In this section, we describe our method of deriving the radial density structure of a circumstellar envelope from the observed intensity profile at 1.3 mm, and then apply this method to each source of Table 2.

The 1.3 mm continuum emission mapped around each YSO is first averaged in circular annuli centered at the peak position. This yields a mean radial intensity profile, $S(\theta)$ , where $\theta$ is the angular radius from source center. Depending on the local environment (cf. Col. 8 of Table 2), a fraction of the map has to be masked in order to avoid including emission from neighboring sources such as the fragments apparent in Figs. 1-2. When the map results from the combination of several coverages, we choose the best coverage to derive the intensity profile and make an accurate modeling (see Appendix). The resulting mean radial profiles are shown in Figs. 3a-g and 11a-l for Taurus embedded YSOs, in Figs. 4a-d and 12a-c for isolated globules, and in Figs. 4e-h and 12e-h for Perseus protostars. The radial profiles of several starless cores are also shown in Figs. 3h, 11m-p and 12d. (Figs. 11 and 12 are only available in electronic form at http://www.edpsciences.org.)

$\begin{figure} \par\includegraphics[angle=270,width=17.5cm,clip]{ms10015f3abcd.eps}\par\includegraphics[angle=270,width=17.5cm,clip]{ms10015f3efgh.eps}\end{figure}$

Figure 3: Radial intensity profiles of the environment of 7 embedded YSOs ( a-g) and 1 starless core ( h). Column density estimates assuming $\k13=0.01\;\mbox{$\mbox{cm}^{2} \, \mbox{g}^{-1}$ }$ and $\mbox{$T_{\mbox{\tiny dust}}$ }=15\$ K are shown on the right axis of each profile (in h): $\k13=0.005\;\mbox{$\mbox{cm}^{2} \, \mbox{g}^{-1}$ }$ and $\mbox{$T_{\mbox{\tiny dust}}$ }=10\$ K). The observed profiles (solid curves) are compared with simulated model profiles (dotted curves) and with the 30 m beam (dashed curve). In a) and c-e), two source profiles are shown, before (upper solid curve) and after (lower solid curve) subtraction of the disk component observed by the IRAM Plateau de Bure interferometer at 1.4 mm (Motte et al. 2001); the latter should correspond to the profile of the envelope alone. The input models are all circularly symmetric with infinite power-law profiles, $I(\theta )\propto \theta ^{-m}$ (see Appendix). Unlike YSO envelope profiles, the profile of the starless core L1535-NE (in h) is not consistent with a single power-law model. See complementary Fig. 11 on-line

These radial intensity profiles trace the underlying source column density profiles (see Sect. 3.1 and Eq. (1) of MAN98), although they also depend on the temperature gradient: $S(\theta)\propto N\h2(\theta)\times T(\theta)$ in the Rayleigh-Jeans approximation of the Planck function. Since one expects both $N\h2(\theta)$ and $T(\theta)$ to behave roughly as power-laws over a wide range of angular radii $\theta$ (see Sect. 1 above and Sect. 4.3 below), it is natural to compare the derived radial profiles with those of circularly-symmetric, power-law intensity models of the form $I(\theta )\propto \theta ^{-m}$ .

As shown by Adams (1991), the same power-law intensity distribution, $\tilde{I}(\theta)\propto \theta^{-m}$ , remains after convolution with a Gaussian beam, provided that the profile is examined at radii greater than one full beamwidth from the center (i.e., $\theta \geq 11\hbox{$^{\prime\prime}$ }$ in our case). There is, however, a major complication due to the dual-beam scanning technique, which is addressed by the simulation analysis presented in Appendix. This analysis indicates that a finite-sized, dual-beam scan/map behaves roughly like a high-pass spatial frequency filter suppressing emission on scales larger than the size of the scan/map. In practice, this entails a loss of flux density at large radii in the simulated profiles compared with the intrinsic profiles (see Fig. 7a in Appendix). In order to properly interpret the intensity profile measured toward a given protostellar envelope, we have run an input grid of power-law envelope models through a complete simulation of the mapping and data-reduction processes adapted to this particular source (see, e.g., Fig. 7b in Appendix). We can then estimate the intrinsic power-law index m of the source radial intensity profile (along with its typical uncertainty) by comparison with our output grid of simulated model profiles (cf. Figs. 3-4). The "best-fit'' power-law indices m are listed in Col. 3 of Table 4 for all the sources with spatially-resolved envelope emission. These indices apply to the main portion of the observed intensity profile (see the range of angular radii listed in Col. 2 of Table 4). We give an estimate of the envelope outer radius $\mbox{$R_{\mbox{\tiny out}}$ }$ in Col. 7 of Table 4 whenever there is evidence that the power-law regime breaks down in the outer part of the intensity profile.

Most of the sources are spatially extended and have radial intensity profiles that can be fitted reasonably well over the majority of their extent by one of our simulated $I(\theta )\propto \theta ^{-m}$ models with m = 0.4-1.8. Several observed profiles appear to steepen or merge into some cloud emission beyond a finite radius, $\mbox{$R_{\mbox{\tiny out}}$ }$ (see, e.g., Figs. 3f, 4d, and 4g). The presence of background emission from the ambient cloud is sometimes noticeable in the maps themselves (see, e.g., TMR1 in Fig. 1k). The mean column density we estimate for such background emission is $N\h2 \sim 2{-}7~10^{21}~\mbox{$\mbox{cm}^{-2}$ }$ around Taurus protostars and Bok globules (assuming $\mbox{$T_{\mbox{\tiny dust}}$ }=15\$ K and 20 K, respectively), compared to a column density of envelope material ranging from $N\h2 \sim 5~10^{21}~\mbox{$\mbox{cm}^{-2}$ }$ to $N\h2 \sim 5~10^{23}~\mbox{$\mbox{cm}^{-2}$ }$ . In Perseus, the background has $N\h2 \sim 10^{22}~\mbox{$\mbox{cm}^{-2}$ }$ , while the envelopes reach $N\h2 \sim 10^{24}~\mbox{$\mbox{cm}^{-2}$ }$ (if $\mbox{$T_{\mbox{\tiny dust}}$ }=20$ K).

By contrast, the peculiar Class I sources with compact emission in the maps (cf. Sect. 3) have intensity profiles roughly consistent with the effective point spread function of the 30 m telescope at 1.3 mm (see, e.g., Fig. 3g). This is reminiscent of the radial intensity profiles observed toward Class II sources (cf. AM94).

Finally, we note here that the radial intensity profiles of the cold cores/condensations of Table 3 are flatter than our simulated $I(\theta )\propto \theta ^{-m}$ models with m = 0.5 in their inner regions (i.e., at $r \lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }6\,000\$ AU, where we measure $m \lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }0.1$ - see the example of L1535-NE in Fig. 3h). This is reminiscent of the pre-stellar cloud cores studied by AWM96 and Ward-Thompson et al. (1999).

Table 4: Results of the radial profile analysis for the spatially resolved envelopes
Adopted $\theta$ $m(\theta)^{(1)}$ $\theta_{\mbox{\tiny isoth}}$ ⁽²⁾ $q(\theta)^{(3)}$ p $R_{\mbox{\tiny out}}$ $\mbox{$M_{\mbox{\tiny c$\star$ }}^{\mbox{\tiny\mbox{$R_{\mbox{\tiny out}}$ }}}$ }^{(4)}$ Additional sources

source name range ( $\hbox{$^{\prime\prime}$ }$ ) =m+1-q (AU) ( $M_\odot$ ) of uncertainty

L1551-IRS5 20 $^{\prime\prime}$ -65 $^{\prime\prime}$ $1.5\pm0.3$ 110 $^{\prime\prime}$ 0.4 $2.1\pm0.4$ $9\,100\pm700$ ? 1.5 $^{\mbox{\tiny 25K}}$

L1551-NE 11 $^{\prime\prime}$ -60 $^{\prime\prime}$ $1\pm0.3$ 51 $^{\prime\prime}$ 0.4 $1.6\pm0.4$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }8\,500$ 1.1 $^{\mbox{\tiny 25K}}$

K04113 11 $^{\prime\prime}$ -55 $^{\prime\prime}$ $1.8\pm1$ 26 $^{\prime\prime}$ 0.4 $2.4\pm1$ $8\,000\pm2\,000$ ? 0.4

L1527 11 $^{\prime\prime}$ -120 $^{\prime\prime}$ $1\pm0.4^*$ 26 $^{\prime\prime}$ $\sim$ 0 $2.0\pm0.5$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }17\,000$ 1.7 outflow cavity

K04166 11 $^{\prime\prime}$ -105 $^{\prime\prime}$ $1\pm0.2$ 11 $^{\prime\prime}$ $-0.2\pm 0.2$ $2.2\pm0.5$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }15\,000$ 1.0

K04169 11 $^{\prime\prime}$ -90 $^{\prime\prime}$ $0.8\pm0.2$ 18 $^{\prime\prime}$ 0 $1.8\pm0.3$ $13\,000\pm1\,500$ ? 1.1

T04191 11 $^{\prime\prime}$ -70 $^{\prime\prime}$ $0.8\pm0.4$ 15 $^{\prime\prime}$ 0 $1.8\pm0.5$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }10\,000$ 0.45

IRAM 04191 11 $^{\prime\prime}$ -100 $^{\prime\prime}$ $0.6\pm0.1$ 8 $^{\prime\prime}$ $-0.2\pm 0.2$ $1.8\pm0.4$ $14\,000\pm 1\,500$ ? 1.4 $^{\mbox{\tiny 12.5K}}$

T04325 11 $^{\prime\prime}$ -80 $^{\prime\prime}$ $1.0\pm0.6^*$ 20 $^{\prime\prime}$ 0 $2.0\pm0.7$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }11\,000$ 0.7 within a dense core

TMR1 11 $^{\prime\prime}$ -60 $^{\prime\prime}$ $0.8\pm0.4$ 40 $^{\prime\prime}$ 0.4 $1.4\pm0.5$ $8\,500\pm 1\,500$ ? 0.4

M04381 11 $^{\prime\prime}$ -30 $^{\prime\prime}$ $1.1\pm0.3^*$ 17 $^{\prime\prime}$ 0 $2.1\pm0.4$ $3\,900\pm700$ ? 0.16 within cloud

M04248 11 $^{\prime\prime}$ -80 $^{\prime\prime}$ $0.7\pm0.5^*$ 13 $^{\prime\prime}$ 0 $1.7\pm0.6$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }11\,000$ 0.75 elliptical envelope

K04181+2654 11 $^{\prime\prime}$ -50 $^{\prime\prime}$ $1.2\pm0.6^*$ 15 $^{\prime\prime}$ 0 $2.2\pm0.7$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }7\,000$ 0.16 elliptical envelope

L1157-MM 11 $^{\prime\prime}$ -45 $^{\prime\prime}$ $1.5\pm0.2$ 22 $^{\prime\prime}$ 0 $2.2\pm0.4^*$ $20\,000\pm2\,000?$ 6.5 deprojection

L483-MM 11 $^{\prime\prime}$ -90 $^{\prime\prime}$ $0.4\pm0.3^*$ 54 $^{\prime\prime}$ $0.2\pm0.2$ $1.2\pm0.6$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }18\,000$ 4. elliptical envelope

L588 11 $^{\prime\prime}$ -55 $^{\prime\prime}$ $1.5\pm 0.5$ ? $0.2\pm0.2$ $2.3\pm0.8$ $11\,000\pm2\,000?$ 0.8

B335 11 $^{\prime\prime}$ -120 $^{\prime\prime}$ $1.2\pm0.3^*$ 20 $^{\prime\prime}$ 0 $2.2\pm 0.4$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }30\,000$ 2.5 outflow interaction

L723-MM 11 $^{\prime\prime}$ -40 $^{\prime\prime}$ $1.7\pm0.3$ 17 $^{\prime\prime}$ 0 $2.5\pm0.5^*$ $12\,000\pm3\,000?$ 1.2 deprojection

B361 11 $^{\prime\prime}$ -40 $^{\prime\prime}$ $1.2\pm0.2$ 18 $^{\prime\prime}$ 0 $2.0\pm0.4^*$ $14\,000\pm2\,000?$ 0.9 deprojection

NGC 1333-IRAS4A 11 $^{\prime\prime}$ -50 $^{\prime\prime}$ $1.8\pm0.3$ 31 $^{\prime\prime}$ $0.2\pm0.2$ $2.6\pm0.6$ $17\,000\pm5\,000?$ 8. $^{\mbox{\tiny 35K}}$

L1448-N 11 $^{\prime\prime}$ -35 $^{\prime\prime}$ $1.7\pm0.2$ 32 $^{\prime\prime}$ 0.4 $2.1\pm0.4^*$ $10\,000\pm1\,500?$ 6 deprojection

L1448-C 11 $^{\prime\prime}$ -25 $^{\prime\prime}$ $1.6\pm0.2$ 27 $^{\prime\prime}$ 0.4 $2.0\pm0.4^*$ $7\,500\pm600?$ 2.5 deprojection

NGC 1333-IRAS2 11 $^{\prime\prime}$ -35 $^{\prime\prime}$ $1.7\pm0.3$ 61 $^{\prime\prime}$ 0.4 $2.0\pm0.5^*$ $12\,000\pm2\,000?$ 2.5 $^{\mbox{\tiny 30K}}$ deprojection

L1448-NW 11 $^{\prime\prime}$ -40 $^{\prime\prime}$ $1.3\pm0.3$ 16 $^{\prime\prime}$ $0.2\pm0.2$ $1.8\pm0.7^*$ $13\,000\pm3\,000?$ 3.5 deprojection

IRAS 03282 11 $^{\prime\prime}$ -40 $^{\prime\prime}$ $1.6\pm0.4^*$ 12 $^{\prime\prime}$ 0 $2.3\pm0.6^*$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }12\,000$ 2.5 $^{\mbox{\tiny 15K}}$ deprojection

HH211-MM 11 $^{\prime\prime}$ -45 $^{\prime\prime}$ $1.3\pm0.5^*$ $<37\hbox{$^{\prime\prime}$ }$ $0.2\pm0.2$ $2.1\pm0.6$ $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ } 13\,000$ 2.5 "cometary'' globule

**Table 4:** Results of the radial profile analysis for the spatially resolved envelopes
Adopted	$\theta$	$m(\theta)^{(1)}$	$\theta_{\mbox{\tiny isoth}}$ ⁽²⁾	$q(\theta)^{(3)}$	p	$R_{\mbox{\tiny out}}$	$\mbox{$M_{\mbox{\tiny c$\star$ }}^{\mbox{\tiny\mbox{$R_{\mbox{\tiny out}}$ }}}$ }^{(4)}$	Additional sources
source name	range		( $\hbox{$^{\prime\prime}$ }$ )		=m+1-q	(AU)	( $M_\odot$ )	of uncertainty
L1551-IRS5	20 $^{\prime\prime}$ -65 $^{\prime\prime}$	$1.5\pm0.3$	110 $^{\prime\prime}$	0.4	$2.1\pm0.4$	$9\,100\pm700$ ?	1.5 $^{\mbox{\tiny 25K}}$
L1551-NE	11 $^{\prime\prime}$ -60 $^{\prime\prime}$	$1\pm0.3$	51 $^{\prime\prime}$	0.4	$1.6\pm0.4$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }8\,500$	1.1 $^{\mbox{\tiny 25K}}$
K04113	11 $^{\prime\prime}$ -55 $^{\prime\prime}$	$1.8\pm1$	26 $^{\prime\prime}$	0.4	$2.4\pm1$	$8\,000\pm2\,000$ ?	0.4
L1527	11 $^{\prime\prime}$ -120 $^{\prime\prime}$	$1\pm0.4^*$	26 $^{\prime\prime}$	$\sim$ 0	$2.0\pm0.5$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }17\,000$	1.7	outflow cavity
K04166	11 $^{\prime\prime}$ -105 $^{\prime\prime}$	$1\pm0.2$	11 $^{\prime\prime}$	$-0.2\pm 0.2$	$2.2\pm0.5$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }15\,000$	1.0
K04169	11 $^{\prime\prime}$ -90 $^{\prime\prime}$	$0.8\pm0.2$	18 $^{\prime\prime}$	0	$1.8\pm0.3$	$13\,000\pm1\,500$ ?	1.1
T04191	11 $^{\prime\prime}$ -70 $^{\prime\prime}$	$0.8\pm0.4$	15 $^{\prime\prime}$	0	$1.8\pm0.5$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }10\,000$	0.45
IRAM 04191	11 $^{\prime\prime}$ -100 $^{\prime\prime}$	$0.6\pm0.1$	8 $^{\prime\prime}$	$-0.2\pm 0.2$	$1.8\pm0.4$	$14\,000\pm 1\,500$ ?	1.4 $^{\mbox{\tiny 12.5K}}$
T04325	11 $^{\prime\prime}$ -80 $^{\prime\prime}$	$1.0\pm0.6^*$	20 $^{\prime\prime}$	0	$2.0\pm0.7$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }11\,000$	0.7	within a dense core
TMR1	11 $^{\prime\prime}$ -60 $^{\prime\prime}$	$0.8\pm0.4$	40 $^{\prime\prime}$	0.4	$1.4\pm0.5$	$8\,500\pm 1\,500$ ?	0.4
M04381	11 $^{\prime\prime}$ -30 $^{\prime\prime}$	$1.1\pm0.3^*$	17 $^{\prime\prime}$	0	$2.1\pm0.4$	$3\,900\pm700$ ?	0.16	within cloud
M04248	11 $^{\prime\prime}$ -80 $^{\prime\prime}$	$0.7\pm0.5^*$	13 $^{\prime\prime}$	0	$1.7\pm0.6$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }11\,000$	0.75	elliptical envelope
K04181+2654	11 $^{\prime\prime}$ -50 $^{\prime\prime}$	$1.2\pm0.6^*$	15 $^{\prime\prime}$	0	$2.2\pm0.7$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }7\,000$	0.16	elliptical envelope
L1157-MM	11 $^{\prime\prime}$ -45 $^{\prime\prime}$	$1.5\pm0.2$	22 $^{\prime\prime}$	0	$2.2\pm0.4^*$	$20\,000\pm2\,000?$	6.5	deprojection
L483-MM	11 $^{\prime\prime}$ -90 $^{\prime\prime}$	$0.4\pm0.3^*$	54 $^{\prime\prime}$	$0.2\pm0.2$	$1.2\pm0.6$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }18\,000$	4.	elliptical envelope
L588	11 $^{\prime\prime}$ -55 $^{\prime\prime}$	$1.5\pm 0.5$	?	$0.2\pm0.2$	$2.3\pm0.8$	$11\,000\pm2\,000?$	0.8
B335	11 $^{\prime\prime}$ -120 $^{\prime\prime}$	$1.2\pm0.3^*$	20 $^{\prime\prime}$	0	$2.2\pm 0.4$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }30\,000$	2.5	outflow interaction
L723-MM	11 $^{\prime\prime}$ -40 $^{\prime\prime}$	$1.7\pm0.3$	17 $^{\prime\prime}$	0	$2.5\pm0.5^*$	$12\,000\pm3\,000?$	1.2	deprojection
B361	11 $^{\prime\prime}$ -40 $^{\prime\prime}$	$1.2\pm0.2$	18 $^{\prime\prime}$	0	$2.0\pm0.4^*$	$14\,000\pm2\,000?$	0.9	deprojection
NGC 1333-IRAS4A	11 $^{\prime\prime}$ -50 $^{\prime\prime}$	$1.8\pm0.3$	31 $^{\prime\prime}$	$0.2\pm0.2$	$2.6\pm0.6$	$17\,000\pm5\,000?$	8. $^{\mbox{\tiny 35K}}$
L1448-N	11 $^{\prime\prime}$ -35 $^{\prime\prime}$	$1.7\pm0.2$	32 $^{\prime\prime}$	0.4	$2.1\pm0.4^*$	$10\,000\pm1\,500?$	6	deprojection
L1448-C	11 $^{\prime\prime}$ -25 $^{\prime\prime}$	$1.6\pm0.2$	27 $^{\prime\prime}$	0.4	$2.0\pm0.4^*$	$7\,500\pm600?$	2.5	deprojection
NGC 1333-IRAS2	11 $^{\prime\prime}$ -35 $^{\prime\prime}$	$1.7\pm0.3$	61 $^{\prime\prime}$	0.4	$2.0\pm0.5^*$	$12\,000\pm2\,000?$	2.5 $^{\mbox{\tiny 30K}}$	deprojection
L1448-NW	11 $^{\prime\prime}$ -40 $^{\prime\prime}$	$1.3\pm0.3$	16 $^{\prime\prime}$	$0.2\pm0.2$	$1.8\pm0.7^*$	$13\,000\pm3\,000?$	3.5	deprojection
IRAS 03282	11 $^{\prime\prime}$ -40 $^{\prime\prime}$	$1.6\pm0.4^*$	12 $^{\prime\prime}$	0	$2.3\pm0.6^*$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }12\,000$	2.5 $^{\mbox{\tiny 15K}}$	deprojection
HH211-MM	11 $^{\prime\prime}$ -45 $^{\prime\prime}$	$1.3\pm0.5^*$	$<37\hbox{$^{\prime\prime}$ }$	$0.2\pm0.2$	$2.1\pm0.6$	$\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ } 13\,000$	2.5	"cometary'' globule

Notes: ⁽¹⁾ Power-law index of the radial intensity profile $\mbox{$S_{\mbox{\tiny 1.3~mm}}(\theta)$ }$ over the angular range of Col. 2. ⁽²⁾ Angular radius beyond which heating by the central source is negligible. ⁽³⁾ Power-law index of the radial temperature profile $T(\theta)\propto \theta^{-q}$ over the angular range of Col. 2. ⁽⁴⁾ Total circumstellar mass within the radius $\mbox{$R_{\mbox{\tiny out}}$ }$ of Col. 7 (see Sect. 3.1 for assumed dust opacity and temperature). Star markers in Col. 3 or Col. 6 indicate that additional sources of uncertainty have been taken into account (see Col. 9).

4.2 Factors influencing the derivation of the envelope radial profile

4.2.1 Disk component

In order to derive the intrinsic profile of the envelope, a central point source corresponding to the possible contribution of a compact, unresolved disk should in principle be subtracted out from the bolometer-array data. However, the disk contribution to the single-dish peak flux density appears to be small for the bona-fide protostars of our sample. Based on interferometric measurements, Motte et al. (2001) estimate $S_{\mbox{\tiny 1.3~mm}}^{\mbox{\tiny ~disk}}/\mbox{$S_{\mbox{\tiny 1.3~mm}}^{\mbox{\tiny ~peak}}$ }\lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }30\%$ for 7 Taurus YSOs and 5 Perseus protostars (see also Sect. 3.2). We have simulated the effect of an unresolved disk component of this magnitude and checked that it does not affect the radial intensity profile measured for $\theta> 11\hbox{$^{\prime\prime}$ }$ (see Fig. 8a in Appendix). We conclude that the disk has a negligible influence on our analysis of the envelope radial profile, even in the case of the Perseus protostars with compact envelopes (cf. Sect. 3.2). To illustrate this point, Figs. 3-4 and Figs. 11-12 show the estimated envelope radial profile before and after subtraction of the disk component when the latter is known.

4.2.2 Background cloud

In the presence of filamentary background cloud emission, an intrinsically spherical envelope can artificially look non spherical. The maps of K04169, M04248, and IRAS 03282 (in Figs. 1d, 1f, and 2d) may provide examples of this. In principle, the power-law index $m_{\mbox{\tiny obs}}$ of the observed intensity profile should then be corrected by a term $\epsilon_{\mbox{\tiny back}}$ to yield the intrinsic index of the source envelope: $m_{\mbox{\tiny env}} = m_{\mbox{\tiny obs}} + \epsilon_{\mbox{\tiny back}}$ , where $\epsilon_{\mbox{\tiny back}} >0$ when the effective background emission is positive. (If there is relatively strong background emission just outside the mapped region, part of it may aliased as a negative signal in the restored image, in which case $\epsilon_{\mbox{\tiny back}} < 0$ .) In order to minimize the magnitude of this effect, we measured the intensity profile of each source in the parts of the map least perturbed by any cloud (or companion) emission. For K04169, M04248, and IRAS 03282, we restricted our analysis to sectors perpendicular to the large-scale filamentary emission apparent in Fig. 1d, Fig. 1f, and Fig. 2d. Any remaining background emission should thus be weak compared to the envelope emission itself. We have simulated the observation of a power-law envelope embedded in a Gaussian background of FWHM size 0.5 pc and peak column density $N\h2 \sim 10^{21}$ - $10^{22}\,\mbox{$\mbox{cm}^{-2}$ }$ . These simulations suggest that $\epsilon_{\mbox{\tiny back}}$ is negligible (i.e., $\epsilon_{\mbox{\tiny back}} \lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }0.1$ ) for most sources (see Appendix). In the case of T04325 which lies in the middle of strong emission from the dense core L1535-NE (cf. Fig. 1j), we estimate $\epsilon_{\mbox{\tiny back}} \simeq 0.3$ .

4.2.3 Non spherical envelopes, perturbation
by the outflow

The millimeter dust emission of several sources is clearly not circularly symmetric but displays an elliptical or even more complex morphology (e.g. L1527, B335, HH211-MM). The asymmetries seen toward L1527 and B335 appear to be directly related to the influence of their bipolar outflows. The map of L1527 (Fig. 1l) shows a clear cross-like pattern probably marking the walls of a bipolar cavity excavated by the outflow, which is reminiscent of what is seen at $730\ \mu$ m toward L1551-IRS5 (Ladd et al. 1995). The dust emission mapped around B335 is elongated perpendicular to the outflow axis (Fig. 2a). These bar-like or cross-like enhancements of 1.3 mm dust emission may originate from compression and/or heating of the envelope by the outflow (cf. Gueth et al. 1997). The resulting uncertainties can be estimated by comparing the radial profiles obtained in different quadrants of the maps (see Col. 3 of Table 4). The radial profile averaged over the northern quadrant of the L1527 envelope and the profile measured perpendicular to the B335 outflow are very close to the corresponding circularly averaged profiles: $m_{\mbox{\tiny perp}} = m_{\mbox{\tiny circ}} \pm 0.2$ . In Table 4, we thus give the power-law indices measured for the circularly-averaged intensity profiles of L1527 and B335, but add an extra $\Delta m =0.2$ term to the uncertainty estimated from the power-law fit.

4.3 Assessment of the dust temperature distribution

Assuming that central heating by the inner accreting protostar dominates the thermal balance of the envelope, the dust temperature is expected to decrease outward. More precisely, one expects a radial temperature profile of the form $T(r) \propto r^{-q}$ with $q \approx 0.4$ in the region where the envelope is optically thin to the bulk of the infrared radiation (e.g. Emerson 1988; Butner et al. 1990). In practice, one thus expects (see, e.g., Terebey et al. 1993):

where the stellar luminosity $\mbox{$L_\star$ }$ can be approached by the bolometric luminosity listed in Table 1. Evidently, this temperature distribution is valid only up to the radius where $\mbox{$T_{\mbox{\tiny dust}}$ }(r, \mbox{$L_\star$ })$ reaches the typical dust temperature of the parent ambient cloud, i.e., $\mbox{$T_{\mbox{\tiny cloud}}$ }\sim 10$ K (appropriate to Taurus-Auriga, Perseus, and most isolated globules, see e.g. Myers & Benson 1983). The preceding equation implies that the radius, $\mbox{$R_{\mbox{\tiny isoth}}$ }$ , beyond which $\mbox{$T_{\mbox{\tiny dust}}$ }(r> \mbox{$R_{\mbox{\tiny isoth}}$ }) \approx \mbox{$T_{\mbox{\tiny cloud}}$ }$ , is approximately given by:

The angular radius $\mbox{$\theta_{\mbox{\tiny isoth}}$ }$ corresponding to $\mbox{$R_{\mbox{\tiny isoth}}$ }$ is listed in Col. 4 of Table 4. According to these estimates, most of the envelopes observed in Taurus and isolated globules are likely to be roughly isothermal (at the temperature of the ambient cloud) on the spatial scales sampled by our maps [i.e., for $r > 1\,500~{\rm AU}\times (d/140~{\rm pc})$ ]. On the other hand, we expect central heating to play a significant role in shaping the intensity profiles of the more luminous protostars of Perseus.

For very low luminosity and/or very young accreting protostars, central heating may be completely negligible, and the envelope thermal structure may be dominated by external heating from cosmic rays and the interstellar radiation field (e.g. Goldsmith & Langer 1978; Neufeld et al. 1995), as is probably the case for pre-stellar cores. Gas-grain collisions are expected to maintain the gas and dust temperatures reasonably well coupled to each other, at least for densities $n\h2 \lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }10^5\, \cm3$ (e.g. Ceccarelli et al. 1996). Calculations taking these processes into account show that externally-heated envelopes/cores are likely to be cooler in their inner regions than their parent $\sim$ 10 K molecular clouds (e.g. Falgarone & Puget 1985; Masunaga & Inutsuka 1999; Evans et al. 2001). Masunaga & Inutsuka (1999) find envelope temperatures of $\sim$ 6 K and $\sim$ 10 K at radii of $\sim$ 1000 AU and $\sim$ 10000 AU, respectively, from the center of a $\sim$ 0.1 $L_\odot$ protostar that has just entered the accretion phase. This would correspond to an effective temperature index $q \approx -0.4$ in the range of radii probed by our 1.3 mm observations. Such an outward increase in the envelope temperature is likely to apply to the lowest luminosity protostars of our Taurus sample, e.g., IRAM 04191 and K04166. We have thus adopted $q=-0.2\pm 0.2$ for these objects (see Col. 5 of Table 4).

4.4 Inferred radial density gradients

For an infinite, spherically-symmetric envelope, a simple asymptotic relation exists between the column density profile (see Figs. 3,4) and the underlying radial density gradient (e.g. Arquilla & Goldsmith 1991; YC91; Adams 1991): $N\h2(\bar{r}) \propto r \times \rho(r)$ . Consequently, in the Rayleigh-Jeans regime, spheroidal envelopes with power-law density and temperature gradients (i.e., $\rho(r)\propto r^{-p}$ and $T(r) \propto r^{-q}$ , respectively) are expected to have specific intensity profiles of the form $I(\bar{r})\propto \bar{r}^{-m}$ with m = p + q -1 as a function of projected radius $\bar{r}$ . The power-law density index, p=m+1-q, derived in this way from the "best-fit'' index of the radial intensity profile, m, is given in Col. 6 of Table 4, assuming the temperature index qlisted in Col. 5. Given the morphologies observed in the plane of the sky (see Figs. 1-2), the assumption that the envelopes are roughly spheroidal should be adequate, at least as a first approximation (see, however, Sect. 4.2.3 above and Col. 9 of Table 4). Even in the case of an ellipsoidal envelope, our approach should still yield a correct estimate for the angle-averaged density profile. On the other hand, the density and temperature gradients in the envelopes may not be correctly described by single power-laws over the full range of radii sampled by our observations. If, instead, the density and temperature gradients are represented by series of broken power-laws, then the preceding, simple formula relating p, m, and q will no longer be strictly valid in the transition regions owing to convergence effects. A correction term must be added: $p = m+1-q-\epsilon_{\mbox{\tiny proj}}$ . In particular, this is the case for a finite-sized sphere where the density drops to 0 beyond some radius $\mbox{$R_{\mbox{\tiny out}}$ }$ (e.g. Arquilla & Goldsmith 1985; YC91). Our maps indicate that $\theta_{\mbox{\tiny out}} \lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }70\hbox{$^{\prime\prime}$ }$ (i.e. $\mbox{$R_{\mbox{\tiny out}}$ }\lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }10\,000$ AU at 140 pc) may be a typical value for the outer radius of protostellar envelopes in Taurus and Bok globules (see Col. 7 in Table 4). The correction term due to deprojection effects is then small ( $\epsilon_{\mbox{\tiny proj}} \approx 0.1$ , see Appendix) and is partly compensated for by the presence of background emission (see Sect. 4.2.2). In these cases, we have thus neglected $\epsilon_{\mbox{\tiny proj}}$ and have conservatively estimated the error bar on p as $\Delta p = \Delta m + \Delta q + 0.1$ . For the protostars with compact envelopes (i.e., $\theta_{\mbox{\tiny out}} \lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }40\hbox{$^{\prime\prime}$ }$ in Col. 2 of Table 4), we have used $\epsilon_{\mbox{\tiny proj}} \approx 0.3$ (or 0.2 when a significant background cloud exists) and $\Delta p = \Delta m + \Delta q +0.2$ (see Col. 6 of Table 4).

Taking the various uncertainties into account (see also Sects. 4.2 and 4.3 above), the power-law index of the radial density gradient is found to be $p \sim 1.5-2$ from $r \sim 1\,000\,-\,5\,000$ AU to $r \sim 10\,000\,-\,50\,000$ AU for the spatially resolved envelopes of Taurus YSOs and Bok globules. In several cases, the power-law density structure appears to break down at a finite radius beyond which the envelope merges with the ambient cloud: $\overline{\mbox{$R_{\mbox{\tiny out}}$ }} \lower.5ex\hbox{$\; \buildrel > \over \sim \;$ } 10\,000$ AU for the Taurus YSOs and $\overline{\mbox{$R_{\mbox{\tiny out}}$ }} \lower.5ex\hbox{$\; \buildrel > \over \sim \;$ } 16\,000$ AU for the Bok globules. The power-law density analysis is more uncertain for the compact protostellar envelopes of Perseus for which we estimate $p=(2-2.5)\pm 0.6$ and $\overline{\mbox{$R_{\mbox{\tiny out}}$ }} \sim 10\,000$ AU. For the "peculiar'' Class I sources which are essentially unresolved in our maps (see Fig. 3g and Col. 7 of Table 2), the radius of any envelope, if present at all, must be much smaller, $\mbox{$R_{\mbox{\tiny out}}$ }\lower.5ex\hbox{$\; \buildrel < \over \sim \;$ }1\,500$ AU. The nature of these peculiar Class I sources is discussed in Sect. 5.2.3 below.