The atmosphere is assumed to be thin, with average opacity k. Opacity is defined to be optical depth divided by density. Thus, by definition, the optical depth of the stellar surface, also called the photosphere, is: d τ τ d r = k ρ ρ τ τ = ∫ ∫ R ∞ ∞ k ρ ρ d r = k ∫ ∫ R ∞ ∞ ρ ρ d r where R is the stellar radius, also known as the position of the photosphere. The pressure at the surface is: P 0 = ∫ ∫ R ∞ ∞ g ρ ρ d r = G M R 2 ∫ ∫ R ∞ ∞ ρ ρ d r = G M τ τ k R 2