Direct collapse black holes (DCBH) have been proposed as a solution to the challenge of assembling supermassive black holes by z > 6 to explain the bright quasars observed at this epoch. The formation of a DCBH seed with MBH ˜ 104-5 M⊙ requires a pristine atomic-cooling halo to be illuminated by an external radiation field that is sufficiently strong to entirely suppress H2 cooling in the halo. Many previous studies have attempted to constrain the critical specific intensity that is likely required to suppress H2 cooling, denoted as Jcrit. However, these studies have typically assumed that the incident external radiation field can be modelled with a blackbody spectrum. Under this assumption, it is possible to derive a unique value for Jcrit that depends only on the temperature of the blackbody. In this study we consider a more realistic spectral energy distribution (SED) for the external source of radiation that depends entirely on its star formation history and age. The rate of destruction of the species responsible for suppressing molecular hydrogen cooling depends on the detailed shape of the SED. Therefore the value of Jcrit is tied to the shape of the incident SED of the neighbouring galaxy. We fit a parametric form to the rates of destruction of H2 and H- that permit direct collapse. Owing to this, we find that Jcrit is not a fixed threshold but can lie anywhere in the range Jcrit ˜ 0.5-103, depending on the details of the source stellar population.