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We present a charge distribution-based emission model that calculates theinfrared spectrum of fullerenes (C$_{60}$). Analysis of the modelled spectrumof C$_{60}$ in various charge states shows that the relative intensity of thefeatures in the 5-10 $\mu$m versus 15-20 $\mu$m can be used to probe theC$_{60}$ charge state in interstellar spectra. We further used our model tosimulate emission from polycyclic aromatic hydrocarbons (PAHs) and C$_{60}$ atfive positions in the cavity of reflection nebula NGC~7023. Specifically, wemodelled the 6.2/11.2 band ratio for circumcoronene and circumcircumcoroneneand the 7.0/19.0 band ratio for C$_{60}$ as a function of the ionizationparameter $\gamma$. A comparison of the model results with the observed bandratios shows that the $\gamma$ values in the cavity do not vary significantly,suggesting that the emission in the cavity does not originate from locations atthe projected distances. Furthermore, we find that the C$_{60}$ derived$\gamma$ values are lower than the PAH-derived values by an order of magnitude.We discuss likely scenarios for this discrepancy. In one scenario, we attributethe differences in the derived $\gamma$ values to the uncertainties in theelectron recombination rates of PAHs and C$_{60}$. In the other scenario, wesuggest that PAHs and C$_{60}$ are not co-spatial resulting in different$\gamma$ values from their respective models. We highlight that experiments todetermine necessary rates will be required in validating either one of thescenarios.

Application of the emission model to PAHs and C$_{60}$ II.Polycyclic Aromatic Hydrocarbon emission model in photodissociation regions