. 14
( 28)


We find the mass by multiplying the density by the
we use the free-fall time as a reasonable estimate
of the time it will take a cloud to collapse.
14 >3 2 11.67 g 2 1105 cm 3 2 16.1
1017 cm2 3 There is one important difference between our
MJ 10
idealized cloud and a real cloud. A real cloud will
1035 g
probably have a higher density in the center. We
76 M can see this as follows. If the cloud is initially of uni-
form density, all points will have the same inward
We could have obtained the same mass directly
acceleration. This means that all particles will cover
from equation (15.5).
the same inward distance dr (where dr 0), in
As we will see below, not all the mass will end up
some time interval dt. We can see how this changes
in the star.
the density for different volume spheres. If the ini-
tial density is 0, then the density of a constant
Once a cloud becomes gravitationally bound, mass collapsing sphere that shrinks from r0 to r is
it will begin to collapse. We would like to be able
(r0 / r)3
to estimate the time for the collapse to take place. 0

We begin by considering a particle a distance r
The change in density d is found by differentiat-
from the center of the cloud. It will accelerate
ing to give
toward the center under the influence of the
(r0 / r4) dr
mass closer to the center than r. The acceleration d 3 0

Using equation (15.11) to eliminate I and I0, we
The fractional change in density, d / , is
d/ 3 dr/r (15.9)
(r0 / r)2
(/ 0) (15.13)
This means that the smaller the initial sphere we
consider, the faster its density will grow. (This explains why a figure skater rotates faster as
she brings her arms in. The 1/r2 dependence of
With a higher density at the center, the free-
fall time for material near the center will be less the angular speed has a dramatic effect.)
than for material near the edge. The material To see what effect this has on collapse, we
from the edge will lag behind the material closer again look at a particle a distance r from the cen-
in. This will enhance the density concentration ter of a collapsing cloud. The acceleration at that
point is still GM(r)/r2. However, the radial acceler-
in the center. The net result is that we end up
with a strong concentration in the center. The ation now has two parts: (1) a(r) is associated with
concentration will eventually become the star, the change in magnitude of the radius, and (2)
but the material from the outer parts of the the acceleration associated with the change of
direction, r 2. Therefore,
cloud will continue to fall in on the star for quite
some time.
GM(r)/r 2 2
a(r) r (15.14a)
Example 15.2 Free-fall time
Solving for a(r) gives
Calculate the free-fall time for the cloud in the
GM(r)/r 2 2
above example. a(r) r (15.14b)

In comparing this with equation (15.6), we see
that the acceleration a(r) is less for a rotating
Using equation (15.8) gives
cloud than for a non-rotating cloud. The effect of
[(6.67 10 8 dyn cm2/g2 )(1.67 24
tff 10 g) the rotation is to slow down the collapse perpen-
(105 cm 3)] 1/2 dicular to the axis of rotation.
The effects of rotation will be most significant
1012 s
when the second term on the right-hand side of
105 yr
3 equation (15.14a) is much greater than the first
term, in which case
While almost a million years might sound like a
long time, it is short compared with the main GM(r)/r 2 2
sequence lifetime of the star that will be formed,
Multiplying both sides by r2 gives
or the age of the galaxy.
r3 2
If the cloud is rotating, then the collapse will
be affected by the fact that the cloud™s angular r3 2
momentum must remain constant. The angular 2
0r0) (r0/r) r0
( (15.15)
momentum L is the product of the moment of
inertia I and the angular speed , Noting that v0 0 r0 , where v0 is the speed of a
particle a distance r0 from the center,
L I (15.10)
GM(r) v0 r0 (r0/r)
For a uniform sphere, the moment of inertia is
We now solve for r/r0, the amount by which the
(2/5)Mr 2
I (15.11)
cloud collapses before the rotation dominates:
If I0 and 0 are the original moment of iner-
r/r0 v0 r0/ GM(r) (15.16)
tia and angular speed, and I and are their val-
ues at some later time, conservation of angular
For the cloud given in the two previous exam-
momentum tells us that
ples, with an initial rotation speed v0 1 km/s,
r/r0 0.6. This means that, by the time the cloud
I0 I (15.12)

More of the material can end up in stars if
the cloud breaks up into a multiple star system.
The angular momentum can be taken up in the
orbital motion, but individual clumps can con-
tinue contracting. This fragmentation process is
probably responsible for the high incidence of
binary systems. If a cloud shrinks to half its ini-
tial size, the average density will go up by a factor
(a) of eight. (The density is proportional to 1/volume,
and the volume is proportional to r3.) From equa-
J ω tion (15.5), we see that the Jeans mass of the
denser cloud will be approximately one-third of
the original Jeans mass. This means that it is pos-
sible for the less massive clumps to be bound and
continue their collapse. The fragmentation
process (Fig. 15.1) may be repeated until stellar
mass objects are reached.
ω1 15.2 Problems in star formation
J Jorb
We would like to know the conditions under
which stars will form. We would like to know
ω3 which types of interstellar clouds are most likely
to form stars, and which locations within the
clouds are the most likely sites of star formation.
We would also like to know whether star forma-
tion is spontaneous or whether it needs some
Fig 15.1. Fragmentation of a collapsing interstellar cloud. outside trigger. When we say a trigger is neces-
(a) The cloud is initially rotating as shown. As it collapses, the sary, we mean that the conditions in a cloud are
angular momentum J is conserved. (b) As the cloud becomes
right for star formation, but something is neces-
smaller, its angular speed must increase to keep the
sary to compress the cloud somewhat to get the
angular momentum ¬xed.The rotation inhibits collapse
process started. Once started, it continues on its
perpendicular to the axis of rotation, and the cloud ¬‚attens.
own. Sources for triggering star formation that
(c) Unable to collapse any further, the cloud breaks up, with
have been suggested are the passage of a super-
the total angular momentum being divided between the spin
nova remnant shock front, or the compression
and orbital angular momenta of the individual fragments.
caused by a stellar wind. (Later in this chapter we
will see how expanding HII regions might act as
reaches half its initial size, the rotation can
triggers, and in Chapter 17 we will discuss density
completely stop the collapse perpendicular to the
waves associated with galactic spiral structure.
axis of rotation. Motions parallel to the axis of
These might also induce star formation.)
rotation are not affected by this, so collapse par-
Once the collapse to form stars starts, we
allel to the axis of rotation can proceed unim-
would like to know how it proceeds, and what frac-
peded, and the cloud will flatten. (We will see
tion of the cloud mass ends up in stars. This is
that the tendency of rotating objects to form
sometimes referred to as the efficiency of star forma-
disks will reappear in many astrophysical situa-
tion. We would also like to know how much of the
tions.) Since the collapse is then only in one
mass that goes into stars goes into stars of various
dimension, it is harder to reach stellar densities.
masses. This is called the initial mass function. By
Thus, the effect of rotation is to keep much of the
“initial” we mean the distribution of stellar
material from becoming a star.

masses at the time that a cloud gives birth to stars.
The actual mass distribution in the galaxy is
altered by the fact that stars of different masses
have different lifetimes.
An important problem in understanding the
evolution of star forming clouds comes from the
angular momentum of the cloud. In the previous
section, we saw that the collapse can be slowed
down or even stopped in a rotating cloud. How a Older
cloud distributes and loses its angular momentum Subgroup
probably affects the efficiency of star formation
and the initial mass function. In addition, it may
account for the high fraction of multiple star sys-
tems and for the formation of planetary systems.
An interesting set of problems is posed by
groupings of stars called OB associations. These are
groups of stars in which it has been suggested Fig 15.2. OB associations and molecular clouds.These
that all O stars form. We refer to these groupings associations often have a few subgroups.The older subgroups
as associations rather than clusters because asso- are more extended, since the associations are unbound and
ciations are not gravitationally bound. They are are expanding.Younger subgroups tend to be more closely
expanding, and eventually dissolve into the back- related to molecular clouds.
ground of stars. We would like to know how an
initially bound cloud can give birth to an like to know how an interstellar cloud knows
unbound grouping of stars. In Chapter 13 we saw what distribution of stellar masses it is supposed
that, if a system in virial equilibrium loses more to make.
than half of its mass without the velocity distri-
bution changing, then the system becomes
15.3 Molecular clouds and
unbound. It is clear that the clusters have lost
star formation
more than half their mass.
Another interesting feature of OB associations
is the existence of subgroups. Some associations
We discussed the properties of molecular clouds
have as many as three or four distinct groupings of
in Chapter 14. They are important for star forma-
stars. The subgroups have different ages, as deter-
tion because they are both cool and dense, rela-
mined from their HR diagrams. Also, the older
tive to the rest of the interstellar medium. In
subgroups seem to be larger, which makes sense
Section 15.1, we discussed the conditions under
if they are expanding. A major question in star
which an interstellar cloud is gravitationally
formation is explaining what appears to be a
bound, and expressed the result as a Jeans length.
sequential wave of star formation through an
For a cloud of a given temperature T and number
association. It is in this context that triggers have
density n, the Jeans length is the minimum size
been most actively discussed. OB associations are
of a gravitationally bound cloud. We approxi-
often near molecular clouds, as shown in
mated the Jeans length (equation 15.4) as
Fig. 15.2. The younger subgroups tend to appear
closer to the molecular clouds. RJ
We would also like to know whether low and
Example 15.3 Jeans length for atomic and
high mass star formation take place in different
molecular clouds
ways or in different environments. It has been
suggested, for example, that low mass stars are Compare the Jeans length for an atomic cloud
(T 100 K, n 1 cm 3) and a molecular cloud
being made all the time, whereas high mass star
(T 10 K, n 103 cm 3).
formation takes place in bursts. We would also

We simply take the ratios, noting also that the
mass per particle in molecular clouds is twice that
in atomic clouds.



This means that a much smaller piece of a molecu-
lar cloud can become gravitationally bound than
for an atomic cloud. It is therefore much easier to
get a bound section in a molecular cloud than in
an HI cloud. This comes about because of both the
higher density and lower temperature.

We find a number of different types of molec-
ular clouds. Their basic properties are summa-
rized in Table 15.1. The simplest are the globules,
as shown in Fig. 14.1. These are sometimes called
Fig 15.3. Dark clouds blocking the light from background
Bok globules, after Bart Bok, who suggested that
stars. Note the intricate shapes.This is a near infrared image
they were potential sites for star formation.
of the star forming region RCW108.We see a few bright
Globules are typically a few parsecs across. They young stars, but mostly irregular dark clouds. (Compare this
generally have a simple, round appearance. This with the simple shape of the globule in Fig. 14.1.) [ESO]
simplicity makes them attractive to study. Their
visual extinctions fall in the range 1 10 mag,
which can be determined by star counting. From to those in globules, but the dark clouds are larger.
CO observations, we find their kinetic tempera- Typical sizes for the dark clouds are in the tens of
tures are about 10 K. From observations of CO and parsecs range. Often a size is hard to define because
CS we estimate their densities at 103 cm 3 and up dark clouds appear to consist of a number of small
to 104 cm 3, and masses in the range 10 to 100 M clouds in an irregular arrangement. There is evi-
. We think that they are in a state of slow gravita- dence that they contain low mass stars.
tional contraction. The largest molecular clouds are called giant
The dark clouds, such as those shown in Fig. 15.3, molecular clouds or GMCs. They are generally elon-
have local conditions (density, temperature) similar gated, with a length of about 50 to 100 pc. Their

Interstellar molecular clouds.
Table 15.1.
Tk n(H2) R M AV
(cm 3)
Type (K) (pc) (M ) (mag) Probes

103 104
Dark cloud 10 5“10 1“6 CO
103“104 102“103
Globule 10 1 5“15 CO, CS
GMC envelope 15 300 50 1“5 CO
105“106 103
Dense core 30“100 1 100 CO, CS, H2CO, NH3
and many others
Protostellar cores 100“200 high J transitions of
various molecules
Energetic ¬‚ows 1000 CO, SiO, H2
Envelope of evolved 2000 SiO masers

Fig 15.4. Molecular clouds in the Orion region. Giant
blocks our view of them. GMCs typically have
molecular clouds are indicated by their contours of emission
masses of a few times 105 M , and seem to come
from the CO molecule at 2.6 mm.This was done with a 1 m
telescope, providing an angular resolution of 12 arc min, in complexes whose masses exceed 106 M . These
which, as the black dot at lower left shows, is quite adequate
complexes are among the most massive entities
for this map.This was part of a survey of CO emission in
in the galaxy. There seems to be a close connec-
our galaxy (which we will talk about more in the next
tion between giant molecular clouds and OB
chapter), so this is plotted in galactic coordinates, which are
associations. It therefore appears that O and B
tilted with respect to the celestial equator, which is shown in
stars form in GMCs. GMCs also have lower mass
a red dashed line in this ¬gure.The Orion Nebula (shown in
stars in addition to the O and B stars.
Fig. 15.28) is at the intersection of the two black arrows in
Within the giant molecular clouds we find
the lower right corner of the ¬gure. At the 500 pc distance
of Orion, 100 pc would subtend 11 degrees. So, the cloud denser regions, called dense cloud cores. These are
containing the Orion Nebula is some 80 pc long.There is an denser and warmer than the surrounding cloud.
OB association around the region of the nebula.There is
Their temperatures are above 50 K. Their densities,
another equally large cloud that extends to the north across
determined from studies of a number of different
the celestial equator.This contains the association connected
molecules, are in the range 105 to 106 cm 3. (Even
to Orion™s belt and includes (as a very small part) the
though we call these clouds “dense”, their densi-
Horsehead Nebula (Fig. 14.3). Note that there are some
ties are comparable to the best vacuums we can
other clouds which extend to the west, into Monoceros,
obtain in the laboratory!) These cores are small,
containing yet another OB association. [Thomas Dame, CFA]
only a few tenths of a parsec across, and have
masses of a few hundred M . Our ability to study
them is limited by the angular resolution of our
densities are about 300 cm 3, a little lower than telescopes, but that is helped by the development
of interferometers working at millimeter wave-
for globules or dark clouds. They are also warmer,
with T lengths. We think that these cores are the places
15 K. Their extent can be traced using
in the GMCs where the star formation is taking
CO observations, like those shown in Fig. 15.4. By
place. (Some dense cores are also found in dark
observing CO in nearby GMCs, where we can see
clouds and globules.)
the dust, we gain confidence in the fact that the
One of the observational challenges in study-
CO tells us where the dust (and molecular hydro-
ing dense cloud cores is to find cores in which
gen) is. We therefore use the CO to trace out
there is unambiguous evidence for collapse. After
GMCs that are so far away that foreground dust

all, if these are in the process of forming stars, We would expect the magnetic effects to be
there should be large inward motions that important when the energy associated with the
should be detectable via Doppler shifts. Material presence of the magnetic field is comparable to
on the side of the core closest to us should be the gravitational energy in magnitude. In cgs
units, the energy density (erg/cm3) associated with
moving away from us, and we should see red-
a magnetic field B is
shifted spectral lines. Material on the far side of
the core should be moving towards us, and we
B2/ 8
uB (15.17)
should see blueshifted spectral lines. We do see
objects with both large red- and blueshifts. The Example 15.4 Magnetic energy
problem is that the Doppler shift measurements For what magnetic field strength B does the mag-
cannot tell us which side of the cloud each part netic energy of a cloud equal the absolute value of
of the emission is coming from. So, an expanding the gravitational potential energy? Assume a spher-
cloud could have the same spectra as a collapsing ical cloud with a radius R 10 pc, and a density of
cloud. The trick lies in having high angular reso- molecular hydrogen n(H2) 300 cm 3.
lution and seeing how the Doppler shift changes
with position. Studies of possible collapsing SOLUTION
clouds using millimeter interferometers are just The magnetic energy UB is the energy density, mul-
beginning to yield results. tiplied by the volume of the cloud:
In our discussions, it is important to remem-
a ba b R3
ber that, while we think in terms of spherical UB
8 3
clouds for simplicity, real clouds have irregular
shapes. Most of the larger clouds (GMCs) appear B2R3
elongated, part of larger filamentary structures. 6
It has even been suggested that the geometry of
The magnitude of the (negative) gravitational
interstellar clouds may be better represented by
potential energy is
fractals. However, we can still form a good insight
into the physical processes that govern star for- 3 GM2
mation using our simplified models. 5R
ca R3 b n1H2 2 12mp 2 d
15.4 Magnetic effects and 3
star formation G
16 2
a b GR5 1n1H2 2 12mp 2 2 2
Astronomers are becoming increasingly aware of 15
the fact that magnetic fields can have an impor-
10 GR5 1n1H2 2 12mp 2 2 2
tant effect on the star formation in an interstellar
cloud. Work in this area has been slow for two Equating these and solving for B, we have
reasons. (1) As we have seen, measurements of the
(60 G)1/2 (2n(H2)mp R)
interstellar magnetic fields are very difficult. Until
we have a good idea of field strengths, it is hard 5
6 10 gauss
to estimate their effects. As we mentioned in the
60 G
last chapter, observations of the Zeeman effect in
HI yield field intensities of tens of microgauss in This is of the order of strengths of fields that have
a number of clouds. (2) Theories that include been measured from HI Zeeman measurements.
magnetic fields are much harder to work out
than those that don™t. However, computer simu- As a molecular cloud collapses, the magnetic
lations of gravitational collapse in clouds with field strength will increase, as illustrated in
substantial magnetic fields are being carried out Fig. 15.5. This is because of the flux freezing, dis-
more routinely. cussed in Chapter 11. (Remember, Faraday™s law

to BR2. If the flux is constant, then BR2 must be
Magnetic Field
constant. This means that

B r 1/R2

From the previous example, we see that the mag-
netic energy UB is proportional to B2R3. This means

UB r (1/R2)2 R3

r 1/R
The gravitational potential energy is
UG r GM2/R

Since the mass of the cloud stays constant as it

UG r 1/R
Therefore, the magnetic and gravitational energies
Fig 15.5. Flux freezing in a collapsing interstellar cloud.
Graphically, Faraday™s law tells us that the number of ¬eld have the same dependence on R as the cloud col-
lines crossing the cloud™s surface stays constant as the cloud lapses. If the magnetic field cannot prevent the
collapses.This means that the ¬eld lines are closer together, initial collapse, then it cannot prevent the further
signifying a stronger ¬eld in (b).
collapse. However, if the magnetic field is impor-
tant in the initial collapse, it will continue to be
requires that the flux through a conducting sur-
face be constant.) This only takes place if the
As a cloud evolves, the ions and neutrals do
cloud is a good conductor. Most interstellar
not always stay perfectly mixed. The ions drift
clouds have sufficient ionization for this to be
with respect to the neutrals. If this happens,
the case. The ionization in cold clouds probably
some of the magnetic flux will escape from the
results mostly from cosmic rays. Most of the mass
cloud, meaning that the field is not as high as
of the cloud is in the form of neutral atoms or
one would calculate from flux freezing. The
molecules. There are roughly 107 neutrals for
process, called ambipolar diffusion, has another
every ion. However, as the cloud collapses, these
effect. As the ions move past the neutrals some
neutral particles carry the charged particles
collisions occur. This converts some of the drift
along with them. The charged particles, in turn,
motion into random motions of the neutrals,
provide the conductivity to insure the flux freez-
meaning an increase in the cloud temperature.
ing. This process allows the magnetic field effec-
Therefore, ambipolar diffusion can serve as a gen-
tively to exert a pressure which can inhibit the
eral heat source in a cloud.
The current picture that has emerged suggests
that there are two ways in which the magnetic
Example 15.5 Flux freezing
support of clouds is overcome. One is by ambipo-
For a spherical cloud with the magnetic flux con-
lar diffusion. This occurs in clouds where the
stant as the cloud collapses, find how the magnetic
magnetic energy is comparable to the gravita-
energy varies with the cloud radius R. Compare
tional energy. Ambipolar diffusion allows for the
this with the gravitational energy.
gradual contraction of the cloud. It is thought that
this process produces low mass stars at a roughly
steady rate throughout the galaxy. In the alterna-
For a uniform cloud, the magnetic flux is the
tive situation enough material is gathered together
product of the field B and the projected area of the
cloud R2. This means that the flux is proportional so that the absolute value of the gravitational

equation (15.19) for R, and substitute that solu-
energy is much greater than the magnetic energy,
tion for one of the R™s in equation (15.20) or
and star formation takes place rapidly. It is
thought that this process makes a mixture of high (15.21), we find that
and low mass stars.
a b
1 dE 1 dR
E dt R dt
15.5 Protostars This tells us that, in any time interval dt, the frac-
tional change in the energy dE/E is equal to the
15.5.1 Luminosity of collapsing clouds fractional change in the radius dR/R. These
As a cloud collapses the gravitational potential results tell us that the rate of collapse can be lim-
energy decreases. This is because the particles ited by the rate at which energy can be radiated.
within the cloud are moving closer to the center. We now look at the luminosity in various
The decrease in potential energy must be offset stages of the collapse. As the collapsing cloud
by energy radiated away or by an increase in the heats, it is still well below normal stellar temper-
kinetic energy. This increased kinetic energy can atures, so most of the radiation is given off in the
show up in two forms: (1) it can go into the faster infrared. Therefore, the opacity of the cloud in
infall of the particles in the collapsing cloud; or the infrared plays an important role in determin-
(2) it can go into heating the cloud. ing the nature of the collapse.
Let™s see what happens to the energy in a col- When the collapse begins, the material is
lapsing cloud. From the virial theorem, we know mostly atomic and molecular hydrogen and
atomic helium. As the collapse continues, half the
E K (15.18)
liberated energy goes into the internal energy of
This tells us that as the cloud collapses its inter- the gas. However, this doesn™t increase the temper-
nal kinetic energy K will increase. However, only ature. Instead, the energy goes into the ionization
half the potential energy shows up as increased of these neutral species. Following this, the liber-
kinetic energy. We can therefore see that the total ated energy goes into heating the gas, and the gas
energy of the collapsing cloud is decreasing. This pressure can eventually slow the collapse. For a
means that the cloud must be radiating energy 1 M protostar, the free-fall phase ends when the
away. The virial theorem tells us that half of the radius is about 500 R . (The radius varies approxi-
lost potential energy shows up as kinetic energy, mately with mass.) During the free-fall stage, the
and half the energy is radiated away. luminosity increases and ’ dR>dt ’ increases.
We can relate the luminosity of a contracting
Example 15.6 Luminosity of a collapsing cloud
cloud to its total energy. The total energy is
For a 1 M protostar that has collapsed to a radius
E ( 3/10)GM /R (15.19) of 500 R , (a) calculate the energy that has been lib-
erated to this point; (b) use this to calculate the
The energy lost in radiation must be balanced by
average luminosity if most of the energy is liber-
a corresponding decrease in E. The luminosity, L,
ated in the last 100 years of the collapse.
must therefore be equal to dE/dt. Differentiating
equation (15.19) gives
a 2 ba b
3 GM
dE dR From the virial theorem, the energy radiated will
be one-half times the current gravitational poten-
dt 10 R dt
tial energy:
We can solve for dR/dt to find the collapse rate for
3 GM2
a ba b
a given luminosity:
a 2b a b
10 R
dR dE
3 16.67 dyn cm2>g2 2 12 1033 g 2 2
(15.21) 8
dt 3 GM dt
1500 2 17 1010 cm2
(Remember, for a collapsing cloud, both dR/dt
1045 erg
and dE/dt are negative numbers.) If we solve 2

The average luminosity is this energy divided luminosity and temperature change. Therefore,
its location on an HR diagram changes. If L(t) is
by the time over which it is radiated:
the luminosity of a star as a function of time and
1045 erg
11002 13
T(t) is the temperature as a function of time, we
107 s2
can plot a series of points and connect them to
1035 erg>s follow the evolution of a star. Such a series of
points is called an evolutionary track. Stars evolve
170 L
so slowly compared with human lifetimes that
we cannot deduce the evolutionary track by
This is the average luminosity over the 100 year
observing one star. However, by observing many
period, but the actual luminosity at the end of the
stars, each at a different stage, we can infer the
period is higher, since |dR/dt| is greatest then.
evolutionary tracks. (We have already used evolu-
Once a cloud is producing stellar luminosities
tionary tracks in our discussion of post main
by gravitational collapse, we call it a protostar.
sequence evolution, in Chapters 10 and 11.)
Once the cloud becomes opaque the radiation can
We can also predict evolutionary tracks from
only escape from near the surface. (When the
theoretical models of protostars and stars. We use
opacity is low a photon can escape from anywhere
basic physics to calculate the physical conditions,
within the volume.) Since energy escapes slowly,
and see how the star™s radius and temperature
the temperature rises quickly. Also, a large tem-
change with time. Since the luminosity is given
perature difference can exist between the center
by L (4 R2)( T 4), we can relate changes in R and
and the edge. Under these conditions, the most
T to changes in L and T. When we calculate model
efficient form of energy transport from the center
tracks, we find that the evolutionary track of a
to just outside is by convection. This point was
protostar depends on its mass. This is not surpris-
first realized in 1961 by the Japanese astrophysi-
ing, since we have already seen that the mass
cist Chushiro Hayashi. During this stage the sur-
determines where a star will appear on the main
face temperature stays roughly constant at about
2500 K. Since the radius is decreasing, and the
Some evolutionary tracks for protostars and
temperature is approximately constant, the lumi-
pre-main sequence stars are shown in Fig. 15.6.
nosity decreases.
Note that the protostars appear above the main
During this stage the central temperature is
sequence. This means that for a given tempera-
still rising. When it is high enough, nuclear reac-
ture, T, protostars are more luminous than main
tions start. The contraction goes on for some time
sequence stars of the same temperature.
in the outer parts, as the pressure builds up in
Protostars are also larger than main sequence
the core. Eventually the pressure in the core is
stars of the same temperature. This is not sur-
sufficient to halt the collapse, and the star is
prising since protostars are still collapsing. Once
ready to settle into its main sequence existence.
the accretion phase stops, but before the main
For a protostar, the continuous spectrum
sequence is reached, we call these objects pre-
peaks in the near infrared. The dust in the col-
main sequence stars.
lapsing cloud surrounding the protostar will
Fig. 15.7 shows a model for the collapse of an
absorb some of the radiation. The dust will be
interstellar cloud into a 1 M protostar. At first
heated, but will not be the same temperature as
the cloud is cool, and then it contracts and heats.
the star. The emission from the dust will be in the
As discussed above, the T 4 increase is greater than
far infrared. From this we see that protostars are
the R2 decrease, and the luminosity of the proto-
best observed in the infrared part of the spectrum.
star increases. The peak luminosity is reached
when the temperature reaches 600 K. As the pro-
15.5.2 Evolutionary tracks for protostars tostar becomes denser, its opacity increases.
When we plot an HR diagram with stars we see Eventually, it is harder for the radiation from the
now, we are plotting the distribution of L and T center to escape, and the luminosity begins to
as they are now. However, as a star evolves, its decrease. During this stage energy transport in

15.6 Regions of recent
15 M
star formation

103 When we study star formation, we find that there
are some very obvious signposts of recent or
3M ongoing star formation. Regions of recent star
formation are important for a number of reasons.
L /L

First, they call our attention to places where star


formation might still be taking place. Second, the

newly formed stars have some effect on their
immediate vicinity, which might promote or
inhibit further star formation. In this section we
will look at some of the most prominent: (a) HII
0.5 M
regions, (b) masers, (c) energetic flows, and
(d) protostellar cores. In each case the object
becomes prominent either because of the unique
3 x 104 104 3 x 103
conditions that accompany star formation or
because of the effect of newly formed stars on the
Fig 15.6. Evolutionary tracks for pre-main sequence stars cloud out of which they were born.
on an HR diagram.Tracks are marked by the mass used in
the model.The dashed line represents the zero age main
15.6.1 HII regions
sequence (ZAMS), the place where stars ¬rst join the main
When a massive star forms it gives off visible and
ultraviolet photons. Photons with wavelengths
shorter than 91.2 nm, in the ultraviolet, have
the star is mostly by convection. The part of the enough energy ( 13.6 eV) to ionize H. The stars
evolutionary track at which the luminosity is that give off sufficient ultraviolet radiation to
decreasing quickly while the temperature cause significant ionization are the O and early B
increases slightly is called the Hayashi track. After stars. When most of the hydrogen is ionized, we
call the resulting part of the cloud an HII region,
this collapse slows, the star begins to approach
the main sequence. Eventually, it reaches the as shown in Fig. 15.8.
luminosity of a main sequence star, though it In equilibrium in an HII region there is a bal-
may vary somewhat before settling down. ance between ionizations and recombinations.

Fig 15.7. Model for the col-
lapse of an interstellar cloud into
a protostar and a pre-main
105 sequence star.

Central T(K)

103 Becomes opaque
to IR
H2 dissociates


102 104 106 108 1010 1012 1014 1016 1018 1020 1022 1024

Density (molecules/cm3)




Fig 15.8. HII regions. (a) The Lagoon Nebula (M8), in
Sagittarius, at a distance of 2 kpc. It is 20 pc across. Notice
the cluster of bright blue-white stars, which produce ionizing
radiation.The ionized gas glows red.The name comes from
the dust lane that cuts across the front, blocking our view of
the gas behind. (b) HST image view of M8. (c) The Eagle
Nebula (M16), in Serpens. (d) HST image of the dust lanes in
M16.The bright edges are regions of recent ionization.
(e) HST image of the Omega Nebula (M17), in Sgr, at a
distance of 2 kpc. Here the ionizing stars are not as obvious,
(c) and are embedded deep within the nebula.

(f )

Fig 15.8. (Continued) (f) The Tri¬d Nebula (M20), in Sgr. It
is named for the three-part appearance produced by the
dust lanes.The blue part on top is starlight re¬‚ected from
associated dust, a re¬‚ection nebula. (g) HST image of M20
(h) The Rosette Nebula (NGC 2244) in Monoceros, named
for its red color and petal-like appearance.The cluster of
blue stars in the center has created a cavity in the center of
the cloud. It is 1.3 kpc away and 15 pc across. (i) The Eta
Carina Nebula (NGC 3372), named for the bright star that
illuminates it. It is 3 kpc from Earth. (i)

Fig 15.8. (Continued) (j) The
central region of the Eta Carina
Nebula. (k) HST image of the
immediate vicinity of Eta Carina.
[(a), (c), (f), (h)“(j)
NOAO/AURA/NSF; (b), (d), (g),
(k) STScI/NASA; (e) ESO]


this reason, HII regions are often referred to as
Stromgren spheres, and the radius of an HII region
is called the Stromgren radius, rS.
We can see how the balance between ion-
izations and recombinations determines the
Stromgren radius. If Nuv is the number of ultravi-
olet photons per second given off by the star capa-
ble of ionizing hydrogen, then this is the number
of hydrogen atoms per second that can be ion-
ized. That is, the rate of ionizations Ri is given by
Ri Nuv (15.23)

The higher the density of protons and elec-
trons, the greater the rate of recombinations. The
recombination rate is given by
Rr ne npV (15.24)
where V is the volume of the HII region and is a
coefficient (which depends on temperature in a
Free electrons and protons collide, forming neu-
known way). For the volume, we can substitute
tral hydrogen atoms. However, the ultraviolet
the volume of a sphere with radius rS. If the only
photons from the star are continuously breaking
ionization is of hydrogen, the number density of
up those atoms to form proton“electron pairs.
electrons must equal that of protons, since both
The balance between these two processes deter-
come from ionizations of hydrogen. Equation
mines how large a particular HII region can be.
(15.24) then becomes
Within the HII region, almost all of the hydrogen
2 3
is ionized. There is a rapid transition at the edge, Rr np (4 rS /3) (15.25)
from almost entirely ionized gas to almost
Equating the ionization and recombination rates
entirely neutral gas. The theoretical reasons for
this sharp transition were first demonstrated by
2 3
the Swedish astrophysicist, Bengt Stromgren. For Nuv np (4 rS /3) (15.26)

Solving for rS gives
Table 15.2. Rates of H-ionizing photons
1/3 1/3 2/3
for main sequence stars.
rS np
(3/4 ) (Nuv) (15.27)
Photons/s ( 1048)
Spectral type
From equation (15.27) we can see that the size
of an HII region depends on the rate at which the
O5 51
star gives off ionizing photons and the density of
O6 17.4
the gas. If the gas density is high, the ionizing
O7 7.2
photons do not get very far before reaching their
O8 3.9
quota of atoms that can be ionized. The rate at
O9 2.1
which hydrogen ionizing photons are given off
B0 0.43
changes very rapidly with spectral type, as indi-
B1 0.0033
cated in Table 15.2, so the HII region around an
O7 star is very different from that around a B0
of hydrogen. When an ultraviolet photon causes
star. Often, O and early B stars are found in very
an ionization, some of the photon™s energy shows
small groupings. In these groupings, the HII
up as the kinetic energy of the free proton and
regions from various stars overlap, and the region
electron. Cooling in an HII region is inefficient,
appear as one large HII region.
since there are no hydrogen atoms and no mole-
The ultraviolet radiation from stars can also
cules. Cooling can only take place through trace
ionize other elements. For example, after hydro-
constituents, such as oxygen. Transitions within
gen, the next most abundant element is helium.
these constituents are excited by collisions with
However, the ionization energy of helium is so
protons and electrons. The collisions transfer
large that only the hottest stars produce significant
kinetic energy from the gas to the internal energy
numbers of photons capable of ionizing helium.
of the oxygen. The oxygen then radiates that
On the other hand, the ionization energy of carbon
energy away. Since the heating is efficient and the
(for removing one electron) is less than that of
cooling is inefficient, the temperature is high.
hydrogen. There are many photons that are capable
HII regions can give off continuous radiation,
of ionizing carbon that will not ionize hydrogen.
which can be detected in the radio part of the
This, combined with the lower abundance of C rel-
spectrum. This radiation results from collisions
ative to H, means that CII regions are generally
between electrons and protons in which the two
much larger than HII regions (see Problem 15.20).
do not recombine. Instead, the electron scatters
There are actually two conditions under
off the proton. In the process the electron
which the boundary for an HII region can exist.
changes its velocity. When a charged particle
One is that which we have already discussed. The
changes it velocity, it can emit or absorb a pho-
cloud continues beyond the range of the hydrogen-
ton. This radiation is called Bremsstrahlung (from
ionizing photons. When this happens, we say that
the German for “stopping radiation”). It is also
the HII region is ionization bounded. The other pos-
called free“free radiation, because the electron is
sibility is that the cloud itself comes to an end
free (not bound to the proton) both before and
while there is still hydrogen-ionizing radiation.
after the collision. The spectrum of free“free
In this case, we say that the HII region is density
radiation (Fig. 15.9) is characterized by the tem-
bounded, since its boundary is determined by the
perature of a gas. The spectrum is not that of a
place where the density is so low that we no
blackbody because the gas is not optically thick.
longer think of the cloud as existing. When an
The spectrum is a blackbody curve multiplied by
HII region is density bounded, hydrogen-ionizing
a frequency dependent opacity. Because the
radiation can slip out into the general interstellar
radiation can be described by the gas tempera-
radiation field. This is an important source of
ture, it is also known as thermal radiation. This
ionizing radiation in the general interstellar
radiation is strongest in the radio part of the spec-
medium (i.e. not near HII regions).
trum. Therefore, we can use radio continuum
The temperature of HII regions is quite high “
about 104 K. HII regions are heated by the ionization observations to see HII regions anywhere in our



22' 00"
on 30"

J2000 Declination
log I

Free-free 23' 00"


24' 00"


25' 00"

log ν
Fig 15.9. Schematic spectra of synchrotron radiation and 5h 35m 27s 24s 21s 18s 15s 12s 9s
free“free emission.The vertical axis is intensity and the
J2000 Right Ascension
horizontal axis is frequency, both on a logarithmic scale.The
log“log representation emphasizes the power-law behavior Fig 15.10. Radio image (made with the VLA) of free“free
of the synchrotron radiation emission from an HII region, the Orion Nebula (for which
optical images appear in Fig. 15.28).This is a higher resolu-
tion image than the single dish version in Fig. 4.25. It shows
galaxy. A map of continuum emission from an
the ¬ne scale structure in the core of the nebula.The image
HII region is shown in Fig. 15.10. (Note: In the
was made with the VLA in D (smallest) con¬guration at
encounter between the electron and proton, the
8.4 GHz, providing 8.4 arc sec resolution.This is a nine-¬eld
proton also accelerates and gives off radiation. (3 3) mosaic.The interferometer picks up less than one-
However, the acceleration of the proton is much half of the total ¬‚ux density because it is insensitive to the
less than that of the electron, by the ratio of their extended emission. Of course, it also gives beautiful detail of
masses. This means that the radiation given off the structure in the nebula. [D. Shephard, R. Maddalena,
J. McMullin, NRAO/AUI/NSF]
by the protons is not very important.)
HII regions also give off spectral line radia-
tion, called recombination line radiation. When an HII regions expand with time. When an HII
electron and proton recombine to form a hydro- region first forms (Fig. 15.11), it must grow to its
gen atom, the electron often ends up in a very equilibrium radius. Even after it reaches this equi-
high state. The electron then starts to drop down. librium size, it will continue to expand. This is
It usually falls one level at a time. Larger jumps because the pressure in the HII region is greater
are also possible, but less frequent. With each than that in the expanding cloud. The higher
jump, a photon is emitted at a frequency corre- pressure results from the higher temperature in
sponding to the energy difference for the partic- the HII region. Remember, the temperature in an
HII region is about 104 K, while that in the sur-
ular jump. (The energies are given by equation
3.6.) For very high states, the energy levels are rounding cloud is less than 100 K. The densities in
close together and the radiation is in the radio the HII region and surrounding cloud are similar.
part of the spectrum. As the electron jumps to As the HII region expands, it can compress the
lower states the lines pass through the infrared material in the surrounding cloud, possibly initiat-
and into the visible. Generally, the electron can ing a new wave of star formation, as illustrated in
go all the way down to the ground state before Fig. 15.12. This is one possibility that has been dis-
the atom is re-ionized. We even see H emission cussed for the triggering of star formation. The gas
as part of this recombination line series. This compressed by an expanding HII region will not
gives HII regions a red glow. (This red glow allows automatically form stars. That is because the gas
us to distinguish HII regions from reflection neb- will be heated as it is compressed. If that heat is not
ulae, which appear blue.) lost, the temperature of the cloud will increase.

amplification “ in the number of photons passing
through a material. In the stimulated emission
process, one photon strikes an atom or molecule,
and two photons emerge. The two photons are in
phase and are traveling in the same direction. The
fact that they are in phase means that their inten-
sities add constructively. Stimulated emission can
only take place if the incoming photon has an
energy corresponding to the difference between
two levels in the atom or molecule, and the atom
or molecule is in the upper of the two levels.
If only a few atoms or molecules are in the cor-
rect state, there will not be a significant increase
in the number of photons. Suppose we designate
the two states in the transition as 1 and 2. The
population of the lower state is n1 and the popu-
Fig 15.11. HII region in a molecular cloud. HII regions
lation of the upper state is n2. The requirement
usually form near the edge.
for amplification is that n2 /n1 be greater than g2
/g1, where the g are the statistical weights. The sit-
The pressure will increase and the gas will expand uation is called a population inversion, since it is
again. The re-expansion of compressed gas can only the opposite to the normal situation. Formally, it
be avoided if the gas can cool as it is compressed. corresponds to a negative temperature in the
Radiation from molecules such as CO in the sur- Boltzmann equation (see Problem 15.22). This is
rounding cloud can help with this cooling process. clearly not an equilibrium situation. The popula-
tion inversion in a particular pair of levels must
15.6.2 Masers be produced by a process, called a pump. The
We have already seen how the process of stimu- pump may involve both radiation and collisions.
lated emission can lead to a multiplication “ or The net effect of the pump process is to put
energy into the collection of atoms or molecules
so that some of that energy can come out in the
form of an intense, monochromatic, coherent (in
Gas phase) beam of radiation.
This was first realized in the laboratory, in the
1950s, by Charles Townes (then at Columbia
University). Townes won the Nobel Prize in physics
for this work. Since microwaves were being ampli-
Expanded fied in the process, the device was called a maser
HII Region (Fig. 15.13), an acronym for microwave amplifi-
cation by stimulated emission of radiation.
Subsequently, lasers were developed for the ampli-
fication of visible light. In any laser or maser, two
things are necessary: (1) a pump to provide the
population inversion, and (2) sufficient path
length to provide significant amplification. In
interstellar space, the path length is provided by
the large side of interstellar clouds. In the labora-
Fig 15.12. The HII region expands, compressing gas tory that path length is provided by mirrors.
deeper within the cloud. If this gas can cool quickly, then it (Laboratory masers are used as amplifiers in some
can collapse to form more stars.
radio telescope receivers.)

∆t = R / c


Fig 15.14. Time variability and source size.The signal from
the farthest point the eye can see must travel an extra
distance R over that from the nearest point the eye can see.

(c) A small size for these sources was also
deduced from rapid variations in their intensity.
Suppose we have a sphere of radius R, as shown
in Fig. 15.14. If the sphere were suddenly to
become luminous, then the first photons to leave
each point on the surface would not reach us
simultaneously. The photons from the edge of
(d) the sphere have to travel a distance R farther
than the photons from the nearest point. These
Fig 15.13. Maser ampli¬cation. In each frame, a molecule in
photons will arrive a time t R/c later than the
the upper level of the maser transition is indicated by a large
circle, and one in the lower level is indicated by a small circle. first photons. Therefore, it will take this time for
(a) All of the molecules are in the upper state, and a photon the light we see to rise from its initial low level
is incident from the left. (b) The photon stimulates emission to the final high value. A similar analysis holds
from the ¬rst molecule, so there are now two photons, in
for the time it would take for us to see the light
phase. (c) These photons stimulate emission from the next
turning off.
two molecules, resulting in four photons. (d) The process
The above analysis tells us that an object™s
continues with another doubling in the number of photons.
brightness cannot vary on a time scale faster
than the size of the emitting region, divided by
Shortly after the development of laboratory c. If we see variations in intensity over a time
masers, an interstellar maser was discovered. It scale of a year, the source cannot be larger than
involved the molecule OH. Four emission lines of a light year across. Interstellar masers were
OH were observed, but their relative intensities found to vary in intensity on an even shorter
were wrong for a molecule in equilibrium. As time scale, of the order of a month, indicating
radio telescopes were developed with better angu- an even smaller size.
lar resolution, the emission was observed to
Example 15.7 Maser size
become stronger and stronger. This means that
Estimate the maximum size of a maser that varies
the emission is probably very intense, but coming
on the time scale of one month. What is the angu-
from a very small area. This behavior was sugges-
lar size of this object at a distance of 500 pc?
tive of an interstellar maser. The next maser dis-
covered was in the water (H2O) molecule, at a
wavelength of 1 cm. As observations with better SOLUTION
The time scale for the variations is
resolution became possible, it was clear that the
objects were giving off as much energy as a 1015 K
t (24 h/day)(3600 s/h)(30 day)
blackbody over that narrow wavelength range in
106 s
which the emission was taking place.

equal the average transverse velocity vT. From
This corresponds to a size of
equation (13.6) we see that the distance is related
R c/ t
to the proper motion and transverse velocity by
1016 cm
d(pc) vT(km/s)/4.74 (arc sec/yr)
5.2 10 AU
Therefore, an accurate study of the motions of
masers allows us to determine the distance to a
The angular size (in arc seconds) is related to R (in
cluster of masers. It is hoped that this will
AU), and the distance d (in pc) by (equation 2.16)
develop into a very powerful distance measuring
technique. This technique works equally well for
103 / 500
5.2 random motions or for the masers being in an
expanding shell. The only requirement is that the
10 arc sec
average velocity along the line of sight is the
In fact, masers are even smaller than this size, same as the average velocity perpendicular to the
and have angular extents much less than 1 arc line of sight.
second. This means that we need radio interfer- Maser emission is also observed in the mole-
ometers to study masers. Very long baseline inter- cule SiO (silicon monoxide). From the regions in
ferometry has been used to study masers. which it is observed, it seems that SiO maser
When we try to understand interstellar emission is associated with mass outflow from
masers we must explain both the pump and the evolved red giant stars. Also, some OH masers are
path length for the gain. Many of the theories associated with similar regions.
require very high densities. For example, we
15.6.3 Energetic Flows
think that the presence of water masers suggests
densities in excess of 108 cm 3. This is much A major recent discovery is that many regions of
denser than even the dense cores that we nor- star formation seem to be characterized by strong
mally see in molecular clouds. We therefore outflows of material. One piece of evidence for
think that masers are associated with objects col- such flows comes from the observation of very broad
lapsing to become protostars. We take the pres- wings on the emission lines of CO (Fig. 15.16). The
ence of H2O or OH masers in a region to indicate widths of these wings range from 10 to 200 km/s.
the possibility of ongoing star formation. The broad wings are usually seen only over a
When we observe masers, we often see them small region where the CO emission is strongest.
in clusters, such as that depicted in Fig. 15.15. A peculiar feature of this emission is that the red-
With radio interferometry, we can measure the shifted wing and blueshifted wing seem to be
positions of the masers very accurately. We can
even measure their proper motions. We can use
Doppler shifts to measure their radial velocities.
However, we expect the motions of a cluster to be
random, so the average radial velocity should

Gas Shell
50 0 50
Velocity (km/s)

Fig 15.16. Spectrum of the 2.6 mm CO emission line
from the core of the molecular cloud behind the Orion
Nebula (Fig. 15.28.) The broad wings extend many tens of
kilometers per second on both sides of the line center.
[Jeffrey Mangum, NRAO/AUI/NSF]
Fig 15.15. Cluster of masers in an expanding shell.


Jet Stellar Wind
Molecular 100“200 km/s
Central Star
flowing away
from Observer
˜150 km/s
Expanding Shell
15 km/s


Fig 15.18. A model for sources with bipolar ¬‚ows and HH
objects.The stellar wind comes out in all directions but is
blocked in most directions by a disk around the star.The
Fig 15.17. Model for a bipolar ¬‚ow. Material coming wind emerges mostly at the poles of the disk.This drives
towards us on the near side of the cloud is blueshifted. material in the surrounding cloud away. Below, the effects of
Material going away from us on the far side of the cloud is the motion on the CO line pro¬les are shown, assuming that
redshifted. If the ¬‚ow is not aligned along the line of sight, the wind to the upper left moves away from the observer
the redshifted and blueshifted emission will appear in differ- and the wind to the lower right moves towards the observer.
ent locations on the sky. [Ronald Snell (University of Massachusetts) Snell, R. A. et al.,
Astrophys. J. Lett., 239, L17, 1980, Figs. 2 & 5]

coming from different parts of the cloud. This Evidence for collimated winds is present in
suggests that we are seeing two jets of gas, one another interesting class of objects that we think
coming partially towards us and the other par- are associated with pre-main sequence stars, the
Herbig“Haro (HH) objects shown in Fig. 15.19; see
tially away from us, as shown in Fig. 15.17. Because
of this structure, we call these objects sources of also Fig. 15.20. They were discovered independ-
bipolar flows. ently by George Herbig of the Lick Observatory
and Guillermo Haro of the Mexican National
Actually, we could also envision a model in
which we are seeing infall rather than outflow. Observatory. HH objects appear as bright nebu-
However, there is evidence we are seeing the losity on optical photographs. Their spectra
effects of a wind striking the surrounding cloud, resemble those of stars, and usually show emis-
heating a small region. These small heated sion lines, but no star is present in the nebulosity
regions show emission in the infrared from H2, (Fig. 15.21). We now think that the wind from a
requiring temperatures of about 103 K. Current pre-main sequence star clears a path through the
theories of these sources involve strong stellar cloud. The part of the cloud where the wind runs
winds, as shown in Fig. 15.18. The star is also sur- into the cloud is heated, and glows. We also see
rounded by a dense disk of material. This disk starlight reflected from the dust, explaining the
blocks the wind in most directions, but allows it stellar spectrum. The exciting star is deep within
to escape along the axis, explaining the bipolar the cloud and is not seen directly.
appearance. Remember, we saw earlier in this These observations indicate that winds are an
chapter that disks are likely to form around the important feature of protostellar evolution for
collapsing star. most stars. For low mass stars such as the Sun,

Fig 15.19. A negative optical


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( 28)