THE ASTROPHYSICAL JOURNAL, 510 : 822È827, 1999 January 10( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DOES ““ q B 1 ÏÏ TERMINATE THE ISOTHERMAL EVOLUTION OF COLLAPSING CLOUDS ?
HIROHIKO MASUNAGA1 AND SHU-ICHIRO INUTSUKA
Division of Theoretical Astrophysics, National Astronomical Observatory, Mitaka, Tokyo 181, Japan ; masunaga=milano.mtk.nao.ac.jp,
inutsuka=yso.mtk.nao.ac.jp
Received 1998 April 1 ; accepted 1998 August 13

ABSTRACT
We examine when gravitationally collapsing clouds terminate their isothermal evolution. According toour previous work, the condition with which isothermality is broken down is classiÐed into three cases,
i.e., when (1) the compressional heating rate overtakes the thermal cooling rate, (2) the optical depth forthermal radiation reaches unity, or (3) the compressional heating rate becomes comparable with theenergy transport rate because of radiative di†usion. In the present paper this classiÐcation is extended tomore general values of the initial cloud temperature T and opacity i, and we determine the criticaldensities with which these conditions are satisÐed. Forinit
plausible values of T and i, we Ðnd that theisothermal evolution ceases when case 1 or 3 is satisÐed, and case 2 has no init
signiÐcance. We emphasizethat the condition of ““ q B 1 ÏÏ never terminates isothermality, but nonisothermal evolutions begin either
earlier or later depending on the initial temperature and opacity. This result contrasts with the conventional idea that opaqueness breaks isothermality. On the basis of the critical density discussed above, theminimum Jeans mass for fragmentation, M , is reconsidered. In contrast to the results by previous
F Ðnd that M can be substantially larger than the typicalauthors that M is insensitive to T and i, weFinit
F M increases with decreasing metallicity,value of D10~2 M depending on T and i. In particular,_init
M P i~1, for low-metal clouds. A cloud with i \ 10~4 cm2 g~1F and T \ 10 K yields M \ 3.7 M .
Finit that are intended
F to simply_
Finally,
our critical densities would be helpful for hydrodynamic simulations
handle the hardening of the equation of state.
Subject headings : hydrodynamics È ISM : clouds È radiative transfer È stars : formation
1. INTRODUCTION

familiar idea that isothermality is violated when the cloud
becomes opaque to its thermal radiation at o D 10~13 gcrit
cm~3. It is a great necessity to critically reexamine
thiscommonly believed idea, which has been often stated in theliterature (Larson 1969 ; Appenzellar & Tscharnuter 1974 ;Winkler & Newman 1980).
In the context of star formation, it is important to determine when isothermal evolution ceases. In the sphericalcollapse of a preprotostellar core, the violation of isothermality leads to the formation of a central adiabatic core (theso-called Ðrst core) whose size and mass are characterizedby o (MMI). This critical density has more direct importancecritin cylindrical collapse, because an isothermally collapsing Ðlament is expected to fragment because of thehardening of the equation of state (Inutsuka & Miyama
1992, 1997, hereafter IM92 and IM97, respectively). IM97
derived o and the mass of a fragmented core as functions
crit temperature and opacity. An important impliof the initialcation by IM97 is that the minimum Jeans mass for fragmentation depends sensitively on T and i, contrary to theinit
conclusions drawn by previous authors(Low & LyndenBell 1976 ; Rees 1976 ; Silk 1977 ; Boss 1988). We will revisit
this topic in more detail in ° 3.
Multidimensional numerical calculations are commonly
carried out under the assumption of an isothermal (orpolytropic) equation of state. This is because the radiativetransfer equation requires an unacceptable computationale†ort to solve exactly in multiple dimensions. Some numerical studies have shown that an isothermally collapsing
cloud in some cases yields a barred structure and/or fragments in the central region (Truelove et al. 1998 and references therein). These results, however, may depend on

The assumption that the early stage of protostellar collapse proceeds isothermally is widely accepted in studies ofstar formation. This isothermality is justiÐed because thethermal emission by dust grains is quite e†ective in the early
evolution, and the released gravitational energy can be
immediately radiated away. Isothermal evolution, however,ceases when thermal radiation no longer cools the cloudagainst the compressional heating. Detailed studies of thethermal processes that violate isothermality involve complicated problems including radiative transfer, so quantitative
investigations have not been done to date.
On the other hand, recent progress in computational
facilities has enabled us to handle such complicated problems rather easily. In our previous work (Masunaga,
Miyama, & Inutsuka 1998, hereafter MMI), numerical calculations for protostellar collapse were carried out with anexact treatment for radiative transfer. MMI showed that thecondition with which isothermality is violated is classiÐed
into three di†erent criteria, as is reviewed in ° 2 below. MMI
found, by both numerical results and analytical estimates,that small di†erences in the cloud temperature and opacity
cause drastic changes in the critical density, o , which isthe central density of the collapsing cloud whencrit
the isothermal evolution is terminated.
We generalize the analysis of MMI in the present paper
and reach a conclusion that the condition of ““ q B 1 ÏÏ neverterminates isothermality in possible ranges of parameters
for actual molecular clouds. This result contrasts with the
1 Also at The Department of Astronomy, The University of Tokyo.

822

DOES ““ q B 1 ÏÏ TERMINATE ISOTHERMAL EVOLUTION ?
whether the isothermal approximation remains valid or not
throughout the evolution, especially in the central region,
where the density can be very high. Furthermore, Truelove
et al. (1997, 1998) pointed out that artiÐcial fragments areobserved in hydrodynamic simulations when cell size isinsufficient to resolve the local Jeans length, which implies
that one should also consider the possibility that physicalfragmentation may not actually occur before isothermality
is broken down. Our criteria for the violation of isothermality are useful for checking validity of the isothermal
approximation. Moreover, it is possible to harden the equation of state (EOS) when the density reaches a certain critical value in numerical simulations for collapsing clouds(e.g., Burkert, Bate, & Bodenheimer 1997). Our results
would be helpful to such calculations, in which the isothermal EOS is employed where o \ oand, a harder EOS
crittakes place elsewhere.
In ° 2 we review o as deÐned by MMI and apply it tocritmore general values of the cloud temperature and opacity.
In ° 3 the minimum Jeans mass for fragmentation is reexamined on the basis of o obtained in ° 2. Our summary
crit in ° 4.
and future tasks are described
2. CRITICAL DENSITIES FOR THE VIOLATIONOF ISOTHERMALITY

In general, molecular clouds are under thermal balance
described by the following relation,! ]! \" ,
(1)
g
ext
th
where ! , ! , and " are the compressional heating rate ofextgas, the g heating
ratethdue to the external sources such ascosmic rays and visual or UV photons from surroundingstars, and the radiative cooling rate, respectively. The isothermal collapse proceeds during ! > " and thus ! Bgthext" , but isothermality is broken down
when
the compresthsion of gas becomes so e†ective as to heat up the cloud
sufficiently against the cooling.
In this section we discuss the critical central density ocritwhen isothermal evolution is terminated in gravitational
collapse, following the manner developed by IM97 andMMI. MMI found that the condition with which isothermality is broken down is classiÐed into three di†erent criteria, as follows.
1. Isothermality is violated when ! \ " if ! overg collapsingth
g cloudwhelms " before the optical depth of thethcore, q, reaches unity. This critical density is denoted by o .
th
2. For q [ 1, the cooling rate is given by the energy
transport rate due to radiative di†usion, " , instead of " .
dif q B 1, the isoth
If " [ ! during q \ 1 and " \ ! when
th
g
dif
g
thermal evolution ceases when q arrives at unity. The
central density when q B 1 is denoted by o .
qF1 survives even
3. If " [ ! when q B 1, isothermality
difgafter q exceeds unity until the central density reaches a critical value o , which deÐnes the central density when !
difgbecomes comparablewith " .
dif
Now we evaluate ! , " , and " to derive o . Noteg th rates aredifdeÐned per unit
crit mass.
that the heating and cooling
For brevity we suppose the local thermodynamical equilibrium (LTE), which admits " to be described simply asthfollows," \ 4i(T )pT 4 ,
th
init
init

(2)

823

where i is the frequency-averaged opacity2 per unit mass,
which is independent of density because of LTE, and p
denotes the Stefan-Boltzmann constant. The temperature is
kept at T at the initial isothermal stage. The compresinit rate for gravitational collapse is (see ° 5.2 insional heating
MMI)! \ Ac2 J4nGo ,(3)gswhere c is isothermal sonic speed and G is the gravitationalsconstant. A numerical constant A is found to be of orderunity and is nearly constant through the evolution (seeIM97 and MMI).
The optical depth, q, of the collapsing cloud is deÐned byP=
io dr .
(4)
0
In order to evaluate the optical depth, one must suppose adensity structure of the collapsing cloud. For a spherical
cloud we employ the isothermal similarity solution
obtained by Larson (1969) and Penston (1969). The density
distribution in their solution is approximated byo \ constant for r \ R and o P r~2 for r [ R , where theboundary radius, R , is bapproximately the Jeansb length corb
responding to the current
central density, o . Hence R iscb
expressed as
q\

2nc
s ,
R \C
(5)
b
J4nGo
c
where a dimensionless constant C is 0.75, in comparison
with numerical results (MMI). Equation (4) is then reduced
to

P

Rb

P A B

=
r ~2
io
dr \ 2io R .
c R
c b
0
Rb
b
Combining equations (5) and (6),qD

io dr ]
c

(6)

4nc
s .
q \ Cio(7)c J4nGoc
For a Ðlamentary cloud we use equation (8) following theformulation by IM97, where the isothermal equilibriumconÐguration is adopted for the density distribution

S

2c2
(8)
s
nGo
c
(eq. [28] in IM97). This approximation is justiÐed because
an isothermal Ðlament in equilibrium can collapse while
preserving the initial conÐguration, i.e., undergo homologous collapse. Moreover, a Ðlament formed by the fragmentation of sheetlike clouds has a line mass only twice as large
as the equilibrium line mass (Miyama, Narita, & Hayashi
1987a, 1987b), and therefore the equilibrium structure is
q\

n
io
4 c

2 In eq. (2) the frequency-averaged opacity i should be replaced by the
Planck mean opacity. In contrast, the Rosseland mean opacity may be
preferred in eqs. (10) and (11) below. We, however, do not discriminate
between them for brevity in this paper because the di†erence in values ofthese two opacities is small.

824

MASUNAGA & INUTSUKA

applicable also to collapsing clouds. One can see that equations (7) and (8) are in accord with each other except fornumerical factors.
The opacity per unit mass is assumed to be constantthroughout the collapsing cloud. This assumption is welljustiÐed when considering the dust continuum opacity,which is independent of the velocity structure of clouds, incontrast to the atomic or molecular line opacity.
We deÐne the r-parameter, which is the ratio of the radiation energy density to the material internal energy, foro \ o as follows :crit
E 4(c [ 1)pT 4
init .
r 4 \(9)crit oeco c2
crit s
The energy transport rate due to radiative di†usion is(MMI)
E
4i(T )pT 4
init
init ,"
D\(10)
dif, MD otq2
dif
when the internal energy of material dominates the radiation energy density (i.e., r \ 1), andcritec2i(T )oc
init
"
D
\ s
,
(11)
dif, RD t(c [ 1)q2
dif
when the radiation energy density dominates (i.e., r [ 1),critor collectively\ min ("
,"
),
(12)
dif
dif, MD dif, RD
where e and E are the internal energy density of Ñuid perunit mass and the radiation energy density per unit volume,
respectively, and c denotes ratio of the speciÐc heats (we set
c \ 5/3 in this paper). The radiative di†usion time t isdeÐned by q2j /c, where j 4 1/io is the mean free pathdifof ap light.
photon and c isp the speed of
Combining equations (2)È(10), we derive the critical densities for the violation of isothermality. For spherical symmetry,
"

A

BA B

i
2 T
6`2a
0
init
,
o \ 4.7 ] 10~15 g cm~3th
0.01 cm2 g~1
10 K

A

B

T
~1~2a
init
,
]
10 K
and

A

B
A B

T
(4~2a)@3
init
,
10 K

A B

T
a
init
.
)\i
init
0 10 K

]

B
A B

T
~2~2a
init
.
10 K

(17)

Note that spherical collapse is decelerated when theequation of state is harder than c \ 4/3, where c is theadiabatic component, i.e., d ln P/d ln o. Hence the violationof isothermality does not necessarily cause an immediateinÑuence on the evolution (see MMI). In contrast, cylindrical collapse should be decelerated even by a slight hardening of the isothermal equation of state. Furthermore,
IM92 and IM97 have shown that a collapsing Ðlament is
expected to fragment if the radial collapse is decelerated.
Therefore o for cylindrical collapse, which is consideredcrit signiÐcance than the spherical case.
below, has more
The critical densities for cylindrical (Ðlamentary) collapse
were obtained by IM97 as

A

BA B

i
2 T
6`2a
0
init
,
o \ 4.7 ] 10~15 g cm~3th
0.01 cm2 g~1
10 K
(18)

A

B

i
~2
0
o
\ 4.7 ] 10~12 g cm~3
qF1
0.01 cm2 g~1
]

A

(14)

A B

T
~1~2a
init
.
10 K

(19)

B
A B
B A B

i
~2@3
0
o
\ 4.7 ] 10~13 g cm~3
dif, MD
0.01 cm2 g~1]AT(4~2a)@3
init
,
10 K

(20)

i
~2 T
~2~2a
0
init
.
o
\ 5.4 g cm~3
dif, RD
0.01 cm2 g~1
10 K
(15)

which correspond to equations (14), (17), and (20) in MMI.
Here the opacity is approximated as
i(T

A

i
~2
0
o
\ 1.7 ] 10~2 g cm~3
dif, RD
0.01 cm2 g~1

Another critical density, o , which was not considered in
difIM97, is estimated as

i
~2@3
0
o
\ 6.9 ] 10~14 g cm~3
dif, MD
0.01 cm2 g~1]

law index in frequency (i.e., i P lb) for the dust opacityl(e.g., Beckwith et al. 1990) and is typically D1È2 for interstellar dust grains. The normalizing factor, i , reÑects
0
metallicity and is D0.01 cm2 g~1 for nearby molecular
clouds.
MMI found that these three critical densities are useful in
predicting when the isothermal evolution is terminated,comparing them with radiation hydrodynamic numericalcalculations with an exact treatment for radiative transfer in
spherical symmetry.
In case of r [ 1, equation (15) above is replaced by thecritfollowing equation,
which is derived using equation (11) :

(13)

i
~2
0
o
\ 2.6 ] 10~13 g cm~3
qF1
0.01 cm2 g~1

A B

Vol. 510(16)

Note that in the typical range of temperature for molecular
clouds, a in equation (16) coincides b, which is the power-(21)
The di†erence in obetween spherical and cylindricalcases appears only incritnumerical factors. In the above equations we assumed that the gas constant k/km 4 c2/T isHs
3.6 ] 107 ergs g~1 K.
Equations (13)È(15), (17), and (18)È(21) show that the critical densities are determined solely by T and i. Thus thecriteria for the violation of isothermalityinit
can be classiÐed by
T and i. Figure 1 depicts the critical density (shown byinit

No. 2, 1999

DOES ““ q B 1 ÏÏ TERMINATE ISOTHERMAL EVOLUTION ?

1

-14
0

-12
-1

-14

-2

-10

-18

-3

-22

-4

-5
0

.5

1

1.5

2

2.5

3

FIG. 1.ÈCritical density for the violation of isothermality (eqs. [18]È
[21]) is shown as contour curves. Note that the vertical axis represents not
i but i(T ) (see eq. [16]). Contour levels begin with log o (g0init
10 critcm~3) \ [24 in the lower left corner and increase as [22, [20, [18, . . . .
The dashed line delineates o \ oand, equivalently, o \ o
(see
th
qF1
dif
qF1
text for detail). The critical density is given by o below the dashed line andthby o above. Shaded domains indicate r [ 1, where r is the ratio ofdifcrit internal energy
crit density andthe radiationenergy density to the materialr \ 1 elsewhere. The star and triangle correspond to the model paramcrit adopted by Larson (1969) and Winkler & Newman (1980), respeceters
tively.

contour lines) as a function of T and i. We have chosen
the range of T to be less than init
1000 K, below which dust
grains survive.initAlthough this range is still wider than thepossible temperature range for actual molecular clouds, it is
helpful to realize the behavior of o more clearly on the
T -i plane. The vertical axis, i, is acritproduct of i and theinit
0 Undertemperaturedependence according to equation (16).
a given value of T , the vertical axis is an indicator of iinit0and hence of metallicity.
For r \ 1, boundaries between cases 1 and 3 (o \crito \ o ) are found to fall onto an equivalent
th
o
and
qF1
dif
qF1
condition, which is delineated by a dashed line in Figure 1.
(The reason is described in the next paragraph.) The criticaldensity is given by o below the dashed line and by o \th for o \ o is divided intocrittwo
o above. The region
difdomains : o \ oand crito \ difo. The condition
crit
dif,
RD
crit toodif, MDof o \ o, however, requires
large a T and icritdif, RDthe typical values for molecular clouds,
init andcomparedwiththerefore the domain for o \ odoes not appear in
crit in thedif,lower
RD left corner, where
Figure 1. The shaded domain
the isothermality is broken before q reaches unity, indicatesaT 4 [ oe (a 4 4p/c), which, however, does not necessarilymean that the radiation energy dominates the Ñuid internalenergy. Unless the external radiation Ðeld is intensive
enough, E is much less than aT 4 in optically thin media.
The model parameters adopted by Larson (1969) and
Winkler & Newman (1980) are indicated by the star andtriangle, respectively, in Figure 1. Both points read o D
crit
10~13 g cm~3, which is in accordance with the central
density when the isothermal evolution terminates in theirnumerical calculations. These points accidentally locate
near the dashed line, on which the isothermal evolutionceases when q \ 1. This seems to be a major reason why
many authors have been misled by the idea that opaquenessterminates isothermality.
Equations (2) and (10) show that " and "areth q \ 1. dif,smoothly connected with each other at
InMDotherwords, o \ oand o\odegenerate into anth
qF1
dif, MD
qF1

825

identical line in the T -i plane (Fig. 1, dashed line). Figureinit
2a schematically shows the thermal evolution of a gravitationally collapsing cloud. If r \ 1, the isothermal evolucrittion ceases when ! reaches " (which corresponds to casegth
1 above), or when ! reaches " (case 3). Although neithergdif" nor "
provides exact estimates for intermediatethdif
optical depth, it is clear that the critical density of o
(case
qF1
2) plays no essential roles when r \ 1.
crit
In contrast to ","is not ensured to be condif,
MD
dif,
RD
nected smoothly with " near q B 1, as is found by comthparing equations (2) and (11). Therefore, o \ o
and
th
qF1
o
\o
should be indicated by two separatelines indif, RD
qF1
the T -i plane, and the domain for o \ o
would
init
crit
qF1
appear between these two lines. Figure 2b draws thethermal evolution for r [ 1, where case 2 appears as wellcrit
as case 1 and 3. However, o\ocorresponds todif, RDqF1
T \ 1.1 ] 1013 K ( !) independentlyof i, so case 2 isinithighly unlikely to occur.
As a conclusion, the critical density for the violation ofisothermality is determined by o or o
, depending onthdif,clouds.
MD
T and i, for gravitationally collapsing
T he criticalinit
densityof ohas no signiÐcance in practice. This fact has
qF1 in all previous works on gravitational colbeen overlooked
lapse of protostellar clouds, including our own papers(IM97 and MMI).
3. THE MINIMUM JEANS MASS FOR FRAGMENTATIONThis section is devoted to discussions on the minimum
Jeans mass of a fragment of clouds. Previous formulationsof the minimum Jeans mass are reconsidered, and we showthat they should be modiÐed on the basis of our newly
derived criteria for the violation of isothermality.

FIG. 2.ÈSchematic picture illustrating the classiÐcation for oisshown. The heating and cooling rates (! , " , and " ) are depictedcritas agthdiffunction of time. The optical depth for thermal radiation, q, is small enoughinitially but increases monotonically as the collapse proceeds. The upper
panel is for the case of r \ 1 and the lower panel for r [ 1. See text for
crit
crit
more detail.

826

MASUNAGA & INUTSUKA

3.1. Comments on the Previous Studies
The minimum Jeans mass for fragmentation, M , isF
determined by the condition that the equation of state
becomes sufficiently hard to stop further fragmentation. In
order to obtain M , Low & Lynden-Bell (1976), Silk (1977),
F
and Kanjilal & Basu (1992) imposed two conditions of! \ " (eq. [3]) and q \ 1 simultaneously. As a result, thegthtemperature of clouds that deÐnes M is determinedF
uniquely, along with M itself. However, as IM97 noted,
F
T and i must be left as free parameters, because the cloud
init
temperature is actually given by environmental factors,
including the intensity of interstellar radiation and the rateof cosmic-ray heating. Hence we estimate the minimum
Jeans mass as a function of both T and i on the basis ofinit
the discussion on oin ° 2 above. In this manner we
critpropose a more generalized formulation for M in ° 3.2.
F
MMI shows that the temperature increases only gradually in case 1 after ! reaches " , until the temperaturegtheventually begins to rise rapidly when q exceeds unity.
Therefore one might wonder whether the condition of ! \g" can actually trigger the fragmentation of sphericalth
clouds.
Tohline (1980) raised serious objections to the previous
models for fragmentation of spherically collapsing clouds.
His results indicate that fragmentation occurs after a cloudis collapsed into a highly Ñattened sheet. A collapsed sheetlike cloud is subject to fragmentation into Ðlaments(Miyama et al. 1987a, 1987b), and therefore the problem ofcloud fragmentation becomes a problem of the evolution ofÐlaments. As was mentioned in the previous section, a collapsing Ðlament is expected to fragment by a slight hardening of the isothermal equation of state, in contrast to
spherical collapse, so even a gradual increase of temperature can trigger fragmentation (IM92, IM97).
Rees (1976) considered the condition for determining MF
asGM2
F D 4nfR2pT 4 ,
(22)
init
Rt
ff
where t is the free-fall time and f is a nondimensional
ff than unity that depends on the detailed physics
factor lessof the cooling and opacity. In contrast to the other authors
mentioned above, Rees (1976) considered the thermalbalance in a collapsing cloud after q exceeds unity, and thus
his criterion (eq. [22]) corresponds to our case 3. This criterion, however, assumes that the released gravitationalenergy is instantaneously radiated away from the cloud
surface. In other words, equation (22) neglects the timerequired to convey the released energy to the cloud surface
by radiative di†usion. However, the energy transport rate
due to radiative di†usion is essentially important in case 3,
as was described in the previous section. Therefore, theright-hand side of equation (22) should be modiÐed intoP

Vol. 510

solution for density structure), and therefore the original
formulation by Rees (eq. [22]) requires great care to use.
3.2. A Revised DeÐnition of the Minimum Jeans Mass
Following IM97, we evaluate M as the mean clump
F
mass of a Ðlament with the e†ective radius of H \fJ2c 2/nGo , setting the mean separation between clumpss
crit
to be 8 ] H :
f
16c3
2
(24)
s
M ^ o nH2(8H ) \
.
F
crit
f
f
G
nGo
crit
Eliminating o in equation (24) by equations (18) andcrit(20), we have

S

M

F, th

\ 3.7 ] 10~2 M

A

B

i
~1
0
_ 0.01 cm2 g~1]

for case 1,
MF, dif, MD\ 3.7 ] 10~3 M

A B

T~(3`2a)@2
init
10 K

A

B

i
1@3
0
_ 0.01 cm2 g~1]

for case 3 with rMF, dif, RD

crit\ 1, and\ 1.1 ] 10~9 M

(25)

A B

T
(5`2a)@6
init
10 K

A

i
0
_ 0.01 cm2 g~1

(26)

BA B

T
(5`2a)@2
init
10 K
(27)

for case 3 with r [ 1.
crit
Figure 3 illustrates
M by contour lines in the T -i
init
plane. The dashed line, Ftracing the bottom of a ravine
drawn by contour, represents M\M
andth ThenF,M
qF1equalsM\M
similarly to FigureF,1.
MD
F, qF1
F above,MF, dif,below
thedashed line and M \ M
F,
th
F
F,
dif,
MD
according to Figure 1. Recalling that Low & Lynden-Bell(1976) and Silk (1977) considered o \ o and o \ oth for M
crit is conqF1
simultaneously, one Ðnds that theircritsolution
Fstrained on the dashed line in Figure 3. Using equations (25)
1

0

-1

-2

-1.5

-1

-2
-2
-1
-3

0
1

-4

"

dif

o dV D

4ipT 44nR2pT 4
init onR3 D
init ,
q2
q(23)

where we used equation (10) and replaced ioR with q.
Equation (23) claims that the right-hand side of equation(22) overestimates the cooling rate by a factor of q. It may bepossible to substitute 1/q for f in equation (22), but in thiscase f cannot be a constant factor, since q varies with o (ort ) and R. Furthermore, q is a function also of both opacity
ff temperature (because T characterizes the similarity
and
init

-5
0

.5

1

1.5

2

2.5

3

FIG. 3.ÈMinimum Jeans mass for fragmentation (eqs. [25]È[27]) is
shown as contour curves. Note that the vertical axis represents not i but
0 thei(T ) (see eq. [16]). Contour levels begin with log M /M \ [2 ininit left corner and increase as [1.5, [1, [0.5,
10 . . .F. The_ dashed line
upper
delineates M\Mand, equivalently, M\MaccordingthF, qF1qF1dashed line
to Fig. 1. TheF,minimum
Jeans mass is given by MF, dif belowF,theF, thand by M
above.
F, difNo. 2, 1999

DOES ““ q B 1 ÏÏ TERMINATE ISOTHERMAL EVOLUTION ?

1

0

-1

-2

-5

-4

-3

-2

-1

0

1

FIG. 4.ÈMinimum Jeans masses for fragmentation under given temperatures. Solid, dotted, and dashed curves correspond to T \ 10, 20,
init
and 30 K, respectively.

and (26), one obtains M deÐned on the dashed line byFi~1@(7`4a)0
M \ 8.4 ] 10~3 M, (28)
F_ 0.01 cm2 g~1AB

which indeed recovers M as deÐned by Low & LyndenBell (1976) and Silk (1977).F Low & Lynden-Bell emphasized
the weak dependence of M on i , as is seen in equation
F
0
(28).
On the other hand, if assuming that T does not vary
init clouds, equasigniÐcantly, as is the case in actual moleculartion (25) shows that the dependence on i is no longer
0
weak. For the typical temperature of molecular
clouds
T D 10È20 K and for the typical dust opacity i ^ 0.01
init g~1, Figure 3 shows M \ M . Hence we choose
cm2
F, th M . Now we conequation (25), which reads MF ^ 10~2Fsider how M varies with the decreasing _dust opacity. As
F in ° 2, lowering i under a Ðxed T correwas mentioned
init (25)sponds to lowering metallicity of the clouds. Equation
tells us that M can be signiÐcantly larger than D10~2 M ,F proportional to i for low-metal clouds.
_
which is inversely
0
Figure 4 delineates M as a function of i under Ðxed initial
F 20, and 30 K. For instance, a cloud
temperatures : T \ 10,
initwith T \ 10 K and i \ 10~4 cm2 g~1 (one-hundredth assmall init
as the typical metallicity) provides M \ 3.7 M .
This fact might be responsible for the cuto†, ifF it exists, _atthe low-mass end of the initial mass function.
4. SUMMARY

Our Ðndings in the present paper are summarized asfollows.

827

1. We derived three criteria for the central density ocritwhen isothermality is violated in gravitationally collapsing
clouds. These criteria are supported by numerical results in
our previous work. We classiÐed o in the T -i plane and
crit
init
found that o or odetermines o in plausible ranges
thdif, MD
critof parameters for actual molecular clouds. Another criterion, o \ o , actually has no importance.
crit
qF1
2. This result indicates that the condition of ““ q B 1 ÏÏnever violates isothermality. Instead, nonisothermal evolutions begin either earlier or later, depending on the initial
temperature and opacity, than the optical depth reaches
unity. The conventional idea that the isothermal approximation is valid during the period when the central density
is less than 10~13 g cm~3 is not supported physically.
3. The minimum Jeans mass for fragmentation, M , isFcharacterized by o , hence M is a function of both T
crit
F
init
and i as well as o . The typical value of M is D10~2 M
critF_in accordance with the commonly believed value, but MF
can increase substantially depending on T and i. In parinit
ticular, molecular clouds with lower metallicity yield alarger M inversely proportionally to metallicity. For
T \ 10 FK, a cloud with i \ 10~4 cm2 g~1, which is
init
smaller
by a hundred than the typical value for nearby
molecular clouds, yields M \ 3.7 M .
F_The critical densities derived here would be helpful tonumerical calculations in which either isothermal or aharder EOS (e.g., c \ 5/3) is chosen corresponding towhether o \ o or o [ o .
crit investigation
crit we suppose that the opacity
In the present
per unit mass is independent of the density (i.e., assumingLTE) and is homogeneous throughout the cloud. This
assumption is justiÐed when the cloud density is so highthat interstellar dust is the dominant coolant, but it is inapplicable for lower densities where the cooling by molecularlines mainly contributes to the opacity. For the molecularline cooling, LTE is not satisÐed in general and the assumption that the opacity per unit mass is homogeneous is violated because the line feature depends on the velocity
structure of clouds. Our future works will be intended togeneralize the present results to be applicable to the molecular or atomic line opacity.
The authors are grateful to the anonymous referee andthe scientiÐc editor, S. Shore, for their helpful criticism,
which has improved the manuscript. H. M. thanks Y. Itohfor helpful comments. The research of H. M. is supported by
the Research Fellowships of the Japan Society for the
Promotion of Science for Young Scientists.

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