1 Introduction

Recent observations suggest that most stars in our Galaxy and other galaxies form in compact clusters. In particular, near-IR imaging surveys of nearby molecular cloud complexes have shown that the star formation activity is typically concentrated within a few rich clusters associated with massive dense cores which constitute only a small fraction of the total gas mass available (e.g. Lada et al. 1993; Zinnecker et al. 1993). These embedded clusters comprise various types of young stellar objects (YSOs) - from still collapsing protostars to young main sequence stars - but are usually dominated in number by low-mass pre-main sequence (PMS) stars, i.e., T Tauri stars. Young clusters provide excellent laboratories for studying the formation and early evolution of stars through the observational analysis of large, genetically homogeneous samples of embedded YSOs. Two key characteristics of these young stellar populations are their luminosity distribution and their mass spectrum, which give important observational constraints on the stellar initial mass function (IMF) in the Galaxy. Observations of young embedded clusters can also help us understand possible links between parent cloud properties and the resulting stellar masses. They are however hampered by: (1) dust extinction from the parent cloud which hides most of the newly formed stars at optical wavelengths, (2) the difficulty to recognize the nature of individual sources (e.g. protostars, T Tauri stars, or background sources), (3) the youth (and thus poorly known intrinsic properties) of most cluster members.

Owing to these difficulties, the census of embedded YSOs provided by IRAS and near-IR studies is far from complete even in the nearest clouds (e.g. Wilking et al. 1989 - hereafter WLY89). Thanks to its high sensitivity and good spatial resolution in the mid-IR, the ISOCAM camera on board ISO (Cesarsky et al. 1996; Kessler et al. 1996) was a powerful tool to achieve more complete surveys for YSOs in all major nearby star-forming regions (see Nordh et al. 1996, 1998; Olofsson et al. 1999; Persi et al. 2000).

The nearby $\mbox{$\rho$ ~Ophiuchi}$ cloud is one of the most actively studied sites of low-mass star formation. Its central region harbors a rich embedded cluster with about 100 members recognized prior to the present work (e.g. WLY89, Casanova et al. 1995). While a dispersed population of optically visible young stars, associated with the Upper-Scorpius OB association, has a typical age of several million years (Myr) (e.g. Preibisch & Zinnecker 1999), the central embedded cluster is recognized as one of the youngest clusters known with an estimated age on the order of $\sim$0.3-1 Myr (e.g. WLY89, Greene & Meyer 1995; Luhman & Rieke 1999). This young cluster has been extensively studied at wavelengths ranging from the X-ray to the radio band. The satellites Einstein and ROSAT have revealed $\sim$70 highly variable X-ray sources associated with magnetically-active young stars, including deeply embedded protostellar sources (Montmerle et al. 1983; Casanova et al. 1995; Grosso et al. 2000). In the near-IR, the cloud has been deeply surveyed from the ground using large-format arrays (Greene & Young 1992; Comerón et al. 1993; Strom et al. 1995; Barsony et al. 1997). Unfortunately, due to difficulties in discriminating between background sources and embedded YSOs without performing time-consuming mid-IR photometry (e.g. Greene et al. 1994) or near-IR spectroscopy (e.g. Greene & Lada 1996; Luhman & Rieke 1999), these recent near-IR surveys have only partially increased the number of classified, recognized members. Finally, while only relatively poor angular resolution IRAS data are available so far in the far-IR (e.g. WLY89), deep imaging surveys at an angular resolution of $10-15\hbox{$^{\prime\prime}$ }$ or better exist at (sub)millimeter wavelengths (e.g., Motte et al. 1998 - hereafter MAN98 - Wilson et al. 1999).

It is thanks to the illuminating example of the $\mbox{$\rho$ ~Ophiuchi}$ embedded cluster that the now widely used empirical classification of YSOs was originally introduced. Three IR classes were initially distinguished based on the shapes of the observed spectral energy distributions (SEDs) between $\sim$2 $\mu$m and $\sim$25-100 $\mu$m (Lada & Wilking 1984, WLY89). Objects with rising SEDs in this wavelength range were classified as Class I, sources with SEDs broader than blackbodies but decreasing longward of $\sim$2 $\mu$m as Class II, and sources with SEDs consistent with (or only slightly broader than) reddened stellar blackbodies as Class III. These morphological SED classes are interpreted in terms of an evolutionary sequence from (evolved) protostars (Class I), to T Tauri stars with optically thick IR circumstellar disks (Class II), to weak T Tauri stars with at most optically thin disks (Class III) (Lada 1987; Adams et al. 1987; André & Montmerle 1994 - hereafter AM94). A fourth class (Class 0) was subsequently introduced by André et al. (1993) to accommodate the discovery in the radio range of cold sources with large submillimeter to bolometric luminosity ratios and powerful jet-like outflows, such as VLA 1623 in ${\rho}$ Oph A (e.g. André et al. 1990; Bontemps et al. 1996). Class 0 objects, which have measured circumstellar envelope masses larger than their inferred central stellar masses, are interpreted as young protostars at the beginning of the main accretion phase (e.g. André et al. 2000). The fact that the $\mbox{$\rho$ ~Ophiuchi}$ central region contains at least two Class 0 protostars (André et al. 1993), as well as numerous ( $\lower.5ex\hbox{$\buildrel > \over \sim$ }$60) pre-stellar condensations (MAN98), demonstrates that it is still actively forming stars at the present time. The distance to the $\mbox{$\rho$ ~Ophiuchi}$ cloud is somewhat uncertain. Usually, a value of 160 pc is adopted (e.g. Chini 1981). However, recent Hipparcos results on the Upper-Scorpius OB association (de Zeeuw et al. 1999) provide a reasonably accurate estimate of $145\pm 2$ pc for the average distance to the stars of the OB association. The embedded cluster is located at the inner edge of the molecular complex on the outskirts of the OB association (e.g. de Geus 1989), and not very far, in projection, from the association center (less than $4\hbox{$^\circ$ }$ apart which corresponds to $\sim$10 pc; see Fig. 9 of de Zeeuw et al. 1999). In this paper, we therefore adopt a distance of $d = 140\pm 10$ pc for the $\mbox{$\rho$ ~Ophiuchi}$ IR cluster which corresponds to a distance modulus $5\times\log_{10}(d/10~\rm pc) = 5.73\pm0.15$.

The layout of the paper is as follows. Section 2 gives observational details (Sect. 2.2) and describes the way the data have been reduced to obtain mid-IR images and extract point-sources (Sect. 2.3) along with the photometric uncertainties and the sensitivity levels (Sect. 2.4). In Sect. 3, the identification of detected sources is discussed (Sect. 3.1) and the selection of a new population of 123 Class II YSOs, as well as 16 Class I and 77 Class III YSOs, is described (Sects. 3.2-3.5). We then derive luminosities for these YSOs and build the corresponding luminosity functions in Sect. 4. In Sect. 5, we model the luminosity functions for Class II and Class III YSOs in terms of the underlying mass function and star formation history. In Sect. 6, we discuss the resulting constraints on the IMF of the $\mbox{$\rho$ ~Ophiuchi}$ cloud down to $\sim$ $0.06\,\mbox{$M_\odot$ }$, (Sects. 6.1-6.2) as well as related implications (Sects. 6.3-6.5).