  [Abridged] Fullerenes have been recently detected in various circumstellar
and interstellar environments, raising the question of their formation pathway.
It has been proposed that they can form by the photo-chemical processing of
large polycyclic aromatic hydrocarbons (PAHs). Following our previous work on
the evolution of PAHs in the NGC 7023 reflection nebula, we evaluate, using
photochemical modeling, the possibility that the PAH C$_{66}$H$_{20}$ (i.e.
circumovalene) can lead to the formation of C$_{60}$ upon irradiation by
ultraviolet photons. The chemical pathway involves full dehydrogenation,
folding into a floppy closed cage and shrinking of the cage by loss of C$_2$
units until it reaches the symmetric C$_{60}$ molecule. At 10" from the
illuminating star and with realistic molecular parameters, the model predicts
that 100% of C$_{66}$H$_{20}$ is converted into C$_{60}$ in $\sim$ 10$^5$
years, a timescale comparable to the age of the nebula. Shrinking appears to be
the kinetically limiting step of the whole process. Hence, PAHs larger than
C$_{66}$H$_{20}$ are unlikely to contribute significantly to the formation of
C$_{60}$, while PAHs containing between 60 and 66 C atoms should contribute to
the formation of C$_{60}$ with shorter timescales, and PAHs containing less
than 60 C atoms will be destroyed. Assuming a classical size distribution for
the PAH precursors, our model predicts absolute abundances of C$_{60}$ are up
to several $10^{-4}$ of the elemental carbon, i.e. less than a percent of the
typical interstellar PAH abundance, which is consistent with observational
studies. According to our model, once formed, C$_{60}$ can survive much longer
than other fullerenes because of the remarkable stability of the C$_{60}$
molecule at high internal energies.Hence, a natural consequence is that
C$_{60}$ is more abundant than other fullerenes in highly irradiated
environments.
