Foreground effect on the $J$-factor estimation of classical dwarf
  spheroidal galaxies
  The gamma-ray observation of the dwarf spheroidal galaxies (dSphs) is a
promising approach to search for the dark matter annihilation (or decay)
signal. The dSphs are the nearby satellite galaxies with a clean environment
and dense dark matter halo so that they give stringent constraints on the 1 TeV
dark matter. However, recent studies have revealed that current estimation of
astrophysical factors relevant for the dark matter searches are not
conservative, where the various non-negligible systematic uncertainties are not
taken into account. Among them, the effect of foreground stars on the
astrophysical factors has not been paid much attention, which becomes more
important for deeper and wider stellar surveys in the future. In this article,
we assess the effects of the foreground contamination by generating the mock
samples of stars and using a model of future spectrographs. We investigate
various data cuts to optimize the quality of the data and find that the cuts on
the velocity and surface gravity can efficiently eliminate the contamination.
We also propose a new likelihood function which includes the foreground
distribution function. We apply this likelihood function to the fit of the
three types of the mock data (Ursa Minor, Draco with large dark matter halo,
and Draco with small halo) and three cases of the observation. The likelihood
successfully reproduces the input $J$-factor value while the fit without
considering the foreground distribution gives large deviation from the input
value by a factor of three.
