  Self-gravity becomes competitive as an angular momentum transport process in
accretion discs at large radii, where the temperature is low enough that
external irradiation likely contributes to the thermal balance. Irradiation is
known to weaken the strength of disc self-gravity, and can suppress it entirely
if the disc is maintained above the threshold for linear instability. However,
its impact on the susceptibility of the disc to fragmentation is less clear. We
use two-dimensional numerical simulations to investigate the evolution of
self-gravitating discs as a function of the local cooling time and strength of
irradiation. In the regime where the disc does not fragment, we show that local
thermal equilibrium continues to determine the stress - which can be
represented as an effective viscous alpha - out to very long cooling times (at
least 240 dynamical times). In this regime, the power spectrum of the
perturbations is uniquely set by the effective viscous alpha and not by the
cooling rate. Fragmentation occurs for cooling times tau < beta_crit / Omega,
where beta_crit is a weak function of the level of irradiation. We find that
beta_crit declines by approximately a factor of two, as irradiation is
increased from zero up to the level where instability is almost quenched. The
numerical results imply that irradiation cannot generally avert fragmentation
of self-gravitating discs at large radii; if other angular momentum transport
sources are weak mass will build up until self-gravity sets in, and
fragmentation will ensue.
