The dust extinction laws and dust properties in M31 are explored with a sample of reddened O-type and B-type supergiants obtained from the Local Group Galaxies Survey (LGGS). The observed spectral energy distributions (SEDs) for each tracer are constructed with multiband photometry from the LGGS, PS1 Survey, UKIRT, PHAT Survey, Swift/UVOT, and XMM-SUSS. We model the SED for each tracer in combination with the intrinsic spectrum obtained from the stellar model atmosphere extinguished by the model extinction curves. Instead of mathematically parameterizing the extinction functions, the model extinction curves in this work are directly derived from the silicate–graphite dust model with a dust size distribution of dn / da ∼ a − α exp ( − a / 0.25 ) , 0.005 < a < 5 μ m . The extinction tracers are distributed along the arms in M31, with the derived Milky Way (MW)–type extinction curves covering a wide range of R V (≈ 2–6), indicating the complexity of the interstellar environment and the inhomogeneous distribution of interstellar dust in M31. The average extinction curve with R V ≈ 3.51 and dust size distribution dn / da ∼ a − 3.35 exp ( − a / 0.25 ) is similar to that of the MW but rises slightly less steeply in the far-UV bands, implying that the overall interstellar environment in M31 resembles the diffuse region in the MW. The extinction in the V band of M31 is up to 3 mag, with a median value of A V ≈ 1 mag. The multiband extinction values from the UV to IR bands are also predicted for M31, which will provide a general extinction correction for future works.

The dust extinction curves toward individual sight lines in M33 are derived for the first time with a sample of reddened O-type and B-type supergiants obtained from the Local Group Galaxies Survey (LGGS). The observed photometric data are obtained from the LGGS, PS1 Survey, UKIRT, PHATTER Survey, Galaxy Evolution Explorer, Swift/UVOT, and XMM-SUSS. We combine the intrinsic spectral energy distributions (SEDs) obtained from the ATLAS9 and Tlusty stellar model atmosphere extinguished by the model extinction curves from the silicate-graphite dust model to construct model SEDs. The extinction traces are distributed along the arms in M33, and the derived extinction curves cover a wide range of shapes (R V ≈ 2–6), indicating the complexity of the interstellar environment and the inhomogeneous distribution of interstellar dust in M33. The average extinction curve with R V ≈ 3.39 and dust size distribution dn / da ∼ a − 3.45 exp ( − a / 0.25 ) is similar to that of the Milky Way but with a weaker 2175 Å bump and a slightly steeper rise in the far-UV band. The extinction in the V band of M33 is up to 2 mag, with a median value of A V ≈ 0.43 mag. The multiband extinction values from the UV to IR bands are also predicted for M33, which will provide extinction corrections for future works. The method adopted in this work is also applied to other star-resolved galaxies (NGC 6822 and WLM), but only a few extinction curves can be derived because of the limited observations.

Dust extinction law is crucial to recover the intrinsic energy distribution of celestial objects and infer the characteristics of interstellar dust. Based on the traditional pair method, an improved pair method is proposed to model the dust extinguished spectral energy distribution (SED) of an individual star. Instead of the mathematically parameterizing extinction curves, the extinction curves in this work are directly from the silicate-graphite dust model, so that the dust extinction law can be obtained and the dust properties can be analyzed simultaneously. The ATLAS9 stellar model atmosphere is adopted for the intrinsic SEDs in this work, while the silicate-graphite dust model with a dust size distribution of $dn/da \sim a^{-\alpha}{\rm exp}(-a/a_c),~0.005 < a < 5~\mu{\rm m}$ for each component is adopted for the model extinction curves. One typical extinction tracer in the dense region (V410 Anon9) and one in the diffuse region (Cyg OB2 \#12) of the MW are chosen to test the reliability and the practicability of the improved pair method in different stellar environments. The results are consistent with their interstellar environments and are in agreement with the previous observations and studies, which prove that the improved pair method is effective and applicable in different stellar environments. In addition to the reliable extinction results, the derived parameters in the dust model can be used to analyze the dust properties, which cannot be achieved by other methods with the mathematical extinction models. With the improved pair method, the stellar parameters can also be inferred and the extinction law beyond the wavelengths of observed data can be predicted based on the dust model as well.