By defining R e as the radius of the isophote containing half of the total luminosity of the galaxy, the half-light radius, de Vaucouleurs profile may be expressed as: ln ⁡ ⁡ I ( R ) = ln ⁡ ⁡ I e + 7.669 [ 1 − − ( R R e ) 1 / 4 ] or I ( R ) = I e e − − 7.669 [ ( R R e ) 1 / 4 − − 1 ] where I e is the surface brightness at R e. This can be confirmed by noting ∫ ∫ 0 R e I ( r ) 2 π π r d r = 1 2 ∫ ∫ 0 ∞ ∞ I ( r ) 2 π π r d r .