  We introduce a new notion of circular colourings for digraphs. The idea of
this quantity, called star dichromatic number $\vec{\chi}^\ast(D)$ of a digraph
$D$, is to allow a finer subdivision of digraphs with the same dichromatic
number into such which are "easier" or "harder" to colour by allowing
fractional values. This is related to a coherent notion for the vertex
arboricity of graphs introduced by Wang et al. and resembles the concept of the
star chromatic number of graphs introduced by Vince in the framework of digraph
colouring. After presenting basic properties of the new quantity, including
range, simple classes of digraphs, general inequalities and its relation to
integer counterparts as well as other concepts of fractional colouring, we
compare our notion with the notion of circular colourings for digraphs
introduced by Bokal et al. and point out similarities as well as differences in
certain situations. As it turns out, the star dichromatic number is a lower
bound for the circular dichromatic number of Bokal et al., but the gap between
the numbers may be arbitrarily close to $1$. We conclude with a discussion of
the case of planar digraphs and point out some open problems.
