The Collapse and Formation of Galaxies

Abstract

The ultimate aim of studies of the dynamics of stellar systems is to gain an understanding of why they have the structures that are observed, how they might have formed, and how they might evolve with time. In the case of galaxies, unlike smaller astronomical systems such as stars and star clusters, the problem of understanding their present structure is inseparable from the problem of understanding their formation since, as has long been clear, the two-body relaxation time in galaxies is much longer than the age of the universe, so that properties such as the radial density distribution and the flattening of elliptical galaxies cannot have changed significantly since they were formed. Other features of galaxies, such as the presence of spiral patterns or nuclear activity, may be of a more transitory nature and not closely related to the formation process, but such phenomena probably involve mainly the gas component of galaxies and may be regarded as perturbations on the underlying stellar system, whose basic structure must still be understood in terms of the formation process.

Comparisons of the RPA, SCRPA, Tamm–Dancoff, and full CI methods by analysis of their transition density matrices, oscillator strengths, and energy moments of oscillator strengths for the electric dipole transitions from the ground state of the B+ ion (frozen K‐shell model)

“…However, it has been shown [66,67] that, for a quasi-2-electron system (such as the frozen K-shell model of B+), the particle-hole space of any reference state (which has no zero-eigenvalue of the l-particle reduced density matrix) is the complete Hilbert space of all states orthogonal to the reference state and corresponding to the same number of particles. Thus, for such a system, jI*,) = I * , ) : and any excited state may be obtained by some particfe-hole operator QL acting on the true ground state (Vg).…”

“…Thus, for such a system, jI*,) = I * , ) : and any excited state may be obtained by some particfe-hole operator QL acting on the true ground state (Vg). Accordingly [66,67] (1) applies only to a quasi-2-particle system. (The following discussion applies, to many-particle systems as well.)…”

“…Now the function ~p,(x, y) and the associated particle-hole operator 0; ma? not be uniquely defined by the above relations [66,67], for we could add to 0; any particle-hole operator that annihilates the ground state and still produce an operator that will generate the state Nevertheless, for any particle-hole operator that satisfies Eqs.…”

“…(1) and (2), the l-particle reduced transition density matrix qcL(x, y) for the transition from state Iqg) to state I * , ) of the N-particle system may be written [53,52,66,67] from the state )qg) in the manner of Eq. (1).…”

“…(8)) is satisfied for all operators {6:} that are solutions to the generalized RPA equations [13,1,6,67]. This is totally equivalent [ 131 to a "Hartree-Fock and double excitations" (full CI less single excitations) description of the ground state, and a Tamm-Dancoff description of the excited states.…”