  Dwarf spheroidal galaxies (dSphs) are promising targets for the gamma-ray
dark matter (DM) search. In particular, DM annihilation signal is expected to
be strong in some of the recently discovered nearby ultra-faint dSphs, which
potentially give stringent constraints on the $\mathcal{O}(1)$ TeV WIMP DM.
However, various non-negligible systematic uncertainties complicate the
estimation of the astrophysical factors relevant for the DM search in these
objects. Among them, the effects of foreground stars particularly attract
attention because the contamination is unavoidable even for the future
kinematical survey. In this article, we assess the effects of the foreground
contamination on the astrophysical $J$-factor estimation by generating mock
samples of stars in the four ultra-faint dSphs and using a model of future
spectrographs. We investigate various data cuts to optimize the quality of the
data and apply a likelihood analysis which takes member and foreground stellar
distributions into account. We show that the foreground star contaminations in
the signal region (the region of interest) and their statistical uncertainty
can be estimated by interpolating the foreground star distribution in the
control region where the foreground stars dominate the member stars. Such
regions can be secured at future spectroscopic observations utilizing a
multiple object spectrograph with a large field of view; e.g. the Prime Focus
Spectrograph mounted on Subaru Telescope. The above estimation has several
advantages: The data-driven estimation of the contamination makes the analysis
of the astrophysical factor stable against the complicated foreground
distribution. Besides, foreground contamination effect is considered in the
likelihood analysis.
