Eddington luminosity

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,

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

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

$${\displaystyle -\frac{\mathrm{\nabla }p}{\rho }=\frac{\kappa }{c}{F}_{\mathrm{r}\mathrm{a}\mathrm{d}}.}$$

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, ${\displaystyle \kappa ={\sigma }_{\mathrm{T}}/m_{\mathrm{p}}}$, where ${\displaystyle {\sigma }_{\mathrm{T}}}$ is the Thomson scattering cross-section for the electron and ${\displaystyle m_{\mathrm{p}}}$ is the mass of a proton. Note that ${\displaystyle F_{\mathrm{r}\mathrm{a}\mathrm{d}}=d^{2}E/dAdt}$ 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 ${\displaystyle \kappa F_{\mathrm{r}\mathrm{a}\mathrm{d}}/c}$, 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

$${\displaystyle L={\int }_S{F}_{\mathrm{r}\mathrm{a}\mathrm{d}}\cdot dS={\int }_S\frac{c}{\kappa }\mathrm{\nabla }\mathrm{\Phi }\cdot dS.}$$

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

$${\displaystyle L=\frac{c}{\kappa }{\int }_S\mathrm{\nabla }\mathrm{\Phi }\cdot dS=\frac{c}{\kappa }{\int }_V{\mathrm{\nabla }}^2\mathrm{\Phi }dV=\frac{4\pi Gc}{\kappa }{\int }_V\rho dV=\frac{4\pi GMc}{\kappa }}$$

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, ${\displaystyle M_{⨀}}$, and solar luminosity,${\displaystyle L_{⨀}}$ as: $${\displaystyle \frac{L_{\mathrm{E}}}{L_{⨀}}=1.3\times {10}^4\frac{1}{\kappa }\frac{M}{M_{⨀}}.}$$ 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 $${\displaystyle \frac{L_{\mathrm{E}}}{L_{⨀}}=3.2\times {10}^4\left(\frac{M}{M_{⨀}}\right)}$$

A star's proximity to the Eddington limit can be characterized by its Eddington parameter, ${\displaystyle {\mathrm{\Lambda }}_e}$, the ratio of its actual luminosity ${\displaystyle L}$ to the limit ${\displaystyle L_{\mathrm{E}}}$: $${\displaystyle {\mathrm{\Lambda }}_E={10}^{-4.5}{q}_e\frac{L/L_{⨀}}{M/M_{⨀}}}$$ 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.

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 photons can interact with nuclei, or even with other photons, to create an electron–positron plasma. In that situation the combined mass of the positive–negative charge carrier pair is approximately 918 times smaller (half of 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-type stars). As massive stars evolve the ratio of their luminosity to their mass increases. When they reach an Eddington parameter close to unity, strong stellar winds from radiation pressure expel the outer hydrogen rich layer, leaving these stars to appear helium enriched.

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 explain a mass loss rate of around 10⁻⁴~10⁻³ solar masses per year, whereas losses of up to ⁠1/ 2 ⁠ solar mass per year are needed to understand the η Carinae outbursts. This can be done with the help of the super-Eddington winds driven by broad-spectrum radiation.