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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 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. 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 in hydrostatic equilibrium, the mean acceleration is zero, where u {\displaystyle u} is the velocity, p {\displaystyle p} is the pressure, ρ {\displaystyle \rho } is the density, and Φ {\displaystyle \Phi } is the gravitational potential. If the pressure is dominated by radiation pressure associated with a radiation flux F r a d {\displaystyle F_{\rm {rad}}} , Here κ {\displaystyle \kappa } 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 κ = σ T / m p {\displaystyle \kappa =\sigma _{\rm {T}}/m_{\rm {p}}} , where σ T {\displaystyle \sigma _{\rm {T}}} is the Thomson scattering cross-section for the electron and m p {\displaystyle m_{\rm {p}}} is the mass of a proton. Note that F r a d = d 2 E / d A d t {\displaystyle F_{\rm {rad}}=d^{2}E/dAdt} is defined as the energy flux over a surface, which can be expressed with the momentum flux using E = p c {\displaystyle E=pc} for radiation. Therefore, the rate of momentum transfer from the radiation to the gaseous medium per unit density is κ F r a d / c {\displaystyle \kappa F_{\rm {rad}}/c} , which explains the right hand side of the above equation. The luminosity of a source bounded by a surface S {\displaystyle S} 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 and Poisson's equation gives where M {\displaystyle M} is the mass of the central object. This is called the Eddington Luminosity. For pure ionized hydrogen, where M ⨀ {\displaystyle M_{\bigodot }} is the mass of the Sun and L ⨀ {\displaystyle L_{\bigodot }} 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 plasma. In other circumstances the pressure balance can be different from what it is for hydrogen. 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 .... Discover the K L Humphreys popular books. Find the top 100 most popular K L Humphreys books.