In the single degenerate(SD) scenario for type Ia supernovae(SNe Ia) ,a mass-accreting white dwarf is expected to experience a supersoft X-ray source(SSS) phase. However,some recent observations showed that the expected number of massaccreting WDs is much lower than that predicted from theory,regardless of whether they are in spiral or elliptical galaxies. In this paper,we performed a binary population synthesis study on the relative duration of the SSS phase to their whole massincreasing phase of WDs leading to SNe Ia. We found that for about 40% of the progenitor systems,the relative duration is shorter than 2% and the evolution of the mean relative duration shows that it is always smaller than 5%,both for young and old SNe Ia. In addition,before the SNe Ia explosions,more than 55% of the progenitor systems were experiencing a dwarf novae phase and no more than 10% were staying in the SSS phase. These results are consistent with the recent observations and imply that both in early-and late-type galaxies,only a small fraction of mass-accreting WDs resulting in SNe Ia contributes to the supersoft X-ray flux. So,although our results are not directly related to the X-ray output of the SN Ia progenitor,the low supersoft Xray luminosity observed in early type galaxies may not be able to exclude the validity of the SD model. On the contrary,it is evidence to support the SD scenario.
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