Free Access
Volume 505, Number 2, October II 2009
Page(s) 695 - 706
Section Stellar structure and evolution
Published online 24 July 2009

Online Material

Appendix A: (Circum)Stellar parameters from SED fits

\mbox{ \epsfig{file=10972A1a.eps, height=6.0cm} \epsfig{file=10972A1b.eps, height=6.0cm} \epsfig{file=10972A1c.eps, height=6.0cm} }\end{figure} Figure A.1:

Comparison between mass accretion rates from the literature and those derived from SED fits for the sample of T-Tauri stars considered in Robitaille et al. (2006). SED fits and determination of parameter ranges were performed as for the $\rho $ Ophiuchi objects discussed in this paper. Panel  a) compares the literature data with results of SED fits using all the available photometry, including optical bands. Panel  b) is analogous, but only photometry longward of 1 $\mu $m was used for the SED fits. Panel  c) compares the results of SED fits with and without optical photometry. Reduced $\chi ^2$ values and mean absolute distances from the bisector, both computed considering uncertainties on the abscissa only, are reported within each panel.

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In this appendix, we describe how we constrained some stellar and circumstellar parameters of the objects in our sample by comparing their SEDs with the theoretical models of Robitaille et al. (2006). These consist of a grid of 200 000 model SEDs that include contributions from the central star, the circumstellar disk, and the envelope, parametrized with 14 parameters. The models that best approximate the observed SEDs were found with the aid of the Web-based tool presented by Robitaille et al. (2007). As stated by Robitaille et al. (2007), and in accord with basic principles, this method does not allow the simultaneous determination of all 14 physical parameters, since the SEDs are often defined by less than 14 independent fluxes. However, depending on the available fluxes, some of the parameters can be constrained more narrowly than others. We are interested here, in particular, in obtaining the range of values compatible with the observed SEDs for: i) the extinction toward our objects; ii) their disk accretion rates.

A.1 The method and its validation

Our procedure follows closely that of Robitaille et al. (2007): from the Web interface we obtain, for each object, a list of the 1000 models that best approximate the observed SEDs, i.e. those with the smallest $\chi ^2$. Our ``best guess'' parameter values and associated confidence intervals are then derived by selecting a set of statistically reasonable models and computing the median and the $\pm$$1\sigma$ quantiles of the parameter values for these models. The statistically reasonable models were defined as those with reduced $\chi^2 < (\chi^2_{\rm best}+3)$, where $\chi^2_{\rm best}$ refers to the best fit model, or if this condition results in less than 10 models, the 10 models with smallest $\chi ^2$. Note that, because the uncertainties on the observed SEDs are not well defined (see below), and the parameter space is sampled only discretely by the adopted grid of models, the statistical significance of the thus derived confidence intervals cannot be easily assessed.

A similar method[*] was tested by Robitaille et al. (2007) by considering a sample of Taurus-Auriga objects for which stellar and circumstellar parameters had been derived independently in the literature and comparing these parameters with those obtained from fitting the SEDs, defined from the optical to millimeter wavelengths. In the case of our heavily absorbed $\rho $ Ophiuchi YSOs, the SEDs lack, with the exception of one star, data in the optical bands, i.e. those more directly affected by the accretion-shock emission. In order to test our ability to constrain the accretion rates in the absence of optical information, we repeated the SED fits of the Taurus-Auriga stars of Robitaille et al. (2007), using the same datapoints to define the SEDs, and both including and excluding the optical magnitudes. The results are shown in Fig. A.1. Panel a), analogous to Fig. 2b in Robitaille et al. (2007), compares the accretion rates derived from the SED fits, including optical data, with independent values from the literature. Panel b) compares the results of the SED fits without the optical magnitudes with the literature data. The agreement between the two quantities is acceptable and may actually be considered better than in the former panel: the reduced $\chi ^2$, computed from the identity relation considering only uncertainties on $\dot{M}_{\rm SED}$, is indeed reduced from $\sim$12 to 1.7. This can in part be attributed to the increased error bars; note, however, that the average of the unsigned differences, abs( $\dot{M}_{\rm SED}-\dot{M}_{\rm Lit.}$), is almost unchanged, 0.49 dex for panel a) and 0.48 dex for panel b). Panel c) compares the $\dot{M}$ from the SED fits with and without optical magnitudes, showing that the two sets of values agree within uncertainties. We conclude that the SEDs defined from IR to millimeter wavelengths are indeed sensitive to the accretion rate, at least in the $\dot{M}$ range covered by the Taurus-Auriga sample: log  $\dot{M}=[-8.5,-6]$.

This is due to the effect of viscous heating affecting the disk thermal structure. To exemplify this effect we plot in Fig. A.2, as a function of accretion rate, the ratio between the IRAC 3 band and the J-band flux, for the Robitaille et al. (2006) models for stars with mass between 0.7 and 1.3 $M_\odot $, age between 1 and 2 Myr (implying little or no circumstellar envelope), and low disk inclination with respect to the line of sight ( $i<60^\circ$). We plot with different symbols models with disk inner radii in different ranges, since the inner hole affects the flux at the IRAC 3 wavelength (5.8 $\mu $m). A relation between the two quantities is seen for models with moderate inner disk holes, apparently characterized by different regimes in three different $\dot{M}$ranges: $\log(\dot{M}/M_{\odot})\la-11$, $-11\la\log(\dot{M}/M_{\odot})\la-9$, and $\log(\dot{M}/M_{\odot})\ga-9$. The factor of $\sim$2 scatter around this relation may likely be attributed to model variations within the specified parameter ranges and to the several other unconstrained model parameters. Similar and even more pronounced trends are apparent in analogous plots using fluxes in longer wavelength IRAC and MIPS bands, with the expected difference that at the longer wavelengths, emitted farther out in the disk, the size of the inner hole has a much smaller effect. The three regimes in Fig. A.2 can be understood as follows: i) for large accretion rates, $\log(\dot{M}/M_{\odot})\ga-9$, the flux in the IRAC band, emitted by the inner disk (R<1 AU), is significantly affected by viscous accretion (D'Alessio et al. 1998,1999); ii) for $-11\la\log(\dot{M}/M_{\odot})\la-9$ disk heating is dominated by the stellar photospheric emission and, consequently, no relation between the IRAC flux and $\dot{M}$ is observed; iii) for $\log(\dot{M}/M_{\odot})\la-11$ we again observe a direct relation between the IRAC 3 flux and $\dot{M}$, which we attribute to the fact that these low accretion rates correspond, in the Robitaille et al. (2006) model grid, to very low disk masses ( $M_{\rm disk}\la10^{-6}~M_\odot$ for the $\sim$1 solar mass stars plotted in Fig. A.2). Since, in the model grid, disk mass and accretion are directly correlated and such low mass disks are optically thin (Robitaille et al. 2006), lower accretion rates imply lower disk mass and lower emission in the IRAC band. The IRAC 3 flux vs. $\dot{M}$ correlation in this regime does not therefore imply that that the mid-IR SED carries direct information on disk accretion.

As a result of this discussion, in the derivation of accretion rates for our $\rho $ Ophiuchi sample from the SED fits, we decided not to use values below $10^{-9}~M_{\odot}$ yr-1. In such cases we instead conservatively assigned upper limits to $\dot{M}$ equal to the maximum between $10^{-9}~M_{\odot}$ yr-1 and the upper end of the $\dot{M}$ confidence interval (see above).

\epsfig{file=10972A2.eps, height=8.8cm} }\end{figure} Figure A.2:

Scatter plot of the ratio between the flux in the IRAC 1 band over that in J, as a function of disk accretion rate, according to the Robitaille et al. (2006) models for a solar mass stars. Each point corresponds to one of the Robitaille et al. (2006) models satisfying the following conditions: mass of the central object between 0.7 and 1.3 $M_\odot $, age between 1 and 2 Myr, and disk inclination with respect to the line of sight <60$^\circ $. Different symbols indicate models with an inner disk radius in one of five ranges as indicated in the legend.

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A.2 The $\rho $ Ophiuchi sample

We collected photometric measurements and uncertainties (when available) for our $\rho $ Ophiuchi sample from several sources: J, H, and $K_{\rm s}$ magnitudes (or upper limits) were taken for almost all objects from 2 MASS[*]; Spitzer IRAC (bands 1-4) and MIPS (bands 1 & 2) photometry was collected from the c2d database[*] (Evans et al. 2003); 1.2 mm fluxes were collected from Stanke et al. (2006) and 1.3 mm fluxes from Andre & Montmerle (1994)[*]. Optical $ {\it UBVR}$ photometry for one object with small absorption (DoAr 25) was taken from Yakubov (1992). Table A.1 lists all the photometric flux densities collected from the literature.

Finally, we complement the photometric data with flux densities from the IRS spectra (cf. Sect.  2.1). We computed flux densities between 10 and 18 $\mu $m, at regular wavelength intervals spaced by 0.5 $\mu $m. Each flux density was taken as the average of the spectral bins in 0.2 $\mu $m intervals centered at the nominal wavelength. For the four stars with two IRS observations, we have taken the average of the two spectra. (In three cases the wavelength-averaged fluxes differ by less than 0.1 dex, while in one case, EL29/GY214, the difference is 0.4 dex. In all cases we verified that the results of the model fits did not change appreciably choosing either of the two spectra). Table A.2 lists the flux densities from the IRS spectra. As stated in Sect. 2.1 our sky subtraction procedure does not take into account diffuse nebular emission. In order to assess the significance of diffuse emission on the object flux densities, we have considered the IRS spectra of the 13 YSOs in our sample observed in the context of the Spitzer legacy program From Molecular Cores to Planet-Forming Disks (``c2d'', Evans et al. 2003). As with the entire c2d sample, the reduced/sky-subtracted IRS spectra have been analyzed (and made publicly available) by the c2d team, using a sophisticated extraction and sky subtraction method based on the modelling of the cross dispersion profiles (Lahuis et al. 2007). We have compared the flux densities derived from the c2d-reduced spectra with those derived from the same spectra reduced by us. We find the spectra to be similar, with both the maximum and the wavelength-averaged discrepancy decreasing with object intensity. The maximum discrepancy falls below 10% for the 9 YSOs with c2d-reduced spectra that have an average flux >0.5 Jy. Based on this comparison, and noting that the c2d objects are representative of our sample in their position with respect to nebulosity seen in IRAC and MIPS maps, we decided to use the IRS-derived fluxes to define the SEDs of the 17 stars with average IRS flux >0.5 Jy.

As suggested by Robitaille et al. (2007), in order to account for systematic uncertainties, underestimation of the measurement errors, and intrinsic object variability over time, a lower limit of 25%, 10%, and 40% was imposed on the uncertainties of optical, NIR/MIR, and millimeter fluxes, respectively.

Figure A.3 exemplifies the ``fitting'' procedure described in Sect. A.1 for three of our YSOs. It shows the SEDs with the best fit models and the distributions of two fit parameters, $A_{\rm V}$ and $\dot{M}_{\rm disk}$, both for the 1000 models with lowest $\chi ^2$ and for the statistically reasonable ones (cf. A.1). SEDs and best fit models for the 28 YSOs in our sample are shown in Fig. A.4.

\epsfig{, width=5.80cm}\epsfig{, width=5.95cm}\epsfig{, width=5.95cm} }\end{figure} Figure A.3:

Examples of SED fits for three objects in our sample with [Ne II] detections. From left to right: DoAr25/GY17, WL20/GY240, and IRS44/GY269. The first is classified as Stage/Class II, the other two as Stage/Class I. The upper row shows the SEDs and the best fit models as produced by the Web interface provided by Robitaille et al. (2006). For the datapoints, detections and upper limits are indicated by circles and triangles, respectively. The lower two rows represent distributions of two fit parameters, $A_{\rm V}$ and $\dot{M}_{\rm disk}$. The empty histograms refer to the 1000 model fits with lowest $\chi ^2$ and the green histograms to the statistically reasonable samples of models defined in Sect. A.1. The solid and dashed vertical lines indicate the median and the 1$\sigma $ dispersion for these latter samples. For the panels in the second row, the symbols close to the upper axis indicate the $A_{\rm V}$ values inferred from the AJ in Table 2 (circles) and from the X-ray-derived $N_{\rm H}$in Table 4 (squares).

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\epsfig{, width=4.60cm}\epsfig{file=10972A4a.p..., width=4.60cm}\epsfig{, width=4.60cm} }\end{figure} Figure A.4:

SEDs and best fit models, as produced by the Web interface provided by Robitaille et al. (2006), for the 28 YSOs in our sample.

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Following visual examination of the SED fits and of the distributions of model parameters used to define the confidence intervals, we decided to modify the input datapoints for two objects: for IRS45/GY273 we excluded the 1.2 and 1.3 mm datapoints from Stanke et al. (2006) and Andre & Montmerle (1994); including these points significantly worsened the quality of the fit and had a significant effect on the values of the parameters. The 1.2 mm flux is >20 times higher than the 1.3 mm flux (an upper limit) and can probably be attributed to an extended source that includes our YSO. For GY289, a source with average IRS flux <0.5 Jy, we decided to include the IRS datapoints because: i) they agree quite well with the MIPS fluxes at similar wavelengths; ii) the quality of the model fit is reasonable ( $\chi^2_{\rm best}\sim 2$) and; iii) the confidence intervals of the model parameters are narrower but compatible with those from the fit performed without these points.

For one object, WL5/GY246, we could not obtain a unique fit with the above procedure. The object was previously classified as a deeply absorbed Class III star with an F7 spectral type (Greene & Meyer 1995), and our SED was defined by J, H, K, Spitzer IRAC 1-4 and 1.2/1.3 mm fluxes. Fits both with and without the mm fluxes, likely contaminated by nearby sources (cf. Stanke et al. 2006; Andre & Montmerle 1994), consistently yield high envelope and/or disk accretion rates, typical of a Class I object, but having little effect on the NIR/MIR part of the SED due to the associated large inner disk radii. The NIR/MIR SED can however be fit equally well by purely photospheric ``Phoenix'' models, as suggested by the same Robitaille et al. (2007) web interface used to fit the star/disk/envelope models. We thus decided to assume that WL5/GY246 is a Class III object and to derive its extinction, effective temperature, and stellar mass using the J, H, and K photometry, the spectral type, and the calibrations tabulated by Kenyon & Hartmann (1995). Uncertainties were estimated from the assumed uncertainty on the spectral type, one subclass, and the range of values obtained by estimating the absorption from the J-H, H-K, and J-K colors.

Table 3, introduced in the main text (Sect. 2.3), lists the outcome of the SED-fit process: the quality of the fit (the $\chi ^2$ of the ``best-fit'' model), the object extinction (the sum of interstellar and envelope extinction), the stellar effective temperature and mass, the disk mass, the disk and envelope accretion rates, the evolutionary Stage. The last quantity was assigned following Robitaille et al. (2007). Stage I: $\dot{M}_{\rm
env}/M_* > 10^{-6}$; Stage II: $\dot{M}_{\rm env}/M_* \le 10^{-6}$ and $M_{\rm disk}/M_* > 10^{-6}$; Stage III: $\dot{M}_{\rm env}/M_* \le 10^{-6}$ and $M_{\rm disk}/M_* \le 10^{-6}$. As indicated in the main text, in order to use a designation more familiar to researchers in the field, we also refer to the ``Stages'' as ``Classes''.

Figures A.5 and A.6 compare the extinction values ($A_{\rm V}$) and stellar  $T_{\rm eff}$ obtained from the SED fits with the same parameters listed in Table 2 for Class II and Class III stars. Given the considerable uncertainties of both determinations, the SED fits yield results similar to those obtained with the method of Natta et al. (2006). A similar comparison with the accretion rates derived from the Pa$\beta$ and Br$\gamma$ NIR line fluxes (in Table 2) is less conclusive due to the large number of upper limits and to the large uncertainties that affect the spectroscopic measurements as well as the SED fits. Seven objects can be used for the comparison, having accretion rate estimates or upper limits from both methods. For only two stars, both methods yield estimates: those for IRS 54 are in good agreement; for WL 16 the spectroscopic estimate is 2.6 dex higher than the value from the SED fits, $\dot{M}\sim 10^{-8}$ $M_\odot $ yr-1. The discrepancy is however reduced to 1.2 dex when comparing the result of the SED fit with the Natta et al. (2006) value. Moreover, the derivation of $\dot{M}$from the Pa$_\beta$ line with the method of Natta et al. (2006, see also Sect. 2.3# is better suited for cool stars and is likely to yield inaccurate results for WL 16 ( $T_{\rm
eff}\sim10^4$ K). An independent estimate by Najita et al. (1996) yielded an upper limit compatible with the SED value: $\dot{M}\la
2\times10^{-7}$ $M_\odot $ yr-1. Three other stars have $\dot{M}$ estimates from the SED fits and upper limits from Table 2: in two cases, IRS 51 and IRS 47, the confidence intervals from the SED fits are consistent with the upper limits; for DoAr 25/GY17, the only star with optical magnitudes, the SED fit yields an accretion rate that is 1.6 dex higher than the upper limit from the Pa$_\beta$ line. Finally, for two stars, WL 10 and WL 11, the spectroscopic estimates are 0.4 dex and 0.1 dex larger than the upper limits from the SED fits. The discrepancy is however reduced to 0.24 dex for WL 10 and disappears for WL 11 if the slightly larger $\dot{M}$ values from Natta et al. (2006) are considered instead of those in Table 2.

A.3 Summary

In this Appendix we have shown that the SED models of Robitaille et al. (2006), although undeniably approximate, can be useful to constrain parameters such as the line-of-sight absorption and the disk accretion rate, even in the absence of optical photometry. Although resulting uncertainties in these parameters are often large, the constraints are by and large compatible with independent determinations obtained with more direct methods.

{\epsfig{file=10972A5.eps, width=8.8cm} }\end{figure} Figure A.5:

Comparison of the $A_{\rm V}$values derived from fitting the SEDs with the Robitaille et al. (2006) models with values derived from 2MASS photometry (cf. Table 2). Objects of different SED Class are indicated by different symbols as shown in the legend.

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\par {\epsfig{file=10972A6.eps, width=8.8cm} }\end{figure} Figure A.6:

Same as Fig. A.5 for the effective temperatures.

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Table A.1:   Flux densities, in mJy, collected from the literature (cf. Sect. A.2) and used for the SED fits.

Table A.2:   Flux densities, in Jy, obtained from the IRS spectra for the SED fits.

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