Broad-band X-ray spectra and timing of the accretion-powered millisecond pulsar Swift J1756.9$-$2508 during its 2018 and 2019 outbursts

The accreting millisecond X-ray pulsar Swift J1756.9$-$2508 went into outburst in April 2018 and June 2019, 8.7 yr after the previous activity period. We investigated the temporal, timing and spectral properties of these two outbursts using data from NICER, XMM-Newton, NuSTAR, INTEGRAL, Swift and Insight-HXMT. The two outbursts exhibited similar broad-band spectra and X-ray pulse profiles. For the first time, we report the detection of the pulsed emission up to $\sim100$ keV observed by Insight-HXMT during the 2018 outburst. We also found the pulsation up to $\sim60$ keV observed by NICER and NuSTAR during the 2019 outburst. We performed a coherent timing analysis combining the data from two outbursts. The binary system is well described by a constant orbital period over a time span of $\sim12$ years. The time-averaged broad-band spectra are well fitted by an absorbed thermal Comptonization model in a slab geometry with the electron temperature $kT_{\rm e}=40$-50 keV, Thomson optical depth $\tau\sim 1.3$, blackbody seed photon temperature $kT_{\rm bb,seed}\sim $0.7-0.8 keV and hydrogen column density of $N_{\rm H}\sim 4.2\times10^{22}$ cm$^{-2}$. We searched the available data for type-I (thermonuclear) X-ray bursts, but found none, which is unsurprising given the estimated low peak accretion rate ($\approx0.05$ of the Eddington rate) and generally low expected burst rates for hydrogen-poor fuel. Based on the history of four outbursts to date, we estimate the long-term average accretion rate at roughly $5\times10^{-12}\ M_\odot\,{\rm yr}^{-1}$ for an assumed distance of 8 kpc. The expected mass transfer rate driven by gravitational radiation in the binary implies the source can be no closer than 4 kpc.