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Astron. Astrophys. 358, 869-885 (2000)

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4. Statistical significance of the results

4.1. Inconsistency of the data with a constant SFH

There is a widespread myth on galactic evolutionary studies about the near constancy of the SFH in the disk. This comes primarily from earlier studies setting constraints to the present relative birthrate (e.g., Miller & Scalo 1979; Scalo 1986). The observational constraints have favoured a value near unity, and that was interpreted as a constant SFH.

This constraint refers only to the present star formation rate. As pointed out by O'Connell (1997) and Rocha-Pinto & Maciel (1997), it is not the same as the star formation history .

A typical criticism to a plot like that shown in Fig. 8 is that the results still do not rule out a constant SFH, since the oscilations of peaks and lulls around the unity can be understood as fluctuations of a SFH that was `constant' in the mean. This is an usual mistake of those who are accustomed to the strong, short-lived bursts in other galaxies.

The ability to find bursts of star formation depends on the resolution. Suppose a galaxy that has experienced only once a real strong star formating burst during its entire lifetime. The burst had an intensity of hundred times the average star formation in this galaxy, and has lasted [FORMULA] yr, which are typical parameters of bursts in active galaxies. Fig. 10 shows how this burst would be noticed, in a plot similar to that we use, as a function of the bin size. In a bin size similar to that used throughout this paper (0.4 Gyr), the strong narrow burst would be seen as a feature with a relative birthrate of 3.5. If we were to convolve it with the age errors, like those we used in Paper I, we could find a broad smeared peak similar to those in Fig. 8. For a biggest bin size (1 Gyr), the relative birthrate of the burst would be lower than 1.5. Hence, a relative birthrate of 1.5 in a SFH binned by 1 Gyr is by no means constant. A great bin size can just hide a real burst that, if occurring presently in other galaxies, would be accepted with no reserves.

[FIGURE] Fig. 10. The evanescence of a strong short-lived star formation burst due to the bin size. A star formation burst, lasting [FORMULA] yr, and with varying intensity (10, 50 and 100 times more intense than the average star formation rate) was considered. The plot shows the value of the relative birthrate at the time of a burst. It can be seen that for an age bin similar to that used throughout this paper, namely 0.4 Gyr, even the most strong and narrow burst would be represented by a feature not exceeding 3.5 in units of relative birthrate.

In the case of our galaxy, the bin size presently cannot be smaller than 0.4 Gyr. This is caused by the magnitude of the age errors. We are then limited to features whose relative birthrate will be barely greater than 3.0, especially taking into consideration that the star formation in a spiral galaxy is more or less well distributed during its lifetime. Therefore, in a plot with bin size of 0.4 Gyr, relative birthrates of 2.0 are in fact big events of star formation.

A conclusive way to avoid these mistakes is to calculate the expected fluctuations of a constant SFH in the plots we are using. We have calculated the Poisson deviations for a constant SFH composed by 552 stars. In Fig. 11 we show the 2[FORMULA] lines (dotted lines) limiting the expected statistical fluctuations of a constant SFH.

[FIGURE] Fig. 11. Star formation rate with counting errors. The error bars correspond to an error of [FORMULA], where N is the number of stars found in each age bin. The dotted lines indicate the 2[FORMULA] variations around a constant SFR for a sample having 552 stars. The labels over the peaks are the same as in Fig. 8.

The Milky Way SFH, in this figure, is presented with two sets of error bars, corresponding to extreme cases. The smallest error bars correspond to Poisson errors ([FORMULA], where N is the number of stars in each metallicity bin). The thinner longer error bar superposed on the first shows the maximum expected error in the SFH, coming from the combination of counting errors, IMF errors and scale height errors. These last two errors were estimated from Figs. 4 and 5. The contribution of the scale height errors are greatest at an age of 3.0 Gyr, due to the steep increase of the scale heights around solar-mass stars. The effect of the IMF errors are the smallest, but grows in importance for the older age bins.

From the comparison of the maximum expected fluctuations of a constant SFH and the errors in the Milky Way SFH, it is evident that some trends are not consistent with a constant history, particularly bursts A and B, and the AB gap. We can conclude that the irregularities of our SFH cannot be caused by statistical fluctuations.

4.2. The uncertainty introduced by the age errors

The age error affects more considerably the duration of the star formation events, since they tend to scatter the stars originally born in a burst. We can expect that this error could smear out peaks and fill in gaps in the age distribution. A detailed and realistic investigation of the statistical meaning of our bursts has to be done in the framework of our method, following the observational data as closely as possible. In the case of the Milky Way, the input data is provided by the age distribution. We have supposed that this age distribution is depopulated from old objects, since some have died or left the galactic plane. Our method to find the SFH makes use of corrections to take into account these effects. However, some features in the age distribution could be caused rather by the incompleteness of the sample. These would propagate to the SFH giving rise to features that could be taken as real, when they are not.

Thus, if we want to differentiate our SFH from a constant one, we must begin with age distributions, generated by a constant SFH, depopulated in the same way that the Galactic age distribution. With this approach, we can check if the SFH presented in Fig. 8 can be produced by errors in the isochrone ages in conjunction with statistical fluctuations of an originally constant SFH.

We have done a set with 6000 simulations to study this. Each simulation was composed by the following steps:

  1. A constant SFH composed by 3000 `stars' was built by randomly distributing the stars from 0 to 16 Gyr with uniform probability.

  2. The stars are binned at 0.2 Gyr intervals. For each bin, we calculate the number of objects expected to have left the main sequence or the galactic plane. This corresponds to the number of objects which we have randomly eliminated from each age bin. The remaining stars (around 600-700 stars at each simulation) were put into an `observed catalogue'.

  3. The real age of the stars in the `observed catalogue' is shifted randomly according to the average errors presented in Fig. 5 of Paper I. After that, the `observed catalogue' looks more similar to the real data.

  4. The SFH is then calculated just as it was done for the disk. From each SFH the following information is extracted: dispersion around the mean, amplitude and age of occurrence of the most prominent peak, amplitude and age of occurrence of the deepest lull.

One of the problems that we have found is that due to the size of the sample, and the depopulation caused by stellar evolution and scale height effects, the SFH always presents large fluctuations beyond 10 Gyr. These fluctuations are by no means real. They arise from the fact that in the observed sample (for the case of the simulations, in the `observed catalogue'), beyond 10 Gyr, the number of objects in the sample is very small, varying from 0 to 2 stars at most. In the method presented in the subsections above, we multiply the number of stars present in the older age bins by some factors to find the number of stars originally born at that time. This multiplying factor increases with age and could be as high as 12 for stars older than 10 Gyr; this way, by a simple statistical effect of small numbers, we can in our sample find age bins where no star was observed neighbouring bins where there are one or more stars. And, in the recovered SFH, this age bin will still present zero stars, but the neighbouring bins would have their original number of stars multiplied by a factor of 12. This introduces large fluctuations at older age bins, so that all statistical parameters of the simulated SFHs were calculated only from ages 0 to 10 Gyr.

In Fig. 12, we present two histograms with the statistical parameters extracted from the simulations. The first panel shows the distribution of dispersions around the mean for the 6000 simulations. The arrow indicates the corresponding value for the Milky Way SFH. The dispersion of the SFH of our Galaxy is located in the farthest tail of the dispersion distribution. The probability of finding a dispersion similar to that of the Milky Way is lower than 1.7%, according to the plot. In other words, we can say, with a significance level of 98.3%, that the Milky Way SFH is not consistent with a constant SFH.

[FIGURE] Fig. 12a and b. Distribution of parameters from 6000 simulations, using the scale heights from Scalo (1986). The left panel shows the dispersions around the mean SFH, while the right panel gives the value of the most prominent peak. In all the plots, the arrow indicates the corresponding value for the Milky Way SFH.

In panel b of Fig. 12, a similar histogram is presented, now for the value of the most prominent peak that was found in each simulation. In the case of the Milky Way, we have B1 peak with [FORMULA]. Just like the previous case, it is also located in the tail of the distribution. From the comparison with the values of the highest peaks that could be caused by errors in the recovering of an originally constant SFH, we can conclude with a significance level of 99.5% that our Galaxy has not had a constant SFH.

The use of Holmberg & Flynn (2000) scale heights in the simulations increases these significance levels to 100% and 99.9%, respectively.

These significance levels refer to only one parameter of the SFH, namely the dispersion or the highest peak. For a rigorous estimate of the probability of finding a SFH like that presented in Fig. 11, from an originally constant SFH, one has to calculate the probability to have neighbouring bins with high star formation, followed by bins with low star formation, as a function of age. This can be calculated approximately from Fig. 13, where we show box charts with the results of the 6000 simulations. Superimposed on these box charts, we show the SFH, now calculated with Holmberg & Flynn (2000)'s scale heights. For the sake of consistency, the simulations shown in the figure also use these scale heights, but we stress that the same quantitative result is found using Scalo's scale heights.

[FIGURE] Fig. 13. Box charts showing the results of the 6000 simulations, using Holmberg & Flynn (2000)'s scaleheights. The horizontal lines in the box give the 25th, 50th, and 75th percentile values. The error bars give the 5th and 95th percentile values. The two symbols below the 5th percentile error bar give the 0th and 1st percentile values. The two symbols above the 95th percentile error bar give the 99th and 100th percentiles. The square symbol in the box shows the mean of the data. Superimposed, the Milky Way SFH is shown. From the comparison with the distribution of results at each age bin, the probability to find each particular event in a constant SFH can be calculated. The numbers besides the major events give the probabilities for their being fluctuations of a constant SFH.

A lot of information can be drawn from this figure. First, it can be seen that a typical constant SFH would not be recovered as an exactly `constant' function in this method. This is shown by the boxes with the error bars which delineate 2[FORMULA]-analogous to those lines shown in Fig. 11. The boxes distribute around unity, but shows a bump between 1 to 2 Gyr, where the average relative birthrate increases to 1.4. This is an artifact introduced by the age errors. In each individual simulation, the number of stars scattered off their real ages increases as a function of the age. In the recovered SFH there will be a substantial loss of stars with ages greater than 15 Gyr, since they are eliminated from the sample (note that originally, these stars would present ages lower than 15 Gyr, and just after the incorporation of the age errors they resemble stars older than it). This decreases the average star formation rate with respect to the original SFH, and the proportional number of young stars increases, because they are less scattered in age due to errors. This gives rise to a distortion in the expected loci of constant SFHs. Note also the increase in the [FORMULA]-region as we go towards older ages, reflecting the growing uncertainty of the chromospheric ages.

The diagram allows a direct estimate of the probability for each feature found in the Milky Way SFH be produced by fluctuations of a constant SFH. The box charts gives the distribution of relative birthrates in each age bin. An average probability for the major events of our SFH are shown in Fig. 13, besides the features under interest. Rigorously speaking, the probability for the whole Milky Way SFH be constant, not bursty, can be estimated by the multiplication of the probability of the individual events in this figure. It can be clearly seen that it is much less than the 2% level we have calculated from only one parameter of the SFH. Particularly, note that the AB gap has zero probability to be caused by a statistical fluctuation. All of theses results show that the Milky Way SFH was by no means constant.

4.3. Flattening and broadening of the bursts

Since the errors in the chromospheric ages are not negligible, a sort of smearing out must be present in the data. Due to this, a star formation burst found in the recovered SFH must have been originally much more pronounced. This mechanism probably affects much more older bursts, since the age errors are greater at older ages and the depopulation by evolutionary and scaleheight effects is more dramatic. We can assume that if we found a feature like a burst at say 8 Gyr ago, this probably was much stronger in order to be preserved in the recovered SFH.

The first aspect we want to show is that the errors produce a significant flattening of the original peaks. To do so, we use simulations of a SFH composed by a single burst over a constant star formation rate. The `burst' is characterized by occurring at age [FORMULA], having intensity c times the value of the constant star formation rate, and lasting 1 Gyr. We want to know the fraction of the burst that is recovered, as a function of age and of the burst intensity.

We have performed 50 simulations for each pair [FORMULA], with around 3000 stars in each simulation. A summary of these simulations is shown in Fig. 14. In all the panels (for varying c), the fraction of the recovered burst is high for recent bursts and falls off smoothly until 8-9 Gyr, when it begins to become constant. This stabilization reflects the predominance of the statistical fluctuations, since the recovered fraction is the same, regardless of the age of occurrence. What happens is that the burst becomes more or less undistinguished from the fluctuations. From this we can conclude that it is more difficult to find bursts older than 8-9 Gyr, irrespective of its original amplitude.

[FIGURE] Fig. 14. Recovered fractions for a SFH composed of a single burst superimposed on a constant rate. The `burst' is characterized by occuring at age [FORMULA], having intensity c times the value of the constant star formation rate, and lasting 1 Gyr. We show the cases for [FORMULA] 1.5, 3.5, 5 and 10. In all the plots, the abscissa indicates the age [FORMULA] where the burst happened. The fraction recovered in the first 2 Gyr of age is greater than unity, due to the same problem that distorted the [FORMULA] loci of the constant SFHs in Fig. 13 (see text).

A second problem in the method is the broadening of the bursts. This depends sensitively on the age at which the burst occurs, and the results are even more dramatic. To illustrate this, another set of simulations was done. We consider now a SFH composed of a single burst, of 1000 stars, lasting 0.4 Gyr. No star formation occurs except during the burst. We vary the age of occurrence from 0.3 Gyr to 6 Gyr ago. Just one simulation was done for each age of occurrence, since we are only looking for the magnitude of the broadening introduced by the errors, so the exact shape of the recovered SFH does not matter. The recovered SFHs are shown in Fig. 15. Only the younger bursts are reasonably recovered. The burst at 6 Gyr can still be seen, although many of its stars has been scattered over a large range of ages.

[FIGURE] Fig. 15. Recovered SFHs for an original SFH composed of a single burst of 1000 stars. The curves show examples of how these bursts are broadened, depending on their age of occurrence, due to age errors.

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Online publication: June 20, 2000