  The unstable foliation, that locally is given by changing horizontal
components of period coordinates, plays an important role in study of
translation surfaces, including their deformation theory and in the
understanding of horocycle invariant measures.
  In this article we show that measures of large dimension on the unstable
foliation equidistribute in affine invariant submanifolds and give an effective
rate. An analogous result in the setting of homogeneous dynamics is crucially
used in the effective density and equidistribution results of
Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang.
