The Red Supergiant Content of M31 and M33

The J and K_s images of M31 were taken as part of the Andromeda Optical and Infrared Disk Survey (androids) aimed primarily at obtaining a uniform surface brightness map across a 5 deg² region of the disk and bulge. The project, calibration, and reduction procedure are described by Sick et al. (2014).

The images were taken with the wide-field NIR camera WIRCam (Puget et al. 2004) on the 3.6 m Canada–France–Hawaii Telescope located atop Maunakea. The data set consists of 70 usable J- and K_s -band image pairs. The data were taken in queue mode during four blocks of time: 9 nights during (UT) 2007 August 1–September 23 (27 fields), 12 nights during 2009 August 6–October 28 (12 fields), 4 nights during 2011 September 18–October 9 (4 fields), and 6 nights during 2012 October 6–29 (27 fields). (In addition, there were images obtained for three fields obtained on 2009 October 29 and one field on 2012 October 10 that we judged unusable due to reduction issues.) Each field was observed in 16 × 47 s exposures (i.e., 752 s total) in J and 26 × 25 s exposures (i.e., 650 s total) in K_s . An equivalent amount of time was spent observing the sky: because of crowding and the large angular extent of M31, the sky subtraction was achieved by nodding the telescope away from the M31 field by one or more degrees. The nodding strategy is discussed extensively in Sick et al. (2014) based upon the first two observing seasons. A customized pipeline, described by Sick et al. (2014), produced the final, stacked images that we used in our analysis. They were made available for our project through coauthor S. C.

The image scale of each image is 03 pixel⁻¹. Although nominally each field covered 215 × 215, we found that there were significant issues along the edges and trimmed each image to the central 4001 × 4001 pixels, i.e., 200 × 200. This had no effect on the actual areal coverage, given the large overlap between adjacent frames. Figure 1 shows the area surveyed, which includes roughly 5 deg². The pipeline had provided an accurate world coordinate system, vastly simplifying our task of combining photometry from so many overlapping frames.

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Photometry was complicated by the fact that the seeing was so good (∼05–06) compared to the image scale (03 pixel⁻¹) that the point-spread functions (PSFs) were generally undersampled. That, combined with the large quantity of fields, made PSF fitting an unattractive option. After numerous experiments, we concluded that we could do little better than standard aperture photometry. We used daofind within the iraf environment to identify (mostly) stellar objects 3σ above the expected noise based upon the equivalent read noise and gain. We then performed aperture photometry with phot using a 3 pixel (09) radius aperture; sky was determined locally using a modal definition in an annulus extending from 30 to 60. Attempts to centroid before performing the photometry resulted in many problems, primarily moving fainter detections toward brighter neighboring objects, or moving a detection from a stellar source to a nearby diffraction spike from a very bright star. Therefore, we performed the photometry centered on the daofind positions.

The photometry for each frame and filter was then matched with sources from the 2MASS catalog (Cutri et al. 2003) in order to determine photometric zero points. Although the most heavily saturated stars had been masked out in our images by the pipeline reduction, we found that stars that matched with 2MASS sources brighter than J = 14.3 or K_s  = 13.3 showed symptoms of nonlinearity issues. Restricting the sample to 2MASS stars fainter than these limits, and with photometric ratings of "A" in the 2MASS catalog, we were then able to calibrate each frame photometrically. Typically 150–200 stars were used per frame, with zero-point standard deviations of 0.01–0.02 mag, and standard deviations of the mean below 0.005 mag.

The calibrated J photometry from all 70 fields were then combined and averaged, and the same process repeated for the K_s -band data. Finally, the J- and K_s -band photometry were matched. Only stars detected in both filters were retained. We removed anything fainter than J = 17.5 or K_s  = 19.5 or brighter than J = 14.3 or K_s  = 13.3.

These bright limits are very similar to what we expect for the brightest RSGs in M31; see, for example, Figure 8 in Neugent et al. (2020b). Nevertheless, we decided to add to this sample any 2MASS stars that were not already in our object list. In order to allow for the irregular field boundaries, we simply required that a 2MASS star be within 1' of a star with existing photometry. This added about 6000 stars, bringing the total to 712,716 stellar sources with photometry. Of course, we expect only a small fraction (a few percent) to be in the color–magnitude range of RSGs after we remove foreground objects.

We show the photometric errors as a function of magnitude in Figure 3. The vast majority of stars have a photometric precision better than 0.02 mag to our faint limit. Due to the large number of stars (over 700,000), we have plotted only every 10th point for clarity.

The J and K_s data for M33 were obtained as part of a study of AGB stars in M33 and are described in detail by Cioni et al. (2008). The data were subsequently used by Javadi et al. (2015) as part of their study of the variability of red giant stars in M33.

The data were taken under program U/05B/7 with the Wide Field Camera (WFCam) on the 3.8 m United Kingdom Infrared Telescope (UKIRT) located on Maunakea. ⁷ The camera consists of four arrays such that with four pointings, the gaps between the arrays can be filled in to prove a spatial coverage of 520 × 520 with a native image scale of 04 pixel⁻¹. (The ensemble of four such pointings is referred to as a "tile.") Five such tiles centered on M33 were observed, with some overlap, resulting in a coverage of 3 deg², as shown in Figure 2. The equivalent exposure times were 150 s in J and 270 s in K_s . (Details of the dithering scheme are given in Cioni et al. 2008.) The data were taken in queue mode on (UT) 2005 September 29–30, October 24, November 5, and December 16. The data were reduced through the standard Cambridge Astronomy Survey Unit (CASU) pipeline, ⁸ which produces a chip-by-chip source list with J and K_s values (Irwin et al. 2004) calibrated from 2MASS along with the source classification (i.e., stellar versus nonstellar) and kindly provided to us by the program's P.I., Michael Irwin. For our analysis we followed the recommendations of the CASU documentation and employed their photometry using a 30 radius aperture; these fluxes had been corrected for sky using a locally interpolated value from their two-dimensional background-following algorithm.

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In order to make sense out of the 160 source lists (4 arrays × 4 pointing per tile × 5 tiles in each of the two bandpasses), we first combined all of the measurements on a filter-by-filter basis. We then averaged the photometry for multiple measurements of the same star. The averaged J and K_s photometry was combined, requiring agreement in the coordinates to 08 (2 pixels). In doing so, we paid careful attention to the classification flags. These are produced by the CASU pipeline, with values denoting "stellar," "nonstellar," "noise," "marginal stellar," and "saturated." ⁹ We found that for objects with multiple detections, there was not good consistency between the values assigned to the same object. After much experimentation, we found that we obtained the most satisfactory source list if we required that an object be classified as "stellar" or "marginally stellar" only in one of the two filters.

Although the original source lists were nominally calibrated with respect to 2MASS photometry, a comparison of our final source list with that of 2MASS showed that small corrections were needed to bring them into good accord; we adjusted the CASU J magnitudes by +0.023 mag and K_s by −0.019 mag. These values are comparable to what were found in the analysis of our own UKIRT WFCam data on M31 (Neugent et al. 2020b). Similar to that study, stars brighter than J = 12 or K_s  = 12 began to deviate systematically from the 2MASS values, although they were not flagged as saturated. Thus, we removed these stars and added in any 2MASS stars in the same region that were not included in the edited UKIRT photometry catalog. The final photometry list included 121,328 stellar objects. As with M31, we expect only a small fraction to be in the color–magnitude range of RSGs and pass our membership criteria.

We show the photometric errors as a function of magnitude in Figure 3. M33 is a "cleaner" field than that of M31, without a large bulge, and with considerably less crowding; hence, the photometric errors are smaller, in general. Again, we see that the majority of our photometric errors are <0.02 mag to our faint limit.

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Prior to the availability of the high-quality proper motions and parallaxes that Gaia provides, the only way to definitively differentiate between foreground Galactic red stars and M31/M33 RSGs was by the use of radial velocities (see, e.g., Massey 1998; Drout et al. 2009; Massey et al. 2009; Massey & Evans 2016). Even so, this method was not effective for stars in the northeast quadrant of M31 as there was too much overlap there between the expected radial velocities of the two groups due to M31's rotational velocity. In addition, stars in the Galactic halo will have radial velocities that are similar to the systemic velocities of these galaxies, as much of that motion is actually the reflex motion of the Sun (see, e.g., Figure 5 in Courteau & van den Bergh 1999), making it hard to weed out the occasional Galactic halo red giant star. Broadband optical photometry also provides a clue, as there is a relatively clean separation between dwarfs and supergiants in B − V at a given V − R (Massey 1998) due to the effects of surface-gravity-dependent line-blanketing in the blue, although it is clear from radial velocity studies that the method is not completely reliable (see Figure 3 in Massey & Evans 2016). Fortunately, Data Release 2 (DR2) of Gaia (Gaia Collaboration et al. 2018a) has revolutionized this process.

Nevertheless, the use of DR2 is not completely straightforward; to do this properly requires taking spatially dependent zero-point issues into account as well as considering the intrinsic dispersion in parallaxes and proper motions within a sample. As with previous applications, we have followed a process similar to that of Gaia Collaboration et al. (2018b), using a covariance analysis to separate members from nonmembers (Aadland et al. 2018; Neugent et al. 2020b), having first established the distributions of parallaxes and proper motions using stars whose Gaia magnitudes and colors make it highly likely that they are members. After cleaning the sample, we find average values for the parallax of −0.033 mas, and 0.028 mas yr⁻¹ and −0.014 mas yr⁻¹ for the proper motion in R.A. and decl., respectively, for M31. For M33, the values are −0.081 mas for the parallax, and 0.103 and 0.034 mas yr⁻¹ for the proper motions. With our expected values of proper motions and parallax defined from the median of this sample, we then compare the Gaia measurements for all of the stars in our sample against these values. A star is considered to be foreground if its proper motion and parallax fall beyond the region that contains 99.5% of the comparison sample, while a star is considered to be a definite member if it falls within 75% of the comparison sample. The membership is considered to be uncertain if the star's value falls between this range.

However, the Gaia Collaboration et al. (2018b) methodology makes sense really only if the membership scatter is dominated by intrinsic dispersion and not by the uncertainties in the individual values. For our M31/M33 sources, that is not the case. Consider simply the proper motions. Massive stars in M31 have radial velocities that extend from nearly −600 to −50 km s⁻¹ due to the galaxy's rotational velocity (see, e.g., Figure 2 in Massey & Evans 2016). Of course, the space motions perpendicular to the radial velocity will be much smaller than this because M31 is seen mostly edge on (inclination 77°, according to van den Bergh 2000). Nevertheless, even if the intrinsic dispersion in the plane of the sky were 500 km s⁻¹, at the distance of M31 (760 kpc; van den Bergh 2000), this translates to an angular proper motion dispersion of 0.14 mas yr⁻¹. By comparison, typical uncertainties in the proper motions in our sample are 1–2 mas yr⁻¹, an order of magnitude larger. In other words, the intrinsic scatter is negligible compared to the individual measurement uncertainties. This is even more true for the parallaxes, where a 20 kpc long galaxy arranged along the line of sight at the same distance of M31 would have an intrinsic dispersion of 3.5 × 10⁻⁵ mas, compared to a typical uncertainty in the parallax measurements of our stars of 1 mas.

That being the case, we applied an additional test for stars which the covariance procedure flagged as nonmembers. If both the parallax and proper motions were within 2.5σ of the expected values (where σ is measurement uncertainty), then our classification was changed from "non-member" to "uncertain." We have color-coded the resulting membership determinations in Figures 4 and 5. We have also used the Besançon Galactic model (Robin et al. 2003) to simulate the expected distribution of the foreground at the corresponding Galactic longitude and latitudes and covering an area of the same size as our surveys; we see that there is excellent agreement in where in the CMD we find foreground contamination.

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The distribution of membership in the CMDs is very similar for the two galaxies. There is a virtual "wall" of foreground stars (shown in red) at J − K_s  ∼ 0.8−0.9. These colors correspond to late-K and early-M dwarfs (Table 7.6 of Cox et al. 2012). At magnitudes fainter than K_s  ∼ 16−16.5, the members (shown in black) and foreground stars begin to blend together, and there is an increasing number of stars with ambiguous results (green). Fortunately, this corresponds to $\mathrm{log}L/{L}_{\odot }$ of 4, below which we lose interest. The triangle of fainter (K_s  < 15.5), redder stars (centered at J − K_s  ∼ 1.5) mostly lack Gaia data (as indicated by blue), but these are the stars that we expect to be AGBs (see, e.g., Neugent et al. 2020b.) The comparison with the Besançon models confirms that although most of these fainter stars lack Gaia data, we are justified in our interpretation that these are indeed members.

Following our work in the LMC (Neugent et al. 2020a) and a small portion of M31 (Neugent et al. 2020b), we now identify the region in the CMD where RSGs are found, as shown in Figures 6 and 7. Our ability to separate the RSGs from AGBs stems from the fact that the Hayashi limit shifts to cooler temperatures at lower masses (Hayashi & Hoshi 1961); thus, the AGBs, which are lower in mass than the RSGs, are found at redder colors. At the same time, the Hayashi limit shifts to cooler temperatures at higher metallicities. Thus, there is more overlap between the RSGs and AGBs in M31 at faint magnitudes than in M33. We give the adopted boundaries in Table 2, where we have retained the K₀ and K₁ notation of Cioni et al. (2006a).

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Table 2. Adopted and Derived Relations

	Relation	Source
Adopted Distance:
	M31: 760 kpc (DM = 24.40 mag)	1
	M33: 830 kpc (DM = 24.60 mag)	1
Reddening Relations:
	AK  = 0.12AV  = 0.686E(J − K)	2
	E(J − K) = AV /5.79	2
RSG Photometric Criteria:
M31:	15.5 < Ks  ≤ 17.0: Ks  ≥ Ks0 and Ks  ≤ Ks1	3
	Ks  ≤ 15.5: J − Ks  ≥ 0.984 and Ks  ≤ Ks1	3
	Ks  ≤ 14.4 and (J − Ks ) ≤ 1.9: J − Ks  ≥ 0.984	3
	Ks0 = 28.62 − 13.33(J − Ks )	3, 4
	Ks1 = 31.95 − 13.33(J − Ks )	3, 4
M33:	15.0 < Ks  ≤ 17.0: Ks  ≥ Ks0 and Ks  ≤ Ks1	3
	Ks  ≤ 15.0: J − Ks  ≥ 0.963 and Ks  ≤ Ks1	3
	Ks  ≤ 14.6 and (J − Ks ) ≤ 1.9: J − Ks  ≥ 0.917	3
	Ks0 = 28.04 − 13.542(J − Ks )	3, 4
	Ks1 = 31.56 − 13.542(J − Ks )	3, 4
Adopted Extinction:
	Ks  > 14.5: AV  = 0.75	3
	Ks  ≤ 14.5 and Ks  ≤ Ks1: AV  = 0.75 − 1.26 × (Ks  − 14.5)	3
	Ks  ≤ 14.5 and Ks  ≥ Ks1: AV  = 0.75 + 5.79 × Δ(J − Ks )	3
	M31: Δ(J − Ks ) = (J − Ks ) − (30.29 − Ks  + 0.686(J − Ks ))/14.02	3
	M33: Δ(J − Ks ) = (J − Ks ) − (30.14 − Ks  + 0.686(J − Ks ))/14.23	3
Conversion of 2MASS (J, Ks ) to Standard System (J, K):
	K = Ks  + 0.044	5
	J − K = (J − Ks  + 0.011)/0.972	5
Conversion to Physical Properties (Valid for 3500–4500 K):
	M31: Teff = 5643.5 − 1807.1(J − K)0	3
	M33: Teff = 5606.6 − 1713.3(J − K)0	3
	BCK  = 5.495 − 0.73697 × Teff/1000	3
	K* = K − AK	⋯
	Mbol = K* + BCK  − DM	1
	$\mathrm{log}L/{L}_{\odot }=({M}_{\mathrm{bol}}-4.75)/-2.5$	⋯

References. (1) van den Bergh (2000); (2) Schlegel et al. (1998); (3) This paper; (4) Cioni et al. (2006a); (5) Carpenter (2001).

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The K₁ line denotes the equation separating RSGs from AGBs. However, for brighter stars, we have expanded the allowed region to redder colors as there are clearly some stars in this region, and yet these stars are much brighter than expected for even the brightest AGB stars. (See discussion in Neugent et al. 2020a.) As described below, we assume these stars are bright RSGs with extra (presumed) circumstellar reddening.

In the Yang et al. (2019) study which pioneered this technique to identify the SMC's RSG population, they defined the low J − K_s cutoff line (K₀) as simply parallel to the K₁ line. We followed a similar procedure in Neugent et al. (2020b) in our study of two M31 fields. However, we have since come to realize that this has no physical basis and runs the risk of excluding K-type RSGs at brighter magnitudes. Furthermore, it tends to remove any RSG binaries from the sample, as these will have their colors affected by their bluer companions. Thus, in our analysis of the LMC RSG population (Neugent et al. 2020a), we used a constant J − K_s cutoff except for the faintest stars, which are still likely contaminated by foreground stars. We have followed a similar procedure here.

Having made our selection of RSGs based on this CMD, we then performed some manual editing. Visual inspection of the Local Group Galaxies Survey (LGGS) R-band images (Massey et al. 2006) was used to eliminate obvious galaxies, a few of which were misidentified as potential RSGs. The CFHT data also included M32 and NGC 205, dwarf elliptical companions to M31, visible in Figure 1 to the south and northwest, respectively. We eliminated any spurious detections in the nearby area, i.e., in the rectangular areas α = 00:42:28.6 to 00:42:55.0, δ = 40:49:25 to 40:54:25 (M32), and α = 00:39:55.0 to 00:40:40.0, δ = 41:35:10 to 41:48:15 (NGC 205). There were also numerous detections near the bulge of M31 which were spurious due to the extra noise introduced by the background, and these were also eliminated from our list; we excluded the rectangular region α = 00:42:30.0 to 00:43:00.5 and δ = 41:13:30 to 41:19:00. M33 had far fewer problems.

The list of RSGs is given in Tables 3 and 4 for M31 and M33, respectively. We include the IDs, V magnitudes, and any spectral information from the LGGS, using the most recent version (Massey & Evans 2016). There are 38,817 photometrically identified RSGs in the M31 list and 7088 in the M33 list. Of these, 31,444 and 4819 are within their galaxy's Holmberg radius. As we emphasize below, one must take care to use proper luminosity cuts in discussing the numbers of RSGs. Of these, 238 had previously been identified by the LGGS as RSGs in M31, while 197 had been identified as RSGs in M33. Additional spectroscopy of both galaxies will be discussed in the companion binary frequency paper (Neugent 2021). Prior to finalizing the lists, we used the matches with the LGGS to remove several QSOs that had been identified in Massey et al. (2019). In addition, both lists originally contained several WN-type Wolf–Rayet stars; presumably, their strong emission lines affected their J − K_s colors, causing them to appear red. We removed these, with the exception of J004453.06+412601.7, which is described in the LGGS as "WN3+M:." Whether or not this star is a line-of-sight coincidence between a WR star and an M dwarf, or an actual WR+RSG binary is discussed in the binary paper (Neugent 2021). We also removed two emission-lined LBVs from the lists for each galaxies. There are a handful of remaining stars whose classifications do not agree with their being RSGs: 32 of the M31 photometrically selected RSGs have previously been classified as OB stars; there are 8 such stars in M33. In some cases, these are listed as having M-type companions; it is possible that the others are binaries as well. These need to be rechecked spectroscopically.

Table 3. Photometrically Selected RSGs in M31

M31 RSG#	α2000	δ2000	ρa	Ks	${\sigma }_{{K}_{s}}$	J − Ks	${\sigma }_{J-{K}_{s}}$	#obs	Gaia b	AV	Teff c	$\mathrm{log}L/{L}_{\odot }$ d	LGGS
												ID	V	Type
6832	00 40 50.843	+41 06 48.30	0.61	17.368	0.038	1.080	0.052	1	2	0.75	3850	3.67	⋯	⋯	⋯
6833	00 40 50.845	+41 45 49.99	1.61	15.420	0.012	1.197	0.030	2	2	0.75	3650	4.38	⋯	⋯	⋯
6834	00 40 50.846	+41 05 40.98	0.58	14.793	0.004	1.031	0.006	1	0	0.75	3950	4.72	J004050.87+410541.1	19.72	RSG
6835	00 40 50.865	+40 39 46.27	0.47	17.260	0.034	1.093	0.048	1	2	0.75	3850	3.70	J004050.84+403946.3	22.07	⋯
6836	00 40 50.874	+40 46 21.81	0.39	17.260	0.037	1.077	0.148	1	2	0.75	3850	3.71	J004050.87+404621.6	22.59	⋯

Notes.

^a Galactocentric distance. Assumes a Holmberg radius of 953, inclination 770, and a position angle of the major axis of 350. At a distance of 760 kpc, a ρ of 1.00 corresponds to 21.07 kpc. ^b Gaia flag: 0 = probable member, 1 = uncertain membership; 2 = no Gaia data. ^c Typical uncertainty 150 K. ^d Typical uncertainty 0.05 dex.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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By far the most uncertain part of this process is what to assume for the extinction toward these stars. In general, studies of OB stars in both M31 and M33 have revealed only modest reddening. In many ways, this is most surprising for M31, given that the galaxy is inclined 77° (van den Bergh 2000) to our line of sight. Spectroscopy and photometry of stars in four OB associations of M31 by Massey et al. (1986) showed color excesses of (B − V) ranging from 0.08 to 0.24, i.e., A_V  = 0.25–0.75. Using the LGGS photometry, Massey et al. (2007) used the "blue plume" of OB stars to estimate the average reddening of OB stars M31, finding a very similar value, E(B − V) = 0.13 (A_V  = 0.4). The same method yielded essentially identical results for M33. By contrast, an analysis of the Panchromatic Hubble Andromeda Treasury (PHAT) photometry by Dalcanton et al. (2015) suggest an average extinction of A_V  ∼ 1 for stars in the disk of M31. We discuss this issue further below, in Section 5.

The reddening situation is further complicated by the fact that RSGs are known to make their own dust, and in general show circumstellar extinction in excess of that of OB stars in the same region (Massey et al. 2005). Model fitting of a sample of M31's RSGs by Massey et al. (2009) yielded direct measurements for A_V for a sample of RSGs in M31. Neugent et al. (2020b) noted that these results showed a luminosity dependent on the amount of extinction and adopted A_V  = 0.75 for stars fainter than K = 14.5, but a linear increase in A_V for stars brighter than that. ¹⁰

They found that such a procedure was not only consistent with the handful of RSGs with direct A_V measurements but also resulted in much better agreement with the evolutionary tracks than would have occurred by adopting a constant A_V (see their Figure 10). Physically, this also makes sense: RSGs above some mass limit (∼20M_⊙) flirt with the Eddington limit during their evolution (Ekström et al. 2012), resulting in large episodic mass loss.

We have used the same procedure here; the relevant relations are given in Table 2. In addition, allowing for very red stars (J − K_s  ∼ 1.5) to be included in the RSG list as long as they were brighter than that expected for AGBs meant that Neugent et al. (2020a) had to apply an additional reddening correction for those, with the assumption that their intrinsic colors were similar to that of other RSGs. Thus, their location in the color excess in J − K_s was computed by moving along a reddening line to the center of the RSG distribution. This resulted in a small number of bright stars having a larger correction for extinction. We adopt similar relations for M33. The relevant equations are given in Table 2. We include the adopted A_V values along with our photometry in Tables 3 and 4.

Table 4. Photometrically Selected RSGs in M33

M33RSG#	α2000	δ2000	ρa	Ks	${\sigma }_{{K}_{s}}$	J − Ks	${\sigma }_{J-{K}_{s}}$	#obs	Gaia b	AV	Teff c	$\mathrm{log}L/{L}_{\odot }$ d	LGGS
												ID	V	Type
854	01 32 19.96	+30 37 18.6	1.04	17.114	0.024	0.981	0.038	4	2	0.75	4100	3.92	⋯	⋯	⋯
855	01 32 20.03	+30 34 18.0	1.01	16.874	0.019	0.922	0.030	4	0	0.75	4200	4.04	J013220.05+303418.1	20.76	⋯
856	01 32 20.32	+30 29 37.9	0.98	16.229	0.011	0.924	0.018	4	0	0.75	4200	4.30	J013220.36+302938.0	19.69	RSG
857	01 32 20.69	+31 21 52.2	2.23	16.124	0.013	0.884	0.019	2	0	0.75	4250	4.36	⋯	⋯	⋯
858	01 32 20.95	+30 24 16.2	0.98	17.397	0.031	0.856	0.051	4	2	0.75	4300	3.87	⋯	⋯	⋯

Notes.

^a Galactocentric distance. Assumes a Holmberg radius of 308, inclination 560, and a position angle of the major axis of 230. At a distance of 830 kpc, a ρ of 1.00 corresponds to 7.44 kpc. ^b Gaia flag: 0 = probable member, 1 = uncertain membership; 2 = no Gaia data. ^c Typical uncertainty 150 K. ^d Typical uncertainty 0.05 dex.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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One of the great advantages in conducting our study using NIR photometry is that we are less sensitive to reddening than would otherwise be the case. The extinction at K is only 12% that of V. Thus, even a 1 mag uncertainty in A_V translates to an uncertainty of 0.12 mag in A_K . The impact this has on the luminosity determination must take into account the uncertainty in J − K_s and hence in the effective temperature as well. Using the relations in Table 2, we find that ΔE(J − K) = 1.46ΔA_K . For an uncertainty of 0.12 mag in A_K , this translates to an uncertainty of 0.18 mag in E(J − K). This would change the temperature by about 300 K and the bolometric correction by 0.23 mag. Thus, the overall effect on the luminosity would be 0.15 dex.

There are several steps needed to transform the (J − K_s , K_s ) photometry into effective temperatures and bolometric luminosities. First, the photometry must be transformed from the 2MASS system to the more standard Bessell & Brett (1988) system, as these will provide the connection with effective temperatures via stellar atmosphere models. The transformation equations from 2MASS to the standard system are given by Carpenter (2001) and repeated in Table 2. Next, the photometry is corrected for reddening and extinction based upon the adopted A_V values derived as explained above. The relationships between A_V , A_K , and E(J − K) come from Schlegel et al. (1998) and are also summarized in Table 2.

The basic transformation equations between these intrinsic colors and the desired values for the effective temperatures and bolometric corrections are based upon the MARCS stellar atmospheres (Gustafsson et al. 1975, 2008; Plez et al. 1992) computed at various metallicities specifically for the low surface gravities of RSGs by B. Plez in our collaborative studies of RSGs in the Milky Way (Levesque et al. 2005), Magellanic Clouds (Levesque et al. 2006), and M31 (Massey et al. 2009). A linear equation between intrinsic (J − K)₀ colors and effective temperature works very well for RSGs. There are small differences associated with the metallicity of the models. Following Neugent et al. (2020b), for M31 we adopted a compromise between the solar-metallicity and 2× solar-metallicity models, as a value of 1.5× solar is more in keeping with recent determinations of the metallicity of a young stellar population (Sanders et al. 2012), with evidence of little or no gradient (see also Zaritsky et al. 1994). M33 has a stronger metallicity gradient, but the absolute values are very poorly established. Thus, we found there was little benefit in providing a correction based on galactrocentric distance. Instead, we adopted the same transformation equation as used in our recent LMC study (Neugent et al. 2020a), corresponding to 0.5× solar, as that is fairly representative of studies of OB stars (Urbaneja et al. 2005). (For a few of the differing views of M33's abundances, see Kwitter & Aller 1981; Zaritsky et al. 1994; Magrini et al. 2007 and Bresolin 2011.)

By contrast, we find that there is negligible dependence on metallicity in the equation for the bolometric correction (BC_K ). Again, the equation is given in Table 2. Unlike the V-band BCs that many astronomers are familiar with, the K-band BCs are a positive quantity. As they are closer to the peak flux of RSGs, K-band corrections also change more slowly than V-band corrections with respect to effective temperature, again minimizing the errors we make by doing our analysis in the NIR.

Finally, we combined the dereddened K values with the BC_K and the true distance modulus (assumed to be 24.40 for M31 and 24.60 for M33, following van den Bergh 2000), yielding the bolometric magnitude. We then converted these to bolometric luminosities relative to the Sun, i.e., $\mathrm{log}L/{L}_{\odot }$.

The resulting effective temperatures and luminosities have uncertainties of about 150 K and 0.05 dex, respectively, where we have done error propagation based not only on the photometric errors but also on an uncertainty of 0.5 mag in the adopted values for A_V .

The use of a linear relationship with K_s for determining A_V results in a good match with the evolutionary tracks as we see in the next section. However, it also has a drawback: a dozen M31 stars have unlikely large extinction values (A_V  = 5 − 10) and unrealistically high luminosities. Several of these stars have ambiguous Gaia results, but most are considered probable members despite their very bright K_s 2MASS values (K_s  = 6 − 9). We have retained these stars in the list. Although they are unlikely to be bona fide RSGs, they are probably quite interesting IR sources. None of these overly bright stars are within the field of the LGGS, suggesting they lie well away from regions of star formation. It is possible that in some cases, the membership assessment is wrong. There is no similar problem for M33.

We note that of the 38,817 photometrically identified RSGs in the M31 list and 7088 in the M33 list, only 7585 and 3911 (respectively) have luminosities $\mathrm{log}L/{L}_{\odot }\geqslant 4.0$. Of these, 6412 and 2858 are within their galaxy's Holmberg radius.

In Figure 8, we show the location of the RSGs in the H-R diagram along with the evolutionary tracks from the Geneva group. For M31, we have used the tracks of Ekström et al. (2012) computed for solar metallicity (z = 0.014). Although a higher metallicity would be more appropriate for M31, such tracks are not yet available. For M33 we have used the unpublished z = 0.006 models, kindly made available to us by Cyril Georgy, Sylvia Ekström, and Georges Meynet. This metallicity corresponds roughly to that of the LMC and is as appropriate as anything as discussed above. Although the Geneva models are computed only for single-star evolution, we are using them as the standard for our comparisons as they include the effects of rotation, which is important particularly at subsolar metallicity, and as their new prescription for the mass-loss rates during the RSG phase have been vetted against observations (Neugent et al. 2020b). Binary evolution has little effect on the location of RSGs in the H-R diagram; this is nicely illustrated in the comparison between the single-star Geneva and BPASS binary evolutionary tracks (Eldridge & Stanway 2016; Eldridge et al. 2017) in Figure 8.3 of Levesque (2017). (See also Figure 7 of Levesque 2018.)

The most striking thing about Figure 8 is how superb the agreement is between the tracks calculated by stellar evolution theory, and the effective temperatures and luminosities derived from our photometry and the MARCS stellar atmospheres. The upturn and vertical evolution shown by the tracks is due to the increase in the mean particle mass μ in the stellar cores as He is converted into C and O; the luminosity is a steep function of μ (see discussion in, e.g., Levesque 2017). Where this happens in terms of effective temperature is dependent upon many things, but particularly the treatment of convection (see, e.g., Figure 9 in Maeder & Meynet 1987). ¹¹ At the same time, the calculation of the spectral energy distribution in the stellar atmosphere calculations for these bloated stars is fraught with approximations as well. Thus, this agreement between these two complex models seems absolutely remarkable.

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The redwards extent of the tracks shift to higher effective temperatures at larger luminosities: at $\mathrm{log}L/{L}_{\odot }$ = 5.5 (∼30M_⊙), these stars are barely cool enough to be called "red" supergiants. This shift is is mostly due to the effects of mass loss during these late stages. That is why for the solar-metallicity tracks in M31, the 32M_⊙ stops at $\mathrm{log}{T}_{\mathrm{eff}}\sim 3.77$ (roughly the temperature of the Sun!)—mass loss causes the evolution to proceed back to higher temperatures. (See discussion in Ekström et al. 2012.) At the lower metallicities shown in the tracks for M33, even the 40M_⊙ will produce RSGs. Our data are consistent with these results, although the high-extinction stars mentioned above makes this a little less clean for M31 as we would care. Additional spectroscopy will resolve this. We note that without our luminosity-dependent extinction correction, such agreement would be poorer, as shown in Figure 10 of Neugent et al. (2020b).