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tokens: ratio of 200. Table 5 Spearman’s Rank Correlation Coefficient of the Logarithmic MLR with Stellar Luminosity, Infrared Magnitude, and Colors Figure 12. The relation between MLR and luminosity. Figure 13. Comparison of the relation between MLR and luminosity with the works of de Jager et al. ( 1988 ), van Loon et al. ( 2005 ), and Beasor Davies ( 2018 ). Unlike the common slope, the intercept, i.e., the coefficient of proportion in the relation, is not the same. As discussed in Section 3.1 , the amount of MLR depends on multiple factors, including the dust species, the dust size, the stellar expansion velocity, and luminosity. The effects of dust species and size are discussed in previous sections, and can be as large as one order of magnitude. The expansion velocity and luminosity should have less effect, probably changing the MLR by a factor of a few. Taking all the factors into account, the MLR may differ by two orders of magnitude for a given observational SED, therefore the comparison of the vertical shift is meaningful only under the same model conditions. Taking T eff into account, the relationships become The fitting results are displayed in Figure 14 together with the value of R 2 , which implies an improved relation in comparison with the luminosity alone. In general, the fitting is reasonably good with the residual , but it deviates somewhat from the measurements, particularly for the silicate dust. This may be understood because MLR depends on other parameters in addition to L and T eff , such as stellar mass, radius or surface gravity (Reimers 1975 ; Beasor et al. 2020 ), and pulsation period (Goldman et al. 2017 ). But these parameters are not conveniently available from observation. Figure 14. The relation between MLR and the luminosity and effective temperature. The red line marks Y = X . 4.4. Relation of MLR with Infrared Brightness After examining the infrared-band brightness, a very good relation is found with the mid-infrared flux in the Spitzer/IRAC4 band at 8 μ m and the MIPS24 band at 24 μ m. Linear fitting yields the following relations with the absolute magnitude, for which the distance modulus is 24.32 for M31 and 24.50 for M33 (Paper I ). The fitting lines are shown in Figures 15 and 16 . The correlation coefficients are all large (−0.7) as listed in Table 5 , which implies a tight relation of the mid-infrared flux with the MLR. This relation is expected from the warm temperature of the circumstellar dust, which emits mainly in the mid-infrared. On other hand, the relation with near-infrared flux is weak and not presented. Figure 15. The relation between MLR and the Spitzer/IRAC4 absolute magnitude. Figure 16. The relation between MLR and the Spitzer/MIPS24 absolute magnitude. 4.5. Relation of MLR with Infrared Color Indexes Examining the infrared colors, it is found that the MLR is sensitive to IRAC1 − IRAC4: As seen in Figure 17 , the scattering around the fitting line is significant, and the correlation of MLR