THE ASTROPHYSICAL JOURNAL, 536 : 406È415, 2000 June 10( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

INFALL SIGNATURES IN SPECTRAL LINE PROFILES OF PROTOSTELLAR ENVELOPESHIROHIKO MASUNAGA1 AND SHU-ICHIRO INUTSUKA2
Division of Theoretical Astrophysics, National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan
Received 1999 April 19 ; accepted 2000 January 20

ABSTRACT
A double-peaked proÐle with a stronger blue peak in a molecular line spectrum is considered strong
evidence for the infall motion in the gas envelope surrounding a protostar. Some past studies performedmodel calculations for reproducing observed spectral proÐles using simpliÐed dynamical models such as
isothermal similarity solutions. However, validity of the similarity solutions for spectral line synthesis
should be examined in comparison with more realistic dynamical models.
In this paper we carry out theoretical modeling of molecular line spectra, adopting a spherically symmetric radiation hydrodynamical model for protostar formation taken from a recent work by theauthors. This study concentrates on how infall motions could account for the asymmetric line proÐles
observed toward protostellar sources. We do not explicitly consider the e†ects of rotation and outÑows
on the generation of line proÐles. In our numerical code for the non-LTE line transfer, the level populations are fully consistent with the radiation Ðeld under an arbitrary physical structure in spherical symmetry, and hence, reliable spectral synthesis has been enabled and compared to the LTE, large velocity
gradient (or Sobolev), and microturbulence approximations.
Contrary to the remarks by Zhou, we do not Ðnd an overestimation of line widths, although the
dynamical evolution resembles the Larson-Penston solution rather than the expansion-wave solution.
Furthermore, the infall motion produces wings extending to v \ ^2 km s~1 in line spectra, in contrast
to previous works where wings could not be produced by infall models. These results imply that simpliÐed infall models, such as the isothermal similarity solutions adopted by previous authors, are notalways suitable to the detailed modeling of line spectra.
Subject headings : hydrodynamics È ISM : clouds È ISM : molecules È line : proÐles Èradiative transfer È stars : formation
1. INTRODUCTION

always modeled successfully. A wing component can be
attributed to outÑows, as is usually done, but no theoreticalmodels are available to Ðt observations including the contamination of outÑows.
In order to reproduce the observed line proÐles, previous
workers typically adopted the isothermal similarity solutions, which consist of a family of solutions including two
well-known representatives, i.e., the ““ LP model ÏÏ and the““ Shu model ÏÏ according to the deÐnition by Zhou (1992).
The LP model contains, in addition to the original solution
derived by Larson (1969) and Penston (1969) for t \ 0, thecontinuous counterpart obtained by Hunter (1977) after t
exceeds zero. Here t \ 0 denotes the instance when thecentral density diverges to inÐnity, or when the central staris supposed to be born. On the other hand, the Shu model
contains a sequence of the hydrostatic equilibrium solutions
for t \ 0 and the expansion-wave collapse solution (Shu
1977) for t [ 0. Zhou (1992) calculated emergent line
spectra for these two models, and he reached the conclusionthat the LP model overestimates the velocity width of CS
lines in comparison with observations.
This seems, however, to arise from an unrealistic condition in the ““ naive ÏÏ LP model, where the infall velocity
increases with radius and converges to a Ðnite value(\3.26c ) at inÐnity. On the other hand, numerical experis radiation hydrodynamic (RHD) calculations
ments by(Larson 1969 ; Masunaga, Miyama, & Inutsuka 1998) havefound that the initial phase of protostellar evolution is
excellently approximated by the LP model under reasonable boundary conditions supposing a Ðxed boundary or aconstant external pressure at a Ðnite radius. This ““ realistic ÏÏLP model has decreasing infalling velocity at a large radius,

Observations have revealed that protostellar cloud cores
often show characteristic proÐles in molecular line spectra.
In particular, a double-peaked proÐle with a blue asymmetry in an optically thick line is often observed, which isconsidered strong evidence for infall motion. The originalidea is traced back to Snell & Loren (1977) and Leung &Brown (1977). Walker et al. (1986) provided spectroscopic
evidence for infall motions around a cold IRAS source in
Ophiuchus through a line proÐle analysis and proposedthat the object is a young protostar under dynamical
growth. Zhou (1992) systematically investigated line features as evidence of infall motions using the isothermal selfsimilar collapse solutions. Detailed modeling of line proÐles
for B335 was done, and a double-peaked spectrum with astronger blue peak was conÐrmed to be a good qualitative
signature of infall (Choi et al. 1995). Recently, some statistical investigations were done for protostellar candidates
including class 0 and class I sources (Gregersen et al. 1997 ;Mardones et al. 1997). Mardones et al. (1997) revealed thatline proÐles in class 0 sources are statistically more skewed
toward the blueshifted velocity than in class I sources,
which implies that line asymmetry is a good indicator of theevolutionary status at early stages of star formation. On theother hand, some previous models for infall motions do notaccount for a wing component (e.g., Gregersen et al. 1997),
and hence spectral proÐles of individual sources are not
1 Present address : Earth Observation Research Center, 1-9-9 Roppongi, Minato-ku, Tokyo 106-0032, Japan ; masunaga=eorc.nasda.go.jp.
2 inutsuka=th.nao.ac.jp.

406

INFALL SIGNATURES IN SPECTRAL LINE PROFILES
and hence it would provide a di†erent spectral proÐle fromthat obtained by Zhou (1992). The expansion-wave collapse
solution is believed to provide a more reasonable model,
but it is still too idealized for detailed spectral synthesis. The
applicability of the isothermal similarity solutions remains
unexamined, and thus a more detailed study on spectral lineproÐles is necessary using an elaborate theoretical model.
In this paper we adopt a theoretical model by Masunaga(1999) and Masunaga & Inutsuka (2000), where RHD calculations were performed in spherical symmetry to establish
a realistic model for protostar formation. In addition, we
have developed a numerical code for non-LTE spectral line
synthesis where the level populations are determined fully
consistently with the radiation Ðeld under arbitrary physical structures in spherical symmetry. Our numerical
scheme, therefore, enables us to yield more reliable results
than the LTE, large velocity gradient (LVG ; or Sobolev),and microturbulence approximations, where some simpliÐcations on the level populations or velocity structure areimposed. Since our collapse model assumes spherical symmetry, this study concentrates on to what extent the infall
motions account for the asymmetric line proÐles apart fromthe e†ects of rotation and outÑows.
2. NUMERICAL SCHEME

We introduce our numerical scheme for spectral line synthesis in this section.
2.1. Basic Equations
The radiative transfer equation is solved under the given
absorption coefficient, a , and the source function, S :llLI
1 [ k2 LI
kl \ [a (I [ S ) ,k kl ]
(1)
l kl
l
Lr
r
Lk
where
hl
a \
r
(k, l)(n B[nB),
l 4n J`1,J
J J,J`1J`1 J`1,J(2)nAJ`1 J`1,J
S \.
(3)
l (n B[nB)
J J,J`1J`1 J`1,J
Here k is direction cosine. The Einstein coefficients for the
dipole radiation emitted by rotational transitions between
the quantum numbers of J and J ] 1 are
32n4k2 J ] 1
d
B
\
,
J`1,J
3h2c 2J ] 3
2J ] 3B\
B,
J,J`1 2J ] 1 J`1,J
A
\
J`1,J

16hB3
,
(J ] 1)3BJ`1,Jc2

where k is the electric dipole moment and B is the rotationd Only the transitions between *J \ ^1 are perconstant.
mitted for the dipole radiation. The rotation constant isdeÐned by
B4

h
,
8n2I

407

where I is the moment of inertia of a molecule. The energy
level corresponding to the Jth angular momentum L \J[J(J ] 1)]1@2+ is given by
E \ BJ(J ] 1)h .
J
Here we consider only linear molecules.
In order to evaluate a and S , population at every energylllevel is solved.
Ln
J \ n (A] J1B)J`1 J`1,J
J`1,J J`1,J
Lt
]n
J1
B
J~1 J,J~1 J~1,J[ n (A] J1
B)
J J,J~1
J,J~1 J,J~1[ n J1B
J J`1,J J,J`1

AB] N ; cc { n { [ n ; cc { ,H2 @ J ,J J
J @ J,J
J
J

where

(4)

P P

=
dk
dlI r
(k, l)(5)kl J`1,J~10
and cc { is the collisional transition rate of J@ ] J. In our
J ,Jcalculations
the left-hand side of equation (4) is set to be
zero (i.e., L/Lt 4 0).
The line proÐle function for static media, r0(l), isJ`1,J
assumed here to be the thermal Doppler broadening
:
J1

J`1,J

\1
2

1

CA

where

BD

l[l
2
1
J`1,J
,
r0
(l) \
exp [
J`1,J
*l
*l Jn
D
D
l
*l \ J`1,J
D
c

S

(6)

2kT
.
m

The proÐle function is shifted in the frequency space as aresult of the bulk motion with radial velocity v ;rll [ J`1,J v k .
(7)
r(k, l) \ r0
r
J`1,J
J`1,J
c

A

B

2.2. Computational Procedure
The computational procedure follows a typical "iteration method. First, the transfer equation (1) is integrated under trial values for n in equations (2) and (3). The
J as trial level populations.
Boltzmann distribution is used
Second, equation (4) is solved by using J1, which isJ`1,Jestimated by the solution of the transfer equation.
Newlyobtained n provides improved values of the absorptionJ
coefficient andsource function, with which equation (1) issolved again. The above procedure is iterated until the selfconsistent solution between the source function and theradiation Ðeld is obtained everywhere in the collapsing
cloud. The criterion for convergence is that the maximum of
relative errors in a and S is below 1%.
l radiativel
Together with thetransfer equations for N transitions from J \ 0 to N, equation (4) is solved for each J
between 0 and N [ 1 along with the normalization equation,N; n \ a Xn ,
J
c H2
J/0

(8)

408

MASUNAGA & INUTSUKA

where X is the fractional abundance of the molecular tracer
relative to molecular hydrogen. A dimensionless factor a iscunity when most molecules occupy rotational levels lower
than J \ N. Otherwise, a should be adjusted appropriatelycto give the correct normalization. Correction by a becomescimportant when T is large enough to pump up the populations to higher levels than J \ N. In our cases such hot
regions are conÐned near the central protostar, where thedensity is so high that the Boltzmann distribution isachieved and LTE can be assumed. We therefore determine
a assuming that n obeys the Boltzmann distribution :
c
J
E
=
E
N
; (2J ] 1) exp [ J .
a \ ; (2J ] 1) exp [ J
c
kT
kT
J/0
J/0
(9)

A BN

A B

Emergent spectra obtained from the converged solutions
are presented in terms of the antenna temperature, T ,Aversus the Doppler velocity. The antenna temperature is/
T ())P ())d)
l
T \ source B,(10)A/ P ())d)4n lwhere P represents a Gaussian beam pattern with a given
FWHM,l which is given as the wavelength divided by thediameter of an imaginary telescope. The brightness temperature, T , is obtained by
B
c2
T \
I ,
(11)
B 2l2k lwhere I is the calculated spectral proÐle. Our numericallcode permits
an o†-center beam convolution in addition to
the centered beam. The aperture efficiency and the beam
efficiency of the telescope are assumed to be unity (100%) in
our calculations.
The Doppler velocity is simply deÐned by
v
f[f
0,
\
(12)
c
f
0
where c is the speed of light, f is the frequency, and f is the0frequency at the line center measured in the rest frame.
3. CALCULATIONS AND RESULTS

3.1. Molecular T racers
We use CS, HCO`, and H13CO` as molecular tracers
for collapsing clouds. Rotational transitions of these molecules have high critical densities, Z105È106 cm~3 in n , at2which the collisional deexcitation rate is balanced withHEinsteinÏs A-coefficient. They therefore trace dense regions inmolecular clouds, which are candidates for star-forming
cloud cores. Typical abundances are 10~9È10~8 relative tomolecular hydrogen for CS and HCO` and smaller bynearly 1/100 for H13CO`. For probable values of column
density of star-forming cores, HCO` lines are typicallyopaque in contrast to H13CO` lines, which are opticallythin because of their low abundance. Molecules are
assumed to be destroyed for T [ 2000 K. Spectral proÐles
do not su†er from the arbitrariness of the destruction temperature since the region where molecules are destroyed
occupies only a negligible volume in the cloud core.
Tables 1 and 2 show the parameters used in calculations
for the molecular tracers. The collisional deexcitation rates,
cc { (J@ [ J), are taken from tabulated data by Green &J ,JVol. 536
TABLE 1
MOLECULAR PARAMETERS 1

MoleculeBa

CS . . . . . . . . . . . . .
HCO` . . . . . . . .
H13CO` . . . . . .

24.4956
44.592943.3772

kb
d
1.958
3.3
3.3

Xc

Nd

4 ] 10~9
1 ] 10~8
1 ] 10~1010
6
6a Rotation constant in GHz.
b Electric dipole moment in debye.
c Fractional abundance relative to H .
2d Levels of J \ 0ÈN are considered in calculations.

Chapman (1978) for CS and from data by Monteiro (1985 ;for J \ 0 to J \ 3) and Green (1975 ; for J \ 4 and J \ 5)for HCO` and H13CO ` , scaled appropriately. The collisional excitation rate, cc {(J@ [ J), is calculated using cc {J ,Jfrom detailed balance : J,JA

B

E
cc { 2J@ ] 1
J,J \exp [ J{,J .
(13)2J ] 1kT
cc {J ,J
Equation (13) assures that the level populations obey the
Boltzmann distribution when the radiative transitions are
negligible compared to the collisional transitions.
Since data of the collisional transition rates are given for
only low temperatures (e.g., ¹30 K in Monteiro 1985), we
extrapolate the deexcitation rates for higher temperatures
proportionally to JT assuming that the cross section
would not vary drastically with temperature in the transition rate, SpvT. Although this assumption is rather crude,
it would cause no signiÐcant error in the level populations
because the density is high enough for the population toachieve the Boltzmann distribution (or LTE) wherever thetemperature exceeds 30 K in our present problems.
3.2. Results
Figure 1 shows molecular line proÐles of HCO` and
H13CO` (J \ 3È2 and 4È3). Four stages of the main accretion phase of the evolution were chosen : at the initial stage(labeled ““ 1 ÏÏ), just after the Ðrst core is formed (““ 6 ÏÏ), at theearly stage (““ 12 ÏÏ), and at the late stage (““ 13 ÏÏ). Curve 12
corresponds to the class 0 or early class I phase and curve
13 to more evolved stages. Refer to Table 3, which is taken
TABLE 2
MOLECULAR PARAMETERS 2

Molecule

Transition

Frequency(GHz)

Resolutiona(arcsec)

Resolutionb(AU)

CS . . . . . . . . . . . . .

J \ 1È0J \ 2È1
J \ 3È2
J \ 4È3
J \ 1È0J \ 2È1
J \ 3È2
J \ 4È3
J \ 1È0J \ 2È1
J \ 3È2
J \ 4È3

48.99115
97.98230
146.97345
195.96460
89.1858
178.3716
267.5574
356.7432
86.7544173.5088
260.2632
347.0176

60
60
60
60
69
35
23
17
71
36
24
19

9000
9000
9000
9000
10350
5250
3450
2550
10650
5400
3600
2850

HCO` . . . . . . . .

H13CO` . . . . . .

a A 10 m telescope is assumed except for CS lines, where a constantresolution is given for comparison with Zhou 1992.
b The object is assumed to be at a distance of 150 pc from an observer,i.e., 1@@ \ 150 AU.

No. 1, 2000

INFALL SIGNATURES IN SPECTRAL LINE PROFILES

409

14
12
12
1
10
6
8

12

10

8

13
6

6

4

4

2

2

0

0
-2

0

2

-2

0

2

-2

0

2

1.4
1

1.2

1

.8

.8
.6
.6
.4
.4
.2
.2

0

0
-2

0

2

FIG. 1.ÈComputational results of spectral line synthesis for HCO` and H13CO` (J \ 3È2 and 4È3) are shown. Line proÐles are given for the initialcondition (““ 1 ÏÏ), just after the Ðrst core is formed (““ 6 ÏÏ), the early stage (““ 12 ÏÏ), and the late stage (““ 13 ÏÏ) of the main accretion phase. Refer to Table 3 for label
numbers.

from Masunaga & Inutsuka (2000), for the elapsed timecorresponding to each label number.
While the kinetic temperature is almost constantthroughout the cloud core at the initial stage, the excitation
temperature, T , decreases with radius near the outerex upper left panel) because the initial density
boundary (Fig. 2,

is too low to collisionally excite levels against the deexcitations by spontaneous emission. A static cloud core withoutwardly decreasing T produces a symmetric doubleex
peaked proÐle for an opticallythick line, which appears in
HCO` J \ 3È2 at the initial stage. For the other lines
shown in Figure 1 the initial proÐles are resolved out.

410

MASUNAGA & INUTSUKA

Vol. 536

1.5label 1label 6

3

1
2

.5
1

0

0
0

1

2
log R [AU]34label 12

3

0

1

2
log R [AU]3label 13

3

2

2

1

1

0

4

0
0

1

2
log R [AU]

3

4

0

1

2
log R [AU]3

4

FIG. 2.ÈExcitation temperature, T , of HCO`(J \ 3È2) (dotted lines) as a function of radius as well as the kinetic temperature (solid lines), or equally, T
ex corresponds to each of the four evolutionary stages indicated by the label numbers shown.
exunder the LTE approximation. Each panel

TABLE 3blue

red

Isovelocity Contour

30

ELAPSED TIMESa

Label

t[t b
0
(yr)

1 .....................
6 .....................
12 . . . . . . . . . . . . . . . . . . . .
13 . . . . . . . . . . . . . . . . . . . .

[1.7522 ] 105[1.2656 ] 100
2.2958 ] 104
1.3788 ] 105

a Taken from Masunaga & Inutsuka 2000.
b The o†set of time, t 4 1.7526 ] 105 yr,
0represents the instance when
the second collapse begins (or the instance when a protostar
is born).

As the evolution proceeds, double-peaked proÐles with ablue asymmetry appear in optically thick lines (HCO`), as
also found by past studies (Zhou 1992 ; Zhou et al. 1993 ;Walker, Narayanan, & Boss 1994). Portraying the mechanism of the spectral line formation of these asymmetric features according to the usual manner, Figure 3 illustrates thevelocity Ðeld of the projected velocity to the line of sight justafter the Ðrst core is formed (curve 6) within the 30 AU ] 30
AU region in the center. The qualitative nature of the velocity Ðeld is similar at the more evolved stages. Examinationof the Ðgure reveals that there are two intersections by the

20

10

obs.
0
2.0

-2.0

-10

1.6

-1.6

-1.2

1.2

-20
0.8km/s

-0.8

-30
-30

-20

-10

0

10

20

30

FIG. 3.ÈThe isovelocity contour lines of the projected velocity to theline of sight just after the Ðrst core is formed (curve 6) within the 30
AU ] 30 AU region in the center. The qualitative nature of the velocityÐeld is similar at the more evolved stages. The solid lines correspond to theblueshifted components, and the dotted lines correspond to the redshiftedcomponents. The velocity interval between the adjacent contour lines is 0.2
km s~1, and the highest level is 2.0 km s~1.

No. 1, 2000

INFALL SIGNATURES IN SPECTRAL LINE PROFILES

line of sight for each contour curve for all redshifted andblueshifted velocities. The Ðrst intersection to an observer
occurs in the outer redshifted component, while the Ðrstblueshifted intersection occurs closer to the central object.
Since the excitation temperature decreases with radius, theblue component in a spectrum should therefore be more
intense than the red component, provided the observed
molecular line is optically thick enough. Spectral proÐles ofan opaque line should have a deep absorption feature at theline center because the bulk of the gas remains cold andalmost at rest near the edge of the cloud. This cold, lowvelocity gas absorbs the radiation from the active region in
the cloud center.
The moment just after the Ðrst core is formed, labeled““ 6,ÏÏ also corresponds to the instance just before the birth ofa protostar (cf. Masunaga & Inutsuka 2000). As an observational counterpart, Onishi, Mizuno, & Fukui (1999)
reported a protostellar condensation in Taurus that is very
close to the moment of the formation of a protostellar core.
The observed spectral proÐles of HCO` J \ 3È2, 4È3 andH13CO` J \ 3È2, 4È3 (Fig. 2 in Onishi et al. 1999) are
quite similar to our calculated proÐles labeled ““ 6 ÏÏ in
Figure 1.
The most intensive feature in Figure 1 is seen in an earlystage of the main accretion phase, when the maximum value
is achieved in the luminosity (Masunaga & Inutsuka 2000).
At the end of the main accretion phase, the peak intensity
decreases again because of the non-LTE e†ect (described
below) and because of the depletion of material in the infalling envelope. On the other hand, line widths continue
growing throughout the evolution. Wings extend to v \ ^2
km s~1 in the main accretion phase. Wing components
survive even after the peak intensity is diminished, because
the infall velocity becomes larger as the gravitational potential is deepened by the growth of the protostellar mass.
The optically thin lines (H13CO`) show single-peakedproÐles with a slight blue asymmetry. The line width
increases monotonically with time similarly to the opticallythick lines (Table 4). An optically thin line is helpful todiscriminate the infall signature from other possible interpretations of the double-peaked proÐles in opaque lines.
Cloud complexes that have two separate velocity comTABLE 4
VELOCITY WIDTHS*V a(km s~1)
MOLECULE

TRANSITION

1

6

12

13

CS . . . . . . . . . . . . .

J \ 1È0J \ 2È1
J \ 3È2
J \ 4È3
J \ 1È0J \ 2È1
J \ 3È2
J \ 4È3
J \ 1È0J \ 2È1
J \ 3È2
J \ 4È3

0.770
0.911
0.854
0.707
0.645
0.725
0.672
0.485
0.384
0.382
0.360
0.358

1.07
1.23
1.28
1.30
0.885
1.06
1.06
1.04
0.615
0.704
0.812
1.027

1.41
1.77
2.00
2.23
1.22
1.69
1.93
2.10
0.810
1.03
1.26
1.53

1.45
2.01
2.58
3.38
1.22
1.85
2.27
2.57
0.885
1.44
2.04
2.54

HCO` . . . . . . . .

H13CO` . . . . . .

a The velocity width, *V , is deÐned by eq. (14). Refer to Table 3 forlabel numbers. See also Figs. 1 and 6.

411

ponents would also show a double-peaked line proÐle, but
the double peaks should appear in both optically thick andthin lines in such a case. A set of double-peaked opaquelines and single-peaked transparent lines with a blue asymmetry is strong evidence for infall motions.
In order to demonstrate the signiÐcance of non-LTEe†ects, we have also performed additional calculations
assuming LTE (Fig. 4). As shown by Figure 1, the LTE peakintensities in the optically thin lines are largely overestimated at every stage throughout the evolution because thekinetic temperature, or the LTE temperature, is systematically higher than the actual excitation temperature in
non-LTE calculations (Fig. 2). For optically thick lines, one
can easily see that an LTE calculation fails to produce ashallower absorption dip in the line center, since LTE calculations cannot reproduce a rapid decrease with radius of T
exnear the cloud edge as observed in the non-LTE results.
Figure 2 also shows that the excitation temperature at theÐnal stage exceeds the kinetic temperature where 1 [log R(AU) [ 2, where the infalling gas is directly irradiatedby hot photons from the central region and thus the molecular energy levels are pumped up to higher states more
frequently than expected by the Boltzmann distribution forthe local kinetic temperature. In a signiÐcant fraction of thecloud core [log R(AU) Z 2], however, the excitation temperature falls below the kinetic temperature because of theattenuation of hot photons and lowered density due to thedepletion of the material. These e†ects cause a substantial
overestimation of the emissivity in LTE calculations compared to the true non-LTE case as well as for the opticallythin lines. This also explains why the LTE peak intensity
keeps growing with time even at the Ðnal stage (curve 13) for
the optically thick line (HCO`) in contrast to the non-LTE
case (compare Fig. 4 with Fig. 1). In the optically thin lines,
the LTE peak intensity decreases with time at the Ðnal stage(curve 13), where the depletion of the infalling material
cancels the overestimation of T . The line proÐle of HCO`
ex(labeled ““ 6 ÏÏ) exhibits red asymmetryin Figure 4 because
the kinetic temperature increases with radius near the cloudedge as a result of the heating by optical photons incident
through the outer boundary (Masunaga & Inutsuka 2000).
We also present in Figure 5 spectral proÐles convolved byan o†-center Gaussian beam by 40A (6000 AU). The lineproÐles are totally di†erent from those convolved by thecentered beam. The peak intensity is weak even at the mostactive phase (curve 12). The lines remain narrow throughout the evolution and wings never come out. These signiÐcant changes in the line properties indicate that thebroadening of lines and the appearance of wings are produced in a conÐned (relatively high density) region near thecenter, and thus they are diluted away in an o†-center
beam. The blue asymmetry becomes weaker in an o†-centerbeam for the HCO` J \ 3È2 line, and it completely disappears from the other lines shown in Figure 5. In general, ano†-center beam suppresses the infall signatures.
We deÐne a line width, *V , according to Zhou (1992) as
follows :*V \ 2[2 ln 2(SV 2T [ SV T2)]1@2 ,(14)

where the angle bracket indicates an average, SqT 4; (qT )/; T . A Gaussian line proÐle yields *V equal to
A by Athis deÐnition. Table 4 tabulates *V for each
FWHM
line proÐle. Line widths increase monotonically with timefor every transition, reÑecting the increasing infall velocity.

412

MASUNAGA & INUTSUKA

Vol. 536

20

15

1
6

15

12
10
13

10

5
5

0

0
-2

0

2

-2

0

2

-2

0

2

3
3
2.5

2

2

1.5

1

1

.5

0

0
-2

0

2

FIG. 4.ÈSpectral line proÐles for HCO` and H13CO` (J \ 3È2 and 4È3) under the LTE approximation are shown. See also Fig. 1.

For comparison with the results by Zhou (1992), we
perform spectral synthesis calculations for CS lines using
the same parameters as adopted by Zhou for the fractional
abundance (X \ 4 ] 10~9) and the resolution (60A at 140pc). We found that the largest line widths throughout theevolution are 1.45, 2.01, and 2.58 km s~1 for CS J \ 1È0,
2È1, and 3È2 lines, respectively, at the end of the main accre-tion phase. As the dynamical evolution resembles the LPmodel rather than the Shu model (Masunaga et al. 1998),
our results should be compared with the ““ LP Ñow ÏÏ in
Zhou (1992). Zhou obtained *V \ 4.03, 4.24, and 4.43 km
s~1 for CS J \ 1È0, 2È1, and 3È2, respectively, at a corresponding evolutionary stage (t \ 1.5 ] 105 yr) for the ““ LPÑow.ÏÏ The line widths obtained by Zhou are signiÐcantly

No. 1, 2000

INFALL SIGNATURES IN SPECTRAL LINE PROFILES

413

1
.2
1
.8
6
.15
12
.6
13
.1
.4

.05
.2

0

0
-2

0

2

.014

-2

0

2

-2

0

2

.0012

.012

.001

.01
.0008
.008
.0006
.006
.0004
.004
.0002

.002

0

0
-2

0

2

FIG. 5.ÈSpectral line proÐles convolved by an o†-center beam by 40@@ for HCO` and H13CO` (J \ 3È2 and 4È3) are shown. See also Fig. 1.

larger than our results. Zhou excluded the LP Ñow becauseof the overestimation of the line width, but our results do
not su†er from such unrealistically broad line proÐles.
Indeed, our results for CS line proÐles (Fig. 6) are totallydi†erent from ZhouÏs ““ LP Ñow ÏÏ proÐles, which showed an
extremely broad absorption at the line center. As predicted
in ° 1, the very large *V found by Zhou (1992) is due to alarge value of the infall velocity (3.3c ) at large radii in the““ naive ÏÏ LP model, while the velocitys declines to zero near
the outer boundary in a ““ realistic ÏÏ LP model that we
obtained, which yields acceptable values for *V .
Another important result of our calculation is the existence of wings extending to v \ ^2 km s~1, in contrast to
some past studies where wings could not be produced by

414

MASUNAGA & INUTSUKA

1.4

Vol. 536

1.4

1.2

1

1

6

1.2

1

12
.8

.8
13

.6

.6

.4

.4

.2

.2

0

0
-4

-2

0

2

4

1.2

-4

-2

0

2

4

-4

-2

0

2

4

.8

1
.6
.8

.6

.4

.4
.2
.2

0

0
-4

-2

0

2

4

FIG. 6.ÈComputational results of the spectral line synthesis for CS (J \ 1È0, 2È1, 3È2, and 4È3) are shown. Line proÐles are given for the initial condition(““ 1 ÏÏ), just after the Ðrst core is formed (““ 6 ÏÏ), the early stage (““ 12 ÏÏ), and the late stage (““ 13 ÏÏ) of the main accretion phase. Refer to Table 3 for label numbers.

infall models (e.g., Gregersen et al. 1997). A main reason
why these past calculations failed to reproduce wings in line
spectra may be attributed to the use of the expansion-wave
collapse solution obtained by Shu (1977). The expansionwave collapse solution consists of an inner free-falling
region and a surrounding hydrostatic envelope, where thelatter gives no contribution to the high-velocity wing.
Moreover, the expansion-wave collapse solution has theminimum mass accretion rate (M0 \ 0.975c3/G at the origin)s
among the family of the isothermal self-similarsolutionsthat include the singular origin. Hence, the growth rate of
the central point mass (or a protostar) is so slow that theNo. 1, 2000

INFALL SIGNATURES IN SPECTRAL LINE PROFILES

gravitational potential grows only gradually, which is lessefficient to produce the high-velocity wings.
Walker et al. (1986) obtained wings at v B ^2 km s~1
assuming a collapse model resembling the expansion-wave
collapse solution, which may imply that the expansionwave collapse solution does not always fail to explain thepresence of wings. However, Walker et al. (1986) modiÐed
the inner portion of the expansion-wave collapse solution toÐt observations, and thus the density and velocity distributions are not treated self-consistently for r \ 1016 cm, where
the wing emission is produced. Moreover, the peak intensities and the depth of the absorption dip show some disagreement between the model and the observation for even
their best-Ðt model. While the simpler models used byearlier workers may be useful in obtaining a ““ Ðrst-order ÏÏunderstanding of conditions in a collapsing cloud core, adetailed analysis and quantitative prediction of line spectra
require more realistic collapse models.
Our results do not exclude possible contamination ofoutÑows in observed spectra. Since 87% of class 0 and I
sources show evidence of driving an outÑow (Mardones etal. 1997), the contamination would be inevitable. Althoughquantitative studies on the contamination of outÑows
require a reliable model for molecular outÑows in the
future, some qualitative or statistical methods are available
to discriminate the signature of outÑows from that of infalls
in line spectra. In order to distinguish the contribution of
infalls from that of outÑows in line spectra of actual objects,
intensity maps in di†erent velocity ranges would be helpful.
While both the blue and red intensities integrated along theline of sight would exhibit similar spatial distributions when
the infall dominates the line formation, the blue and red
wing intensities might be peaked on o†-center positions indi†erent directions in the map (Gregersen et al. 1997).
Rotation of the protostellar envelopes would also a†ect
the line formation in actual protostellar objects. Intensity
maps would be helpful again for separating the signature ofrotation from that of infall in observed emissions. Walker etal. (1994) and Narayanan & Walker (1998) performed LTE
radiative transfer calculations using nonspherical, threedimensional models for spectral line synthesis. They reported that the ““ blue bulge ÏÏ appears across the projectedrotational axis in the intensity maps. The assumption ofLTE, however, can cause errors in spectral line synthesis ofprotostellar objects, as shown in ° 3.2 above. A non-LTE
calculation for multidimensional models is required toobtain a more quantitative understanding of the e†ects thatrotation has on spectral line formation in collapsing cloud
cores.
4. SUMMARY

The results obtained in this paper are summarized asfollows.

415

1. We have developed a spherically symmetric, numericalcode for non-LTE line transfer problems and have applied
it to protostar formation. Our formulation uses neither the
LVG (or Sobolev) nor microturbulence approximations
and is therefore applicable to arbitrary distributions fordensity, temperature, and velocity. For dynamical models,
RHD calculations are used, where the density, temperature,and velocity distributions are obtained in a self-consistentmanner.
2. We conÐrm that the computational results show
double-peaked proÐles with a stronger blue peak for optically thick molecular lines. Optically thin lines show singlepeaked proÐles with a slight blue asymmetry. These
qualitative features are consistent with past studies.
3. The peak is most intensive at an early stage of themain accretion phase, which corresponds to the genuine
class 0 phase (see Masunaga & Inutsuka 2000). While thepeak intensity is lowered in later stages, line widths increase
monotonically with time. The LTE approximation leads toqualitatively di†erent trends from the true non-LTE resultsin spectral proÐles. An o†-center beam is found to suppress
the infall signatures. Contrary to the remarks by Zhou
(1992), we did not Ðnd an overestimation of line widths
when the dynamical evolution resembles the LP model
rather than the Shu model. Modest values of line widths are
obtained under a realistic boundary condition.
4. The infall motion also produces wings extending tov \ ^2 km s~1 in line spectra. Wings could not be produced in some previous works where the expansion-wave collapse solution was adopted as the infall model. Our results
show that the presence of wings in line spectra does notalways mean the existence of outÑows.
As a consequence of the present paper, we found that an
RHD model reproduces typical features in line spectra ofclass 0 and I sources more naturally than the isothermal
self-similar solutions. SimpliÐed infall models, such as theisothermal similarity solutions adopted by previous
authors, are not always suitable to the detailed modeling ofline spectra.

The authors greatly appreciate the helpful comments of
Yasuo Fukui and Toshikazu Onishi. The authors are also
very grateful to the anonymous referee for instructive comments, which helped us polish the manuscript and led us toa better understanding of past investigations related to thisstudy. The research of H. M. is supported by the Research
Fellowships of the Japan Society for the Promotion of
Science for Young Scientists.

REFERENCES
Choi, M., Evans, N. J., II, Gregersen, E. M., & Wang, Y. 1995, ApJ, 448,
Monteiro, T. S. 1985, MNRAS, 214, 419
742
Narayanan, G., & Walker, C. K. 1998, ApJ, 508, 780
Green, S. 1975, ApJ, 201, 366
Onishi, T., Mizuno, A., & Fukui, Y. 1999, PASJ, 51, 257
Green, S., & Chapman, S. 1978, ApJS, 37, 169
Penston, M. V. 1969, MNRAS, 144, 425
Gregersen, E. M., Evans, N. J., II, Zhou, S., & Choi, M. 1997, ApJ, 484, 256
Shu, F. H. 1977, ApJ, 214, 488
Hunter, C. 1977, ApJ, 218, 834
Snell, R. L., & Loren, R. B. 1977, ApJ, 211, 122
Larson, R. B. 1969, MNRAS, 145, 271
Walker, C. K., Lada, C. J., Young, E. T., Maloney, P. R., & Wilking, B. A.
Leung, C. M., & Brown, R. L. 1977, ApJ, 214, L73
1986, ApJ, 309, L47
Mardones, D., Myers, P. C., Tafalla, M., Wilner, D. J., Bachiller, R., &Walker, C. K., Narayanan, G., & Boss, A. P. 1994, ApJ, 431, 767
Garay, G. 1997, ApJ, 489, 719
Zhou, S. 1992, ApJ, 394, 204
Masunaga, H. 1999, Ph.D. thesis, Univ. Tokyo
Zhou, S., Evans, N. J., II, Kompe, C., & Walmsley, C. M. 1993, ApJ, 404,
Masunaga, H., & Inutsuka, S. 2000, ApJ, 531, 350
232
Masunaga, H., Miyama, S. M., & Inutsuka, S. 1998, ApJ, 495, 346