Eddington Luminosity

Eddington Luminosity

Every luminous object powered by accretion has a natural upper limit — the Eddington luminosity. Above this limit, the outward radiation pressure on the surrounding ionised hydrogen plasma exceeds the inward pull of gravity, and the accreting material is blown away.

Derivation

For a fully ionised hydrogen plasma, radiation exerts pressure via Thomson scattering off free electrons. Balancing radiation pressure against gravity for a proton-electron pair at radius $r$:

$\frac{L \kappa_{\text{es}}}{4\pi r^2 c} = \frac{G M}{r^2}$

Solving for $L$:

$L_{\text{Edd}} = \frac{4\pi G M c}{\kappa_{\text{es}}}$

where:

• $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻²
• $c = 2.998 \times 10^8$ m/s
• $\kappa_{\text{es}} = 0.020$ m² kg⁻¹ (electron scattering opacity for solar composition)

Physical Significance

The Eddington luminosity sets the maximum brightness of X-ray binaries, determines mass accretion rates onto black holes, and limits the growth of supermassive black holes in the early universe.

Your Task

Implement three functions. All constants must be defined inside each function.

• eddington_luminosity_W(M_kg) — returns $L_{\text{Edd}}$ in watts
• eddington_luminosity_solar(M_solar) — returns $L_{\text{Edd}} / L_\odot$
• mass_from_eddington(L_W) — returns $M$ in kg given luminosity in watts

Use $G = 6.674 \times 10^{-11}$, $c = 2.998 \times 10^8$ m/s, $\kappa_{\text{es}} = 0.020$ m² kg⁻¹, $L_\odot = 3.828 \times 10^{26}$ W, $M_\odot = 1.989 \times 10^{30}$ kg.