# De Vaucouleurs' law

de Vaucouleurs' law (also called the de Vaucouleurs profile) describes how the surface brightness $I$ of an elliptical galaxy varies as a function of apparent distance $R$ from the center:[1]

$\ln I(R) = \ln I_{0} - k R^{1/4}.$

By defining Re as the radius of the isophote containing half the luminosity (i.e., the radius of the inner disk contributing half the brightness of the galaxy), de Vaucouleurs' law may be written:

$\ln I(R) = \ln I_{e} + 7.669 \left[ 1 - \left( \frac{R}{R_{e}} \right)^{1/4} \right]$

or

$I(R) = I_{e} e^{-7.669 \left[ (\frac{R}{R_{e}})^{1/4} - 1 \right]}$

where Ie is the surface brightness at Re. This can be confirmed by noting

$\int^{R_e}_0 I(R) r dr = \frac{1}{2} \int^{\infty}_0 I(R) r dr .$

de Vaucouleurs' law is a special case of Sersic's law, with Sersic index n=4. A number of density laws that approximately reproduce de Vaucouleurs' law after projection onto the plane of the sky include Jaffe's model and Dehnen's model.