  We have experimentally studied the electronic $3p\leftarrow 3s$ excitation of
Na atoms attached to $^3$He droplets by means of laser-induced fluorescence as
well as beam depletion spectroscopy. From the similarities of the spectra
(width/shift of absorption lines) with these of Na on $^4$He droplets, we
conclude that sodium atoms reside in a ``dimple'' on the droplet surface and
that superfluid-related effects are negligible. The experimental results are
supported by Density Functional calculations at zero temperature, which confirm
the surface location of sodium on $^3$He droplets, and provide a detailed
description of the ``dimple'' structure. The calculated shift of the excitation
spectra for the two isotopes is in good agreement with the experimental data.
