Abstract
The asphericity of a flexible monodisperse f-star polymer in d=3 is calculated in the framework of the standard two-parameter model in the asymptotic limit of infinite molecular weight. In contrast to previous analytical investigations we employ an asphericity measure which takes into account that the size and shape of the polymer coils are strongly correlated. We consider ideal non-interacting (NEV) star polymers as well as molecules with excluded-volume (EV) interactions by means of renormalized perturbation theory. We show that in both cases the mean asphericity parameter takes a universal numerical value when the molecular weight tends to infinity. The same holds for the corresponding asymptotic distribution functions, which are also proved to be universal functions. These universal properties depend only on the topology, i.e. the number of arms f. For NEV stars we obtain a decrease of the asphericity with increasing number of arms. The numerical values are in excellent agreement with results of previous computer simulations. From the extrapolation of the epsilon -expansion results for EV stars we conclude that the influence of the EV on the shape asymptotics qualitatively depends on the number of arms. For f=1 and j=2, i.e. a linear chain, we reproduce a previous finding that the EV enlarges the asphericity of the chains by about 5%. For f>3, however, we observe a decrease of the asphericity due to the EV which is intuitively appealing because of the strong steric repulsion for large values of f. These results are also in good agreement with previous computer experiments.