarf is very hot when it forms, and it will gradually cool as it radiates its energy away. This means that its radiation, which initially has
8127-473: The DAV , or ZZ Ceti , stars, including HL Tau 76, with hydrogen-dominated atmospheres and the spectral type DA; DBV , or V777 Her , stars, with helium-dominated atmospheres and the spectral type DB; and GW Vir stars , sometimes subdivided into DOV and PNNV stars, with atmospheres dominated by helium, carbon, and oxygen. GW Vir stars are not, strictly speaking, white dwarfs, but are stars that are in
8256-799: The Kerr–Newman solution is κ = r + − r − 2 ( r + 2 + a 2 ) = M 2 − Q 2 − J 2 / M 2 2 M 2 − Q 2 + 2 M M 2 − Q 2 − J 2 / M 2 , {\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2\left(r_{+}^{2}+a^{2}\right)}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},} where Q {\displaystyle Q}
8385-573: The Lady of the Isle of Sleep , promoting the morals of Majipoor by sending dreams to its inhabitants; while a hereditary King of Dreams on the distant continent of Suvrael punishes wrongdoers by visiting them with nightmares . The post of King of Dreams is created at the end of King of Dreams , while Valentine Pontifex ends with the creation of a fifth Power, representing the Piurivar. Majipoor receives
8514-556: The Nobel Prize for this and other work in 1983. The limiting mass is now called the Chandrasekhar limit . If a carbon-oxygen white dwarf accreted enough matter to reach the Chandrasekhar limit of about 1.44 solar masses (for a non-rotating star), it would no longer be able to support the bulk of its mass through electron degeneracy pressure and, in the absence of nuclear reactions, would begin to collapse. However,
8643-473: The Urca process . This process has more effect on hotter and younger white dwarfs. Because neutrinos can pass easily through stellar plasma, they can drain energy directly from the dwarf's interior; this mechanism is the dominant contribution to cooling for approximately the first 20 million years of a white dwarf's existence. As was explained by Leon Mestel in 1952, unless the white dwarf accretes matter from
8772-484: The radius of the Sun ; this is comparable to the Earth's radius of approximately 0.9% solar radius. A white dwarf, then, packs mass comparable to the Sun's into a volume that is typically a million times smaller than the Sun's; the average density of matter in a white dwarf must therefore be, very roughly, 1 000 000 times greater than the average density of the Sun, or approximately 10 g/cm , or 1 tonne per cubic centimetre. A typical white dwarf has
8901-454: The selection effect that hotter, more luminous white dwarfs are easier to observe, we do find that decreasing the temperature range examined results in finding more white dwarfs. This trend stops when we reach extremely cool white dwarfs; few white dwarfs are observed with surface temperatures below 4000 K , and one of the coolest so far observed, WD J2147–4035 , has a surface temperature of approximately 3050 K. The reason for this
9030-472: The shell theorem , the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton . Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius , and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example,
9159-597: The 1940s. By 1950, over a hundred were known, and by 1999, over 2000 were known. Since then the Sloan Digital Sky Survey has found over 9000 white dwarfs, mostly new. Although white dwarfs are known with estimated masses as low as 0.17 M ☉ and as high as 1.33 M ☉ , the mass distribution is strongly peaked at 0.6 M ☉ , and the majority lie between 0.5 and 0.7 M ☉ . The estimated radii of observed white dwarfs are typically 0.8–2%
9288-436: The 1960s failed to observe this. The first variable white dwarf found was HL Tau 76 ; in 1965 and 1966, and was observed to vary with a period of approximately 12.5 minutes. The reason for this period being longer than predicted is that the variability of HL Tau 76, like that of the other pulsating variable white dwarfs known, arises from non-radial gravity wave pulsations. Known types of pulsating white dwarf include
9417-478: The CNO cycle may keep these white dwarfs hot on a long timescale. In addition, they remain in a bloated proto-white dwarf stage for up to 2 Gyr before they reach the cooling track. Although most white dwarfs are thought to be composed of carbon and oxygen, spectroscopy typically shows that their emitted light comes from an atmosphere that is observed to be either hydrogen or helium dominated. The dominant element
9546-413: The Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation , which gives the formula g = G M r 2 {\displaystyle g={\frac {GM}{r^{2}}}} where M is the mass of the object, r is its radius, and G is the gravitational constant . If ρ = M / V denote the mean density of
9675-437: The Earth, and hence white dwarfs. Willem Luyten appears to have been the first to use the term white dwarf when he examined this class of stars in 1922; the term was later popularized by Arthur Eddington . Despite these suspicions, the first non-classical white dwarf was not definitely identified until the 1930s. 18 white dwarfs had been discovered by 1939. Luyten and others continued to search for white dwarfs in
9804-516: The Piurivar, were conquered by humans many thousands of years ago and are now confined to a reservation. The planet is ruled by an unusual tetrarchy of Powers: an adoptive Coronal rules in a highly visible and symbolic manner from his palace atop Castle Mount; the previous Coronal retires to become the Pontifex , the head of the bureaucracy in an underground Labyrinth ; the Coronal's mother becomes
9933-416: The Schwarzschild solution, take k a {\displaystyle k^{a}} to be the time translation Killing vector k a ∂ a = ∂ ∂ t {\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}} , and more generally for the Kerr–Newman solution take k a ∂
10062-451: The absolute luminosity and distance, the star's surface area and its radius can be calculated. Reasoning of this sort led to the realization, puzzling to astronomers at the time, that due to their relatively high temperature and relatively low absolute luminosity, Sirius B and 40 Eridani B must be very dense. When Ernst Öpik estimated the density of a number of visual binary stars in 1916, he found that 40 Eridani B had
10191-442: The accretion of material onto an oxygen–neon–magnesium white dwarf. Type Iax supernovae , that involve helium accretion by a white dwarf, have been proposed to be a channel for transformation of this type of stellar remnant. In this scenario, the carbon detonation produced in a Type Ia supernova is too weak to destroy the white dwarf, expelling just a small part of its mass as ejecta, but produces an asymmetric explosion that kicks
10320-433: The atmospheres of some white dwarfs. Around 25–33% of white dwarfs have metal lines in their spectra, which is notable because any heavy elements in a white dwarf should sink into the star's interior in just a small fraction of the star's lifetime. The prevailing explanation for metal-rich white dwarfs is that they have recently accreted rocky planetesimals . The bulk composition of the accreted object can be measured from
10449-399: The behavior of real structures. In relativity, the Newtonian concept of acceleration turns out not to be clear cut. For a black hole, which must be treated relativistically, one cannot define a surface gravity as the acceleration experienced by a test body at the object's surface because there is no surface, although the event horizon is a natural alternative candidate, but this still presents
10578-457: The binary orbit. This was done for Sirius B by 1910, yielding a mass estimate of 0.94 M ☉ , which compares well with a more modern estimate of 1.00 M ☉ . Since hotter bodies radiate more energy than colder ones, a star's surface brightness can be estimated from its effective surface temperature , and that from its spectrum . If the star's distance is known, its absolute luminosity can also be estimated. From
10707-461: The classification of DB3, or, if warranted by the precision of the temperature measurement, DB3.5. Likewise, a white dwarf with a polarized magnetic field , an effective temperature of 17 000 K , and a spectrum dominated by He I lines that also had hydrogen features could be given the classification of DBAP3. The symbols "?" and ":" may also be used if the correct classification is uncertain. White dwarfs whose primary spectral classification
10836-424: The collapse. If a white dwarf star accumulates sufficient material from a stellar companion to raise its core temperature enough to ignite carbon fusion , it will undergo runaway nuclear fusion, completely disrupting it. There are three avenues by which this detonation is theorised to happen: stable accretion of material from a companion, the collision of two white dwarfs, or accretion that causes ignition in
10965-405: The coolest known white dwarfs. An outer shell of non-degenerate matter sits on top of the degenerate core. The outermost layers, which are cooler than the interior, radiate roughly as a black body . A white dwarf remains visible for a long time, as its tenuous outer atmosphere slowly radiates the thermal content of the degenerate interior. The visible radiation emitted by white dwarfs varies over
11094-407: The core of the star will collapse and it will explode in a core-collapse supernova that will leave behind a remnant neutron star, black hole , or possibly a more exotic form of compact star . Some main-sequence stars, of perhaps 8 to 10 M ☉ , although sufficiently massive to fuse carbon to neon and magnesium , may be insufficiently massive to fuse neon . Such a star may leave
11223-411: The crystallization theory, and in 2004, observations were made that suggested approximately 90% of the mass of BPM 37093 had crystallized. Other work gives a crystallized mass fraction of between 32% and 82%. As a white dwarf core undergoes crystallization into a solid phase, latent heat is released, which provides a source of thermal energy that delays its cooling. Another possible mechanism that
11352-404: The current age of the known universe (approximately 13.8 billion years), it is thought that no black dwarfs yet exist. The oldest known white dwarfs still radiate at temperatures of a few thousand kelvins , which establishes an observational limit on the maximum possible age of the universe . The first white dwarf discovered was in the triple star system of 40 Eridani , which contains
11481-430: The current view is that this limit is not normally attained; increasing temperature and density inside the core ignite carbon fusion as the star approaches the limit (to within about 1%) before collapse is initiated. In contrast, for a core primarily composed of oxygen, neon and magnesium, the collapsing white dwarf will typically form a neutron star . In this case, only a fraction of the star's mass will be ejected during
11610-489: The differential equation − 1 2 ∂ ∂ r ( − 1 + 2 M r ) = κ . {\textstyle -{\frac {1}{2}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .} Therefore, the surface gravity for the Schwarzschild solution with mass M {\displaystyle M}
11739-479: The discovery that all the stars of very faint absolute magnitude were of spectral class M. In conversation on this subject (as I recall it), I asked Pickering about certain other faint stars, not on my list, mentioning in particular 40 Eridani B. Characteristically, he sent a note to the Observatory office and before long the answer came (I think from Mrs. Fleming) that the spectrum of this star
11868-533: The end point of stellar evolution for main-sequence stars with masses from about 0.07 to 10 M ☉ . The composition of the white dwarf produced will depend on the initial mass of the star. Current galactic models suggest the Milky Way galaxy currently contains about ten billion white dwarfs. If the mass of a main-sequence star is lower than approximately half a solar mass , it will never become hot enough to ignite and fuse helium in its core. It
11997-417: The equation of state can then be solved to find the structure of the white dwarf at equilibrium. In the non-relativistic case, we will still find that the radius is inversely proportional to the cube root of the mass. Relativistic corrections will alter the result so that the radius becomes zero at a finite value of the mass. This is the limiting value of the mass – called the Chandrasekhar limit – at which
12126-407: The event horizon). For the Schwarzschild case, this value is mathematically well behaved for all non-zero values of r and M . When one talks about the surface gravity of a black hole, one is defining a notion that behaves analogously to the Newtonian surface gravity, but is not the same thing. In fact, the surface gravity of a general black hole is not well defined. However, one can define
12255-435: The former relationship have been found to be rocky planets. Thus, for rocky planets, density grows with mass as ρ ∝ M 1 / 4 {\displaystyle \rho \propto M^{1/4}} . For gas giant planets such as Jupiter, Saturn, Uranus, and Neptune, the surface gravity is given at the 1 bar pressure level in the atmosphere. It has been found that for giant planets with masses in
12384-483: The hundred star systems nearest the Sun. The unusual faintness of white dwarfs was first recognized in 1910. The name white dwarf was coined by Willem Jacob Luyten in 1922. White dwarfs are thought to be the final evolutionary state of stars whose mass is not high enough to become a neutron star or black hole . This includes over 97% of the stars in the Milky Way . After the hydrogen - fusing period of
12513-536: The mass-produces twice as much force. Newtonian gravity also follows an inverse square law , so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light , which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space. A large object, such as
12642-524: The newly devised quantum mechanics . Since electrons obey the Pauli exclusion principle , no two electrons can occupy the same state , and they must obey Fermi–Dirac statistics , also introduced in 1926 to determine the statistical distribution of particles that satisfy the Pauli exclusion principle. At zero temperature, therefore, electrons can not all occupy the lowest-energy, or ground , state; some of them would have to occupy higher-energy states, forming
12771-406: The non-relativistic formula T = p / 2 m for the kinetic energy, it is non-relativistic. When the electron velocity in a white dwarf is close to the speed of light , the kinetic energy formula approaches T = pc where c is the speed of light, and it can be shown that there is no stable equilibrium in the ultrarelativistic limit . In particular, this analysis yields the maximum mass of
12900-399: The nonrelativistic Fermi gas equation of state, which gives where R is the radius, M is the total mass of the star, N is the number of electrons per unit mass (dependent only on composition), m e is the electron mass , ℏ {\displaystyle \hbar } is the reduced Planck constant , and G is the gravitational constant . Since this analysis uses
13029-563: The object, this can also be written as g = 4 π 3 G ρ r {\displaystyle g={\frac {4\pi }{3}}G\rho r} so that, for fixed mean density, the surface gravity g is proportional to the radius r . Solving for mass, this equation can be written as g = G ( 4 π ρ 3 ) 2 / 3 M 1 / 3 {\displaystyle g=G\left({\frac {4\pi \rho }{3}}\right)^{2/3}M^{1/3}} But density
13158-406: The occasional starship , but is generally considered a backwater planet. Metals of all sorts are scarce, since the planet has a very light crust. Technology is pervasive but often of ancient origin and no longer fully understood. For example, draft animals called "mounts" are used for farming and transport, but the animals were artificially created in the distant past by genetic manipulation, and
13287-477: The predicted companion. Walter Adams announced in 1915 that he had found the spectrum of Sirius B to be similar to that of Sirius. In 1917, Adriaan van Maanen discovered van Maanen's Star , an isolated white dwarf. These three white dwarfs, the first discovered, are the so-called classical white dwarfs . Eventually, many faint white stars that had high proper motion were found, indicating that they could be suspected to be low-luminosity stars close to
13416-526: The presently known value of the limit was first published in 1931 by Subrahmanyan Chandrasekhar in his paper "The Maximum Mass of Ideal White Dwarfs". For a non-rotating white dwarf, it is equal to approximately 5.7 M ☉ / μ e , where μ e is the average molecular weight per electron of the star. As the carbon-12 and oxygen-16 that predominantly compose a carbon–oxygen white dwarf both have atomic numbers equal to half their atomic weight , one should take μ e equal to 2 for such
13545-412: The pressure. This electron degeneracy pressure supports a white dwarf against gravitational collapse. The pressure depends only on density and not on temperature. Degenerate matter is relatively compressible; this means that the density of a high-mass white dwarf is much greater than that of a low-mass white dwarf and that the radius of a white dwarf decreases as its mass increases. The existence of
13674-438: The product of mass loss in binary systems or mass loss due to a large planetary companion. If the mass of a main-sequence star is between 0.5 and 8 M ☉ , its core will become sufficiently hot to fuse helium into carbon and oxygen via the triple-alpha process , but it will never become sufficiently hot to fuse carbon into neon . Near the end of the period in which it undergoes fusion reactions, such
13803-456: The progenitor star would thus become a surface magnetic field of c. 100 × 100 = 1 million gauss (100 T) once the star's radius had shrunk by a factor of 100. The first magnetic white dwarf to be discovered was GJ 742 (also known as GRW +70 8247 ), which was identified by James Kemp, John Swedlund, John Landstreet and Roger Angel in 1970 to host a magnetic field by its emission of circularly polarized light. It
13932-471: The range up to 100 times Earth's mass, their gravity surface is nevertheless very similar and close to 1 g , a region named the gravity plateau . Most real astronomical objects are not perfectly spherically symmetric. One reason for this is that they are often rotating, which means that they are affected by the combined effects of gravitational force and centrifugal force . This causes stars and planets to be oblate , which means that their surface gravity
14061-432: The recently discovered planet, Gliese 581 c , has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it
14190-429: The relatively bright main sequence star 40 Eridani A , orbited at a distance by the closer binary system of the white dwarf 40 Eridani B and the main sequence red dwarf 40 Eridani C . The pair 40 Eridani B/C was discovered by William Herschel on 31 January 1783. In 1910, Henry Norris Russell , Edward Charles Pickering and Williamina Fleming discovered that, despite being a dim star, 40 Eridani B
14319-523: The result is that the planet or star in question can be treated as a near-perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge . Examples of such rapidly rotating stars include Achernar , Altair , Regulus A and Vega . The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to
14448-402: The rigorous mathematical literature. The fine structure of the free boundary of white dwarfs has also been analysed mathematically rigorously. The degenerate matter that makes up the bulk of a white dwarf has a very low opacity , because any absorption of a photon requires that an electron must transition to a higher empty state, which may not be possible as the energy of the photon may not be
14577-417: The star is allowed to rotate nonuniformly, and viscosity is neglected, then, as was pointed out by Fred Hoyle in 1947, there is no limit to the mass for which it is possible for a model white dwarf to be in static equilibrium. Not all of these model stars will be dynamically stable. Rotating white dwarfs and the estimates of their diameter in terms of the angular velocity of rotation has been treated in
14706-492: The star's envelope in the AGB phase and may also contain material accreted from the interstellar medium . The envelope is believed to consist of a helium-rich layer with mass no more than 1 ⁄ 100 of the star's total mass, which, if the atmosphere is hydrogen-dominated, is overlain by a hydrogen-rich layer with mass approximately 1 ⁄ 10 000 of the star's total mass. Although thin, these outer layers determine
14835-404: The stars Sirius (α Canis Majoris) and Procyon (α Canis Minoris) were changing their positions periodically. In 1844 he predicted that both stars had unseen companions: If we were to regard Sirius and Procyon as double stars, the change of their motions would not surprise us; we should acknowledge them as necessary, and have only to investigate their amount by observation. But light
14964-450: The strengths of the metal lines. For example, a 2015 study of the white dwarf Ton 345 concluded that its metal abundances were consistent with those of a differentiated , rocky planet whose mantle had been eroded by the host star's wind during its asymptotic giant branch phase. Magnetic fields in white dwarfs with a strength at the surface of c. 1 million gauss (100 teslas ) were predicted by P. M. S. Blackett in 1947 as
15093-406: The surface gravity for a black hole whose event horizon is a Killing horizon. The surface gravity κ {\displaystyle \kappa } of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if k a {\displaystyle k^{a}} is a suitably normalized Killing vector , then
15222-664: The surface gravity is defined by k a ∇ a k b = κ k b , {\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},} where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that k a k a → − 1 {\displaystyle k^{a}k_{a}\to -1} as r → ∞ {\displaystyle r\to \infty } , and so that κ ≥ 0 {\displaystyle \kappa \geq 0} . For
15351-495: The surface gravity of Earth could be expressed in cgs units as 980.665 cm/s , and then taking the base-10 logarithm ("log g ") of 980.665, giving 2.992 as "log g ". The surface gravity of a white dwarf is very high, and of a neutron star even higher. A white dwarf's surface gravity is around 100,000 g ( 10 m/s ) whilst the neutron star's compactness gives it a surface gravity of up to 7 × 10 m/s with typical values of order 10 m/s (that
15480-506: The technologies involved have been forgotten. Many great engineering works are described, but these were also usually created long ago. Many modern technologies, such as TV and radio , seem to be nonexistent or of very limited use. The average Majipooran lives a peasant lifestyle, and agriculture is a common occupation. There are several races which inhabit Majipoor in the millions, along with various aliens visiting from off-world. Ancient laws dictate that no race may exclusively occupy
15609-420: The thermal evolution of the white dwarf. The degenerate electrons in the bulk of a white dwarf conduct heat well. Most of a white dwarf's mass is therefore at almost the same temperature ( isothermal ), and it is also hot: a white dwarf with surface temperature between 8000 K and 16 000 K will have a core temperature between approximately 5 000 000 K and 20 000 000 K . The white dwarf
15738-426: The vast majority of observed white dwarfs. If a star is massive enough, its core will eventually become sufficiently hot to fuse carbon to neon, and then to fuse neon to iron. Such a star will not become a white dwarf, because the mass of its central, non-fusing core, initially supported by electron degeneracy pressure, will eventually exceed the largest possible mass supportable by degeneracy pressure. At this point
15867-436: The white dwarf can no longer be supported by electron degeneracy pressure. The graph on the right shows the result of such a computation. It shows how radius varies with mass for non-relativistic (blue curve) and relativistic (green curve) models of a white dwarf. Both models treat the white dwarf as a cold Fermi gas in hydrostatic equilibrium. The average molecular weight per electron, μ e , has been set equal to 2. Radius
15996-464: The years by various authors, such as peeling surface gravity and Kodama surface gravity. As of current, there is no consensus or agreement on which definition, if any, is correct. Semiclassical results indicate that the peeling surface gravity is ill-defined for transient objects formed in finite time of a distant observer. White dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter . A white dwarf
16125-407: Was A. I knew enough about it, even in these paleozoic days, to realize at once that there was an extreme inconsistency between what we would then have called "possible" values of the surface brightness and density. I must have shown that I was not only puzzled but crestfallen, at this exception to what looked like a very pretty rule of stellar characteristics; but Pickering smiled upon me, and said: "It
16254-455: Was first confirmed in 2019 after the identification of a pile up in the cooling sequence of more than 15 000 white dwarfs observed with the Gaia satellite. Low-mass helium white dwarfs (mass < 0.20 M ☉ ), often referred to as extremely low-mass white dwarfs (ELM WDs), are formed in binary systems. As a result of their hydrogen-rich envelopes, residual hydrogen burning via
16383-486: Was of spectral type A, or white. In 1939, Russell looked back on the discovery: I was visiting my friend and generous benefactor, Prof. Edward C. Pickering. With characteristic kindness, he had volunteered to have the spectra observed for all the stars – including comparison stars – which had been observed in the observations for stellar parallax which Hinks and I made at Cambridge, and I discussed. This piece of apparently routine work proved very fruitful – it led to
16512-435: Was suggested to explain the seeming delay in the cooling of some types of white dwarves is a solid–liquid distillation process: the crystals formed in the core are buoyant and float up, thereby displacing heavier liquid downward, thus causing a net release of gravitational energy. Chemical fractionation between the ionic species in the plasma mixture can release a similar or even greater amount of energy. This energy release
16641-553: Was used to prospect for oil near the city of Egbell (now Gbely , Slovakia .) In 1924, the torsion balance was used to locate the Nash Dome oil fields in Texas . It is sometimes useful to calculate the surface gravity of simple hypothetical objects which are not found in nature. The surface gravity of infinite planes, tubes, lines, hollow shells, cones, and even more unrealistic structures may be used to provide insights into
#8991