  The recently discovered satellite dwarf galaxy Ursa Major III provides a
promising opportunity to explore the signatures resulting from dark matter (DM)
annihilation, due to its proximity and large J-factor. Owing to the absence of
an excess of $\gamma$-ray signatures originating from Ursa Major III,
observations of $\gamma$-rays, such as those from Fermi-LAT, can be utilized to
set constraints on the DM annihilation cross section. In this study, we
determine the DM density profile, and consider the relationship between DM
density and velocity dispersion at different locations within Ursa Major III
through Jeans analysis. We calculate the J-factor of Ursa Major III for s-wave
annihilation, along with the effective J-factors for p-wave and Sommerfeld
enhanced annihilation scenarios. Utilizing these derived J-factors, we set
stringent constraints on DM annihilation cross sections in three scenarios.
Given the substantial impact of member star identification on the J-factor of
Ursa Major III, we further calculate J-factors with the condition of excluding
the largest velocity outlier. Our analysis reveals a notable reduction in the
median value and an increase in the deviation of J-factors, thereby leading to
considerably weaker constraints.
