The Eddington luminosity, also referred to as the Eddington limit, is the maximum
luminosity
Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a st ...
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
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
. 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 hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s such as
quasars.
Originally, Sir
Arthur 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
''Bremsstrahlung'' (), from "to brake" and "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typicall ...
) interaction.
Derivation
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 the same throughout the star.
From
Euler's equation
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
in
hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
, the mean acceleration is zero,
where
is the velocity,
is the pressure,
is the density, and
is the
gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
. 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
Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequen ...
cross-section for the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
and
is the mass of a proton. Note that
is defined as 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
may be expressed with these relations as
Now assuming that the opacity is a constant, it can be brought outside of the integral. Using
Gauss's theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface (mathematics), surface to the ''divergence'' o ...
and
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
gives
where
is the mass of the central object. This is called the Eddington Luminosity. For pure ionized hydrogen,
where
is the mass of the Sun and
is the luminosity of the Sun.
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.
Different limits for different materials
The derivation above for the outward light pressure assumes a
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
plasma. In other circumstances the pressure balance can be different from what it is for hydrogen.
In an evolved star with a pure
helium
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
atmosphere, the electric field would have to lift a helium nucleus (an
alpha particle
Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be produce ...
), 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
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
or
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
, 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).
Super-Eddington luminosities
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
Eta Carinae (η Carinae, abbreviated to η Car), formerly known as Eta Argus, is a stellar system containing at least two stars with a combined luminosity greater than five million times that of the Sun, located around distant in t ...
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 burst
In gamma-ray astronomy, gamma-ray bursts (GRBs) are immensely energetic explosions that have been observed in distant galaxies. They are the most energetic and luminous electromagnetic events since the Big Bang. Bursts can last from ten milli ...
s,
nova
A nova (plural novae or novas) is a transient astronomical event that causes the sudden appearance of a bright, apparently "new" star (hence the name "nova", which is Latin for "new") that slowly fades over weeks or months. Causes of the dramati ...
e and
supernova
A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or when ...
e 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
X-ray binaries are a class of binary stars that are luminous in X-rays.
The X-rays are produced by matter falling from one component, called the ''donor'' (usually a relatively normal star), to the other component, called the ''accretor'', which ...
and
active galaxies
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that has a much-higher-than-normal luminosity over at least some portion of the electromagnetic spectrum with characteristics indicating that the luminosity is not pr ...
are able to maintain luminosities close to the Eddington limit for very long times. For accretion-powered sources such as accreting
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s or
cataclysmic variable
In astronomy, cataclysmic variable stars (CVs) are stars which irregularly increase in brightness by a large factor, then drop back down to a quiescent state. They were initially called novae (), since ones with an outburst brightness visible to ...
s (accreting
white dwarf
A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
s), 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 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 radiat ...
s (ULXs).
For accreting
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s, not all the energy released by accretion has to appear as outgoing luminosity, since energy can be lost through the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
, 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.
Other factors
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:
*Porosity. A problem with steady winds driven by broad-spectrum radiation is that both the radiative flux and gravitational acceleration scale with ''r''
−2. The ratio between these factors is constant, and in a super-Eddington star, the whole envelope would become gravitationally unbound at the same time. This is not observed. A possible solution is introducing an atmospheric porosity, where we imagine the stellar atmosphere to consist of denser regions surrounded by lower density gas regions. This would reduce the coupling between radiation and matter, and the full force of the radiation field would only be seen in the more homogeneous outer, lower density layers of the atmosphere.
*Turbulence. A possible destabilizing factor might be the turbulent pressure arising when energy in the
convection zone
A convection zone, convective zone or convective region of a star is a layer which is unstable due to convection. Energy is primarily or partially transported by convection in such a region. In a radiation zone, energy is transported by radiation ...
s builds up a field of supersonic turbulence. The importance of turbulence is being debated, however.
*Photon bubbles. Another factor that might explain some stable super-Eddington objects is the
photon bubble effect. Photon bubbles would develop spontaneously in radiation-dominated atmospheres when the radiation pressure exceeds the gas pressure. We can imagine a region in the stellar atmosphere with a density lower than the surroundings, but with a higher radiation pressure. Such a region would rise through the atmosphere, with radiation diffusing in from the sides, leading to an even higher radiation pressure. This effect could transport radiation more efficiently than a homogeneous atmosphere, increasing the allowed total radiation rate. In
accretion disc
An accretion disk is a structure (often a circumstellar disk) formed by diffuse material in orbital motion around a massive central body. The central body is typically a star. Friction, uneven irradiance, magnetohydrodynamic effects, and other ...
s, luminosities may be as high as 10–100 times the Eddington limit without experiencing instabilities.
Humphreys–Davidson limit
![Lbv](https://upload.wikimedia.org/wikipedia/commons/e/ec/Lbv.png)
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.
See also
*
Hayashi limit
Hayashi limit is a theoretical constraint upon the maximum radius of a star for a given mass. When a star is fully within hydrostatic equilibrium—a condition where the inward force of gravity is matched by the outward pressure of the gas—the st ...
*
List of most massive stars
This is a list of the most massive stars that have been discovered, in solar masses ().
Uncertainties and caveats
Most of the masses listed below are contested and, being the subject of current research, remain under review and subject to const ...
References
External links
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*
External links
Surpassing the Eddington Limit
{{Star
Astrophysics