Modeling X-ray reflection spectra from returning radiation: application to 4U 1630$-$47

Abstract: Returning radiation is thought to play a key role in disk illumination of black hole X-ray binaries in the high-soft state, yet it has not been fully incorporated into XSPEC reflection models. We present a new table model for reflection spectroscopy that, for the first time, self-consistently accounts for the returning radiation. To isolate this effect, we adopt the standard disk-corona configuration but disable the corona, allowing the reflection spectrum to be produced solely by self-irradiation of the disk. Applying our model to the black hole X-ray binary 4U 1630$-$47, we report a rapidly spinning black hole ($a_* \sim 0.99$), a disk inclination of $i \sim 53^\circ$, a mass accretion rate of $\dot{M}{\rm BH} \sim 15\% \, {\rm \dot{M}{Edd}}$, and an electron density of $n_{\rm e} \sim 10^{21} \,\mathrm{cm^{-3}}$ to reproduce the observed reflection features. The model also yields a source distance of $D\sim 8.2 \, {\rm kpc}$, slightly below the commonly adopted value of $10 \, {\rm kpc}$ in the literature. Compared to the widely used relxillNS, our model naturally produces a harder high-energy reflection spectrum, fitting the data without invoking a Comptonized component.