Context: The Einasto model has become one of the most popular models for
describing the density profile of dark matter haloes. There have been
relatively few comprehensive studies on the dynamical structure of the Einasto
model, mainly because only a limited number of properties can be calculated
analytically. Aims: We want to systematically investigate the photometric and
dynamical structure of the family of Einasto models over the entire model
parameter space. Methods: We used the SpheCow code to explore the properties of
the Einasto model. We systematically investigated how the most important
properties change as a function of the Einasto index $n$ . We considered both
isotropic models and radially anisotropic models with an Osipkov-Merritt
orbital structure. Results: We find that all Einasto models with $n<{\textstyle \frac{1}{2}}$
have a formal isotropic or Osipkov-Merritt distribution function that is
negative in parts of phase space, and hence cannot be supported by such orbital
structures. On the other hand, all models with larger values of $n$ can be
supported by an isotropic orbital structure, or by an Osipkov-Merritt
anisotropy, as long as the anisotropy radius is larger than a critical value.
This critical anisotropy radius is a decreasing function of $n$ , indicating
that less centrally concentrated models allow for a larger degree of radial
anisotropy. Conclusions: Studies of the structure and dynamics of models for
galaxies and dark matter haloes should not be restricted to completely
analytical models. Numerical codes such as SpheCow can help open up the range
of models that are systematically investigated. This applies to the Einasto
model discussed here, but also to other proposed models for dark matter haloes,
including different extensions to the Einasto model.