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Following Paczy\'{n}ski & Zi\'{o}lkowski (1968) and Han et al. (1994), weassume that the envelope of an asymptotic giant branch (AGB) or a first giantbranch (FGB) star is lost when the binding energy of the envelope is equal tozero ($\Delta W=0$) and the core mass of the AGB star or the FGB star at thepoint ($\Delta W=0$) is taken as the final mass. Using this assumption, wecalculate the IFMRs for stars of different metallicities.We find that the IFMRsdepends strongly on the metallicity, i.e. $Z=0.0001, 0.0003, 0.001, 0.004,0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08$ and 0.1. From $Z=0.04$, the finalmass of the stars with a given initial mass increases with increasing ordecreasing metallicity. The difference of the final mass due to the metallicitymay be up to 0.4 $M_{\odot}$. A linear fit of the initial-final massrelationship in NGC 2099 (M37) shows a potential evidence of the effect ofmetallicity on the IFMR. The IFMR for stars of $Z=0.02$ obtained in the papermatches well with those inferred observationally in the Galaxy. For $Z\geq0.02$, helium WDs are obtained from the stars of $M_{\rm i}\leq 1.0 M_{\odot}$and this result is upheld by the discovery of numerous low-mass WDs in NGC 6791which is a metal-rich old open cluster. Using the IFMR for stars of $Z=0.02$obtained in the paper, we have reproduced the mass distribution of DA WDs inSloan DR4 except for some ultra-massive white dwarfs. The trend that the meanmass of WDs decreases with effective temperature may originate from theincrease of the initial metallicities of stars. We predict that metal-richlow-mass stars may become under-massive white dwarfs.

Initial-Final Mass Relationship for Stars of Different Metallicities