Astron. Astrophys. 330, 990-998 (1998)

4. The stellar population of the Serpens cloud

While theoretical work on PMS evolution is progressing, an observational effort to understand star forming history in a cloud is needed in order to address the IMF question.

4.1. The luminosity functions

The Initial Mass Function (IMF) is of fundamental interest to several fields of astronomy. Current estimates of the IMF are based on observations of stars in the solar neighbourhood (Salpeter 1955, Scalo 1986). PMS stars undergo considerable luminosity evolution with poorly known time scales before they arrive at the zero age main sequence (ZAMS). Thus the observed luminosity function is a result of the IMF, the PMS evolution and the star forming history. The K-band luminosity function (KLF) of the identified Serpens sources and of all stars observed toward the central part of the Serpens cloud core is presented in Fig. 6. The KLF is displayed as histograms of the number of sources versus the apparent K magnitude, with a bin size of 1 mag.

Fig. 6. K magnitude distribution of the identified Serpens sources and of all detected NIR sources in the field.

The comparison of the distributions shows that all sources with are Serpens objects, as well as of the sources with . When considering only the distribution of all stars detected at K, irrespectively of their nature, we note the presence of a peak occurring at mag and a decrease in the observed number of sources with decreasing K brightness beyond mag. This apparent turnover is probably due to our K detection limit (K=16.3). On the other hand, the KLF of the identified Serpens sources presents a turnover beyond K=12 that is well below our K detection limit and cannot be considered as an artefact. Such a trend has been called a turnover because KLFs derived from Miller & Scalo (1979) or Salpeter (1955) IMF show the number of sources increasing with decreasing K brightness. However, these KLFs were derived using a mass-luminosity relationship appropriate for main sequence stars. As we demonstrate in the next section, turnovers in luminosity functions are not necessarily inconsistent with Miller & Scalo (1979) or Salpeter (1955) mass functions when the cluster members are not yet on the main sequence. In a study of several star forming regions, Zinnecker et al. (1993) also find and discuss similar results. We have determined the cumulative number of sources per square degree brighter than a given K magnitude detected by Casali & Wainscoat (quoted in Eiroa & Casali 1992, hereafter CW92) in a field close to Serpens (galactic coordinates l = 40 , b = -4 ) and normalized to an area equal to that covered by our K image. That curve can be fitted by . This means that around 260 sources should be detected with in the area we surveyed. However, the mean extinction of the cloud is about = 10 mag (Zhang et al., 1988) and, therefore, the number of sources in the line of sight to Serpens is expected to be lower. The results are presented in Fig. 7 which plots the cumulative number of sources brighter than a given K magnitude versus that magnitude.

Fig. 7. Cumulative number of stars brighter than a given K magnitude. Identified Serpens sources are plotted as crosses while filled circles represent all sources detected in the field. The dashed line represents the expected number of background stars after the empirical star count survey of Casali & Wainscoat, normalized to a 19 square arcmin field. The continuous line is the least square fit in the range .

The CW92 line is plotted as a dashed line and normalized to an area equal to that covered by our K image with an extinction of =1.1 (equivalent to =10, using the RL85 standard extinction law). In the range , the relation between and the apparent K magnitude for all sources appears linear and a least square analysis was performed on these data. The resulting fit is with a correlation coefficient of 0.99. This linear relation is not satisfied in the case of the identified Serpens sources. The deficiency of stars with reflects the apparent turnover in the K-distribution of stars.

This plot shows another interesting point. There is an excess of stars with with respect to the number of expected sources from the star count survey of CW92. Since the majority of these stars is related to the molecular cloud, this evidences a clustering process in Serpens.

4.2. KLF modeling

A fundamental consequence of the theory of stellar evolution is that the life history of a star is almost entirely predetermined by its initial mass. Consequently, to understand the star formation history and the consequent luminosity evolution of an embedded population of young stars such as the one we observed, requires a detailed knowledge of both the initial distribution of stellar masses at birth and how this quantity varies through space and time. Since the stars are young, a time-dependent main sequence mass-luminosity relation must be used to determine the stellar mass. In this aim, we modeled the predicted form of the K luminosity function using Miller & Scalo (1979) IMF and theoretical PMS mass-luminosity relationships from the isochrones of D'Antona & Mazzitelli (1994). Our first objective was to directly compare synthetic KLFs with observations to place constraints on the star formation history and the underlying mass function of the Serpens molecular cloud.

We then evaluated the following equation:

where is the slope of the mass-K luminosity relation and is the underlying stellar mass function given by the half-gaussian form of the Miller & Scalo (1979) IMF:

where C₀ = 106.0, C₁ =1.09 and C₂ = -1.02. The PMS evolutionary tracks give, for each age ranging from 7 yr to 10⁸ yr, and each mass from 0.02 to 2.5 , the effective temperature and luminosity of the star. For simplicity we considered stars as blackbodies and the luminosities were converted into K magnitude. The tracks used in our models were derived assuming Alexander et al. (1989) opacities and the Canuto and Mazzitelli (1990, 1992) convection model. For statistical purposes, KLFs were constructed using intervals of one magnitude bin. We did not take into account the infrared excess emission and we assumed that our PMS stars were diskless. In any case, the infrared excess present in some of the objects we observed is smaller than the bin size of the observed and modeled KLFs.

4.3. Comparison of models with the observations

In Fig. 8a and 8c, we have plotted the mass-m_k luminosity relation that we used at two different ages (0.3 and 2 Myr.), while Fig. 8b and 8d show the corresponding KLFs for a coeval cluster of low-mass stars. As pointed out by Zinnecker et al. (1993), features as peaks in the KLF are strongly correlated to the sharp inflection at the corresponding point in the mass-K luminosity relation. This point of inflection is due to the deuterium burning in the contracting PMS star and it is seen to move towards lower masses as the cluster ages. A physical interpretation of this phenomenon is that while deuterium is burning strongly in a star of given mass, D-burning is ending in higher mass stars and has yet to begin in stars of lower masses, for a sample of stars born at the same time. This leads to an increase in the value of and a peak in the luminosity function at the corresponding K magnitude. That is why features and turnovers in KLFs are not necessarily due to features in the IMF; rather they may often reflect the complex process of PMS evolution.

Fig. 8. Time dependant mass-mk luminosity relation, as derived from evolutionary tracks of d'Antona & Mazzitelli (1994) for 0.3 Myr (a) PMS stars and model 2.2 m (mk) luminosity function for a coeval cluster of low-mass stars of 0.3 Myr (b). Figs. c and d: same as a and b for 2 Myr PMS stars.

The modeled KLFs were compared to the KLF observed for the Serpens molecular cloud. None of the coeval models fits the shape of the observed KLF in a satisfactory way. This led us to construct KLFs for clusters of different ages and to compare the resulting KLF with the observations. The best fit to the data is obtained with two bursts of star formation at different epochs. One is 10⁵ yr old, the other is around 3 Myr old. Fig. 9 presents histograms of the model KLF superimposed on the observed KLF for the identified Serpens sources, normalized to 100 stars. To attempt to account for the effects of uniform foreground extinction, we extincted our models by 1.5 mag at K, which introduced a shift in the model KLF toward larger magnitudes, without changing its shape. We can note some differences between the two curves, but the general shape, the location and the intensity of the peak of the observed luminosity function are quite well reproduced by the model. However, these results are qualitative since we could not take into account the effects of differential extinction seen toward the cloud (because we lack spectroscopic data), and the relatively poor statistic number of sources identified as cloud members (55 stars), may partly explain the differences observed between the two histograms.

Fig. 9. Comparison of model and Serpens K luminosity function, normalized to 100 stars. To account for the effects of uniform foreground extinction, we extincted our model KLF by 1.5 mag at K.

4.4. Influence of the IMF upon KLFs

In order to evaluate the influence of the IMF upon the resulting K luminosity functions, we have used different IMFs. We still kept the general shape of the half-gaussian form of MS79 IMF (given by Eq.  2), but we examined how the KLF does evolve by changing the number of stars at the two extremes of the curve. The expressions used can be summarized as follows:

The results of these models are presented in Fig. 10, which plots the shapes of each IMF used, while Fig. 11 shows the corresponding luminosity functions.

Fig. 10. IMFs resulting from Eq. 3 to Eq. 5. For , Eq. 3 and Eq. 4 are equivalent.

Fig. 11. KLFs resulting from Eq. 2 to Eq. 5

We can learn from Fig. 11 that on the one hand, increasing the value of (i.e. increasing the number of high-mass stars) leads to an increase of the number of bright stars. On the other hand, Eq.  3produces a KLF which increases the number of stars with fainter magnitudes (i.e. low-mass stars). However, changing the value of does not alter significantly the resulting KLFs, which remains within the error bars.

4.5. Star formation efficiency

The Star Formation Efficiency (SFE) is defined by: , and represents the mass of gas converted into stellar mass in a molecular cloud. A realistic estimate of this parameter is not simple since: i) the cloud mass depends on the density tracers used and on the spatial resolution of the observations, and ii) the estimate of the stellar masses cannot be inferred directly from the observations. We used the value of 1450 given by White et al. (1995) to estimate the cloud mass. This lower limit of comes from their high resolution observations of the Serpens Nebula.

To estimate the mass of stars, we used our KLF model that gives the number of PMS stars for each magnitude bin (i.e. interval of mass). We found a total of 16.5 for the 55 sources identified as Serpens objects which gives a mean stellar mass of 0.3 and then, a SFE of . This estimate is comparable with those obtained towards other dark cloud complexes, forming low- to intermediate-mass stars; Ophiuchus: (Wilking et al. 1989); Taurus: (Kenyon et al. 1990); L1641: (Evans & Lada 1991); L1630: (Lada 1990). These low limits to the SFE reflect the star forming activity that has occurred to date (Evans & Lada 1991, Leisawitz et al. 1989).

An upper limit of the SFE in the Serpens cloud can be estimated by the assumption that all the 138 objects detected belong to Serpens, and that they are 0.3 stars in average. This leads to a SFE of . This is lower than the value obtained by Eiroa & Casali (1992), who found a SFE in the range. To estimate the cloud mass, they used the value of 450 based on the measurements by Loren et al. (1979). Their lower limit of the SFE was obtained by considering that stars with have , stars with have , and stars with have . The upper limit is obtained with the assumption that all the detected objects belong to Serpens and are . These values are significantly larger than ours, because: i) the stellar masses may have been overestimated, and ii) the low-resolution of Loren et al.'s (1979) observations were made with large beams and may present a factor of uncertainty of 3-5 for the estimated mass due to uncertainties in radiative transfer effects and geometry.

We conclude that the SFE in the Serpens cloud lies in the range . This is in good agreement with the value obtained by White et al. (1995), who found a SFE of , with a total stellar mass of 37 , and confirm that the SFE in this dark cloud is no more than a few percent.

© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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