Eddington luminosity

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.

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 takes into account other radiation processes such as bound–free and free–free radiation interaction.

Derivation

The maximum possible 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 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,

$${\displaystyle {\frac {du}{dt}}=-{\frac {\nabla p}{\rho }}-\nabla \Phi =0}$$

where ${\displaystyle u}$ is the velocity, ${\displaystyle p}$ is the pressure, ${\displaystyle \rho }$ is the density, and ${\displaystyle \Phi }$ is the gravitational potential. If the pressure is dominated by radiation pressure associated with a radiative energy flux ${\displaystyle F_{\rm {rad}}}$,

$${\displaystyle -{\frac {\nabla p}{\rho }}={\frac {\kappa }{c}}F_{\rm {rad}}\,.}$$

Here ${\displaystyle \kappa }$ is the opacity of the stellar material, defined as the fraction of radiation energy flux absorbed by the medium per unit density and unit length. For ionized hydrogen, ${\displaystyle \kappa =\sigma _{\rm {T}}/m_{\rm {p}}}$, where ${\displaystyle \sigma _{\rm {T}}}$ is the Thomson scattering cross-section for the electron and ${\displaystyle m_{\rm {p}}}$ is the mass of a proton. Note that ${\displaystyle F_{\rm {rad}}=d^{2}E/dAdt}$ is defined as the energy flux over a surface, which can be expressed with the momentum flux using ${\displaystyle E=pc}$ for radiation. Therefore, the rate of momentum transfer from the radiation to the gaseous medium per unit density is ${\displaystyle \kappa F_{\rm {rad}}/c}$, which explains the right-hand side of the above equation.

The luminosity of a source bounded by a surface ${\displaystyle S}$ may be expressed with these relations as

$${\displaystyle L=\int _{S}F_{\rm {rad}}\cdot dS=\int _{S}{\frac {c}{\kappa }}\nabla \Phi \cdot dS\,.}$$

Now assuming that the opacity is a constant, it can be brought outside the integral. Using Gauss's theorem and Poisson's equation gives

$${\displaystyle L={\frac {c}{\kappa }}\int _{S}\nabla \Phi \cdot dS={\frac {c}{\kappa }}\int _{V}\nabla ^{2}\Phi \,dV={\frac {4\pi Gc}{\kappa }}\int _{V}\rho \,dV={\frac {4\pi GMc}{\kappa }}}$$

where ${\displaystyle M}$ is the mass of the central object. This result is called the Eddington luminosity. 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, under the conditions in stellar atmospheres, typically are free protons. When the outward electric field is sufficient to levitate the protons against gravity, both electrons and protons are expelled together.

The formula can be expressed in terms of solar mass, ${\displaystyle M_{\bigodot }}$, and solar luminosity,${\displaystyle L_{\bigodot }}$ as:^: 262 $${\displaystyle {\frac {L_{\mathrm {E} }}{L_{\bigodot }}}=1.3\times 10^{4}{\frac {1}{\kappa }}{\frac {M}{M_{\bigodot }}}.}$$ When the radiation pressure is dominated by photon-electron scattering, the opacity can be approximated by ${\displaystyle \kappa =0.20(1+X)}$ for a gas with mass fraction ${\displaystyle X}$. For pure ionized hydrogen, ${\displaystyle X=0.7}$ giving $${\displaystyle {\frac {L_{\mathrm {E} }}{L_{\bigodot }}}=3.2\times 10^{4}\left({\frac {M}{M_{\bigodot }}}\right)}$$

A star's proximity to the Eddington limit can be characterized by its Eddington parameter, ${\displaystyle \Lambda _{e}}$, the ratio of its actual luminosity ${\displaystyle L}$ to the limit ${\displaystyle L_{\mathrm {E} }}$: $${\displaystyle \Lambda _{E}=10^{-4.5}q_{e}{\frac {L/L_{\bigodot }}{M/M_{\bigodot }}}}$$ where ${\displaystyle q_{e}}$ is the average number of electrons per atomic mass unit in the gas. The numerator here is the star's relative luminosity driving gas off the star and the denominator gives its relative gravitational strength pulling the gas in.

Super-Eddington accretion

Accretion at rates significantly above the Eddington limit, eg ${\displaystyle \approx 10L_{\textrm {Edd}}/c^{2},}$ are called super-Eddington accretion. The emergent luminosity from such accretion flow saturates at a few times ${\displaystyle L_{\textrm {Edd}}}$ due to photon trapping, where photons created by accretion processes are reabsorbed by the dense flow.

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 (ULXSs).

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.

One example of super-Eddington accretion is the radio-loud quasar ID830 (eFEDS J084222.9+0010000), detected at redshift 3.43. Observational data from magnesium-II emission line indicates a central black hole mass of approximately 440 million solar masses. Calculations derived from its X-ray luminosity indicate an Eddington ratio of roughly 12.8. ID830 has a high ratio of UV to X-ray emission suggesting a highly luminous X-ray corona and an active radio jet. The system may represents a transitional super-Eddington phase following a major accretion burst. Most similar objects are considered to be blazars rather than super-Eddington sources.

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 regions of lower-density gas. This would reduce the coupling between radiation and matter, and the full force of the radiation field would be seen only 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 zones 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. Accretion discs may exhibit luminosities as high as 10–100 times the Eddington limit without experiencing instabilities.

Humphreys–Davidson limit

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.

The H–D limit for cool supergiants is placed at around 320,000 L_☉.

See also

• Hayashi limit
• List of most massive stars
• M82 X-1
• M82 X-2

References

Further reading

External links

• Surpassing the Eddington Limit.