4 Cluster integrated colours

According to the discussed procedure, we are now in the position of predicting integrated colors for each given assumption about the cluster age and/or chemical composition. As a first step, let us test whether the synthetic cluster procedure is able to reasonably reproduce the integrated colours observed in actual clusters. On very general ground, one expects that synthetic clusters well reproducing the CM diagram distribution of stars should automatically produce reasonable colours. To discuss this point Table 2 compares observed colours for our test clusters, M 30 and 47 Tuc, with the result of theoretical simulations for the selected choices about the cluster age.

Figure 5: Predicted stochastic fluctuations of integrated colours for a population of 15 Gyr and Z=0.006 as a function of the integrated cluster visual magnitude, error bars refer to one $\sigma$. Filled circles show the maximum/minimum values obtained for each colour

 

 

Table 2: Observed and expected integrated colours for the two prototype globulars M 30 and 47 Tuc. M_V^0.5 is the cluster V absolute magnitude at half-light radius (r^0.5)

Cluster	U-B		B-V		V-R		V-I
M 30 ( MV0.5=-6.6)	0.02	(.02)	0.59	(.04)	0.41	(.02)	0.85	(.04)
Synthetic	-0.02	(.02)	0.61	(.03)	0.43	(.02)	0.86	(.03)
47 Tuc ( MV0.5=-8.6)	0.34	(.01)	0.85	(.01)	0.51	(.01)	1.10	(.02)
Synthetic	0.40	(.01)	0.91	(.01)	0.53	(.01)	1.09	(.03)
Syn.+PAGB	0.34		0.87		0.52		1.07

Observational data are from the compilation by Harris (1999), as dereddened according to the relations given by Reed et al. (1988). On theoretical side, we followed Scalo (1986) assuming a Salpeter exponential law down to $M_{\rm inf} = 0.3~M_{\odot}$ and a flatter function down to $0.1~M_{\odot}$. However, neither moving $M_{\rm inf}$ up to $0.6~M_{\odot}$ or down to $0.2~M_{\odot}$ (see Table 3) nor changing the IMF slope (see Table 4) causes significant variations of the integrated colours (see also Kurth et al. 1999).

 

 

Table 3: The effects of changing the slope of the mass distribution of low mass stars for two assumptions of the integrated M_V magnitude: $M_{\rm inf}$ is the lower limit of the Salpeter law (x=1.35). From $M_{\rm inf}$ down to $0.1~M_{\odot}$ a flatter distribution with x=0.2 is assumed

$M_{\rm inf}$ ($M_{\odot}$)	MV	U-B	B-V	V-R	V-I
0.2	$-4.17\pm 0.10$	$0.43 \pm 0.05$	$0.92 \pm 0.05$	$0.54 \pm 0.03$	$1.09 \pm 0.08$
0.4	$-4.16\pm 0.105$	$0.43 \pm 0.05$	$0.91 \pm 0.05$	$0.53 \pm 0.03$	$1.07 \pm 0.08$
0.5	$-4.15\pm 0.11$	$0.43 \pm 0.05$	$0.91 \pm 0.05$	$0.53 \pm 0.03$	$1.07 \pm 0.08$
0.6	$-4.13\pm 0.11$	$0.43 \pm 0.05$	$0.91 \pm 0.05$	$0.53 \pm 0.035$	$1.06 \pm 0.08$
0.2	$-7.38 \pm 0.03$	$0.43 \pm 0.01$	$0.93 \pm 0.01$	$0.55 \pm 0.01$	$1.12 \pm 0.02$
0.4	$-7.37 \pm 0.03$	$0.43 \pm 0.02$	$0.92 \pm 0.02$	$0.54 \pm 0.01$	$1.10 \pm 0.02$
0.5	$-7.36 \pm 0.03$	$0.43 \pm 0.01$	$0.92 \pm 0.01$	$0.54 \pm 0.01$	$1.10 \pm 0.02$
0.6	$-7.35 \pm 0.03$	$0.43 \pm 0.01$	$0.92 \pm 0.01$	$0.54 \pm 0.01$	$1.10 \pm 0.02$

 

 

Table 4: The effects of changing the slope of the IMF for $M>M_{\rm inf}=0.3~M_{\odot}$: a flatter (x=0.35) and steeper (x=2.35) IMF than the Salpeter law are assumed

x	MV	U-B	B-V	V-R	V-I
0.35	$-7.52\pm 0.04$	$0.43\pm 0.015$	$0.92 \pm 0.02$	$0.54\pm 0.015$	$1.10\pm 0.04$
1.35	$-7.35 \pm 0.03$	$0.43 \pm 0.01$	$0.92 \pm 0.01$	$0.54 \pm 0.01$	$1.10 \pm 0.02$
2.35	$-7.52\pm 0.15$	$0.42\pm 0.015$	$0.91\pm 0.02$	$0.54 \pm 0.01$	$1.10\pm 0.04$

Because of the statistic nature of the approach, one expects that theoretical results are affected by intrinsic fluctuations whose amplitude decreases when increasing the number of stars populating the synthetic cluster. To cope with such an evidence, numerical simulations have been performed by populating the synthetic cluster till reaching the observed cluster V magnitude at half-light radius.

For each case the random procedure has been repeated 10 times and Table 2 gives for each colour the obtained mean value together with the corresponding root mean square. Taking into account the uncertainty in M 30 observational data (Reed 1985), one can conclude that our synthetic approach is reproducing the colours of the actual cluster within observational uncertainties and theoretical statistical fluctuations. On the contrary, one finds that our predictions for 47 Tuc give fairly redder U-B and B-V colours, whereas both observational and theoretical data appear rather firmly established. This puzzling evidence is easily understood when recalling that Lloyd Evans (1974) discovered near the cluster center a hot luminous post-asymptotic giant branch, for which Dixon et al. (1995) give a temperature of $\sim$ 10 000 K and log $g\le 2.5$. Assuming a mass of $0.6~M_{\odot}$ one thus derives log  $L/L_{\odot}\sim 3.5$. As shown in the last row of Table 2, adding the contribution of a similar object to the integrated cluster light, theory and observation appear in good agreement. Such an occurrence reinforces the already discussed evidence that integrated colours of even rich galactic globular like 47 Tuc are still affected by the stochastic presence of luminous stars in the fast phase of PAGB evolution.

Beyond such a rare occurrence, one is in all cases facing an uncertainty in predicted colour as due to statistical fluctuation in the population of bright stars. As already discussed in Paper I, one can generalize the numerical experiments presented in the previous section for M 30 and 47 Tuc by investigating the expected fluctuations of the predicted colours as a function of the integrated cluster V-magnitude. This because statistical fluctuations in the population of bright stars depend on their number, which - in turn - governs the total cluster luminosity. Figure 5 gives a quantitative estimate of such an uncertainty, showing the range of colours predicted by our computations over ten different random simulations for any given assumption about the cluster luminosity.

Figure 6: Observed integrated colours of galactic globular clusters (Harris 1999). Theoretical predictions by this work and other authors are also shown (BC 1996 refers to Bruzual and Charlot models see Leitherer et al. 1996)

As expected, one finds that the statistically uncertainty decreases when the integrated V-magnitude of the cluster increases, being reduced to few hundreds of magnitude only for cluster luminosities larger than $M_{V}^{\rm tot}\sim -9$ mag, i.e. for a magnitude reached by few very rich galactic globulars, like 47 Tuc ( $M_{V}^{\rm tot}=-9.42$) or NGC 2419 (-9.58), NGC 6388 (-9.82), NGC 6402 (-9.02), etc. On the contrary, data in Fig. 5 shows that the colours of the large majority of galactic globulars can be sensitively affected by intrinsic uncertainties. Note the increased uncertainty in V-I colours, due to the stochastic contribution of cool luminous AGB stars.

 

 

Table 5: Synthetic integrated colours computed assuming a constant number of He-burning stars ( $N_{\rm HB}=300$), a Salpeter IMF and t=15 Gyr for three different values of the Reimers parameter $\eta$ ( $\sigma _{\rm M}=0.04$)

Z	MV	U-B	B-V	V-R	V-I
$\eta=0.2$
0.0001	$-7.92 \pm 0.04$	$0.028 \pm 0.015$	$0.65 \pm 0.02$	$0.46 \pm 0.01$	$0.91\pm 0.02$
0.0003	$-7.86 \pm 0.06$	$0.07 \pm 0.02$	$0.71 \pm 0.03$	$0.47 \pm 0.02$	$0.94 \pm 0.04$
0.001	$-7.81 \pm 0.05$	$0.19 \pm 0.02$	$0.81 \pm 0.02$	$0.50 \pm 0.01$	$1.01 \pm 0.02$
0.006	$-7.37 \pm 0.04$	$0.45 \pm 0.01$	$0.94 \pm 0.02$	$0.56 \pm 1.02$	$1.16 \pm 0.05$
0.02	$-7.05 \pm 0.03$	$0.70 \pm 0.01$	$1.04 \pm 0.01$	$0.61 \pm 0.02$	$1.32 \pm 0.07$
$\eta =0.4$
0.0001	$-7.76 \pm 0.05$	$-0.01 \pm 0.015$	$0.62 \pm 0.02$	$0.44 \pm 0.01$	$0.88 \pm 0.02$
0.0003	$-7.74 \pm 0.03$	$0.05 \pm 0.01$	$0.65 \pm 0.01$	$0.43 \pm 0.005$	$0.87 \pm 0.01$
0.001	$-7.78 \pm 0.05$	$0.18 \pm 0.02$	$0.76 \pm 0.02$	$0.48 \pm 0.01$	$0.98 \pm 0.03$
0.006	$-7.35 \pm 0.03$	$0.43 \pm 0.01$	$0.92 \pm 0.01$	$0.54 \pm 0.01$	$1.10 \pm 0.02$
0.02	$-7.24 \pm 0.02$	$0.70 \pm 0.01$	$1.04 \pm 0.01$	$0.61 \pm 0.02$	$1.28 \pm 0.07$
$\eta=0.6$
0.0001	$-7.64 \pm 0.06$	$-0.06 \pm 0.02$	$0.61 \pm 0.02$	$0.43 \pm 0.01$	$0.865 \pm 0.025$
0.0003	$-7.56 \pm 0.04$	$0.003 \pm 0.02$	$0.63 \pm 0.02$	$0.43 \pm 0.01$	$0.87 \pm 0.02$
0.001	$-7.52 \pm 0.09$	$0.10 \pm 0.02$	$0.70 \pm 0.02$	$0.45 \pm 0.01$	$0.92 \pm 0.02$
0.006	$-7.32 \pm 0.05$	$0.39 \pm 0.02$	$0.87 \pm 0.02$	$0.52 \pm 0.01$	$1.06 \pm 0.03$
0.02	$-7.07 \pm 0.02$	$0.69 \pm 0.01$	$1.03 \pm 0.01$	$0.60 \pm 0.01$	$1.27 \pm 0.05$

Before closing this discussion it is worth mentioning that the dynamical evolution of a cluster could affect the distribution of stars and then influence the integrated colours in the region of our interest ( r < r^0.5). As a simple insight on the problem, we address the attention on two major effects of the long-term evolution driven by two-body relaxation: i) a re-distribution of stars, due to the mass segregation (King et al. 1995), and ii) a process of star loss. Both the effects cause a flattening of the IMF as the dynamical evolution of the cluster proceeds, since the evaporation rate due to two-body relaxation is larger for low mass than for high mass stars as a consequence of mass segregation (Spitzer 1987; Bolte 1989; Vesperini & Heggie 1997). Concerning our model (at r < r^0.5) the displacement of lower masses from central region to outer parts of globulars and/or the evaporation of stars can be simulated by subtracting low mass stars. This means that the influence of these mechanisms on integrated colours can be evaluated by analyzing the models computed for different assumptions about the slope of the IMF at low masses (Table 3). The result is that in poorest clusters ( $M_{V}\sim -4.1$ mag) as in richest ones ( $M_{V}\sim -7.3$ mag) the reduced number of stars with mass lower than $0.6~M_{\odot}$ does not affect the integrated colours. This is easy to understand if one considers that the contribution of stars with $M \lesssim 0.6~M_{\odot}$ to the total V light is only of the order of 2 - 3$\%$ in the extreme case $M_{\rm inf}=0.2~M_{\odot}$. An additional loss of stars due to dynamics can be caused by the disc shocking, which however needs a sizeable mass segregation to produce a differential escape of stars with different masses. For our purpose this case can again be re-conducted to a variation of the IMF at low masses, with similar results. Of course, a detailed treatment of the dynamical processes is more appropriate to study the problem, since peculiar spatial distribution of stars and/or star losses may affect the integrated colours and their statistical fluctuations, but it is beyond the purpose of this paper.

 

 

Table 6: Synthetic integrated colours as in Table 3 but for different ages ($\eta$ = 0.4 and $\sigma _{\rm M}=0.04$)

Z	MV	U-B	B-V	V-R	V-I
age = 15 Gyr
0.0001	$-7.76 \pm 0.05$	$-0.01 \pm 0.015$	$0.62 \pm 0.02$	$0.44 \pm 0.01$	$0.88 \pm 0.02$
0.0003	$-7.74 \pm 0.03$	$0.05 \pm 0.01$	$0.65 \pm 0.01$	$0.43 \pm 0.005$	$0.87 \pm 0.01$
0.001	$-7.78 \pm 0.05$	$0.18 \pm 0.02$	$0.76 \pm 0.02$	$0.48 \pm 0.01$	$0.98 \pm 0.03$
0.006	$-7.35 \pm 0.03$	$0.43 \pm 0.01$	$0.92 \pm 0.01$	$0.54 \pm 0.01$	$1.10 \pm 0.02$
0.02	$-7.24 \pm 0.02$	$0.70 \pm 0.01$	$1.04 \pm 0.01$	$0.61 \pm 0.02$	$1.28 \pm 0.07$
age = 12 Gyr
0.0001	$-7.84 \pm 0.05$	$0.002 \pm 0.02$	$0.60 \pm 0.02$	$0.43 \pm 0.02$	$0.86 \pm 0.03$
0.0003	$-7.73 \pm 0.05$	$0.04 \pm 0.01$	$0.66 \pm 0.02$	$0.44 \pm 0.01$	$0.88 \pm 0.02$
0.001	$-7.78 \pm 0.04$	$0.15 \pm 0.015$	$0.77 \pm 0.02$	$0.48 \pm 0.01$	$0.97 \pm 0.02$
0.006	$-7.56 \pm 0.04$	$0.39 \pm 0.015$	$0.90 \pm 0.015$	$0.53 \pm 0.01$	$1.09 \pm 0.02$
0.02	$-7.23 \pm 0.01$	$0.64 \pm 0.01$	$1.01 \pm 0.01$	$0.60 \pm 0.01$	$1.28 \pm 0.05$
age = 10 Gyr
0.0001	$-7.91 \pm 0.04$	$-0.002 \pm 0.015$	$0.61 \pm 0.03$	$0.43 \pm 0.02$	$0.86 \pm 0.04$
0.0003	$-7.98 \pm 0.03$	$0.03 \pm 0.01$	$0.66 \pm 0.02$	$0.44 \pm 0.01$	$0.88 \pm 0.02$
0.001	$-7.73 \pm 0.03$	$0.15 \pm 0.01$	$0.76 \pm 0.02$	$0.47 \pm 0.01$	$0.96 \pm 0.02$
0.006	$-7.44 \pm 0.03$	$0.36 \pm 0.01$	$0.88 \pm 0.015$	$0.53 \pm 0.01$	$1.08 \pm 0.04$
0.02	$-7.17 \pm 0.03$	$0.60 \pm 0.01$	$0.99 \pm 0.02$	$0.58 \pm 0.02$	$1.26 \pm 0.07$
age = 8 Gyr
0.0001	$-8.13 \pm 0.03$	$-0.005 \pm 0.01$	$0.62 \pm 0.01$	$0.435 \pm 0.01$	$0.87 \pm 0.02$
0.0003	$-7.97 \pm 0.03$	$0.02 \pm 0.01$	$0.66 \pm 0.01$	$0.43 \pm 0.01$	$0.87 \pm 0.02$
0.001	$-7.58 \pm 0.04$	$0.13 \pm 0.015$	$0.74 \pm 0.02$	$0.465 \pm 0.01$	$0.96 \pm 0.025$
0.006	$-7.49 \pm 0.03$	$0.32 \pm 0.01$	$0.86 \pm 0.01$	$0.51 \pm 0.01$	$1.06 \pm 0.03$

Bearing this in mind, Fig. 6 compares current predictions as obtained for a cluster age of 15 Gyr with observational data for 147 galactic globulars from Harris (1999). The rather satisfactory agreement makes us confident about the reliability of the simulations we will discuss further on. The same Fig. 6 compares present with other theoretical predictions as given in the current literature. We regard the good agreement with the results recently presented by Kurt et al. (1999) on the basis of Padua isochrones (Fagotto 1994 and references therein) as an evidence that the still existing difference in the evolutionary results (see, e.g., Castellani et al. 2000 for a discussion on that matter) are of minor relevance as far as integrated cluster colours is concerned.

Figure 7 shows that present results appear in reasonable agreement with the two colour (U-B), (B-V) diagram for globulars in both the Galaxy and in the Large Magellanic Clouds without the intervention of the artificial shift invoked in Girardi et al. (1995). The same figure shows that the distribution of Large Magellanic Clouds clusters appears well fitted by predictions for young clusters already presented in Paper I.

Figure 7: Two colour diagram of observed integrated colours of LMC (filled circles) and galactic stellar clusters (diagonal crosses). Theoretical integrated colours by this work models with a fixed age t=15 Gyr and different chemical composition (solid line) and by Paper I models of ages ranging from 30 Myr to 5 Gyr for two different metallicities Z=0.02 (dashed line) and Z=0.006 (dotted line) are also plotted

The "solidity" of the previous results vis-a-vis the assumptions about the efficiency of mass loss has been investigated by repeating the previous computations but with the two new assumptions $\eta=0.2$ or 0.6. The results, as presented in Table 5, show that the case $\eta=0.2$runs against observations (Fig. 8), whereas the increase of $\eta$ from 0.4 to 0.6 has little effect on the predicted colours: only U-B colours for metal poor clusters appear marginally affected by the increase of mass loss, as expected from the evidence that such a colour is governed by the hot HB populations which, in turn, sensitively depends on the amount of mass loss. One may notice that the inadequacy of the 0.2 case would not be revealed by the two colour diagram in Fig. 7, since data in Fig. 8 show that the location of old clusters in this diagram tends to degenerate with respect of such a parameter. Conversely, one may conclude that predictions concerning (U-B) vs. (B-V) colours have a peculiar solidity against uncertainties in the amount of mass loss.

Figure 8: Observed integrated colours of galactic globular clusters (Harris 1999) are compared to models (t=15 Gyr) of with different labelled assumption on $\eta$

Finally, Fig. 9 and Table 6 show theoretical colours for selected assumptions about the cluster age, in the range 8 to 15 Gyr. As expected, one finds that decreasing the age the clusters tend to become redder. However, the amount of such a variation is so small that, taking into account the already discussed statistical uncertainties, one concludes that integrated colours are of little use to constrain the cluster ages within the above reported limits.