Osservatorio Astrofisico di Arcetri, INAF, Largo E. Fermi 5, 50125 Firenze, Italy

Abstract
Aims. The aim of this paper is to provide a measurement of the mass accretion rate in a large, complete sample of objects in the core of the star forming region $\rho$ Oph.
Methods. The sample includes most of the objects (104 out of 111) with evidence of a circumstellar disk from mid-infrared photometry; it covers a stellar mass range from about 0.03 to 3 $M_\odot$ and it is complete to a limiting mass of $\sim$ 0.05 $M_\odot$ . We used J and K-band spectra to derive the mass accretion rate of each object from the intensity of the hydrogen recombination lines, Pa $\beta$ or Br $\gamma$ . For comparison, we also obtained similar spectra of 35 diskless objects.
Results. The results show that emission in these lines is only seen in stars with disks, and can be used as an indicator of accretion. However, the converse does not hold, as about 50% of our disk objects do not have detectable line emission. The measured accretion rates show a strong correlation with the mass of the central object ( $\dot M_{\rm acc}$ $\propto M_\star^{1.8\pm0.2}$ ) and a large spread, of two orders of magnitude at least, for any interval of $M_\star$ . A comparison with existing data for Taurus shows that the objects in the two regions have similar behaviour, at least for objects more massive than $\sim$ 0.1 $M_\odot$ . The implications of these results are briefly discussed.

Even if accretion disks have been part of the accepted paradigm of star formation for many years, many of their physical properties are poorly known, and the physical mechanism of angular momentum transfer, which determines the disk evolution, is still unclear. The physical quantity that controls the accretion phase is the mass accretion rate through the disk $\dot M_{\rm acc}$ . This quantity can be derived only indirectly, by fitting models to observed quantities such as the UV excess emission and/or the profiles and intensity of lines believed to form in the accreting gas. Measurements of $\dot M_{\rm acc}$ are now available for a large number of stars in Taurus (e.g., Muzerolle et al. 2005, and references therein). The results have shown that $\dot M_{\rm acc}$ is a strong function of the mass of the central object, roughly ${\propto} M_\star^2$ , and that a large dispersion is present (about two orders of magnitude) for objects with the same $M_\star$ . Both results are a challenge for accretion disk models, as discussed, e.g., by Muzerolle et al. (2003) and Natta et al. (2004).

Measurements of accretion rates in other star forming regions are scarce in comparison, mostly limited to very low mass objects (Muzerolle et al. 2003, 2005). In a study of very low mass objects and brown dwarfs in Ophiuchus, Natta et al. (2004) found that they are actively accreting with $\dot M_{\rm acc}$ higher by at least one order of magnitude than objects of similar mass in Taurus. This could be due to a difference in age, since the Ophiuchus BDs are very young objects, younger than their Taurus counterparts, but could also be due to different environmental conditions.

While it is clearly necessary to improve the physical models of accretion disks, at the same time it is important to study large and if possible complete samples of stars in a variety of star forming regions, differing in age and global properties.

We report in this paper the results of a project aimed at measuring the mass accretion rate of a large sample of pre-main sequence objects, ranging from a few solar masses to few tens of Jupiter masses, in the star forming region $\rho$ Oph. The core of $\rho$ Oph is perfectly suited for such a study, as it is rich in pre-main sequence stars, which include intermediate mass objects, T Tauri stars (TTS) and brown dwarfs (BDs). Its stellar content has been studied, e.g., by Luhman & Rieke (1999, LR99 in the following, and references therein to previous work), Natta et al. (2002) and, more recently, by Wilking et al. (2005). Moreover, $\rho$ Oph is very different from Taurus, younger and more compact, and it will allow us to explore the accretion properties of pre-main sequence stars under different conditions, following the results of Natta et al. (2004).

Ophiuchus has been observed in two mid-IR bands with ISO by Bontemps et al. (2001, BKA01 in the following), who detected 199 sources in the $\rho$ Oph core. Of these, 111 were classified, on the basis of their IR colors, as Class II objects, i.e., visible young stellar objects with evidence of disks. They provide a sample of systems with disks complete to a limiting mass of about 0.05 $M_\odot$ . In a spectroscopic study of the very low luminosity objects of the BKA01 sample, Natta et al. (2002) confirmed that they were BDs with mid-IR excess, very likely from a circumstellar disk; as mentioned, these BDs show significant differences in accretion properties from their analogs in Taurus.

The disadvantage of observing $\rho$ Oph is its high extinction, which makes veiling measurements in the UV and visual impossible except for a few objects. The most effective way to determine $\dot M_{\rm acc}$ for the Ophiuchus sample is therefore to use the luminosity of hydrogen recombination lines, such as Pa $\beta$ and/or Br $\gamma$ . The relation between IR line luminosity and accretion luminosity, independently measured from the UV excess, was established by Muzerolle et al. (1998b) for TTS, and by Calvet et al. (2004) for intermediate mass objects. Natta et al. (2004) extended it to very low mass objects, where $\dot M_{\rm acc}$ was determined by fitting the observed H $\alpha$ profiles with the predictions of magnetospheric accretion models.

In this paper, we present the results of a spectroscopic IR survey of Ophiuchus objects. In Sect. 2, we describe the properties of the observed sample, which includes almost all (104 out of 111) the Class II objects and a subset (35 objects out of 77) of the diskless systems (Class III), also from the BKA01 survey, that we will use for comparison. The observations, data reduction and method of analysis are discussed in Sect. 3. The results are presented in Sect. 4 and discussed in Sect. 5. Section 6 summarizes our conclusions.

2 Characteristics of the observed sample

2.1 The sample

The most complete survey of young stellar objects in the $\rho$ Oph Main Cloud (L 1688) is that obtained in two mid-IR bands (6.7 and 15.3 $\mu$ m) with ISOCAM (BAK01). Based on the near and mid-IR colors, the objects were divided in Class I (accreting protostars), Class II and tentative Class II (pre-main sequence stars with IR excess typical of disks, like classical T Tauri stars or CTTS), and Class III/tentative Class III (objects with colors typical of stellar photospheres, like weak-line T Tauri stars or WTTS). BAK01 estimate that their Class II sample of 111 objects is complete to a limiting luminosity $L_\star$ $\sim$ 0.03 $L_\odot$ , corresponding approximately to 0.05 $M_\odot$ . The Class III sample is only complete to $\sim$ 0.2 $L_\odot$ (about 0.15 $M_\odot$ ). Note that not all the Class III objects have been confirmed as $\rho$ Oph members. Barsony et al. (2005) have recently confirmed the accuracy of the ISOCAM results with ground-based 10 $\mu$ m observations of a large subset of the BKA01 sources.

Our sample includes 104 of the 111 Class II/tentative Class II objects (Class II for simplicity in the following) listed by BAK01 in the $\rho$ Oph core. Most of the spectra (96) were obtained in the J band; the remaining 8, of objects too weak in J, in the K band; one object has been observed at both wavelengths. As a comparison sample, we observed 35 of the 77 Class III and tentative Class III (in the following, Class III) objects, 31 in the J band and 4 in K. The objects and their properties are listed in Tables C.1 and C.2.

2.2 Stellar parameters

The stellar properties (i.e., spectral type, luminosity, mass and radius) of the BAK01 sample are well known only for a handful of objects. The main difficulty comes from the large uncertainties that affect spectral types, due to the combination of high extinction and large veiling, even at near-IR wavelengths (e.g., LR99 and references therein; Doppman et al. 2003; Wilking et al. 2005). LR99, using K-band low resolution spectra, provide spectral types for 37 of our Class II objects. However, 23 of them have uncertainties of almost one spectral class.

Given the uncertainties, and considering that most of our objects do not have any spectral classification, we have decided to adopt a statistical approach, following BAK01.

First, we compute the extinction toward each object from the observed (J-H)-(H-K) colors, as given by 2MASS, corrected to CIT system, adopting the Ophiuchus extinction law of Kenyon et al. (1998) and the locus of CTTS defined by Meyer et al. (1997). The result can be expressed as:

The stellar luminosity is computed from the J magnitude and A_J, using a bolometric correction similar to that adopted by BAK01:

When only H and K magnitudes were available (17 Class II and 1 Class III objects), we estimated the stellar luminosity using Eqs. (2) and (4) of BKA01.

There are 5 objects (4 Class II and 1 Class III) that have companions clearly seen in our spectra, but which are not resolved in the 2MASS photometry. All the companions have a good detection of the continuum; the flux ratio between the primary and the secondary is always larger than a factor of 3. Two of the companions ( $\rho$ Oph-ISO 068b and $\rho$ Oph-ISO 072b) have been detected in the K-band $\rho$ Oph multiplicity survey of Ratzka et al. (2005), with flux ratios to the primary of 0.19 and 0.16, respectively. We have accordingly not corrected the 2MASS magnitudes of the primaries for the contribution of the companions, because the corrections to the derived parameters would have been within the uncertainties. The secondary components have no detectable Pa $\beta$ emission, and we will omit them from our analysis in the following; their properties are summarized in Table C.3.

To determine stellar radii and masses, we make the assumption that the star formation in Ophiuchus is coeval, and that all the objects lie on a single isochrone in the HR diagram. With this assumption, we can derive stellar mass, temperature and radius from the measured $L_\star$ . This procedure is reasonable for the Ophiuchus core, whose age estimates range between 0.5 and 1 Myr, with very few stars older than that (BAK01; LR99). In the following, we adopt the D'Antona & Mazzitelli (1997 and 1998 web updates; DM98 in the following) evolutionary tracks for an age of 0.5 My. The uncertainties introduced by the assumption of coeval star formation and the differences expected if other evolutionary tracks were used are discussed in Appendix A.

3 Observations and data analysis

3.1 Observations and data reduction

Near infrared moderate resolution J and K band spectroscopic observations of all targets in our sample were obtained at the ESO Observatories in Chile. The objects were either observed using the SofI instrument at the NTT 2.2 m telescope (June 2004, Visitor Mode) or the ISAAC instrument at the Antu 8.2 m VLT unit telescope (Spring 2004, Service Mode), as specified in Tables C.1 and C.2. Detailed descriptions of both these instruments are available on the ESO web pages. For all the objects that were observable at J-band, with SofI we used the 0.6 arcsec slit and the Blue low resolution grims, resulting in a spectral resolution of approximately $\lambda/\Delta\lambda\sim 1000$ and a spectral coverage from $\sim$ 0.95 to ${\sim} 1.64~\mu$ m; with ISAAC we employed the short-wavelength low resolution spectral mode with central wavelength $1.25~\mu$ m and $0.6\hbox{$^{\prime\prime}$ }$ slit width, giving a spectral resolution of $\lambda/\Delta\lambda\sim 900$ and a spectral coverage limited to the J-band. A number of objects were only observable at K-band, for these we either used the SofI Red low resolution grism with similar spectral resolution as for the Blue grism observations and spectral coverage from $\sim$ 1.6 to $2.5~\mu$ m, or the ISAAC short wavelength low resolution mode with central wavelength $2.2~\mu$ m, which offers a similar spectral resolution as the J-band observations and a spectral coverage limited to the K-band. Integration times varied from about 0.5 to 2 h on source, depending on the expected brightness of the objects and observing conditions (in Visitor Mode).

During the Visitor Mode observations at the NTT telescope, we acquired several telluric standard stars per night at varying airmasses; each Observing Block from our programme executed in Service Mode at the VLT was preceded or followed by a telluric standard observed with the same instrument mode and at a similar airmass as our target stars. Spectroscopic flat fields and arcs were obtained during daytime either before or after our observations. Standard methods were employed to calibrate our data. We did not attempt to obtain flux calibrated spectra; all our spectra are wavelength calibrated using OH airglow lines and corrected on an arbitrary intensity scale for telluric absorption and instrument response using the telluric standard star observations.

Correction for telluric absorption and instrumental response was obtained observing at similar airmasses early type stars (early B or O) of known spectral type from the telluric standards lists of ISAAC. These stars all have Pa $\beta$ or Br $\gamma$ absorption which were manually removed from the spectra before applying the correction.

Most of the spectra are of excellent quality; the detection limits of the Pa $\beta$ or Br $\gamma$ equivalent width are in general of the order of 0.5-1 Å. Variations around this limit are mainly related to the signal to noise ratio achieved on the photospheric continuum of the individual objects. The signal to noise ratio depends on the telescope/instrument used, the observing conditions, the integration time and the apparent magnitude of the object. It is not necessarily a function of the object intrinsic luminosity because the extinction can be very different and because we tried as much as possible to observe two objects at the same time by properly aligning the slit, so that some relatively bright source near a faint one may have been observed with ISAAC and a long integration time. However, most of the lower luminosity objects have been observed with ISAAC and, expecting lower line intensities, with a higher signal to noise ratio; thus, the line detection limits for low luminosity objects are generally lower than for intermediate luminosity ones.

The sample studied in this paper includes also the 9 BKA01 sources for which Natta et al. (2004) obtained J and K band spectra with ISAAC. We have taken the Natta et al. (2004) J band spectra and reanalyzed them in the same manner used for the others.

3.2 Method

The luminosity of Pa $\beta$ and Br $\gamma$ are computed from the measured equivalent widths of the emission lines and the broad-band J and K fluxes, corrected for extinction, determined as described in Sect. 2.2. No correction for underlying photospheric absorption was applied, since the expected equivalent width is small ( $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 0.5 Å; Wallace et al. 2000) for objects with $T_{\rm {eff}}$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 5000 K, which represent the quasi-totality of our sample (see Table C.1) and would not change the results.

There are 12 Class II (11 of them have no Pa $\beta$ detection) for which it was not possible to determine line fluxes, due to lack of J magnitudes; they will not be included in the following discussion. Similarly, we will not consider further the one Class II ( $\rho$ Oph-ISO 035) with weak Pa $\beta$ in absorption.

The accretion luminosity of each Class II object is derived from the empirical correlation between $L_{\rm acc}$ and the luminosity of Pa $\beta$ or Br $\gamma$ , derived by Natta et al. (2004) and Calvet et al. (2004), respectively (see also Muzerolle et al. 1998b):

$\displaystyle \log{L_{\rm acc}/L_\odot}$	=	$\displaystyle 1.36~ \log{L({\rm P}_\beta)/L_\odot} + 4$	(3)
$\displaystyle \log{L_{\rm acc}/L_\odot}$	=	$\displaystyle 0.9~ (\log{L({\rm Br}_\gamma)/L_\odot}+4)\> -0.7.$	(4)

The reliabilty of our procedure was verified by applying it to a sample of well studied pre-main sequence stars, covering roughly the same range of masses, for which reliable values of the stellar parameters (i.e., mass and radius) and of the accretion rate could be found in the literature. Using literature measurements of the Pa $\beta$ intensity and of $L_\star$ , we derived for each object mass and accretion rate as done for the Ophiuchus objects, and compared them to the "real'' values. Details can be found in the Appendix B.

We have applied a similar procedure to the Class III objects; the results are shown in Table C.2.

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4706fig1.ps}\end{figure}$	Figure 1: Equivalent width of the Pa $\beta$ emission line for all objects with known $L_\star$ . Circles are Class II sources, filled circles are detections, empty circles with arrows upper limits. Squares are Class III sources. Two objects with $L_\star$ < 0.01 $L_\odot$ and Pa $\beta$ undetected are not plotted.
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4 Results

4.1 Equivalent widths

Emission in the near-IR hydrogen recombination lines has been detected in 45% of Class II sources, 46 of the 96 observed in Pa $\beta$ and 1 out of 9 observed in Br $\gamma$ . In contrast, no Class III source shows emission in these hydrogen lines; 8 Class III objects have Pa $\beta$ or Br $\gamma$ in absorption and for the others we do not see the lines. The measured equivalent widths are given in Tables C.1 and C.2. Figure 1 shows the Pa $\beta$ equivalent width as function of $L_\star$ .

Six Class III objects have Pa $\beta$ in absorption with equivalent widths $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'076$ }}\mathchar''7218$ }}$ 1 Å, i.e., larger than one can expect in late-type stars (Wallace et al. 2000). They are likely earlier type stars, and this is certainly the case of $\rho$ Oph-ISO 180, which is classified A7 by Wilking et al. (2005) and of $\rho$ Oph-ISO 113, earlier than F8 according to LR99. For these six stars, as already mentioned, the method used to estimate A_J and all the derived stellar parameters is not correct; therefore, we omit their stellar parameters from Table C.2.

The comparison between the Class II and Class III samples clearly shows that emission in the near-IR hydrogen lines, in contrast to that in optical lines such as H $\alpha$ is restricted to objects with circumstellar disks, and can therefore be used as a reliable accretion indicator. However, one should keep in mind that the opposite is not necessarily true, as about 50% objects with disks have no detected emission.

The fraction of Class II objects with detected Pa $\beta$ emission varies from 56% for $L_\star$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'076$ }}\mathchar''7218$ }}$ 1 $L_\odot$ to 42% for $0.03\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''... ...bin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}1$ $L_\odot$ . Very low luminosity objects (7 objects with $L_\star$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 0.03 $L_\odot$ ) have a marginally higher detection rate ( $\sim$ 57%), due in part to the sensitivity limit of our measurements, which is higher for lower luminosity objects (see Sect. 3.1), but also due to the incompleteness of the BKA01 survey for very low luminosity sources, which are detected only when they have a large mid-IR excess, very likely indicative of higher accretion rates.

4.2 Accretion luminosity

Figure 2 shows the accretion luminosity of Class II objects computed from the IR line luminosity as a function of $L_\star$ .

For any given $L_\star$ , there is a large range of measured $L_{\rm acc}$ (about 50), which does not seem to vary significantly with $L_\star$ ; because of our sensitivity limit, this is probably just a lower limit to the actual range of $L_{\rm acc}$ . One can also see that for the majority of objects $L_{\rm acc}$ / $L_\star$ < 0.1, but there is a significant fraction of cases with $L_{\rm acc}$ $\sim$ $L_\star$ .

4.3 Mass accretion rate

Figure 3 shows the mass accretion rate $\dot M_{\rm acc}$ of Class II sources as function of $M_\star$ . There is a clear trend of increasing $\dot M_{\rm acc}$ with increasing $M_\star$ . Not including upper limits, we find using ASURV (Feigelson and Nelson 1985) $\dot M_{\rm acc}$ $\propto$ $M_\star$ $^{1.8\pm 0.2}$ ; the slope does not change if we include the upper limits in the analysis.

Superimposed on this trend, there is a large spread of $\dot M_{\rm acc}$ for any value of $M_\star$ , of two orders of magnitude at least. Within statistical fluctuations, the objects are distributed quite uniformly in this range.

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4706fig2.ps}\end{figure}$

Figure 2: Accretion luminosity from the IR lines as function of $L_\star$ for Class II objects. Dots show $L_{\rm acc}$ measurements from Pa $\beta$ (filled: detections, empty: upper limits); diamonds measurements from Br $\gamma$ (filled: detections, empty: upper limits); The dotted lines show the locus of $L_{\rm acc}$ / $L_\star$ = 0.01, 0.1 and 1, as labelled. Crosses are objects in Taurus (see text for references).

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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4706fig3.ps}\end{figure}$	Figure 3: Mass accretion rate derived from the IR lines as function of $M_\star$ . Symbols as in Fig. 2. The dashed line shows the relation $\dot M_{\rm acc}$ $\propto$ $M_\star$ ^1.8, derived from a statistical analysis using ASURV; the dotted line plots, for comparison, the relation $\dot M_{\rm acc}$ $\propto$ $M_\star$ .
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5 Discussion

The results summarized in Fig. 3 describe the accretion properties of the largest sample of Class II stars in any single star-forming region studied so far. The sample contains more than hundred objects with evidence of disks, and is complete in the mass interval from $\sim$ 0.03 to about 3 $M_\odot$ . The corresponding accretion rates vary from ${\sim} 10^{-11}$ to ${\sim} 10^{-6}$ $M_\odot$ /yr, with a strong dependence of $\dot M_{\rm acc}$ on $M_\star$ ( $\dot M_{\rm acc}$ $\propto$ $M_\star^{1.8\pm 0.2}$ ). For any $M_\star$ , there is a large dispersion of values of $\dot M_{\rm acc}$ , of two orders of magnitude at least, which does not seem to change with $M_\star$ . Note that the real spread is likely bigger, because of the many upper limits in our survey.

5.1 Ophiuchus and Taurus

One of the aims of our study was to compare the accretion properties in Ophiuchus with those of objects in Taurus. The Taurus results are shown by crosses in Figs. 2 and 3. The accretion luminosity and mass accretion rate have been derived from the UV and optical veiling and/or by fitting with magnetospheric accretion models the H $\alpha$ profile. This second method is the only possible one for very low mass objects and BDs, since veiling cannot be detected below a limiting value $\dot M_{\rm acc}$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 10^-10 $M_\odot$ /yr. The results are from Gullbring et al. (1998), Muzerolle et al. (1998b, 2003, 2005), White & Ghez (2001), White & Basri (2003), and Calvet et al. (2004); note that, for homogeneity, we have re-determined $M_\star$ using DM98 tracks for all objects.

The methods used to derive $L_{\rm acc}$ and $\dot M_{\rm acc}$ in the two regions are therefore different, since in Ophiuchus $\dot M_{\rm acc}$ is derived from the luminosity of the hydrogen recombination lines. However, the relations (Eqs. (3) and (4)) we used have been "calibrated'' mostly using Taurus objects (see, e.g., Muzerolle et al. 1998b; Calvet et al. 2004; Natta et al. 2004), so that we do not expect any systematic difference in the Ophiuchus-Taurus comparison due to the different methods.

The two figures show that the accretion properties of the two star forming regions are very similar. Muzerolle et al. (2005) derive $\dot M_{\rm acc}$ $\propto$ $M_\star$ ^2.1 for their sample (mostly in Taurus, with additional brown dwarfs from other star-forming regions), neglecting upper limits. Within the errors, this relation is identical to what we obtain in Ophiuchus. If we concentrate in Fig. 3, we can see that not only the slope of the relation of $\dot M_{\rm acc}$ with $M_\star$ , but also the range of values is very similar. In particular, the two samples have similar values of the maximum $\dot M_{\rm acc}$ for any given $M_\star$ , and similar spread of $\dot M_{\rm acc}$ values, at least for $M_\star$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'076$ }}\mathchar''7218$ }}$ 0.06-0.08 $M_\odot$ .

For lower $M_\star$ , most Taurus BDs have very low accretion rates, 1-2 orders of magnitude lower than similar objects in Ophiuchus. As already discussed, the fact that we do not find these very low accretors in Ophiuchus most likely reflects the incompleteness of the BAK01 sample at very low masses, and selects objects with comparatively strong mid-IR fluxes. Natta et al. (2002) showed that the BAK01 sample of brown dwarfs has relatively large luminosity, and is probably very young. As discussed in Sect. 4.1, a fraction larger than for more luminous objects has detected Pa $\beta$ . All this indicates that there may be low $\dot M_{\rm acc}$ BDs which are missing from the Ophiuchus sample. It is, in a way, more surprising that very few, if any, of the brown dwarfs in Taurus have high $\dot M_{\rm acc}$ , while higher mass objects in the two regions have very similar accretion properties. It is possible that this difference between the two regions at the very low end of the $M_\star$ distribution contains important information, that needs further investigation. This is, however, beyond the scope of this paper.

5.2 Variability

All pre-main sequence stars are variable objects, and, in particular, all the accretion indicators in TTS and BDs show large variability.

Variability does not affect the correlation of $\dot M_{\rm acc}$ with $M_\star$ , as the $\rho$ Oph sample is sufficiently large that individual fluctuations cannot change it. It may be more important when we consider the spread of $\dot M_{\rm acc}$ values for any given $M_\star$ . Recently, Scholz & Jayawardhana (2005) have studied the variability of accretion indicators (mostly H $_\alpha$ ) for six young brown dwarfs; they claim that the accretion rate in some of their objects varies by at least one order of magnitude, and that this variability may account for the large spread in the $\dot M_{\rm acc}$ - $M_\star$ correlation.

We have estimated the magnitude of the spread in $\dot M_{\rm acc}$ for individual objects by looking at the results of Gatti et al. (2006), who have recently obtained J-band spectra of a small (14 objects) subset of our Ophiuchus sample. The Gatti et al. sample includes both TTS and BDs, observed one to two years later than the spectra discussed in this paper. The two data sets show variations in the Pa $\beta$ equivalent width of a factor of two at most (in both directions), with only one exception, where the Pa $\beta$ equivalent width has increased by a factor of three over the time interval between the two sets of observations. For the same objects, we have also looked in the literature for variations of the broad-band J magnitude, used to compute the line flux (Sect. 3.2). The variation of $\dot M_{\rm acc}$ , computed taking the maximum variations in the J magnitude and in the Pa $\beta$ equivalent width, is of a factor $\sim$ 4. This is much smaller than the dispersion of points in Fig. 3 and would not change significantly any of our conclusions.

A detailed analysis of the variability of the IR emission lines and continuum, in analogy to what has been done for H $_\alpha$ (e.g., Johns-Krull & Basri 1997), is certainly needed. However, from the results obtained so far, it seems unlikely that the dispersion of $\dot M_{\rm acc}$ values can be accounted for by variability alone, and that, if averaged over a sufficiently long period of time, one would find that all the $\rho$ Oph stars of a given mass accrete at the same rate.

5.3 Viscous disks

The $\dot M_{\rm acc}$ dependence on $M_\star$ is difficult to understand in terms of disk physics, as discussed, e.g., by Muzerolle et al. (2003), Natta et al. (2004), Calvet et al. (2004). In a standard steady accretion disk model, $\dot M_{\rm acc}$ is proportional to the disk mass divided by the time scale for viscous evolution. In an $\alpha$ -disk (Shakura & Sunyaev 1973), the viscosity depends on the ratio $\Omega/c_{\rm s}^2$ , where $\Omega$ is the keplerian angular velocity and $c_{\rm s}$ the sound speed; then, $\dot M_{\rm acc}$ $\propto M_{\rm d} \times M_\star^{-1/2} \times T_{\rm d}$ , where $M_{\rm d}$ and $T_{\rm d}$ are disk mass and temperature, respectively. With the further assumptions that $M_{\rm d}\propto M_\star$ (e.g., Natta et al. 2000), and that the disk heating is dominated by the stellar irradiation, this gives, to zero order, $\dot M_{\rm acc}$ $\propto M_\star^{1/2} \times T_\star$ . For PMS stars, the relation between $T_\star$ and $M_\star$ is rather shallow (approximately $T_\star \propto M_\star^{0.4}$ for $M_\star$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'076$ }}\mathchar''7218$ }}$ 0.1 $M_\odot$ , and much flatter for lower masses; see, e.g., DM98) and we expect $\dot M_{\rm acc}$ to increase roughly as $M_\star^\gamma$ , with $\gamma \mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}1$ . The relation will be even flatter if the contribution of the stellar radiation to the disk heating is negligible.

It is possible that $\alpha$ (or, more generally, the efficiency of momentum transfer) depends, in turn, on $M_\star$ . If viscosity is the result of magneto-rotational instabilities (MRI) (see, e.g., Balbus & Hawley 1991), the disk gas should be sufficiently ionized. Muzerolle et al. (2003) suggest that the steep correlation of $\dot M_{\rm acc}$ with $M_\star$ can be explained if the disk ionization is controlled by the X-ray radiation from the star, since the X-ray luminosity is not constant over the mass spectrum, but is observed to increase with $M_\star$ .

X-ray observations of Ophiuchus have been recently carried out with Chandra and XMM satellites by Imanishi et al. (2001) and Ozawa et al. (2005). Both studies detected a significant fraction of Class II sources (70 and 48 % respectively); they found that the X-ray spectral properties, as well as the relationship between $L_{\rm X}$ and $L_{\rm bol}$ of class II sources are similar to those of class III sources, but did not investigate the behaviour of X-ray luminosity with stellar mass. To our knowledge, the only study addressing the relation between mass and X-ray luminosity for young stars over a large range of luminosities and masses is in Orion. The COUP Chandra observations of Orion show that $L_{\rm X}$ scales approximately as $M_\star$ ^1.1-1.4in the interval 0.1-2 $M_\odot$ (Preibisch et al. 2005). However, it is not clear that this variation of $L_{\rm X}$ is sufficient to produce the observed $\dot M_{\rm acc}$ - $M_\star$ correlation, and more detailed MRI models, which include X-ray ionization, are required. If the X-ray emission of the central star is controlling accretion, the large spread of $L_{\rm X}$ observed in the COUP data could also explain the large spread of $\dot M_{\rm acc}$ for any given $M_\star$ .

Viscous disk models predict that $\dot M_{\rm acc}$ decreases with time (e.g., Hartmann et al. 1998). Calvet et al. (2000) estimate $\dot M_{\rm acc}$ $\propto t^{-1.5}$ , with a large uncertainty, from a sample of TTS in Taurus, Chamaeleon and Ophiuchus. Neither the similarity of accretion rates between Ophiuchus and Taurus nor the very large spread observed in both regions support age as a main factor in the determination of $\dot M_{\rm acc}$ . If the Calvet et al. (2000) rate is correct, the difference in age between Taurus and Ophiucus should give on average a difference in $\dot M_{\rm acc}$ of a factor $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'076$ }}\mathchar''7218$ }}$ 3, of which we have no evidence. In addition, the Ophiuchus $\dot M_{\rm acc}$ range of more than two orders of magnitude corresponds to an age range of at least a factor 20, much too large when compared to the HR location of the objects (see, e.g., LR99).

The time evolution of viscous disks is influenced by the presence of close companions (see Calvet et al. 2000). Companions truncate the circumstellar disk at a radius which depends on the binary separation. As the disk evolves, more and more matter expands outside the truncation radius, with the effect of decreasing the disk mass and $\dot M_{\rm acc}$ . A sample of objects with the same initial value of $\dot M_{\rm acc}$ but companions at different distances will show with time an increasing spread of $\dot M_{\rm acc}$ values.

This effect, however, is not seen in the Taurus TTS (White & Ghez 2001), where the accretion rate is similar for single and primary stars with companions as close as 10 AU. At the age of Ophiuchus, only very close companions have had time to reduce $\dot M_{\rm acc}$ by a significant factor (separation $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 30 AU or $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 0.2 arcsec for an age of 10⁶ years according to Calvet et al. 2000). There have been a number of multiplicity surveys of Ophiuchus, some capable of detecting very close binaries. Three Class II objects (i.e., objects with a mid-IR detected circumstellar disk) have companions closer than $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'074$ }}\mathchar''7218$ }}$ 0.25 arcsec (Barsony et al. 2005; Ratzka et al. 2005); one has detected Pa $\beta$ , while in the other two cases the line has not been detected. The observational evidence of a correlation between the accretion rate and the presence of very close companions is clearly inconclusive. At this stage, it cannot be quantitatively confirmed nor dismissed, and should be investigated further.

5.4 Initial conditions

Although all the effects discussed so far can play a role and need further investigation, it is possible that differences in the initial conditions, i.e., in the physical properties of the molecular cores from which the star+disk system forms, determine the TTS disk properties, and in particular the behaviour of $\dot M_{\rm acc}$ disussed in this paper.

The self-similar viscous disk models of Hartmann et al. (1998) show that the accretion rate is proportional to the disk mass at t=0, i.e., when accretion onto the disk stops, and, in the early phases ot the evolution, to its t=0 outer radius, which in turns depend on the core properties. Alexander & Armitage (2006) have started exploring how this can introduce a $\dot M_{\rm acc}$ $\propto$ $M_\star ^2$ correlation at a later time.

More realistic models that follow the formation and evolution of circumstellar disks (Hueso & Guillot 2005) illustrate clearly how different core properties (in particular, different rotation velocities) can create a large spread of $\dot M_{\rm acc}$ for objects with the same $M_\star$ and age.

Models that compute the evolution of disks starting from the core infall phase over a large range of parameters are required, if we want to estimate the effect of the initial conditions on the relation of $\dot M_{\rm acc}$ with $M_\star$ and on its scatter. The observations presented in this paper, and the similar results for Taurus, provide an excellent test of such models. Note that the the fact that disk accretion properties in Taurus and Ophiuchus are very similar, while the two regions have large differences in their environment, should put strong constraints on these models, which will be interesting to explore fully.

6 Summary and conclusions

In this paper, we report the results of a near-IR spectroscopic survey of a large sample of very young objects in the $\rho$ Oph core. The sample includes all Class II objects, i.e., objects with evidence of circumstellar disks from mid-IR photometry (BKA01). This sample covers the mass range between about 0.03 to 3 $M_\odot$ ; according to BKA01, it is complete to a limiting magnitude of about 0.03 $L_\odot$ , or 0.05 $M_\odot$ . We have also observed a significant fraction of Class III objects, i.e., with no mid-IR excess emission, covering a similar range of luminosities.

In contrast to the Balmer lines, the near-IR hydrogen recombination lines are seen in emission only in Class II objects. Of all our Class III sample, none has detected Pa $\beta$ emission. This confirms our assumption (Natta et al. 2004) that the near-IR lines can provide an immediate indication of the accreting properties of young stars, even when only relatively low resolution spectra are available.

We have derived the mass accretion rate $\dot M_{\rm acc}$ from the luminosity of the hydrogen recombination lines, mostly from Pa $\beta$ but in few cases from Br $\gamma$ . In total, we obtain measurements of $\dot M_{\rm acc}$ for 45 Class II objects, and upper limits for 39.

Our results show that $\dot M_{\rm acc}$ increases sharply with $M_\star$ ( $\propto M_\star^{1.8\pm0.2}$ ). We also find a large range of values of $\dot M_{\rm acc}$ for any given value of $M_\star$ (a spread of roughly two order of magnitudes, independent of $M_\star$ ). As discussed in the text, this is likely a lower limit to the true dispersion.

When compared to accretion measurements in Taurus (see Muzerolle et al. 2005, and references therein), we find that the two regions look very similar, at least for objects with $M_\star$ $\mathbin{\lower 3pt\hbox {$\rlap{\raise 5pt\hbox{$\char'076$ }}\mathchar''7218$ }}$ 0.1 $M_\odot$ . For both Taurus and Ophiuchus, the dependence of $\dot M_{\rm acc}$ on $M_\star$ , the upper envelope of the $\dot M_{\rm acc}$ distribution (i.e., the largest values of $\dot M_{\rm acc}$ that any object of a given mass seems able to sustain), and the range of $\dot M_{\rm acc}$ values for any given $M_\star$ , are very similar. At lower mass, the accretion rates of the Ophiuchus objects are much larger than their Taurus analogs.

The observed behaviour of $\dot M_{\rm acc}$ does not have an obvious explanation. The correlation of $\dot M_{\rm acc}$ with $M_\star$ may be due to a dependence of the disk physics on the properties of the central star. Muzerolle et al. (2003) suggest as a cause the effect of the X-ray emission from the central star on the disk ionization and angular momentum transfer. It is also possible that the correlation reflects the properties of the pre-stellar cores, from which the star and disk form. Both possibilities need to be explored further.

The large spread of values of $\dot M_{\rm acc}$ for any $M_\star$ may be a side-product of the same mechanisms that produce the correlation between these two quantities, as discussed in Sect. 5. In addition, other effects may play an important role, for example the dynamical action of close companions, or the intrinsic variability of the accretion process.

Appendix A: Testing the assumption of coeval star formation

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4706fig4.ps}\end{figure}$

Figure A.1: Same as Fig. 3 for different ages and evolutionary tracks. The top-left panel is for DM98 0.5 My (as in Fig. 3), the bottom-left for DM98, 1 My, the top right is for Siess (2000) evolutionary tracks at 0.5 My, the bottom right at 1 My. In each panel, the two dotted lines ( $\dot M_{\rm acc}$ $\propto$ $M_\star ^2$ ) have been drawn to guide the eye in the comparison.

The assumption of coeval star formation, albeit quite reasonable for a region like $\rho$ Oph, introduces errors in our results. The same is true of the choice of any specific set of evolutionary tracks. However, it turns out that both kinds of errors are unimportant, when dealing with a large sample of objects as in our case.

Figure A.1 shows the analog of Fig. 3, reproduced on the top-left panel, computed using the DM98 isochrone for 1 My and the evolutionary tracks of Siess (2000) for 0.5 My and 1 My, respectively. Older tracks give slightly lower values of $\dot M_{\rm acc}$ , especially for more massive objects, while the range of $M_\star$ remains practically the same. Adopting different evolutionary tracks does not change the results. The main consequence of assuming coeval star formation is to reduce slightly the real spread of $\dot M_{\rm acc}$ for any given value of $M_\star$ .

Appendix B: Taurus: a test of the method

A validation of the method used to compute the two quantities $\dot M_{\rm acc}$ and $M_\star$ and an estimate of the errors can be obtained by applying the same procedure to a sample of objects with known stellar parameters and accretion rates.

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{4706fig5.ps}\end{figure}$

Figure B.1: Top panel: $\dot M_{\rm acc}$ vs. $M_\star$ for objects in Taurus for which $\dot M_{\rm acc}$ has been obtained from veiling and/or by fitting the H $\alpha$ profiles with magnetospheric accretion models (crosses and squares). Data from Gullbring et al. (1998), Muzerolle et al. (1998b, 2003, 2005), White & Ghez (2001), White & Basri (2003), Calvet et al. (2004). Squares identify objects for which we could find in the literature Pa $\beta$ observations. Diamonds are very low mass objects in Ophiuchus and Chamaeleon with $\dot M_{\rm acc}$ estimates from H $\alpha$ profile fitting (from Natta et al. 2004). Bottom panel: both $M_\star$ and $\dot M_{\rm acc}$ have been derived following the method we use for the Ophiuchus objects in this paper (see text). As in Fig. 3, the two dotted lines show the run of the $\dot M_{\rm acc}$ $\propto$ $M_\star$ ^1.8 and $\dot M_{\rm acc}$ $\propto$ $M_\star$ relationships.

The only sample for which this is possible is Taurus, which has been studied extensively over a large range of masses We have taken all the Taurus objects for which we could find in the literature reliable stellar parameters and accretion rates, measured from veiling and/or by fitting the observed H $\alpha$ profiles with magnetospheric accretion models (Muzerolle et al. 1998a, 2003, 2005; Calvet et al. 2004). For those with published Pa $\beta$ fluxes or equivalent widths, we have followed the same procedure used for the $\rho$ Oph sample. We have first computed $L_{\rm acc}$ from L(Pa $\beta$ ), and determined the stellar parameters $M_\star$ / $R_\star$ and $M_\star$ from $L_\star$ , assuming coeval star formation at 1 My and the DM98 evolutionary tracks. $\dot M_{\rm acc}$ is then derived from $L_{\rm acc}$ and $M_\star$ / $R_\star$ .

The results are summarized in the Fig. B.1. The top panel shows the complete sample of Taurus objects for which we could find measurements of $\dot M_{\rm acc}$ in the literature. The squares are those for which also Pa $\beta$ data exist; because none of the BDs in Taurus has a published J-band spectrum, we have added the BDs in Ophiuchus and Chamaeleon for which Natta et al. (2004) have measured $\dot M_{\rm acc}$ from model fitting of the H $\alpha$ profiles. The bottom panel shows the same plot when both $M_\star$ and $\dot M_{\rm acc}$ are derived from the observed $L_\star$ and Pa $\beta$ luminosity, as for the $\rho$ Oph stars.

The results indicate that our procedure does not introduce systematic trends in the results. The trend of $\dot M_{\rm acc}$ increasingsharply with $M_\star$ is reproduced in our method, and also the range of $\dot M_{\rm acc}$ for a given $M_\star$ is similar, even if, as expected, the assumption of coeval star formation underestimates its spread slightly.

Accretion in the $\rho$ -Ophiuchi pre-main sequence stars^,

1 Introduction