Self-consistent dynamical models with a finite extent -- I. The uniform
  density sphere
  The standard method to generate dynamical models with a finite extent is to
apply a truncation in binding energy to the distribution function. This
approach has the disadvantages that one cannot choose the density to start
with, that the important dynamical quantities cannot be calculated
analytically, and that a fraction of the possible bound orbits are excluded a
priori. We explore another route and start from a truncation in radius rather
than a truncation in binding energy. We focus on the simplest truncated density
profile, the uniform density sphere. We explore the most common inversion
techniques to generate distribution functions for the uniform density sphere,
corresponding to a large range of possible anisotropy profiles. We find that
the uniform density sphere cannot be supported by the standard isotropic,
constant anisotropy or Osipkov-Merritt models, as all these models are
characterised by negative distribution functions. We generalise the Cuddeford
inversion method to models with a tangential anisotropy and present a
one-parameter family of dynamical models for the uniform density sphere. Each
member of this family is characterised by an anisotropy profile that smoothly
decreases from an arbitrary value $\beta_0\leqslant0$ at the centre to
completely tangential at the outer radius. All models have a positive
distribution function over the entire phase space, and a nonzero occupancy of
all possible bound orbits. This shows that one can generate nontrivial
self-consistent dynamical models based on preset density profile with a finite
extent.
