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SWIFT J1749.4-2807: A neutron or quark star?

We investigate an unique accreting millisecond pulsar with X-ray eclipses, SWIFT J1749.4$-$2807 (hereafter J1749), and try to limit the binary system by various methods including that of the Roche lobe, the mass-radius relations of both a main sequence (MS) and a white dwarf (WD) companion stars, as well as the measured mass function of the pulsar. The calculations are based on the assumption that the radius of the companion star has reached its Roche radius (or at 90%), but the pulsar's mass has not been assumed to be a certain value. Our results are as follows. The companion star should be a MS. For the case that the radius equals to its Roche one, we have a companion star with mass $M\simeq 0.51 M_{\odot}$ and radius $R_{\rm c}\simeq 0.52R_{\odot}$, and the inclination angle is $i\simeq 76.5^{\circ}$; for the case that the radius reaches 90% of its Roche one, we have $M\simeq 0.43M_{\odot}$, $R_{\rm c}\simeq 0.44R_{\odot}$ and $i\simeq 75.7^{\circ}$. We also obtain the mass of J1749, $M_{\rm p}\simeq 1M_\odot$, and conclude that the pulsar could be a quark star if the ratio of the critical frequency of rotation-mode instability to the Keplerian one is higher than $\sim 0.3$. The relatively low pulsar mass (about $\sim M_\odot$) may also challenge the conventional recycling scenario for the origin and evolution of millisecond pulsars. The results presented in this paper are expected to be tested by future observations.