Effect of inhomogeneity of the Universe on a gravitationally bound local
  system: A no-go result for explaining the secular increase in the
  astronomical unit
  We will investigate the influence of the inhomogeneity of the universe,
especially that of the Lema{\^i}tre-Tolman-Bondi (LTB) model, on a
gravitationally bound local system such as the solar system. We concentrate on
the dynamical perturbation to the planetary motion and derive the leading order
effect generated from the LTB model. It will be shown that there appear not
only a well-known cosmological effect arisen from the homogeneous and isotropic
model, such as the Robertson-Walker (RW) model, but also the additional terms
due to the radial inhomogeneity of the LTB model. We will also apply the
obtained results to the problem of secular increase in the astronomical unit,
reported by Krasinsky and Brumberg (2004), and imply that the inhomogeneity of
the universe cannot have a significant effect for explaining the observed
$d{\rm AU}/dt = 15 \pm 4 ~{\rm [m/century]}$.
