  Local structure can have important effects on luminosity distance
observations, which could for example affect the local estimation of the Hubble
constant based on low red-shift type Ia supernovae. Using a spherically
symmetric exact solution of the Eistein's equations and a more accurate
expansion of the solution of the geodesic equations, we improve the low
red-shift expansion of the monopole of the luminosity distance in terms of the
curvature function. Based on this we derive the coordinate independent low
red-shift expansion of the monopole of the luminosity distance in terms of the
monopole of the density contrast. The advantage of this approach is that it
relates the luminosity distance directly to density observations, without any
dependency on the radial coordinate choice. We compute the effects of different
inhomogeneities on the luminosity distance, and find that the formulae in terms
of the density contrast are in good agreement with numerical calculations, in
the non linear regime are more accurate than the results obtained using linear
perturbation theory, and are also more accurate than the formulae in terms of
the curvature function.
