X-ray flashes powered by the spindown of long-lived neutron stars

X-ray flashes (XRFs) are a class of high-energy transients whose nature is still open to question. Similar in many aspects to common gamma-ray bursts (GRBs), their strong X-ray emission is accompanied by very low or absent emission in the gamma-ray band. Despite this key difference, a number of indications have consolidated the idea that XRFs and GRBs share a common origin, including a number of potential XRF/supernova associations and the consistency of some XRFs with the Amati relation for long GRBs. However, the difficulties in explaining XRFs as off-axis or intrinsically weak GRBs still cast doubts on this interpretation. Here we explore the possibility that some XRFs are instead powered by the spindown of a long-lived neutron star (NS) formed in a binary NS (BNS) merger or, possibly, in a core-collapse supernova. Focusing on XRF 020903 and a few other cases observed by HETE-2, we show that their lack of gamma-ray emission, spectral properties, duration and X-ray luminosity find a natural explanation within our hypothesis. Moreover, we point out that the agreement of XRF 020903 with the Amati and Ghirlanda relations for long GRBs is respectively only marginal and problematic. Assuming a BNS merger origin for the long-lived NS, we use XRF observations to estimate a lower limit on the rate of BNS mergers accompanied by a potentially observable XRF signal. Within the reach of the advanced LIGO and Virgo gravitational wave detectors, we find $>0.02\text{--}0.05{yr}^{-1}$. Finally, we discuss the implications of a supernova association for the XRF events considered.