We study the absorption line shape caused by Frenkel excitons in one-dimensional ring-shaped molecular aggregates, such as circular light-harvesting systems, subjected to dynamic disorder with a finite correlation time. We focus on dichotomic noise and show that for arbitrary orientations of the molecular transition dipoles relative to the ring, the absorption spectrum may be calculated exactly by solving two very similar sets of 2N coupled linear equations of motion, where N is the number of molecules in the ring. These sets are a factor of N smaller than in methods developed previously, which allows us to evaluate the exact line shapes for N up to 12 and study size dependence of the spectra. Previous exact calculations were limited to hexamers (N=6). Moreover, in contrast to earlier work, we take into account long-range dipolar transfer interactions between all molecules in the ring. We find that the dipole orientation and the long-range interactions strongly affect the dependence of the spectrum on the ring size. This holds true particularly for the exchange narrowing of the linewidths in the fast-fluctuation regime.

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