The X-ray bursts within the 2010 outburst of the accreting millisecond X-ray pulsar SAX J1748.9-2021

With the observations from $\mathit{\text{Rossi}}\mathit{\text{X-ray}}\mathit{\text{Timing}}\mathit{\text{Explorer}}$, we search and study the X-ray bursts of accreting millisecond X-ray pulsar SAX~J1748.9-2021 during its 2010 outburst. We find 13 X-ray bursts, including 12 standard type-\uppercase\expandafter{\romannumeral1} X-ray bursts and an irregular X-ray burst which lacks cooling tail. During the outburst, the persistent emission occurred at $\sim$(1-5)$\mathrm{\%}{\dot{\mathrm{M}}}_{\mathrm{E}\mathrm{d}\mathrm{d}}$. We use a combination model of a blackbody (BB), a powerlaw, and a line component to fit the persistent emission spectra. Another BB is added into the combination model to account for the emission of the X-ray bursts due to the thermonuclear burning on the surface of the neutron star. Finally, we modify the combination model with a multiplicative factor $f_{\mathrm{a}}$, plus a BB to fit the spectra during the X-ray bursts. It is found that the $f_{\mathrm{a}}$ is inversely correlated with the burst flux in some cases. Our analysis suggests that the ignition depth of the irregular X-ray burst is obviously smaller than those of the type-\uppercase\expandafter{\romannumeral1} X-ray bursts. We argue that the detected type-\uppercase\expandafter{\romannumeral1} X-ray bursts originate from helium-rich or pure-helium environment, while the irregular X-ray burst originates from the thermonuclear flash in a shallow ocean.