. 13
( 28)


are blown into the interstellar medium either via
the effects of stellar winds, or as part of a plane-
tary nebula. This is another example of the cycli-
cal processing of material between stars and the
interstellar medium.
Once the grains are in clouds, they can collect
particles from the gas and grow. There are some
limits. For example, once a layer of molecular
Fig 14.10. On top is the arrangement for doing optical or
hydrogen (H2) one molecule thick forms on the
uv absorption spectroscopy on an interstellar cloud. Below is
grains, no more hydrogen will stick. Grains will
a schematic absorption spectrum.
be destroyed, or diminished in size by a number

strong radio continuum emission. The cool inter-
These simple molecules are not generally stable in
stellar gas must be observed via radio spectral
the laboratory. CH is charged and would combine
with a negative ion or electron under laboratory
The first interstellar radio line to be observed
conditions. CH and CN have an outer electronic
was from atomic hydrogen, but it was not a tran-
shell with only one electron (as does H), making
sition in which an electron jumps from one orbit
them chemically reactive. The presence of these
to another. As we have said, these transitions are
unstable molecules in the interstellar gas sug-
in the visible and ultraviolet parts of the spec-
gests densities much lower than in the typical
trum. For the radio transition, the hydrogen stays
in the ground electronic state. This is illustrated
In these early studies no hydrogen absorption
in Fig. 14.11. Both the electron and proton have
lines were observed. This is not because there was
intrinsic angular momentum, called spin. We
no hydrogen present. The temperatures in inter-
have already seen that this spin can have two
stellar clouds are generally low, and most of the
hydrogen is in the ground state. Therefore, the
only H absorption lines possible are the Lyman
lines in the ultraviolet. Now that ultraviolet obser-
vations are possible from satellites, astronomers
can study these absorption lines.
You might wonder how we know that the
absorption lines are coming from the interstellar
gas and not from the stars themselves. After all,
we have already seen the large number of absorp- Spins Aligned
Spins Opposite
tion lines present in stars. One distinguishing (a)
feature is that the interstellar lines are much
narrower than the stellar absorption lines. By
narrower we mean they cover a smaller range of
frequency (or wavelength). Interstellar lines have
Doppler broadenings that correspond to a few
kilometers per second. If the Doppler broadening
is produced by random thermal motions, this
ν = 2.5 x 109 MHz
suggests a temperature of about 100 K. Also, sys-
(» = 121.6 nm)
tematic studies show that, on the average, the
absorption lines are stronger when detected in
the light of more distant stars. The more distant
the star is, the more interstellar material there is
between us and the star. The narrow interstellar
lines do not appear in the spectra of all stars. This
ν = 1420 MHz
suggests that the interstellar gas is clumpy, just (» = 21 cm)
as the interstellar dust is clumpy.
(Expanded by
factor of 105 )
14.4.2 Radio studies of atomic hydrogen
Much of what we know about the interstellar
medium comes from radio observations. We have Fig 14.11. (a) Origin of the 21 cm line.The splitting comes
already seen that supernova remnants, plane- from a magnetic interaction which depends on the spin
tary nebulae and pulsars are sources of radio directions of the electron and the proton.The energy is
emission. These are generally hot sources, or higher when the spins are parallel and lower when they are
antiparallel. (b) Energy level diagram showing the splitting of
sources with high energy electrons that produce
the hydrogen ground state (n 1).The splitting is greatly
a high radio luminosity. However, most of the
exaggerated in this ¬gure.
interstellar gas is cool and does not produce a

possible orientations. We refer to them as ˜up™
and ˜down™. This means that the electron and pro- Continuum
ton spins can be either parallel or antiparallel.
The relative orientation of the spins affects the
magnetic force between the electron and the pro-
ton. The state with the spins parallel has slightly
more energy than the state with the spins

antiparallel. The atom can undergo transitions
between these two states. The energy difference


Emission Line
corresponds to a frequency of about 1400 MHz, or

n or
a wavelength of 21 cm. This is generally referred

to as the 21 cm line.
If we take the energy of the transition h

and divide by Boltzmann™s constant, the quan-

tity h k gives the temperature necessary to see
collisional excitation of the hydrogen upper
state. This is about 0.07 K. This means that even
at the low temperatures of interstellar space
Fig 14.12. Conditions for radio absorption and emission
there will be sufficient energy to excite transi-
lines. If a radio continuum source is viewed through an inter-
tions between these two states in hydrogen. The
stellar cloud, then radio absorption lines can be seen against
21 cm line is easily observable under interstellar
the continuum source. (This also requires the continuum
conditions. The possibility of detecting this line source to appear hotter than the cloud at the wavelength of
was discussed in Leiden (Netherlands) in the observation.) If there is no background continuum source,
early 1940s by Henk van de Hulst. After that there then only emission lines can be seen.
was a race among Australian, Dutch and
American groups to detect the line. The first
both the excitation temperature and the optical
detection of the 21 cm line from interstellar
depth. The excitation temperature enables us to
hydrogen was in the early 1950s by a group at
Harvard, led by Edwin Purcell, who won the Nobel calculate the kinetic temperature of the gas, an
important quantity. The optical depth can be con-
Prize in physics for his work.
verted into a column density for atomic hydro-
Since that time there have been extensive
gen. If we know the column density and size of a
observations of the 21 cm line by radio astro-
cloud, we can also find the average local density
nomers all over the world. It is probably fair to say
of hydrogen. So you can see that the 21 cm line
that it was the dominant tool for studying the
observations provide astronomers with an impor-
interstellar medium and galactic structure
tant tool for studying the physical conditions in
through the 1960s, and continues to be very use-
many interstellar clouds.
ful. In these studies, the line was observed in both
One important feature of the radio observa-
emission and absorption. The conditions for
tions is that interstellar dust is transparent at
emission or absorption lines are shown in
radio wavelengths. Therefore we can use radio
Fig. 14.12. In order for the line to be in absorp-
telescopes to detect objects across the galaxy, far
tion, there must be a background continuum
beyond what we can see optically in the presence
source whose brightness temperature at 21 cm is
of dust. Since we can use it to observe clouds any-
greater than the excitation temperature of the
where in the galaxy, the 21 cm line is a very use-
atoms in the particular cloud being observed.
ful tool for studying galactic structure. Also, since
Under most conditions the excitation tempera-
it is a spectral line, we can observe its Doppler
ture of the 21 cm line is close to the kinetic tem-
shift and learn about motions throughout our
perature of the clouds.
galaxy. We will see how these studies are used in
By studying both absorption and emission
Chapter 16.
lines in a given region it is possible to deduce

We have already seen (in Chapter 6) that some
energy levels shift in the presence of a magnetic
field, the Zeeman effect. The levels involved in
Left the 21 cm line fall into this category. The stronger
Right the magnetic field, the greater the shift. This
SUM means that we can use the Zeeman shift in the
Intensity (Arb. Units)

DIFF 21 cm line to measure the strength of interstellar
magnetic fields. The experiment is difficult
because the Zeeman shift is much less than the
width of the normal 21 cm line. However, oppo-
site polarizations are shifted in opposite direc-
tions. Since we can detect different polarizations
separately, we can subtract one polarization™s
spectrum from the other, leaving a very small sig-
nal, as shown in Fig. 14.13. The experiment is also
difficult because a small difference in the
response of the telescope to the two polarizations
can mimic the effects of a Zeeman shift. Despite
these difficulties, recent experiments have suc-
Frequency (Arb. Units) ceeded in measuring fields of the order of tens of
(a) microgauss in a growing number of interstellar
clouds. Fields of this strength may sound very
.04 weak, but they are strong enough to influence
the evolution of these clouds, as we will discuss
in the next chapter.
Antenna Temperature

By making maps of the 21 cm emission
astronomers have been able to form a good pic-
0 ture of the cloud structure in the interstellar gas.
These maps show an irregular cloud structure,
similar to that shown in the dust clouds. Typical
clouds have the following physical parameters:
temperature, 100 K; hydrogen density nH
1“10 cm 3, lengths of tens of parsecs; hydrogen
1021 cm 2. The clouds
column densities up to
’60 ’40 ’20 0 20 40 60
fill about 5% of the volume of interstellar space,
Radial Velocity
meaning that the average density of atomic
hydrogen in interstellar space is of the order of
0.1 cm 3; One interesting recent finding is the
Fig 14.13. Zeeman effect in atomic hydrogen. (a) Idealized
situation.The line shift is a very small fraction of the linewidth. presence of large HI shells, which stir up the
However, opposite polarizations are shifted in opposite direc-
interstellar medium throughout the galaxy
tions.When we subtract one polarization from the other, we
(Fig. 14.14).
are left with a very distinctive pattern. (Note that the differ-
The regions between the clouds are not empty.
ence curve is approximately the derivative of the original
Studies of the line profiles of the 21 cm line show
curve.This follows directly from the de¬nition of a derivative.)
very broad, faint wings. This is interpreted as
(b) Spectra from a real source. In this ¬gure, the two smooth
coming from a small amount of very hot gas.
curves that have very little noise are the spectra for opposite
Temperatures of about 104 K have been estimated
polarizations.The difference is shown on an expanded scale,
so it looks noisy.The dashed line is the best ¬t to that differ- for this low density gas between the clouds. We
ence spectrum. [(b)Carl Heiles (University of California at will see later in this chapter that the low density
Berkeley)/Heiles, C., Astrophys. J., 336, 808, 1989, Fig. 1a]
means it is very hard for the gas to lose energy

1 — 1021

NH (cm’2)
y (kpc)



Fig 14.15. Hydrogen column density NH as a function of
10 0 10
visual extinction AV .Though there is some scatter, there is a
x (kpc)
good correlation in the two quantities as long as AV is less
than one magnitude.
Fig 14.14. Images made from 21 cm observations, showing
the large scale structure, projected in a galactic coordinate
system, with the plane of the Milky Way acting as the equator. For clouds that are near enough to be seen
The map shows the locations of the largest HI shells, called
optically, it was found that the 21 cm emission
˜supershells™.The larger circles indicate larger shells.To give
often follows the optical obscuration of the
an idea of galactic scale, these are superimposed on estimates
dust. This suggests that the gas and dust are
of where pieces of spiral arms may lie.We will talk more
well mixed. This idea was tested in detail by see-
about the spiral structure of the galaxy in Chapter 16. [Naomi
ing the degree to which the hydrogen column
McClure-Grif¬ths,Australia Tel. National Facility/McClure-
density NH correlates with the visual extinction
Grif¬ths, N. M. et al., Astrophys. J., 578, 189, 2002, Fig. 18]
AV. The results of these studies are shown in
Fig. 14.15. There is some scatter in the data, but
and cool (a situation somewhat similar to the it is clear that NH and AV are related. The gen-
solar corona). It has been noted that if we com- eral ratio NH/AV is approximately 1021 atoms
pare the pressure within a cloud Pcl with the pres- cm 2/1 mag.
sure in the intercloud medium Pic, we find This ratio was found to hold as long as the
extinction is less than 1 mag. When the extinc-
Pcl nclTcl
tion becomes higher, the relationship no longer
Pic nicTic
holds. This was a mystery for many years. Two
110 cm 2 1102 K 2
possible solutions were proposed. One was that
10.1 cm 2 1104 K 2
3 with a lot of dust, very little radiation can pene-
trate to heat the cloud. It is possible that the
hydrogen is so cold that the emission lines are
just very weak. The other possibility is that under
The pressure in the clouds and intercloud
the higher extinction conditions, pairs of hydro-
medium is approximately the same. Some theo-
gen atoms combine to form molecular hydrogen,
reticians have proposed a picture of the interstel-
H2. Molecular hydrogen obviously has a very dif-
lar medium known as the two-phase model, in
ferent structure than atomic hydrogen, and has
which this equality of pressures is not a coinci-
no equivalent of the 21 cm line. In fact, cold
dence, but follows from the ways in which the gas
molecular hydrogen has no emission or absorp-
can cool. The two-phase model is now considered
tion lines in the radio or visible parts of the spec-
overly simplified and has been replaced by more
trum. The recent discovery of a large number of
dynamic pictures of the interstellar medium.

interstellar molecules, including H2, tells us that If we equate the formation and destruction
the latter explanation is correct. rates, we can solve for the equilibrium CO

11 s 2tdis
14.5 Interstellar molecules 15 3
nCO 10 cm
5 3
3 10 cm
14.5.1 Discovery
The discovery of optical absorption lines of CH,
Since nH 10 cm 3 the fractional abundance
CH and CN raised the possibility that molecules
of CO, nCO/nH, is about 3 10 6. This is low enough
might be an important constituent of the inter-
that it did not raise the hopes of finding very
stellar gas. However, it was thought that the
complex molecules. We have even been very opti-
densities were too low for chemistry to proceed
mistic by assuming that every collision between
very far. In fact, the existence of these three
C and O leads to a CO molecule.
unstable species supported that general notion. If
However, radio searches for small molecules
chemistry had proceeded very far, these species
were carried out, with some of the initial candi-
would have been incorporated into more complex
dates being chosen by the availability of conven-
ient radio transitions. In the 1960s three simple
To see how these arguments worked, we can
molecules were found, OH (at a wavelength of
estimate the rate at which molecules form in a
18 cm), H2O (at a wavelength of 1 cm), and NH3 (at
cloud. Let™s take the example of C and O coming
a wavelength of 1 cm). The abundances of these
together to form the simple molecule CO. The
molecules were surprisingly high, and astronomers
rate of formation of CO per unit volume is
were encouraged to carry out searches for other
given by
molecules. In 1969 one of the most important
molecular discoveries took place. CO was found
Rform nCnOv (14.17)
at a wavelength of 2.6 mm, by a group at Bell
where nC and nO are the C and O densities, respec-
Laboratories, led by Arno Penzias and Robert Wilson
tively, v is the relative speed of the atoms, and
(who shared the Nobel Prize in physics for their
is the cross section for a collision. We take to be
earlier discovery of the cosmic background radia-
the geometric cross section (the approximate size
tion, to be discussed in Chapter 21). They used the
of an atom, 10 16 cm 2) and v to be the average
NRAO millimeter telescope shown in Fig. 4.28(a).
thermal speed at a temperature of 100 K (about
This was the first molecule to be found at mil-
105 cm/s). Finally, since both C and O have cosmic
limeter wavelengths. Remember, at shorter wave-
abundances of about 10 3 that of H, we take each
lengths we can produce good angular resolution
of their densities to be 10 3 nH. This will give us a
with modest sized telescopes. (Of course, the tele-
factor of nH in the rate, so the density is very
scope surfaces require greater precision and must
10 cm 3 the rate becomes
important. If nH
be placed at dry sites.) The abundance of CO is
10 15 cm 3/s.
also very high, with CO densities of about 1 cm 3,
We have to compare this with the rate at
much higher than our previous estimate. As we
which CO is destroyed. One destruction mecha-
will see, the 2.6 mm line of CO has taken its place
nism is photodissociation. An ultraviolet pho-
alongside the 21 cm line as one of the important
ton strikes the molecule with a sufficient
tools in studying the cool interstellar gas.
energy to break it apart. An unprotected CO
Following these initial discoveries, a large
molecule can live an average of 10 3 years
number of interstellar molecules were found.
(3 1010 s) in the interstellar radiation field.
Over 100 have been discovered to date. They are
The dissociation rate per molecule is the inverse
listed in Table 14.1. There are many familiar
of the lifetime. The dissociation rate per unit
molecules, such as formaldehyde (H2CO), methyl
volume is the density of CO molecules divided
alcohol (CH3OH), and ethyl alcohol (CH2CH3OH).
by the lifetime:
There are some unfamiliar molecules. Some of
these are charged species, such as HCO , and
Rdis nCO/tdis (14.18)
Interstellar molecules, arranged by number of atoms.
Table 14.1.
2 3 4 5 6 7 8 9 10 11 13

H 2S H2C2O C3H2O
SiN c-SiC2
HLi H3

others have unpaired electrons and are chemi-
cally active in the laboratory, such as CCH. Even
carbon chain molecules of moderate length (such
H Atom
as HC11N) have been found. There are even some
simple ring molecules. Many of these molecules
were discovered by observations at millimeter
wavelengths on telescopes such as that shown in
Fig. 4.28(a).
The discovery of so many interstellar mole-
Dust Grain A B
cules was obviously a surprise. How could the pre-
dictions that molecules could not form have been C D
so wrong? One answer is that the clouds in which
the molecules have been found are not the same
H2 Molecule
clouds that were studied at 21 cm. They have
higher densities and visual extinctions, and
lower temperatures. The higher densities mean
that chemical reactions take place faster (remem-
ber the formation rate goes roughly as the square
of the overall density). The higher visual extinc-
tions provide shielding from the ultraviolet radi-
Fig 14.16. Formation of H2 on a grain surface.
ation that dissociates the molecules.
We don™t see 21 cm emission from these
surface, the energy can be efficiently transferred
clouds because the atomic hydrogen has been
to the grain, resulting in a slight increase in grain
converted to molecular hydrogen. As we have
temperature. The fact that the dust plays an
already seen, the molecular hydrogen has no
important role in the formation of H2 and the
radio or optical spectrum. Since hydrogen is the
protection of H2 once it is formed, results in an
most abundant element, we classify interstellar
interesting situation. When a cloud has a visual
clouds by the form in which the hydrogen is
extinction of less than one magnitude, almost all
found. For example, clouds in which the hydro-
of the hydrogen is atomic. When the extinction is
gen is mostly atomic are called HI clouds. Clouds
greater than 1 mag, almost all of the hydrogen is
in which the hydrogen is mostly ionized are
molecular. This explains the breakdown in the
called HII regions (to be discussed in the next chap-
relationship between NH and AV above 1 mag.
ter). Clouds in which the hydrogen is mostly
Despite the important role that dust plays in
molecular are called molecular clouds.
the formation of the most abundant molecule H2,
14.5.2 Interstellar chemistry most of the interstellar chemistry cannot proceed
in this way. Many of the molecules are formed in
Since the discovery of so many interstellar mole-
the gas. At the beginning of this section we cal-
cules, considerable effort has gone toward a bet-
culated a very low rate for two atoms to collide in
ter understanding of interstellar chemistry. It
the gas to form a molecule. However, the densi-
appears that some of the chemical reactions take
ties in molecular clouds are at least 103 times
place on grain surfaces. The grain surface pro-
those we used for our estimate, and the reaction
vides a place for two atoms to migrate around
rates go as the square of the density. Therefore,
until they find each other. They also provide a
the reaction rates in molecular clouds are much
sink for the binding energy of a molecule. The
faster than our initial calculation suggests. There
example of molecular hydrogen is shown in Fig.
is also another factor that increases the cross sec-
14.16. If two H atoms formed in the gas phase, the
tion for collisions if one of the reactants is an ion
particular properties of the H2 molecule would
and the other is a neutral. Such a reaction is
keep it from radiating away the excess energy
called an ion“molecule reaction.
before the molecule flew apart. On a grain

+ rium abundances of various molecules. These are
the abundances for which the rates of destruc-
tion and formation are equal. These theories have
+ been quite successful at predicting the abun-
dances of most of the simpler (especially two- and
three-atom) molecules. More work is still needed
for the heavier molecules. In addition, it may be

that many interstellar clouds are not old enough
+ to have reached an equilibrium situation. If that is
the case, the abundances should still be changing.
Motion of
14.5.3 Observing interstellar molecules
When we observe interstellar molecules, we are
“ not observing transitions in which electrons jump
from one level to another. Such transitions do
exist for molecules, as they do for atoms. However,
+ they require energies of the order of at least a few
- electron volts and are in the visible part of the
+ spectrum. These transitions are not easily excited
in the cool interstellar medium. Another type of
Fig 14.17. Dipole in an electric ¬eld. In this case, the
transition in molecules, involving lower energies,
electric ¬eld is provided by the positive charge, and weakens
is vibrational. We can think of a molecule as con-
with distance from that charge.The negative end of the
dipole is closer to the positive charge, so an attractive force sisting of a number of balls connected by springs.
felt by the negative end is greater than the repulsive force felt The springs can stretch and bend at certain fre-
by the positive end.The dipole is thus attracted to the
quencies, with certain energies. Transitions
charge. (The same thing would happen with a negative charge.)
between vibrational states are possible. The ener-
gies associated with vibrational transitions usu-
ally place the resulting photons in the infrared.
To see how the rate is enhanced, let™s consider
This is still too energetic for the cool clouds.
the case of a positive ion (as shown in Fig. 14.17). We
There is another type of transition, with even
have already said that the grains must be nega-
lower energies. It involves the rotation of the mole-
tively charged, so the gas must be positively
cules. The rotational motion is also quantized and
charged. In addition, Table 14.1 shows that many
transitions among rotational states can take place.
positive ions have been detected. The neutral atoms
The photons associated with these transitions are
can still have an electric dipole moment, even
generally in the radio part of the spectrum. To see
though it has no net charge. The dipole will tend to
what the energy levels look like in this case, we con-
line up with the electric field of the ion. Since the
sider a diatomic molecule (such as CO), rotating
ion is positive, the negative end of the dipole will
end-over-end about its center of mass. The rota-
end up closer to the ion. The negative end of the
tional inertia is I. If the molecule is rotating with
dipole will therefore feel an attractive force which
an angular speed , the energy is given by
is slightly greater than the repulsive force felt by
11 2 2I 2
E (14.19)
the positive end, which is farther away. The dipole
will feel a net attractive force. This attractive force
The angular frequency can be expressed in terms
significantly increases the effective cross section of
of the angular momentum L as
the reactants and speeds the reaction.
Theoreticians have tried to identify the chem- LI (14.20)
ical reactions that might be important in the
Using this, equation (14.19) becomes
interstellar medium. They then carry out model
L2 2 I
calculations in which they calculate the equilib- E (14.21)

major difference between I(CO) and I(CS) are from
The condition for the quantization of angular
momentum is different than the one we saw for the difference in the masses of the O and S. In
electrons in an atom. If J is an integer, called the addition, the bond lengths are slightly different in
rotational quantum number, then L2 is related to J by the two molecules. Note that the closest spacing is
1 2 1h 2 2 2
for the first two energy levels ( J 0 and J 1). As
L2 J1 J (14.22)
we go to higher values of J, the energy levels are
(Note: for large values of J this is not very dif- farther apart. This means that at low tempera-
ferent from the condition L Jh/2 for orbiting tures only a few of the lowest energy levels are
electrons.) If we put equation (14.22) into equa- populated. For example, the 2.6 mm transition in
tion (14.21), the energy becomes which CO is most commonly observed is the
1 2 1h 2 2 2
J 1 S 0. The values of rotational inertia for
J1 J
many simple molecules are such that the lowest
E (14.23)
transitions lie in the millimeter part of the radio
spectrum. That is why so many molecules were
For any given molecule, the energy levels are
discovered at millimeter wavelengths.
determined by the rotational inertia. If I is large,
Adding more atoms to a molecule can compli-
the energy levels will be close together. If I is
cate the spectra. If we still have a linear molecule
small, the energy levels will be farther apart. The
(for example, in HCN the three atoms are in a
levels for CO and CS are shown in Fig. 14.18. The
line), then the energy levels are essentially the
same as the diatomic case, with the appropriate
3 value for I. If molecules are not linear, then the
spectra are more complicated, since we have to
allow for rotation about more than one axis, but
there are similarities to the linear case.
If we want to look at a new interstellar mole-
cule, we need to know the wavelengths at which
it can emit. For the most part, we rely on accurate
laboratory measurements of molecular spectra.
Once the wavelengths of a few transitions have
been measured, those of other transitions can be
2 calculated very accurately (using expressions
such as equation (14.23)). There are some mole-

3 cules that have been found in interstellar space
without prior laboratory study. These were found
accidentally, in the course of searches for other
molecules. In some cases the interstellar medium
provides us with a unique opportunity to study
molecules that are not stable in the laboratory.
The most important feature of interstellar
1 molecules is that they provide us with a way of
obtaining information about the physical condi-
tions in the molecular clouds. If we take the
CO CS energy corresponding to the 2.6 mm photon, and
divide by k, we find an equivalent temperature of
Fig 14.18. Rotational energy levels for two diatomic
5.5 K. This means that rotational transitions in
molecules, CO and CS.The states are designated by the
molecules are excited even at low temperatures.
rotational quantum number J.The differences between the
Also, the factor e E kT in the Boltzmann equation
two molecules arise from the differences in rotational inertias
resulting from different masses for O and S, and different is most sensitive to changes in temperature and
bond lengths for the two molecules. density when E is of the order of kT.

state. The emitted photon has an energy equal to
Before After
the energy difference between the states.
There is an additional radiative process that is
important “ stimulated emission. This is emission of
a photon, stimulated by the presence of another
photon. Suppose we consider only two energy
states. The molecule starts in the higher energy
state. When the molecule is struck by a photon,
whose energy is equal to the energy difference
between the states, the molecules cannot absorb
the photon, since the molecule is already in the
higher state. However, the presence of that pho-
ton can cause the molecule to drop to the lower
(b) state, emitting a second photon. In the process of
stimulated emission one photon comes in and
two photons go out. The two photons have the
same wavelength, are in phase with each other,
and travel in the same direction. As will see
below, it is stimulated emission that is responsi-
ble for amplification in masers and lasers.
Molecules can also be induced to make tran-
Fig 14.19. Types of interactions between radiation and sitions by collisions with other particles. In a
matter. (a) Absorption. A photon is absorbed, leaving the molecular cloud most of the matter is in H2,
atom or molecule in an excited state. In this case the
which we don™t usually directly observe.
excited state is denoted by a larger symbol for the atom or
However, this H2 makes its presence felt by forc-
molecule. (b) Emission.The atom or molecule starts in the
ing transitions in other molecules, as illustrated
excited state, and spontaneously makes a transition to the
in Fig. 14.20. The process works in both direc-
lower state, giving off a photon of the appropriate energy.
tions. An H2 molecule can strike a CO molecule,
(c) Stimulated emission.The atom or molecule is in the
for example. In the process, the H2 can lose
excited state. A photon whose energy would be right to
kinetic energy, while the CO is excited to a
produce an absorption if the atom or molecule were in the
lower state, strikes the atom or molecule, inducing a down- higher energy state, or the H2 molecule can gain
ward transition and the emission of a photon.There are now kinetic energy with the CO going to a lower
two photons.They are at the same frequency, traveling in the
energy state.
same direction and in phase.
In order to use observations of interstellar
molecules to tell us about cloud physical condi-
tions, we must be able to calculate the rates at
To see how we use molecules as probes of the which these various processes occur under differ-
physical conditions in interstellar clouds, we ent conditions. It is then necessary to carry out
must first know the ways in which a molecule large calculations to model the conditions in a
can go from one rotational state to another. One cloud. In these models we require that the popu-
set of processes involves the interaction with radi- lation of each level stays constant. The rate at
ation, either by emission or absorption of pho- which molecules can reach any state must equal
tons. Radiative processes are illustrated in the rate at which they leave that state. We use the
Fig. 14.19. We are already familiar with absorp- model calculations to predict the strengths of
tion and emission of photons. For absorption, the various molecular lines. We then compare those
photon must have the energy corresponding to a predictions with observations. The models are
transition between the molecule™s initial state to adjusted until agreement is found. The model is
a higher energy state. For emission, the molecule then used to predict the results of new observa-
goes from a higher energy state to a lower energy tions, and the process continues.

H2 Molecule that of hydrogen, while most other molecules
have abundances that are about 10 9 times that
of hydrogen (a part per billion) or less. The CO is
CO Molecule easy to excite and to observe. It is particularly use-
ful in tracing out the extent of molecular clouds.
Also, observations of CO allow us to estimate
(a) cloud masses and kinetic temperatures. Some
other molecules that are very useful are carbon
sulfide (CS) and formaldehyde (H2CO). These
other molecules are rarer and harder to excite
than CO. We only see them in very dense parts of
clouds. Therefore, we can use these molecules to
tell us about the densities in these clouds.
Determining the masses of molecular clouds
(b) is more difficult than for atomic clouds. When
we look at atomic clouds through the 21 cm line
we are looking at the primary constituent of
those clouds “ HI. So, when we measure HI col-
umn densities from the strength of the 21 cm
line, we can convert those column densities
directly into cloud masses. Also, since the optical
depth of that line is generally small, the strength
Fig 14.20. Excitation of a CO molecule via a collision with of the line is close to proportional to the column
an H2 molecule. (a) In this case the H2 strikes the CO.
density. When we look at molecular clouds, say
(b) The H2 is moving slower, having lost some kinetic energy,
by observing CO, we are observing a trace con-
and the CO is rotating faster with energy that it gained in
stituent, something present in a part per million
the collision. (c) The CO emits a photon (spontaneously)
relative to H2, and that abundance may vary
and drops back to the lower state. (A transition to the
according to local chemical conditions. So, even
lower state could have also been caused by a collision with a
if we could measure the total mass of CO in a
different H2 molecule, causing that H2 molecule to speed up,
taking away the rotational energy of the CO.) molecular cloud, we wouldn™t know by what fac-
tor to multiply that to determine the H2 mass. In
addition, for the most part, the transitions that
In studying molecular lines we can also learn we observe in CO are not optically thin, so it is
about the velocities within a cloud by studying hard to relate their intensity directly to the CO
the line profiles. An interesting feature of molec- column density.
ular lines is that they are almost always wider Analyzing molecular excitation to give H2
(cover a wider frequency range) than would be densities in different parts of a cloud can be use-
expected for a line in which the only Doppler ful in studying local conditions within a cloud.
broadening is from random thermal motions. These do work, but involve extensive observa-
This implies that we are seeing the effects of tions, making maps of different lines and differ-
other internal motions in the cloud. These ent molecules, taking advantage of the fact that
motions can include collapse, expansion or rota- different transitions are excited in different tem-
tion. We may also be seeing the effects of turbu- perature and density regimes.
lent stirring of the gas. This stirring may be There is one technique that is potentially use-
driven by mass loss from both old and young ful. In the last chapter, we discussed using the
stars. virial theorem to determine the masses of clus-
Probably the most usefully studied interstellar ters of stars. This might also work for molecular
molecule is carbon monoxide, CO. It is very abun- clouds. Even though the H2 is mostly invisible,
dant. Its abundance is about 10 4 to 10 5 times the CO in the cloud still feels its gravitational

effects. So, a more massive cloud would be able for ion“molecule reactions in cool clouds.
to support larger internal motions and still Therefore, observing the same molecule with dif-
remain bound. We can use the CO emission to ferent isotopic substitutions can tell us about
measure both the extent of the cloud and the how that molecule was made.
internal motions, from the widths the spectral
lines. There is always the problem that we don™t
14.6 Thermodynamics of the
know if a cloud is dynamically relaxed, or even
interstellar medium
bound. After all, we have seen earlier in this
chapter that the interstellar medium is a very
turbulent place. If we want to use virial masses, The temperature of any object is determined by
we need to find clouds that seem to be in rela- the balance between heating and cooling. There is
tively quiet regions. generally some temperature for which the rates of
heating and cooling will be the same, allowing the
Example 14.5 Virial mass of a molecular cloud temperature to stay constant. In this section we
Find the virial mass of a molecular cloud with
will look at the heating and cooling processes.
vr 3.0 km/s and R 10 pc.
When we talk about a heating process we mean
one that tends to increase the kinetic energy of the
gas. When we talk about a cooling process we
From equation (13.52) we have mean one that tends to decrease the kinetic
15 2 13.0 105 cm s 2 2 110 pc 2 13.18 1018 cm pc 2 energy.
16.67 dyn cm2 g2 2
M One way for an interstellar cloud to be heated
is by the absorption of photons. These photons
1038 g
2.1 come from a nearby star, or from the combined
light of many distant stars. A photon entering a
105 M
cloud is not, by itself, a mechanism for heating
Objects this massive are called giant Molecular the cloud. We must have a way of converting the
clouds and will be discussed more in the next energy of the photon into kinetic energy in the
chapter. gas. The most important mechanisms for photon
heating are as follows.
Another interesting aspect of interstellar mol-
(1) Heating the dust. A photon strikes a dust grain
ecules involves the substitution of various iso-
with the photon energy going towards increas-
topes. For example, the most common form of
carbon is 12C and most of the CO is in this form. ing the grain temperature. The hot grain is
then struck by an atom or molecule in the gas,
However, some CO is formed with the rarer
species 13C. Making such a substitution changes and it transfers some of its energy to that atom
the rotational inertia, I, for the molecule, shift- or molecule.
(2) Excitation of atoms or molecules. A photon
ing the spectral lines. The shifts are quite large,
strikes an atom or molecule, leaving it in an
and are easy to detect. By contrast, the wave-
excited state. The excited atom or molecule
lengths of electronic transitions (in C atoms for
undergoes a collision with another atom or
example) depend on the electron nucleus
molecule. The first atom or molecule drops
reduced mass, and the shift is very small when
we go from 12C to 13C. We can therefore measure back to the lower energy state, and the energy
shows up as an increased kinetic energy for
the amounts of different isotopes in the interstel-
the second atom or molecule. It is important
lar medium. Since all of the heavy elements come
that the collision takes place before the first
from stars, these measurements can tell us about
atom or molecule has had time simply to emit
the ways in which earlier generations of stars
a photon and drop to the lower state, since the
have enriched the interstellar medium. It also
photon can escape, leaving the cloud with no
turns out that changing isotopes changes the
additional energy.
chemical reaction rates. This is particularly true

(1) Emission from grains. An atom or molecule
(3) Ionization. The incoming photon strikes an
strikes a grain, with the atom or molecule
atom or molecule, ejecting an electron. The
losing kinetic energy and the grain becoming
electron can then transfer its kinetic energy
hotter. The grain can then radiate this excess
to the rest of the gas through collisions.
energy away. It must radiate away before it is
These collisions must take place before the
struck by another atom or molecule that
electron recombines with an ion, releasing a
might take back the energy.
(2) Excitation. One atom or molecule strikes
(4) Photoelectric effect. In the process, an incom-
another, with the first losing kinetic energy
ing photon strikes a grain surface, causing the
and the second being driven into an excited
ejection of an electron. The electron™s kinetic
state. The second one then emits a photon
energy is then available to heat the gas.
and drops back to its lower state. Of course,
The above processes also work for heating by
the emission of the photon must take place
streams of high energy particles, known as cosmic
before another collision forces the second one
rays, which permeate the interstellar medium.
back to the lower state.
The sources of cosmic rays will be discussed in
(3) Ionization. One atom or molecule strikes
Chapter 19.
another, with the second being ionized. The
The interstellar medium can also be heated by
electron then recombines, accompanied by
the direct injection of mechanical energy from
the emission of a photon, before it can collide
high velocity flows. For example, a supernova
with another particle in the gas.
remnant, traveling at high speeds, will transfer
In the heating and cooling processes, differ-
some of its kinetic energy to material it overruns.
ent atoms and molecules play important roles in
Stellar winds can accomplish the same things.
different density and temperature regimes. For
When we look at large scale maps of interstellar
example, in the cool molecular clouds, much of
clouds, we see evidence for many loops and ˜bub-
the cooling comes from radiation by CO. In very
bles™. These suggest that the interstellar medium
hot regions the cooling can come from unusual
is constantly being stirred up by processes such as
emission lines in certain ions (discussed in
supernova explosions.
Chapter 15). This multitude of processes allows us
We now look at cooling processes. We must
to have a wide variety of temperatures in the
remember that a cooling process must take
interstellar medium, ranging from cool (10 K)
kinetic energy from the gas and remove that
high density regions to hot (104 K) low density
energy from the cloud. These processes are just
the inverse of the heating processes.

Chapter summary

In this chapter we looked at various components extinction is wavelength dependent, producing
of the interstellar medium. We saw how they are interstellar reddening. We saw what could be
observed, and we looked at the physical processes deduced about grain sizes from extinction curves.
that are important in their current state and We obtain information on grain composition from
evolution. infrared spectra. We saw how the equilibrium tem-
Though only 1% of the interstellar mass, the peratures of grains are determined by a balance
dust is the most easily visible part of the interstel- between radiation absorbed and radiation emitted.
lar medium. We detect dust by its blocking of The interstellar gas can be observed in the
starlight, known as extinction. Warm dust can be optical and ultraviolet parts of the spectrum, but
detected by emission in the infrared. The extinc- radio observations are most useful in studying
tion consists of both scattering and absorption. The the cool gas. Extensive studies have been made

using the 21 cm line of hydrogen. However, the (10“50 K), we are usually observing transitions
clouds revealed by these studies do not have high from one rotational state to another. At these low
enough densities for them to be the likely sites of temperatures and densities, most of the chemical
star formation. reactions are probably between ions and neutral
Star formation probably takes place in cooler, species.
denser, molecular clouds. In these clouds most of We also looked at how interstellar clouds are
the hydrogen is in the form of H2. We cannot heated and at how they cool. In heating, any
observe the H2 from the ground, except when it is energy input must eventually be converted into
heated in a few small regions. Instead, we study kinetic energy in the gas. In cooling, the kinetic
molecular clouds by emission from trace mole- energy of the gas must be converted into energy,
cules, such as CO. At these low temperatures such as radiation, that can leave the cloud.


14.15. What are the advantages of using radio
14.1. When we look at an image like Fig. 14.1,
observations (as opposed to optical) in
how do we know that something is blocking
studying the interstellar gas?
the light of distant stars, rather than there
14.16. What are the mechanisms that we have
simply being fewer stars in one direction?
discussed for broadening interstellar lines?
14.2. How will scattering affect the temperature of
14.17. What does the two-phase model of the
the struck dust grain? How will absorption?
interstellar medium try to explain?
14.3. If we look at a bright nebula, how might we
14.18. Compare the advantages of studying the
know that it is a reflection nebula?
21 cm line of H with the 2.6 mm line of CO.
14.4. What does counting stars tell us about the
Discuss both the observational advantages
extinction in a cloud?
and the differences in the physical informa-
14.5. When we use star counts, what are the
tion that we obtain.
relative advantages of looking in magnitude
14.19. Why did early molecular discoveries (CH,
ranges and taking total numbers of stars?
CH , CN) discourage further searches?
14.6. What is the significance of the constancy of
the ratio of total-to-selective extinction? 14.20. How do ion“molecule reactions help in
14.7. What is the evidence that there is more interstellar chemistry?
than one type of interstellar dust grain? 14.21. Why is it hard to make H2 directly in the
14.8. Why is it not likely that interstellar dust gas phase?
grains are all spherical? 14.22. As you increase the mass of the atoms in a
14.9. What do we mean by a dilute blackbody? diatomic molecule, what happens to the
*14.10. How would the equilibrium temperature of rotational energy levels?
a dust grain change if the albedo were less 14.23. Explain how we use observations of a mole-
in the infrared than in the visible? cule such as CO to tell us about conditions
14.11. Why do we not see H absorption from in molecular clouds.
interstellar HI clouds? 14.24. In studying molecular clouds, what are the
14.12. What is the explanation for the disappear- advantages of studying a variety of mole-
ance in the correlation between HI column cules?
density and visual extinction above one 14.25. Explain how CO can act to cool molecular
magnitude? clouds. Go through the steps of how the
14.13. What is the evidence that interstellar gas energy would transform from the kinetic
and dust are well mixed? energy of the gas to a form where it can
14.14. How can we study the Zeeman shift in escape the cloud.
interstellar HI when it is only a small 14.26. Why do low density gases have difficulty
fraction of the linewidth? cooling?


14.9. How far must a dust grain be from a 104 K
(For all of these problems, where necessary assume the
star for it to have a temperature of 1000 K?
normal ratio of total-to-selective extinction.)
*14.10. Suppose the albedo of a dust grain is constant
14.1. Suppose we observe an A0 V star to have an
in the infrared (where it will emit most of its
apparent (visual) magnitude of 15.7. The star
energy), at a value a(IR), and is constant over
is in a cluster whose distance is known to be
the visible and ultraviolet (where it will absorb
950 pc. (a) What is the extinction between us
most of its energy), with a value a(V). It is near
and the star? (b) If we had not known the
a star whose spectrum is that of a blackbody
distance to the cluster, and not taken the
at temperature T*, and whose radius is R*. The
extinction into account, by how much would
grain is a distance d from the star. Derive an
our distance calculation have been off?
expression for the dust temperature.
14.2. We observe a star that is 1000 pc away. How
*14.11. Suppose a dust grain has an albedo a( ). It is
much extinction would there be if we
near a star whose spectrum is that of a
calculated a distance of 2000 pc when we
blackbody at temperature T*, and whose
didn™t take extinction into account?
radius is R*. The grain is a distance d from
14.3. Suppose we observe an A0 V star to have mV
the star. Derive an expression for the dust
16.0, mB 17.0. (a) What is the extinction
temperature. (You will have to leave your
between us and the star, and how far away is
answer in terms of an integral, since the
the star? (b) What are the extinction and
form of a( ) is not given.)
distance if mB 18.0?
14.12. What is the range of distances from a BO V
14.4. Suppose we observe a K0 V star to have
star for which the dust temperature is
mV 16.0, mB 17.0. (a) What is the extinction
between 50 and 1000 K?
between us and the star, and how far away is
*14.13. In our discussion of the temperature of a
the star? (b) What are the extinction and
dust grain near a star, we did not account
distance if mB 18.0?
for the fact that the dust grains near the star
14.5. We are observing a cluster whose distance is
would block some light from reaching dust
known to be 850 pc. The visual extinction
grains far from the star. Show how this
between us and the cluster is 2 mag. What
effect modifies the results.
would be the mV and mB for (a) an O5 star
14.14. Show that, in a gas, the number of particles
and (b) a G5 star?
hitting a surface per second per unit surface
14.6. Prove that, for the same angular momen-
area is nv, where n is the number density of
tum, end-over-end rotation for a cigar-
particles per volume and v is the speed of
shaped object has a lower energy than rota-
the particles.
tion about the long axis.
14.15. Compare the width (both in km/s and nm) of
14.7. Suppose we have a grain with a radius of
10 5 cm and a charge of e. An electron the Na D line (589.6 nm) for thermal broad-
ening in an interstellar cloud (50 K) and a
starts from far away at rest. It is attracted
stellar atmosphere (5000 K).
and eventually hits the surface. (a) How fast
14.16. What frequency resolution would be needed
is it going when it hits the surface? (b) How
to observe the 21 cm line with a velocity
does this compare with the the average
resolution of 0.1 km/s?
thermal speed of the electrons if they are at
14.17. For an excitation temperature of 100 K, what
100 K?
is the ratio of populations for the two levels
14.8. Use the results of Example 14.4 to give a
in the 21 cm transition? (Take the statistical
scaling relationship that can be used to
weight of the lower level to be 1 and the
calculate the dust temperature (in Kelvin)
upper level to be 3.)
when the stellar temperature is given in units
of 104 K, and the distance from the star is 14.18. How does the angular resolution of a 100 m
diameter telescope at 21 cm compare with
given in stellar radii.

that of a 12 m diameter telescope at 2.6 mm? value of cos , where is the angle through
What is the significance of these numbers? which the light is scattered, and 0 for
*14.19. Estimate the rate at which two H atoms can forward scattering. Find the value of the
form an H2 molecule on a grain surface. phase function in the following limiting
Assume that all atoms hitting a grain stick, cases: (a) all forward scattering, (b) all rear-
and that as soon as two H atoms are on the ward scattering, (c) random scattering with
grain surface, they immediately form a all directions equally likely.
molecule. 14.23. For rotational energy levels of diatomic
14.20. Suppose we have an electric dipole made up molecules the most likely transitions are
of two charges, q and q, a distance d apart. those for which J changes by 1. Find an
The dipole is placed a distance r from a charge expression for the energies of these allowed
Q. (a) Find an expression for the net force on transitions, as a function of J and I.
the dipole. (b) How is this related to chemical 14.24. Calculate the mass of an HI cloud with a
reactions in the interstellar medium? circular appearance, with a radius of 10 pc
14.21. For the CO, J 1 S 0 transition (just and an average H column density of
1021 cm 2.
considering the two lowest levels), what is
the ratio of the populations of the two levels 14.25. Calculate the virial mass for a molecular
for temperatures of 10, 20, 30 K? The statistical cloud with vr 5 km/s and R 20 pc.
weights of the two states are 1 and 3. 14.26. For a 30 pc radius molecular cloud to have a
virial mass of 106 M , what must be the
14.22. For scattering of light by interstellar grains
we define the phase function as the average value of vr ?

Computer problems

the Boltzmann equation with statistical weights
14.1. If we are far from an individual star, we can treat
1. Plot the graph of the fraction in the upper state
the interstellar radiation field as a dilute black-
body of temperature 104 K, with a dilution factor vs. T for T ranging from 0 to 1000 K.
of 10 14. This means that the spectrum of the radi- 14.5. For CO and CS, what rotational transitions
ation looks like that of a 104 K blackbody, with the would be observable in the visible part of the
intensity reduced by a factor of 10 14. Use this to spectrum?
For the CO, J
14.6. 1 S 0 transition (just consider-
estimate the temperature of an interstellar grain
ing the two lowest levels), make a graph of the
in such an environment.
ratio of the populations of the two levels for tem-
14.2. Plot the graph of the temperature of a dust grain
peratures ranging from 0 to 1000 K. The statistical
vs. distance from an O5 star, with the graph cov-
weights of the two states are 1 and 3.
ering distances from 1000 AU to 1 pc.
14.7. For the lowest three energy levels of CO, draw a
14.3. Draw diagrams like Fig. 14.13(a) where the split-
graph of the fraction in the J 1 state vs. T for T
ting between the two components is a smaller and
ranging from 0 to 1000 K. The statistical weights
a larger fraction of the linewidth. (The red and
of the three states are 1, 3 and 5.
blue curves are both gaussians with the same
14.8. Calculate virial masses for molecular clouds
peak intensity and linewidth.)
1, 3, 5 and 10 pc, and vr 1, 2 and
with R
14.4. Assume that the populations of the 21 cm transi-
5 km/s.
tion, in a gas of temperature T, are described by
Chapter 15

Star formation

In Chapter 14, we discussed the contents of the The condition for gravitational binding (total
interstellar medium, the material out of which energy negative) is then
new stars must be formed. In this chapter, we will
13>5 2GM2>R 13>2 2 1M>m 2kT
identify those parts of the interstellar medium
that are involved in star formation, and see what Dividing both sides by GM and multiplying by
we know, and what we have to learn, about the (5/3) gives
star formation process.
1M>R2 15>2 2 1kT>Gm 2 (15.1)

15.1 Gravitational binding The mass and radius of a cloud are not inde-
pendent, since they are related to the density
M/(4 /3)R3. We might therefore like to use
In Chapter 13, we talked about gravitational
equation (15.1) to estimate the smallest size cloud
binding for clusters of stars. The same concepts
of a given , m and T for which the cloud is grav-
apply to interstellar clouds, with the stars in the
itationally bound. This quantity is called the
cluster being replaced by the particles that
Jeans length, RJ. James Jeans obtained essentially
make up the cloud (either H or H2 ). The gravita-
the same result with a more sophisticated analy-
tional potential energy is now due to the inter-
sis. We therefore eliminate M in equation (15.1),
action among all of the particles in a cloud. For
and change the inequality to an equality, since
a uniform spherical cloud, the gravitational
we are looking for the value of R that is on the
potential energy is (3/5)GM2 /R. The kinetic
boundary between bound and unbound. This
energy is still related to the rms velocity disper-
sion, but with a large number of particles,
14 >3 2R3 >RJ
which can easily be related to the cloud temper- 5kT>2Gm
ature, so the kinetic energy is (3/2)(M/m)kT,
Solving for RJ,
where M is the total mass of the cloud and m is

115kT>8 Gm 2 1>2
the mass per particle.
RJ (15.2)
The clouds are kept together by the gravita-
tional attraction amongst all of the particles in Note that (15/8 )1/2 0.77, which is close to
the cloud. If the gravitational forces that hold the unity. As the geometry of the cloud changes, the
cloud together are greater than the forces driving exact value of the constant will change, but it
it apart, we say the cloud is gravitationally bound. will still be close to unity. We then write
We can think of the random thermal motions in
1kT>Gm 2 1>2
the gas as resisting the collapse. RJ (15.3)

We can rewrite this in terms of n, the number of is given by
particles per unit volume (n /m), as
1kT>Gm2n 2 1>2 G(4 /3)r3 /r2
RJ (15.4)

(4 /3)Gr (15.6)
We can also use equation (15.1) to give us the
minimum mass for which a cloud of given , T If the acceleration of this particle stayed con-
and m will be bound. This minimum mass is stant with time, then the free-fall time, the time
called the Jeans mass. It is the mass of an object for it to fall a distance r, would be
whose radius is RJ, so
2r 1>2
c d
14 >3 2R3 tff (15.7)
MJ a1r2

14 >3 2 1kT>Gm 2 3>2 1>2
c d
14 >3 2 1Gr 2
41kT Gm 2 3 2 12

41kT Gm 2 3 2 1nm2 Note that the constant (3/2 )1/2 0.7, which
we can approximate as unity, since we are mak-
ing an estimate of the time. This gives
Example 15.1 Jeans length and mass
1>1G 2 1>2
Find the Jeans length and mass in a cloud with tff (15.8)
105 H atoms per centimeter cubed and a tempera-
The free-fall time is independent of the starting
ture of 50 K.
radius. Therefore, all matter in a constant density
cloud has approximately the same free-fall time.
However, as the cloud collapses, the density
We use equation (15.4) to find RJ:
increases. The collapse proceeds faster. The free-
11.38 erg >K2 150 K2
16 1>2
c d
fall time for the original cloud is therefore an
16.67 10 8dyn cm2>g2 2 11.67 g2 2 1105cm2
RJ 24
10 upper limit to the actual collapse time. However,
the result is not very different, since most of the
1017 cm
time will be taken up in the early stages of the col-
0.2 pc lapse, when the acceleration is not appreciably dif-
ferent from the one we have calculated. Therefore,


. 13
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