5 The luminosity function of the bulge

Figure 16: The bulge near-IR LF as obtained from the SOFI data.

The disk-decontaminated CMD shown in Fig. 5c has been used to construct the bulge luminosity function (LF) from the RGB tip down to $\sim$1 mag below the main sequence turnoff, in each of the three bands: J,H and $K_{\rm s}$, with the results being shown in Fig. 16. The number counts in the SOFI-SMALL field (J>16, J₀>15.61) have been normalized to those in the SOFI-LARGE field, according to the ratio of their areas.

For an easier comparison with the LFs of other objects, or different bulge regions, the LFs presented here are always shown as a function of dereddened magnitude. The adopted average E(B-V) is 0.45, as derived from the comparison with the fiducial loci of the CMD of NGC 6528 (Fig. 9). The relations between the absorptions in different bands have been assumed to be $A_V=3.1\times E$(B-V), $A_I=0.479\times A_V$, $A_J=0.282\times A_V$, $A_H=0.190\times A_V$, and $A_K=0.114\times A_V$ (Cardelli et al. 1989).

The broad peak at $J_0\sim13.2$ in Fig. 16 is the HB clump, slightly "bimodal'' because merged with the RGB bump, as already discussed in Sect. 4. The main sequence turnoff is located at $J_0=17.05\pm0.20$, as determined from Fig. 5. The steep decrease in the number of stars that could be expected just above the turnoff, is in fact smeared in this LF due to depth effect, differential reddening, metallicity dispersion, photometric errors and blending effects (see below).

Figure 17: The bulge LF from this work complemented with the one obtained in Paper II from NICMOS data.

Figure 17 shows the comparison, in J and H, between the SOFI LF and the very deep NICMOS LF from Paper II. A simple normalization by the ratio of the field areas brings the NICMOS LF in perfect agreement with the SOFI one, based on a field more than 100 times larger. Note that the disk contribution to the NICMOS LF was subtracted using Kent (1992) model LF of disk and bulge (cf. Paper II for details).

Figure 18: Comparison with previously published bulge LFs. Counts from different sources have been scaled according to the ratios of the field area and to the different surface brightness, if referring to another bulge region. Histogram: this work; filled squares: data from Tiede et al. (1995); open circles, data from Frogel & Whitford (1987); filled triangles: data from the 2MASS sky survey.

The upper panel of Fig. 18 shows the comparison between the SOFI LF and the LF by Tiede et al. (1995; TFT) in the same bulge region, but on a smaller area (4056 arcsec²). The disk contribution has been subtracted from the TFT star counts using the ratio between bulge and disk stars computed from Figs. 5a, b. The upper panel of Fig. 18 also shows the LF obtained by Frogel & Whitford (1987; FW in the figure label) for the M giants in Baade's Window. The latter has been normalized both for the different area and the surface brightness difference between the two fields.

The lower panel of Fig. 18 shows the comparison with the counts from the 2MASS survey, also normalized only for the ratio of the field areas. The 2MASS counts plotted here were extracted from a region of 927 arcmin², centered on the SOFI field. In the range where they are complete, i.e., for J₀<15 the 2MASS counts agree perfectly well with the SOFI counts, and are consistent with TFT and FW counts.

Figure 19: Comparison with the K-band LF by DePoy et al. (1993). Open symbols refer to counts affected by incompleteness at the faint end and by saturation at the bright end.

Finally, Fig. 19 shows the comparison with the K-band bulge LF from DePoy et al. (1993). The latter was obtained from the photometry of a 604 square arcmin fields towards Baade's Window, and included disk stars. The two LF where normalized according to the different field area and surface brightness, and the foreground disk contamination, estimated from the present data, has been subtracted from the Depoy LF.

   

Table 3: Complete near-IR bulge luminosity function^a.

J0	log NJ	H0	log NH	K0	log NK
7.98	-0.242	7.61	0.117	7.27	-0.008
8.23	-0.184	7.86	-0.008	7.52	0.168
8.48	-0.485	8.11	0.144	7.77	0.256
8.73	-0.133	8.36	0.293	8.02	0.418
8.98	0.027	8.61	0.504	8.27	0.469
9.23	0.621	8.86	0.526	8.52	0.418
9.48	0.469	9.11	0.536	8.77	0.481
9.73	0.547	9.36	0.594	9.02	0.557
9.98	0.594	9.61	0.805	9.27	0.776
10.23	0.566	9.86	0.794	9.52	0.800
10.48	0.811	10.11	0.962	9.77	0.904
10.73	1.007	10.36	1.062	10.02	0.926
10.98	1.027	10.61	1.138	10.27	1.144
11.23	1.136	10.86	1.149	10.52	1.095
11.48	1.128	11.11	1.210	10.77	1.260
11.73	1.279	11.36	1.352	11.02	1.250
11.98	1.352	11.61	1.383	11.27	1.428
12.23	1.415	11.86	1.447	11.52	1.416
12.48	1.633	12.11	1.553	11.77	1.500
12.73	1.623	12.36	1.733	12.02	1.568
12.98	1.857	12.61	2.003	12.27	1.784
13.23	2.068	12.86	2.075	12.52	2.006
13.48	2.013	13.11	2.090	12.77	2.085
13.73	2.143	13.36	2.114	13.02	2.121
13.98	2.114	13.61	2.100	13.27	2.188
14.23	1.959	13.86	1.959	13.52	1.991
14.48	2.033	14.11	1.982	13.77	1.996
14.73	1.944	14.36	1.892	14.02	1.978
14.98	2.068	14.61	2.072	14.27	1.919
15.23	2.140	14.86	2.111	14.52	2.033
15.48	2.188	15.11	2.083	14.77	2.090
15.73	2.412	15.36	2.487	15.02	2.179
15.98	2.756	15.61	2.665	15.02	2.187
16.23	2.790	15.86	2.818	15.27	2.422
16.48	2.957	16.11	2.918	15.52	2.740
16.73	3.130	16.36	3.071	15.77	2.813
16.98	3.228	16.61	3.191	16.02	2.916
17.23	3.376	16.86	3.320	16.27	3.080
17.48	3.518	17.11	3.478	16.52	3.218
17.73	3.666	17.36	3.598	16.77	3.285
17.98	3.612	17.61	3.676	17.02	3.504
18.23	3.591	17.86	3.737	17.27	3.661
18.48	3.676	18.11	3.803	17.52	3.760
18.73	3.764	18.36	3.890	17.77	3.802
18.98	3.814	18.61	3.977	18.02	3.826
19.23	3.854	18.86	4.013	18.27	3.904
19.48	3.901	19.11	4.008	18.52	3.926
19.73	3.952	19.36	4.033	18.77	3.971
19.98	4.001	19.61	4.094	19.02	3.903
20.23	4.039	19.86	4.138
20.48	4.075	20.11	4.162
20.73	4.141	20.36	4.213
20.98	4.219	20.61	4.290
21.23	4.246	20.86	4.335
21.48	4.232	21.11	4.348
21.73	4.262	21.36	4.385
21.98	4.343	21.61	4.433
22.23	4.400	21.86	4.417
22.48	4.410	22.11	4.356
22.73	4.401	22.36	4.361
22.98	4.397	22.61	4.432
23.23	4.413	22.86	4.452
23.48	4.449	23.11	4.386
23.73	4.492	23.36	4.290
23.98	4.535

^a Horizontal lines mark the boundaries between 2MASS, SOFI and NICMOS based data, from top to bottom.

From the combination of the data from 2MASS, SOFI, and NICMOS, a composite LF for the Galactic bulge can be constructed, using the best data for each luminosity range. The result is reported in Table 3, that lists the star counts for J, H, and $K_{\rm s}$ bands, while Table 4 lists the optical V and I LF. Note that NICMOS data are available only in the J and H bands. All the counts have been normalized to the area of $8\farcm3\times8\farcm3$ mapped by the SOFI-LARGE field: i.e., the 2MASS counts have been divided by 13.43, the SOFI-SMALL counts have been multiplied by 4.6, and the NICMOS counts have been multiplied by 609. These scaling factors can be used to calculate the Poissonian errors associated with the counts in each bin. The numbers in Table 3 have been corrected for both disk contamination and incompleteness, although the latter is only significant for the few faintest bins of the NICMOS LF, and it is always $\la$$50\%$.

5.1 Simulated CMD and theoretical LF

In order to compare our observations to corresponding theoretical predictions we have developped a simulator which generates the CMD of a stellar population with a single age and a wide metallicity spread. In this way, we neglect the presence of an age spread, which is justified since the location of the RGBs of (relatively) old stellar populations is much more sentitive to metallicity than to age. The code results from a development of the CMD simulator used in Greggio et al. (1998), adapted to describe single age Stellar Populations with a wide metallicity spread. A thorough description of it can be found in Rejkuba (2002, Ph.D. Thesis). We give here a short report, and specify the ingredients used in our simulations.

A Monte Carlo procedure is used to extract mass and metallicity of a simulated star, which gets then positioned on the H-R diagram via interpolation among isochrones. The determined bolometric magnitude and effective temperature are transformed into monochromatic magnitudes by interpolating within bolometric correction tables. Incompleteness and photometric errors, as measured on the real frames are applied. In particular this includes the brightening effect due to blending that was discussed in Sect. 2.1. The procedure is iterated until the number of objects observed in some CMD region is reached.

The isochrones data base consists of Cassisi & Salaris (1997) models, implemented with Bono et al. (1997), plus some additional models explicitly computed for this application with the code described in Cassisi & Salaris (1997) and Cassisi et al. (2002). The metallicity range goes from Z= 0.0001 to Z= 0.04, and the helium abundance varies in locksteps with Z following $Y\simeq 0.23+2.5Z$.

Theoretical bolometric corrections (BCs) have been obtained by convolving the model atmospheres computed by Castelli et al. (1997) with the Landolt V and I and the SOFI $J, H, K_{\rm s}$ filter passbands and fixing the zero point such that BC_V=-0.07 for the model of the Sun, and all the colors are zero for the model of Vega. However, the brightest stars in our sample are cooler than $T\sim4000$ K, limit below which current model atmospheres are known to fail to reproduce the observed spectra, due to the inappropriate treatment of molecules. Therefore, the empirical BCs by Montegriffo et al. (1998) where used instead of the theoretical ones for temperatures T $\la$ 4000 K. The dependence of the BCs on the metallicity parameter is taken into account in the simulations.

The results of the artificial stars experiments described in Sect. 2 have been used to assign the detection probability and the photometric error (i.e. the difference between input and output magnitude) to the simulated stars. The synthetic CMD has thus the same observational biases as that obtained from the measured frames.

A number of simulations have been computed for ages ranging from 8 to 15 Gyr, and by adopting:
(i) an IMF slope of -1.33, as found in Paper II;
(ii) the metallicity distribution derived in Sect. 4;
(iii) an average distance modulus of (m-M)₀=14.47 and reddening of E(B-V) =0.42. Each obtained synthetic CMD has been further dispersed for depth ( $\rho(r)\propto r^{-3.7}$) and variable reddening ($\Delta E$(B-V) $=\pm 0.15$) effects.

Figure 20 (left panel) shows the synthetic CMD for a 13 Gyr old population with the characteristics described above, and a comparison with the bulge near-IR CMD observed with SOFI is shown in the right panel.

The simulation code was run until the number of stars with M_J<1.2reached the number of stars sampled by SOFI-LARGE in this CMD region. Then it was run again until extracting as many stars with M_J>1.2 as in the SOFI-SMALL field.

The main purpose of the comparison in Fig. 20 is to show how much the simulation code is able to reproduce the observed CMD, including the observational biases introduced by the dispersions in distance, reddening and metallicity, as well as blending.

Figure 20 shows that indeed the main features of the observed CMD are well reproduced by the simulator. Noticeable exceptions are the morphology of the HB clump, and of the lower RGB. In fact, the observed HB is significantly less defined than the simulated one, and also the color width of the RGB is underestimated by about a factor of two in J-K. In principle, this mismatch can be caused by several effect, although none of them, alone, seems convincing to us. Larger spread in the observed CMD can be due to an underestimate of one of the following effects:

(i) Distance spread: the bulge density law could be flatter than $\rho\propto r^{-3.7}$, and/or the bulge being a bar may increase the distance dispersion along the line of sight. Note that adopting a larger distance dispersion would smear the HB clump, making is more similar to the observed one, but would not have any appreciable effect on the spread in color of the RGB. Hence, a larger distance spread alone would not be sufficient to make the simulated CMD identical to the observed one.

(ii) Differential reddening. Larger values of this parameter would broaden the RGB of the simulated CMD, but also would cause the HB to appear tilted along the reddening line, which is not seen in the observed CMD of the bulge, while it is prominent in e.g., NGC 6553 (Zoccali et al. 2001a). Also, the differential reddening needed to account for the spread in color seen in the near-IR CMD would imply a too large spread in the optical CMD. An attempt was made to correct for differential reddening following the method used for NGC 6553 (Zoccali et al. 2001a), but failed due to the small scale of the reddening variations across the field. Indeed, in the case of NGC 6553, one finds reddening variations of the order of $\delta E(V-I)\sim 0.1$on scales of only $\sim$20''.

(iii) Photometric errors. Larger SOFI-LARGE photometric errors than adopted in the simulation would certainly help smearing the simulated RGB and HB respectively in color and magnitude, but can hardly solve both problems at once. To smear enough the simulated HB the photometric error should be $\sim$0.3-0.4 mag, which would produce a too broad RGB compared to the observed CMD (see Fig. 20).

(iv) Problems in the theoretical models. The RGB temperatures (hence colors) are strongly dependent on the mixing-length parameter, which needs to be empirically calibrated, and there are no perfect calibrators. The models used in the present simulations were calibrated by fitting the RGB of template globular clusters from Frogel et al. (1983) for [Fe/H] $\la-0.7$, and at solar metallicity by demanding to the solar model to have the solar radius. Ideally, it would have been preferable to have a homogeneous calibration over the whole metallicity range, but the solar metallicity clusters would have introduced in the calibration the uncertainly in their reddening. As a net result, the adopted mixing length is some $10\%$ larger at [Fe/H] = 0 than at [Fe/H] $\la-0.7$, which has the effect of compressing somewhat the relative spacing of the RGBs as a function of metallicity. This effect goes indeed in the direction of reducing the color dispersion in the simulated RGBs. Moreover, the simulation is also affected by the uncertainty in the bolometric corrections and color-temperature transformations. All in all, Salaris et al. (2002) estimate an uncertainty of $\sim$0.10-0.15 mag in the range of optical colors spanned by theoretical RGBs, which can also be taken as indicative of the uncertainty in the range of the near-IR colors.

Figure 20: Left panel: simulated CMD for a 13 Gyr old population with the MD determined in Sect. 4. Right panel: the bulge CMD for the SOFI-LARGE field.

While the origin of the dicrepancy of Fig. 20 is still partly unclear, we proceed to the construction of the theoretical LF from the simulated CMD, keeping in mind that the region around the HB and lower RGB is presently not well reproduced by our simulations. The same code with the same inputs was then used to generate a much larger number of stars compared to the simulation shown in Fig. 20, in order to construct a smooth LF from the upper RGB down to the limit of the NICMOS photometry and the result is shown in Fig. 21. The simulation includes photometric error and blending effects, and is meant to match the observed LF after correction for incompleteness.

 

 

Table 4: Optical bulge luminosity function.

V0	log NV	I0	log NI
11.73	-1.110	10.46	-0.381
11.98	-1.106	10.71	-0.038
12.23	-0.406	10.96	0.501
12.48	0.010	11.21	0.646
12.73	0.345	11.46	0.773
12.98	0.663	11.71	0.918
13.23	0.910	11.96	1.072
13.48	1.024	12.21	1.153
13.73	1.204	12.46	1.269
13.98	1.370	12.71	1.252
14.23	1.552	12.96	1.402
14.48	1.707	13.21	1.498
14.73	1.855	13.46	1.559
14.98	2.079	13.71	1.685
15.23	2.138	13.96	1.950
15.48	2.089	14.21	2.143
15.73	2.058	14.46	2.087
15.98	2.133	14.71	2.090
16.23	2.130	14.96	1.953
16.48	2.119	15.21	1.864
16.73	2.240	15.46	2.001
16.98	2.346	15.71	2.036
17.23	2.467	15.96	2.119
17.48	2.718	16.21	2.213
17.73	2.906	16.46	2.368
17.98	3.126	16.71	2.480
18.23	3.232	16.96	2.685
18.48	3.334	17.21	2.937
18.73	3.398	17.46	3.122
18.98	3.408	17.71	3.226
19.23	3.457	17.96	3.371
19.48	3.453	18.21	3.414
19.73	3.454	18.46	3.500
19.98	3.475	18.71	3.529
20.23	3.477	18.96	3.542
20.48	3.466	19.21	3.583
20.73	3.462	19.46	3.606
20.98	3.436	19.71	3.601
21.23	3.457	19.96	3.611
21.48	3.422	20.21	3.654
21.73	3.422	20.46	3.609
21.98	3.456	20.71	3.614
22.23	3.400	20.96	3.417
22.48	3.359	21.21	3.287

Figure 21 finally shows the comparison between the observed $J, H, K_{\rm s}$ LFs (dots) and the theoretical LFs generated from the simulated CMD described above (lines). The observed LFs result from the combination of all the available data, namely 2MASS + SOFI + NICMOS all scaled to the SOFI-LARGE area, as in Tables 3. The simulated LFs refer to 8 and 13 Gyr old populations. Observed and theoretical LFs were normalized to the total number of stars with MAG₀<12 in all bands. There is overall agreement between the theoretical and the observed LFs, though the HB clump + RGB bump appears much sharper in the simulation than in the observed LF, as expected from Fig. 20.

Figure 21: The complete bulge LF in the three near-IR bands (dots), as resulting from the combination of the 2MASS, SOFI and NICMOS data (from Table 3). A theoretical LF for an age of 13 Gyr (solid line) and 8 Gyr (dotted line) is shown as a solid line for comparison.

This comparison also shows that formally the 8 Gyr isochrone gives a slightly better fit to the data above the turnoff, which is the part of the LF sensitive to age. However, the same effect causing the smearing of the HB would have also made shallower the drop off of the luminosity function just above turnoff, hence making the bulge population to appear younger than it is when comparing to the simulated LFs. Hence, we believe that before having identified the origin of the additional RGB and HB dispersion the comparison of simulated and observed LFs cannot set more stringent constraints on the age of the bulge stars other than being $\sim$10 Gyr.

5.2 The bulge spectral energy distribution

Figure 22: The spectral energy distribution of the bulge summing the contribution of all individual star (filled circles). The error bars here show what would be the systematic displacement of all the points if an uncertainty of E(B-V) $=\pm 0.1$ is allowed. Superimposed are the spectral energy distributions of template galaxies from Mannucci et al. (2001).

Integrating the LFs in Tables 3 and 4, one can determine the total luminosity of the sampled bulge stellar population in each band. This has been performed using the whole database (NICMOS, SOFI, 2MASS for the infrared and WFI for the optical), then normalizing the results to the area of the SOFI-LARGE field. For the J and H bands the NICMOS data allow to include the contribution of all stars down to $\sim$ $0.15~M_\odot$. The K-band contribution of stars fainter than the SOFI limit has been estimated from the theoretical M/L ratio of lower main sequence stars and adopting -1.33 for the slope of the IMF, and found to be just a fraction $\sim$10^-5 of the total K-band luminosity. This result is not surprising given the flat bulge IMF. For the same reason, we integrated the V and I LFs of Table 4 safely neglecting the contribution of the lower MS stars. The resulting sampled luminosities are:

$$\begin{eqnarray*}&& L_{V}=1.00\times 10^5~L_{V,\odot} \\ && L_{I}=1.13\times 10... ...times 10^5~L_{H,\odot} \\ && L_{K}=4.20\times 10^5~L_{K,\odot} \end{eqnarray*}$$

where the solar magnitudes have been taken from Allen (2000). To derive monochromatic fluxes, the integrated magnitudes in each band have been transformed to AB magnitudes, and the monochromatic flux has been calculated as log $F_\nu = -0.4~ m_{AB} + 19.44$ (Oke & Gunn 1983), with $F_\lambda = F_\nu c/\lambda^2$. Figure 22 shows the resulting spectral energy distribution (SED) of the galactic bulge, compared to the template spectra of elliptical, Sa and Sc galaxies from Mannucci et al. (2001). The Galactic bulge follows quite well the Sc template, which is not surprising given that the Milky Way is indeed an Sbc galaxy. For the optical part of the spectrum, the Mannucci et al. templates rely on those by Kinney et al. (1996), which in the case of Sc galaxies come from the integrated spectra of a region $10''\times 20''$ wide at the center of the two galaxies NGC 2403 and NGC 598, whose radii are 10.6 and 35.2 arcmin respectively, i.e., the sampled regions are located well inside the bulge of these galaxies. Visual inspection of a DSS image of NGC 2403 suggests that the aperture used to derive the spectrum of this galaxy almost certainly includes a few site of active star formation. Their effect is however minimal in the V and I bands, while dominating the spectrum only in the IUE ultraviolet.

On the other hand, the average value of the Mg₂ index of the ellipticals in the Kinney et al. sample is 0.314, which compares to Mg₂=0.23 (Puzia et al. 2002) for the Galactic bulge. The difference is likely due to the well populated subsolar metallicity component on the bulge MD (Greggio 1997). At optical wavelenghts, this component provides more relative flux than the high Z one, which results in both a lower composite Mg₂ index, and in a bluer SED. Actually, the computation of the theoretical Mg₂ index for a composite stellar population with the bulge MD as determined here, using solar scaled models, yields even lower values (Mg₂=0.16-0.19 for 10-15 Gyr, respectively). An extensive discussion of the effects of alpha enhancement on the theoretical Lick indices is presented in Maraston et al. (2003) and Thomas et al. (2002).

5.3 The sampled luminosity-star number connection

The number of stars with mass in the range 0.15 to 1 $M_\odot$in the observed SOFI field can be obtained by integrating the bulge IMF with the appropriate scale factor A:

$$% N_{\rm PRED}=A\int_{0.15}^1 M^{-1.33}{\rm d}M,$$ (6)

with $A\propto L_{\rm T}$ (Renzini 1998), where $L_{\rm T}$ is the bolometric luminosity sampled by the SOFI field. For given age, the ratio $A/L_{\rm T}$ is a weak function of both metallicity and IMF. Its dependence on age is relatively stronger, in fact, from the models by Maraston (2002), at [M/H] =-0.2 and for IMF slope of -1.33, $A/L_{\rm T}$ ranges from 0.82 to 1.1 for an age varying from 10 to 13 Gyr.

The total bolometric luminosity sampled by the SOFI-LARGE field can be obtained applying the appropriate bolometric corrections to individual stars in the sample. To evaluate that, we have run the simulation code to obtain the bolometric and monocromatic luminosities of a stellar population with the observed metallicity distribution, obtaining: $L_{\rm T}=1.39 L_V=1.16 L_I=0.70 L_J=0.53 L_H=0.51 L_K$ with the coefficients changing by at most $3\%$ when varying the age from 10 to 13 Gyr. No observational errors were applied in this run.

Using the five monochromatic luminosities with the above relation and averaging the results we obtain a total luminosity of $L_{\rm T}=177~000~L_\odot$, hence $N_{\rm PRED}=380~000$ and 510 000, respectively for 10 and 13 Gyr. These numbers compare with the 424 000 $\pm$ 81 000 stars in the SOFI-LARGE field, the error in the latter being dominated by the Poisson noise in the number of objects in the NICMOS field. Predicted and observed numbers are quite consistent (see also Paper I), since these theoretical estimates are expected to be accurate to within $\sim$$10\%$.

5.4 The M/L ratios

Integrating the IMF from 0.15 to 100 $M_\odot$, and adopting the prescription in Paper II for the mass of the stellar remnants (those above $\sim$$1~M_\odot$) one derives the total stellar mass in the bulge SOFI-LARGE sample. Using the luminosities in the various bands given above, one then determines the corresponding M/L ratios. This gives:

$$\begin{eqnarray*}&& M/L_{V}=3.67 \\ && M/L_{I}=3.25 \\ && M/L_{J}=1.28 \\ && M/L_{H}=1.00 \\ && M/L_{K}=0.87 \end{eqnarray*}$$

with M/L_K in very nice agreement with the dynamical value, $M/L_{K}\simeq 1$ (Kent 1992).