Astronomy and Astrophysics – Astrophysics
Scientific paper
2001-09-07
Mon.Not.Roy.Astron.Soc. 348 (2004) 482
Astronomy and Astrophysics
Astrophysics
30 pages, 11 figures, submitted to the Monthly Notices of the Royal Astronomical Society
Scientific paper
10.1111/j.1365-2966.2004.07361.x
A matrix method is formulated in a Lagrangian representation for the solution of the characteristic value problem governing modes of oscillation and instability in a collisionless stellar system. The underlying perturbation equations govern the Lagrangian displacement of a star in the six-dimensional phase space. This matrix method has a basis in a variational principle. The method is developed in detail for radial oscillations of a spherical system. The basis vectors required for the representation of the Lagrangian displacement in this case are derived from solutions of the Lagrangian perturbation equations for radial perturbations in a homogeneous sphere. The basis vectors are made divergence free in the six-dimensional phase space in accordance with the requirement of Liouville's theorem that the flow of the system in the phase space must be incompressible. The basis vectors are made orthogonal with respect to a properly chosen set of adjoint vectors with the aid of a Gram-Schmidt procedure. Some basis vectors are null vectors in the sense that their inner products with their own adjoint vectors vanish. The characteristic frequencies of the lowest radial modes are calculated in several approximations for members of a family of spherical models which span a wide range of central concentrations. The present formulation of the matrix method can be generalized for nonradial modes in spherical systems and for modes in axisymmetric systems.
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