4 Results

4.1 Colour-magnitude diagram

Figure 3 shows the CMD for all 5448 stars of sample 1, which matched the above mentioned quality criteria. One can see the upper three magnitudes of the sequence of RGB and AGB stars, above this are bright red supergiants (RSG). At $(V-i)_0\approx 0^{\rm mag}$ one can see a component of relatively blue stars. This arises from the fact, that our field lies "near" a spiral arm of M 31, where star formation takes place. Field3 of Brewer et al. (1995) lies in the spiral arm and shows an even stronger blue component. Field 5 of this paper gives also an estimate of an upper limit of the galactic foreground contribution, since it lies more than 32kpc from the center of M 31. It turns out, that the foreground contributes nothing for (V-i)₀ bluer than about 1.

Figure 3: Colour-magnitude diagram for all stars of sample 1. Also plotted is a 1 kpc-reddening-vector for absorption within the disc of M 31 (as described in the text)

   
4.2 Colour-colour diagram

For the 2546 stars of sample 2, which matched our quality criteria, a colour-colour diagram (CCD) is drawn in Fig. 4. The bifurcation of $\approx$1 $^{\rm mag}$ on the righthand side, due to the separation of the M- and C-stars, is obvious. Photometric variability and a varying amount of M 31 internal reddening could contribute to the scatter of these two sequences.

We defined the selection criteria for M/C-stars as follows: The border in (V-i)₀ was chosen in accordance with our synthetic photometry (see Fig. 2). We determined a scatter for all early stars [ $(V-i)_0 < 0.7^{\rm mag}$] in (TiO-CN)₀ with $\sigma=0.2^{\rm mag}$. The criteria for (TiO-CN)₀ were then chosen in order to separate the areas for M/C-stars by 2.5$\sigma$ of this scatter. Brewer et al. (1995) confirmed these latter selection criteria with explicit spectroscopic observations.

The following selection criteria for M/C-star candidates were used (and are drawn in the CCD of Fig. 4):

$\bullet$ M: (V-i)₀>1.62 $^{\rm mag}$, (TiO-CN) $_0>0.15^{\rm mag}$,
$i_0>18.5^{\rm mag}$ (no RSGs, galactic contamination),
$M_{\rm bol}< -3.5^{\rm mag}$ (tip of the RGB, see Sect. 4.3)
$\bullet$ C: (V-i)₀>1.62 $^{\rm mag}$, (TiO-CN) $_0< -0.3^{\rm mag}.$

Stars with (V-i)₀>1.62 $^{\rm mag}$ and (TiO-CN)₀ between the selection areas could be candidate S-stars (spectroscopically confirmed by Brewer et al. 1996). The criteria must be kept constant for investigations of further fields or galaxies, in order to be consistent in comparing the numbers of selected stars.

Figure 4: Colour-colour diagram for the stars of sample 2, and the selection areas for M- and C-stars (as described in the text). Synthetic photometry is plotted as well (M-stars with open circles, C-stars with asterisks). Again a 1 kpc-reddening-vector for absorption is plotted (as described in the text)

Figure 5 shows a CMD for the stars of sample 2. The 286 M-stars and 47 C-stars from the selection areas of Fig. 4 are drawn with open circles and asterisks, respectively. They lie on the upper end of the RGB- and the AGB-sequence. Moreover, the identified Cepheids are indicated by filled squares.

Figure 5: Colour-magnitude diagram for all stars of sample 2. All stars in the selection areas (M-stars with open circles, C-stars with asterisks) and the identified Cepheides (filled squares) are indicated. In the upper right corner the 1 kpc-reddening-vector is plotted (as described in the text)

4.3 Luminosity function

The bolometric LF is one of the important concepts used for comparing AGB models with observations. It gives clues to many topics (star formation history (SFH), stellar evolution, inital mass function, mass loss rates, ...) and led to the refinement of the AGB theory in the past (e.g. bright C-star mystery $\rightarrow$different mass loss laws, semi-convection, overshooting, hot bottom burning, ...).

To compare our results with those of Brewer et al. (1995), we calculated a bolometric LF for the AGB star candidates of sample 1, which were selected using the following criteria:

$\bullet$ (V-i)₀ > 1.62 $^{\rm mag}$
$\bullet$ i₀ > 18.5 $^{\rm mag}$
$\bullet$ -7 $^{\rm mag}$ > $M_{\rm bol}$ > -1 $^{\rm mag}$.

To calculate the absolute bolometric magnitude $M_{\rm bol}$ of late-type stars we used a distance modulus for M 31 of $(m-M)=24.43^{\rm mag}$ (Freedman & Madore1990) and the bolometric correction (BC) given by Bessel & Wood(1984):

$M_{\rm bol} = i_0 + BC - 24.43$
$BC = 0.3 + 0.38\cdot(V-i_{\rm C})_0 -0.14\cdot(V-i_{\rm C})_0^2$.

This Cousins-system BC is valid for M-stars, which have 1<(V-I)<6. It was - in accordance with Brewer et al.(1995) - used for C-stars too, although it is not fully correct in this case. We can use this Cousins-irelation, because our Gunn-i observations are calibrated to the Cousins-system (see Sect.3.2).

By comparing some characteristics of the diagrams (blue component, Cepheids and RSGs in the CMD; number of bright AGB stars and steepness of the LF) of different fields, one can draw conclusions for the SFHs (see e.g. Grebel 1999). Figure 6 shows the bolometric LF for all AGB candidates. A comparison of it with Fig. 9 of Brewer et al.(1995) reveals, that it is very similar to the LFs of their fields 3 and 4, for which active SF is concluded for the last few Gyrs.

Figure 7 shows the i₀-LF of the selected M/C-stars (of Fig. 4), where the histogram of the M-stars is scaled by a factor of 0.1 to simplify the comparison. C-stars have a lower mean i₀-magnitude, which may reflect the fact, that stars can change their spectral type from M to C by the third dredge-up, while they ascend the AGB. The bolometric LF of the C-stars (calculated with a BC, which we derived from data of C-stars in Bessel & Wood 1984) seems to be consistent with the one of Groenewegen (1999).

Figure 6: Bolometric LF for the AGB candidates of sample 1, selected as described in the text

Figure 7: LFs for the selected M- (286 stars) and C-stars (47); the former was multiplied by a factor of 0.1 to simplify the comparison

4.4 C-stars

Taking into account that (due to the short evolutionary stage) optically detectable C-stars have a relatively small scatter in their i-magnitudes, they can be used to estimate the distance of the population. The results of Brewer et al.(1995) showed that neither different star formation histories nor different metallicities strongly influence the mean i-magnitudes of a population of C-stars, which makes them a useful standard candle. The observations of Richer (1981) and Richer et al. (1985) resulted in a mean absolute i-magnitude of $\langle M_{I}\rangle =-4.75\pm 0.47^{\rm mag}$ for a sample of LMC C-stars. Combining this with the mean apparent magnitude $\langle m_{I}\rangle=19.96\pm 0.4^{\rm mag}$ of the above selected C-star candidates (Fig. 7) results in a distance modulus of $(m-M)=24.71\pm 0.62^{\rm mag}$, which is in agreement with other distance determinations, e.g. the one by Freedman & Madore (1990).

   
4.5 Stars without V photometry

For our observational test run the exposure time of the V frames was too short. Therefore, we have 515 red stars, which are too weak in the V band to have reliable measurements. However, we can still use photometry in the other three filters to select AGB candidates, as Fig. 8 demonstrates. The lower limit in i can be choosen for example as the magnitude of the tip of the RGB ( $i_0=20.8^{\rm mag}$, see Brewer et al. 1995), while the upper limit is the magnitude of the end of the RGB/AGB-sequence in the CMD. The borders in (TiO-CN)₀ are the same as for the stars with photometry in all four filters. Doing this, we identify another 14 C- and 221 M-star candidates. This rough selection does not consider variable magnitudes (can be above 1 $^{\rm mag}$ in i on the AGB), but could be very useful for observations of very distant galaxies, as the observing time for the V-filter can be saved.

With all C/M-stars from the Sects. 4.2 and 4.5, we get a C/M-ratio of $\approx$0.12. This result is again similar to the "raw'' C/M-ratio of the neighbouring field 4 of Brewer et al. (1995).

Figure 8: Diagram to select AGB candidates from stars without V photometry. In the lower right corner the 1 kpc-reddening-vector for absorption within the disc of M 31 is shown