Eddington luminosity

The Eddington luminosity or Eddington limit is the luminosity at which radiation pressure overcomes the force of gravity, making it possible for luminous objects to blow themselves apart. It sets an effective limit on how much light an object can emit without destroying itself. The Eddington limit is reached in some of the most energetic objects in the universe, including supermassive stars and X-ray binary stars.

The Eddington limit is defined by

L_Eddington = 4 π G M_star m_H/ σ_T (eq. 1)

or

L_Eddington = 1.25 × 10³⁸ × (M_star/M_Sun) ergs per second (eq. 2)

where G is the gravitational constant, m_H is the mass of a hydrogen atom, and σ_T is the Thomson scattering cross section. Any object more luminous than its Eddington limit will begin blowing off its own matter until (most likely) the luminosity falls below the limit again.

The theory behind the Eddington luminosity is straightforward. The force of gravity per unit mass is defined by

f_gravity = G M₁ / R² (eq. 3)

where M₁ is the mass of the gravitator (like a star), and R is the distance between the gravitator and whatever object is being pulled in (like a hydrogen atom).

Electromagnetic radiation also exerts a pressure, the radiation pressure. The force of radiation per unit mass is defined by

f_radiation = κ F / c, (eq. 4)

where κ is the opacity of the absorbing material (hydrogen atoms), F is the flux of radiation coming from the source, and c is the speed of light.

Assume that the star is giving off light in all directions, and that the pressure of the photons works in the opposite direction of the force of gravity. The Eddington limit is reached when the force of radiation equals the force of gravity, and the limit is exceeded when the radiation force is larger than the force of gravity. So in the limiting case,

G M₁ / R² = κ F / c = (κ/c) × L_Eddington / (4 π R²) (eq. 5)

or

L_Eddington = 4 π G M₁ c / κ (eq. 6)

Now we have to decide on what the opacity, κ, is. Every Eddington limit system we observe in the universe is very hot. Any matter on or near these objects will also be very hot, and therefore will be ionized. Furthermore, most of the matter in the universe is hydrogen, so it's a reasonable assumption that most of the matter doing the absorbing here will be hydrogen, too. The opacity of ionized hydrogen is caused entirely by Thomson scattering, so we can assume that

κ = σ_T/m_H (eq. 7)

which by substitution into equation 6 gives us equation 1.

The Eddington limit is important in a few areas of astrophysics. In ordinary stars, the limit is only reached when stars are very hot. For main sequence stars, only the most massive, bright, hot stars go anywhere near the Eddington limit. These stars can be more than fifty times the mass of our Sun, and such behemoths are very rare; out of a hundred billion stars in our galaxy, only a handful are supermassive stars, and most die in giant supernova explosions within a million years of their birth. However, there is an upper limit to the mass a star can have, and the mass is set by the Eddington limit. Stars form by accreting matter from the nebulae (like the Orion Nebula) where they are born. If the star becomes so massive and hot that the Eddington limit is reached, a strong stellar wind will push away any infalling matter rather than accreting it. This limit was first theorized by Roberta Humphreys and Kris Davidson of the University of Minnesota, and is now known as the Humphreys-Davidson limit for stellar mass.

X-ray binary stars can also reach the Eddington limit under certain conditions. X-ray binaries are neutron stars or stellar black holes which are in a binary system with a much more normal star. The neutron star or black hole cannibalizes matter from their companions when the normal star overflows its Roche lobe. When this happens, the matter falls from the companion, and into an accretion disk around the compact object. This accretion disk can get so hot that it emits x-rays (hence the name). Since the accretion disk is very hot, it is also very bright. Sometimes, the accretion disk gets so bright that it briefly exceeds the Eddington limit, and pushes matter out of the accretion disk. When this happens, accretion slows down, the disk cools, and the luminosity of the accretion disk drops back below the Eddington limit. Several X-ray binaries are known to have luminosities near the Eddington limit, including the brightest in our sky, Scorpius X-1.