The Eddington luminosity, also referred to as the Eddington limit, is the maximum luminosity a body (such as a star) can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. The state of balance is called hydrostatic equilibrium. When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers. Since most massive stars have luminosities far below the Eddington luminosity, their winds are mostly driven by the less intense line absorption. The Eddington limit is invoked to explain the observed luminosity of accreting black holes such as quasars.
Originally, Sir Arthur Stanley Eddington took only the electron scattering into account when calculating this limit, something that now is called the classical Eddington limit. Nowadays, the modified Eddington limit also counts on other radiation processes such as bound-free and free-free radiation (see Bremsstrahlung) interaction.
The limit is obtained by setting the outward radiation pressure equal to the inward gravitational force. Both forces decrease by inverse square laws, so once equality is reached, the hydrodynamic flow is different throughout the star.
where is the velocity, is the pressure, is the density, and is the gravitational potential. If the pressure is dominated by radiation pressure associated with a radiation flux ,
Here is the opacity of the stellar material which is defined as the fraction of radiation energy flux absorbed by the medium per unit density and unit length. For ionized hydrogen , where is the Thomson scattering cross-section for the electron and is the mass of a proton. Note that is defined at the energy flux over a surface, which can be expressed with the momentum flux using for radiation. Therefore the rate of momentum transfer from the radiation to the gaseous medium per unit density is , which explains the right hand side of the above equation.
The luminosity of a source bounded by a surface is
where is the mass of the central object. This is called the Eddington Luminosity. For pure ionized hydrogen
The maximum luminosity of a source in hydrostatic equilibrium is the Eddington luminosity. If the luminosity exceeds the Eddington limit, then the radiation pressure drives an outflow.
The mass of the proton appears because, in the typical environment for the outer layers of a star, the radiation pressure acts on electrons, which are driven away from the center. Because protons are negligibly pressured by the analog of Thomson scattering, due to their larger mass, the result is to create a slight charge separation and therefore a radially directed electric field, acting to lift the positive charges, which are typically free protons under the conditions in stellar atmospheres. When the outward electric field is sufficient to levitate the protons against gravity, both electrons and protons are expelled together.
In an evolved star with a pure helium atmosphere, the electric field would have to lift a helium nucleus (an alpha particle), with nearly 4 times the mass of a proton, while the radiation pressure would act on 2 free electrons. Thus twice the usual Eddington luminosity would be needed to drive off an atmosphere of pure helium.
At very high temperatures, as in the environment of a black hole or neutron star, high energy photon interactions with nuclei or even with other photons, can create an electron-positron plasma. In that situation the combined mass of the positive-negative charge carrier pair is approximately 918 times smaller (the proton to electron mass ratio), while the radiation pressure on the positrons doubles the effective upward force per unit mass, so the limiting luminosity needed is reduced by a factor of ≈918×2.
The exact value of the Eddington luminosity depends on the chemical composition of the gas layer and the spectral energy distribution of the emission. A gas with cosmological abundances of hydrogen and helium is much more transparent than gas with solar abundance ratios. Atomic line transitions can greatly increase the effects of radiation pressure, and line driven winds exist in some bright stars (e.g., Wolf-Rayet and O stars).
The role of the Eddington limit in today’s research lies in explaining the very high mass loss rates seen in for example the series of outbursts of η Carinae in 1840–1860. The regular, line driven stellar winds can only stand for a mass loss rate of around 10−4–10−3 solar masses per year, whereas mass loss rates of up to 0.5 solar masses per year are needed to understand the η Carinae outbursts. This can be done with the help of the super-Eddington broad spectrum radiation driven winds.
Gamma-ray bursts, novae and supernovae are examples of systems exceeding their Eddington luminosity by a large factor for very short times, resulting in short and highly intensive mass loss rates. Some X-ray binaries and active galaxies are able to maintain luminosities close to the Eddington limit for very long times. For accretion-powered sources such as accreting neutron stars or cataclysmic variables (accreting white dwarfs), the limit may act to reduce or cut off the accretion flow, imposing an Eddington limit on accretion corresponding to that on luminosity. Super-Eddington accretion onto stellar-mass black holes is one possible model for ultraluminous X-ray sources (ULXs).
For accreting black holes, not all the energy released by accretion has to appear as outgoing luminosity, since energy can be lost through the event horizon, down the hole. Such sources effectively may not conserve energy. Then the accretion efficiency, or the fraction of energy actually radiated of that theoretically available from the gravitational energy release of accreting material, enters in an essential way.
The Eddington limit is not a strict limit on the luminosity of a stellar object. The limit does not consider several potentially important factors, and super-Eddington objects have been observed that do not seem to have the predicted high mass-loss rate. Other factors that might affect the maximum luminosity of a star include:
Observations of massive stars show a clear upper limit to their luminosity, termed the Humphreys-Davidson limit after the researchers who first wrote about it. Only highly unstable objects are found, temporarily, at higher luminosities. Efforts to reconcile this with the theoretical Eddington limit have been largely unsuccessful.
Blitzars are a hypothetical type of astronomical object in which a spinning pulsar rapidly collapses into a black hole. They are proposed as an explanation for fast radio bursts (FRBs). The idea was proposed in 2013 by Heino Falcke and Luciano Rezzolla.Bright giant
The luminosity class II in the Yerkes spectral classification is given to bright giants. These are stars which straddle the boundary between ordinary giants and supergiants, based on the appearance of their spectra.CN star
A CN star is a star with strong cyanogen bands in its spectrum. Cyanogen is a simple molecule of one carbon atom and one nitrogen atom, with absorption bands around 388.9 and 421.6 nm. This group of stars was first noticed by Nancy G. Roman who called them 4150 stars.Infrared dark cloud
An infrared dark cloud (IRDC) is a cold, dense region of a giant molecular cloud. They can be seen in silhouette against the bright diffuse mid-infrared emission from the galactic plane.Iron star
In astronomy, an iron star is a hypothetical type of compact star that could occur in the universe in the extremely far future, after perhaps 101500 years.
The premise behind iron stars states that cold fusion occurring via quantum tunnelling would cause the light nuclei in ordinary matter to fuse into iron-56 nuclei. Fission and alpha-particle emission would then make heavy nuclei decay into iron, converting stellar-mass objects to cold spheres of iron. The formation of these stars is only a possibility if protons do not decay. Though the surface of a neutron star may be iron, according to some predictions, it is distinct from an iron star.
Unrelatedly, the term is also used for blue supergiants which have a forest of forbidden FeII lines in their spectra. They are potentially quiescent hot luminous blue variables. Eta Carinae has been described as a prototypical example.Lambda Boötis star
A Lambda Boötis star is a type of peculiar star which has an unusually low abundance of iron peak elements in its surface layers. One possible explanation for this is that it is the result of accretion of metal-poor gas from a circumstellar disc, a second possibility is the accretion of material from a hot Jupiter suffering from mass loss. The prototype is Lambda Boötis.Lead star
A lead star is a low-metallicity star with an overabundance of lead and bismuth as compared to other products of the S-process.Nir Shaviv
Nir Joseph Shaviv (Hebrew: ניר יוסף שביב, born July 6, 1972) is an Israeli‐American physics professor, carrying out research in the fields of astrophysics and climate science. He is a professor at the Racah Institute of Physics of the Hebrew University of Jerusalem, of which he is now its chairman.
He is best known for his solar and cosmic-ray hypothesis of climate change. In 2002, Shaviv hypothesised that passages through the Milky Way's spiral arms appear to have been the cause behind the major ice-ages over the past billion years. In his later work, co-authored by Jan Veizer, a low upper limit was placed on the climatic effect of CO2.His best known contribution to the field of astrophysics was to demonstrate that the Eddington luminosity is not a strict limit, namely, that astrophysical objects can be brighter than the Eddington luminosity without blowing themselves apart. This is achieved through the development of a porous atmosphere that allows the radiation to escape while exerting little force on the gas. The theory was correctly used to explain the mass-loss in Eta Carinae's giant eruption, and the evolution of classical nova eruptions.According to German TV magazine Monitor he is one of the stars of the climate skepticism movement.OB star
OB stars are hot, massive stars of spectral types O or early-type B that form in loosely organized groups called OB associations. They are short lived, and thus do not move very far from where they formed within their life. During their lifetime, they will emit much ultraviolet radiation. This radiation rapidly ionizes the surrounding interstellar gas of the giant molecular cloud, forming an H II region or Strömgren sphere.
In lists of spectra the "spectrum of OB" refers to "unknown, but belonging to an OB association so thus of early type".Q star
A Q-Star, also known as a grey hole, is a hypothetical type of a compact, heavy neutron star with an exotic state of matter. The Q stands for a conserved particle number. A Q-Star may be mistaken for a stellar black hole.Seyfert galaxy
Seyfert galaxies are one of the two largest groups of active galaxies, along with quasars. They have quasar-like nuclei (very luminous, distant and bright sources of electromagnetic radiation) with very high surface brightnesses whose spectra reveal strong, high-ionisation emission lines, but unlike quasars, their host galaxies are clearly detectable.Seyfert galaxies account for about 10% of all galaxies and are some of the most intensely studied objects in astronomy, as they are thought to be powered by the same phenomena that occur in quasars, although they are closer and less luminous than quasars. These galaxies have supermassive black holes at their centers which are surrounded by accretion discs of in-falling material. The accretion discs are believed to be the source of the observed ultraviolet radiation. Ultraviolet emission and absorption lines provide the best diagnostics for the composition of the surrounding material.Seen in visible light, most Seyfert galaxies look like normal spiral galaxies, but when studied under other wavelengths, it becomes clear that the luminosity of their cores is of comparable intensity to the luminosity of whole galaxies the size of the Milky Way.Seyfert galaxies are named after Carl Seyfert, who first described this class in 1943.Starfield (astronomy)
A starfield refers to a set of stars visible in an arbitrarily-sized field of view, usually in the context of some region of interest within the celestial sphere. For example: the starfield surrounding the stars Betelgeuse and Rigel could be defined as encompassing some or all of the Orion constellation.Stellar-wind bubble
A stellar-wind bubble is a cavity light years across filled with hot gas blown into the interstellar medium by the high-velocity (several thousand km/s) stellar wind from a single massive star of type O or B. Weaker stellar winds also blow bubble structures, which are also called astrospheres. The heliosphere blown by the solar wind, within which all the major planets of the Solar System are embedded, is a small example of a stellar-wind bubble.
Stellar-wind bubbles have a two-shock structure. The freely-expanding stellar wind hits an inner termination shock, where its kinetic energy is thermalized, producing 106 K, X-ray emitting plasma. The hot, high-pressure, shocked wind expands, driving a shock into the surrounding interstellar gas. If the surrounding gas is dense enough (number densities or so), the swept-up gas radiatively cools far faster than the hot interior, forming a thin, relatively dense shell around the hot, shocked wind.Stellar atmosphere
The stellar atmosphere is the outer region of the volume of a star, lying above the stellar core, radiation zone and convection zone.Stellar mass
Stellar mass is a phrase that is used by astronomers to describe the mass of a star. It is usually enumerated in terms of the Sun's mass as a proportion of a solar mass (M☉). Hence, the bright star Sirius has around 2.02 M☉. A star's mass will vary over its lifetime as additional mass becomes accreted, such as from a companion star, or mass is ejected with the stellar wind or pulsational behavior.Supernova impostor
Supernova impostors are stellar explosions that appear at first to be a supernova but do not destroy their progenitor stars. As such, they are a class of extra-powerful novae. They are also known as Type V supernovae, Eta Carinae analogs, and giant eruptions of luminous blue variables (LBV).Ultraluminous X-ray source
An ultraluminous X-ray source (ULX) is an astronomical source of X-rays that is less luminous than an active galactic nucleus but is more consistently luminous than any known stellar process (over 1039 erg/s, or 1032 watts), assuming that it radiates isotropically (the same in all directions). Typically there is about one ULX per galaxy in galaxies which host them, but some galaxies contain many. The Milky Way has not been shown to contain a ULX. The main interest in ULXs stems from their luminosity exceeding the Eddington luminosity of neutron stars and even stellar black holes. It is not known what powers ULXs; models include beamed emission of stellar mass objects, accreting intermediate-mass black holes, and super-Eddington emission.Yellow giant
A yellow giant is a luminous giant star of low or intermediate mass (roughly 0.5–11 solar masses (M)) in a late phase of its stellar evolution. The outer atmosphere is inflated and tenuous, making the radius large and the surface temperature as low as 5,200-7500 K. The appearance of the yellow giant is from white to yellow, including the spectral types F and G. About 10.6 percent of all giant stars are yellow giants.
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