Top-down formation of fullerenes in the interstellar medium

[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.