The observed rotation curves of low surface brightness (LSB) galaxies play an essential role in studying dark matter, and indicate that there exists a central constant density dark matter core. However, the cosmological N-body simulations of cold dark matter predict an inner cusped halo with a power-law mass density distribution, and cant reproduce a central constant-density core. This phenomenon is called cusp-core problem. When dark matter is quiescent and satisfies the condition for hydrostatic equilibrium, using the equation of state can get the density profile in the static and spherically symmetric space-time. To solve the cusp-core problem, we assume that the equation of state is independent of the scaling transformation. Its lower order approximation for this type of equation of state can naturally lead to a special case, i.e. $p=zetarho+2epsilon V_{rot}^{2}rho$, where $p$ and $rho$ are the pressure and density, $V_{rot}$ is the rotation velocity of galaxy, $zeta$ and $ epsilon$ are positive constants. It can obtain a density profile that is similar to the pseudo-isothermal halo model when $epsilon$ is around $0.15$. To get a more widely used model, let the equation of state include the polytropic model, i.e. $p= frac{zeta}{rho_{0}^{s}}rho^{1+s}+ 2epsilon V_{rot}^{2}rho$, we can get other kinds of density profiles, such as the profile that is nearly same with the Burkert profile, where $s$ and $rho_{0}$ are positive constants.
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