Surveys of clusters younger than ~20 Myr offer both advantages and disadvantages for studies of ^(M) at subsolar masses (see [M6]). On the positive side, low-mass stars are at their most luminous during this phase of pre-main sequence contraction (Section 3.4.3) A 0.3 M0 star has a luminosity of ~0.1 L0 at age 10 Myr, as compared with a luminosity of ~0.013 L0 at 100 Myr, or ^0.008 L0 on the main sequence. These objects can therefore be identified at substantially greater distances from the Sun than is possible for their older counterparts in Pleiades-like clusters.
Young clusters offer the further advantage of being largely free of the effects of dynamical evolution. Some degree of mass segregation exists in even the youngest systems, with the highest-mass stars often more centrally concentrated than the average cluster member. Given typical crossing times of between a few x105-106yr, this concentration is more likely to be a result of the conditions prevailing during the earliest stages of star formation rather than a consequence of dynamical relaxation. Solar-type and lower-mass stars appear to follow nearly identical radial density distributions (as in the Pleiades), thereby minimising potential biases due to incomplete areal coverage.
Youth, however, also carries its disadvantages. Since the cluster members are still in the pre-main sequence contraction stage, and no empirical mass estimates are yet available (Section 9.4.2), theoretical calculations offer the only method of estimating a mass-luminosity relationship. Applying these calibrations demands reliable age estimates, which are complicated by the fact that clusters do not form instantaneously: that is, cluster members span a range of ages. The resultant uncertainty in individual ages leads to corresponding uncertainty in mass, which is highest at these young ages when stars evolve so rapidly.
Further complications can arise from the presence of circumstellar disks in some systems. The material in those disks usually dominates the energy distribution in the mid- and far-infrared, and, given sufficiently high temperatures, may even make a significant contribution to the observed flux at shorter wavelengths, leading to an overestimate of both the photospheric bolometric luminosity and the mass. Finally, and most importantly, protoclusters are still embedded within the remains of the parent molecular cloud. The highly-variable obscuration caused by dust within that cloud is a serious impediment to an accurate interpretation of observations, at least at optical wavelengths.
Dust obscuration amounts to a loss of a few magnitudes at visual wavelengths for a typical few-Myr-old star cluster (such as IC 348, [L2]), and can reach levels of Av ~ 20-50 magnitudes, or more, in the denser regions of embedded protoclusters such as NGC 2024 (in Orion) or p Ophiuchi. The absorption is variable on scales of ^1,000-10,000 AU, or ~1-10arcsec for the nearer clusters. Under such circumstances, optical surveys are capable of identifying only the most luminous and least obscured cluster members, and are therefore poorly suited to providing catalogues for statistical analysis. Fortunately, the scattering properties of interstellar dust lead to significantly less absorption at longer wavelengths. In particular, the absorption at 2.2 AK, is almost a factor of 10 less than that at 0.5 Typical cloud temperatures are between 50 and 100 K, so dust emission peaks at A ~ 60 but is negligible at near-infrared wavelengths.
Given these circumstances, star-formation regions have long been recognised as interesting targets for infrared observations. However, while even the earliest scans led to notable discoveries - such as massive protostars like the Becklin-Neugebauer object and the surrounding Kleinmann-Low nebula in Orion [B2], [K1] - initial surveys were limited to either bright sources or small solid angles, and often both. It is only with the development in the mid-1990s of large-format infrared arrays and high-sensitivity spectrographs that it has become possible to undertake studies capable of detecting protostars with masses below 0.1 M0 over the entire area of major star-formation regions. As a result, the full potential of infrared studies remains to be realised. Nonetheless, preliminary results are intriguing.
In general, investigations of the stellar mass function in young, star-forming regions have followed two broadly complementary lines of attack: statistical analysis of deep star-counts, and more detailed source-by-source analyses. Those methods are described in more detail in the following two sections, and Section 9.7.3 uses observations of the stellar-mass members of IC 348 and the Orion Nebula Cluster (ONC) as practical examples of their application. The last few years have seen detailed surveys of a number of young clusters and associations, with particular attention focused on the substellar-mass members. Section 9.7.4 summarises the main results from those studies, and considers the overall implications for the initial mass function.
Near-infrared number-magnitude counts offer a straightforward means of probing the stellar content in obscured, star-forming regions. The luminosity function of cluster members can be determined statistically by comparing source counts centred on the cluster against counts made within nearby, off-cluster fields. If the cluster distance is known, the apparent luminosity function <(mK) can be transformed to $(Mk). Since this technique is based on direct imaging, it offers the possibility of obtaining a complete census of even the lowest-luminosity cluster members through a series of simple and efficient observations. With the current generation of infrared detectors, data can be obtained covering an entire cluster in a matter of only a few nights on an intermediate-sized telescope.
There are, however, complications in deriving a mass function from the resultant K-band luminosity function:
• The (mass, MK) relationship must be appropriate for the age of the cluster, tc, and therefore requires both an accurate estimate of tc and reliable pre-main sequence mass-luminosity relationships.
• The cluster stars are unlikely to be exactly coeval. Rather than the cluster forming in a single burst at time T = T0 — tc, where T0 is the present time, individual stars span a range of ages, ti = tc ± At.
• While working in the near-infrared minimises the effects of obscuration, differential reddening (either foreground or within the cluster itself) is likely to be present at the 0.1-1.0 magnitude level in AK. Moreover, there may be significant differences between the total obscuration in cluster and off-cluster fields: dust within the young cluster usually leads to higher reddening of background stars. This can produce systematic errors in <(mK).
• Emission from circumstellar disks can contribute significantly to the flux in the thermal infrared (A > 2 in young protostars.
• Source counts alone cannot distinguish single and multiple stars.
• All of these effects combine to make it difficult to distinguish cluster members from field stars.
Many of these problems can be addressed: optical and near-infrared colours can be used to map differential reddening, while excess radiation at longer wavelengths (above the predicted photospheric flux) measures possible contributions from hot circumstellar dust.
The usual technique is to compute the expected K-band luminosity function (KLF) based on an estimated initial mass function, age and star-formation history. The last two parameters can be determined to some extent from photometric and/or spectroscopic data, although a degree of guess-work is also often required. The predicted luminosity function is matched against the observed KLF (in the observational plane), and the input parameters adjusted until reasonable concordance is achieved. Initial analyses [Z2] were based on single-burst star-formation models, but more recent studies [L2], [L3] have adopted more complex (and more realistic) star-forming histories. In general, this approach is effective at ruling out inappropriate models, but only identifies consistent (rather than unique) solutions.
An alternative method of studying young clusters is to use spectroscopic and photometric observations to estimate bolometric luminosities and effective temperatures for each cluster member. Given these data, each star can be placed on the H-R diagram, making due allowance for foreground reddening and circumstellar dust emission, while eliminating foreground and background field stars. Comparison with pre-main sequence evolutionary tracks permits the estimation of masses and ages on a star-by-star basis (see Figure 9.8), whilst the mass function and starformation history follow from summation of the individual results.
In principle, this approach offers higher precision than the statistical KLF analysis. There are, however, significant practical obstacles - notably in obtaining spectroscopic data of the requisite accuracy for the faintest, and most highly obscured, cluster members. Simultaneous observations of tens of candidate cluster members, using optical or near-infrared multi-object spectrographs can go some way towards addressing the latter problem, but photon scarcity limits the full-scale application of this method.
9.7.3 Two case studies: IC 348 and the Orion Nebula Cluster
To illustrate the relative merits of these two types of analysis we consider their application to IC 348 and the Orion Nebula Cluster (the Trapezium), two well-
studied, young star-forming regions. Observations of these clusters demonstrate the complementary nature of the techniques.
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