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Appendix: Simulations of dual-beam scanning
observations with the bolometer array

We have developed a program that simulates, for any given input model envelope, the dual-beam maps that would be observed by the various channels of the bolometer array in the on-the-fly scanning mode. For each source, the simulation procedure uses the same "mapping geometry'' as in the real observations: same map size, integration steps (in both azimuth and elevation), absolute coordinates of map center, observing hour angle, wobbler throw, and array geometry. The procedure also accounts for any possible offset of the source with respect to map center and includes a convolution with the beam of the telescope. The simulated dual-beam maps are then reduced in exactly the same way as the real data, including baseline subtraction, flagging of any bad channel, restoration to equivalent single-beam Az-El maps, and combination/projection onto a RA-Dec grid (see, e.g., Emerson et al. 1979; Broguière et al. 1995).

These simulations allow us to assess the loss of signal which results from observing extended envelopes with the dual-beam method. As an example, Fig. 7a shows the transformation of a $I(\theta )\propto \theta ^{-1}$ input envelope model (dotted line) after convolution with the 30 m beam (dashed-dotted line) and the various observation/data reduction steps (solid line). For such an extended envelope model, the beam convolution does not modify the slope of the profile at angular radii $\theta>\ $HPBW (i.e., $11\hbox{$^{\prime\prime}$ }$ here), in agreement with the conclusions of Adams (1991). However, the observation/data reduction technique does change the shape of the radial profile (see hatched region in Fig. 7a), due to the finite size of the maps. This is because all spatial scales larger than the azimuth extent $\Delta Az$of the maps are filtered out. The slope of the simulated output profile depends only weakly on the value of the wobbler throw as long as the latter remains small compared to the map size (in azimuth). (Note that the chop throw would be the most determinant factor in the case of on-off/jiggle-stare maps.) The array geometry and the offset of the source model with respect to map center have little influence on the simulated profile providing the map is big enough that the source is effectively imaged as a positive and a negative signal by each bolometer. Baseline subtraction also has a negligible effect as long as only low-order polynomials are taken out, namely a baseline of order 0-2 applied in time on the global image, plus a simple DC level (0-order baseline) on each map row.


  \begin{figure}
\par\includegraphics[angle=270,width=16.3cm,clip]{ms10015f8.eps}\end{figure} Figure 8: a) Grid of simulated profiles corresponding to a $I(\theta )\propto \theta ^{-1}$ input envelope model including an unresolved central source (or disk) characterized by $S_{\mbox{\tiny disk}} / \mbox{$S_{\mbox{\tiny env}}^{\mbox{\tiny peak}}$ }=
0$, 1, 2 and 10. b) Simulated profiles corresponding to finite-sized isothermal envelope models with $\rho (r) \propto r^{-2}$ inside an outer radius corresponding to $\theta_{\mbox{\tiny out}}=45\hbox{$^{\prime\prime}$ }$ and $70\hbox {$^{\prime \prime }$ }$

Figure 7b presents a grid of simulated model profiles based on the map geometry corresponding to our observations of L1527 (see Fig. 3c). The input models have power-law profiles of the form $I(\theta )\propto \theta ^{-m}$ with m=0.1 to 2.5 beyond an inner radius of $\theta_{\mbox{\tiny in}}=0.5\hbox{$^{\prime\prime}$ }$. (Note that the simulated profiles for m < 2 are not sensitive to the exact value of $\theta_{\mbox{\tiny in}}$ as long as it is much smaller than the beam radius, i.e. $\theta_{\mbox{\tiny in}}<\;$HPBW/10.) As expected, the shallower the slope of the input model, the larger the fractional loss of emission due to the dual-beam observing mode. It is thus crucial to estimate the level of this effect through simulations before a reliable comparison between observed profiles and input models can be made. Such a comparison is more uncertain in the case of envelopes with steep density gradients (e.g. $m\ge2$ in Fig. 7b), as this leads to radial intensity profiles which closely follow the shape of the beam. Although the main beam of the 30 m telescope is a relatively well known Gaussian with FWHM $\ \simeq 11\hbox{$^{\prime\prime}$ }$ at 1.3 mm, the effective error beam associated with dual-beam on-the-fly bolometer observations is more poorly characterized. In particular, the beam model proposed by Guélin (1992) from spectroscopic observations at 1 mm appears to be inconsistent with our own estimation based on bolometer mapping of strong point sources (see dashed line in Fig. 7b). Fortunately, this is not a serious problem here since the radial intensity profiles of most of our sources are well resolved and cannot be fitted by steep ($m\geq2$) power-laws.

In order to simulate the effect of an inner disk component, we have added a central point source to a $I(\theta )\propto \theta ^{-1}$power-law envelope model (see Fig. 8a). The resulting intensity profile follows the shape of the error beam at large radii when $S_{\mbox{\tiny disk}} / \mbox{$S_{\mbox{\tiny env}}^{\mbox{\tiny peak}}$ }\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }5$. (For a $I(\theta)
\propto \theta^{-1.5}$ input envelope model, the same occurs when $S_{\mbox{\tiny disk}} / \mbox{$S_{\mbox{\tiny env}}^{\mbox{\tiny peak}}$ }\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }3$.) Although such large "disk-to-envelope'' ratios could apply to T Tauri stars (e.g. HL Tau), they are not observed in the case of bona-fide protostars which typically have $S_{\mbox{\tiny disk}} / \mbox{$S_{\mbox{\tiny env}}^{\mbox{\tiny peak}}$ }\lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }30\%$ (see, e.g., Looney et al. 2000; Motte et al. 2001). In most cases, the disk should thus have a negligible influence on our analysis of the envelope radial profiles at angular radii $\theta>\ $HPBW.

We have also embedded a $I(\theta )\propto \theta ^{-1}$ envelope model in a Gaussian background cloud with FWHM$\ =0.5$ pc and a cloud-to-envelope peak flux ratio of $S^{\mbox{\tiny
peak}}_{\mbox{\tiny cloud}} / \mbox{$S_{\mbox{\tiny env}}^{\mbox{\tiny peak}}$ }\sim 5\%$, corresponding to a typical ambient column density of $N\h2\sim 10^{21}-10^{22} \, \mbox{$\mbox{cm}^{-2}$ }$. The emission of such an extended Gaussian cloud is essentially filtered out by the dual-beam observing technique. Thus, the apparent power-law index $m_{\mbox{\tiny obs}}$ of the radial profile should only be corrected by a small term $\epsilon_{\mbox{\tiny back}} \lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }0.1$ to yield the intrinsic index of the envelope profile, i.e., $m_{\mbox{\tiny env}} = m_{\mbox{\tiny obs}} + \epsilon_{\mbox{\tiny
back}}$ (see Sect. 4.2.2).

Finally, we have simulated the dual-beam observation of a finite-sized envelope with an outer radius $\mbox{$R_{\mbox{\tiny out}}$ }$ similar to the values measured from Figs. 3 and  4 (see Table 4). Typically, our observations suggest $\mbox{$R_{\mbox{\tiny out}}$ }\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }
6\times HPBW\simeq 30\%\:\Delta Az$ in Taurus and Bok globules, and $\mbox{$R_{\mbox{\tiny out}}$ }\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }3.5~HPBW \simeq
18\%\:\Delta Az$ in Perseus. According to YC91, the slopes of the intensity profiles measured on such angular scales (i.e., HPBW $\
< \theta < 3.5-6\times HPBW$) should be corrected by $\epsilon_{\mbox{\tiny
proj}} > 0.5$ when estimating the intrinsic slopes of the density profiles (i.e., $p = m+1-q-\epsilon_{\mbox{\tiny proj}}$), in order to properly account for projection effects. Figure 8b shows that, due to dual-beam filtering, the effective correction term is much smaller in our case: $\epsilon_{\mbox{\tiny proj}} \lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }0.1$in Taurus and Bok globules and $\epsilon_{\mbox{\tiny proj}} \simeq
0.3$ in Perseus.


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