The double or extended main-sequence turnoffs (MSTOs) in the color-magnitude diagram (CMD) of intermediate-age massive star clusters in the Large Magellanic Cloud are generally interpreted as age spreads of a few hundred Myr. However, such age spreads do not exist in younger clusters (i.e., 40-300 Myr), which challenges this interpretation. The effects of rotation on the MSTOs of star clusters have been studied in previous works, but the results obtained are conflicting. Compared with previous works, we consider the effects of rotation on the MS lifetime of stars. Our calculations show that rotating models have a fainter and redder MSTO with respect to non-rotating counterparts with ages between about 0.8 and 2.2 Gyr, but have a brighter and bluer MSTO when age is larger than 2.4 Gyr. The spread of the MSTO caused by a typical rotation rate is equivalent to the effect of an age spread of about 200 Myr. Rotation could lead to the double or extended MSTOs in the CMD of the star clusters with ages between about 0.8 and 2.2 Gyr. However, the extension is not significant; and it does not even exist in younger clusters. If the efficiency of the mixing were high enough, the effects of the mixing would counteract the effect of the centrifugal support in the late stage of evolution; and the rotationally induced extension would disappear in the old intermediate-age star clusters; but younger clusters would have an extended MSTO. Moreover, the effects of rotation might aid in understanding the formation of some "multiple populations" in globular clusters.

Although the nature of the progenitor of Type Ia supernovae (SNe Ia) is still unclear, the single-degenerate (SD) channel for the progenitor is currently accepted, in which a carbon-oxygen white dwarf (CO WD) accretes hydrogen-rich material from its companion, increases its mass to the Chandrasekhar mass limit, and then explodes as a SN Ia. The companion may be a main sequence or a slightly evolved star (WD + MS), or a red-giant star (WD + RG). Incorporating the effect of mass-stripping and accretion-disk instability on the evolution of WD binary, we carried out binary stellar evolution calculations for more than 1600 close WD binaries. As a result, the initial parameter spaces for SNe Ia are presented in an orbital period-secondary mass (log P i , M i 2 ) plane. We confirmed that in a WD + MS system, the initial companion leading to SNe Ia may have mass from 1 M ⊙ to 5 M ⊙ . The initial WD mass for SNe Ia from WD + MS channel is as low as 0.565 M ⊙ , while the lowest WD mass from the WD + RG channel is 1.0 M ⊙ . Adopting the results above, we studied the birth rate of SNe Ia via a binary population synthesis approach. We found that the Galactic SNe Ia birth rate from SD model is 2.55 − 2.9 × 10 −3 yr −1 (including WD + He star channel), which is slightly smaller than that from observation. If a single starburst is assumed, the distribution of the delay time of SNe Ia from the SD model may be a weak bimodality, where WD + He channel contributes to SNe Ia with delay time shorter than 10 8 yr and WD + RG channel to those with age longer than 6 Gyr.

Using reconstructed opacities, we construct solar models with low heavy-element abundance. Rotational mixing and enhanced diffusion of helium and heavy elements are used to reconcile the recently observed abundances with helioseismology. The sound speed and density of models where the relative and absolute diffusion coefficients for helium and heavy elements have been increased agree with seismically inferred values at better than the 0.005 and 0.02 fractional level respectively. However, the surface helium abundance of the enhanced diffusion model is too low. The low helium problem in the enhanced diffusion model can be solved to a great extent by rotational mixing. The surface helium and the convection zone depth of rotating model M04R3, which has a surface Z of 0.0154, agree with the seismic results at the levels of 1 $\sigma$ and 3 $\sigma$ respectively. M04R3 is almost as good as the standard model M98. Some discrepancies between the models constructed in accord with the new element abundances and seismic constraints can be solved individually, but it seems difficult to resolve them as a whole scenario.Comment: 10 pages, 1 figur