781
Progress of Theoretical Physics, Vol. 42, No. 4, October 1969
Thermal and Dynamical Evolution of Gas Clouds
of Various Masses
Tsuguo HATTORI, Takenori NAKANO* and Chushiro HAYASHI
Department of Physics, Kyoto University, Kyoto
*Research Institute for Fundamental Physics, Kyoto University, Kyoto
(Received June 14, 1969)
The evolution of gas clouds of masses, 104, 102 , 1 and 10- 2 M~_;, with the population I
composition is investigated both for transparent and opaque stages by comparing the rates
of cooling, heating, contraction and expansion. As heating and cooling processes in the transparent stage, we consider the thermal radiation from grains, the line-emission by H 2 , C+, Si+,
C and 0 including the effect of self-absorption, the ionization of atoms by interstellar starlight and cosmic-ray particles, and the absorption of star-light by grains. In the opaque stage
cooling is due to the diffusion of thermal photons.
The results show that in the transparent stage a cloud much greater than 10- 2 Me contracts freely and cools down to about lOoK if its mean density is greater than a certain critical value which depends on the mass of the cloud. In the opaque stage the cloud undergoes
nearly adiabatic free-fall. On the other hand, if a cloud of mass nearly equal to or smaller
than 1 X 10- 2 Me is formed with a mean density greater than a certain critical value, it establishes
immediately gravitational equilibrium and the subsequent evolution is the Kelvin contraction.
§ 1.
Introduction
Recently, some OH-line emission clouds have been discovered near the compact HII regions in the galactic radio sources W 49 and IC 1795. 1),:!) Their small
diameters suggest that they might be of relatively high densities and thus they
may be interpreted as protostars in an early stage of dynamical contraction, 3 l
whose evolution has been investigated by Hayashi and Nakano 4l (to be referred
to as HN).
According to HN, a gas cloud of one solar mass contracts freely to become
a protostar if its mean density is greater than about 2 X 10- 18g/ c:m 3 • The protostar begins to be opaque to thermal radiation when the mean density and the
mean temperature become about 4 X 10- 14g/ cm 3 and 15°K, respectively. This
initial stage of the opaque protostar is determined mainly by the cooling effect
of grains, almost irrespective of the previous history of evolution. Because at
this stage the pressure is too low to maintain gravitational equilibrium, the protostar continues free-fall contraction until finally the protostar flares up and the
quasi-hydrostatic equilibrium is established throughout the star.
The evolution of protostars of the other masses, 10 2Me and 10- 2Me, has
been investigated by Hayashi, 5l but the study is limited only to the stage when
they are opaque to interstellar star-light. The complete investigation which covers
782
T.
Hattori~
T. Nakano and C. Hayashi
all the evolutionary stages has ·been postponed until now.
After the work of HN some cooling processes have been found to be important by several authors. Dalgarno and Rudge 6l showed that the H atom
collision is as efficient as the electron collision in the excitation of c+ and Si+
Ions. Callaway and Dugan 7l calculated the excitation cross section of C and 0
atoms by collisions with I-I atoms and found that these processes followed by
photon emission are efficient cooling processes in some regions of density and
temperature.
In this paper we shall re-examine the evolution of the gas clouds of various
masses by taking into account the cooling processes mentioned above in addition
to those considered in HN. As the mass of the cloud we shall choose 10 21110 ,
M 0 and 10- 21110 in order to cover the range of the stellar mass. Furthermore,
in connection with the formation of the star clusters, we shall also investigate
the evolution of a cloud of 10 41\116 . As in I-IN, the effects of the surrounding
medium on the clouds will be neglected except for the heating by interstellar
star-light and cosmic-ray particles.
In § 2, the processes of cooling and heating will be summarized. As the
cooling processes we shall take into account the line-emission by H 2 molecules,
c+ and Si+ ions and C and 0 atoms, as well as the thermal radiation from
grams. Furthermore, the self-absorption of line-emission photons will be taken
into account. As the heating sources we shall consider the interstellar star-light
and cosmic-ray particles.
In § 3, the evolution of the clouds which are transparent to thermal radiation will be investigated by comparing the rate of cooling or heating with the
rate of free contraction or free expansion.
In § 4, the evolution in the early opaque stage will be investigated by comparing the rate of cooling due to the diffusion of photons with the rate of freefall contraction.
In § 5, we shall evaluate the uncertainty in the evolution which arises from
the uncertainties in the abundance of H 2 molecules and in the intensity of cosmicray particles. Finally, discussions will be made of the difference between our
treatment and those of the other people.
§ 2.
Processes of cooling and heating
In this section we shall summarize the processes of cooling and heating in
the clouds of population I composition. We shall abopt the relative abundances 8l
given by
(nu+2nu
2
):
n(C): n(O): n(Si) =10 6 : 400:890:32.
(1)
At present we have no accurate knowledge of the abundance and para-to-ortho
ratio of H 2 molecules in the interstellarclouds. Stoeber and Williams 9l showed
that the photodissociation of H 2 is very eiiicient and its number density is about
Therrnal and Dynamical Evolution of Gas Clouds
783
10- 7cm- 3 m the usual interstellar clouds, while Heiles 10 ) found several dense
clouds in which most of the hydrogen is probably H 2 • We shall assume in the
following that all of the H 2 molecules are of para-type and nu 2 == O.lnu in the
stage transparent to thermal radiation. In view of the uncertainty mentioned
above, we shall also study the two other cases, nu 2 = nu and nu 2 == 0, in § 5.
The cooling processes to be considered are thermal radiation from interstellar
grains and line emission by H 2 molecules, c+ and Si+ ions, and C and 0 atoms.
The heating processes are the absorption of star-light by grains and by C and Si
atoms, which are formed by the recombination of ions with electrons, and the
ionization of hydrogen atoms by cosmic-ray particles. Furthermore, since we
are considering a wide range of the density, the effect of self-absorption of lineemission photons will be taken into account.
1) General treatment of cooling by line emission
We consider an element (atom, molecule or ion) which has a low-lying excited state. The element excited by collision with other particles is de-excited
partly by collision and partly by emission of a photon. Some of the emitted
photons are absorbed by the element in the ground state, some induce the photon
emission from the clement in the excited state and others leave the cloud.
In terms of the radiative transition rate, A, of the element from the excited
state to the ground state, the absorption cross section for the line-emission photon
is expressed as 11 l
(2)
where }, is the wavelength, LIA. is the line width, and g1 and g 2 are the statistical
weights of the ground and excited states, respectively. As LIA. -vve take the Doppler width
LIA. =
A.
(2n!?-T) 1; 2,
mc 2 I
(3)
where m is the mass of the element. The cross section for induced emissiOn
is then given by (g 1 jg 2) (J. When the colliding particles which excite and deexcite the element have a Maxwellian velocity distribution of temperature T,
the collisional excitation rate r and the de-excitation rate r' have a relation
(4)
where E is the energy difference between the two states.
In the steady state condition, the number density of the photons N and the
number densities n 1 and n 2 of the element in the ground and excited states, respectively, are determined by
(5)
784
T. Hattori, T. Nakano and C. Hayashi
(6)
(7)
where td is the mean escape time of the photons from the cloud and n is the
total number density of the element which is considered to be a constant. When
the radius of the cloud R is smaller than the mean free path of the photons,
the mean escape time td is given by
(8)
When R is larger than the mean free path, td is equal to the diffusion time of
the photons through the cloud, i.e.
(9)
N.
By eliminating n 2 from Eqs. (5) and (6), we have a quadratic equation for
The solution is expressed in good approximation as
where
(11)
IS the mean free time of the photons.
per unit time is given by
Then, the cooling rate per unit volume
A=EN/td.
(12)
When the condition, A+ r'> (r'- rg 1 jg 2 ) td/t, Is fulfilled, the cooling rate is
given simply by
A
A=--= En 1 r-----,
A+r'
(13)
which is a usual expression without self-absorption. One of the cases in which
the above condition is fulfilled is td<t, i.e. the cloud is transparent to the photons.
The other case is A/r'>tdjt>1, which can also be expressed as
1-
(~--~---) td/t <1 .
A+r'
(14)
This inequality means that, if the photon is absorbed, It 1s emitted again with
the probability A/ (A+ r') which is nearly equal to unity, and finally most of
them leave the cloud.
In the other extreme case, A+ r' (r'- rg 1 jg 2) td/t, we have from Eqs. (2),
(4), (10) and (11)
<
N=
8 n(ei>;~cr_l)-lil.-4LIJ,,
which Is equal to the number density in the Planck distribution.
(15)
785
Thermal and Dynamical Evolution of Gas Clouds
2)
Hydrogen molecules
The hydrogen molecule excited to the J-th rotational level emits a photon
of energy
EJ,J-z
with a probability
=
(16)
0.0147 (2J -1) e V
o£1~>
A
J,J
_2 = 7.52 x 10- 13 J(J -l) ( 2 J - I t sec- 1 •
2J + 1
(17)
From Eqs. (2) and (3) the absorption cross section of an H 2 molecule for this
photon is given by
nCa>
-1 11 X 10-24 J(J -1) (2J -1)
•
2J-3
T-1/2
cmz,
(18)
VJ,J-2-
where T is in °K. For the 28,a photons which are emitted by molecules of
J = 2, the protostar of mass 111 becomes opaque at the density
(19)
as long as T is much lower than E 2 , 0 / k = 512°K. For photons from the levels
of larger J the critical density is higher because the populations of these levels
are smaller. Since at high densities as given by Eq. (19) the cooling by grains
is much more efficient than H 2 molecules as will be shown below, we are allowed
to neglect the effect of self-absorption.
Under steady state condition the number density nJ of H 2 molecules in the
J-th state is determined by
vo J-z,J)u + 1lu vo J-z,J)n + nJ+2 {nu( vo J+z,J)H + 1Zu vo J+2,J)H + AJ+2,J}
nJ {nn ( ( VO J,J+2)H + ( vo J,J-2)n) + 1lu
VO J,J+2)H + ( vo J,J-z)HJ + AJ,J-2}
nJ-2 {nn(
=
2(
2}
2(
2 ( (
2
2
(20)
and
(21)
where v is the relative velocity of the colliding particles, oJ,P Is the collision
cross section for the transition from J to J', and the bracket () denotes the
mean value. Then, the cooling rate is given by
(22)
At temperatures below 150°K most of H 2 molecules are m the ground state
J = 0, and the principal excitation is to the J = 2 state. In this case Eq. (22)
is written simply as
(23)
T. Flattori, T. Nakano and C.
786
~Iayashi
where
(24)
In this paper the values of Zu have been computed by using the values of
<vGo,2)H and <v0" 0 , 2)u calculated by Allison and Dalgarno. 13 )
At temperatures above 150°K the contribution of higher rotational levels
must be taken into account. The cooling rate at low densities, nu:S10 2cm- 3, has
been calculated by Takayanagi and Nishimura. 14 ) At higher densities where the
collisional de-excitations are much faster than the radiative transitions, the population of each level is approximately given by the Bolt%mann distribution. In this
case the cooling rate is simply given by
2
2
(25)
where
G (T) = ~7=_2_[t J~!- ~){(2_{=:-_I)/~l exp_{=_0!_~~c_!__-=_§)§_'}_,o/6kT}] .
~Y=o [ (2J + 1) exp {- J(J + 1) E2,o/6l~T}]
5
(Z 6)
The values of G (T) are g1ven m Table I.
Table I. The values of G(T).
200
300
1
400
500
1000
20
160
---'--~-4.7------~------~i------
G(T)
3)
1.5
Carbon and silicon zons
i) Ionization degree. Some of the carbon and silicon atoms are ionized by
the interstellar star-light. The ionization degree is determined by the balance
between photo-ionization and recombination, i.e.
(27)
where (} is the photo-ionization cross section of the atom, a is the recombination
coefficient, and nr, na, and ni are the number densities of the ionizing photon,
neutral atom and ion, respectively. The number density of electron is taken to
be equal to the ion density, i.e. ne=n(C+) +n(Si+).
For carbon which is much more abundant than silicon we have 15 )
rJ = 1.1 X 10- 17 cm 2,
a= 1.7 X 10- 17 ( Ta/10 4)-o.s cm 3 sec-\
(28)
where Tc is the electron temperature. For Tc = 30°K the ionization of carbon
is complete at p~2 X 10- 20 g/ cm 3 •
ii) Cooling by the ions. The c+ and Si+ ions have low-lying excited
states. Some of the ions excited by electron or H atom impact return to the
Thermal and Dynamical E·volution of Gas Clouds
787
ground state by emitting a photon. Following Seaton11 )' 16 ) and Dalgarno and
Rudge 6) we shall take into account the 2j}P112 - 2P 312 transition for c+ and the 3p
2
P 112 - 2P 312 transition for Sj+.
The c+ ion in the excited state emits a photon of energy E (C+) = 0.0079 e V
with the probability .fl (C_ 1_) = 2.36 X 10- 6 sec- 1• 11 ) The self-absorption cross section
is then given by
The de-excitation rate by I-I atom and electron impact 1s g1vcn bl),u)
(30)
The cooling rate by c+ ions is then given by Eq. (12) with the help of Eqs.
( 4), (8), (9), (10) and (11).
As is seen in 1) of this section, the effect of self-absorption becomes efficient
when the condition .fl+r'~ (r'-rgl/gz)ta/t is fulfilled. For c+ ions this critical
density is given by
p
=2 X Jo-" l(i~:Z.68T,"'/~ (1 :::~
&(M/iT)l "I' (1t) -'I'g/ em'.
(31)
For Si+ ions in the excited state which have a excitation energy E(Si+)=0.036
e V and transition probability A (Si+) = 2.13 X lo-- 4 sec-\ 11 ) we have the self-absorption cross section
(32)
The collisional de-excitation rate is given byG),n)
rI
cs· +) =
1
(4-. 57. X 10- 10nu I 1 . 30 X Io--"""1e- 1/ 2nc ) sec - 1 .
I
(33)
The effect of self-absorption becomes efficient at
iii) Heating by interstellar star-light. The photo-ionization of neutral atoms
gives rise to energy gain to the gas. This heating rate has been calculated by
Schatzman. 17 ) In this paper we shall use the empirical fornmla given by Takayanagi and Nishimura 14 )
(35)
where the factor (1 + 5 X 10 19 p)- 1 has been introduced to include the effect of
partial ionization.
iv) Electron temjJerature. The electron temperature Te Is determined by
the balance of the energy gain and loss of electrons, 14 ) i.e.
(36)
788
T. Hattori, T. Nakano and C. Hayashi
where n 1 and n 2 are the number densities of c+ or Si + in the ground and excited states, and re and re' are the excitation and de-excitation rates by electron
impact, respectively, and Tc 1t is the heating rate by cosmic-ray particles described
below. The third term on the right-hand side of Eq. (36) represents the energy
exchange rate between H atoms and electrons through elastic scattering. The
coefficient r which has been calculated by Spitzer and Savedoff 18 ) is numerically
given by 14l
(37)
The difference between Te and T is negligibly small m all the regions of
p and T considered in this paper.
4) Carbon and oxygen atoms
The collisional excitation of carbon atoms from the ground state 3P 0 to the
3
P 1 or 3P 2 state by H atoms may be an efficient cooling process. 7 l Carbon atoms
can exist only at high densities where the recombination of c+ ions with electrons is very efficient, i.e. p>10- 19 g/ emS, or the cloud is sufficiently opaque to
the star-light which ionizes carbon atoms, i.e.
(38)
where we have used 10 cm 2/g as the opacity due to grains.
The carbon atom in the 3P 2 state emits a photon of energy E 2 (C) = 0.0052 e V
with a probability A 2 (C) = 2.8 X 10-- 7 sec- 1 • 12 ) The self-absorption cross section for
this photon is given by
(39)
The collisional excitation cross section has been calculated by Calla way and
Dugan.7) From their result we have the collisional de-excitation rate
rz ' (C) =2.1
X
10 -11 n 11 rT 3jl0
l
w
E 2 (C)} s;1o fx+----..Jx e
1
o t
The integral has been numerically evaluated.
for this result we have
l~T
-X
dx sec -1 .
Using the approximate expression
(40)
The error in this approximation does not exceed 2.5 per cent.
As is easily seen, the self-absorption is very efficient at all densities at
which carbon atoms can exist and, consequently, the cooling by C atoms is negligibly small compared to that by grains described below. The contribution of
the 3P 1 state is also negligible.
The oxygen atom in the ground state 3P 2 is excited to the 3P 1 or 3P 0 state
by H atom impact. The atom in the 3P 1 state emits a photon of energy E 1 (0)
= 0.020 e V with a probability A 1 (0) = 8.9 X 10- 5 sec-- 1 • 12 ) The self-absorption cross
section is then given by
789
Thermal and Dynamical Evolution of Gas Clouds
(41)
Using the cross section calculated by Callaway and Dugan we obtain the collisional de-excitation rate
(42)
m the same approximation as Eq. (40) has been derived.
absorption becomes efficient at
p
~
6 X 10 _ 22 [
T
{0.84 7 + kT j E (0)}
7
3/ 10 (1-
1
(MJ1 em .
Mc•x
gj
· ]"1 ---·
e-E,cOJ!'~r)
·
The effect of self2 7
2
(43)
For the 3P 0 state of oxygen atoms vYhich has the excitation energy
Eo (0) = 0.028 e V and the transition probability A 0 (0) = 1.7 X 10- 5 sec-\ 12 ) we have
the self-absorption cross section
(5 0 (0)
=
2.0 X 10- 18 T- 112 cm 2 •
(44)
The collisional de-excitation rate is approximately given by
(45)
The effect of self-absorption becomes efficient at the density nearly equal to
twice the value of Eq. ( 43).
5)
Grains
The internal energy of grains obtained by collision with hydrogen atoms
and molecules is emitted as thermal radiation. The cooling rate for this process
is given by 4l' 5l
Aa = na {nu(vn) + nH (Vn )}nra 2k (T- Tu),
2
2
(46)
where (vn) and (vn) are thermal velocities of hydrogen atoms and molecules,
and na, ru and Ta are the number density, the mean radius and the temperature
of grains, respectively. Following Gaustad 19 ) ru is taken as 2 X 10- 5 em for ice
grains (Ta:S100°K) and 6 X 10- 6 em for mineral grains (Ta2:100°K). The number
density nu is taken as 1.0 X 10- 13 (nn + 2nu 2 ) both for ice and mineral grains.
The grain temperature Tu is determined by the balance of energy gain and
loss, i.e.
(47)
where /Cvis is the opacity of grains for the visual light, I is the 1cnergy flux of
interstellar star-light, and /Cp is the Planck mean opacity of grains at Tu which
has been tabulated in HN. When the cloud is transparent to the star-light, we
take I= I 0 = 2 X 10- 2 erg/ cm 2 sec. When the cloud is opaque to the star-light,
we take the mean value of the flux in the cloud, 1.e.
(48)
where r = ICvisPR
IS
the optical thickness of the cloud.
T. Hattori, T. Nakano and C. I-Iayashi
790
6)
Cosm.ic-ray particles
Hayakawa, Nishimura and Takayanagi 20 l tried to account for the mean kinetic
temperature, 125°K, of the interstellar cloud by extrapolating the energy spectrum
of the cosmic-ray particles dmvn to 10 MeV. As in HN \Ve shall adopt one
tenth of their value, i.e
(49)
In the following we shall give some remark on the uncertainty in the value
21
of Tcu· Recently, Spitzer and Tomasko l reconsidered the heating by this pxocess.
By extrapolating the intensity of low energy cosmic-rays measured at Earth and
IVIars to the outside of the planetary system, they obtained a lower limit to the
heating rate which is equal to 0.023 times the value of Eq. (49). On the other
hand, they estimated the upper limit by assuming that the energetic particles are
ejected from supernovae. The velocity of expansion of a Type I supernova shell
1s now believed to be about 20,000 km/sec, which corresponds to a proton kinetic
energy of 2 MeV. The upper limit obtained by them is about 4 times the value
of Eq. (49).
The range of a 2 MeV proton in hydrogen is 3 X 10- 3g/ cm 2• 22 l Then the
energetic particles cannot enter deep into the high density cloud. In such a
situation the heating rate of Eq. ( 49) should be multiplied by exp (- r' /2),
where r' is the radius of the cloud in units of the range of a 2 MeV proton.
On the other hand, the range of a relativistic particle is much greater. For
instance, it is 5 X 10 2g/ cm 2 for a 1 Ge V proton. 23 l When a few MeV protons are
shut off, the heating comes from relativistic cosmic-Tays. In this case the heating
Tate is about two orders of magnitude smaller than Eq. (49).
§ 3.
Evolution of clouds in the tran.spm:ent stage
In this section we shall investigate the evolutionary feature of the gas clouds
which are transparent to thermal radiation in the same manner as in HN.
The time-scales of cooling, heating, free-fall contraction and free expansion
of the cloud are given, respectively, by
tc = 3/?pT/2;J.Jnn (A-T),
(50)
th = 3/?pT/2p;nn (T --A),
(51)
t 1 = (32nGpj3)- 112 ,
(52)
te = (fJ.mn/81zTY 12 (3M/ 4npYI3 ,
(53)
where tt is the mean molecular weight, 1\1[, p and T are the mass, mean density
and mean temperature of the cloud, and A and T are the totals of the cooling
and heating rates described in § 2, respectively.
By comparing the time-scales tc, th, t 1 and tc with one another, we can find
791
Thermal and Dynmnical Evolution of Gas Clouds
2.5
T=
1
\
Ag
logT(Kl
,}·'
\.II
1.0
p '~,
\ ',,
I
'
I
I
'
I
'
\
I
'
0.5 .
-24
\
\
' ... ------
-----~-
-
....
''
-22
'~-20
-18
log .P (g/cm
-16
3
-14
'
-12
)
Fig. 1. The evolutionary feature of a cloud of 1 M0 in the density-temperature diagram. Each
solid curve represents the state of te=t1 , te=tc, te=th, t 1 =tc or t 1 =th. The dashed curves
divide the diagram into seven regions according to the predominant process of cooling or heating. On the boundary between the cooling region and the heating region the total of the cooling
rates just counteracts that of the heating rates. On the dot-dashed curve -r= 1 the cloud becomes
opaque to thermal radiation. If the cloud of 1 M0 is born with a density lower than that of
the critical point P, it finally expands nearly along the curve te=th, and returns to the interstellar medium. On the other hand the cloud which is born with a higher density finally contracts
nearly along the curve t 1 =tc to the point A. The evolutionary path after passing the point
A is shown in Fig. 4.
the evolutionary feature in the p- T plane. The results for the clouds of mass
1, 10 2 and 10 4 M 0 are shown in Figs. 1, 2 and 3, respectively. The time-scales
t 1 and te defined by Eqs. (52) and (53) are meaningless when t 1 is nearly equal
to te because at this state the cloud makes neither free contraction nor free
expansion. However, in these figures we have used symbolically the expression
t 1 =te for the lines on which the pressure force counteracts the gravity. For
these lines we have used the equilibrium state of the polytrope of index 1.5.
The equilibrium line of the polytrope does not depend sensitively on the polytropic index. It is to be noticed that when the numerical constants in Eqs. (52)
and (53) are slightly changed, the relation t 1 = te corresponds exactly to the
equilibrium lines in Figs. 1, 2 and 3. 5 )
The equilibrium line t 1 = te divides the p- T diagram into two regions; regions
of expansion (above) and contraction (below). However, in the region far above
the curve tc = te or tc = t 1 , cooling is much faster than expansion or contraction,
and then evolution proceeds down-vvard nearly vertically in the p-T plane. On
792
T. 1-Iattori, T. Nakano and C. Hayashi
2.5
///,/~\
~/
T=1
I
\
I
I
\
\
I
I
2.0
\
\
\
1.0
A '""'
o:s
-24
-18
log .fJ ( g/cm 3 )
-20
-22
-16
-14
Fig. 2. The evolutionary feature of a cloud of 102 Me in the density-temperature diagram.
explanation is the same as in Fig. 1.
T=
The
1
I
2.5
I
I
/
/
_,. _____ ...-:::\
/
1.5
\
\
\
\
\
\
\
\
\I \
1.0
\
I
J;_i
I
I
I
I
I
I
05
L._..__ _
\
\
\
\
\
\\
\ r::R ''-------------__J_24_ __,______2,'--'-;-----"---_2--'-0--'--'----_-'-18----'----_-16L>.___
I-'-4-----'
____L____ _ _
Jog .fJ ( g/cm 3 )
Fig. 3. The evolutionary feature of a cloud of 104 Me in the density-temperature diagram.
explanation is the same_as in Fig. 1.
The
Thennal and Dynamical Evolution of Gas Clouds
793
the other hand, in the region far below the curve th = te or th =-= t 1 , evolution
prc_:>ceeds upward. It is, therefore, to be noticed that on the equilibrium line
te = t 1 the cloud is only temporarily in gravitational equilibrium.
If a gas cloud is born with density lower than that of the point P in Fig.
1, 2 or 3, it approaches the curve te = th regardless of its initial temperature,
and finally returns to the interstellar medium along this curve. On the other
hand, the cloud born with the density between the points P and A approaches
the curve tc = t 1 , and finally contracts to the point A, almost irrespective of its
previous history. The evolution after passing the point A where the protostar
becomes opaque to thermal radiation, is nearly adiabatic free-fall contraction as
will be shovvn in the next section. In this sense the point A may be called the
initial state of an opaque protostar.
The general feature of evolution described above is common to all gas
clouds much greater than 10- 2 Mev for which the point A lies far below the
equilibrium line t 1 =te. The critical density for contraction and expansion and
the characteristics of the point A for each mass are given in Tables II and III,
respectively. For 1 Mev the density of the point P is nearly equal to the result
of HN, because the newly considered cooling processes are not efficient near
this point.*l
The evolutionary feature of the gas cloud of 10- 2 Mev is quite different
from those of more massive clouds, as will be described in the next section both
for the transparent and opaque stages.
§ 4.
Evolution of protostars in the opaque stage
As the contraction proceeds, the protostar becomes opaque to thermal radiation which is emitted by grains. This state for each mass is shown by the curve
r = 1 in Figs. 1, 2 and 3.
In the opaque stage, the energy loss is determined by the diffusion of photons
throughout the protostar. As has been shown in the previous section, the contraction of the protostars much greater than 10-- 2 M 0 is essentially a free-fall
near the point A. Then the evolutionary path in the p-T diagram is determined
by4)
dlog T
dlog p
},~---=1-¢,
(54)
where
(55)
*) In Figs. 1, 9, 10 and 11 of HN there are mistakes in drawing the slope of the line t 1 =te.
However, this mistake hardly shifted the position of the critical point P.
794
T. Hattori, T. Nalwno and C. Hayashi
l is a thermodynamical constant which is equal to (r -1)-r,
r
being the ratio
of the specific heats, and !Gn is the Rosseland mean opacity of grains which 1s
tabulated in HN. As has been shown in HN, the radiation temperature Tr is
nearly equal to T in most of the region of p and T under consideration.
The state of ¢ = 1 has the same characteristics as that of tc = t 1 in the transparent stage. In Figs. 1, 2 and 3 this state is represented by the curves denoted
by tc=t1 . As we proceed to the right from the curve tc=t1 , the value of ¢
decreases rapidly and the evolutionary path is given by an adiabat asymptotically. Putting the opacity in the form JC 0 Ta and Tr= T, we obtain the evolutionary path
y)a-3 1 ]' )a-3 4
(
\1/2
- (--;;~1~~= -3-~;:JCJ:J~Ts 6~)
({}i?:
3
-3-=-a-=fiX/6-
{pcs-a)/A-11/6- Pocs-a)/'--11/6},
(56)
where To and Po are the values at the initial state, for instance, at the point A
in Figs. 1, 2 or 3. The evolutionary paths for each mass are shown in Fig. 4.
In this computation we have assumed that hydrogen is all in molecular form
because of high density and we have taken a= 2.4. 4)
As is seen in Fig. 4, the evolutionary path and then tbe mean entropy of
the protostar are insensitive to the stellar mass. This facilitates the study
of evolution in the later stages. For example, there ·vvill be a possibility that
a ne.,-,v protostar is born by the fragmentation of a large cloud which has so
far been considered. If this
protostar is already opaque at
the time of fragmentation, it
evolves along the same adiabat
/
/
2.0
/
as the parent cloud. This evolutionary path deviates only
/
slightly from that of an opaque
/
/
/
protostar
of the same mass
1.5
vvhich was transparent when
it was born. Then it is sufficient to investigate the evolu1.0
tion along the latter path as
-10
-14
-12
-16
shown in Fig. 4.
3
log .fJ ( 9/cm )
I£ the cloud is expanding
Fig. 4. Evolutionary paths of the protostars in the early
freely, the evolutionary path
opaque stage. The solid curves represent the paths
is described by Eq. (54) with
for the case of the adopted grain density ng = 1.0 X 10- 13
/
/
/
/
/
/
/
--
(nn + 2n 112 ). The dashed curves are for the case
ng = 1.0 X 10- 14 (nu + 2nu 2 ) .
Each dot on the path
represents the stage when the protostar becomes
opaque to thermal radiation, 1.e. the initial stage of
the opaque protostar.
31
16rro.c(/J?nn) ~ (. 3 \
6_
9
Y- -21~
4n1\1)
1
X
_]~'~~(~~---)\
513
!CnP
T
4
(57)
113
795
Thennal and Dynmnical Evolution of Gas Clouds
. :D
2.5
\:
·~
I
I
'
I
:\
I
I
2.0
8
I
I
I
I
I
I
'
I
I
I
I
,..
~~.L--
................
1.5 //'
',,
''
c;;,,
r:;
1.0
''
' ......... ,,_
~
........._
II
Te-i
T=
' \
II
I
'\
\
I r
1 1 cR
~
0.5
'"' I
'\.............. ,
'
'
p ------------,- ...... ,
II
-20
-18
-14
-16
3
log p ( g/cm )
'
-12
Fig. 5. The evolutionary feature of a cloud of l0- 2M0 in the density-temperature diagram.
The cloud which is born in the region to the left of the dashed curve PCD is destined
finally to expand along the curve te = th. On the other hand, the cloud which is born
in a region between the curve PCD and the line t 1 =te, it expands at first and settles
down to the equilibrium line t 1 =te· Afterwards it contracts slowly along this line
emitting radiation.
instead of Eq. (55). The state of ¢ = -1 has the same characteristics as that of
tc = te in the transparent stage.
The evolutionary feature of the gas cloud of 10- 2 M 0 which is shown in
Fig. 5 is quite different from those of more massive clouds shown in Figs. 1, 2
and 3. The point A lies near the equilibrium line t 1 = te. If the cloud of such
a small mass is formed by the fragmentation of a larger cloud, it lies above the
line t 1 = te in Fig. 5, because the parent cloud cannot cool down to this line as
is seen in Figs. 1, 2 and 3. If the cloud is born in a region to the left of the
dashed curve PCD in Fig. 5 (the line PC is an adiabat for the gas of nH = 0.1 nH,
and CD is a line of constant density), it expands finally along the curve th = te
and returns to the interstellar medium. If it is born in a region between the curve
PCD and the line t 1 = te, it expands at first and settles down to the equilibrium
line t 1 = te. A:fterwards it contracts slowly along this line*) emitting radiation.
This is the so-called Kelvin contraction. Then the critical density for contraction and expansion is equal to the density on the line CD in Fig. 5.
2
*l On the part EF of this line, the cooling time is smaller than the free fall time, so that the
evolutionary path deviates slightly from the equilibrium line.
796
T. Hattori, T. Nakano and C. Hayashi
§ 5.
Discussion
In view of the uncertainty in the abundance of H 2 molecules, we have further
investigated the cases of
=
and lln 2 = 0. As has been shovvn in § 3, cooling
2
by H 2 molecules is efficient only at low densities. Thus, it is for the clouds
greater than about 10 4M 0 that the general evolutionary feature is affected by
the H 2 abundance. The comparison of the cooling and heating processes is made
in Fig. 6 for the cloud of 10 4 M 0 for the three cases of the H 2 abundance,
nn = 0, 0.1 and 1.0. The position of the critical point P for contraction and
expansion (i.e., the cross point of the equilibrium line t 1 = te and the curve A= T)
Is only slightly affected by the H 2 abundance, as shovn1 in Fig. 6 and in Table II.
The intensity of energetic particles is also uncertain. We have investigated
the evolutionary feature of the cloud of
10 4 M 0 assuming that the heating rate
3.0 ,---,----,-------,.---,----.,. .------,
is two orders of magnitude smaller than
Eq. (49). The result is shown in Fig.
\00-.
-/ ()·
7. Comparison with Fig. 3 shows that
0--?-"l.
/
2.5
the difference is noticeable only for densi///~0~
~/-~\101.
ties below 10- 21 g/ ems. The density of
the critical point P is 6 X 10- 25g/ ems
which is one order of magnitude smaller
2.0
than the value found in § 3. However,
the point P for the clouds smaller than
10 21\10 is not much affected by the inten1.5
sity of energetic particles.
Further, there is uncertainty in the
concentration of the grains; all of the
1.0
heavy elements may not be attached to
-25 -24
-23
-21
-22
-20
the grains. The evolutionary path in the
early opaque stage which has been comFig. 6. The comparison of the cooling and
puted for the case nu = 10- 14
and that
heating processes for a cloud of 104 M0
13
for three cases of the H2 abundance.
for the case nu = 10- nn as studied in
The solid curves represent the case
2
§§ 3 and 4 are shown in Fig. 4. At
=O.lnn. The dashed curves are for the
low
densities the paths for the two cases
case of n 1h=nn, and the dot-dashed curves
are comparatively different from each
are for nn 2 = 0.
JZn nn
nn/
L..___
_ L . __ _. t _ __
_ j _ _ _ . . L _ J . . . __
___j
nn
nn
Table II. The values of log p, where p is the critical density for contraction and expansion
in g/cm 3, for ng=10- 13 n, ne=5Xl0- 4 (nu+2n 1h) and FeR equal to the value of Eq. (49).
II
I0 4M0
I0 2M0
IM0
I
ro- 2M0
~~~-----~------~---,---~--~-------------~-------
-------------------~-
For nu 2 =nn
;:: :::::~·Inn
~
-23.5
-20.4
-17.7
-14.3
=~~:~
=:~:~
=~~::
=~!:~
Thermal and Dynamical Evolution of Gas Clouds
797
3.0 .-----.-----,.----,-~.----,---r--------'..-"'~----.-----:------r---r---,---.------,
1/
\\
..... /
..-""'
T=l
\11-,_/''
I
\
2.5
\
I
\
2.0
\
\
\
\
I,
I,
I\
I
t
A
\
\
I
I
\
\
I
I
I
05 L___
t
\
\
rcR ',
-_-_-_-_-_-_J...--_-_-_-_-_-L..::...._ _
-18
-16
1L....,---1__j_.J.____j__ ___L__ __L.__ _...L__....l-..1.
-24
-22
-20
log
.? (g/cm
3
____l _ ____j__ ___.l
-14
)
Fig. 7. The evolutionary feature of a cloud of 104111(0) in the density-temperature diagram with
the heating rate by cosmic-ray particles w-z times the value of Eq. (49).
other, but with a slight increase of the density the difference IS mostly diminished.
Recently Nishida 24 l investigated the evolution of a protostar of 1 M 0 in a
manner similar to ours. His treatment and result differ from ours in the following
respects. Nishida did not take into account the effect of self-absorption for cooling by line emission. He assumed that the carbon and silicon are nearly completely ionized even at
10- 20 gj cm 3 in his calculation of the rate of heating
through the photo-ionization of these elements. The line t 1 = te which he drew
simply by equating Eqs. (51) and (52) differs fairly greatly from the equilibrium
lines of the polytropes.
The evolutionary feature of a protostar of 1 M 0 investigated by Ananaba
and Gaustad 25 l is quite different from that of HN and this paper. They pointed
out three reasons. The first reason is that, for the protostar which is initially
at rest as assumed by them, the time-scale of contraction at early stages is much
greater than that given by Eq. (52). However, this does not yield such a large
difference. Further, it can be easily seen that the other two reasons come from
misunderstanding of HN. The true reason lies in the opacity value. Ananaba
and Gaustad assumed that for T <150°K the opacity takes a constant value 2.4
cm 2/ g. They started their calculation from the state of T= 50°K and p = 1.7
16
3
X 10- g/ cm •
Because of their high opacity value, the protostar is sufficiently
P>
T. rlattori, T. Nakano and C. 1-Iayashi
798
opaque to thermal radiation and the energy loss is determined by the diffusion
rate of radiation throughout all the stages of their calculation. However~ the
opacity due to ice grains decreases rather greatly with decrease of the temperature for T<I00°K as shown in Table I of HN and, consequently, for their initial
state the protostar is transparent and the treatment of Ananaba and Gaustad is
not allowed.
Table III. Initial states (the points A) of opaque protostars, for ng=l0-13n, ,u=1.5, Kp=0.033
and KR=0.010 cm 2/g at 15oK.
M/1~~----~~--~~g p(g/cm3)
__
w-z
1
10 2
104
-12.6
--13.3
-14.0
-14.7
17
14
12
11
log (R/ R0)
log(L/L(O))
3.6
4.5
5.4
6.3
-2.3
-0.8
+0.7
+2.4
References
P. G. Mezger, J. Schraml and Y. Terzian, Astrophys. J. 150 (1967), 807.
P. G. Mezger, W. Altenhoff, ]. Schraml, B. F. Burke, E. C. Reifenstein III and T. L. Wilson,
Astrophys. J. 150 (1967), L 157.
3) P. G. Mezger and B. J. Robinson, Nature 220 (1968), 1107.
4) C. Hayashi and T. Nakano, Prog. Theor. Phys. 34 (1965), 754.
5) C. Hayashi, Ann. Rev. Astron. Astrophys. 4 (1966), 171.
6) A. Dalgarno and M. R. H. Rudge, Astrophys. J. 140 (1964), 800.
7) J. Callaway and A. F. Dugan, Phys. Rev. 163 (1967), 26.
8) L. H. Aller, Abundance of the Elements (Interscience Publishers, Inc., New York, 1961).
9) T. P. Stecher and D. A. Williams, Astrophys. J. 149 (1967), L29.
10) C. E. Heiles, Astrophys. ]. 151 (1968), 919.
11) M. J. Seaton, Ann. d'astrophys. 18 (1955), 188.
12) L. Spitzer, Jr., Astrophys. ]. 109 (1949), 337.
13) A. C. Allison and A. Dalgarno, Proc. Phys. Soc. 90 (1967), 609.
14) K. Takayanagi and S. Nishimura, Publ. Astron. Soc. Japan 12 (1960), 77.
15) C. W. Allen, Astrophysical Quantities (Athlone Press, London, 1963).
16) M. J. Seaton, Rev. Mod. Phys. 30 (1958), 979.
17) E. Schatzman, Rev. Mod. Phys. 30 (1958), 1009.
18) L. Spitzer, Jr. and M. P. Savedof£, Astrophys. J. 111 (1950), 593.
19) J. E. Gaustad, Astrophys. ]. 138 (1963), 1050.
20) S. Hayakawa, S. Nishimura and K. Takayanagi, Publ. Astron. Soc. Japan 13 (1961), 184.
21) L. Spitzer, Jr. and M. G. Tomasko, Astrophys. ]. 152 (1968), 971.
22) W. 'Whaling, Handbuch der Physil~ (Springer-Verlag, Berlin) 34 (1958), 193.
23) E. Fermi, Nuclear Physics (University of Chicago Press, 1950).
24) M. Nishida, Publ. Astron. Soc. Japan 20 (1968), 162.
25) S. E. Ananaba and J. E. Gaustad, Astrophys. J. 153 (1968), 95.
1)
2)