A statistical hardness classification of ultraluminous X-ray sources using long-term Swift/XRT monitoring

Abstract

We characterize long-term hardness behavior in 29 ultraluminous X-ray sources (ULXs) using over 3500 archival Swift/XRT observations. We classify observations as soft-band dominated, hard-band dominated, or uncertain from the hardness ratio (HR; 1.5--10/0.3--1.5 keV count rates) and its confidence bounds. We derive $f_{\rm soft}$ and $f_{\rm hard}$ and define the Crossing Index (CI) as the fraction of consecutive confidently classified observations on opposite sides of $HR=1$. We analyze these descriptors with rule-based classification and unsupervised hierarchical clustering. The rule-based analysis identifies eight soft-band-dominated, five hard-band-dominated, and 16 mixed-band systems; clustering yields a similar three-group structure. Known or candidate accretor types tend to occupy band-dominated groups, whereas unknown accretors are more common in the mixed-band group. All four sources with reported long-term X-ray modulation discussed in an orbital or super-orbital context are classified as mixed-band. The three-cluster solution is preferred in 25 of 29 leave-one-out trials, while bootstrap tests find the band-dominated groups more stable than the mixed-band group. As the classification uses observed count rates, the groups represent Swift/XRT signatures rather than intrinsic spectral states. Response-folding tests show that modeled HR depends on absorption column density and photon index. Bayesian reanalysis of a representative low-count source reduces CI from 56.3\% to 38.5\% while preserving its mixed-band classification. Random downsampling produces no systematic shift in CI; halving the observations changes median CI by at most $\sim10\%$ across four sources. Larger samples with independent absorption constraints can test links among these observational classes, accretor type, long-term periodic modulation, and other X-ray properties.