Eddington Luminosity & Eddington Ratio Calculator

An accreting black hole or neutron star can’t shine arbitrarily brightly forever: past a certain luminosity, the outward push of its own light on the infalling gas should overwhelm the inward pull of its gravity. That threshold — the Eddington limit — sets a natural brightness scale for every accreting compact object, from X-ray binaries a few times the Sun’s mass to supermassive black holes billions of times heavier. This calculator turns a mass into that limit, and an observed luminosity into how close to (or past) it a real source is sitting.

Where the limit comes from

Picture a cloud of ionized hydrogen around a compact object. Gravity pulls each proton inward; outgoing photons scatter off free electrons (Thomson scattering) and, because protons and electrons are electrostatically tied together, that scattering effectively pushes the proton–electron pairs outward too. Balance the two forces on one proton’s worth of gas and solve for the luminosity that achieves it:

G is the gravitational constant, m_p the proton mass, c the speed of light, and σ_T the Thomson cross-section. Every constant here is fixed, so L_Edd depends on nothing but mass — which is what makes it such a useful benchmark: two black holes of the same mass have the same Eddington limit regardless of anything else about them.

The hidden assumption. This derivation assumes fully ionized hydrogen — one proton’s worth of mass dragged along per scattering electron. A different composition (a solar mix, or pure ionized helium) raises the mass-per-electron and lowers L_Edd by roughly that same factor. This calculator implements the standard pure-hydrogen case exactly as written above.

The Eddington ratio

Given an object’s actual (observed or estimated) luminosity L, its Eddington ratio is simply

λ_Edd < 1 means the source is sub-Eddington — comfortably below its limit. λ_Edd ≈ 1 means it’s radiating close to the theoretical maximum for its mass. The worked example this calculator opens with: a 10 M☉ black hole has L_Edd ≈ 1.26 × 10³⁹ erg/s; observed at 6.3 × 10³⁸ erg/s, that’s λ_Edd ≈ 0.5 — half its Eddington luminosity.

When λ_Edd exceeds 1

Real accreting sources routinely get observed above λ_Edd = 1 — ultraluminous X-ray sources (ULXs) are the clearest example, some reaching λ_Edd in the tens. That isn’t a contradiction of physics, it’s a sign the idealized assumptions behind the formula above have broken down: real accretion flows aren’t spherically symmetric, radiation can escape preferentially along low-density channels (geometric beaming), and the flow can become “photon-trapped” or clumpy (porous, via photon bubbles) in ways that let more matter fall in than the smooth, steady picture allows. The Eddington limit is a genuinely useful benchmark — just not a hard ceiling nature always respects.

Reading the ratio meter and the mass–luminosity chart

• The ratio meter places λ_Edd on a fixed log scale from 0.01× to 100×, coloured green (sub-Eddington), amber (near the limit), and red (super-Eddington) — so “0.5” registers instantly as “comfortably in the green” rather than requiring you to remember what counts as safe.
• The mass–luminosity chart plots L_Edd against mass in log-log space, where the M¹ scaling in the formula becomes a straight line of slope exactly 1. The object’s Eddington limit always sits on that line; its observed luminosity, if you supply one, shows up as a second point directly above the line (super-Eddington) or below it (sub-Eddington) — turning “how far from the limit” into a distance you can see.