Constraining soft and hard X-ray irradiation in ultraluminous X-ray sources

Most ultraluminous X-ray sources (ULXs) are argued to be powered by supercritical accretion onto compact objects. One of the key questions regarding these objects is whether or not the hard X-rays are geometrically beamed toward the symmetric axis. We propose to test the scenario using disk irradiation, to see how much the outer accretion disk sees the central hard X-rays. We collect a sample of 11 bright ULXs with an identification of a unique optical counterpart, and model their optical fluxes considering two irradiating sources: soft X-rays from the photosphere of the optically thick wind driven by supercritical accretion, and if needed in addition, hard X-rays from the Comptonization component. Our results indicate that the soft X-ray irradiation can account for the optical emission in the majority of ULXs, and the fraction of hard X-rays reprocessed on the outer disk is constrained to be no more than $\sim$$10^{-2}$ in general. Such an upper limit is well consistent with the irradiation fraction expected in the case of no beaming. Therefore, no stringent constraint on the beaming effect can be placed according to the current data quality.