3 Diagnostics

Some examples are given with a synthetic population to elucidate the method and its associated merit function as described in Sect. 2. It is further shown that the residual points of the data not matched can provide an indication about the parameter to be improved. A diagnostic diagram is much easier to interpret due to the relative small number of residual points than a CMD with the simulation of the best fit which may have 5000 or more points. It might be advantageous for other methods such as ratios, maximum likelihood, multivariate analysis or Bayesian inference to display the residuals from the best fit obtained. In this way one can obtain an independent visual impression of the performance of different methods and even obtain hints about possible improvements.

The test population used as reference in the simulations has the following specifications:

• a metallicity range, spanning Z=0.005-0.030;
• an age range from 8-9 Gyr;
• an initial mass function with a Salpeter slope;
• an exponentially decreasing star formation rate with a characteristic time scale of 1 Gyr.

This population, displayed in Fig. 1, was found to contribute significantly to the CMD of Baade's Window (Ng et al. 1996a). In the simulation the test population is placed at 8 kpc distance. The observational limits are $V_{\rm lim}=22\rm ^m$ and $I_{\rm lim}=21\rm ^m$.In each simulation $N\!=\!5000$ stars are considered. For simplicity an extinction and crowding free case is considered with Gaussian distributed photometric errors amounting to $\sigma \lesssim 0\hbox{$.\!\!^{\rm m}$}05$ per passband. This is then compared with models, in which one of the specified parameters is varied. For the $\chi^2$-merit function a 3$\sigma$ limit around each point (c_i,m_i) is adopted for the range of the error ellipse. These variations are then followed by comparison with different age, metallicity, initial mass function and star formation history.

  

Table 1: Description of the diagnostic statistics for the various models displayed in Figs. 2$\,$-$\,$4, see text for additional details

Table 1 shows the results of the various simulations performed and discussed below, while Fig. 2 displays the global merit function of these simulations. Figures 3a$\,$-$\,$i display the diagnostics diagrams, resulting from the comparison between the "observed" and synthetic CMDs.

  

Figure 1: The (V, V-I) colour-magnitude diagram of the original stellar population (Z=0.005-0.030, t=8-9 Gyr) placed at 8 kpc, see text for additional details. Note that this simulation reflects a reddening free case and that the shape of the red horizontal branch is due to a large spread in metallicity

3.1 No variation

The first step is to demonstrate what values one obtains for $F_\chi,F_{\rm P}\;{\rm and}\;F$ with an acceptable model population. Such a population is obtained from a different realization of the test population by modifying the seed of the random number generator.

  

Figure 2: The global merit for various simulations. The letters a$\,$-$\,$bf i refer to a model for which the result are given in Table 1. The solid and long dashed lines denote respectively the 1$\sigma$ and 3$\sigma$ area, where acceptable simulations for the stellar populations ought to be located. Model d could be acceptable. However, Fig. 3d hints that there is still a systematic discrepancy between the observations and simulations

The values of the merit functions for this simulation (model a) are listed in Table 1 and are consistent with the expectation that for 5000 points about 71 points will not be matched in each of the observed and simulated dataset. Figure 2 shows further that this population has in the ($F_{\rm P},F_\chi$)-plane an average $\sigma$close to 1. Figure 3a shows that the residual points (open circles for "observations" and crosses for simulations) are randomly distributed over the original (shaded) population. This is indicative that the model a simulation is an acceptable replacement for the original population.

  

Table 2: Formal 1$\sigma$ uncertainties determined with Eq. (10) for model a (see Table 1 and Sect. 3.1), where $[Z]=\log~Z/Z_\odot$,$\alpha$ is the index of the power-law initial mass function and $\beta$ is the index of the exponential star formation rate, specifying its characteristic time scale

Table 2 gives the formal 1$\sigma$ uncertainties determined with Eq. (10). The estimated error in the distance gives an uncertainty in the distance modulus of about $0\hbox{$.\!\!^{\rm m}$}06$. This is a realistic value, because it is in close agreement with the $0\hbox{$.\!\!^{\rm m}$}07-0\hbox{$.\!\!^{\rm m}$}08$ uncertainty in the distance modulus obtained, for example, by Gratton et al. (1997) for globular clusters. The uncertainty in the extinction is about d$A_V\!\simeq0\hbox{$.\!\!^{\rm m}$}06$, which is also in good agreement with the best estimates for the reddening of the globular clusters mentioned above. They yield $E(B-V)=0\hbox{$.\!\!^{\rm m}$}02$, which is equivalent with an extinction of about $0\hbox{$.\!\!^{\rm m}$}06$with a standard reddening law. The $5-10\%$ uncertainties in the age limits indicate that at a 1$\sigma$ level the lower and upper age limit are likely not the same, but the whole population might on the other hand be almost indistinguishable from a population with a single age of about 8.7 Gyr. In addition, a 10% uncertainty is not very different from the 12% random errors estimated for the ages of globular clusters (Gratton et al. 1997).

The uncertainties in the metallicity range hint that an unambiguous determination of the presence of a metallicity range can be established. The estimated errors range from 0.1-0.2 dex, which is once more a realistic value if one considers that the uncertainty in the metallicity scale is of the order of 0.1 dex. Most remarkable is the very small error obtained for the index of the slope of the power-law IMF. This is mainly due to the fact that small differences in the slope tend to result in large number of residuals of main sequence stars for the Poisson merit function. It strongly suggests that the method described in this paper potentially could reveal crucial information about the IMF. This is in high contrast with the estimated uncertainty for the index of an exponentially decreasing star formation rate. Due to the relative small age range the uncertainty is quite large. As a consequence, a population with a constant star formation rate cannot be discarded. On the other hand, Dolphin (1997) has demonstrated that in case of a considerably larger age range stronger constraints on the star formation rate could be obtained.

3.2 Distance

The distance of a stellar aggregate should be determined as accurately as possible, because uncertainties in the distance modulus can result in a different age for the stellar population, see Gratton et al. (1997) for a detailed discussion. For the next simulation (model b) the distance of the synthetic population is modified to 8.5 kpc. Figure 3b displays the diagnostic diagram with the residuals.

A fraction of the residuals, located on the main sequence or the sub-giant branch, are fainter than the original input population (shaded area). This feature provides a strong hint that the adopted population is not located at the proper distance. It results in a considerably larger value for the Poisson merit function. This leads to a rather large value for the global merit and gives an indication that the parameters for the matching stellar population are not yet optimally tuned. This furthermore appears in Fig. 2, which shows that the simulation for model b is not in the $1\sigma -3\sigma$ range of acceptable solutions.

3.3 Extinction

A synthetic population (model c) is made with an average extinction of $A_V=0\hbox{$.\!\!^{\rm m}$}10$,to demonstrate the effects of differences in the extinction. In addition, a random scatter is added to the extinction, amounting to $0\hbox{$.\!\!^{\rm m}$}02$.Figure 3c shows the corresponding diagnostic diagram with two residual sequences (original versus extinction modified), each located respectively near the blue and red edge of the original input population (shaded area). The sequences are also shifted from each other along the reddening line. A comparison between Figs. 3b and 3c further shows that there are distinct differences between the distribution of the residuals when the distance or extinction are not optimally tuned.

3.4 Photometric errors

In this simulation the effect of overestimating the photometric errors (model d) in the simulations is demonstrated. The photometric errors as a function of magnitude are increased by 50%. The results are displayed in Fig. 3d. The figure shows that the residuals for the modified population are mainly located outside the shaded region of the original population, while the residuals from the original population are almost centered on the shaded area.

Figure 2 indicates that model d might be an acceptable solution, but Fig. 3d hints that the parameters are not yet optimally tuned. Figure 3d further displays that the distribution of the residuals is significantly different from the residuals shown in any of the other panels of Fig. 3. It is therefore possible to identify an inadequate description of the photometric errors from the residuals and improve with this information the description of the simulated errors.

3.5 Crowding

In crowded fields single stars can lump together and thus form an apparent (not always resolved) binary on an image (Ng 1994; Ng et al. 1995; Aparicio et al. 1996). This results in the "disappearance" of some of the fainter stars. One can mimic this effect easily in the Monte-Carlo simulations of a synthetic CMD. One should avoid to include in these simulations the completeness factors as obtained from artificial star experiments. In crowding limited fields one obtains from the simulations of apparent binaries the completeness as a function of magnitude. This apparent, binary induced completeness function should be compared with the completeness factors obtained from artificial stars experiments.
  



Figure 3: Diagnostics diagrams, resulting from the various simulations given in Table 1. Frame a) shows the diagram for a different realization, while non of the major input parameters are varied; in frame b) the distance is varied; frame c) shows the diagram when some extinction is added; frame d) displays the residuals when different photometric errors are adopted; frame e) represents the case where the amount of crowding is overestimated; frame f) when a younger age is adopted for the stellar population; frame g) when the upper metallicity limit is underestimated; frame h) shows the effect for a different slope of the power-law initial mass function; and frame i) shows the case when the star formation rate is increasing towards younger age, instead of decreasing. The residual "observed" points which are not fitted are indicated by open circles, while the residual synthetic points are indicated by crosses. In each frame the shaded area shows approximately the part of the CMD covered by the original population. In addition the global merit F is reported in each frame

The crowding simulations (model e) are performed under the assumption that 5% of the observed stars in the CMD are (apparent) binaries. Note that $\sim$em10% of the total number of simulated stars are involved in the crowding simulations. The actual number is slightly lower, because the simulations allow for the small possibility of the formation of multiple star lumps. The results are displayed in Fig. 3e. The diagnostics show, except for a significantly larger scatter, some similarity with those from Fig. 3d. However, the difference is that one part of the residuals from Fig. 3d forms a thinner sequence on top of the lower main sequence band, while the other part of the residuals is located close to the lower main sequence band. In the case of crowding the central band is broader and the remaining residuals are not located near the lower main sequence band.

3.6 Age

The galactic bulge might contain a rather young stellar population (Ng et al. 1995; Kiraga et al. 1997), however the results by Bertelli et al. (1995) and Ng et al. (1996) contradict such a suggestion. The results by Ng et al. (1995) might be induced by the limited metallicity range used in the simulations, while the results from Kiraga et al. (1997) do not appear to allow for a different interpretation. The next simulation (model f) is therefore made with a population which has an age in the range 4-5 Gyr and Fig. 3f displays the diagnostic diagram. At first sight this appears to be similar to the diagram of Fig. 3c, but there is a pronounced difference: the residual sequences are not parallel and furthermore one of the two sequences is slightly brighter and bluer than the main sequence turn-off of the original population.

In Sect. 3.1 it is noted that ages can be determined to an accuracy of at least 10%. An analysis of a deep bulge CMD with the method discussed in this paper might well constrain the actual age of the major stellar populations in the galactic bulge.

3.7 Metallicity

In the following simulation (model g) the upper metallicity limit of the synthetic population is decreased to Z=0.020. Figure 3g displays the diagnostic CMD and shows two almost parallel sequences as in Fig. 3c, where the extinction is varied. The similarity is due to the fact that extinction and metallicity differences show a similar behaviour in (V, V-I) CMDs. Photometry in other passbands should be explored to avoid this degeneracy. Ngem&emBertelli (1996) demonstrate that near-infrared (J, J-K) photometry would resolve unambiguously the degeneracy between extinction and age-metallicity.

3.8 Initial mass function

The sensitivity of the slope of a power-law initial mass function is demonstrated by changing the slope of the original population from $\alpha\!=\!2.35$ to 1.35 (model h). Figure 3h displays the corresponding diagnostic CMD. The residuals for the simulation with a shallower slope are dominating the upper part of the diagnostic diagram, while the residuals from the original population are concentrated near the faint detection limit. The large number of residuals give rise to an increase of the value obtained for the Poisson merit function. As mentioned in Sect. 3.1 this behaviour provides a strong constraint in the determination of the slope of the power-law IMF. Note in addition that the distribution of the residuals is quite different from the other panels in Fig. 3. One should realize however that no strong constraint for the power-law IMF can be obtained when main sequence stars below the turn-off are not available for analysis.

3.9 Star formation history

The final demonstration (model i) invokes a synthetic population generated for an increasing star formation rate with a characteristic time scale of 1 Gyr. The uncertainties given in Table 2 (see also Sect. 3.1) already suggest that a study of the star formation rate cannot be done reliably when a small age range is considered. However, the differences in the index of the exponential star formation rate are for this simulation suitably chosen, such that some differences will show up. The residuals of the simulation with an increasing star formation rate are displayed in Fig. 3i. The diagnostic CMD shows two parallel sequences comparable to the sequence in Figs. 3c and 3g. This partly provides an indication of the difficulties involved in studies of the star formation rate. It further shows once more that (V, V-I) CMDs are not an optimal choice to study differences in star formation histories, because it will be difficult to distinguish differences in the extinction, metallicity, star formation history and even small age differences from one another. Additional photometry in other passbands ought to be used instead.