6 Discussion

6.1 The initial mass function in $\mbox{$\rho$ ~Ophiuchi}$

Since the $\mbox{$\rho$ ~Ophiuchi}$ molecular cloud is still actively forming stars, the final IMF of the cluster cannot be directly measured. The masses of the stars already formed in the cluster provide only a "snapshot'' of the local IMF (cf. Meyer et al. 2000). However, assuming that the mass distribution of formed stars does not change significantly with time during the cluster's history (an assumption made in the models of Sect. 5.1), any snapshot of the mass distribution taken on a large, complete population of PMS objects should accurately reflect the end-product IMF.

Compared to previous investigations of the IMF in the $\mbox{$\rho$ ~Ophiuchi}$ cloud based exclusively on near-IR data (e.g. Comerón et al. 1993; Strom et al. 1995; Williams et al. 1995; Luhman & Rieke 1999), the use of mid-IR photometry has allowed us to consider a much larger sample of young stars (see, e.g., Fig. 6). Since our sub-sample of Class II YSOs is fairly large (123 objects) and complete down to low luminosities, it provides an excellent opportunity to derive improved constraints on the $\mbox{$\rho$ ~Ophiuchi}$ IMF down to low masses.

In a statistical sense at least, Class II YSOs are believed to represent a specific phase of PMS evolution which follows the (Class 0 and Class I) protostellar phases, and precedes the Class III phase (see Sect. 1). Due to their short lifetime ( $\lower.5ex\hbox{$\buildrel < \over \sim$ }$$10^5\,$yr), protostars make up only a small fraction of a young cluster's population (cf. Fletcher & Stahler 1994), and can be neglected in the global mass function. Furthermore, in contrast to PMS stars, protostars have not yet reached their final stellar masses.

Class III objects are more numerous and should thus contribute significantly to the global mass distribution. Furthermore, it has been suggested that some YSOs evolve quickly to the Class III stage, perhaps as early as the "birthline'' for PMS stars (e.g. Stahler & Walter 1993), and spend virtually no time in the Class II phase (cf. André et al. 1992; Greene & Meyer 1995). Since such objects cannot be identified through IR observations, their exact number and mass distribution will not be known until the results of deep X-ray (and follow-up) surveys are available (see Sect. 3.5). However, providing the (range of) evolutionary timescale(s) from Class II to Class III is independent of mass, both classes of PMS objects should have identical mass functions. The results of Sect. 5.3 do seem to support this view, as they suggest that the mass functions of the Class II and Class III samples do not differ down to $\mbox{$M_\star$ }\sim 0.17\,\mbox{$M_\odot$ }$.

We therefore conclude (and will assume in the following) that the mass distribution of Class II YSOs determined in Sect. 5.1 and shown in Fig. 8 currently represents our best estimate of the IMF in the $\mbox{$\rho$ ~Ophiuchi}$ embedded cluster down to $\mbox{$M_\star$ }\sim 0.055\,\mbox{$M_\odot$ }$. As discussed in Sect. 6.2 below, this mass function applies to stellar systems rather than single stars.

6.2 Influence of binary stars

A large proportion ( $\lower.5ex\hbox{$\buildrel > \over \sim$ }$50%) of field stars are in fact multiple (e.g. binary) systems (e.g. Duquennoy & Mayor 1991). This is also true for young PMS stars, and there is a growing body of evidence that the binary fraction is even larger in a young PMS cluster like $\mbox{$\rho$ ~Ophiuchi}$ than for main sequence stars in the field (e.g. Leinert et al. 1993; Simon et al. 1995). Since the present study is based on ISOCAM/near-IR observations which do not have enough angular resolution to separate most of the expected binaries, a significant population of low-mass stellar companions are presumably missing from the mass function derived above. These low-mass companions are hidden by the corresponding primaries.

To estimate the magnitude of this binary effect, we show three simple models in Fig. 8 which assume a population of hidden secondaries corresponding to a binary fraction f of 50%, 75%, and 100%, respectively. In each case, we start from a primary mass function with the two-segment power law form derived in Sect. 5.1. We then add a population of secondaries with masses distributed uniformly (in logarithmic units) between a minimum mass of $0.02\,\mbox{$M_\odot$ }$ and the mass of the primary. We thus assume that the component masses are uncorrelated and drawn from essentially the same mass function (cf. Kroupa et al. 1993).

The global mass functions resulting from addition of companions to the primary mass function are shown in Fig. 8. It can be seen that the global mass functions are similar in form to the original primary mass function. Neither the position of the break point ( $\mbox{$M_\mathrm{flat}$ }$) nor the slope in the high-mass range are affected by the addition of companions. However, the slope in the low-mass range ($\alpha_1$) steepens as the binary fraction increases. Indeed, the power-law index between $0.055\,\mbox{$M_\odot$ }$ and $\mbox{$M_\mathrm{flat}$ }=0.55\,\mbox{$M_\odot$ }$ changes from $\alpha_1 = -0.15$ to $\alpha_1 = -0.31$ for $f=50\%$, $\alpha_1 = -0.37$ for $f=75\%$, and $\alpha_1 = -0.42$ for $f=100\%$.

In summary, if we account for uncertainties in the binary fraction ( $50\%\, \lower.5ex\hbox{$\buildrel < \over \sim$ }\, f\, \lower.5ex\hbox{$\buildrel < \over \sim$ }\, 100\%$), our best estimate of the single-star mass function in $\mbox{$\rho$ ~Ophiuchi}$ is well described by a two-segment power-law with a low-mass index $\alpha=-0.35\pm0.25$ down to $0.055\,\mbox{$M_\odot$ }$, a high-mass index $\alpha_2 = -1.7$, and a break (flattening) occurring at $\mbox{$M_\mathrm{flat}$ }\sim 0.55\pm0.25\,\mbox{$M_\odot$ }$.

Figure 9: Individual luminosity functions for the four sub-clusters associated with the dense cores Oph A, Oph B, Oph EF, and L1689S (see Fig. 1). These are displayed with 0.4 dex bins and scaled by a factor 1/2 for direct comparison with the 0.2 dex bin total luminosity function (light grey histogram in the background). The light solid curves correspond to the model of Fig. 7a scaled to the number of stars in each sub-cluster.

6.3 Luminosity/mass functions in individual cloud cores

Our sample of Class II YSOs is large enough that we can study the properties of the four sub-clusters associated with the dense cores Oph A, Oph B, Oph EF, and L1689S (see Fig. 1). All 123 Class II sources but 5 belong to these 4 sub-clusters. The luminosity functions of Class II objects in the individual sub-clusters are displayed in Fig. 9 along with the "best-fit'' model of Sect. 5.1 (solid curve). They all agree reasonably well in shape with both the total luminosity function and the model: all four luminosity functions are essentially flat over two orders of magnitude in luminosity and appear to have a peak at $\mbox{$L_\star$ }\sim 1.5\,L_\odot$. The agreement is particularly good for the sub-cluster with the largest number of stars, Oph A (see Fig. 9a), but even the smallest sub-cluster, L1689S, tends to reproduce the shape of the global luminosity function on a smaller scale (Fig. 9d). In contrast, for instance, the luminosity function derived for the Chamaeleon I cloud based on ISOCAM data (Persi et al. 2000) differs markedly from the $\mbox{$\rho$ ~Ophiuchi}$ luminosity functions. It does not exhibit any peak at 1.5$\,L_\odot$ and is consistent with an older ($\sim$3 Myr) PMS population having a similar underlying mass function (see Kaas & Bontemps 2001).

The similarity of the individual luminosity functions suggests similar distributions of stellar ages and stellar masses in each of the four sub-clusters.

6.4 Global aspects of star formation in $\mbox{$\rho$ ~Ophiuchi}$

After correction for unresolved binaries (assuming a binary fraction $f=75\%$), the total number of Class II sources (including companions) down to 0.055 $\,\mbox{$M_\odot$ }$ is $\sim$145. Assuming a Class III to Class II number ratio of 19/22 as found by X-ray surveys (Grosso et al. 2000 - see Sect. 3.5), we infer the presence of $\sim$125 Class III stars (including associated companions) in the same mass range. The typical number ratio of Class Is (plus Class 0s) to Class IIs is 18/123 suggesting an additional $\sim$21 embedded YSOs. Altogether, we therefore estimate that there are currently $N_\star \sim 291$ YSOs down to $\sim$ $0.055\,\mbox{$M_\odot$ }$ including $\sim$19% of brown dwarfs. The average and median masses of these objects are $\sim$ $0.35\,\mbox{$M_\odot$ }$and $\sim$ $0.20\,\mbox{$M_\odot$ }$ respectively. The total mass of condensed objects (including brown dwarfs) in the cluster is thus estimated to be $\mbox{$M_\star$ }^\mathrm{clust}=291\times0.35\sim102\,\mbox{$M_\odot$ }$. (The brown dwarfs with $0.055\,\mbox{$M_\odot$ }< \mbox{$M_\star$ }< 0.08\,\mbox{$M_\odot$ }$ contribute only $\sim$$4\%$ of this mass.) Restricting ourselves to L1688 (thus subtracting the $\sim$$10\%$ contribution from L1689) whose average radius is approximately 0.4 pc (cf. CS contours in Fig. 1), we find $N_\star^\mathrm{L1688} \sim 262$, $\mbox{$M_\star$ }^\mathrm{L1688} \sim 92\,\mbox{$M_\odot$ }$, $n_\star^\mathrm{L1688} \sim 980$ stars/pc³, and $\rho_\star^\mathrm{L1688} \sim 340\,\mbox{$M_\odot$ }$/pc³, where $n_\star$ and $\rho_\star$ are the stellar number density and stellar mass (volume) density of the cluster, respectively.

Adopting a conservative value of 2 Myr for the cluster age, the total mass of $\mbox{$M_\star$ }^\mathrm{clust}=102\,\mbox{$M_\odot$ }$ translates into an average star formation rate of $5.1\times 10^{-5}\,\mbox{$M_\odot$ }/$yr, corresponding to one new YSO (of typical mass 0.20 $\,\mbox{$M_\odot$ }$) every $\sim$4000 yr. Lastly, we can derive the star formation efficiency (SFE) in L1688, defined as SFE $=\mbox{$M_\mathrm{star}$ }/(\mbox{$M_\mathrm{star}$ }+\mbox{$M_\mathrm{gas}$ })$. The total molecular gas mass, $\mbox{$M_\mathrm{gas}$ }$, of L1688 has been estimated to range between $550\,\mbox{$M_\odot$ }$ (from C¹⁸O measurements - Wilking & Lada 1983) and $1500\,\mbox{$M_\odot$ }$ (from CS(2-1) data - Liseau et al. 1995). Using $\mbox{$M_\star$ }^\mathrm{L1688}= 92\,\mbox{$M_\odot$ }$, we thus get SFE $^\mathrm{L1688} \sim 6-14$%, which is somewhat lower than previous estimates ($\geq$$22\%$ - WLY89). Note, however, that active star formation in L1688 appears to be limited to the three sub-clusters/dense cores Oph A, Oph EF, and Oph B (see Fig. 1 and Loren et al. 1990), where the local star formation efficiency is significantly higher: $SFE \sim 31\%$, using a total core mass of $200\,\mbox{$M_\odot$ }$ (Loren et al. 1990).

Figure 10: Comparison of the pre-stellar mass spectrum measured by MAN98 for 58 protocluster condensations (dark histogram with statistical error bars) with the YSO mass function derived in Sect. 5.1 for 123 Class II systems (light histogram with error bars accounting for both statistical and age uncertainties).

6.5 Comparison with the $\mbox{$\rho$ ~Ophiuchi}$ protocluster condensations

In an extensive 1.3$\,$mm dust continuum imaging survey of L1688 with the IRAM 30 m telescope (11 $^{\prime\prime}$ resolution), MAN98 could identify 58 compact starless condensations. Molecular line observations (e.g. Belloche et al. 2001) indicate that the condensations are gravitationally bound and thus likely pre-stellar in nature. MAN98 noted a remarkable similarity between the mass spectrum of these pre-stellar condensations and the IMF of Miller & Scalo (1979).

In Fig. 10, we compare the pre-stellar mass spectrum determined by MAN98 with the mass distribution of Class II YSOs derived in Sect. 5.1. (As such, both distributions are uncorrected for the presence of close binary systems.) It can be seen that there is a good agreement in shape between the two mass spectra. A Kolmogorov-Smirnov test performed on the corresponding cumulative distributions confirms that they are statistically indistinguishable at the 95% confidence level. This supports the suggestion of MAN98 that the IMF of embedded clusters is primarily determined by cloud fragmentation at the pre-stellar stage of star formation. The fact that both the pre-stellar and the YSO spectrum of Fig. 10 present a break at roughly the same mass $\sim$ $0.5\, \mbox{$M_\odot$ }$ is quite remarkable. A small, global shift of the masses by only $\sim$30% upward or downward in one of the spectra would make them differ at the 2$\sigma$ statistical level. Although in absolute terms, both sets of masses are probably uncertain by a factor of $\sim$2 (due to uncertainties in the 1.3$\,$mm dust opacity and in the cluster age, respectively), this suggests that the protocluster condensations identified at 1.3$\,$mm may form stars/systems with an efficiency larger than $\sim$50-70%.