. 9
( 28)


when the nucleus (Z, A) captures a neutron:
1Z, A 2 n S 1Z, A 12 (9.29)
Fig 9.6. The CNO cycle. Solid arrows represent reactions.
Symbols over these arrows indicate emitted particles.
What happens next depends on whether the
Dashed lines indicate when a created particle participates in
rate of neutron capture is slow or rapid, com-
another reaction.
pared to the rate of beta decay. If neutron capture
is slow, we call the sequence of reactions an s-
particle before breaking up. If this did not hap-
process (Fig. 9.7). In this situation the new nucleus
pen, the buildup of heavier elements would be
blocked. The combination of the 4He and 8Be gives
+n = neutron capture
4 8
Be S 12C
He (9.22b)
e- = beta decay
The triple-alpha process is also important as
solar mass stars age and leave the main sequence
(discussed in Chapter 10).
In massive stars, there is another scheme that
is important in converting four protons into one
He nucleus. It is called the CNO cycle. The cycle is

indicated graphically in Fig. 9.6, and the steps are:
12 1
H S 13N
C (strong force) (9.23)
13 13
N SC e (beta decay) (9.24)
13 1
H S 14N
C (strong force) (9.25)
14 1
H S 15O
N (strong force) (9.26)
S 15N
O e (beta decay) (9.27)
15 1
H S 12C 4
N He (strong force) (9.28)
We see that the net result is the conversion of
four protons into one 4He nucleus plus two Fig 9.7 The r- and s-processes.The horizontal axis indi-
positrons, two neutrinos, and three photons. All cates increasing proton number Z, and the vertical axis indi-
nuclei created as intermediate products are used cates increasing neutron number N. In an r-process, a neu-
in the next step. In addition, the last step returns tron is captured, and in an s-process the capture is followed
the 12C we need to start the cycle, so the cycle can by a beta decay.The n next to an arrow indicates neutron
go again. In a sense, we can think of the 12C as a capture.The e next to an arrow indicates a beta decay.

section that tunneling by those nuclei with higher
(Z, A 1) will beta decay before it can capture
than average energy leads to a strong temperature
another neutron:
1Z, A 12 S 1Z 1, A 2
dependence on the reaction rates. For example, the
e (9.30)
rates of some important reactions depend on tem-
perature as T7 or even higher powers.
If the neutron capture is rapid we call the
sequence of reactions an r-process (Fig. 9.7). The We say that a collapsing object makes a tran-
nucleus (Z, A 1) will capture another neutron sition from protostar to star when its primary
before it beta decays: source of energy generation is fusion via the p“p
1Z, A 12 n S 1Z, A 22
chain, rather than gravitational collapse. The
changeover is not a sudden one, as the material
In either case, the resulting nucleus can closer to the center heats faster. Eventually,
either beta decay or capture a neutron, depend- enough energy is generated internally for the col-
ing on the relative rates. When we have a string of lapse to halt. When this happens, the star reaches
nuclei for which neutron capture is favorable, the a stable condition. It is on the main sequence.
r-process allows the buildup of neutron rich As nuclear reactions take place in the star, the
nuclei. This will go on until so many neutrons composition of the star is actually changing. This
are added that a beta decay breaks the chain. The change could affect the spectral type and lumi-
r- and s- processes can explain the abundances of nosity of the star, while the basic structure doesn™t
many of the heavier nuclei. (It should be noted change very much. These changes result in a main
that these are not equilibrium processes.) sequence that is a band on the HR diagram rather
The various nuclear processes that we have dis- than a thin line. However, there is a line that we
cussed are responsible for the presence of the heavy can identify as connecting the points on the HR
elements around us. We will see in later chapters diagram where stars of each spectral type first
how this material is spread into interstellar space. appear on the main sequence. We call this line the
The net result is to produce the abundances shown zero-age main sequence, or ZAMS. This line is shown
in Fig. 9.8. Nuclear physics determines which ele- relative to the main sequence band in Fig. 9.9.
ments are the most abundant.
When we discuss stellar structure in the next
section, we will treat the nuclear physics as some-
thing that is known. We assume that once we know
the composition of some region and the tempera-
Main Sequence
ture, we can specify which nuclear reactions are
important. Moreover, we assume that we know how
Absolute Magnitude

the reaction rates depend on temperature. This is a
very important point. We have already seen in this
Relative Abundance

108 +10

Spectral Type

Fig 9.9 Zero-age main sequence.The main sequence on
10 20 30 40 50 60 70 80 90 100
this HR diagram appears as a band, since stars on the main
Atomic Number (Z)
sequence become slightly brighter as their composition
Fig 9.8 Cosmic abundances of the elements as a function changes.The lower edge of this band represents the points
of atomic number Z. where stars ¬rst appear on the main sequence, the ZAMS.

Dividing both sides by dr gives us the rate at
9.4 Stellar structure which we add mass as we go farther out from the
center of the star:
4 r2 1r2
The basic philosophy of stellar structure studies
dM dr (9.33)
is that stars obey the laws of physics, so we
should be able to predict and explain their struc- This condition is called mass continuity, and
ture by applying those laws. To do this, we must simply tells us how the rate of change of M(r),
identify the basic physical processes that are mass interior to r, is related to the density at r.
important in stars, such as nuclear physics for M(r) is important because, for a spherical object,
energy generation. We must also be able to per- the gravitational force on an object a distance r
form large numbers of intricate calculations. from the center only depends on M(r).
This latter facility is provided by modern com-
9.4.1 Hydrostatic equilibrium
Once we carry out stellar structure calcula- The material in a shell of radius r is pulled toward
tions, we can compare the predictions of the the- the center of the star by the gravitational attrac-
ories with observations. For example, if we put tion of all the mass interior to that shell.
1 M of material into model calculations, we Something must support the matter in the shell
should come out with a star whose radius, tem- or else the star will collapse. That something is
perature and luminosity match those of the Sun. the pressure difference between the bottom of the
If we put in different amounts of material, shell and the top of the shell. This condition is
our calculations should reproduce the main called hydrostatic equilibrium (Fig. 9.10). Hydrostatic
sequence. We should also find stars in the same equilibrium applies to the Earth™s atmosphere as
mass range as those on the main sequence. well as in the oceans and in a glass of water. The
Stellar models should allow us to predict stellar weight of each layer of the fluid is supported by
evolution. They should also be able to tell us the pressure difference between the bottom and
how changes in composition lead to changes in the top.
structure. We can see how much of a pressure difference
Stars are easier to analyze than some other is needed by considering a small cylinder, of
astronomical objects because they have simple height dr and area dA, as shown in Fig. 9.10. The
shapes. They are spheres. We also assume that mass of the cylindrical element is
1r2 dr dA
their structure is spherically symmetric. That is,
dm (9.34)
the conditions “ temperature, density and com-
position “ only depend on distance from the cen-
ter, not the direction in which you are going away P(r+dr)dA
from the center. We can rotate the star through
any angle about any axis through the center and
not change the result. (This is not strictly true if
the star is rotating or has a strong magnetic
field.) To study a star we divide it into spherical
shells, each of thickness dr, as shown in Fig. 9.1. If
the density a distance r from the center is (r),
then the mass contained in a shell of radius r and r
thickness dr is

1r2 dV
dM (9.32a)
Fig 9.10 Hydrostatic equilibrium.The distance from the
4 r2 dr) is the volume of the shell.
where dV (
center of the star is r, and the thickness of the shell is dr.The
This gives
density in the shell is (r).We consider the forces on a cylin-
4 r2 1r2 dr
der of height dr and end area dA.The pressure at r is P(r).
dM (9.32b)

The gravitational force depends on M(r). In any both sides by dA, add dP to both sides, and then
particular model, we can find M(r) by integrating divide both sides by dr to give
3GM1r2 r2 4 1r2
equation (9.33):
dP dr (9.43)

1r¿ 2r¿ 2 dr¿
This is sometimes called the equation of hydrostatic
M1r2 4 (9.35)
equilibrium. We can rewrite it more simply by not-

ing that the quantity GM(r)/r2 is equal to the local
The gravitational force is given by
acceleration of gravity g(r), so
GM1r2 dm r2
FG (9.36)
1r2 g1r2
dP dr (9.44)
We use the minus sign ( ) to indicate that the
The equation of hydrostatic equilibrium tells
force is directed downward. Taking dm from
us that the denser the fluid is, the more rapidly P
equation (9.34) gives
changes with r. This is because a denser fluid
3GM1r2 r2 4 1r2 dr dA
FG (9.37) means a higher mass shell, and a stronger gravi-
tational force pulling it in. This requires a larger
We now look at the force exerted on the top
pressure difference between the top and the bot-
and bottom of the cylinder by the pressure of the
tom to support it. Also, the greater g(r) is, the
fluid. The difference between the upward force
greater the gravitational force is pulling the shell
on the bottom and the downward force on the
in. This means that a larger g(r) also requires a
top is called the buoyant force, FB. If P(r) is the pres-
faster rate of change in the pressure.
sure at the bottom of the cylinder and P(r dr) is
Example 9.4 Central pressure of the Sun
the pressure at the top, then
Use the equation of hydrostatic equilibrium to
FB P1r2 dA P1r dr2 dA estimate the central pressure of the Sun by consid-
3P1r2 dr2 4 dA
ering the whole Sun as one shell.
P1r (9.38)

We have used the fact that force is SOLUTION
pressure area. We define the pressure differ- If we consider a whole star to be one shell, then
ence between the top and the bottom, dP, as dR R, the radius of the star, and dP PC , the cen-
tral pressure (taking the pressure at the surface to
P1r P1r2
dP dr2 (9.39)
be zero). The equation of hydrostatic equilibrium
then gives
We see that the pressure will decrease as r
1GM R2 2
increases (since the pressure at the bottom of the PC R
shell must be greater than the pressure at the
where is the average density, and is approxi-
top), so dP is a negative number. Equation (9.38)
mately M/R3, so
then becomes
1GM R2 1M R3 2
5P1r2 3P1r2 dP 4 6 dA
GM2 R4
dP dA (9.40)
Substituting for the Sun gives
The condition for hydrostatic equilibrium is
16.67 dyn cm2>g2 2 12 1033 g 2 2
FG FB 0 (9.41)
1010 cm2 4
Substituting from equations (9.37) and (9.40) 1016 dyn cm2
The actual value, obtained from stellar models, is
3GM1r2 r2 4 1r2 dr dA dP dA 0 (9.42) about 20 times this value
Another equation that we use in modeling
We are interested in the rate at which the
stars is the equation of state. The state of a gas is
pressure changes with radius dP/dr, so we divide

described by the pressure, density and tempera- the energy entering any shell per second must
ture. The equation of state relates those three equal the energy leaving that shell per second.
quantities. We can write it in the general form If radiation transport dominates, we can calcu-
f 1 , T2
late the required temperature distribution, T(r). We
P (9.45)
let f(r) be the flux of radiation through a surface at
radius r. If the surface emits like a blackbody, then
The actual form of the function depends on the
nature of the gas. For an ideal gas, the equation of
T1r2 4
f1r2 (9.47)
state has the simple form
1 m2 kT
We can find the rate at which f (r) changes with
P (9.46)
where m is the mass per particle. For a gas of rel- 4 T1r2 3
df dT (9.48)
ativistic particles, the equation of state is differ-
If we interpret df as the small change in f due to
ent, and we will discuss it in Chapter 10.
a small change in T, dT, then
Example 9.5 Central temperature of the Sun
4 T1r2 3 dT
df (9.49)
Use the result of the previous example and the
equation of state for an ideal gas to estimate the
However, the change in flux passing through a
central temperature of the Sun.
given layer of the star must depend on the ability
of the layer to absorb radiation. In Section 6.2 we
saw that this is given by the absorption coeffi-
To use the ideal gas law, we need to estimate the cient . In stellar structure it is more convenient
density. We simply use the average density, which to deal with the absorption coefficient per den-
is the mass divided by the volume. Since the hydro- sity of material. We therefore let (r) be the opac-
gen is completely ionized, there are an equal num- ity per unit mass at r. This means that (r) (r)
ber of electrons and protons, so the average mass gives the fraction of radiation absorbed per cen-
per particle is (1/2)mp. timeter. Using these definitions
¿1r2 1r2f1r2 dr
mP df (9.50)
The minus sign ( ) tells us that the flux is
3 2 R3
mP14 decreased by the absorption.

10.5 2 11.67 g 2 11 1016 dyn cm2 2 14 3 2 17
1010 cm2 3
12 1033 g 2 11.38 erg K 2

107 K

This is a little larger than the model results, with
T ˜ 1.5 107 K. We define the luminosity of a given layer as
the flux f multiplied by the surface area of the
9.4.2 Energy transport
4 r2 f1r2
L1r2 (9.51)
In making a stellar model we must also consider
how energy gets from the inside of the star to the If we use equations (9.49) and (9.50) to eliminate
outside. In general, energy can be transported by df and equation (9.51) to eliminate f(r), we have
conduction, convection and radiation. In stellar
¿1r2 1r2 L1r2 dr
interiors conduction is not generally important. 4 T1r2 3 dT (9.52)
4 r2
The energy transport must be such that the tem-
perature T(r) does not change with time. If it did, Solving for the luminosity gives
the star would not be stable for the multibillion
16 r2 T1r2 3 dT
c d
year lifetime for stars like the Sun. For the tem-
¿1r2 1r2 dr
L1r2 (9.53)
perature distribution to be constant with time,

This tells us how the rate of energy flow and potential energies for bound systems, called
the virial theorem, which tells us that E K.
depends on the rate at which the temperature
changes with r. In general, dT/dr is negative (the Suppose we now add energy. This increases E,
making it less negative. This makes K less positive,
temperature drops with distance from the cen-
ter), so the luminosity is positive. A more exact so the total thermal kinetic energy decreases. The
calculation gives essentially the same result as gas cools. Therefore, if the star produces too
equation (9.53), but with the 16 replaced by 64/3. much energy for its equilibrium configuration, it
From equation (9.53) we can see that the opac- can expand and cool to adjust. The above argu-
ity per unit mass (r) is very important in deter- ment doesn™t apply to degenerate gases, since
mining the energy transfer, and therefore the their thermal energy is essentially independent
structure of the star. The opacity depends on the of the temperature. Therefore, as we encounter
composition of the star. Accurate stellar structure degenerate stars, or parts of stars, we will see that
calculations require good knowledge of the opac- explosions are possible.
ity as a function of composition and temperature.
In addition to energy transport, we must also
9.5 Stellar models
consider energy generation. If energy is gener-
ated in a particular shell, then the energy leaving
that shell will exceed the energy entering the In the previous section we saw a group of equa-
shell by the amount of energy generated. We let tions that describe the physics that governs stel-
(r) be the energy generated per unit mass within lar structure: (1) mass continuity, (2) hydrostatic
the shell at radius r. The increase in luminosity in equilibrium, (3) equation of state, (4) energy
that layer, due to energy generation, is then transport, and (5) energy generation.
4 r 2 1r2 (r) dr
The inputs to stellar models are the mass of
dL (9.54)
the star and the composition. We must also spec-
ify the nuclear physics, which gives the energy
The rate at which the luminosity changes
generation as a function of these conditions. We
with radius is then
4 r2 1r2 (r)
must also put in the information about the opac-
dL dr (9.55)
ities. For the purposes of calculation, we break
To carry out model calculations we must be the star into spherical shells. We must solve for
able to specify (r) as a function of composition the distribution of density (r) and temperature
and temperature. This is where the input from T(r) which satisfy the conditions imposed by the
nuclear physics is important. equations of stellar structure. This is such a com-
If, at any r, the radiative temperature gradient plicated process that realistic stellar models are
becomes too large, then convection will set in. all calculated on computers.
The quantitative effect of convection is discussed Stellar model calculations can also be used to
in Section 23.3.2. Convection is an adiabatic predict stellar evolution. As nuclear reactions
process, no energy is lost from parcels of material take place in the hot central core of the star, the
as they move outward. At that point the tempera- composition changes. This changes the energy
ture gradient becomes the one appropriate for generation rate and the opacity, meaning the
adiabatic processes. This is called the adiabatic structure will change. We do a model calculation
lapse rate. for the initial composition. We then determine
We might wonder why stars don™t explode the rate at which various nuclei are produced or
with all the energy they produce. The answer is destroyed. We then know what the composition
that their stability comes from their negative will be some time, say 50 000 years, later. We
heat capacity. (Heat capacity is the energy required now calculate a model with the altered composi-
to raise the temperature by a given amount.) To tion, and then repeat the process. We follow the
see this, we look at the total energy E U K evolution of the star in these time steps. We
(where K is now the kinetic energy of the thermal choose the time steps so that the composition
motions of particles in the gas). In Chapter 13, we changes somewhat, but not too much, during
will derive a relationship between total, kinetic each time step.

relatively low energy, and this makes it particu-
2.0 larly hard to detect.
There is, however, a source of higher energy
Log Density

1.0 neutrinos. Once 3He is formed, most of it reacts
to form 4He, as discussed in Section 9.2. However,

Log T
in a small fraction of cases, the following reac-
tion can take place:
’1.0 5.0
3 4
He S 7Be
He (9.56)
’2.0 4.0
The 7Be then captures a proton:
0 7
p S 8B
Be (9.57)
The boron then beta decays, emitting a neutrino
Fig 9.11 Temperature and density as a function of distance
in the process:
from the center of the Sun, as calculated from the solar
model which best agrees with the global properties (radius, 8
B S 8Be e (9.58)
surface temperature) of the Sun.
The 8Be then breaks apart into two alpha
To see the results of a model calculation, we 8
Be S 4He 4
He (9.59)
look at a model for the Sun. The distribution of
The neutrino emitted in the beta decay of the 8B
temperature and density is shown in Fig. 9.11.
has enough energy to provide us with some
chance of detecting it.
9.6 Solar neutrinos We can detect this neutrino using an isotope
of chlorine, 37Cl. About 25% of all naturally
Though our model for the Sun gives the correct occurring chlorine is in the form of this isotope.
radius and temperature, there are certain aspects When struck with a sufficiently high energy neu-
we cannot check directly. Almost all of the direct trino, it can absorb the neutrino:
information we receive from the Sun comes from 37
S 37Ar
Cl e (9.60)
photons emitted in the solar atmosphere. We can-
This particular isotope of argon is radioactive,
not directly observe photons that are emitted in
and its decay can be detected in normal particle
the nuclear reactions in the center. Those pho-
detectors in the laboratory. If we start with a tank
tons are quickly absorbed and their energy takes
about 107 years to reach the surface. We have no of chlorine (and no radioactive argon), and we end
up with a small amount of 37Ar, we can determine
direct observations of the solar core now.
There is one opportunity to make a direct the rate at which neutrinos were hitting the tank.
observation of the solar core. Neutrinos created This is the basic idea behind an experiment
in nuclear reactions in the core escape at the conducted by R. Davis of the Brookhaven National
speed of light, virtually unattenuated. They reach Laboratory. The source of liquid chlorine is per-
us 8.5 min after they are created. We could use chloroethylene (cleaning fluid). Since neutrinos
our stellar models to predict the rate at which interact infrequently, a large quantity of “detec-
tor” is used “ about 105 gallons. The argon is pro-
neutrinos are emitted, and then try to measure
duced as an inert gas, so it will form bubbles in
the flux of neutrinos at the Earth. This would be
the fluid, allowing it to be removed. The experi-
a direct test of the solar model. The problem is
ment is run for some period of time, typically a
that neutrinos are very hard to detect. If the
month. The gas is then collected, and is measured
whole Sun cannot absorb many neutrinos, a
to gauge 37Ar activity.
detector on Earth will absorb even fewer. The
Even if some radioactive argon is found, we
neutrino produced in the first reaction of the
proton“proton chain (p p ’ d e ) has a don™t know that it results from solar neutrinos.

There are also high energy particles in the Earth™s model. It comes from helioseismology (discussed
atmosphere, called cosmic rays, which can pro- in Chapter 6), in which the oscillations of the Sun
duce a similar result. The Earth can shield the are studied. The solar model (temperature, den-
tank from cosmic rays but not from neutrinos. sity and composition vs. radius) can be used to
The tank is therefore placed 1.5 km underground predict the details of these oscillations. The
in the Homestake gold mine, near Lead, South agreement between theory and experiment is
Dakota. Another source of possible contamina- very good, meaning that the solar model is prob-
tion is natural radioactivity from the rocks in the ably not the cause of the solar neutrino problem.
mine. For this reason, the tank is surrounded by It has been suggested that the Sun may go
a larger tank containing water. The water blocks through a cycle in its energy generation, and that
the high energy particles from radioactive decays right now it is generating less than the average
in the mine, but doesn™t block neutrinos. You can amount of energy. In this cycle, at some point the
see from this brief description that this experi- core cools, reducing the rate of nuclear reactions.
ment is a very difficult one. The pressure decreases and the core contracts. As
The results of the experiment from some 20 the core contracts, it converts gravitational
years observing have been astounding. It is con- potential energy into kinetic energy, and begins
venient to express the rate of detections in solar to heat. As the core becomes hotter the rate of
neutrino units or SNU. The standard model of the nuclear reactions increases. The pressure increases
Sun predicts that we should be measuring 8 SNU. and the core expands. The cycle then starts again.
The experiment, which detects about one event If this is the answer, then the neutrino observa-
every two days, yields 2.6 SNU, a number that is tions are giving us a good view of what is happen-
only about one-third of this value. This leads ing in the Sun now. In this picture, it is possible for
to what we call the solar neutrino problem. the cycle to take place with very little variation in
Astrophysicists have so much faith in our under- the solar luminosity. Since photons are scattered
standing of stellar interiors that such a large dis- many times before they can go from the center to
crepancy is an indication of a severe problem. the photosphere, the light from the photosphere
A number of solutions have been suggested. reflects the average energy production over the
One possibility is that there is something wrong Kelvin time. It should be pointed out that no
with the experiment itself. Possibly, the detector mechanism for such oscillations has been found.
is not as sensitive to neutrinos as originally It has been suggested that we don™t know as
thought. However, various aspects of the experi- much as we thought we did about neutrino
ment have been checked and refined over the physics. We now know that there are three differ-
years, and there is a general feeling that the ent neutrino types, each with its own antineu-
experiment is correct. trino. (We™ll discuss particle physics more in
If the experiment is correct, then the solar Chapter 21.) The type of neutrino that we have
model must be examined. It is possible that some been discussing is called the electron neutrino,
of the inputs are not correct. For example, the because it always appears in reactions with elec-
nuclear reaction rates have a very strong temper- trons (or positrons). This is the type of neutrino
ature dependence. If we have that dependence produced in the Sun, and the type that will inter-
act with 37Cl. However, some theories have sug-
slightly wrong, then significant changes can
gested that neutrinos can change their identities.
occur in the solar model. It is also possible that
If this is the case, then a neutrino can be created as
the opacities (as a function of composition and
an electron neutrino in the Sun, but change its
temperature) are slightly off. The solar neutrino
identity by the time it reaches the Earth.
problem has stimulated work in these areas. We
According to this idea, as few as one-third of the
now have improved nuclear physics data and
neutrinos produced in the Sun might be capable of
opacity data. A better solar model has been calcu-
being absorbed by the 37Cl. These identity changes,
lated, with a much smaller uncertainty, but the
or neutrino oscillations, are also related to the sug-
predicted neutrino flux remains essentially the
gestions that neutrinos have a very small (but not
same. There is an additional check on the solar

zero) mass. The experimental evidence for this is The second set of new experiments involves
being studied, and new experiments are underway. larger versions of traditional detectors used in ele-
In the meantime two new types of solar neu- mentary particle physics experiments. These
trino experiments are being done. The first involve large amounts of water as detectors, with
involves gallium as a detector. Since the 8B neu- each reaction producing a small flash of light, and
trino comes from a relatively unimportant that light is detected by an array of photomultipli-
branch in the Sun™s nuclear reaction chain, it ers. The first set of experiments was carried out by
may be that there is a small error in our calcula- a Japanese group at Kamiokande. The experiment
tions that just happens to be magnified for this was originally set up as a particle physics experi-
minor branch. There is another reaction that also ment, but it turned out that it was also capable of
produces a detectable neutrino. The reaction is detecting solar neutrinos. An important feature of
not as important as the proton“proton chain, but this detector is that it can measure the direction
it is more significant than the branch involving from which the neutrinos are arriving, and the
B. This reaction is called the p“e“p reaction: group were able to verify that the neutrinos are,
indeed, coming from the Sun (and not, for exam-
p e pSd (9.61)
ple, from some background contamination).
Following their original success, they built a sec-
It occurs once for every 400 direct p“p reac-
ond version, known as SuperKamiokande, more
tions. It is rare because it is much harder to bring
closely designed for observing solar neutrinos. The
three particles together at the same time than it
results of these observations is that observed flux
is to bring two particles together.
is about half the theoretical predictions. For their
The neutrino produced by the p“e“p reaction
is not as high in energy as the 8B decay neutrino, work on this problem, Davis and M. Koshiba, who
and is not absorbed by 37Cl. However, it has a headed the Japanese project, shared the 2002
Nobel Prize in Physics.
higher energy than the p“p neutrino. The p“e“p
At this point, it appears that the best solu-
neutrino can be absorbed by gallium. As with
tion to the solar neutrino problem involves the
chlorine, large quantities of gallium are needed
neutrinos. If they have a very small mass, and
for the experiments. Two such experiments are
can change identities, then the theory and
being carried out, a European“Israeli“USA collab-
experiment can be brought into agreement. In
oration known as GALLEX (30 tons of gallium),
2002, a group in Sudbury, Canada (Sudbury
and a Russian“USA collaboration known as SAFE
Neutrino Observatory, SNO, shown in Fig. 9.12),
(60 tons of gallium). These experiments yield
provided strong evidence that neutrino oscilla-
about one event per day. The measured rates are
tions are the answer. An analysis of their results
about half of the predicted values.

Fig 9.12 The large detector of
the Sudbury (Canada) Neutrino
Observatory (SNO). [Photo
courtesy of Sudbury Neutrino

suggests that the neutrino mass is in the 10 2 eV nos created in the Sun to change their identity by
the time they reach our detectors. The next major
range. (Remember, for comparison, the mass of
the proton is almost 109 eV.) The analysis also test of these ideas would be an experiment to
detect directly the p“p neutrinos (by far the most
shows that there is a significant chance (of the
abundant, and lowest energy).
order of 50% with a large uncertainty) for neutri-

Chapter summary

In this chapter we looked at the processes respon- supported by the pressure difference between its
sible for the structure of main sequence stars. bottom and top. We also saw that the temperature
We started by looking at energy sources. distribution is determined by the requirement
Nuclear reactions are the only source capable of that the temperature of each layer be constant.
giving stars their inferred lifetimes. We saw that Once the basic laws of stellar structure are
the temperatures required for nuclear reactions outlined, stellar models can be computed, gener-
to take place are in excess of 107 K, even with tun- ally using computers. In a model, we start with a
neling to help bring the nuclei together. certain mass and composition, and calculate the
The basic source of energy on the main equilibrium configuration.
sequence is the conversion of hydrogen to We saw how much of stellar structure seems
helium. In low mass stars this takes place prima- to be understood, but encountered the puzzle of
rily via the proton“proton chain. In more massive the solar neutrino experiment. The neutrinos
stars, with higher central temperatures, other allow us to see what the core of the Sun is doing
cycles, such as the CNO cycle, are important. now, and it does not appear to be doing as much
We also looked at the basic processes that as models predict. The answer to this problem
govern stellar structure. We saw that normal stars seems to lie in neutrons changing their identity
are in hydrostatic equilibrium, with each layer in their trip from the Sun.

9.1. (a) Why is 1 eV/atom a reasonable estimate 9.7. Why are the rates of certain fusion reactions
for the energy available in chemical reac- very sensitive to temperature?
tions? (b) Is the estimate likely to be wrong by 9.8. Why are the r- and s-processes important?
more than a factor of ten in either direction? 9.9. (a) What are the parameters that we put into
Explain. (c) If the estimate is wrong by a fac- a stellar model? (b) What calculations do we
tor of ten in either direction, will it change perform? (c) How do we test the results?
the conclusion that the Sun cannot exist on 9.10. (a) What do we mean by spherical symmetry?
chemical reactions? (b) Why will a rotating star not be spherically
9.2. It has been said that if we did not know that symmetric?
E mc2, then we would not know about *9.11 In equation (9.26) we have used the form of
nuclear energy. Discuss this. the gravitational force between two point par-
ticles. These particles have masses M(r) and dm
9.3. Explain the factors that place upper and lower
and are a distance r apart. However, M(r) rep-
limits on the number of neutrons that go into a
nucleus with some specific number of protons. resents an extended mass and dm represents a
9.4. Why can™t a free proton beta decay into a shell, so neither represents a point. How can
neutron? we use the simple formula? (Hint: Treat the
justification for M(r) and dm separately.)
9.5. What are the similarities between gamma
emission by nuclei and visible light emission *9.12 What effect would a slight increase in opac-
by atoms? ity at all layers have on the structure of a
9.6. Why is there no Coulomb barrier to fission if star?
there is one to fusion? 9.13 Explain how we simulate stellar evolution.

9.14 Why is the solar neutrino problem so collapse as a stellar energy source. Why isn™t
important? this a contradiction?
9.15 When we considered explanations of the *9.16 The equilibrium structure of a star is
solar neutrino experiment, we said that the ultimately determined by its mass and com-
Sun may be generating some energy now position. Show that the structure of a star
through gravitational collapse. However, ear- determines the rate of energy generation and
lier in the chapter, we ruled out gravitational not the other way around.

*9.10. Suppose we have Z protons and have to distrib-
9.1. What is the gravitational potential energy of
(a) the Sun, (b) a 1 M black hole? ute them into two nuclei, one with Z1 protons
and the other with Z2 protons (Z Z1 Z2).
9.2. (a) What is the gravitational potential energy
of an interstellar cloud, with a density of 1000 (a) What arrangements give the maximum
H atoms/cm3 and radius of 10 pc? (b) What is and minimum Coulomb repulsion between
its kinetic energy if its temperature is 10 K? the two nuclei? (b) What does this tell you
*9.3. Find the gravitational potential energy of a about the types of fusion that are most likely
sphere with a 1/r2 density distribution. Take to take place in stars?
the total mass of the sphere to be M, and let 9.11. (a) How close can two protons get if one is at
0/r out to a radius, R.
the density (r) rest and the other has a kinetic energy equal
to the average energy at T 107 K? (b) What is
Express your final answer in terms of M and R.
9.4. Estimate the lifetime of a 10 M star on the the wavelength of the moving proton, and
main sequence to give off energy stored from how does it compare with the minimum sepa-
gravitational collapse. ration between the two protons? (c) Repeat
9.5. Calculate the mass corresponding to the bind- the calculations for a proton with ten times
ing energy of an H atom. What fraction of the the average energy at this temperature.
mass of the atom is this? *9.12. Suppose the density of a star is given by
9.6. What is the rate at which the Sun is convert-
r 6 r0
1r2 • 0 1r0
ing mass to energy? 2
r2 r0 6 r 6 R
9.7. (a) What is the difference in mass between
the neutron and the proton, expressed in
MeV? (b) How does this relate to the energy (a) Find an expression for M(r). (b) If the mass
available in the beta decay of a neutron? of the star is 1 M and R R , and r0 0.1 R ,
9.8. Calculate the binding energy of a 4He nucleus. what is the value of 0?
9.9. How much energy per proton is given off in 9.13. For the density distribution in the previous
the p“p chain? (Express your answer in MeV.) problem, find P(r).

Computer problems

9.1. Calculate the gravitational potential energies for is in the core and available for nuclear reactions.
stars in the middle of each spectral class on the (Hint: Use the mass“luminosity relationship.) Plot
main sequence (O5, B5, . . .). your results for the range of masses encountered
9.2. Write a routine that will calculate the gravita- on the main sequence.
tional potential energy for uniform density ellipti- 9.4. Make a table showing the Kelvin time for stars in
cal objects which have two axes the same. Evaluate the mid-range of each spectral type (O5, B5, . . .).
the potential energy for the two following cases, 9.5. Make a graph of the electrical potential energy
both clouds with 100M of material: (a) an oblate between two protons as a function of their separa-
tion, r. Let r range from the size of a nucleus to ten
(flattened) ellipsoid with semi-axes 10, 10 and 5 pc;
(b) a prolate (elongated) ellipsoid with semi-axes 5, times that.
5, and 10 pc.
9.3. Estimate the nuclear reaction lifetime of a star as a
function of its mass. Assume that 10% of the mass
Chapter 10

Stellar old age

We have already seen that the mass of a star is the this takes place in a shell at the outer edge of the
core (Fig. 10.1). We refer to this as a hydrogen-
most important property in determining a star™s
burning shell, where the word “burning” refers to
structure. For a main sequence star the mass
determines the size and temperature. The life- nuclear reactions, rather then chemical burning.
time of a star on the main sequence depends on As the core contracts, the rate of energy genera-
the available fuel and the rate at which that fuel tion in the shell increases. The shell can easily
is being consumed “ the luminosity. Both of these give off energy at a greater rate than the core did
quantities depend on the star™s mass, so the life- during the star™s normal lifetime.
time on the main sequence also depends on the While all of this is happening in the interior,
mass. When the star uses up its basic supply of the outer layers of the star are changing. Energy
fuel, its ultimate fate also depends on its mass. In transport from the core is radiative, and is limited
fact, the mass and the initial composition of a by the rate at which photons can diffuse through
star completely determine its structure and evo- the star. The outer layers of the star become hotter
lution. This can be proven mathematically on the and expand. As the gas expands, it cools. The star™s
basis of the physical equations involved. This radius has increased, but its temperature has
result is known as Vogt™s theorem. decreased, so the luminosity increases slightly.
The behavior of the star™s track on the HR diagram
is shown in Fig. 10.2. The track moves to the right
10.1 Evolution off the main (cooler), and the star appears as a subgiant.
sequence There is a mechanism that keeps the surface
temperature from becoming too low. The rate of
10.1.1 Low mass stars photon diffusion increases as the absolute value
We first look at stars whose mass is less than of dT/dr increases. Remember, dT/dr is negative,
about 5 M . Eventually a star will reach the point so we are saying that the greater the temperature
where all the hydrogen in the core has been con- difference between some point on the inside and
verted to helium. For a low mass star, the central the surface, the greater the energy flow between
temperature will not be high enough for the those two points. (In winter, the larger tempera-
helium to fuse into heavier elements. There is ture difference between the inside and outside of
still a lot of hydrogen outside the core, but the your house results in a faster heat loss, and higher
temperature is not high enough for nuclear reac- fuel bills.) If the surface temperature of the star
tions to take place. The core begins to contract, falls too much, the photon diffusion is faster,
converting gravitational potential energy into delivering more energy to the surface, raising
kinetic energy, resulting in a heating of the core. the surface temperature. Therefore, as the radius
The hydrogen just outside the core is heated to continues to increase, the surface temperature
the point where it can fuse to form helium, and remains approximately constant. The luminosity

H Envelope H Envelope
106 30M

104 15M
He 9M

102 3M

Regions of
Nuclear 1M
Fig 10.1. Star with an H-burning shell. (a) The temperature
in the star is not hot enough to fuse the helium in the
center, but is hot enough to keep the H in the shell burning.
104 104 103
3 3
(b) In this star, the temperature is hot enough to keep both
burning. (Remember, by “burning” we are talking about
nuclear reactions.) Fig 10.2. Evolutionary tracks away from the main
sequence on an HR diagram. Each track is marked by the
mass for the model.The dashed line is the zero-age main
therefore increases, and the evolutionary track
sequence (ZAMS).
moves vertically. The star is then a red giant.
By the time the star becomes a red giant, the
gas. We will discuss degenerate gases in Section
energy transport in the envelope is convective.
10.4, but for now we note that the equation of
This is because of the large value of dT/dr. The
state is very different for a degenerate gas. In an
analogous situation on Earth involves the heating
ideal gas, when the triple-alpha process starts, the
of the atmosphere. Sunlight heats the ground,
extra energy generated causes an increase in pres-
and then infrared radiation from the ground heats
sure, which causes the gas to expand, slowing the
the air. (This explains why the air is cooler at high
reaction rate. This keeps the reactions going
altitudes; it is farther from the direct heat source,
slowly. In a degenerate gas the pressure doesn™t
the ground.) In this situation, we say that the
depend on temperature and no such safety valve
energy transport is radiative. However, if dT/dr
exists. The conversion of helium to carbon takes
becomes larger, then dP/dr, the rate at which the
place very quickly. We call this sudden release of
pressure falls, also becomes large. The air that is
energy the helium flash. The energy released causes
heated near the ground expands slightly, and
a brief increase in stellar luminosity.
becomes very buoyant, being driven upward by
Following the helium flash the energy produc-
the pressure difference between the bottom and
tion decreases. The core is no longer degenerate,
top of any parcel of air. The hot air rising, being
and steady fusion of helium to carbon takes place.
replaced by cool air falling, known as convection,
This region is surrounded by a shell in which
becomes the dominant mode of energy transport.
hydrogen is still being converted into helium. At
We now look at the evolution of the core while
this point the star reaches the horizontal branch
the star is becoming a red giant. The temperature
of the core climbs to 108 K. This is hot enough for on the HR diagram. The outer layers of the star
are weakly held to the star, since they are so far
the triple-alpha process to take place (equations
from the center. The star begins to undergo mass
9.11 and 9.12), fusing the helium into carbon. The
loss. The subsequent evolution depends on the
density is so high that the material no longer
amount of mass that is lost.
behaves like an ideal gas. This is called a degenerate

Eventually all the helium in the core is con-
verted into carbon and oxygen. The temperature (Not to Scale)
is not high enough for further fusion, and the
103 R
core again begins to contract. A helium-burning
shell develops, and the rate of energy production Core (10-2 R )
again increases. The envelope of the star again
expands. On the HR diagram the evolutionary
track ascends the giant branch again, reaching
what is called the asymptotic giant branch. Stars on
H Burning
the asymptotic giant branch are more luminous
than red giants. The star can briefly become large He Burning
enough to become a red supergiant at this stage. C Burning
The star can also undergo oscillations in the rate Ne Burning
of nuclear energy generation. O Burning
Si Burning
10.1.2 High mass stars
Fe Burning
More massive stars live a shorter lifetime on the
main sequence than do lower mass stars. As with
the lower mass stars, the main sequence lifetime
for higher mass stars ends when the hydrogen in
the core is used up. The core then begins to con-
tract, and the temperature for helium fusion to Fig 10.3. Shells in the core of a high mass star as it
heavier elements is quickly reached. The helium evolves away from the main sequence. (a) The core is only a
fusion takes place before the core can become small fraction of the total radius. (b) In the core, there is a
succession of shells of different composition. Each shell has
degenerate. Therefore, in contrast with the helium
exhausted the fuels that are still burning in shells farther out.
flash in lower mass stars, the helium burning in
more massive stars takes place steadily. At this
point, the star has a helium-burning core with a
10.2 Cepheid variables
hydrogen-burning shell around it (Fig. 10.3).
When the helium in the core is exhausted, the
10.2.1 Variable stars
temperature is high enough for the carbon and
oxygen to fuse into even heavier elements. At this If we monitor the brightnesses of certain stars,
time, we have a carbon- and oxygen-burning core, we find that many oscillate with time. These are
known as variable stars. The periods of variability
surrounded by a helium-burning shell, which in
turn is surrounded by a hydrogen-burning shell. range from seconds to years. We have already
As heavier elements are built up, the core devel- seen that eclipsing binaries appear as variables.
ops more layers. However, many stars have luminosity variations
As the luminosity of the core increases, the associated with physical changes in the stars them-
outer layers of the star expand. The atmosphere selves (rather than simply by eclipsing one another).
cools with the expansion, but the size increases Since we will be using specific stars as exam-
sufficiently for the luminosity to increase. At ples, we will briefly mention systems for naming
this point the envelope is convective, and the normal and variable stars. The bright stars are
temperature gradient is limited by the adia- named, in order of brightness within their con-
batic lapse rate. So the envelope must grow to a stellation, by a Greek letter, followed by the Latin
large size to accommodate the large tempera- genitive form of the constellation name. An exam-
ture difference between the core and the sur- ple is Orionis (abbreviated as Ori). Some of the
face. Eventually, the radius of the star reaches brightest stars are also known by their ancient
about 103 R . At this point the star is called a names. For example, Ori is Betelgeuse. Variable
red supergiant. stars are listed in order of discovery within a

given constellation. The first is designated R (e.g. Temperature (K)
50,000 10,000 5,000 3,500
R Ori), the next S, and so on to Z. After that, two let-
Classical cepheids
ters are used, starting with RR, RS to RZ, then SS to Ia
5 10
SZ, and so on, until ZZ is reached. Then comes AA
W Virginis stars
through AZ, BB through BZ, and so on to QZ. (The
letter J is never used because of possible confusion III

Absolute visual magnitude
0 10
with I.) This gives a total of 334 variable stars per

ue RR Lyrae stars
constellation. Beyond that, numbers starting with e
335, preceded by a V (for variable), are used (e.g.
V335 Ori, V336 Ori, etc.) v
For any particular star, we are interested in pro- 2

ducing a light curve, a graph of its magnitude as a
function of time. Studies of variable stars often 4
require very long term monitoring. In some cases,
it is possible to recover information on a star™s vari- Spectral type
ability from plate archives. When photographic
Fig 10.4. HR diagram, showing the locations of various
plates are taken at an observatory, the astronomer
types of variable stars.
who took them is often required to return the
plates when that astronomer™s work has been com-
longer, and are called long period variables. Any indi-
pleted. The astronomer may be interested in only
vidual Mira variable may show fluctuations in its
one star on the plate, but it contains a record of
period. The stars change their brightness by about
many stars. With the advent of CCD observations,
6 mag, or over a factor of 250 in brightness. For
archives are no longer being kept in the same man-
example, the apparent magnitude of Mira ranges
ner. Observations of many variable stars can be so
from 9 mag to 3 mag. These changes in brightness
time consuming that it has become an area of
are accompanied by changes in spectral type. Mira
astronomy where amateur observers have been
changes from M9 to M5. This means that a tem-
able to make major contributions, generally coor-
perature change is accompanying the luminosity
dinated by the American Association of Variable
Star Observers (AAVSO). (In measuring light curves,
Cepheid variables are named after the prototype
we often measure time in Julian days, the number
of days since noon on January 1 4713 BC, or modi- Cephei. Its period is 5.4 days and its apparent
fied Julian days, the number of days since the magnitude varies from 3.6 to 4.3 mag. In general,
beginning of the Besselian year 1950 (see Appendix Cepheids have periods from 1 to 100 days. We know
F for a further discussion of timekeeping). of more than 1000 in our galaxy. Another familiar
We distinguish different types of variable stars Cepheid is Polaris (the North Star), which changes
by such things as their period and the magnitude by only 0.1 mag, from 2.5 to 2.6. (For a long time,
range. A particular class of variable is generally astronomers did not know that Polaris is a variable,
named after the prototype of the class, either the and it was used as a reference for measuring the
first or most prominent star with the distinguish- magnitudes of other stars.) Cepheids have masses
of approximately 6 M , and radii of about 25 R .
ing properties of the class. In this section, we look
at a few examples of the most important types of
10.2.2 Cepheid mechanism
variables. Different types of variables appear in
When we study the spectral lines in Cepheids, we
different parts of the HR diagram, as shown in Fig.
can detect Doppler shifts that vary throughout the
10.4. These bright stars were named before their
light cycle (Fig. 10.5). The Doppler shifts go through
variable nature was known, so they do not follow
a cycle in the same period as the light. This means
the naming convention discussed above.
Mira variables are named after the prototype that the surface of the star is moving. The size of
the star changes as the luminosity changes. The
(a star also known as O Ceti). These stars have
spectral type also changes throughout the cycle.
periods of about three months to two years, or even

ceding section. To see how a Cepheid oscillates,
lets consider the oscillations of a normal star.
These oscillations are radial. They involve inward
and outward motions of the outer layers of the

star. Suppose we are able to perturb a star by
decreasing its radius R. The density then increases,
and the pressure increases. The excess pressure
will make the outer layers expand back. However,
(a) just as a swing overshoots its lowest point as it
returns from its maximum height, the star can
7000 overshoot its equilibrium radius R0. Now the star
is larger than its equilibrium radius, and the
pressure decreases, allowing material to fall back.

This process then repeats itself.
In the above analysis, we have ignored the
effects of opacity. In a normal star, the opacity
decreases as the temperature increases. Now, we
again start the perturbation by reducing R below
R0. This causes P and T to increase. The increase in
T decreases the opacity. The reduction of the opac-
ity allows some of the excess pressure to be relieved
R Rmin

by allowing heat to flow out of the denser regions
as radiation. This reduces the tendency of the star
to overshoot. If we had started with a perturbation
in which R R0, P and T would have decreased. The
opacity would have increased, and the tendency to
fall back too fast would be reduced. The result of
the opacity is to quench the oscillation.
For a narrow range of conditions, the opacity
vr (km s)

increases as the temperature increases. The source
of the opacity is the ionization of He to form
He . If we now start with a perturbation in which
R R0, the pressure and temperature increase. Now
0 1 2 3 4 5 6 7 8
the opacity also increases, so the excess pressure
Time (days)
is not relieved, except by driving the star back.
(d) The tendency to overshoot is enhanced. Similarly,
with R R0, the pressure and temperature
Fig 10.5. Radial velocity and light curves for Cep, the
decrease. The opacity also decreases, reducing
prototypical Cepheid. (a) Apparent magnitude as a function
the pressure even further. The material falls back
of time within the period. (b) Temperature as a function of
quickly and overshoots. This oscillation can con-
time. (c) Radius, relative to the minimum radius, as a function
of time. (d) Radial velocity of the surface as a function of tinue indefinitely, rather than being quenched.
time. Note that the radial velocity is one quarter cycle (90 ) These are the conditions that produce a Cepheid
out of phase with the radius.

10.2.3 Period“luminosity relation
The luminosity change is then associated with
An important feature of Cepheids is that they pro-
changes in the surface temperature and in the
vide us with a method of measuring distances.
The method involves a period“luminosity relation
A star may become a Cepheid variable when it
(Fig. 10.6). This relation was discovered by Henrietta
reaches the stage described at the end of the pre-

P“L relation

Relative number


type I

2 Type II
RR lyrae
1 10
1 10 100
Period (days)
Period (days)
Fig 10.7. Distribution of periods for Cepheids. Note that
Fig 10.6. Period“luminosity relation for Cepheids.
there are two distinct groupings.

Leavitt, who was studying Cepheids in the Large
an interesting result. The distribution has two
and Small Magellanic Clouds, two small galaxies
peaks in it. This suggests that there are actually
near the Milky Way. The advantage of studying
two different types of Cepheids. The group with
Cepheids in either of these galaxies is that the
the shorter periods are typical of those studied in
Cepheids are all at the same distance. For the
the Magellanic Clouds, and are called classical
Small Magellanic Cloud it was found that there is
Cepheids. The group with longer periods are called
a relationship between the period of the Cepheid
type II Cepheids or W Virginis stars (named after their
and its mean apparent magnitude. Since all of
prototype). Type II Cepheids are found in globular
the stars are at essentially the same distance, this
clusters in our galaxy (see Chapter 13 for a discus-
means that there is a relationship between the
sion of clusters). In general, a type II Cepheid is
period and the mean absolute magnitude.
1.5 mag fainter than a classical Cepheid of the
If we know the exact relationship between
same period. Also, the period“luminosity relation
period and absolute magnitude, then, when we
is slightly different for the two types.
observe a Cepheid, we can measure its period and
The original calibration of the Cepheid dis-
convert that into an absolute magnitude. We can
tance scale was carried out for type II Cepheids,
always measure the apparent magnitude. The dif-
since we can study them in our galaxy. However,
ference m “ M is the distance modulus, and gives
when we look at a distant galaxy, we can more
us the distance. This technique is important
easily study the brighter classical Cepheids.
because Cepheids are bright enough to be seen in
Therefore, the Cepheids studied in other galaxies
other galaxies, providing us with distances to
were 1.5 mag brighter than assumed. This means
those galaxies.
that the galaxies are farther away than originally
However, before we can use the period“
luminosity relationship, it must be calibrated. We
need independent methods of measuring dis- Example 10.1 Cepheid distance scale
tances to some Cepheids. This is difficult, since By how much does the calculated distance to a
there are none nearby. Statistical studies have been galaxy change when we realize that we are looking
used to achieve this calibration. More recently, the at classical, rather than type II, Cepheids?
Hiparcos satellite, which was designed to provide
more accurate trigonometric parallaxes than had SOLUTION
previously been available, made great strides on We have already seen that the Cepheids originally
this problem. studied are 1.5 mag brighter than assumed. This
If we plot a histogram indicating how many increases the distance modulus, m ” M, by 1.5 mag.
Cepheids have various periods (Fig. 10.7), we find By equation (2.18), this increases the distance by a

factor of 10(1.5/5) 2. Thus, these galaxies are twice being driven away. The actual mechanism for driv-
ing material away is still not fully understood. It may
as far away as originally thought. The difference
involve pressure waves moving radially outward. It
between the two types of Cepheids was realized in
may also involve radiation pressure. Photons carry
the 1950s, and people talked about the size of the
universe doubling.
Another type of variable star that is useful in
distance determinations is the RR Lyrae variable.
These are found in globular clusters and are
sometimes called cluster variables. They have short
periods, generally less than one day. The absolute
magnitudes of all RR Lyrae stars are very close to
zero. Actually, they fall between zero and unity,
and obey a weak period“luminosity relation of
their own. The absolute magnitudes were estab-
lished by using clusters whose distances were
known from other techniques. Once the absolute
magnitudes are calibrated, we can use RR Lyrae
stars as distance indicators.
It should not be surprising that stars with
pulsations have period“luminosity relations.
For radial oscillations, we expect the period to
be roughly equal to (G ) 1/2, where is the aver-
age density of the star. We can understand this
qualitatively by noting that a star pulsating
under its own gravity is like a large pendulum.
The period of a pendulum is 2 (L/g)1/2. For a star,
GM/R2, so the period is approxi-
L R and g
mately (GM/R3) 1/2, and M/R3 is approximately the
density. Therefore, since the period is related to
(which is approximately M/R3), and the luminos-
ity is related to the radius, the period should be
related to the luminosity (see Problem 10.4).
(G ) 1/2 is also approximately the period of a satel-
lite orbiting near the surface of a mass M, or the
period of a small mass dropped through a hole in
a larger mass. In short, if gravity dominates,
(G ) 1/2 is the time scale.

10.3 Planetary nebulae
We have already said that the outer layers of a red Fig 10.8. Images of planetary nebulae. (a) HST image of
giant are held together very weakly. Remember, the Ring Nebula (M57), in the constellation Lyra. It is at a
the gravitational force on a mass m in the outer distance of 1 kpc, and is about 0.3 pc across.This image
layer is GmM/R2, where M is the mass of the star and reveals elongated dark clumps of material at the edge of the
nebula. (b) The Dumbbell Nebula (M27), in a ground-based
R is its radius. As the star expands, M stays con-
image.This is 300 pc away and is 0.5 pc across, in the con-
stant, so the pull on the outer layer falls off as 1/R2.
stellation of Vulpecula.
Since the outer layer is weakly held, it is subject to





energy and momentum. (Remember, the momen-
Fig 10.8. (Continued) (c) HST image of the Helix Nebula,
which is a little more than 100 pc away. Notice the dark tum of a photon of energy E is E/c.) When photons
knots with glowing edges.These knots may be the result of from inside the star strike the gas in the outer lay-
faster clumps overtaking the main part of the nebula. (d) The
ers, and are absorbed, their momentum is also
Hourglass Nebula, in an HST image.This is approximately
absorbed. By conservation of momentum, the shell
2.5 kpc away.This picture was from three separate images,
must move slightly outward.
taken in the light of ionized nitrogen (represented in red),
We do observe shells that are ejected. They are
hydrogen (green) and doubly ionized oxygen (blue). (e) HST
fuzzy in appearance in small telescopes, just like
image of NGC 3132, at a distance of 0.8 kpc with a diameter
planets; when originally observed, they were
of 200 pc.The gases are expanding from the central star at a
called planetary nebulae (Fig. 10.8). (Their name has
speed of 15 km/s. (f) HST image of the Egg Nebula (CRL
2688) which is about 0.1 pc across, and is believed to be a nothing to do with their properties, but with
star making the transition to a planetary nebula (a proto- their appearance as viewed with small telescopes.)
planetary nebula). [(a), (c)“(f ) STScI/NASA; (b) ESO]
From the photograph in Fig. 10.8, we see that

rial than the line of sight through the center (Fig.
10.9). Thus, the center appears to be quite faint.
V When we look at spectral lines in planetary nebu-
lae, we see two Doppler shifts. One line is red-
shifted and the other is blueshifted. The
blueshifted one comes from the part of the shell
that is moving towards us, and the redshifted line
comes from the part of the shell that is moving


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