47 research outputs found
Two-integral distribution functions for axisymmetric stellar systems with separable densities
We show different expressions of distribution functions (DFs) which depend
only on the two classical integrals of the energy and the magnitude of the
angular momentum with respect to the axis of symmetry for stellar systems with
known axisymmetric densities. The density of the system is required to be a
product of functions separable in the potential and the radial coordinate,
where the functions of the radial coordinate are powers of a sum of a square of
the radial coordinate and its unit scale. The even part of the two-integral DF
corresponding to this type of density is in turn a sum or an infinite series of
products of functions of the energy and of the magnitude of the angular
momentum about the axis of symmetry. A similar expression of its odd part can
be also obtained under the assumption of the rotation laws. It can be further
shown that these expressions are in fact equivalent to those obtained by using
Hunter and Qian's contour integral formulae for the system. This method is
generally computationally preferable to the contour integral method. Two
examples are given to obtain the even and odd parts of their two-integral DFs.
One is for the prolate Jaffe model and the other for the prolate Plummer model.
It can be also found that the Hunter-Qian contour integral formulae of the
two-integral even DF for axisymmetric systems can be recovered by use of the
Laplace-Mellin integral transformation originally developed by Dejonghe.Comment: 1 figur
Two-integral distribution functions for axisymmetric systems
Some formulae are presented for finding two-integral distribution functions
(DFs) which depends only on the two classical integrals of the energy and the
magnitude of the angular momentum with respect to the axis of symmetry for
stellar systems with known axisymmetric densities. They come from an
combination of the ideas of
Eddington and Fricke and they are also an extension of those shown by Jiang
and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of
the system is required to be expressed as a sum of products of functions of the
potential and of the radial coordinate. The solution corresponding to this type
of density is in turn a sum of products of functions of the energy and of the
magnitude of the angular momentum about the axis of symmetry. The product of
the density and its radial velocity dispersion can be also expressed as a sum
of products of functions of the potential and of the radial coordinate. It can
be further known that the density multipied by its rotational velocity
dispersion is equal to a sum of products of functions of the potential and of
the radial coordinate minus the product of the density and the square of its
mean rotational velocity. These formulae can be applied to the Binney and the
Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also
found under the assumption of the laws of the rotational velocity
A Simple Proof of Inequalities of Integrals of Composite Functions
In this paper we give a simple proof of inequalities of integrals of
functions which are the composition of nonnegative continous convex functions
on a vector space and vector-valued functions in a weakly compact
subset of a Banach vector space generated by -spaces for Also, the same inequalities hold if these vector-valued functions
are in a weakly* compact subset of a Banach vector space generated by
-spaces instead