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Astron. Astrophys. 361, 73-84 (2000)

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6. The results

In this section, the fields will be analyzed separately, deriving the reddening along the line of sight and the possible solutions. For the best fitting solutions the simulated CMDs are displayed in Fig. 4. In the simulations of F3 and F4 a young spiral-arm-like population is also included, as described in Sect. 7. After discussing the single results for each field in the following subsections, they will be compared to derive the best fitting solution for all the fields. Since the studied fields are located at low Galactic latitude, we expect them to be more sensitive to the star formation rate and to the scale height than to the scale lenght of the disk components. For the same reason the parameters of the thick disk component would not be strongly constrained.

[FIGURE] Fig. 4a-d. The V-(V-I) CMD of the best solution for each of the fields is shown as example: F1 a , F2 b , F3 c , F4 d . F1 and F2 simulations include disc components (faint dots), while in the case of F3 and F4 the spiral arm is as well plotted (starred dots). In each plot the line indicates the observational edge of the main sequence. The star formation of the thin disc components is decreasing in time (see text for details).

6.1. Discussing the CMD: extinction determinations

Ng et al. (1995), Bertelli et al. (1995) proved that the slope of the main sequence in the CMD of the disk population is mainly governed by the extinction along the line of sight. At each magnitude V the bluest stars on the main sequence can be interpreted as the envelope of the main sequence turnoffs of the population having absolute magnitude [FORMULA], shifted towards fainter magnitudes and redder colours by the increasing distance and corresponding extinction. Starting from an initial guess, the amount of extinction at increasing distances is adjusted until a satisfactory agreement between the main sequence blue edge location in the data and in the theoretical simulations is reached. The comparison between data and simulations is made using a [FORMULA] test. The results are given in Fig. 5.

[FIGURE] Fig. 5. The reddening along the line of sight, as determined for the four fields. For comparison, the values of Mendez & van Altena (1998)are included in the plots.

We point out that F2 and F3 show a relatively modest increase of the interstellar absorption along the line of sight ([FORMULA]A_v less than 0.5 mag) between 1.5 and 2 kpc distance from the Sun for F2 and between 2 and 5 kpc for F3. Whereas this increment in F2 is probably due to some local density increase of the interstellar matter, the increment in F3 can be explained on larger scales since the direction of F3 at l=292 is crossing the inner spiral arm, if the description of the spiral pattern given by Taylor & Cordes (1993)is adopted. At 1.5 Kpc distance from the Sun, the spiral arm is reached (see Sect. 7), at a height of 40 pc above the Galactic plane. At about 5 Kpc distance, the spiral arm is left at a height of 150 pc above the plane. For this field the increase of extinction is well correlated with the intersection of the spiral arm pattern. In Field F4 (l=305) which does also point towards the inner arm, no special increment of the extinction is noticeable. This is not entirely surprising, since the the fields are chosen on the basis of their integrated colours as having low total absorption (see Sect. 2).

Finally, we would like to address the question whether the trends in the extinction versus the distance shown in Fig. 5 are real. It is quite difficult to give an estimate of the uncertainty on these determinations. However, the simulations show that by changing the extinction of [FORMULA] mag at a given distance d a significant shift in the location of the main sequence edge at magnitudes fainter than [FORMULA] is produced. We can safely assume that the [FORMULA] determination of Fig. 5 cannot have an internal error larger than 0.2 mag.

We point out that these determinations of extinction are dependent on the adopted age and metallicity range of the population. To estimate the uncertainty due to the combined effect of different age and metallicity distributions, we derive the extinction separately for the three disk components, namely the young thin, the old thin and the thick disk described in Sect. 6.1. The determination of the extinction along the line of sight [FORMULA] turns out to be different from the adopted values at maximum of 0.2. To further check these results, we make a comparison with the values derived from reddening maps by Mendez & van Altena (1998). Taking into account the errors on Mendez & van Altena determinations (0.23 mag in [FORMULA], with Av= 3.2 [FORMULA]), the agreement is reasonable up to distances of about 8 Kpc, at (l,b)=(292.4,1.63) (field F3), and (305.0,-4.87) (field F4). At (l,b)=(265.54, -3.08) (field F1) and (276.4,-0.54) (field F2) our and Mendez & van Altena (1998)Av determinations are consistent up to 3-4 Kpc, our values being higher of a factor [FORMULA] at larger distances.

When the maximum reddening in each direction is compared with the maps by Schlegel et al. (1998) derived using DIRBE data, the agreement is excellent, in spite of the fact the authors claim their values should not be trusted for [FORMULA].

6.2. Discussing the CMD: the star formation rate

To infer the star formation rate (SFR) of the thin disk component, simulated CMD and LF are calculated with constant, increasing or decreasing rate and then compared with observational CMDs using a [FORMULA] test. The thick disc is assumed to have a SFR constant from 11 Gyr to 8 Gyr. Due to the low galactic latitude of the observed fields, it is not possible to distinguish between constant or slightly increasing/decreasing rate for this component.

From the analysis of the Hipparcos data Bertelli et al. (1999)derive a disk SFR constant from 10 Gyr to 4.5 Gyr, and then increasing by a factor of 1.5-2 from 4.5 Gyr to 0.1 Gyr. However it is not straightforward that the SFR found in the solar neighborhood is representative of the whole disk, as has already been suggested by Bertelli et al. (1999).

To assess this point, we simulate the CMDs and luminosity functions of the fields using the parameterization of the SFR derived from the Hipparcos data by Bertelli et al. (1999). While the luminosity functions are not inconsistent with this SFR, the CMDs of F3 and F4 cannot be reproduced, since too many young stars brighter than V=19.5 and bluer than the main sequence observational edge are produced. This is evident comparing the simulated and the observed CMDs of F3 and F4 in Fig. 6 and Fig. 7. The extinction cannot be responsible of this discrepancy: if the extinction along the line of sight at closer distances is increased to match the observational location of the blue edge of the main sequence at brighter magnitudes, then the faint main sequence turns out to be too red.

[FIGURE] Fig. 6a and b. Simulation of the CMD of F3 obtained using the Hipparcos parameterized SFR as described in the text a . The line shows the observational edge of the main sequence. For sake of easy comparison the observational CMD of F3 (see Fig. 3) is as well shown b .

[FIGURE] Fig. 7a and b. Simulation of the CMD of F4 obtained using the Hipparcos parameterized SFR as described in the text a . The line shows the observational edge of the main sequence. For sake of easy comparison the observational CMD of F4 (see Fig. 3) is as well shown b .

The case of F1 and F2 is substantially different. In these fields the blue edge of the main sequence is reproduced using the Hipparcos SFR (see Fig. 8 for F2). However, too many evolved stars are expected in F2 (740 stars brighter than V=22) in comparison with the data (80 stars). No additional information is coming from F1. Due to the poor statistic, the observed and the expected number of stars in this field (63 and 74 respectively) are compatible inside the errors.

[FIGURE] Fig. 8a and b. Simulation of the CMD of F2 obtained using the Hipparcos SFR as described in the text. The line shows the observational edge of the main sequence. To allow a easy comparison, the observational CMD of F2 (see Fig. 3) is as well shown b .

However, it cannot be excluded that this result is dependent on the adopted parameterization of the solar neighborhood star formation. To make a further check we use the observed Hipparcos population, and we distribute it along the line of sight in the disk. The resulting CMD is presented in Fig. 9 for F3. The previous conclusions are substantially unchanged. Similar result can be reached in the case of F4, not shown for conciseness.

[FIGURE] Fig. 9a and b. Simulation of the CMD of F3 obtained using the Hipparcos population as described in the text. The line shows the observational edge of the main sequence. For comparison the observational CMD of F3 (see Fig. 3) is shown b .

From this investigation we conclude that the solar neighborhood cannot be considered representative of the properties of the whole disk. An analogous discussion can be made at varying SFR. The simulations show that any assumption of an increasing or even constant SFR yields a too high number of young stars on the blue side of the main sequence. Actually, the most convincing result is obtained with a SFR constant from 10 Gyr to 2 Gyr, then declining by a factor of 10 between 2 Gyr and 0.1 Gyr (see Fig. 4).

As a final comment, the CMDs of F3 and F4 show an additional sprinkle of stars brighter that V [FORMULA] 15.5 mag and bluer than the mean location of the main sequence (see Fig. 3). These stars cannot be reproduced unless a young burst of star formation well confined in distance is assumed. This feature will be proved to be consistent with the presence of a spiral arm (see Sect. 7).

6.3. The results for F3

Various combinations of the disk parameters, at changing scale height and scale length of the thick disk, scale length of the thin disk. Both spatial distributions (sech2 and double exponential) for the disk are tested. The observational luminosity functions are compared with the simulations, using a [FORMULA] square test. Fig. 10 presents the comparison of the observational luminosity function with four simulations.

[FIGURE] Fig. 10a-d. The luminosity function of F3 is displayed with the four selected solutions. Solutions a),b),c) are calculated using a sech2 mass distribution, solution d) is making use of the exponential mass distribution: a  [FORMULA], [FORMULA]=0.22 kpc, [FORMULA]= 1.0 kpc; b  [FORMULA], [FORMULA]=0.21 kpc, [FORMULA]= 1.5 kpc; c  [FORMULA], [FORMULA]=0.22 kpc, [FORMULA]= 1.0 kpc; d  [FORMULA], [FORMULA]=0.20 kpc, [FORMULA]= 1.0 kpc.

For every set of models the most convincing solutions for decreasing stars formation are listed in Table 2 together with the [FORMULA] where [FORMULA] and [FORMULA] are the observed and the expected number of stars per magnitude bin, and [FORMULA] is the number of degrees of freedom (which is equal to the number of bins minus 1, since the in the simulations we impose that the total number of model stars is equal to the observed total number of stars)


Table 2. Solutions for F3

As expected, the scale height of the thick disk hz is poorly constrained by the data. Any value of hz in the range 700-1500 pc, turns out to be acceptable, higher values resulting in a slightly lower percentage of the thick disk component. The most convincing fits are obtained for a thick disk percentage of 2-3%. The sech2 mass distribution seems to be favored by the data, resulting in more convincing luminosity function fits. Values of thin disk hz of 220[FORMULA]30 pc seem to be more consistent with the data.

Imposing a cut of the disk at 14 Kpc, the fit is not substantially improved, as is expected due to the low Galactic latitude.

6.4. The results for F4

Table 3 presents the solutions and Fig. 11 shows the comparison of the observational luminosity function with four simulations. Analogously to F3, the sech2 mass distribution is slightly favored, again resulting in higher percentages of the thick disk component. Compared to Field 3, the percentage of thick disk is generally higher in this field: to reproduce the data with constant SFR at least 3-4% of thick disk is needed. The data are not sensitive to the scale lenght and scale height of the thick disk.

[FIGURE] Fig. 11a-d. The luminosity function of Field 4 is displayed with four selected solutions for sech2 mass distribution: a  [FORMULA], [FORMULA], [FORMULA]; b  [FORMULA], [FORMULA], [FORMULA]; c  [FORMULA], [FORMULA], [FORMULA]; d  [FORMULA], [FORMULA], [FORMULA].


Table 3. Solutions for F4

Concerning the scale height of the think disk, the most convincing solutions are for 270 [FORMULA] 30 pc. Scale lenght values as low as 1.1 Kpc result is a less good fit of the luminosity function, while all the values in the range 1.2-1.5 Kpc are consistent with the data.

Analogously to F3, introducing a cut of the disk at 14 Kpc, we do not improve the fit of the luminosity function.

6.5. Hunting for common solutions

The scale height of the thin disk derived for F4 are in the mean higher than the ones derived for F3. However they are consistent at the 2-[FORMULA] level, if using the best value of 280[FORMULA]30 pc for F4 and 220[FORMULA]30 for F3. The most convincing common solution with decreasing star formation rate is found about 2-4% of thick disk, a scale height around [FORMULA] and the sech2 distribution.

Due to the poor statistics, F1 and F2 (see Table 1) do not set further constraints on the determination of the scale height and lenght of the disk. The errors are relatively large and all the solutions, we found for the fields F3 and F4 do also fit the fields F1 and F2. Hence we can only conclude, that F1 and F2 are consistent within the uncertainties with the other fields.

6.6. Discussing the mass distribution

We use the value of the central value of the mass distribution [FORMULA] (see Eqs. 2 and 3) to derive the local mass density. [FORMULA] can be derived for each field imposing the total number of stars in a selected region of the CMD. So, we expect that inhomogeneities of the mass distribution reflect in different constants in different fields.

Taking into account all the disk components, there is a slight evidence that the total mass density might be higher of a factor 1.5-2 in F4 and F1 than in F2 and F3, the main difference residing in the mass of the component older than 2 Gyr. However it is not clear whether this effect is real then reflecting inhomogeneities of the disk on small scale, or it must simply be interpreted as due to the uncertainty on the mass determination. From this constant, we can derive the mass density in the local neighborhood in stars more massive than 0.1 [FORMULA], calculated using Kroupa (2000) IMF. For the best solutions, we derive a total local star density of 0.025 [FORMULA] pc -3 in F3 and 0.036 in F4, 0.034 in F1 and 0.022 in F2. These results are in agreement with previous determinations of the local density. Oort (1960) find 0.18 [FORMULA] pc -3 as total local mass density, where the total mass density in stars and interstellar matter would not exceed 0.08 [FORMULA] pc -3. Lower values are derived by Creze et al. (1998) who estimate that the local mass density in stars can be 0.042-0.048 [FORMULA] pc -3, including 0.015 in stellar remnants, while the total dynamical mass density cannot exceed 0.065-0.10 [FORMULA] pc -3.

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© European Southern Observatory (ESO) 2000

Online publication: September 5, 2000