Eddington Luminosity Limit & Black-Hole Accretion Rate Formula

A model-dependent reference scale

The classical Eddington balance compares inward gravity with outward radiation force for steady, spherical flow and electron-scattering opacity. For ionized hydrogen, LEdd = 4πGMmpc/σT is approximately 1.26×10³¹ W per solar mass. This worksheet adopts that rounded coefficient.

It is not a universal ceiling on luminosity or mass supply. Composition, opacity, geometry, radiation trapping and outflows change the interpretation; super-Eddington flows exist. The simple calculation does not simulate an accretion disk.

Keep luminosity conversions consistent

B2 is mass in solar masses. B8 gives luminosity in watts. B9 divides that same output by the IAU nominal solar luminosity, 3.828×10²⁶ W. The resulting coefficient is approximately 32,915.36 solar luminosities per solar mass.

For B2=100,000,000, B8 is 1.26×10³⁹ W and B9 approximately 3.291536×10¹² solar luminosities. These extra displayed digits do not improve the physical precision of the rounded starting coefficient.

=IF(B5<>"OK",B5,B8/3.828E26)

Calculate reference rest-mass inflow

Set radiative efficiency ε in B3 as a fraction, for example 0.1 rather than 10. From L=ε Ṁ c², B10 divides luminosity by εc², then converts kg/s to solar masses per Julian year using 31,557,600 seconds/year and an adopted solar mass of 1.98847×10³⁰ kg.

The default gives about 2.2249 solar masses/year. Changing ε to 0.3 divides this by three, giving about 0.7416, without changing LEdd. Those efficiencies are illustrative inputs; this sheet does not infer black-hole spin or validate a disk-efficiency model.

=IF(B5<>"OK",B5,IF(B6<>"OK",B6,B8/(B3*299792458^2)*31557600/1.98847E30))

Distinguish supplied mass from retained growth

B10 is the rest-mass inflow associated with the chosen radiative efficiency, not automatically the rate at which the black hole gains mass. In a simplified accounting model where radiation removes fraction ε of rest-mass energy and there are no other losses, retained growth is (1−ε)Ṁ. B11 displays that distinction.

At ε=0.1 the illustrative retained rate is about 2.0024 solar masses/year. Winds, jets, advection and other effects are not included.

=IF(ISNUMBER(B10),(1-B3)*B10,B10)

Preserve small rates and reject invalid inputs

B5 requires positive numeric mass; B6 requires numeric efficiency strictly between zero and one. Luminosity remains available when only efficiency is invalid, because it does not depend on efficiency. Rate outputs show a message instead of dividing by zero.

For a 10-solar-mass example with ε=0.1, the inflow is about 2.2249×10⁻⁷ solar masses/year. The old four-decimal rounding erased this to zero. The formulas now retain their raw results; use scientific display formatting for small values.

Try doubling mass: both luminosities and both rates double at fixed efficiency. Doubling efficiency halves rest-mass inflow but does not simply halve retained growth, since its retained fraction also changes.

Further reading

• University of Maryland — Accreting sources and Eddington balance
• IAU 2015 resolutions — Nominal solar luminosity
• Abolmasov and Chashkina — On the Eddington limit for relativistic accretion discs