Chapter 7

Stellar models and stellar stability
In the previous chapters we have reviewed the most important physical processes taking place instellar interiors, and we derived the differential equations that determine the structure and evolution ofa star. By putting these ingredients together we can construct models of spherically symmetric stars.
Because the complete set of equations is highly non-linear and time-dependent, their full solution
requires a complicated numerical procedure. This is what is done in detailed stellar evolution codes,
the results of which will be described in later chapters. We will not go into any detail about thenumerical methods commonly used in such codes – for those interested, some of these details may befound in Chapter 24.2 of Maeder or Chapter 11 of Kippenhanhn.
The main purpose of this chapter is to briefly analyse the differential equations of stellar evolution
and their boundary conditions, and to see how the full set of equations can be simplified in somecases to allow simple or approximate solutions – so-called simple stellar models. We also address thequestion of the stability of stars – whether the solutions to the equations yield a stable structure ornot.

7.1 The differential equations of stellar evolution
Let us collect and summarize the differential equations for stellar structure and evolution that we havederived in the previous chapters, regarding m as the spatial variable, i.e. eqs. (2.6), (2.11), (5.4), (5.17)
and (6.41):∂r
1=∂m 4πr2 ρ

(7.1)

Gm
∂P
1 ∂2 r=−−∂m4πr44πr2 ∂t

(7.2)

∂l
∂s
= ǫnuc − ǫν − T
∂m
∂t

(7.3)

∂T
Gm T
=−
∇
∂m
4πr4 P

with


lP
3κ




 ∇rad = 16πacG mT 4
∇=



 ∇ad + ∆∇!
X
X
∂Xi Ai mu−(1 + δi j ) ri j +rkl,i [+ mixing terms]
=
∂t
ρ
j
k,l
97

if ∇rad ≤ ∇ad(7.4)

if ∇rad > ∇adi = 1...N(7.5)

Note that eq. (7.2) is written in its general form, without pre-supposing hydrostatic equilibrium. In
eq. (7.3) we have replaced ∂u/∂t −(P/ρ2 )∂ρ/∂t by T ∂s/∂t, according to the combined first and second
laws of thermodynamics. Eq. (7.4) is generalized to include both the cases of radiative and convectiveenergy transport. The term ∆∇ is the superadiabaticity of the temperature gradient that must follow
from a theory of convection (in practice, the mixing length theory); for the interior one can take∆∇ = 0 except in the outermost layers of a star. Finally, eq. (7.5) has been modified to add ‘mixing
terms’ that describe the redistribution (homogenization) of composition in convective regions. There
are N such equations, one for each nucleus (isotope) indicated by subscript i.
The set of equations above comprise 4 + N partial differential equations that should be solved
simultaneously. Let us count the number of unknown variables. Making use of the physics discussed
in previous chapters, the functions P, s, κ, ∇ad , ∆∇, ǫnuc , ǫν and the reaction rates ri j can all be
expressed as functions of ρ, T and composition Xi . We are therefore left with 4+N unknown variables(r, ρ, T , l and the Xi ) so that we have a solvable system of equations.
The variables r, ρ, T , l and Xi appearing in the equations are all functions of two independent
variables, m and t. We must therefore find a solution to the above set of equations on the interval
0 ≤ m ≤ M for t > t0 , assuming the evolution starts at time t0 . Note that M generally also depends ont in the presence of mass loss. A solution therefore also requires specification of boundary conditions(at m = 0 and m = M) and of initial conditions, for example Xi (m, t0 ).

7.1.1

Timescales and initial conditions

Let us further analyse the equations. Three kinds of time derivatives appear:• ∂2 r/∂t2 in eq. (7.2), which describes hydrodynamical changes to the stellar structure. Theseoccur on the dynamical timescale τdyn which as we have seen is very short. Thus we cannormally assume hydrostatic equilibrium and ∂2 r/∂t2 = 0, in which case eq. (7.2) reduces tothe ordinary differential equation (2.13). Note that HE was explicitly assumed in eq. (7.4).
• T ∂s/∂t in eq. (7.3), which is often written as an additional energy generation term (eq. 5.5):
ǫgr = −T∂s∂u P ∂ρ=− + 2∂t∂t ρ ∂t

It describes changes to the thermal structure of the star, which can result from contraction(ǫgr > 0) or expansion (ǫgr < 0) of the layers under consideration. Such changes occur on thethermal timescale τKH . If a star evolves on a much longer timescale than τKH then ǫgr ≈ 0
and the star is in thermal equilibrium. Then also eq. (7.3) reduces to an ordinary differential
equation, eq. (5.7).
• ∂Xi /∂t in eqs. (7.5), describing changes in the composition. For the most abundant elements –the ones that affect the stellar structure – such changes normally occur on the longest, nuclear
timescale τnuc .
Because normally τnuc ≫ τKH ≫ τdyn , composition changes are usually very slow compared to theother time derivatives. In that case eqs. (7.5) decouple from the other four equations (7.1–7.4), which
can be seen to describe the stellar structure for a given composition Xi (m).
For a star in both HE and TE (also called ‘complete equilibrium’), the stellar structure equations (7.1–7.4) become a set of ordinary differential equations, independent of time. In that case it issufficient to specify the initial composition profiles Xi (m, t0 ) as initial conditions. This is the case forso-called zero-age main sequence stars: the structure at the start of the main sequence depends only98

on the initial composition, and is independent of the uncertain details of the star formation process, avery fortunate circumstance!
If a star starts out in HE, but not in TE, then the time derivative represented by ǫgr remains in theset of structure equations. One would then also have to specify the specific entropy profile s(m, t0 )as an initial condition. This is the case if one considers pre-main sequence stars. Fortunately, as weshall see later, in this case there is also a simplifying circumstance: pre-main sequence stars start out
as fully convective gas spheres. This means that their temperature and pressure stratification is nearly
adiabatic, so that s can be taken as constant throughout the star. It then suffices to specify the initial
entropy.

7.2 Boundary conditions
The boundary conditions for the differential equations of stellar evolution constitute an important partof the overall problem. Not all boundary conditions can be specified at one end of the interval [0, M]:some boundary conditions are set in the centre and others at the surface. This means that direct
forward integration of the equations is not possible, and the influence of the boundary conditions on
the solutions is not easy to foresee.

7.2.1

Central boundary conditions

At the centre (m = 0), both the density and the energy generation rate must remain finite. Therefore,both r and l must vanish in the centre:
m=0:

r = 0 andl = 0.

(7.6)

However, nothing is known a priori about the central values of P and T . Therefore the remaining two
boundary conditions must be specified at the surface rather than the centre.
It is possible to get some idea of the behaviour of the variables close to the centre by means of a
Taylor expansion. Even though Pc and T c are unknown, one can do this also for P and T , writing forexample
" #
" 2 #
dP
1 2 d PP = Pc + m
+ ...
+ m
dm c 2dm2 c
and making use of the stellar structure equations for dP/dm, etc, see Exercise 7.5.

7.2.2

Surface boundary conditions

At the surface (m = M, or r = R), the boundary conditions are generally much more complicated thanat the centre. One may treat the surface boundary conditions at different levels of sophistication.
• The simplest option is to take T = 0 and P = 0 at the surface (the ‘zero’ boundary conditions).
However, in reality T and P never become zero because the star is surrounded by an interstellar
medium with very low, but finite density and temperature.
• A better option is to identify the surface with the photosphere, which is where the bulk of theradiation escapes and which corresponds with the visible surface of the star. The photosphericboundary conditions approximate the photosphere with a single surface at optical depth τ = 32 .
We can write
Z ∞
Z ∞
τph =
κρ dr ≈ κph
ρ dr,
(7.7)
R

R

99

where κph is an average value of opacity over the atmosphere (all layers above the photosphere).
If the atmosphere is geometrically thin we also have
Z
GM
dP
GM ∞
= − 2 ρ ⇒ P(R) ≈ 2
ρ dr.
(7.8)
dr
R
R
R
Since τph = 23 and T (R) ≈ T eff we can combine the above equations to write the photosphericboundary conditions as:
m = M (r = R)

:

P=

2 GM
3 κph R2

and

L = 4πR2 σT 4 .

(7.9)• The problem with the photospheric boundary conditions above is that the radiative diffusion
approximation on which it is based breaks down when τ <∼ a few. The best solution is therefore
to fit a detailed stellar atmosphere model to an interior shell (at τ > 23 ) where the radiative diffusion approximation is still valid. This is a more complicated and time-consuming approach,and in many (but not all) practical situations the photospheric conditions are sufficient.

7.2.3

Effect of surface boundary conditions on stellar structure

It is instructive to look at the effect of the surface boundary conditions on the solution for the structureof the outer envelope of a star. Assuming complete (dynamical and thermal) equilibrium, the envelope
contains only a small fraction of the mass and no energy sources. In that case l = L and m ≈ M. It isthen better to take P, rather than m, as the independent variable describing depth within the envelope.
We can write the equation for radiative energy transport as
dT
κl
T
3
L κ
= ∇rad =≈ const ·3
dP P
16πacG mT
M T3(7.10)

Figure 7.1. Schematic diagram of log T versus log P illustrating the different types of envelope structure
solutions, as discussed in the text. Figure from Kippenhahn & Weigert.

100

log T (K)7

Figure 7.2. Structure of detailed stellar models on
the zero-age main sequence in the log P, log T diagram. Each curve is for a different mass, from top
to bottom: 16 M⊙ , 8 M⊙ , 4 M⊙ , 2 M⊙ , 1 M⊙ , 0.5 M⊙
and 0.25 M⊙ . The + symbols on each curve indicate,for increasing P, the part of the envelope containing
0.01M, 0.1M and 0.5M (so most of the T and P variation in the envelope occurs in the outer 1 % of themass). The dotted (blue) parts of each curve indicate
radiative regions of the star, the solid (red) parts indicate convective regions.

ZAMS models, Z = 0.02

6

5

Stars with M ≤ 1 M⊙ have low T eff and therefore
convective envelopes. The depth of the convective
envelope increases strongly with decreasing surface
temperature (and thus with decreasing mass); the0.25 M⊙ star is completely convective. On the other
hand, more massive stars have higher T eff and mostly
radiative envelopes, except for small convective layers near the surface caused by partial ionization.

4
5

10

15

log P (dyn/cm2)

We also approximate the opacity by a simple law, κ = κ0 Pa T b (this can represent e.g. Kramers opacity,a = 1, b = −4.5; or electron scattering, a = b = 0). We can then integrate dT/dP to giveT 4−b = B (P1+a + C)

(7.11)

where B ∝ L/M = constant and C is an integration constant, determined by the boundary conditions.
For the Kramers opacity, which is a reasonable approximation for stellar envelopes of moderate temperatures, we find T 8.5 = B (P2 + C).
The different possible solutions are characterized by the√value of C, and the various possibilities
are illustrated in Fig. 7.1. At large enough pressure, P ≫ C, all solutions approach T ∝ P2/8.5 ≈P0.235 . This corresponds to an actual temperature gradient, ∇ = d log T/d log P = 0.235 < ∇ad ≈ 0.4,
which is consistent with the assumed radiative transport. There is a fundamental difference between
solutions with C ≥ 0 and C < 0, however.
Radiative envelopes correspond to solutions with C ≥ 0. In the special case C = 0 the slopeof the solution ∇ remains equal to 0.235 when P, T → 0. This corresponds to the ‘zero’
boundary conditions discussed above. For C > 0 the solutions lie above the C = 0 line,
and the slope decreases (∇ < 0.235) as the surface is approached. This corresponds to more
realistic, e.g. photospheric, boundary conditions with large enough T eff , demonstrating thatstars with relatively hot photospheres have radiative envelopes. In practice this is the casewhen T eff >∼ 9000 K. Fig. 7.1 demonstrates that such envelope solutions quickly approach the‘radiative zero’ structure, C = 0. This means that the envelope structure is insensitive to theassumed surface boundary conditions, and in practice the photospheric BC’s are sufficient.
Convective envelopes correspond to solutions with C < 0. In this case the solutions lie below theC = 0 line, and their slope increases as log P decreases. This is shown by the dotted line in
Fig. 7.1. However, the assumption of radiative transport breaks down when ∇ > ∇ad ≈ 0.4
and convection sets in. Therefore, stars with cool photospheres have convective envelopes, inpractice when T eff <∼ 9000 K. The actual temperature stratification in the envelope is close toadiabatic, ∇ = ∇ad , until the surface is approached. Before this happens, however, partial ionization decreases ∇ad below 0.4 and gives rise to a much shallower slope. Since the different
101solution lie close together near the surface, but further apart in the interior, the structure of aconvective envelope is sensitive to the surface boundary conditions. This means that the structure also depends on the uncertain details of near-surface convection (see Sec. 5.5). A smallchange or uncertainty in T eff can have a large effect on the depth of the convective envelope!
For small enough T eff the whole star can become convective (leading to the Hayashi line in the
H-R diagram, see Sect. 9.1.1).
The approximate description given here is borne out by detailed stellar structure calculations, as
demonstrated in Fig. 7.2. Also note that if we assume electron scattering instead of Kramers opacity,
the description remains qualitatively the same (the radiative zero solution then has ∇ = 0.25 insteadof 0.235).

7.3 Equilibrium stellar models
For a star in both hydrostatic and thermal equilibrium, the four partial differential equations for stellar
structure (eqs. 7.1–7.4) reduce to ordinary, time-independent differential equations. We can further
simplify the situation somewhat, by ignoring possible neutrino losses (ǫν ) which are only important invery late stages of evolution, and ignoring the superadiabaticity of the temperature gradient in surface
convection zones. We then arrive at the following set of structure equations that determine the stellar
structure for a given composition profile Xi (m):dr1=dm 4πr2 ρ(7.12)

Gm
dP
=−
dm
4πr4

(7.13)dl= ǫnucdm(7.14)

Gm T
dT
=−
∇
dm
4πr4 P

with


3κ
lP




 ∇rad = 16πacG mT 4
∇=



 ∇ad

if ∇rad ≤ ∇ad(7.15)

if ∇rad > ∇ad

We note that the first two equations (7.12 and 7.13) describe the mechanical structure of the star, andthe last two equations (7.14 and 7.15) describe the thermal and energetic structure. They are coupled
to each other through the fact that, for a general equation of state, P is a function of both ρ and T .
Although simpler than the full set of evolution equations, this set still has no simple, analytic
solutions. The reasons are that, first of all, the equations are very non-linear: e.g. ǫnuc ∝ ρT ν withν ≫ 1, and κ is a complicated function of ρ and T . Secondly, the four differential equations arecoupled and have to be solved simultaneously. Finally, the equations have boundary conditions at
both ends, and thus require iteration to obtain a solution.
It is, however, possible to make additional simplifying assumptions so that under certain circumstances an analytic solution or a much simpler numerical solution is possible. We have alreadydiscussed one example of such a simplifying approach in Chapter 4, namely the case of polytropic
models in which the pressure and density are related by an equation of the form
P = K ργ .
Since in this case P does not depend on T , the mechanical structure of a stellar model can be computed
in a simple way, independent of its thermal and energetic properties, by solving eqs. (7.12) and (7.13).
102

Another approach is to consider simple scaling relations between stellar models with different
masses and radii, but all having the same (or a very similar) relative density distributions. If a detailed
numerical solution can be computed for one particular star, these so-called homology relations can beused to find an approximate model for another star.

7.4 Homology relations
Solving the stellar structure equations almost always requires heavy numerical calculations, such as
are applied in detailed stellar evolution codes. However, there is often a kind of similarity between
the numerical solutions for different stars. These can be approximated by simple analytical scaling
relations known as homology relations. In past chapters we have already applied simple scaling
relations based on rough estimates of quantities appearing in the stellar structure equations. In thissection we will put these relations on a firmer mathematical footing.
The requirements for the validity of homology are very restrictive, and hardly ever apply to realistic stellar models. However, homology relations can offer a rough but sometimes very helpful
basis for interpreting the detailed numerical solutions. This applies to models for stars on the main
sequence and to so-called homologous contraction.
Definition Compare two stellar models, with masses M1 and M2 and radii R1 and R2 . All interior
quantities in star 1 are denoted by subscript ’1’ (e.g. the mass coordinate m1 ), etc. Now consider
so-called homologous mass shells which have the same relative mass coordinate, x ≡ m/M, i.e.
x=

m1
m2
=
M1
M2

(7.16)The two stellar models are said to be homologous if homologous mass shells within them arelocated at the same relative radii r/R, i.e.
r1 (x) r2 (x)
=
R1
R2

or

r1 (x) R1=
r2 (x) R2(7.17)

for all x.
Comparing two homologous stars, the ratio of radii r1 /r2 for homologous mass shells is constant. In
other words, two homologous stars have the same relative mass distribution, and therefore (as weshall prove shortly) the same relative density distribution.
All models have to obey the stellar structure equations, so that the transition for one homologous
model to another has consequences for all other variables. We start by analysing the first two structure
equations.
• The first stellar structure equation (7.12) can be written for star 1 as
M1
dr1
=
dx
4πr1 2 ρ1(7.18)

If the stars are homologous, then from eq. (7.17) we can substitute r1 = r2 (R1 /R2 ) and obtain#"M2
ρ2 M1  R2 3
dr2
.
=
·
dx
4πr2 2 ρ2 ρ1 M2 R1(7.19)
103

We recognize the structure equation for the radius of star 2 (i.e. eq. 7.18 with subscript ‘1’ replaced by ‘2’), multiplied by the factor in square brackets on the right-hand side. This equationmust hold generally, which is only the case if the factor in square brackets is equal to one forall values of x, that is ifρ2 (x) M2  R2 −3.
(7.20)=
ρ1 (x) M1 R1This must hold at any homologous mass shell, and therefore also at the centre of each star. The
factor M R−3 is proportional to the average density ρ̄, so that the density at any homologous
shell scales with the central density, or with the average density:
ρ(x) ∝ ρc ∝ ρ̄

(7.21)

Note that, therefore, any two polytropic models with the same index n are homologous to eachother.
• We can apply a similar analysis to the second structure equation (7.13) for hydrostatic equilibrium. For star 1 we havedP1
GM1 2 x=−dx
4πr1 4(7.22)

so that after substituting r1 = r2 (R1 /R2 ) we obtain
"
#
#
"
dP1
dP2  M1 2  R2 4
GM2 2 x  M1 2  R2 4
·
=
,
=−
·
dx
M2
R1
dx
M2
R1
4πr2 4(7.23)

where the second equality follows because star 2 must also obey the hydrostatic equilibrium
equation. Hence we have dP1 /dx = C dP2 /dx, with C equal to the (constant) factor in square
brackets. Integrating we obtain P1 (x) = C P2 (x) + B, where the integration constant B = 0
because at the surface, x → 1, for both stars P → 0. Thus we obtain
P2 (x)M2=
P1 (x)
M1

!2

R2
R1

!−4

,

(7.24)

at any homologous mass shell. Again this must include the centre, so that for all x:
P(x) ∝ Pc ∝

M2
.
R4

(7.25)The pressure required for hydrostatic equilibrium therefore scales with M 2 /R4 at any homologous shell. Note that we found the same scaling of the central pressure with M and R from ourrough estimate in Sect. 2.2, and for polytropic models of the same index n.
We can combine eqs. (7.20) and (7.24) to show that two homologous stars must obey the following
relation between pressure and density at homologous points,
!2/3
!4/3
ρ2 (x)M2
P2 (x)=,(7.26)
P1 (x)M1
ρ1 (x)
orP(x) ∝ M 2/3 ρ(x)4/3 .

(7.27)
104

7.4.1

Homology for radiative stars composed of ideal gas

In order to obtain simple homology relations from the other structure equations, we must make additional assumptions. We start by analysing eq. (7.15).
• First, let us assume the ideal gas equation of state,
P=

R
ρT.
µ

Let us further assume that in each star the composition is homogeneous, so that µ is a constantfor both stars, though not necessarily the same. We can then combine eqs. (7.20) and (7.24) toobtain a relation between the temperatures at homologous mass shells,
T 2 (x) µ2 M2 R2=T 1 (x) µ1 M1 R1

!−1

or

T (x) ∝ T c ∝ µMR(7.28)• Second, we will assume the stars are in radiative equilibrium. We can then write eq. (7.15) as
3M κld(T 4 )=−
dx
16π2 ac r4(7.29)

This contains two as yet unknown functions of x on the right-hand side, κ and l. We must therefore make additional assumptions about the opacity, which we can very roughly approximate
by a power law,
κ = κ0 ρa T b .

(7.30)For a Kramers opacity law, we would have a = 1 and b = −3.5. However, for simplicity let us
assume a constant opacity throughout each star (but like µ, not necessarily the same for bothstars). Then a similar reasoning as was held above for the pressure, allows us to transformeq. (7.29) into an expression for the ratio of luminosities at homologous points,
T 2 (x)T 1 (x)!4l2 (x) M2 κ2 R2=
l1 (x) M1 κ1 R1!−4⇒

l2 (x)µ2=
l1 (x)
µ1

!4

M2
M1

!3

κ2
κ1

!−1

(7.31)

making use of eq. (7.28) to obtain the second expression. This relation also holds for the surface
layer, i.e. for the total stellar luminosity L. Hence l(x) ∝ L and
L∝

1 4 3
µ M
κ

(7.32)

This relation represents a mass-luminosity relation for a radiative, homogeneous star with constant opacity and ideal-gas pressure.
Note that we obtained a mass-luminosity relation (7.32) without making any assumption about themode of energy generation (and indeed, without even having to assume thermal equilibrium, because
we have not yet made use of eq. 7.14). We can thus expect a mass-luminosity relation to hold not
only on the main-sequence, but for any star in radiative equilibrium. What this relation tells us is thatthe luminosity depends mainly on how efficiently energy can be transported by radiation: a higher
105

opacity gives rise to a smaller luminosity, because the nontransparent layers work like a blanket
wrapped around the star. In practice, for a star in thermal equilibrium (e.g. on the main sequence) the
power generated by nuclear reactions Lnuc adapts itself to the surface luminosity L, and thereby also
the central temperature needed to make the nuclear reactions proceed at the rate dictated by L.
Note, however, that the simple mass-luminosity relation (7.32) depends on the assumption of
constant opacity. If we assume a Kramers opacity law, the mass-luminosity also depends (weakly) on
the radius. It is left as an exercise to show that, in this caseµ7.5 M 5.5.
(7.33)
R0.5
This means that if the opacity is not a constant, there is a weak dependence of the luminosity on themode of energy generation, through the radius dependence (see Sec. 7.4.2).
L∝

7.4.2

Main sequence homology

For stars that are in thermal equilibrium we can make use of the last structure equation (7.14) to derive
further homology relations for the radius as a function of mass. We then have to assume a specificform for the energy generation rate, sayǫnuc = ǫ0 ρT ν

(7.34)

so that eq. (7.14) can be written asdl= ǫ0 M ρT ν
(7.35)dx
By making use of the other homology relations, including the mass-luminosity relation eq. (7.32), we
obtain for a homogeneous, radiative star with constant opacity and consisting of an ideal gas:R ∝ µ(ν−4)/(ν+3) M (ν−1)/(ν+3)(7.36)The slope of this mass-radius relation therefore depends on ν, that is, on the mode of nuclear energy
generation. For main-sequence stars, in which hydrogen fusion provides the energy source, there are
two possibilities, see Table 7.1.
We can also obtain relations between the central temperature and central density and the mass ofa star in thermal equilibrium, by combining the homology relations for the radius (7.36) with thosefor density and temperature (7.20 and 7.28):
ρc ∝ µ3(4−ν)/(ν+3) M 2(3−ν)/(ν+3)
7/(ν+3)Tc ∝ µM(7.37)

4/(ν+3)(7.38)

Again, the result depends on the mode of energy generation through the value of ν. For main-sequencestars the possibilities are tabulated in Table 7.1.
The mass-luminosity and mass-radius relations (7.32) and (7.36) can be compared to the observed
relations for main-sequence stars that were presented in Chapter 1, and to the results of detailed stellar
structure calculations. This comparison is deferred to Chapter 9, where the main sequence is discussed
in more detail.
Table 7.1. Homology relations for the radius, central temperature and central density of main-sequence stars

pp-chainν≈4

CNO cycleν ≈ 18R ∝ M 0.43

R ∝ µ2/3 M 0.81

106

T c ∝ µM 0.57

T c ∝ µ1/3 M 0.19

ρc ∝ M −0.3

ρc ∝ µ−2 M −1.4

7.4.3

Homologous contraction

We have seen in Chapter 2 that, as a consequence of the virial theorem, a star without internal energy
sources must contract under the influence of its own self-gravity. Suppose that this contraction takes
place homologously. According to eq. (7.17) each mass shell inside the star then maintains the same
relative radius r/R. Writing ṙ = ∂r/∂t, etc., this means thatṙ(m) Ṙ= .
r(m) R
Since in this case we compare homologous models with the same mass M, we can replace x by themass coordinate m. For the change in density we obtain from eq. (7.20) thatρ̇(m)Ṙ= −3 ,
ρ(m)R(7.39)and if the contraction occurs quasi-statically, i.e. slow enough to maintain HE, then the change inpressure follows from eq. (7.24),
Ṙ 4 ρ̇(m)Ṗ(m)= −4 =.
P(m)R 3 ρ(m)(7.40)

To obtain the change in temperature for a homologously contracting star, we have to consider theequation of state. Writing the equation of state in its general, differential form eq. (3.48) we caneliminate Ṗ/P to get
 ρ̇
Ṙ
1 4
1
Ṫ
=
− χρ
=
(3χρ − 4) .
(7.41)
T
χT 3
ρ χT
R
Hence the temperature increases as a result of contraction as long as χρ < 43 . For an ideal gas,with χρ = 1, the temperature indeed increases upon contraction, in accordance with our (qualitative)
conclusion from the virial theorem. Quantitatively,
1 ρ̇
Ṫ
=
.
T
3ρ
However, for a degenerate electron gas with χρ = 53 eq. (7.41) shows that the temperature decreases,
in other words a degenerate gas sphere will cool upon contraction. The full consequences of thisimportant result will be explored in Chapter 8.

7.5 Stellar stability
We have so far considered stars in both hydrostatic and thermal equilibrium. But an important question that remains to be answered is whether these equilibria are stable. From the fact that stars canpreserve their properties for very long periods of time, we can guess that this is indeed the case. But
in order to answer the question of stability, and find out under what circumstances stars may become
unstable, we must test what happens when the equilibrium situation is perturbed: will the perturbation be quenched (stable situation) or will it grow (unstable situation). Since there are two kinds ofequilibria, we have to consider two kinds of stability:• dynamical stability: what happens when hydrostatic equilibrium is perturbed?
• thermal (secular) stability: what happens when the thermal equilibrium situation is perturbed?
1077.5.1

Dynamical stability of stars

The question of dynamical stability relates to the response of a certain part of a star to a perturbationof the balance of forces that act on it: in other words, a perturbation of hydrostatic equilibrium. We
already treated the case of dynamical stability to local perturbations in Sec. 5.5.1, and saw that in thiscase instability gives rise to convection. In this section we look at the global stability of concentric
layers within a star to radial perturbations, i.e. compression or expansion. A rigorous treatment of
this problem is very complicated, so we will only look at a very simplified example to illustrate theprinciples.
Suppose a star in hydrostatic equilibrium is compressed on a short timescale, τ ≪ τKH , so thatthe compression can be considered as adiabatic. Furthermore suppose that the compression occurs
homologously, such that its radius decreases from R to R′ . Then the density at any layer in the star
becomes!−3
R′′ρ→ ρ =ρ
R
and the new pressure after compression becomes P′ , given by the adiabatic relation!γad!−3γad
P′
ρ′
R′
=
=
.
P
ρ
R
The pressure required for HE after homologous contraction is
!
!4/3
!−4
ρ′
P′
R′
=
=
P HE
ρ
R
Therefore, if γad > 34 then P′ > P′HE and the excess pressure leads to re-expansion (on the dynamicaltimescale τdyn ) so that HE is restored. If, however, γad < 43 then P′ < P′HE and the increase of pressure
is not sufficient to restore HE. The compression will therefore reinforce itself, and the situation isunstable on the dynamical timescale. We have thus obtained a criterion for dynamical stability:
γad > 34

(7.42)

It can be shown rigorously that a star that has γad > 43 everywhere is dynamically stable, and if γad = 43
it is neutrally stable. However, the situation when γad < 43 in some part of the star requires further
investigation. It turns out that global dynamical instability is obtained when the integral
Z 
4P
dm
(7.43)γad −3 ρover the whole star is negative. Therefore if γad < 43 in a sufficiently large core, where P/ρ is high,the star becomes unstable. However if γad < 34 in the outer layers where P/ρ is small, the star as awhole need not become unstable.
Cases of dynamical instability
Stars dominated by an ideal gas or by non-relativistic degenerate electrons have γad = 53 and are
therefore dynamically stable. However, we have seen that for relativistic particles γad → 34 and stars
dominated by such particles tend towards a neutrally stable state. A small disturbance of such a star
could either lead to a collapse or an explosion. This is the case if radiation pressure dominates (athigh T and low ρ), or the pressure of relativistically degenerate electrons (at very high ρ).
108

A process that can lead to γad < 43 is partial ionization (e.g. H ↔ H+ + e− ), as we have seenin Sec. 3.5. Since this normally occurs in the very outer layers, where P/ρ is small, it does notlead to overall dynamical instability of the star. However, partial ionization is connected to driving
oscillations in some kinds of star.
At very high temperatures two other processes can occur that have a similar effect to ionization.
These are pair creation (γ + γ ↔ e+ + e− , see Sect. 3.6.2) and photo-disintegration of nuclei (e.g.
γ + Fe ↔ α). These processes, that may occur in massive stars in late stages of evolution, also leadto γad < 43 but now in the core of the star. These processes can lead to a stellar explosion or collapse(see Chapter 13).

7.5.2

Secular stability of stars

The question of thermal or secular stability, i.e. the stability of thermal equilibrium, is intimately
linked to the virial theorem. In the case of an ideal gas the virial theorem (Sect. 2.3) tells us that the
total energy of a star is
Etot = −Eint = 12 Egr ,(7.44)

which is negative: the star is bound. The rate of change of the total energy is given by the difference
between the rate of nuclear energy generation in the deep interior and the rate of energy loss in theform of radiation from the surface:Ėtot = Lnuc − L(7.45)In a state of thermal equilibrium, L = Lnuc and Etot remains constant. Consider now a small perturbation of this situation, for instance Lnuc > L because of a small temperature fluctuation. This leads
to an increase of the total energy, δEtot > 0, and since the total energy is negative, its absolute value
becomes smaller. The virial theorem, eq. (7.44), then tells us that (1) δEgr > 0, in other words the star
will expand (δρ < 0), and (2) δEint < 0, meaning that the overall temperature will decrease (δT < 0).
Since the nuclear energy generation rate ǫnuc ∝ ρT ν depends on positive powers of ρ and especially
T , the total nuclear energy generation will decrease: δLnuc < 0. Eq. (7.45) shows that the perturbation
to Etot will be quenched and the state of thermal equilibrium will be restored.
The secular stability of nuclear burning thus depends on the negative heat capacity of stars composed of ideal gas: the property that an increase of the total energy content leads to a decrease of
the temperature. This property provides a thermostat that keeps the temperature nearly constant andkeeps stars in a stable state of thermal equilibrium for such long time scales.
We can generalise this to the case of stars with appreciable radiation pressure. For a mixture ofideal gas and radiation we can write, with the help of eqs. (3.11) and (3.12),
P Pgas Prad 2=+= 3 ugas + 13 urad .
ρρρ

(7.46)

Applying the virial theorem in its general form, eq. (2.24), this yields
2 Eint,gas + Eint,rad = −Egr(7.47)and the total energy becomes
Etot = −Eint,gas = 12 (Egr + Eint,rad ).

(7.48)The radiation pressure thus has the effect of reducing the effective gravitational potential energy. If
β = Pgas /P is constant throughout the star, then eq. (7.48) becomes
Etot = 21 β Egr(7.49)
109

This is negative as long as β > 0. The analysis of thermal stability is analogous to the case of an ideal
gas treated above, and we see that stars in which radiation pressure is important, but not dominant,are still secularly stable. However, if β → 0 then the thermostatic effect no longer works.
Thermal instability of degenerate gases
In the case of a degenerate electron gas, the pressure and the internal energy are independent of thetemperature (Sec. 3.3.5). The mechanical structure of an electron-degenerate star – or the degeneratecore of an evolved star – is therefore independent of the thermal-energetic structure (Sec. 7.3). If
the same perturbation Lnuc > L discussed above is applied to a degenerate gas, the resulting energy
input will have no effect on the electron pressure and on the stellar structure. Therefore there will be
no expansion and cooling. Instead, there will be a temperature increase because the ionized atomic
nuclei still behave as an ideal gas, and the energy input will increase their thermal motions. Thus theeffect of the perturbation will be δT > 0, while δρ ≈ 0.
Because of the strong sensitivity of the nuclear energy generation rate to T , the perturbation will
now lead to an increase of Lnuc , and thermal equilibrium will not be restored. Instead, the temperature
will continue to rise as a result of the increased nuclear energy release, which in turn leads to further
enhancement of the energy generation. This instability is called a thermonuclear runaway, and it
occurs whenever nuclear reactions ignite in a degenerate gas. In some cases it can lead to the explosionof the star, although a catastrophic outcome can often be avoided when the gas eventually becomes
sufficiently hot to behave as an ideal gas, for which the stabilizing thermostat operates. This can beseen from eq. (7.41), valid in the case of homologous expansion, which we can write as
 δρ
δT
1 4
=
− χρ
.
(7.50)T
χT 3
ρ
As soon as the gas is heated enough that it is no longer completely degenerate, χT > 0 and some
expansion will occur (δρ < 0), while χρ decreases below 53 . From eq. (7.50) we see that when χρdrops below the critical value of 43 , δT changes sign and becomes negative upon further expansion.
We shall encounter several examples of thermonuclear runaways in future chapters. The mostcommon occurrence is the ignition of helium fusion in stars with masses below about 2 M⊙ – thisphenomenon is called the helium flash. Thermonuclear runaways also occur when hydrogen gas
accumulates on the surface of a white dwarf, giving rise to so-called nova outbursts.
The thin shell instability
In evolved stars, nuclear burning can take place in a shell around an inert core. If such a burning
shell is sufficiently thin the burning may become thermally unstable, even under ideal-gas conditions.
We can make this plausible by considering a shell with mass ∆m inside a star with radius R, locatedbetween a fixed inner boundary at r0 and outer boundary at r, so that its thickness is d = r − r0 ≪ R.
If the shell is in thermal equilibrium, the rate of nuclear energy generation equals the net rate of heatflowing out of the shell (eq. 7.14). A perturbation by which the energy generation rate exceeds therate of heat flow leads to expansion of the shell, pushing the layers above it outward (δr > 0). This
leads to a decreased pressure, which in hydrostatic equilibrium is given by eq. (7.40),
δPδr= −4 .
(7.51)Pr
The mass of the shell is ∆m = 4πr0 2 ρ d, and therefore the density varies with the thickness of theshell as
δd
δr r
δρ
=− =−
.
(7.52)
ρdr d110

Eliminating δr/r from the above equations yields a relation between the changes in pressure anddensity,
δP
d δρ
=4
.
P
r ρ

(7.53)

Combining with the equation of state in its general, differential form eq. (3.48) we can eliminate δP/P
to obtain the resulting change in temperature,
 δρ
δT
1  d
4 − χρ
=.
(7.54)TχT r
ρ
The shell is thermally stable as long as expansion results in a drop in temperature, i.e. whend> χρ
(7.55)
rsince χT > 0. Thus, for a sufficiently thin shell a thermal instability will develop. (In the case of an
ideal gas, the condition 7.55 gives d/r > 0.25, but this is only very approximate.) If the shell is very
thin, the expansion does not lead to a sufficient decrease in pressure to yield a temperature drop, evenin the case of an ideal gas. This may lead to a runaway situation, analogous to the case of a degenerategas. The thermal instability of thin burning shells is important during late stages of evolution of stars
up to about 8 M⊙ , during the asymptotic giant branch.
4

Suggestions for further reading
The contents of this chapter are also (partly) covered by Chapter 24 of Maeder, where the question ofstability is considered in Section 3.5. A more complete coverage of the material is given in Chapters
9, 10, 19, 20 and 25 of Kippenhahn.

Exercises
7.1 General understanding of the stellar evolution equations
The differential equations (7.1–7.5) describe, for a certain location in the star at mass coordinate m, thebehaviour of and relations between radius coordinate r, the pressure P, the temperature T , the luminosity
l and the mass fractions Xi of the various elements i.
(a) Which of these equations describe the mechanical structure, which describe the thermal-energetic
structure and which describe the composition?
(b) What does ∇ represent? Which two cases do we distinguish?

(c) How does the set of equations simplify when we assume hydrostatic equilibrium (HE)? If weassume HE, which equation introduces a time dependence? Which physical effect does this timedependence represent?

(d) What do the terms ǫnuc and T ∂s/∂t represent?
(e) How does the set of equations simplify if we also assume thermal equilibrium (TE)? Which equation introduces a time dependence in TE?
(f) Equation (7.5) describes the changes in the composition. In principle we need one equation forevery possible isotope. In most stellar evolution codes, the nuclear network is simplified. This
reduces the number of differential equations and therefore increases speed of stellar evolution
codes. The STARS code behind Window to the Stars only takes into account seven isotopes. Whichdo you think are most important?

111

7.2 Dynamical Stability(a) Show that for a star in hydrostatic equilibrium (dP/dm = −Gm/(4πr4 )) the pressure scales withdensity as P ∝ ρ4/3 .

(b) If γad < 4/3 a star becomes dynamically unstable. Explain why.
(c) In what type of stars γad ≈ 4/3?

(d) What is the effect of partial ionization (for example H ⇆ H+ + e− ) on γad ? So what is the effect ofionization on the stability of a star?
(e) Pair creation and photo-disintegration of Fe have a similar effect on γad . In what type of stars (andin what phase of their evolution) do these processes play a role?
7.3 Mass radius relation for degenerate stars(a) Derive how the radius scales with mass for stars composed of a non-relativistic completely degenerate electron gas. Assume that the central density ρc = aρ̄ and that the central pressure
Pc = bGM 2 /R4 , where ρ̄ is the mean density, and a and b depend only on the density distribution
inside the star.
(b) Do the same for an extremely relativistic degenerate electron gas.
(c) The electrons in a not too massive white dwarf behave like a completely degenerate non-relativisticgas. Many of these white dwarfs are found in binary systems. Describe qualitatively what happens
if the white dwarf accretes material from the companion star.
7.4 Main-sequence homology relations
We speak of two homologous stars when they have the same density distribution. To some extent main
sequence stars can be considered as stars with a similar density distribution.
(a) You already derived some scaling relations for main sequence stars from observations in the first
set of exercises: the mass-luminosity relation and the mass-radius relation. Over which mass range
were these simple relations valid?
(b) During the practicum you plotted the density distribution of main sequence stars of different
masses. For which mass ranges did you find that that the stars had approximately the same density
distribution.
(c) Compare the L-M relation derived form observational data with the L-M relation derived fromhomology, eq. (7.32). What could cause the difference? (Which assumptions may not be valid?)(d) Show that, if we replace the assumption of a constant opacity with a Kramers opacity law, themass-luminosity-radius relation becomes eq. (7.33),µ7.5 M 5.5.
R0.5(e) Substitute a suitable mass-radius relation and compare the result of (d) with the observational data
in Fig. 1.3. For which stars is the Kramers-based L-M relation the best approximation? Can you
explain why? What happens at lower and higher masses, respectively?
L∝

7.5 Central behaviour of the stellar structure equations(a) Rewrite the four structure equations in terms of d/dr.
(b) Find how the following quantities behave in the neighbourhood of the stellar center:- the mass m(r),- the luminosity l(r),- the pressure P(r),- the temperature T (r).

112

Chapter 8

Schematic stellar evolution –consequences of the virial theorem
8.1 Evolution of the stellar centre
We will consider the schematic evolution of a star, as seen from its centre. The centre is the pointwith the highest pressure and density, and (usually) the highest temperature, where nuclear burning
proceeds fastest. Therefore, the centre is the most evolved part of the star, and it sets the pace ofevolution, with the outer layers lagging behind.
The stellar centre is characterized by the central density ρc , pressure Pc and temperature T c andthe composition (usually expressed in terms of µ and/or µe ). These quantities are related by theequation of state (EOS). We can thus represent the evolution of a star by an evolutionary track in the(Pc , ρc ) diagram or the (T c , ρc ) diagram.

8.1.1

Hydrostatic equilibrium and the P c -ρ c relation

Consider a star in hydrostatic equilibrium (HE), for which we can estimate how the central pressure scales with mass and radius from the homology relations (Sec. 7.4). For a star that expands orcontracts homologously, we can apply eq. (7.26) to the central pressure and central density to yield
Pc = C · GM 2/3 ρc 4/3(8.1)

where C is a constant. This is a fundamental relation for stars in HE: in a star of mass M that expands
or contracts homologously, the central pressure varies as central density to the power 34 . The valueof the constant C depends on the density distribution in the star. Note that we found the same relation
for polytropic stellar models in Chapter 4, eq. (4.18), where C = Cn depends on the polytopic index.
However, the dependence on n, and hence on the density distibution, is only very weak. For polytropic
models with index n = 1.5 – 3, a range that encompasses most actual stars, C varies between 0.48 and
0.36. Hence relation (8.1) is reasonably accurate, even if the contraction is not exactly homologous.
In other words: for a star of a certain mass, the central pressure is almost uniquely determined by thecentral density.
Note that we have obtained this relation without considering the EOS. Therefore (8.1) defines auniversal relation for stars in HE that is independent of the equation of state. It expresses the fact thata star that contracts quasi-statically must achieve a higher internal pressure to remain in hydrostatic
equilibrium. Eq. (8.1) therefore defines an evolution track of a slowly contracting (or expanding) starin the Pc -ρc plane.
1138.1.2The equation of state and evolution in the P c -ρ c plane

By considering the EOS we can also derive the evolution of the central temperature. This is obviously
crucial for the evolutionary fate of a star because e.g. nuclear burning requires T c to reach certain(high) values. We start by considering lines of constant T , isotherms, in the (P, ρ) plane.
We have encountered various regimes for the EOS in Chapter 3:• Radiation dominated: P = 13 aT 4 . Hence an isotherm in this region is also a line of constant P.
• (Classical) ideal gas: P =R
ρT . Hence an isotherm has P ∝ ρ.
µ• Non-relativistic electron degeneracy: P = KNR (ρ/µe )5/3 (eq. 3.35). This is independent of temperature. More accurately: the complete degeneracy implied by this relation is only achieved
when T → 0, so this is in fact the isotherm for T = 0 (and not too high densities).
• Extremely relativistic electron degeneracy: P = KER (ρ/µe )4/3 (eq. 3.37). This is the isotherm
for T = 0 and very high ρ.
Figure 8.1 shows various isotherms schematically in the log ρ – log P plane. Where radiationpressure dominates (low ρ) the isotherms are horizontal and where ideal-gas pressure dominates they
have a slope = 1. The isotherm for T = 0, corresponding to complete electron degeneracy, has slopeof 35 at relatively low density and a shallower slope of 34 at high density. The region to the right andbelow the T = 0 line is forbidden by the Pauli exclusion principle, since electrons are fermions.
The dashed lines are schematic evolution tracks for stars of different masses M1 and M2 . According to eq. (8.1) they have a slope of 34 and the track for a larger mass lies at a higher pressure than thatfor a smaller mass.
Several important conclusions can be drawn from this diagram:• As long as the gas is ideal, contraction (increasing ρc ) leads to a higher T c , because the slopeof the evolution track is steeper than that of ideal-gas isotherms: the evolution track crosses
isotherms of higher and higher temperature.

Figure 8.1. Schematic evolution in the log ρ – log P plane. Solid lines are isotherms in the equation of state;
the dashed lines indicate two evolution tracks of different mass, which have a slope of 34 . See the text for an
explanation.

114

Note that this is consistent with what we have already concluded from the virial theorem for an
ideal gas: Ein = − 21 Egr . Contraction (decreasing Egr , i.e. increasing −Egr ) leads to increasing
internal (thermal) energy of the gas, i.e. to heating of the stellar gas!
• Tracks for masses lower than some critical value Mcrit , e.g. the track labeled M1 , eventually
run into the line for complete electron degeneracy because this has a steeper slope. Hence forstars with M < Mcrit there exist a maximum achievable central density and pressure, ρc,max and
Pc,max , which define the endpoint of their evolution. This endpoint is a completely degeneratestate, i.e. a white dwarf, where the pressure needed to balance gravity comes from electrons
filling the lowest possible quantum states.
Because complete degeneracy corresponds to T = 0, it follows that the evolution track must
intersect each isotherm twice. In other words, stars with M < Mcrit also reach a maximum
temperature T c,max , at the point where degeneracy starts to dominate the pressure, after whichfurther contraction leads to decreasing T c . ρc,max , Pc,max and T c,max all depend on M and increase with mass.
• Tracks for masses larger than Mcrit , e.g. the one labeled M2 , miss the completely degenerateregion of the EOS, because at high ρ this has the same slope as the evolution track. This change
in slope is owing to the electrons becoming relativistic and as their velocity cannot exceed c,they exert less pressure than if there were no limit to their velocity. Hence, electron degeneracy
is not sufficient to counteract gravity, and a star with M > Mcrit must keep on contracting andgetting hotter indefinitely – up to the point where the assumptions we have made break down,e.g. when ρ becomes so high that the protons inside the nuclei capture free electrons and aneutron gas is formed, which can also become degenerate.
Hence the evolution of stars with M > Mcrit is qualitatively different from that of stars with
M < Mcrit . This critical mass is none other than the Chandrasekhar mass that we have alreadyencountered in Chapter 4 (eq. 4.22)
MCh =

5.836
M⊙ .
µe 2

(8.2)

It is the unique mass of a completely degenerate and extremely relativistic gas sphere. A star withM ≥ MCh must collapse under its own gravity, but the electrons become extremely relativistic – and,if M is equal to or not much larger than MCh , also degenerate – in the process.

8.1.3

Evolution in the T c -ρ c plane

We now consider how the stellar centre evolves in the T c , ρc diagram. First we divide the T , ρ plane
into regions where different processes dominate the EOS, see Sec. 3.3.7 and Fig. 3.4, reproduced inFig. 8.2a.
For a slowly contracting star in hydrostatic equilibrium equation (8.1) implies that, as long as thegas behaves like a classical ideal gas:RT c ρc = C GM 2/3 ρc 4/3
µ

→

Tc =

CG
µ M 2/3 ρc 1/3
R

(8.3)

(Compare to Sec. 7.4.3.) This defines an evolution track in the log T , log ρ plane with slope 13 . Stars
with different mass evolve along tracks that lie parallel to each other, those with larger M lying at
higher T c and lower ρc . Tracks for larger mass therefore lie closer to the region where radiation
115

10

10
radiation

6

6
ideal gas

4
−5

degenerate
NR
ER

0

5

MCh

8

log Tclog Tc

8

100
10
1.0

4
−5

10

0.1

log ρc0510log ρcFigure 8.2. The equation of state in the log T c – log ρc plane (left panel), with approximate boundaries between
regions where radiation pressure, ideal gas pressure, non-relativistic electron degeneracy and extremely relativistic electron degeneracy dominate, for a composition of X = 0.7 and Z = 0.02. In the right panel, schematic
evolution tracks for contracting stars of 0.1 – 100 M⊙ have been added.

pressure is important: the larger the mass of a star, the more important is the radiation pressure.
Furthermore, the relative importance of radiation pressure does not change as a star contracts, because
the track runs parallel to the boundary between ideal gas and radiation pressure.1
As the density increases, stars with M < MCh approach the region where non-relativistic electron
degeneracy dominates, because the boundary between ideal gas and NR degeneracy has a steeper
slope than the evolution track. Inside this region, equating relation (8.1) to the NR degenerate pressure
gives:
ρc 1/3
KNR 5/3 = C GM 2/3
µe

→

CG
ρc =
KNR

!3

µe 5 M 2(8.4)

When degeneracy dominates the track becomes independent of T c , and the star moves down along a
track of constant ρc . This is the ρc,max we found from the Pc , ρc diagram. The larger the mass, the63higher this density. (When the electrons become relativistic at ρc >∼ 10 g/cm , the pressure increases
less steeply with density so that the central density for a degenerate star of mass M is in fact larger
than given by eq. 8.4).
Equations (8.3) and (8.4) imply that, for a star with M < MCh that contracts quasi-statically, T c
increases as ρc 1/3 until the electrons become degenerate. Then a maximum temperature is reached,and subsequently the star cools at a constant density when degenerate electrons provide the pressure.
The schematic evolution tracks for 0.1 and 1.0 M⊙ given in Fig. 8.2 show this behaviour. This canbe compared to eq. (7.41) for homologous contraction (Sec. 7.4.3), which indicates that the slope ofan evolution track in the log T – log ρ plane is equal to ( 34 − χρ )/χT . This equals 31 for an ideal gas,
but changes sign and becomes negative once χρ > 34 . When degeneracy is almost complete, χρ = 53
and χT → 0 such that the slope approaches infinity. The maximum temperature is reached when the1

It is easy to show for yourself that the evolution track for a star in which radiation pressure dominates would have thesame slope of 13 in the log T , log ρ plane. However, such stars are very close to dynamical instability.

116

ideal gas pressure and degenerate electron pressure are about equal, each contributing about half ofthe total pressure. Combining eqs. (8.3) and (8.4) then implies that the maximum central temperaturereached increases with stellar mass as (see Exercise 8.2)
T c,max =

C 2G2
4/3
µ µ5/3.
e M
4R KNR(8.5)

For M > MCh , the tracks of contracting stars miss the degenerate region of the T -ρ plane, because
at high density the boundary between ideal gas and degeneracy has the same slope as an evolution
track. The pressure remains dominated by an ideal gas, and T c keeps increasing like ρc 1/3 to very
high values (> 1010 K). This behaviour is shown by the schematic tracks for 10 and 100 M⊙ .

8.2 Nuclear burning regions and limits to stellar masses
We found that stars with M < MCh reach a maximum temperature, the value of which increases withmass. This means that only gas spheres above a certain mass limit will reach temperatures sufficiently
high for nuclear burning. The nuclear energy generation rate is a sensitive function of the temperature,
which can be written asǫnuc = ǫ0 ρλ T ν(8.6)

where for most nuclear reactions (those involving two nuclei) λ = 1, while ν depends mainly on
the masses and charges of the nuclei involved and usually ν ≫ 1. For H-burning by the pp-chain,ν ≈ 4 and for the CNO-cycle which dominates at somewhat higher temperature, ν ≈ 18. For Heburning by the 3α reaction, ν ∼ 40 (and λ = 2 because three particles are involved). For C-burningand O-burning reactions ν is even larger. As discussed in Chapter 6, the consequences of this strong
temperature sensitivity are that• each nuclear reaction takes place at a particular, nearly constant temperature, and• nuclear burning cycles of subsequent heavier elements are well separated in temperature
As aR star contracts and heats up, nuclear burning becomes important when the energy generated,
Lnuc = ǫnuc dm, becomes comparable to the energy radiated away from the surface, L. From thismoment on, the star can compensate its surface energy loss by nuclear energy generation: it comes
into thermal equilibrium. The first nuclear fuel to be ignited is hydrogen, at T c ∼ 107 K. From thehomology relation (7.38) we expect that the central temperature at which hydrogen fusion stabilizes
should depend on the mass approximately as
T c = T c,⊙ (M/M⊙ )4/(ν+3)(8.7)

where T c,⊙ ≈ 1.5 × 107 K, for a composition like that of the Sun (µ = µ⊙ ). We can estimate theminimum mass required for hydrogen burning by comparing this temperature to the maximum central
temperature reached by a gas sphere of mass M, eq. (8.5). By doing this (and taking C = 0.48 for ann = 1.5 polytrope) we find a minimum mass for hydrogen burning of 0.15 M⊙ .
Detailed calculations reveal that the minimum mass for the ignition of hydrogen in protostars
is about 2 times smaller than this simple estimate, Mmin = 0.08 M⊙ . Less massive objects become
partially degenerate before the required temperature is reached and continue to contract and coolwithout ever burning hydrogen. Such objects are not stars according to our definition (Chapter 1) but
are known as brown dwarfs.
We have seen earlier that the contribution of radiation pressure increases with mass, and becomes4
dominant for M >∼ 100 M⊙ . A gas dominated by radiation pressure has an adiabatic index γad = 3 ,117

which means that hydrostatic equilibrium in such stars becomes marginally unstable (see Sec. 7.5.1).
Therefore stars much more massive than 100 M⊙ should be very unstable, and indeed none are known
to exist (while those with M > 50 M⊙ indeed show signs of being close to instability, e.g. they lose
mass very readily).
Hence stars are limited to a rather narrow mass range of ∼ 0.1 M⊙ to ∼ 100 M⊙ . The lower
limit is set by the minimum temperature required for nuclear burning, and the upper limit by therequirement of dynamical stability.

8.2.1

Overall picture of stellar evolution and nuclear burning cycles

As a consequence of the virial theorem, a self-gravitating sphere composed of ideal gas in HE must
contract and heat up as it radiates energy from the surface. The energy loss occurs at a rate
L = −Ėtot = Ėin = − 21 Ėgr ≈

Egr
τKH

(8.8)

This is the case for protostars that have formed out of an interstellar gas cloud. Their evolution, i.e.
overall contraction, takes place on a thermal timescale τKH . As the protostar contracts and heats up
and its central temperature approaches 107 K, the nuclear energy generation rate (which is at firstnegligible) increases rapidly in the centre, until the burning rate matches the energy loss from thesurface:
L = −Ėnuc ≈

Enuc
τnuc

(8.9)

At this point, contraction stops and T c and ρc remain approximately constant, at the values needed forhydrogen burning. The stellar centre occupies the same place in the T c -ρc diagram for about a nuclear
timescale τnuc . Remember that for a star of a certain mass, L is essentially determined by the opacity,i.e. by how efficiently the energy can be transported outwards.
When H is exhausted in the core – which how consists of He and has a mass typically ∼10% of thetotal mass M – this helium core resumes its contraction. Meanwhile the layers around it expand. This
constitutes a large deviation from homology and relation (8.1) no longer applies to the whole star.
However the core itself still contracts more or less homologously, while the weight of the envelope
decreases as a result of its expansion. Therefore relation (8.1) remains approximately valid for thecore of the star, i.e. if we replace M by the core mass Mc . The core continues to contract and heat up
at a pace set by its own thermal timescale,
Lcore ≈ Ėin,core ≈ − 12 Ėgr,core ≈Egr,core
τKH,core(8.10)

as long as the gas conditions remain ideal. It is now the He core mass, rather than the total mass ofthe star, that determines the further evolution.
Arguments similar to those used for deriving the minimum mass for H-burning lead to the existence a minimum (core) mass for He-ignition, This is schematically depicted in Fig. 8.3, whichsuggests that this minimum mass is larger than 1 M⊙ . However, the schematic tracks in Fig. 8.3 havebeen calculated for a fixed composition X = 0.7, Z = 0.02, which is clearly no longer the valid since
the core is composed of helium. You may verify that a He-rich composition increases the maximum
central temperature reached for a certain mass (eq. 8.5). Detailed calculations put the minimum massfor He-ignition at ≈ 0.3 M⊙ . Stars with a core mass larger than this value ignite He in the centre
when T c ≈ 108 K, which stops further contraction while the energy radiated away can be supplied
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10

log Tc

Si−b
O−b
C−b
He−b

8
H−b

6

100
10

Figure 8.3. The same schematic evolution tracks as
in Fig. 8.2, together with the approximate regions in
the log T c – log ρc plane where nuclear burning stages
occur.

1.0

4
−5

0.1

0

5

10

log ρcby He-burning reactions. This can go on for a length of time equal to the nuclear timescale of He
burning, which is about 0.1 times that of H burning. In stars with a He core mass < 0.3 M⊙ the core
becomes degenerate before reaching T c = 108 K, and in the absence of a surrounding envelope itwould cool to become a white dwarf composed of helium, as suggested by Fig. 8.3. (In practice,
however, H-burning in a shell around the core keeps the core hot and when Mc has grown to ≈ 0.5 M⊙
He ignites in a degenerate flash.)After the exhaustion of He in the core, the core again resumes its contraction on a thermal
timescale, until the next fuel can be ignited. Following a similar line of reasoning the minimum(core) mass for C-burning, which requires T ≈ 5 × 108 K, is ≈ 1.1 M⊙ . Less massive cores are
destined to never ignite carbon but to become degenerate and cool as CO white dwarfs. The minimum core mass required for the next stage, Ne-burning, turns out to be ≈ MCh . Stars that develop
cores with Mc > MCh therefore also undergo all subsequent nuclear burning stages (Ne-, O- and Siburning) because they never become degenerate and continue to contract and heat after each burning
phase. Eventually they develop a core consisting of Fe, from which no further nuclear energy can besqueezed. The Fe core must collapse in a cataclysmic event (a supernova or a gamma-ray burst) andbecome a neutron star or black hole.
The alternation of gravitational contraction and nuclear burning stages is summarized in Table 8.1,
together with the corresponding minimum masses and characteristic temperatures and energies. The
schematic picture presented in Fig. 8.3 of the evolution of stars of different masses in the T –ρ diagram
can be compared to Fig. 8.4, which shows the results of detailed calculations for various masses.
To summarize, we have obtained the following picture. Nuclear burning cycles can be seen as longlived but temporary interruptions of the inexorable contraction of a star (or at least its core) under
the influence of gravity. This contraction is dictated by the virial theorem, and a result of the factthat stars are hot and lose energy by radiation. If the core mass is less than the Chandrasekhar mass,then the contraction can eventually be stopped (after one or more nuclear cycles) when electron
degeneracy supplies the pressure needed to withstand gravity. However if the core mass exceeds
the Chandrasekhar mass, then degeneracy pressure is not enough and contraction, interrupted by
nuclear burning cycles, must continue at least until nuclear densities are reached.
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Figure 8.4. Detailed evolution tracks in the log ρc - log T c plane for masses between 1 and 15 M⊙ . The
initial slope of each track (labelled pre-main sequence contraction) is equal to 13 as expected from our simple
analysis. When the H-ignition line is reached wiggles appear in the tracks, because the contraction is then nolonger strictly homologous. A stronger deviation from homologous contraction occurs at the end of H-burning,
because only the core contracts while the outer layers expand. Accordingly, the tracks shift to higher density
appropriate for their smaller (core) mass. These deviations from homology occur at each nuclear burning
stage. Consistent with our expectations, the most massive star (15 M⊙ ) reaches C-ignition and keeps evolving
to higher T and ρ. The core of the 7 M⊙ star crosses the electron degeneracy border (indicated by ǫF /kT = 10)
before the C-ignition temperature is reached and becomes a C-O white dwarf. The lowest-mass tracks (1 and2 M⊙ ) cross the degeneracy border before He-ignition because their cores are less massive than 0.3 M⊙ . Basedon our simple analysis we would expect them to cool and become He white dwarfs; however, their degenerate
He cores keep getting more massive and hotter due to H-shell burning. They finally do ignite helium in an
unstable manner, the so-called He flash.

Suggestions for further reading
The schematic picture of stellar evolution presented above is very nicely explained in Chapter 7 ofPrialnik, which was one of the sources of inspiration for this chapter. The contents are only briefly
covered by Maeder in Sec. 3.4, and are somewhat scattered throughout Kippenhahn & Weigert, seesections 28.1, 33.1, 33.4 and 34.1.

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Table 8.1. Characteristics of subsequent gravitational contraction and nuclear burning stages. Column (3)gives the total gravitational energy emitted per nucleon since the beginning, and column (5) the total nuclear
energy emitted per nucleon since the beginning. Column (6) gives the minimum mass required to ignite acertain burning stage (column 4). The last two columns give the fraction of energy emitted as photons andneutrinos, respectively.
phase

T (106 K)

total Egr /nmain reactions

total Enuc /n

Mmin

grav.
nucl.
grav.
nucl.
grav.
nucl.
grav.
nucl.
grav.

0 → 1010 → 30
30 → 100
100 → 300
300 → 700
700 → 1000
1000 → 1500
1500 → 2000
2000 → 5000

∼ 1 keV/n

1

6.7 MeV/n∼ 10 keV/n

H → 4 He

0.08 M⊙4∼ 100 keV/n

He → 12 C, 16 O

≈ 7.4 MeV/n

0.3 M⊙

12∼ 150 keV/n

C → Mg, Ne≈ 7.7 MeV/n

1.1 M⊙

16

O → S, Si
Si → . . . → Fe

≈ 8.0 MeV/n≈ 8.4 MeV/n

1.4 M⊙∼ 400 keV/n

γ (%)
100
∼95
100
∼100
∼50
∼0

ν (%)
∼5
∼0
∼50
∼100
∼100
∼100
∼100

Exercises
8.1 Homologous contraction (1)(a) Explain in your own words what homologous contraction means.
(b) A real star does not evolve homologously. Can you give a specific example? [Think of core versus
envelope](c) Fig. 8.3 shows the central temperature versus the central density for schematic evolution tracks assuming homologous contraction. Explain qualitatively what we can learn form this figure (nuclear
burning cycles, difference between a 1 M⊙ and a 10 M⊙ star, ...)(d) Fig. 8.4 shows the same diagram with evolution tracks from detailed (i.e. more realistic) models.
Which aspects were already present in the schematic evolution tracks? When and where do theydiffer?
8.2 Homologous contraction (2)In this question you will derive the equations that are plotted in Figure 8.2b.
(a) Use the homology relations for P and ρ to derive eq. (8.1),
Pc = CGM 2/3 ρc4/3
To see what happens qualitatively to a contracting star of given mass M, the total gas pressure can beapproximated roughly by:!γ
ρR(8.11)P ≈ Pid + Pdeg = ρT + Kµµe
where γ varies between 35 (non-relativistic) and 43 (extremely relativistic).
(b) Combine this equation, for the case of NR degeneracy, with the central pressure of a contracting
star in hydrostatic equilibrium (eq. 8.1, assuming C ≈ 0.5) in order to find how T c depends on ρc .
(c) Derive an expression for the maximum central temperature reached by a star of mass M.

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8.3 Application: minimum core mass for helium burning
Consider a star that consists completely of helium. Compute an estimate for the minimum mass forwhich such a star can ignite helium, as follows.
• Assume that helium ignites at T c = 108 K.

• Assume that the critical mass can be determined by the condition that the ideal gas pressure andthe electron degeneracy pressure are equally important in the star at the moment of ignition.
• Use the homology relations for the pressure and the density. Assume that Pc,⊙ = 1017 g cm−1 s−2and ρc,⊙ = 60 g cm−3 .

122