Red Supergiants in M31 and M33. I. The Complete Sample

We remove the foreground dwarfs using the near-infrared color–color diagram. The study of Bessell & Brett (1988) found that the intrinsic color indexes have clear bifurcations on the J − H/H − K two-color diagram for giant and dwarfs. Dwarfs have higher surface gravity than giants or supergiants; the collision rate between atoms is higher, and molecules are easier to form (Allard & Hauschildt 1995). This makes molecules form at relatively high temperatures in dwarfs, causing absorption in the H band and darkening the H-band brightness and eventually leading to smaller J − H and bigger H − K than giants.

The borderline between dwarfs and giants are redefined intentionally and specifically, though Bessell & Brett (1988) already obtained the intrinsic color indexes. For this purpose, the high-accuracy photometric data are chosen with the error of J-, H-, and K-band photometry less than 0.05 mag and N_Flag=3, which means the object is identified as "stellar" in all three bands. The J − H/H − K diagrams for these accurate photometries are shown in Figure 4 for M31 and M33. It can be seen that there is a very clear boundary between giants and dwarfs. With the increase of H − K, J − H increases to about 0.7 and then turns down for dwarfs; meanwhile, the J − H of red giants and supergiants starts from about 0.7 and increases monotonically. The trends and values coincide very well with the result of Bessell & Brett (1988).

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The dividing line is defined quantitatively. In a step of 0.01 in the range of H − K from 0.05 to 0.3, the point of the maximum surface density is calculated on the dwarf branch, and then the piecewise function is used to fit those points. When H − K ≤ 0.13, the function is linear; for H − K > 0.13, the function is quadratic. This piecewise function represents the relation of the J − H and H − K of dwarfs. Because the foreground extinction is pretty small (an average A_V ∼ 0.17; Schlegel et al. 1998; Schlafly & Finkbeiner 2011; Cordiner et al. 2011), these colors basically represent the intrinsic color indexes of dwarfs. We shift the piecewise functions up by 0.12 and 0.09 mag by eye-check and take them as the dividing lines between giants and dwarfs in the M31 and M33 fields, where the difference in the shift in the vertical axis is caused by the different foreground extinction to the two galaxies, i.e., A_V ∼  0.17 mag and 0.11 mag for M31 and M33, respectively, according to SFD98 (Schlegel et al. 1998). The function forms and coefficients of the adopted dividing lines are listed in the first rows of Table 1.

Table 1.  Function and the Coefficients of Its Dividing Lines in Figures 4–6

	Color Criteria	Magnitude Criteria	Coefficients
			M31	M33
UKIRT	H − K < 0.10	All sources
	H − K ≤ 0.13	J − H < a(H − K) + b	a = 2.265, b = 0.329	a = 2.837, b = 0.258
	H − K > 0.13	$J-H\lt a{\left(H-K\right)}^{2}+b(H-K)+c$	a = −1.384, b = 0.241, c = 0.616	a = −1.441, b = 0.241, c = 0.620
PS1	r − z < 0.30	All sources
	r − z ≤ 1.10	$z-H\lt \tfrac{{{KP}}_{0}{e}^{m(r-z-{X}_{0})}}{K+{P}_{0}({e}^{m(r-z-{X}_{0})}-1)}+C$	K = 1.265, P0 = 0.409, m = 5.357	K = 1.382, P0 = 0.484, m = 5.027
			X0 = 0.106, C = 0.712	X0 = 0.087, C = 0.610
	r − z > 1.10	$z-H\lt a{\left(r-z\right)}^{2}+b(r-z)+c$	a = 0.143, b = −0.283, c = 2.103	a = 0.137, b = −0.233, c = 2.067
LGGS	V − R < 0.50	All sources
	All sources	$B-V\lt \tfrac{{{KP}}_{0}{e}^{m(V-R-{X}_{0})}}{K+{P}_{0}({e}^{m(V-R-{X}_{0})}-1)}+C$	K = 2.242, P0 = 0.434, m = 3.547	K = 2.572, P0 = 0.475, m = 2.959
			X0 = 0.005, C = − 0.461	X0 = −0.126, C = −0.666

Note. In fact, the coefficients listed in this table are based on the fitting results of the maximum surface density of the dwarf branch, which need to be moved up as the dividing line. The values of shift are described in Section 3.1.1 and 3.1.2.

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With the dividing line determined, the criterion is applied to the entire initial sample to remove foreground dwarfs. In addition, we remove sources with H − K < 0.1 that is apparently bluer than RSGs. This action certainly also removes the blue and yellow supergiants in the galaxy, which are absent in the final catalog of the members in M31 and M33. Finally, 414,490 and 77,091 foreground dwarfs are removed in the M31 and M33 fields, i.e., 33.3% and 37.9% of the initial samples.

The above method is expected to remove all of the foreground dwarfs in the sample, but the uncertainty of color indexes would move some sources around the borderline. The completeness and pollution rate of the selected member stars are estimated by Monte Carlo simulation. First, the sources with "N_Flag=3" and σ_J,H,K < 0.05 mag are taken as a no-error "perfect" sample so that their locations in the J − H/H − K diagram can absolutely decide being a dwarf or a giant. Then, we perform 5000 simulations for a random error with a two-dimensional Gaussian distribution for each source whose width is four times the error of J − H and H − K. Finally, the UKIRT/NIR criterion is used to divide dwarfs and giants to compute the completeness and pollution rate of giants when the JHK photometric errors are limited to less than 0.2 mag. For M31 and M33, the simulation results show that the completeness of the selected stars is about 93%, and the pollution rate is about 9%.

Actually, the pollution rate calculated by this method will be overestimated, and the completeness will be underestimated. On one hand, we multiply the errors of the sources whose JHK photometric errors are less than 0.05 mag by 4 to simulate the case where the JHK photometric errors are limited to be less than 0.2 mag, which means increasing the photometric errors systemically. On other hand, the distribution of sources with JHK photometric errors less than 0.05 mag is already scattering due to the photometric errors. Therefore, the true pollution rate of the selected giants is smaller than the simulation value, and the completeness is larger than the simulation value.

Although the J − H/H − K diagram works very efficiently and is applicable to all the sample stars, the identification deserves to be checked by other methods. One purpose is to confirm the identification, and the other is to further remove some foreground stars that are very close to the borderline in the NIR color–color diagrams (CCD) but may be significantly distinguishable in optical bands in particular for some relatively blue stars.

• 1.  
  The r − z/z − H diagram. By convoluting the spectrum of the MARCS model with the transmission functions of the r, z, and H filters, Yang et al. (2020c) found that the r − z/z − H diagram can distinguish dwarfs from giants well. We introduce the r − z/z − H diagram only as an auxiliary method to remove foreground dwarfs, because the information used to distinguish dwarfs from giants here, H-band photometry, has already been used in the J − H/H − K CCD (see Section 3.1.1), and only those foreground dwarfs with good photometric quality will be taken into account.The PS1/DR2 data are used, where the forced mean PSF magnitude is taken for its consideration of both photometric depth and accuracy. The data flags of PS1/DR2 are complex and only those with good measurements are used. The cross-match between UKIRT and PS1 by a radius of 1'' resulted in 184,750 and 37,304 stars in M31 and M33 with good photometry. Similar to the method in Section 3.1.1, we select the "good-measurement" sources with r-, z-, and H-band photometric errors less than 0.05 mag to determine the borderline between dwarfs and giants in the r − z/z − H diagram. The positions of the maximum surface density of the dwarf branch on r − z/z − H diagram are calculated in a step of 0.01 of r − z from 0 to 2.5, and then a piecewise function is used to fit those points. When r − z ≤ 1.1, the sigmoid function, which is also known as the logistic function, is used for fitting; otherwise, the function is a quadratic curve. As shown in Figure 5, the piecewise functions are shifted up by 0.12 and 0.10 mag for M31 and M33 to become the dividing lines between giants and dwarfs. It should be noted that a slight difference in the intrinsic color indexes r − z or z − H appears between our results and the MARC model though they are in general agreement. The functions used are listed in the second rows of Table 1.The criterion is applied to the PS1 sources with "good measurement," and the photometric error in the r, z, and H bands is less than 0.1 mag. The sources below the dividing line or with r − z < 0.3 are removed. This removes 102,174 and 17,648 dwarfs in the M31 and M33 fields, respectively, in which 93,938 (91.9%) and 15,568 (88.2%) are also removed by the NIR CCD. In other words, an additional 8,236 and 2,080 stars are removed, which are mostly around the borderline in the NIR CCD or in the area close to the center of the galaxy where the photometry is of poor quality.
• 2.  
  The B − V/V − R diagram. Massey (1998) proved that the B − V/V − R diagram can separate RSGs from the foreground dwarfs. The low-surface-gravity RSGs are redder in B − V at a given V − R than the high-surface-gravity dwarfs because the B band covers many metallic lines to become sensitive to surface gravity. Although Massey et al. (2009), Drout et al. (2012), and Massey & Evans (2016) already determined the dividing line to separate RSGs in M31 and M33 from the foreground dwarfs, we still renew it to derive a quantitative relation of B − V and V − R for dwarfs stars like in previous calculations for J − H/H − K and r − z/z − H. The method is the same as in Section 3.1.1. The stars whose B-, V-, and R-band photometric errors less than 0.05 mag are chosen to define the borderline by the points of the maximum surface density on the B − V/V − R diagram in steps of 0.01 in the range of V − R from -0.25 to 1.30. The logistic function is taken to fit these points and shifted up by 0.12 and 0.10 mag for M31 and M33, respectively, respectively, as the dividing lines shown in Figure 6. The function forms and coefficients of dividing lines are listed in the third rows of Table 1.The LGGS criteria work very well for bright sources when Massey et al. (2009) and Massey & Evans (2016) limit the sources by V brighter than 20 mag, i.e., photometric error less than 0.01. But for faint sources with a slightly large photometric error, to distinguish dwarfs from giants is very hard. As shown in the left panel of Figure 7, a large portion of the foreground dwarfs selected by the NIR CCD are located in the B − V/V − R region of dwarfs, confirming the consistency between optical and NIR criteria. But when the photometric error increases to 0.05 mag, many of the NIR-selected foreground dwarfs fall into the giants region of the LGGS diagram as shown in the right panel of Figure 7, needless to say that many of the objects have photometric uncertainty of  ∼0.1 mag. This can be understood that the difference between giants and dwarfs in the B − V/V − R diagram is too small to tolerate any significant photometric error. Comparing Figure 6 with Figure 4 and Figure 5 demonstrates clearly that the difference of colors in the optical (mostly ∼0.1 mag) is much less significant than in near-infrared (mostly ∼0.2 mag). The cross-match of the LGGS catalog with UKIRT by a radius of 1'' results in 92,441 (7.4% of the initial NIR sample) and 45,279 (22.3%) associations. These associations are a small part of the sample, and taking the risk of mixing dwarfs and giants together into account, the LGGS criteria are not used to remove dwarfs in this work.

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As mentioned in Section 2, sources brighter than 17.94 and 18.22 in the K band are complete. With the lower limits of RSGs in M31 and M33 being K = 17.62 and 18.11, the samples are considered to be complete, i.e., there are 5498 and 3055 RSGs in M31 and M33, respectively. RSGs in M31 and M33 are listed in Table 4 and 5 with R.A. and decl. coordinates, magnitudes, magnitude errors, astrometric information, etc. One thing to note is that our preliminary selection of stars is not very strict in that the stars labeled "−7" for bad pixel are included and that only two of the JHK bands labeled "stellar" are required. If all three JHK bands are required to be "stellar," then the number of RSGs is reduced to 3268 and 2804 for M31 and M33, respectively. The number of reduction for M31 is significant and very minor for M33, again due to the much more crowded field in M31. We think these numerals must have underestimated the sample. Moreover, if the sources with two of the three bands labeled "−7" are removed, the number of RSGs is further reduced to 3154 and 2635, respectively.

Comparing the location of various types of evolved stars in the J − K/K diagram with the limiting magnitude of 2MASS, it can be inferred that some RSGs and AGBs are detectable by 2MASS, but not all of them. If the quality flag is further restricted to "AAA," then only bright AGBs and RSGs in M31 with K < 16 can be recognized by the 2MASS photometry. For M33, the situation is reduced because the main 2MASS point-source catalog has a brighter limiting magnitude of K ∼ 15 for the "AAA" sources.

The pollution rate of the RSG sample is estimated from the control field. First, the UKIRT fields are divided into geometric regions of M31, M32, M110, and M33 and the control fields. The region of a galaxy is demarcated by an ellipse whose major and minor axes are determined by the $B=25\ \mathrm{mag}/{\mathrm{arcsec}}^{2}$ isophotes in the B band shown in Figure 11. The fields beyond the geometric regions of the galaxies are classified into control fields. The major axes, minor axes, and position angles of the galaxies are listed in Table 6.

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The sky area of galaxies is calculated by S = πab, where a and b are the half-major and half-minor axes. The sky area of the control fields is the total sky area minus the sky area of the galaxies. The sky area of different regions and the number of RSGs are listed in Table 6. If all of the RSGs in the control fields are foreground stars, or fake RSGs, the pollution rate can be estimated. First, the fake RSGs per sky area are calculated by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{fake}}={N}_{\mathrm{control}\,\mathrm{fields}}^{\mathrm{RSGs}}/{S}_{\mathrm{control}\mathrm{fields}},\end{eqnarray} \tag{ 5 }$$

where ${N}_{\mathrm{control}\,\mathrm{fields}}^{\mathrm{RSGs}}$ is the number of RSGs in the control fields and S_{control fields} is the sky area of the control fields. Then, the number of fake RSGs in the region of a galaxy is

$$\begin{eqnarray}&&{N}_{\mathrm{fake}}={{\rm{\Sigma }}}_{\mathrm{fake}}\times {S}_{\mathrm{galaxy}},\end{eqnarray} \tag{ 6 }$$

where S_galaxy is the sky area of the galaxy and N_fake is the number of fake RSGs. For M31, S_galaxy = S_M31 + S_M110 because M32 is inside M31. The pollution rate is then

$$\begin{eqnarray}&&P={N}_{\mathrm{fake}}/{N}_{\mathrm{galaxy}},\end{eqnarray} \tag{ 7 }$$

where N_galaxy is the number of RSGs within the galaxy's area. For M31, N_galaxy = N_M31 + N_M32 + N_M110. The pollution rate turns out to be 1.30% and 0.49% for the catalog of RSGs in M31 and M33, respectively. Indeed, the fake RSGs in the control fields located around the rim of the galaxies as shown in Figure 11, which indicates some of them are actually the members of M31 or M33. In another word, M31 and M33 extend to a larger area than the labeled ellipse defined by $25\ \mathrm{mag}/{\mathrm{arcsec}}^{2}$ isophotes. The true pollution rate should be smaller than the above derived values.

The spatial distribution of the selected RSGs is shown in Figure 11 with the GALEX ultraviolet image as the background to display the massive-star regions. The locations coincide very well with the spiral arms, which are expected for RSGs as massive stars. This structure supports our identification of RSGs.

The number of RSGs is 5225 within 2.567 deg² of M31 and 3001 within 0.644 deg² of M33. Apparently, the surface density of RSGs is not the same. It is well known that metallicity influences the time a massive star spends in the RSG stage. In order to characterize the density and the massive-star formation rate, the number of RSGs is normalized to the stellar mass of the galaxy. In addition to M31 and M33, we supplement the other galaxies (SMC, LMC, and MW) whose RSG samples are systematically studied. The number of RSGs in SMC and LMC is 1405 and 2974 from Yang et al. (2019, Y+19 for short) and Yang et al. (2020a, Y+20b for short). It should be noted that the sample of RSGs in the Milky Way galaxy is incomplete because of the interstellar extinction itself and the complexity caused by it. A recent search in the Galaxy found 889 nearby RSGs (Messineo & Brown 2019, MB19 for short); meanwhile, Gehrz (1989) predicted at least 5000 RSGs. Both values are displayed in Figure 12, and the former one indicates the lower limit. The adopted stellar masses are 3.1 × 10⁸M_⊙, 1.5 × 10⁹M_⊙, 5.2 × 10¹⁰M_⊙, 2.6 × 10⁹M_⊙, and 1.0 × 10¹¹M_⊙ for the SMC, LMC, MW, M31, and M33, respectively (Besla 2015; Fattahi et al. 2016). The metallicity from the SMC and LMC to M33 and M31 is increasing, from subsolar metallicity to supersolar metallicity. Specifically, the value of $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ is 8.13 (Russell & Dopita 1990), 8.37 (Russell & Dopita 1990), 8.70 (Esteban & Peimbert 1995), 8.75 (Garnett et al. 1997), and 9.00 (Zaritsky et al. 1994) for the SMC, LMC, MW, M33, and M31, respectively.

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The RSG density per 10⁸M_⊙ is presented in Figure 12. With increasing metallicity, the RSG density per stellar mass decreases rapidly. When the metallicity $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ increases by 0.9 dex, the number of RSGs per stellar mass decreases by about 60 times. This can be understood as metallicity affecting the time stars spent in different evolutionary stages. Previous studies have shown that metallicity affects the ratio of blue-to-red supergiants (B/R) and Wolf-Rayet stars to RSGs (W-R/RSGs). When metallicity increases by 0.9 dex, the B/R ratio and the W-R/RSG ratio increase by about 7 times (Maeder et al. 1980) and 100 times (Massey 2002) respectively. Our conclusion that the density of RSGs decreases with increasing metallicity is consistent with this scenario. On the other hand, metallicity cannot be the only factor to influence the density of RSGs. M33 is much more metal-rich than LMC, but holds very similar RSG density. In fact, M33 is the only SAcd-type galaxy in them; the strong star-forming activity may account for the higher massive-star formation rate. Among these galaxies, M31 should be the most similar to our galaxy in both type and metallicity. Taking M31 as the reference, the number of RSGs in our galaxy should be ∼2800 after scaling by stellar mass. But this is only the lower limit of the number of RSGs in our galaxy. Because the metallicity of the Milky Way is lower than that of M31, the number of RSGs per stellar mass should be higher than that of M31. As shown in Figure 12, the Gehrz (1989) prediction of the number of 5000 RSGs in our galaxy agrees well with the overall trend of the number of RSGs per stellar mass with metallicity, but is still lower than the overall trend. A detailed study of this problem will be presented in our future work.