  We consider a coarse-grained (CG) model with pairwise interactions, suitable
to describe low-density solutions of star-branched polymers of functionality
$f$. Each macromolecule is represented by a CG molecule with $(f+1)$
interaction sites, which captures the star topology. Potentials are obtained by
requiring the CG model to reproduce a set of distribution functions computed in
the microscopic model in the zero-density limit. Explicit results are given for
$f=6,12$ and $40$. We use the CG model to compute the osmotic equation of state
of the solution for concentrations $c$ such that $\Phi_p = c/c^* \lesssim 1$,
where $c^*$ is the overlap concentration. We also investigate in detail the
phase diagram for f=40, identifying the boundaries of the solid intermediate
phase. Finally, we investigate how the polymer size changes with $c$. For
$\Phi_p\lesssim 0.3$ polymers become harder as $f$ increases at fixed reduced
concentration $c/c^*$. On the other hand, for $\Phi_p\gtrsim 0.3$, polymers
show the opposite behavior: At fixed $\Phi_p$, the larger the value of $f$, the
larger their size reduction is.
