  This paper presents high-order Runge-Kutta (RK) discontinuous Galerkin
methods for the Euler-Poisson equations in spherical symmetry. The scheme can
preserve a general polytropic equilibrium state and achieve total energy
conservation up to machine precision with carefully designed spatial and
temporal discretizations. To achieve the well-balanced property, the numerical
solutions are decomposed into equilibrium and fluctuation components which are
treated differently in the source term approximation. One non-trivial challenge
encountered in the procedure is the complexity of the equilibrium state, which
is governed by the Lane-Emden equation. For total energy conservation, we
present second- and third-order RK time discretization, where different source
term approximations are introduced in each stage of the RK method to ensure the
conservation of total energy. A carefully designed slope limiter for spherical
symmetry is also introduced to eliminate oscillations near discontinuities
while maintaining the well-balanced and total-energy-conserving properties.
Extensive numerical examples -- including a toy model of stellar core-collapse
with a phenomenological equation of state that results in core-bounce and shock
formation -- are provided to demonstrate the desired properties of the proposed
methods, including the well-balanced property, high-order accuracy, shock
capturing capability, and total energy conservation.
