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At distance $r$, isotropic luminosity $L$ produces radiative acceleration $a_{\mathrm{rad}}=\kappa L/(4\pi r^{2}c)$ in material of opacity $\kappa$. In fully ionized hydrogen, one Electron supplies a Thomson scattering cross-section ${\sigma }_T$ for approximately one Proton mass of material, so $\kappa ={\sigma }_T/m_p$. Balancing radiation against the Newtonian gravitational field of the black hole gives the Eddington luminosityBelow this luminosity gravity can confine optically thin ionized hydrogen; above it, electron scattering alone accelerates the gas outward.

An electron--positron pair has mass $2m_e$ and total scattering cross-section $2{\sigma }_T$, so its opacity is ${\sigma }_T/m_e$. ThereforeThe limit is lower because the radiative force per unit inertial mass is larger in a pair plasma.

For a cloud with arbitrary constant opacity per unit mass,Suppose the luminosity jumps to a constant $L>L_{\mathrm{Edd}}$ while the cloud is at rest at $r_0$. With $\Gamma =L/L_{\mathrm{Edd}}$, its radial acceleration isMultiplication by $\dot{r}$ and integration from $r_0$ to infinity gives conservation of energyThis result assumes that $L$ and $\kappa$ remain constant and that the cloud stays optically thin.

The proton is the positively charged spin-one-half baryon found in atomic nuclei. Its mass is approximately $1836$ times the Electron mass.