3. The Sun, Inside and Out
The Sun, Inside and Out
The Sun is not a typical star for at least three reasons. First, it
is a G2 spectral type star.1 Second, it is not part of a multiplestar
system, and third, we live on a planet that orbits around
it. The fact that we orbit a star such as the Sun (Figure 3.1) is
perhaps not especially surprising since, as discussed in Chapter
1, the Weak Anthropic Principle stipulates that the time for us
to evolve TUS must satisfy the inequality TUS = 4.5 x 109 < TMS,
the Sun’s main-sequence lifetime. The greatest mass that our Sun
might possibly have had would correspond to the condition TMS =
TUS, which implies a mass no greater than about 1.3 M. Any star
with a mass larger than this limit will have TMS < TUS. We will
discuss the relationship between the parent star and the habitable
planetary zone, where life can exist, in more detail in the next
chapter.
The most common type of star is smaller, less massive,
less hot at the surface, and of lower luminosity than the Sun.
These are the M dwarf stars,1 which constitute some 75 percent
of all the stars within our galaxy. The G2 spectral-type stars,
such as our Sun, make up about 8 percent of the entire stellar
inventory. In addition, the available data on star companionship
suggests that of order 80 percent of stars reside within multiple
systems so the Sun, being a single star, is in the minority group.
The Sun is certainly unique in that it is the only known star
to be supporting an evolving system of life forms. Of the other
billions of Sun-like stars in the Milky Way galaxy, the number that
might be harboring extraterrestrial life (intelligent or otherwise) is
completely unknown (recall Drake’s Equation in Chapter 1). We
will pick up the discussion on extrasolar planetary systems2 again
in Chapter 4. For this chapter, however, we will look more closely
at the internal workings of the Sun. What makes the Sun tick and
why does it puff up into a luminous giant with increasing age?
Indeed, by fully understanding the canonical structure of the Sun
71
72 Rejuvenating the Sun and Avoiding Other Global Catastrophes
Figure 3.1. The Sun in ultraviolet light. This image from the SOHO satellite
observatory shows a massive solar prominence (upper right) that has been
produced by the lifting of hot plasma above the Sun’s surface by twisted
magnetic fields. (Image courtesy of ESA/NASA)
and how it changes with time, our solar-rejuvenating descendants
can attempt to devise methods that will stop its gigantism in its
tracks.
Star Basics
In a number of ways stars are very simple objects: they are massive,
hot-gas spheres with a long-lasting internal energy source. This
being said, there are many subtle complexities associated with the
internal workings of a star and the manner in which its physical
properties change with time. This chapter will not present the
differential equations that describe stellar interiors nor describe in
detail the physics associated with stellar evolution.3 The intention
here is to provide a straightforward order of magnitude overview of
the physical properties of stars and tease out the main relationships
between key quantities.
The Sun, Inside and Out 73
Dynamical
equilibrium Material body
Radiation
Absorption &
reemission
Energy
supply
‘Visible’ star
Figure 3.2. Schematic diagram of Eddington’s two-component star model.
The material body is made of electrons, ions, and atoms, and the radiative
body is composed of electromagnetic radiation. Neither body would be
stable for long if it were not linked to the other via atomic interactions.
In the book The Internal Constitution of the Stars, published
in 1926, Arthur Eddington4 described a star as being two bodies,
superimposed (Figure 3.2). In this manner a star may be thought
of as a material body (the atoms, protons, neutrons, and electrons)
that make up its physical mass, and a radiative body composed
of ‘ethereal’ photons (electromagnetic radiation). The two bodies –
the material and the radiative – are symbiotic structures, and one
without the other would soon become something that is not a
star. The material body must be stable enough to prevent gravitational
collapse, and the radiative body must be constrained so that
its constituent photons do not leak out of the material body too
quickly. For example, the time for a photon to travel unimpeded
from the center of the Sun to its surface is TUP = R/c = 3.2
seconds. Yet, as we shall explain below, the actual photon dwell
time within the Sun is of order 5 x 1012 times larger than TUP and
this, it turns out, is essential if the material body is to remain
stable against gravitational collapse.
The Dynamical Timescale
A fundamental observation concerning the Sun is that it is neither
significantly expanding nor significantly contracting with time.
This simple observation tells us that the Sun must be stable against
74 Rejuvenating the Sun and Avoiding Other Global Catastrophes
gravitational collapse and that it has a long-lasting internal energy
supply. To better understand this situation, we can ask what the
dynamic collapse time (TDYN ) of the Sun would be if all of the
forces opposing gravity suddenly vanished. To answer this consider
a small mass mblob of material at the Sun’s surface (Figure 3.3).
The gravitational interaction between this blob of material and
the rest of the Sun will be FGrav = G M mblob/R
2, where G is
the universal gravitational constant. Now, Newton’s second law
of motion tells us that this force will cause the blob of material
to accelerate towards the Sun’s center such that F = mblob ablob
= FGrav, where ablob is the acceleration. Since the characteristic
time of the collapse is TDYN we can describe the acceleration as
ablob ≈ (change in speed/TDYN ≈ (change in distance/TDYN )/TDYN
= R/(TDYN )
2. Substituting this expression for ablob into Newton’s
second law formula reveals the characteristic collapse time:
TDYN = R3
/GM
1/2 (3.1)
which, upon substitution for the Sun’s radius and mass reveals
TDYN = 26.6 minutes—yes, minutes! If the material body of the
Sun were not held in check by forces opposing gravity it would
collapse into a black hole in about half an hour. Clearly this has
FP
‘Blob’ of mass m
FG
Radius = R*
Star of mass M*
Figure 3.3. The dynamic collapse time is a measure of how long it would
take an isolated blob of gas at the Sun’s surface to travel to the center
under unimpeded gravitational collapse. The unimpeded collapse of the
Sun would produce a black hole with an event horizon some 6-km across
in about half an hour.
The Sun, Inside and Out 75
not come about in the 4.56 billion years since the Sun formed, and
the question now is why.
The short dynamical time does not mean that a star cannot
change with time; it simply asserts a minimum time for any
change. One can think of dynamical time as the sound crossing
time of a star—this being the time required to ‘communicate’
across the star’s diameter that a change has actually taken place.
Hydrostatic Equilibrium
The size of a star is determined by a number of competing forces.
The gravitational force FGrav continually works towards making a
star as small as possible, while the outward pressure of the star’s
hot gas interior (Fhot) works toward making it as large as possible.
The dynamical balance condition that determines the size of a star
requires that at all points within its interior Ftotal = FGrav + Fhot = 0.
The balance condition comes about because if Ftotal > 0, then the
star would expand, while if Ftotal < 0, the star would contract. This
condition of hydrostatic equilibrium is very useful and indicates
that the pressure and temperature of a star must increase inward
towards the center.
To see why this is so, let’s first consider the variation in
pressure. The surface gas pressure (PS) must, by necessity, be zero,
since beyond the star’s surface there is no more gas.5 To estimate
the central pressure (PC) consider the star to be split into two
hemispheres (Figure 3.4), each of mass M/2, where M is the total
mass of the star under consideration. The gravitational centers of
these two hemispheres will be a distance R apart, where R is the
radius of the star, and the area over which the two hemispheres
‘interact’ is A = R2. Pressure is defined as being the force acting
per unit area. The force holding our split star together is the gravitational
attraction between the two hemispheres FGrav = G (M/2)
(M/2)/R2, and the area over which this force is spread is A. Hence,
the central pressure will be approximately:
PC ≈ FGrav/A = G/4 M2
/R4 (3.2)
For the Sun, Equation (3.2) indicates PC ∼ 9 x 1013 Pascal. (This
value of PC is actually about a factor of six too big when compared
76 Rejuvenating the Sun and Avoiding Other Global Catastrophes
Mass = M* / 2
Surface area
π R2
Radius R
Mass = M* / 2
Figure 3.4. To estimate the central pressure of a star, consider two
hemispheres, each containing half the mass of the star, with centers a
distance R apart interacting over a surface area of R2.
with the results from detailed numerical calculations.) So, we now
have our result: the pressure inside of a star increases inward, from
zero at the surface to ∼ 1014 (M/M 2/(R/R4 Pa at the center.
That the pressure increases towards the center of a star
makes sense, since the hydrostatic equilibrium condition requires
that successive regions within a star must support the weight of
overlying layers. As the entire weight of the overlying star must be
supported by the central region, it makes sense that the pressure
there must be highest.
The Pressure Law
The gas in the interior of a star provides the pressure that supports
the weight of overlying layers because it is a hot gas. Indeed, the
ideal gas equation links the pressure P at any point in the star to
the local temperature T, the density , and the number of particles
per unit volume n. The relationship is P = n kT, where k is the
Boltzmann constant.6
To determine the number of particles per unit volume, we
must first describe the basic particle makeup of stellar material.
Astronomers usually express the composition of a star in terms of
the mass fractions of hydrogen (X), helium (Y), and all the other
elements lumped together (Z). Astronomers also call the elements
other than hydrogen and helium metals or heavy elements. By
definition X + Y + Z = 1 and, for example, if 74 percent of the mass
The Sun, Inside and Out 77
of a star is in the form of hydrogen, while 25 percent is in the
form of helium, then 1 percent must be in the form of all other
elements, and X = 0.74, Y = 0.25, and Z = 0.01. Now, given the high
temperatures that prevail in stellar interiors the various atoms
will be fully ionized, with all their electrons stripped away. In this
manner, the stellar gas can be thought of as a ‘soup’ of positively
charged ions and freely moving electrons. Table 3.1 indicates the
number of ions and electrons that will be produced when the stellar
material is fully ionized.
From Table 3.1 it can be seen that each ionized hydrogen atom
will contribute one electron and one ion to the number density.
Each ionized helium atom, on the other hand, will contribute
three particles, two electrons and one nucleus, to the number
density. The ion contribution from the heavy elements is going to
be very small, first because Z itself is small, and second because
the atomic mass number7 A is generally large. The next element in
the Periodic Table after helium is lithium,and this has an atomic
number of A = 7. Consequently, this term is usually ignored in the
final sum for the particle number density. The number of electrons
contributed by the ionized heavy elements will typically be half
the atomic mass (i.e., /2). Bringing together the various terms
described in Table 3.1, the number density of particles in a fully
ionized gas is:
n =
mH
2X +
3
4
Y +
Z
2
=
mH
(3.4)
The term introduced in Equation (3.4) is called the mean
molecular weight, and the ideal gas equation corresponding
becomes: P = (k/ mH ) T. At this stage we note that, for a star
composed entirely of hydrogen, X = 1 and =½. The initial
Table 3.1. The number of ions and electrons produced as a result of
complete ionization. The term corresponds to the average value of
the atomic mass number7 of the heavy elements, is the density of stellar
material, and mH is the mass of the hydrogen atom.
Number of ions Number of electrons
Hydrogen X ( /mH) X ( /mH)
Helium Y ( /4mH) Y ( /4mH)2
Metals Z ( / mH) ≈ 0 Z( /mH) (/2)=Z( /2mH)
78 Rejuvenating the Sun and Avoiding Other Global Catastrophes
composition of the Sun, as determined by the chemical analysis
of meteorites,8 is believed to be X = 0.71, Y = 0.27, and Z = 0.02,
with a corresponding mean molecular weight of = 0.613.
The Central Temperature
The ideal gas equation indicates that the pressure will increase
provided the product T increases, and since the pressure required
to support the weight of overlying layers increases with increasing
depth, the temperature of a star must also increase inward towards
the center. Again, to first order approximation, let us consider a
star of mass M and radius R made entirely of hydrogen. Let us also
assume a uniform density model. In this case, = M/
4/3 R3].
Using this value of the density with the ideal gas equation and
substituting for PC in Equation (3.2), an expression for the central
temperature TC can be derived:
TC = G mH/3k M/R (3.5)
For the Sun, Equation (3.5) indicates a central temperature of TC ≈
5 x 106 K. This value is about a factor of three on the low side
compared to more detailed model calculations that find TC ≈15 x
106 K.
We now have a series of expressions that characterize the
properties of the material body of a star. Next we need to consider
the radiative body, and then we need to identify the ways in which
the material and the radiative bodies (recall Figure 3.2) interact.
Photon Diffusion Time
The temperature at the center of the Sun is some 15 million degrees
Kelvin. At such high temperatures the characteristic wavelength of
electromagnetic radiation is ≈ 2 x 10−10 meters, well inside the Xray
to
-ray region of the electromagnetic spectrum. The radiation
emitted at the surface of the Sun (where TS = 5780 K), however,
has a characteristic wavelength of ≈ 5 x 10−7 meters, in the
visible part of the electromagnetic spectrum. Clearly, something
The Sun, Inside and Out 79
has happened to the radiation during its journey from the Sun’s
center to its surface. Indeed, rather than taking an unimpeded
straight-line path from the center to the surface (in a travel time
of TUP), the photon zigzags along a random walk path. The typical
distance l traveled by a photon between interactions with a particle
(the so-called mean free path length) amounts to l ∼ 0.1 to 1.0-mm
throughout most of the Sun’s interior. The mean free path is
actually related to the density of stellar material and the opacity
(which is a measure of how much the stellar material hinders
the passage of radiation through it) by the relationship l = 1/( ).
The opacity is a complex term to quantify; it basically depends,
however, upon the temperature, density, and composition of the
stellar material. There is no simple or universal formula that
describes the opacity of stellar material. (There are, however, a
few approximations that apply under specific circumstances, as
described below.) This being said, we need not worry about the
details here. Indeed, the timescale of the random walk process by
which the photons generated at the center of a star diffuse outward
can be well approximated by a simple fixed-step random-walk
model (Figure 3.5).
In a random-walk process the direction of motion changes
abruptly after the photon has traveled a distance l, and the radial
displacement from the center D after n steps will be approximately
D2 = n l2. So the typical number of random-walk steps experienced
by a photon as it moves from the center to the surface of a star is
n=(R/l)
2. Here we have simply set the radial displacement D = R,
the star’s radius. Each step in the random-walk process takes the
photon a time Tstep = l/c to accomplish, where c is the speed of
light, and consequently the photon diffusion time TPD is:
TPD = nTstep = R2
/
lc (3.6)
In passing from the center of the Sun to its surface, each photon
typically undergoes n = 5 x 1025 interactions, taking of order TPD =
5 x 105 years to complete the journey.
It is the temperature gradient within a star – T/ R – that
allows the energy generated at the center to flow outward. Characteristically,
for the Sun, T/ R ≈ (TC – TS)/R = 0.02 K/m. That
is, the temperature drops by two one-hundredths of a degree for
80 Rejuvenating the Sun and Avoiding Other Global Catastrophes
Photosphere
Random-walk
path of photon
Displacement D
after N interactions
Star of radius R
Photon escapes into
space when D = R
Figure 3.5. The random-walk path of a photon moving from the center of a
star outward. After N random direction steps the radial displacement from
the center is D. Once the radial displacement D of the photon is equal to
the radius of the star R, it will escape into space, carrying away energy.
The region where photons escape into space is called the photosphere.
every meter moved outward towards its surface. This is a small
temperature gradient within the Sun, but it is enough to allow the
energy to flow from the center to the surface, where it is lost into
space. The important point at this stage, however, is that since
the photons typically move less than a millimeter between interactions,
they essentially ‘sense’ a radiation field corresponding to
that of a blackbody radiator of temperature T(r), where r is the
distance from the center. This is why for the Sun the characteristic
wavelength of the radiation emitted into space corresponds to that
of a blackbody radiator of temperature TS = 5780 K, rather than
one of TC = 15 million K.
Energy Transport
There are three possible modes by which energy can be transported:
conduction, convection, and radiation. The first of these modes,
conduction, occurs when energy is transferred by direct collisions
and is correspondingly important within evolved stars that have
The Sun, Inside and Out 81
regions of very high density (i.e., white dwarfs). The convective
mode of energy transport requires the mass motion of a hot gas,
while radiative transport involves the steady (albeit via a randomwalk)
outward flow of photons. It is the local temperature gradient
at each point within a star that actually determines the mode by
which the energy is transported outward. When the temperature
varies rapidly with distance (that is, when the temperature gradient
is steep), the energy will be transported by convection; otherwise,
the energy transport is via radiation. In the case of the Sun the socalled
convective stability condition dictates that energy transport
is by convection over the outer third of its radius.9 Within the
inner two-thirds of the Sun (by radius) the energy transport is via
radiation.
The sizes of the convective and radiative zones within a star
vary according to its age and its initial mass. Although the Sun
has a radiative core and a convective envelope, stars with masses
greater than about 1.2 M have convective cores and radiative
envelopes (see Figure 3.6); stars less than about 0.3 M in mass
have fully convective interiors. The reason for the changeover in
Convective
energy
transport
Radiative
energy
transport
Convective
energy
transport
0.5 1.0 1.5 2.0
0.5
0.0
1.0
m / M
Mass of star (M)
Sun’s
interior
Figure 3.6. Schematic diagram of the relative size and type of energy
transport zones within hydrogen-burning stars of different mass. The y-axis
indicates the fractional mass, varying between the center at m/M = 0
to the surface at which m/M = 1. The shaded regions indicate those
zones that are convective. Above a mass of 1.2 M stars have convective
cores and radiative envelopes; below 0.3 M stars have fully convective
interiors.
82 Rejuvenating the Sun and Avoiding Other Global Catastrophes
energy transport relates to the mode of energy generation itself and
the opacity. For low-mass stars the internal temperatures are low
enough [a result that follows from Equation (3.5)] that the gas is
only partially ionized, which is a situation conducive to convective
energy transport. Above 1.2 M the so-called CN fusion reaction
network switches on; this mode of energy generation is highly
temperature sensitive, again a situation conducive to the onset of
convective motion.
The Surface Temperature of a Star
If a star is in a steady state, neither appreciably heating up nor
cooling down with time, the energy radiated into space at its
surface must be exactly compensated for by the energy generated
deep within its interior. We can express this condition in terms
of the surface and central energy flux. At the surface the photon
leaves the star at the speed of light c, and consequently the surface
energy flux will be (a T4
S ) c, where a is the radiation density
constant, and TS is the star’s surface temperature. Now, this
surface energy flux must be matched by the energy flux at the star’s
center, which has moved through the star at an effective speed VE
determined by the photon diffusion process. That is, (a T4
S ) c ≈
(a T4
C) VE , where TC is the central temperature. This expression
rearranges to give a relationship between the surface and central
temperatures:
TS =
VE/c
1/4 TC (3.7)
The effective diffusion speed of the photon is given by VE = c/n,
where n in this case is taken to be the direct number of mean-free
path lengths between the star’s center and its surface, and since
n 1, Equation (3.7) tells us, as expected, that TS TC. For the
Sun we have with a direct walk n = (R/l) ≈ 6.9 x 1011 (taking l
= 1-mm) and TC = 4 x 106 K from Equation (3.5). Hence, we find
a surface temperature of TS ≈ 4400 K. The Sun’s actual surface
temperature is 5,800 K.
The Sun, Inside and Out 83
Stellar Luminosity
The luminosity L of a star is defined as the total amount of electromagnetic
energy radiated into space per second. Taking a star to
be a spherical blackbody radiator of radius R and surface temperature
TS, the luminosity can be determined through the StefanBoltzmann
law:
L = 4 R2 TS
4 (3.8)
Now, if we substitute for TS from Equation (3.7) and use the
identity = a c/4, where c is the speed of light and a is the radiation
density constant, so Equation (3.8) can be rearranged as:
L ≈
aTC
4
V /R2
/
lc = radiant energy content /TPD (3.9)
where V = (4/3) R3 is the volume of the star. Equation (3.9) tells
us, therefore, that the luminosity of a star is governed by the rate
at which the radiant energy content of the star is lost into space
via photon diffusion.10 For a given amount of energy within a star,
Equation (3.9) indicates that the longer the photon diffusion time,
the smaller the luminosity. Equation (3.6) further tells us that the
photon diffusion time increases as the mean-free path l between
photon interactions with stellar material decreases. In other words,
if the opacity of the stellar material increases (so that l decreases),
the luminosity of the star will also decrease.
Although the mass and radius of a star’s material body determines
its central temperature TC (as indicated by Equation3.5), it is
the mean-free path of the photons that determines the star’s surface
temperature and luminosity. If the mean-free path length of the
photons within a star is reduced (by, for example, increasing the
opacity ), then the luminosity will also be reduced as a result of the
increased photon diffusion time (thanks to Equation 3.9). Likewise,
the surface temperature will be reduced because of the reduction in
the effective photon diffusion speedVE (as described byEquation 3.7).
Energy Generation
The Sun radiates energy into space at a rate of L = 3.85 x 1026 joules
per second (or watts), and yet it does not cool off with time (i.e.,
TS is constant on timescales well in excess of TPD). This indicates
84 Rejuvenating the Sun and Avoiding Other Global Catastrophes
that the Sun must have an internal energy source. Further, since
on a timescale of at least several TPD the Sun’s radius is neither
increasing nor decreasing, it is correspondingly not suffering any
net loss or net gain of internal energy. That is, there must be an
exact balance between the energy lost and radiated into space at
the surface of the Sun, and the energy generated deep within its
interior.
A star can tap two main energy sources during its lifetime.
It can contract and thereby feed off the gravitational potential
energy liberated, or it can feed from the energy liberated by nuclear
fusion. While feeding on its gravitational energy a star will become
physically smaller and hotter at its center, but its composition
will remain unchanged. If it generates internal energy via nuclear
fusion reactions, however, then it will maintain a near constant
radius and central temperature, but its internal composition will
change with time. A star can, in effect, turn on and turn off the
gravitational potential energy source as required. That is, if the
nuclear fusion reactions are unable to provide the energy to power
a star, it will begin to contract on the so-called Kelvin-Helmholtz
timescale (TKH) which is given by the ratio of the gravitational
potential energy of the star divided by its luminosity (i.e., its rate
of energy loss). Correspondingly, for stars of mass M, radius R and
luminosity L,
TKH ≈ GM2
/R /L (3.10)
where G is the universal gravitational constant. For the Sun,
TKH ≈ 2 x 107 years, and the contraction rate would amount to
about 75 meters per year if the Sun were powered entirely by
contraction. The release of gravitational potential energy not only
results in the star becoming smaller; it also causes the central
temperature to increase, as indicated by Equation 3.5 when R is
reduced for a fixed mass M. This effect is important since, for
example, at the end of a star’s main-sequence phase (i.e., with the
exhaustion of hydrogen in its core; see below), it is the contraction
of the central regions that causes the central temperature and
density to increase, a situation that continues until the triple-alpha
reactions—during which helium is converted into carbon—can
commence.
The Sun, Inside and Out 85
Nuclear Fusion
Stars such as the Sun generate energy within their central regions
by the nuclear fusion reactions that transform four hydrogen nuclei
(4 protons) into a helium nucleus (containing 2 protons and 2
neutrons). Schematically, 4H ⇒ 4He + energy (see Figure 3.7). Such
fusion reactions will run efficiently once the temperature exceeds
Tfusion ∼ 10 million K. The region over which this temperature
condition is met within the Sun can be determined from the central
temperature and the average temperature gradient. Specifically, it
is required that TC – [ T/ R]r > Tfusion which indicates that r <
(TC − Tfusion)/[ T/ R] = 5 x 106/0.02 ∼ 2.5 x 108 meters ∼ R/3.
Hence, the energy radiated at the surface of the Sun is entirely
generated within the inner third (by radius) of its interior.
The conversion of hydrogen into helium proceeds in the
Sun via the proton-proton (PP) chain,11 and the details of the
reaction network were described by Hans Bethe in the late 1930s.
Specifically, Bethe realized that during the first step of the chain
two things must happen. First, the two protons must approach
one another so closely that there is a non-zero probability that
they will overcome their mutual electrostatic repulsion. This
condition is determined by the so-called Gamow factor (since the
P
Positron annihilates with an electron (energy)
+ P Neutrino exits star (loss of energy)
D + e+ +νe
+ P Gamma ray photon (energy)
3He + γ
+3He
4He + 2P
Figure 3.7. Schematic flow of the PP chain. The end result of the chain is
that 4P ⇒ 4He + 2e+ +2 e + energy. Here e+ is a positron, the antimatter
equivalent of the electron, while e is an electron neutrino; both are
generated during the inverse beta decay step. The energy liberated per
conversion is just over 4 x 10−12 joules.
86 Rejuvenating the Sun and Avoiding Other Global Catastrophes
details of quantum mechanical ‘tunneling’ were first determined
by George Gamow in 1928). Second, at the same time that the
protons undergo this very close approach, one proton must undergo
an inverse beta decay to produce a neutron. In this manner a
deuterium nucleus, consisting of a bound proton and neutron, can
be formed. In fact, stars only exist for long periods of time because
the 2He nucleus, consisting of two protons, is unstable and because
the inverse beta decay occurs only rarely to produce deuterium.
If it were not for the inverse beta decay requirement dramatically
slowing down the initial P+P reaction, the Sun would have
converted all of its central H into 4He in a matter of a few hundred
thousand years.12 The average time interval required for a single
proton to fuse with another proton, to produce deuterium, is of
order 10 billion years. This time is, in fact, the nuclear (or mainsequence)
timescale, and it is the characteristic time after which
all the protons in the energy generation region of a star will have
fused with other protons. In any given second, however, of order
1038 protons successfully undergo fusion reactions within the Sun,
and the energy generated by these reactions will be radiated into
space after a time TPD.
The rate at which nuclear energy is generated per unit mass
per second (r) at radius position r within a star is determined by the
local composition, temperature, and density: (r) = {composition,
T(r), (r)}. The full details of the energy generation calculation need
not be followed here, but suffice it to say that a power law representation
can be applied such that:
r = 0 X2 T (3.11)
where X is the hydrogen mass fraction of the star, (r) is the density,
T(r) is the temperature, and 0 and are constants. For stars such
as the Sun, where TC ≈ 15 million Kelvin, the exponent has a
value of about 4. The energy generation rate is clearly sensitive to
changes in temperature, due to the size of the exponent , and any
process that reduces the central temperature and density will result
in a decrease in the energy generation. Likewise, as one would
expect, as the hydrogen in the core is gradually converted into
helium, the central value of X will decrease and, ultimately, when
X(core) = 0, the hydrogen fusion reactions will stop altogether.
The Sun, Inside and Out 87
The Mass Luminosity Law
As indicated above, the temperature gradient in the Sun is of
order TC/R ∼ 0.02 K/m, and a photon typically travels less than
a millimeter between interactions with the surrounding stellar
material. In this manner we can treat the radiation field at radius
r within a star as that due to a blackbody radiator of temperature
T(r). So, let us consider the flow of radiation through a thin shell
of thickness r. At the base of the shell (at radius r), the StefanBoltzmann
law tells us that the radiative flux F(r) = T4 (watts/m2).
At the top of the shell the temperature is T(r + r), and the flux will
be F(r) + F = (T + T)
4 ≈ (T 4 + 4T3 T). Here we are ignoring
terms of ( T)
2 and higher powers, as they will be very small. Now,
T is negative because the temperature must decrease outward
from the center of the star, so the energy flux absorbed within the
shell must be F = 4 T 3 T. This absorption of energy is related
to the opacity (r) of stellar material and, by definition, the energy
flux absorbed across a region of width r will be F = - (r) (r) F(r)
r. Now, again by definition, the energy flowing across a shell of
radius r each second (the luminosity) is L(r)=4 r2 F(r), so equating
our two terms for F we have: L(r) = - (4 r2) 4 T 3 [ T/ r]/ (r)
(r). To make headway now, we need to express the opacity (r)
in terms of the composition, density, and temperature at radius r.
For Sun-like and lower mass stars the opacity can be expressed in
terms of the so-called Kramer’s power law with (r) = 0 (1 + X)
T −3 5, where 0 is a constant. For higher mass, higher temperature
stars the opacity switches to that of electron scattering, which
varies as es = 0.02(1 + X) m2/kg, independent of the temperature
and density.
At this stage we will combine the expressions for the
luminosity, opacity, and the temperature gradient to determine
how the luminosity varies with stellar mass M, the hydrogen mass
fraction X, and the mean molecular weight μ. When the Kramer’s
opacity law holds true, the luminosity varies as the mass to the
fifth power:
L = LKR
7 5
1+X
M5 (3.12)
88 Rejuvenating the Sun and Avoiding Other Global Catastrophes
where LKR is a constant13. Equation (3.11) is the mass-luminosity
relationship that approximately holds true for stars in the mass
range 0.5 < M/M < 2.0. For higher mass stars, where the opacity is
mostly due to electron scattering, the luminosity varies according
to the mass cubed. The data derived from binary star observations
indicates that stellar luminosity varies as the approximate
fourth power of the star mass (L ∼ M4), which indicates that the
simplified arguments being presented in this chapter are, in fact,
dimensionally correct and do indeed offer a reasonable description
of stellar characteristics.
The key physical feature missing from the models described so
far is that for the mode of energy transport. We have assumed that
all the energy is carried by radiation, whereas detailed computer
models indicate that stars can have extensive convection zones,
and this will have an important effect upon their internal structure
and the exact form of the mass-luminosity relationship. For stellar
masses smaller than about 0.5 M more than 50 percent of the
star’s outer envelope by mass is convective; for masses greater than
about 20 M more than 50 percent of the star’s interior, again by
mass, is convective.14 All the above being said, the main point that
will be of interest to future star engineers is that the luminosity
of a star can be reduced by lowering its mass.
A Journey Through the HR Diagram
The Hertzsprung-Russell (HR) diagram is the historical battleground
between theory and observation. The diagram displays the
luminosity and temperature relationship of stars, and it reveals
that important correlations exist between the two quantities
(Figure 3.8). Of prime importance is the delineation of the main
sequence that stretches along the diagonal from the hot, high
luminosity (high-mass) stars to the low luminosity, cool (lowmass)
stars. Indeed, over 90 percent of the observed stars fall on
the main-sequence diagonal in the HR diagram. Those stars not on
the main-sequence fall in either the red giant or the white dwarf
regions.
Detailed computer models have shown that the mainsequence
is delineated by those stars that are generating internal
The Sun, Inside and Out 89
White dwarf stars
Sun
R = 1/100 R
R = R
R = 100 R
Red giant region
Main sequence
Bright
⇑
Luminosity
Faint
Hot ⇐ Temperature Cool
O B A F G K M
Spectral type
Figure 3.8. A schematic HR diagram. The lines of constant radius can
be placed in the diagram according to the Stefan-Boltzmann relationship
expressed in Equation (3.8).
energy through the conversion of hydrogen into helium. The red
giant region, on the other hand, is populated by those stars that
are generating internal energy through the conversion of helium
into carbon.15 Further, the white dwarf region is populated by old,
low-mass stars that are in fact simply cooling off,16 their days of
producing internal energy through fusion reactions having ended.
These corpse stars, given enough time, will eventually become
zero luminosity, zero temperature black dwarfs.
Detailed computer models also indicate that the manner in
which the surface temperature and luminosity of a star vary with
time is dependent upon its initial mass. Stars end their formation
stage by initiating hydrogen fusion reactions and correspondingly
settle onto the main sequence in the HR diagram. All stars17
go through a hydrogen fusion stage on the main sequence and a
helium fusion stage in the red giant region (Figure 3.9). Stars more
massive that 8 M can initiate fusion reactions beyond that of
the triple alpha reaction, but they eventually end their days as a
supernova—literally blowing themselves apart after the formation
of an iron-rich central core. A neutron star18 remnant may or may
90 Rejuvenating the Sun and Avoiding Other Global Catastrophes
Star formation produces star of mass Min
H burning–main sequence stars
He burning–giant stars
Advanced nuclear
burning
Planetary
nebular
Min < 0.1 M
Min < 50 M Min > 50 M
Min < 8 M Min > 8 M
(no H burning)
Brown Dwarf
Processed material
returned to ISM
Accretion from
Binary companion Supernova
Collapsar
Type II SN (GRB + Black hole)
(Neutron star)
White Dwarf
Type I SN
Figure 3.9. Evolutionary pathways followed by stars according to their
initial mass. The horizontal block arrows indicate stages where mass is
either lost into space or accreted by the star if it chances to be in a binary
system.
not be produced during the rapid supernova phase; astronomers
are still debating the exact details. Stars with an initial mass of
less than 8 M are unable to initiate fusion reactions beyond that
of helium burning and consequently become white dwarfs after
undergoing a visually dramatic planetary nebular stage.
The Journey of the Canonical Sun
As the Sun ages it steadily consumes the hydrogen within its
core—the region encompassing the inner third (by radius) of its
interior. Indeed, it is the change in the core’s composition that
drives the star’s evolution towards a hotter, more luminous, and
larger configuration (Figure 3.10). As the hydrogen stored in the
Sun’s central core is depleted by PP fusion reactions, so X ⇒0 and
μ increases from 0.613 (its zero-age main-sequence value) to 1.316
at core hydrogen exhaustion. Equation (3.12) indicates that this
The Sun, Inside and Out 91
Log(L / L )
4
3.7 3.6 3.5
0.0
1.0
2.0
3.0
PN and to WD stage
Pre-main sequence track
Sun (now)
End MS
Red giant
branch
Horizontal
branch
Asymptotic giant
branch
Thermal pulsing
1
2
3
5
6
Log T
Figure 3.10. The Sun’s journey through the HR diagram. The time intervals
between the numbered points are given in Table 3.2.
composition change should result in the luminosity increasing by
a factor of order (1.316/0.613)7 5 ∼ 300. Such a dramatic increase
in the Sun’s luminosity during its main-sequence lifetime will
not actually occur, because it is only the inner core that experiences
the compositional change; the outer envelope maintains the
original solar composition. It is the chemical discontinuity at the
core boundary, however, that will eventually cause the Sun to
become a red giant, and if nothing is done about it, it is at this
stage that the Sun will cause all life on Earth to become extinct
(as we will discuss in the next chapter).
We need not follow the detailed story of the Sun’s formation
here. Needless to say, however, it formed through the gravitational
collapse of a low density, low temperature, and extended cloud of
gas. The time for the Sun to reach Point 1 in Figure 3.10 (its socalled
zero-age main-sequence position) is determined by the rate
at which material is accreted at the center of the solar nebula. The
92 Rejuvenating the Sun and Avoiding Other Global Catastrophes
various detailed models for the Sun’s formation suggest a protoSun
stage lasting about 10 million years. At Point 1 (Figure 3.10)
the PP chain reactions begin in the core, and the Sun becomes
a bona fide star. Table 3.2 indicates that on the zero-age mainsequence
the Sun is actually slightly less luminous and slightly
smaller than is currently observed.19 After 4.5 billion years the
Sun, as we currently see it, is about middle-aged, with half of the
hydrogen within its central core having been consumed. At Point 2
in Figure 3.10, the hydrogen within the core has all been consumed.
This point is reached some 11 billion years (see Table 3.2) after the
PP reactions first started. After central hydrogen exhaustion, the
Sun begins moving rightward in the HR diagram, becoming cooler
and larger. At Point 3 the luminosity and radius begin to increase
rapidly, and the Sun starts to ascend the red giant branch. Deep in
the interior of the Sun the temperature and density of the central
core are now increasing rapidly, with energy being generated in a
thin hydrogen ‘burning’ shell above the dormant core. Eventually,
the temperature in the central core becomes hot enough (about
100 million degrees Kelvin) for helium fusion reactions15 to begin.
Indeed, the onset of helium burning is a veritable hot, explosive
flash.
At the peak of the helium flash the Sun will find itself at the
tip of the red-giant branch (Point 4), where it will be some 2,349
times more luminous than at present, and some 166 times larger
(see Table 3.2). At this stage the planet Mercury will be destroyed,
Table 3.2. Characteristics of the solar evolutionary track shown in
Figure 3.10. The ages given in the second column are in units of billions
of years. The luminosity and radius values given in the third and fifth
columns are expressed in units of the Sun’s current luminosity and radius.
(Table data based upon the model calculations by Sackmann, Boothroyd,
and Kraemer; see Reference 20).
Stage Time (t9) L/L T (K) R/R
1 0 0 0.70 5586 0 897
Now 4 5 1.00 5779 1 00
2 10 91 2.21 6517 1 58
3 11 64 2.73 4902 9 5
4 12 233 2349 3107 165 8
5 12 234 41 4724 9 5
6 12 345 130 4375 20
The Sun, Inside and Out 93
consumed within the Sun’s bloated outer envelope. We will discuss
this stage in greater detail in the next chapter. The Sun spends a
very short amount of time at the red-giant tip (Point 4), perhaps
a few hundred thousand years, eventually dropping substantially
in luminosity to a position on the so-called horizontal branch,
where it will begin the steady consumption of helium within
its core (point 5). At this stage the Sun will be about 40 times
more luminous than at present. The core helium burning phase
is not as long lasting as the main-sequence phase, and within a
hundred million years the Sun’s central helium supply will become
exhausted (Point 6). At this stage a number of complex internal
processes are set up. The Sun now begins to ascend the asymptotic
giant branch and hydrogen and helium fusion reactions are
taking place within rings (or shells) around the carbon-rich central
core. The hydrogen shell source is situated above the helium
shell source, and the two ‘furnaces’ turn on and off in a complex
series of interactions. At this stage the Sun undergoes what are
called thermal pulses—periods of rapid expansion and contraction
(in time intervals of several hundred days) accompanied by large
swings in luminosity and temperature. Such stars are distinguished
observationally as long-period Mira variables, named after the
prototype system omicron Ceti (Mira).
Various numerical models have been developed by
astronomers to describe the advanced thermal pulsing stage of
the Sun.20 However, these models do not, as of yet, offer a clear
consensus on what might happen to Earth. Some models suggest
the Sun will expand beyond 215 R thus engulfing Earth and
bringing its history to a final close. Other models suggest that the
Sun won’t expand quite so much and, consequently, Earth as a
physical body will survive. Life, however, will have long perished
because of the Sun’s increased luminosity. The key unknown at
this stage is exactly how much mass the Sun might lose during
the red-giant and asymptotic giant phases. Present observations
suggest that low-mass stars, such as the Sun, lose of order 0.1 to
0.2 solar masses in the form of a stellar wind during the post mainsequence
phase. The numerical models including mass loss find
that the Sun might not expand to the extent that Earth is engulfed
during the thermal pulsing stage. Part of the reason why Earth,
again as a physical object, survives when mass loss is included is
94 Rejuvenating the Sun and Avoiding Other Global Catastrophes
because its orbital semi-major axis actually increases—a topic we
shall look at again in the next chapter.
With the onset of thermal pulsing the Sun is beginning to
truly die, and it is rapidly running out of fuel in those regions that
are hot enough to power the hydrogen and helium shell sources.
The Sun may be dying at this stage, but it will go out in a blaze
of glory. The thermal pulsing results in the core and the envelope
parting company, and over a period of several tens of thousands of
years the outer envelope will be cast off into the surrounding interstellar
medium. The ultraviolet photons produced by the hot, nowexposed
carbon-rich core, however, begin to ionize the hydrogen at
the innermost edge of the expanding gas envelope, and this leads
to the formation of a planetary nebula. William Herschel, who first
described such nebulae in the late 18th century, thought that such
nebulae reminded him of planetary disks, and astronomers have
continued to use his misnomer ever since.
The central, carbon-rich core evolves rapidly during the
planetary nebula phase, and while initially very luminous (L ∼
2500 L and extremely hot (T ∼ 30,000 K), it soon enters the
white dwarf region in the HR diagram (see Figure 3.8). As a white
dwarf16 the future Sun will gradually cool off and slowly fade
out. Indeed, the cooling time of a white dwarf is immense. Since
white dwarfs are not generating any energy within their interiors
their total energy reserve is essentially the thermal energy of their
constituent particles. To order of magnitude, the energy reserve
that a white dwarf has to radiate into space is EWD ≈ nkT, where
n is the number of particles, k is the Boltzmann constant, and
T is a typical internal temperature. For a one solar mass white
dwarf assumed to be composed entirely of carbon, n ∼ 1056; taking
T = 107 K we then have EWD ∼1.4 x 1040 joules. Further, adopting
a typical luminosity of LWD = 10−3 L the cooling time will be
Tcool = EWD/LWD ∼ 3.5 x 1016 seconds = 109 yrs. The cooling time
will actually be much greater than a billion years since as the white
dwarf cools its luminosity decreases and consequently the thermal
energy is radiated away less rapidly. Indeed, a more detailed calculation
indicates that several tens of billions of years are required
for a white dwarf to become a black dwarf.
The ultimate end state of the Sun will be that of a black dwarf,
a cold object about the same size as Earth, with zero luminosity
The Sun, Inside and Out 95
supported against gravitational collapse by its constituent degenerate
electrons.16 For the canonical Sun the deep future promises
an infinitely21bleak outlook.
The Reasons for Gigantism
Before continuing with this chapter we should look at why
stars become giants as they age. This being said, the exact
physical reasons for why a star puffs up to become a giant
when the central hydrogen supplies are depleted are not fully
agreed upon by astronomers. Everyone agrees that it happens;
indeed, it is an observational fact. But it appears that many subtle
effects come into play in order to produce the tendency towards
gigantism.
The various detailed computer models indicate that two
distinctly different things begin to happen when all of the hydrogen
within a star’s core has been converted into helium. First, the core
can no longer provide energy via PP fusion reactions to heat the
stellar interior, and consequently gravitational contraction of the
core (not the whole star) begins. The contraction timescale of the
core will be that of the Kelvin-Helmholtz timescale described in
Equation (3.10). As the core contracts it both heats up and acquires
a higher density. This is good, since it will eventually result in the
onset of helium fusion reactions.
At the same time the core begins to contract as a result of
core hydrogen exhaustion, the outer envelope of the star begins
to expand. In addition, hydrogen fusion reactions begin in a shell
source surrounding the hydrogen-depleted core. It is generally
argued that one of the principal reasons that the envelope expands
at this stage is because of the composition difference between the
core and the envelope. We can see that this should result in the star
expanding by looking at the ideal gas equation. As described above,
the pressure P is related to the density, composition (through
the mean molecular weight ), and the temperature T, such that
P = (k/ mH) T. Across the core-envelope boundary,22 the pressure
and temperature must be constant; the density on the other hand
must vary in step with the mean molecular weight, such that /
is constant across the boundary. Since the mean molecular weight
96 Rejuvenating the Sun and Avoiding Other Global Catastrophes
varies from 0.6 in the core to 1.3 in the envelope, the density
must correspondingly decrease by a factor of 0.6/1.3 ≈ 0.5 across
the core-envelope boundary. This reduction in the density dictates
that the star must expand in order to accommodate the amount
of material situated above the core.23 In addition to the coreenvelope
composition jump effect, experiments with numerical
stellar models have shown that a star will swell up as a result
of becoming more centrally condensed and as a result of having
a hydrogen-burning shell situated above an electron degenerate
core.24
The importance of the core-envelope composition jump for
producing gigantism is further exemplified by the fact that stars
less than about 0.25 M in mass do not undergo a red-giant phase.
The reason for this is that the interiors of these very low-mass stars
are almost fully convective (recall Figure 3.6). The evolution of a
0.12 M star is shown in Figure 3.11, and in contrast to the Sun’s
evolutionary track (Figure 3.10), this star evolves to higher rather
Trad + 402 Gyr
Trad = 5742 Gyr
Log L / L
-3
-4
-5
Pure cooling
sequence
begins
Pre-main
sequence
track
Main
sequence
Helium
white dwarf
H-shell
burning
H-core burning
Trad + 539 Gyr
5000 4000 3000 2000 Temperature
Figure 3.11. HR diagram showing the evolutionary path of a 0.12 M star.
The fully mixed interior of very low-mass stars (see Figure 3.6) prevents
the core-envelope chemical discontinuity from coming about, and consequently
these stars do not undergo a red-giant phase following hydrogen
exhaustion. Figure based upon calculations by Gregory Laughlin and coworkers:
The Astrophysical Journal, 482, 420–432 (1997).
The Sun, Inside and Out 97
than lower temperatures with age, and there is no large change in
the radius.
At time Trad (see Figure 3.11) a small radiative core develops,
and a small compositional gradient begins to grow with the
continued depletion of hydrogen. The core, however, soon becomes
inert, and a hydrogen-burning shell source develops. Eventually,
all of the hydrogen that can possibly be consumed by the PP fusion
reactions is used up (Trad + 402 Gyr in Figure 3.11), and the star
becomes a helium white dwarf, slowly cooling off into obscurity
and reversing its direction of evolution in the HR diagram.
A Negative Feedback System
In this chapter we have attempted to provide an overview of
the internal workings of stars. Perhaps the key point is that in
general one can think of a star as a negative feedback system (see
Figure 3.12) that is capable of finding an equilibrium size, surface
temperature, and luminosity in accordance with its mass, its mode
of energy generation, and its composition. If any one or all of the
latter quantities are changed, then the equilibrium values for the
temperature, size, and luminosity also change.
Figure 3.12 shows the essential arrangement of interconnections
between surface and central stellar quantities. The line
linking the central temperature to the opacity and the surface
temperature in the figure corresponds to Equation (3.7). The
relationships between the central and surface temperatures, the
temperature gradient, and the radius establishes the feedback
mechanism responsible for enablement of hydrostatic equilibrium,
whereby the pressure gradient is capable of supporting the
weight of overlying layers at each point inside of the star. The
link between the central temperature and composition establishes
the energy generation rate and luminosity as described by
Equation (3.9) and Equation (3.11). The energy generated at the
center of the star flows down the temperature gradient, from the
hot center to the cooler surface, and is eventually radiated into
space. The time for the radiation to traverse the temperature
gradient (TPD) is given by Equation (3.6).
98 Rejuvenating the Sun and Avoiding Other Global Catastrophes
Composition Mass Mass loss
Luminosity
Central temperature
Composition
change
Radius
Temperature
gradient
Pressure
gradient
Energy
generation
Surface temperature
Temperature
gradient
Opacity
Figure 3.12. A schematic interaction chart showing the relationships
between various stellar quantities. The dashed box in the lower right
contains those quantities that are involved in establishing hydrostatic
equilibrium. The box in the upper left contains those quantities that are
responsible for driving stellar evolution. The dashed line indicating mass
loss is also a mechanism for driving internal readjustment.
Once the mass and composition have been specified the
radius, surface temperature, and luminosity of a star are established
according to the operation of a stabilizing negative feedback
mechanism. To see how this works, imagine that for some reason
the central energy generation rate suddenly increases. Equation
(3.11) tells us that such an increase must have come about because
of an increase in the central temperature. Now an increase in
the central temperature will cause an increase in the central
pressure (thanks to the perfect gas law), and this in turn will
result in an increase in the temperature and pressure gradients.
An increase in the pressure gradient, however, will cause the star
to expand. For a fixed stellar mass Equation (3.5) indicates that
as the radius increases so the central temperature decreases and,
hence, according to Equation (3.11), the energy generation rate
will be reduced. In this negative feedback manner, an increase in
the central temperature results in the star readjusting its internal
structure such that the increase is damped out. Conversely, if the
nuclear energy generation rate suddenly decreased, the response
The Sun, Inside and Out 99
of the negative feedback mechanism would be to cause the star
to shrink, thereby increasing the central temperature and consequently
causing an increase in the energy generation rate. By the
use of negative feedback mechanisms a star is able to remain
stable against collapse (that is, in hydrostatic equilibrium), and it
is able to generate exactly the right amount of energy in its central
regions to compensate for the energy that it loses into space at its
surface.
The reason why stars must evolve with time is illustrated in
Figure 3.12. As a result of the generation of energy through nuclear
fusion reactions (i.e., via the PP chain; see Figure 3.7), the composition
of the central regions is changed with time. Specifically, the
composition changes from one that is initially hydrogen-rich to
one that is helium-rich and hydrogen-depleted. This change in the
central composition will correspondingly result in a change in the
mean molecular weight μ[given in Equation (3.4)] and the opacity
of the stellar material. Because of the changes in these latter two
quantities the star will have to find a new equilibrium temperature
gradient resulting in a new luminosity, radius, and surface
temperature. The mass-loss term shown in Figure 3.12 will also
drive stellar change since, literally, the mass of the star is reduced
over time—a point future asteroengineers will most definitely take
note of.
Fundamental Constants
“From a drop of water a logician could infer
the possibility of an Atlantic or a Niagara without
having seen or heard of one or the other.”
Arthur Conan Doyle
In this chapter we have been mostly concerned with describing
what goes on in the interiors of stars. Now, however, we will
address the issue of why stars have the actual characteristics that
they do. It turns out, as you will see, that the observed properties
of the stars are determined by the fundamental constants of
physics.
100 Rejuvenating the Sun and Avoiding Other Global Catastrophes
We have already indicated (see Note 3 in Chapter 1) that the
speed of light c is a fundamental constant. It is a limiting speed, and
we can observe no object that travels faster than c = 2.99792458 x
108 m/s in our universe. Other fundamental constants of our
universe (in all their measured glory) are: G = 6.6742 x 10−11
m3/kg/s2, the universal gravitational constant; h = 6.6260693 x
10−34 Js, Planck’s constant (fundamental to the quantum world);
the proton mass mp = 1.67262171 x 10−27 kg, and the unit of
electrical charge e = 1.60217653 x 10−19 C. Both of the latter two
constants are important in the description of atomic structure.
If you change any one of these fundamental constants then you
literally generate a universe with properties distinctly different
from our own. There is, in fact, a rather restricted range of
variation among the fundamental constants that will allow for
the formation of stars and the existence of life. Other possible
universes (sometimes called ‘’world ensembles”) might be entirely
void of stars; still other universes might be full of very low-mass
stars that are incapable of undergoing helium fusion reactions to
produce the carbon atoms25 essential for the emergence of life.
From the fundamental constants just given one can construct
two dimensionless constants: = e2/c ≈ 1/137 and G = Gm2
p/c
≈ 5 x 10−39 (here we have used the standard notation that = h/2 ).
These two constants – the electromagnetic fine scale constant and
the gravitational fine scale constant, respectively – when combined
with fundamental mass and length terms, appear to account for
such observations as why the universe is as big as it is now, why
stars are as massive as they are, and why atoms have their various
properties.26
In the following discussion we are going to use a standard
dodge, and rather than account for exact values, we will consider
only the order of magnitude argument. That is, what is really
important is the determination of whether a quantity is of order 10
in size, 100, 10−7, 1023, and so on. In this manner we will mostly be
using the ‘∼’ sign rather than the strict ‘=’ sign in the calculations
that follow. This method, in which small constant factors such as
2 or are ignored, is an example of the Fermi calculation approach
discussed earlier (see Note 1 of Chapter 1). The idea is that by the
end of a series of multiplications and divisions most of these small
constant terms will cancel each other out to produce a number
The Sun, Inside and Out 101
of order unity. This method usually works, but there are times
when one has to be a little careful, so a dose of due diligence is
also required.
The Quantum World of the Electron
To answer the question, ‘’Why are there stars?’ we must first take
a brief excursion into the small-scale quantum world of the atom
and, more specifically, look at how the electrons inside a star
interact with one another. On the scale of the atom we must first
abandon our everyday notions of what particles are. We are now
in the quantum realm, where entities such as electrons, atomic
nuclei, and photons have simultaneous wave-like and particle-like
properties. The Nobel Prize-winning physicist Louis de Broglie
(1892–1987) described this wave-particle duality by arguing that
the wavelength of a particle is related to its momentum p through
the relationship p = h/ , where h is Planck’s constant. Further, the
energy of motion K of our particle is related to its momentum by
the relationship K = p2/2m = h2/2m 2, where m is the particle mass.
Where this becomes important to our story is that if a particle, say
an electron, is confined to a region of size d, then its associated
de Broglie wavelength must satisfy the condition ≤ d. Since,
however, a maximum wavelength corresponds to a minimum
momentum, the energy of a confined entity (i.e., our electron) must
be at least K0 ∼ h2/2 me d2, where me ≈ mp/1000 is the electron
mass. In other words, an electron cannot be completely at rest even
when it is confined. This behavior is important since it results
in an outward pressure that tends to resist further confinement.
Nobel Prize-winning physicist Wolfgang Pauli (1900–1958) introduced
the quantum mechanical idea of the non-overcrowding of
electrons27 and, accordingly, if there are N electrons in a volume
of space V with a characteristic dimension d, then the minimum
energy of each electron is K0 ∼ h2/2 me d2, where d3 = (V/N). An
electron gas constrained according to the Pauli exclusion principle
is said to be degenerate. This effect is important for stars since
the overcrowding pressure results in degenerate electrons being
able to support the star against gravitational collapse.28 We will
102 Rejuvenating the Sun and Avoiding Other Global Catastrophes
come back to this point in a few moments, but first let’s re-cast
Equation (3.5), our expression for the central temperature of a star,
in terms of the average inter-particle separation d.
Collapsing Gas Clouds
For a perfect gas,6 Boyle’s law provides a relationship between
the pressure P, the temperature T, the volume V, and the total
number of particles N in the gas, with P V/T = N k, where (again
in all its derived glory) k = 1.38065 x 10−23 (J/K) is the Boltzmann
constant. In our order of magnitude approach the volume V ∼ R3,
where R is the characteristic dimension defining the volume. So
our first expression for the gas pressure is, accordingly, P ∼ N T
k/R3. Further, from Equation (3.2) we have that the central pressure
varies to order of magnitude as PC ∼ G M 2/R4, where M = N mp is
the mass of the gas now assumed to be composed of hydrogen—
as is appropriate for stars. In addition, we also note that if the
average separation between the particles in our gas is d, then N d3
∼ volume ∼ R3. By equating the two expressions for the pressure
we find a relationship for the central temperature such that TC = G
N2/3 ( c/k d). Now, the first thing to notice about this expression
is that it tells us that the central temperature must increase as
the typical separation d between the gas particles decreases. The
second point to notice is that once d is ‘fixed,’ the only other
variable term is the number of particles in the gas N. All of the
other terms are fundamental constants.
Let us now follow the gravitational collapse of a large, cold,
pure hydrogen gas cloud. As the collapse proceeds, the volume
of the cloud becomes smaller and the gas cloud heats up, since
the spacing d between particles necessarily decreases. Eventually
the temperature will become sufficiently high that the hydrogen
will become ionized, resulting in equal numbers of protons and
electrons being produced. Most of the mass of the cloud resides in
the protons, since mp ∼ 1000 me. The electrons, however, can be
squeezed together only so much during the collapse before Pauli’s
exclusion principle comes into play. Once the electrons become
degenerate they can generate sufficient overcrowding pressure to
halt the collapse of the gas cloud. So what we need now is
The Sun, Inside and Out 103
to find an estimate of dmin, the minimum separation distance
between electrons at which degeneracy becomes important. At
this separation the temperature of the gas cloud will have reached
a maximum value Tmax.
Why Stars Are Massive
All material objects possess gravitational potential energy EG, with
EG ∼ - G M 2/R, where R and M are the radius and mass. As
we have seen earlier, if no forces oppose the attractive gravitational
force, then a body will collapse on the dynamical collapse
timescale given in Equation (3.1). For our collapsing hydrogen
cloud, however, the electrons will eventually become degenerate,
and the overcrowding pressure resulting from the Pauli
exclusion principle will halt the collapse. The point at which the
contraction stops is determined by the condition that the gravitational
potential energy per electron is comparable to the minimum
kinetic energy of the electron K0. This condition will allow us to
find dmin. When K0 ∼ (G M 2/R)/N, we find by substitution that
dmin ∼ 1/ G N2/3 (me c/).29 So now we have an expression for
the separation distance between electrons at the moment where
degeneracy sets in. This result also allows us to determine the
central temperature at the onset of degeneracy as Tmax ∼ 2
G N4/3
(me c2/k).
We are now nearly at the point where an estimate of the
minimum mass for a star can be established. Indeed, what has
been found is that Tmax is determined solely by the value of N,
the total number of particles in the original gas cloud, all other
terms in its evaluation being fundamental constants. As we saw
earlier in this chapter, a compact, hot cloud of gas becomes a
star (at least in name) once the central regions are hot enough
for steady nuclear fusion reactions to begin, and this requires that
TC ∼ TNUC ∼ 107 K. We also saw earlier in this chapter that once
nuclear ‘burning’ begins the collapse of a star is halted, since it
need not contract any more to replenish the energy radiated into
space at its surface. In this respect the value of Tmax, or more
specifically N, determines the outcome of the collapse of a gas
cloud. If Tmax < TNUC then nuclear reactions will not be initiated
104 Rejuvenating the Sun and Avoiding Other Global Catastrophes
in the cloud, and it will collapse until electron degeneracy sets in,
at which point the cloud will simply begin to radiate its internal
energy into space and cool off. If, on the other hand, Tmax > TNUC
then nuclear fusion reactions will have begun before degeneracy
sets in, and the cloud will become a star. To a first approximation
an estimate for the ‘stardom’ condition can be set as Tmax = TNUC
which requires that N > 1056, which in turn indicates that the
minimum mass for a star is Mmin = 1056 mp ∼ 0.1 M What this
result tells us is that only assemblages of at least 1056 particles (i.e.,
hydrogen atoms) can possibly turn into stable stars with hydrogen
fusion reactions occurring in their cores. Smaller assemblages with
N < 1056 particles will form stable degenerate bodies (i.e., brown
dwarfs) held together by their own gravity, but supported by the
electron overcrowding pressure resulting from the Pauli exclusion
principle.
The Sun contains N ∼ 1057 particles, so it sits appropriately
above Mmin, but we now have to ask the next obvious question:
‘’What is the greatest mass that a star can have?’ We won’t
go through all the details here, but it turns out26 that as Tmax
increases, so the contribution of radiation pressure Prad becomes
more and more important, and once the mass of a star is greater
⇑
Tmax
N
Prad >> Pgas
107 K
STARS
Brown dwarfs
1059 1054 1056 Sun
(Mmax (Jupiter) (Mmin) )
Figure 3.13. Schematic plot of the maximum temperature Tmax obtained
during collapse, against the number of particles N contained in the
collapsing cloud. During collapse, each cloud moves vertically upwards in
the diagram until it intersects the curve corresponding Tmax. No stars can
form in the shaded region to the right of the diagram due to the dominance
of radiation pressure. Likewise, no stars can form for N < 1056 since for
these objects Tmax < TNUC ∼ 107 k.
The Sun, Inside and Out 105
than ∼ 100 M (corresponding to N > 1059 particles) an instability
sets in with the result that a would-be star is disrupted.
Somewhat loosely, one can say that massive clouds ‘bounce,’ since
material is eventually driven outward, back into space, by the
strong radiation pressure. A summary of the possible star-forming
collapse scenarios is shown in Figure 3.13.
A Constraint on Planet Building
In addition to explaining the characteristic masses of stars, the
fundamental constants also set a constraint upon the existence of
planets. We won’t derive the full result here because it is rather
complex, but what Professor Brandon Carter30 has argued is that
if G were just slightly greater than its deduced value (based,
remember, on fundamental constants), then all stars would be
fully convective, low-temperature dwarfs. Further, if G were just
slightly smaller than its actual value, then all stars would be hot
with fully radiative interiors.
This is indeed a remarkable result. If all stars were fully
convective, low-temperature red dwarfs, then there would be
no supernovae and no production of carbon or other elements
essential to life. If all stars were hot and radiative, then it is
presently unclear if there would be any planets. The reason for
this latter claim is a little complex, but relates to the idea that
the spin rate of newly forming stars is linked to the development
of strong magnetic fields within their outer convective layers.
This magnetic braking effect is certainly observed. Hot, massive
stars with radiative envelopes spin rapidly, while cool, low-mass
stars (like the Sun) with convective outer layers spin slowly.
Not only this, current theories of planet formation require that
the material within the accretion disk around a newly forming
star sheds its angular momentum as it spirals inward. The most
likely way of doing this is via the formation of spin-axis aligned
jets constrained by magnetic fields. Indeed, such jets have been
observed in a number of newly forming star systems. In addition,
Luisa Rebull and co-workers at NASA’s Spitzer Science Center31
recently conducted a study of some 900 stars in the Orion Nebula
and found that slow-spinning young stars are five times more likely
106 Rejuvenating the Sun and Avoiding Other Global Catastrophes
to have disks (in which planets might form) than fast-spinning
young stars. There are many tenuous and currently unclear threads
in the arguments just presented, but as Carter points out, ‘’If this
is correct, then a stronger gravitational [fine structure] constant
would be incompatible with the formation of planets and, hence,
presumably of observers.”
Our discussion of the physics of stellar interiors is now
complete. For those who have struggled through the mathematical
arguments, well done! For those who want to see more details, then
take another look at Reference 3. Hopefully, in the meantime, we
have made clear some of the remarkable and elegant ideas underlying
the modern-day theory of stellar structure and evolution. In
the next chapter – before we move on to consider how a future
star engineer might try to manipulate the properties of our Sun –
we will look at the costs and consequences of not intervening in
the Sun’s aging process.
Notes and References
1. The spectral classification scheme is essentially a means of arranging
and recognizing similar stars according to their surface temperature
(as illustrated in Figure 3.8). The classification is based upon the
characteristics of specific absorption lines recorded in stellar spectra.
Importantly, the ‘strength’ of an absorption line varies with temperature.
The classification scheme runs according to the designations O,
B, A, F, G, K, and M (each designation having a set of subtypes—i.e.,
A1, A2…A5). The O and B stars are the hottest stars with atmospheric
temperatures in excess of 15,000 K. The K and M stars have the
coolest atmospheres, with temperatures varying between 5,000 and
3,000 K respectively. The Sun is a G spectral type star. More specifically,
it is a G2 spectral type star, indicating that it has a temperature
of about 6,000 K.
2. The highly recommended and well-produced Extrasolar Planet
Encyclopedia can be found at: http://exoplanet.eu/.
3. A good introductory book on stellar structure and evolution is that
by R. C. Smith, Observational Astrophysics (Cambridge University
Press, Cambridge, 1995). C. J. Hansen and S. D. Kawaler provide a
detailed technical description of the stars in their Stellar Interiors:
The Sun, Inside and Out 107
Physical Principles, Structure and Evolution (Springer-Verlag, New
York, 1994). In a paper entitled Order-of-magnitude “theory” of stellar
structure, [American Journal of Physics, 55 (9), 804–810, (1987)],
George Greenstein develops, in a highly readable fashion, a series of
analytic formulas for a model star. A number of the equations that
are developed in this chapter and used in the next are explained in
the paper: A novel stellar model: ‘a sacrifice before the lesser shrine
of plausibility’ [M. Beech, Astrophysics and Space Science, 168, 253–
261, (1990)].
4. A. S. Eddington, The Internal Constitution of the Stars, Cambridge
University Press, Cambridge (1926).
5. With these boundary conditions we are ignoring the fact that the Sun
has a complex outer structure composed of the chromosphere and
the corona. For the construction of most mathematical stellar models
this is not a real problem, since the outer regions of a star contain
relatively small amounts of mass.
6. The perfect gas equation can be used when the particles in a gas
do not interact with one another. Although stars do have regions
of very high density, the temperature of the stellar gas is so high
that the atoms are broken down into their constituent nuclei and
electrons. The typical spacing between components in the stellar gas
is then much larger than the sizes of its individual components and,
consequently, particle interactions are actually quite rare.
7. The atomic mass number refers to the total number of protons and
neutrons in the atomic nucleus. In many atomic species there are
equal numbers of protons and neutrons in the nucleus and, consequently,
the number of electrons associated within a neutral atom
will be half the atomic mass number.
8. The relative abundances of many of the Sun‘s constituent elements
can be determined directly from the study and modeling of its
spectrum. The actual abundances of many elements, however, are
typically determined through the laboratory study of primitive (that
is, unprocessed by heat) carbonaceous chondrite meteorites.
9. Detailed computer models indicate that the Sun is centrally
condensed, with 50 percent of its mass being contained within a
region enveloping just the central quarter of its radius. The outer
convection zone, while extending over a third of the Sun‘s radius,
contains about 5 percent of the Sun‘s actual mass.
10. For those who check the algebra there is actually a factor of threequarters
missing from Equation (3.9), but for this order of magnitude
argument we have taken this to be sufficiently close to unity and can
ignore the difference.
1
The Sun, Inside and Out 109
17. If we define stars as being objects that can initiate hydrogen fusion
reactions at some stage in their evolution, then a minimum mass
of 0.08 M is required to acquire star status. Brown dwarfs with
masses, MBD, such that 15 MJupiter < MBD < 0.08 M are neither stars
nor large Jovian-like planets, but intermediate objects destined to
eventually become low-mass black dwarfs. Brown dwarfs can undergo
a deuterium fusion reaction phase (D + P ⇒ 3He +
), while Jupiter-like
planets generate internal energy through gravitational contraction.
18. Neutron stars are objects that contain a solar mass of material in a
sphere of radius 20 km or so. The exceptionally high density encountered
within neutron stars results in the neutrons becoming degenerate
and this quantum mechanical effect can support the star against
gravitational collapse. There is, however, an upper limit to the mass
of a stable neutron star and this is estimated to be between 2 and 3
solar masses.
19. Ironically, the greenhouse warming of Earth‘s atmosphere that is
currently of great concern was of vital importance when the Sun was
younger and less luminous. Without some additional atmospheric
warming Earth‘s early oceans would have frozen over, substantially
altering the global climate and presumably delaying the onset time
for the first emergence of life.
20. See, for example, I-J. Sackmann, A. J. Boothroyd, and K. E. Kraemer,
Our Sun III: present and future, The Astrophysical Journal, 418,
457–468 (1993); P. Schrder, R. Smith, and K. Apps, Solar evolution and
the distant future of Earth, Astronomy and Geophysics, 42, 6.26–6.29
(2001).
21. One, of course, should never say infinite. Who knows what strange
and currently unknown physics dictates the very long-term properties
of matter. Nonetheless, it is currently thought that black dwarfs
should remain stable for time periods many orders of magnitude
greater than the current age of the universe.
22. In this argument we are assuming that there is a step-function, or
jump, in the quantity / at the core-envelope boundary. Detailed
computer models, however, indicate that the composition varies over
an extended zone. This, however, does not substantially alter the
argument.
23. The reduction in the pressure scale height – the height over which
the pressure falls by a factor of e ≈ 2.718 – implies that the Sun‘s
radius must increase for the pressure to vanish at its surface.
24. I have considered the effects of extreme central mass concentration
in the article The formation of red-giants, Astronomy and Astrophysics,
156, 391–392, (1986). One of the results from this study
110 Rejuvenating the Sun and Avoiding Other Global Catastrophes
was that the radius of a red-giant is determined (at least in part)
by the mass of its central core. Peter Eggleton and R. C. Cannon
[A conjecture regarding the evolution of dwarf stars into red-giants,
Astrophysical Journal, 383, 757–760 (1991)] have further argued that
stars swell up in response to the development of a composition
gradient produced by a hydrogen-shell burning source situated at the
outer boundary of a star‘s inert core. A. P. Whitworth has ‘experimented’
with detained numerical models and in his article, Why redgiants
are giant [Monthly Notices of the Royal Astronomical Society,
236, 505–544, (1989)] finds that in addition to a molecular weight
jump, increased central mass concentration and the presence of a
hydrogen-burning shell, the opacity variation in a star‘s envelope is
highly important in the production of an extended radius. See also the
more recent article by Daiichiro Sugimoto and Masayuki Fujimoto,
Why Stars Become Red-Giants, Astrophysical Journal, 538, 837–853,
(2000).
25. All of the carbon and oxygen atoms that exist in our universe were
made inside of massive stars. The carbon atoms are produced through
the triple-alpha reaction 3 4He ⇒12 C + energy (see Note 15). The
only reason this reaction actually ‘works,’ however, is because of the
presence of an excited state of 12C at the end point of the triple-alpha
reaction. That this coincidence exists is entirely remarkable and, as
Mario Livio and co-workers comment in their paper, The anthropic
significance of the existence of the excited state of 12C [Nature, 340,
281–284 (1989)], “[As] is consistent with the anthropic principle, the
energy of the resonant level of 12C is required to have the value it
does, to ensure carbon production and the consequent development
of carbon-based life.”
26. One of the very best shorter reviews on this topic is by B. J. Carr and
M. J. Rees, The anthropic principle and the structure of the physical
world. Nature, 278, 605–612 (1979). See also the less mathematical
article by John Gribbin and Martin Rees, Cosmic coincidence, New
Scientist magazine, 13 January, 51–54 (1990)
27. Technically, the Pauli exclusion principle applies to so-called spin
½ particles, which includes electrons and neutrons, but not protons.
Pauli‘s exclusion principle is the reason why atoms have different
properties. Without the action of the exclusion principle all electrons
would reside in the lowest energy ground state and chemistry as we
know it would not exist.
28. There is a limit to the support that degenerate electrons can provide.
If the characteristic speed of the electrons becomes relativistic (i.e.,
V ∼ c), then their energy rather than varying as p2/2me will vary as p
The Sun, Inside and Out 111
c, with the result that a limiting mass MC is imposed. If M > MC then
collapse becomes inevitable. It turns out, as shown by Subrahmanyan
Chandrasekhar (see Note 16), that MC ∼ mp/ G
3/2 ∼ 1 M.
29. From the condition K0 ∼ (G M2/R)/N we have 2/ (2 me dmin
2) ∼ G
N 2/3 m2
p/dmin which then rearranges to the expression given in the
text. Note, however, that the 2 has been dropped from the expression
since it is a small number.
30. Brandon Carter, Large number coincidences and the anthropic
principle in cosmology, in Confrontaion of Cosmological Theories
With Observational Data. M. S. Longair (Ed.). IAU Symposium
No. 63. Reidel, Holland (1974). pp. 291–298. Carter shows that normal
stars fall between the two extremes of being either cool and fully
convective, or hot and radiative throughout their interiors, provided
12 (me/mp)
5 ∼ G, where is the electromagnetic fine scale constant.
This condition, in our universe, is only just satisfied with the two
sides having numerical values of 2.3 x 10−39 and 5 x 10−39, respectively.
31. L. Rebull et al., A correlation between pre-main sequence stellar
rotation rates and IRAS excesses in Orion, Astrophysical Journal 646,
297–303 (2006).
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