Eddington boundary

The Eddington limit or Eddington limit (after the British physicist Sir Arthur Stanley Eddington ) is the natural limitation of the luminosity of a star or black hole in astrophysics .

definition

The Eddington limit denotes the greatest flow of energy that can be transported through a hydrostatic gas stratification by means of radiation before the radiation pressure overcomes the hydrostatic pressure . The radiation pressure comes about through the scattering of the radiation by free electrons , the Thomson scattering . The luminosity of a certain object at which the hydrostatic pressure would be overcome by the radiation pressure is called Eddington luminosity .

The Eddington limit is therefore the maximum luminosity that a star can have in hydrostatic equilibrium without becoming unstable and shedding its outer layers. Regardless of this, stellar winds occur even with stars well below the limit. Strictly speaking, Eddington's observation only applies approximately to stars that emit a stellar wind and are therefore not in hydrostatic equilibrium; however, the Thomson scattering in stars has the property that it does not depend on the depth in the star. Because the energy-releasing region inside the star is much smaller than the region that is influenced by the stellar wind, the Eddington limit is still a reasonable limit. It should be noted, however, that the Eddington limit is derived one-dimensional and independent of time - that is, it is both possible that a star only temporarily exceeds the limit without being destroyed, and that a two-dimensional interplay of stellar wind and radiation as a whole allows a luminosity above the limit. The latter is taken into account , for example, for the outbreaks of η Carinae .

The Eddington boundary is a function of the mass of the object that the surrounding material accretes.

Derivation

The hydrostatic equilibrium is established when there is an equilibrium of forces between the outwardly directed force due to Thomson scattering and the inwardly directed gravitational force . In this derivation it is assumed that the material consists of ionized hydrogen and is distributed spherically symmetrically . ${\displaystyle F_{S}}$ ${\displaystyle F_{G}}$

The following applies to: ${\displaystyle F_{S}}$

${\displaystyle F_{S}={\frac {\sigma _{T}S}{c}}}$

${\displaystyle \sigma _{T}}$is the cross-section of Thomson scattering in electrons. The Thomson scattering on the protons (H nuclei) can be neglected because it is significantly less likely. is the radiation flux . The relationship between radiant flux and luminosity is (assuming that the intensity of the radiant flux is the same in all directions): ${\displaystyle S}$ ${\displaystyle L}$

${\displaystyle L=4\pi r^{2}S}$

For valid according to the Newton's law of gravitation : ${\displaystyle F_{G}}$

${\displaystyle F_{G}={\frac {GMm_{p}}{r^{2}}}}$

The mass of the electron was neglected because it is significantly smaller than that of the proton . If you insert the relationship between radiant flux and luminosity into the formula for and equate it , you get after forming: ${\displaystyle m_{p}}$ ${\displaystyle F_{S}}$ ${\displaystyle F_{G}}$

${\displaystyle L_{\text{Edd}}={\frac {4\pi GMm_{p}c}{\sigma _{T}}}\approx 33.000{\frac {M}{M_{\odot }}}L_{\odot }}$

It is

• ${\displaystyle L_{\text{Edd}}}$ the maximum of the luminosity that can be caused by accretion,
• ${\displaystyle M}$the mass of the compact object ,
• ${\displaystyle M_{\odot }}$the solar mass ,
• ${\displaystyle L_{\odot }}$the luminosity of the sun .

The maximum luminosity is therefore a function of the mass of the object.

Eddington accretion rate

Significantly, the Eddington limit also in the accretion of matter onto a compact object, such as a black hole , because if the luminosity exceeds the Eddington limit, the associated radiation pressure is so high that the collapsing material is pressed outward. At the same time, the energy supply is cut off, so that the luminosity drops below the Eddington limit and the material can flow in again. This process can be repeated periodically .

The following applies to the eddingtion accretion rate due to the equivalence of mass and energy : ${\displaystyle {\dot {m}}_{\text{Edd}}}$

${\displaystyle {\dot {m}}_{\text{Edd}}={\frac {L_{\text{Edd}}}{\eta c^{2}}}}$

Here is the efficiency of the radiation production during accretion. ${\displaystyle \eta }$

literature

• Helmut Scheffler, Hans Elsässer : Physics of the stars and the sun . 2nd revised and expanded edition. BI Wissenschaftsverlag, Mannheim 1990, ISBN 3-411-14172-7 .

Web links

• Information on Eddington luminosity (PDF file; 100 kB)