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Lecture 2/5
Stellar evolution
Ulf Torkelsson
1
The Hayashi zone and pre-main-sequence evolution
Stars form from collapsing clouds of interstellar gas. At first this collapse is essentially free fall,
and the temperature increases as the density of the gas increases. Initially the cloud is transparent,
but it becomes opaque at a temperature of a few hundred K and a density of 10−10 to 10−9 kg m−3 .
The gas is at this point mainly molecular hydrogen, but as the collapse continues the temperature
increases and firstly the hydrogen molecules dissociate, then the hydrogen atoms ionise and finally
the helium atoms ionise. Although the collapse continues during each one of these events the
temperature remains constant during each one of them. Inbetween these events and afterwards
the temperature increases as the collapse progresses, until the pressure is large enough to balance
the gravitational attraction. We then have a protostar in hydrostatic equilibrium.
We can estimate the size of the protostar by assuming that all the gravitational potential energy
that has been released goes into the dissociation of the molecules and the ionisation of the atoms
M X
Y
GM 2
χH2 + XχH + χHe ,
=
(1)
α
Rps
mH 2
4
where the χ:s stand for the dissociation and ionisation potentials. We assume that Y = X − 1 and
α = 1/2, so we get that
M
50
Rps
=
.
(2)
R
1 − 0.2X M
We can then determine a mean temperature from the virial theorem
T̄ =
α GM µmH
= 6 × 104 K.
3 kRps
(3)
At this temperature the opacity is so high that the star is fully convective. The temperature
distribution is then close to adiabatic, so the stellar matter follows the polytropic law
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P = Kρ1+ n ,
(4)
K n = Cn Gn M n−1 R3−n ,
(5)
where
where
Cn =
4π
Rnn−3
.
n
(n + 1) Mnn−1
(6)
The only free parameter here is the radius R, which we can determine by coupling the convective
star to a radiative photosphere at r = R. We integrate the equation of hydrostatic equilibrium at
the stellar surface
GM
dP
= −ρ 2 ,
(7)
dr
R
from r = R to the point at which the pressure is zero, which we for simplicity assume is infinity
Z
GM ∞
PR = 2
ρdr.
(8)
R
R
4
The temperature at the stellar surface is the effective temperature that yields L = 4πR2 σTeff
, and
by definition the optical depth at the photosphere is on the order of 1
Z ∞
Z ∞
1=
κρdr = κ̄
ρdr.
(9)
R
R
1
We assume that κ̄ is the opacity at r = R and is a power law of density and temperature
Z ∞
a b
κ0 ρR Teff
ρdr = 1.
(10)
R
We can then write the pressure as
GM −a −b
ρ T .
R2 κ0 R eff
We can now turn this into a linear logarithmic equation
PR =
log PR = log M − 2 log R − a log ρR − b log Teff + constant.
(11)
(12)
We get a second linear logarithmic equation from the polytropic relation
n log PR = (n − 1) log M + (3 − n) log R + (n + 1) log ρR + constant,
(13)
a third expression for PR using the equation of state of an ideal gas PR = RρR Teff /µ
log PR = log ρR + log Teff + constant,
(14)
and finally we get a logarithmic relation for the luminosity
log L = 2 log R + 4 log Teff + constant.
(15)
We can use these equations to eliminate log R, log ρR and log PR to obtain
log L = A log Teff + B log M + constant,
(16)
where
A
B
(7 − n) (a + 1) − 4 − a + b
0.5 (3 − n) (a + 1) − 1
(n − 1) (a + 1) + 1
.
= −
0.5 (3 − n) (a + 1) − 1
=
(17)
(18)
This equation describes the Hayashi track in the (log Teff , log L) diagram. Let us for simplicity
assume that a = 1, and the polytropic index must be in the interval 1.5 ≤ n < 3 so that
A =
B
=
9 − 2n + b
2−n
2n − 1
−
2−n
(19)
(20)
must describe an extremely steep line in the theoretical HR-diagram, since A is large. The fact
that B has the opposite sign as A means that an increasing mass is shifting the track to the left.
Furthermore we notice that the slope is shifting sign at n = 2.
We now introduce γ̄ as the average of γ =dln P/dln ρ, and γ̄a is the average of the adiabatic
exponent. In a fully convective star γ̄ = γ̄a , but if the star has ionisation zones the γ < γa in these
zones and consequently γ̄ < γ̄a and n > na . Only a superadiabatic zone can give us γ̄ > γ̄a , but
such a zone is unstable to convection and thus forbidden. The forbidden zone n < na is located to
the right of the Hayashi track. Thus we see that it is impossible for the star to be located to the
right of the Hayashi track.
Initially the protostar will descend along its Hayashi track. Its effective temperature is constant,
while its luminosity decreases as R2 . As the ionisation is completed in the interior of the star the
opacity drops and the heat transport in the interior of the star becomes radiative, which also
increases the effective temperature of the star. As the temperature increases further the nuclear
reactions will start step by step. The protostellar evolution occurs on the Kelvin-Helmholtz time
scale, since the main energy source is the gravitational potential energy. For a solar-mass star this
time scale is tens of millions of years, while for a 9M star it is only 105 years.
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2
Main-sequence stars
Main-sequence stars burn hydrogen to helium in their cores. Light stars use the p-p-chain, while
stars that are heavier than the Sun use the CNO-cycle. The released energy is transported outwards
through radiation and convection. The Sun has radiative transport in its core and an outer
convection zone that starts at around 0.7R . This outer convection zone becomes deeper for
lighter stars, and stars that are lighter than 0.3M are fully convective. On the other hand the
hydrogen burning becomes dominated by the CNO-cycle the heavy stars develop a convective core
because the steeper temperature dependence of the CNO-cycle confines the nuclear burning to a
smaller volume, and thus the entire flux must get out of a smaller area. However heavy stars also
have higher effective temperatures. which leads to that the outer convection zone disappears.
An important consequence of convection is that the material in the convection zone is mixed
and the composition is thus uniform in the convection zone.
The Sun is essentially transparent to the neutrinos that are released during the hydrogen
burning at its core. Thus if one could detect these neutrinos there would be a very direct way to
probe what is going on at the centre of the Sun, but for the same reason as the Sun is transparent
it is also very difficult to detect the neutrinos. There are, though, some ways in which we can
detect some of these neutrinos. The first experiment that detected the neutrinos was started by
Davis in the 1960s. He used the reaction
ν +37 Cl → e− +37 Ar,
(21)
by filling a tank with 600 tons of cleaning fluid. This experiment detected only a third to half of
the expected number of neutrinos, but this reaction is only sensitive to neutrinos with energies
above 0.8 MeV, which mainly come from the rare reaction
8
B →8 Be + e+ + ν.
(22)
Other later experiments detected more neutrinos, but still less than the expected number of neutrinos. For a long time this was considered to be a problem for our models of the Sun, but in
the late 1990s it was realised electron neutrinos oscillate between the different forms of neutrinos
(electron neutrinos, muon neutrinos and tau neutrinos), but only electron neutrinos were detected
in most of these experiments. The neutrino oscillations were first discovered by the Japanese experiment Super-Kamiokande. Super-Kamiokande detects not only neutrinos from the Sun, but also
neutrinos that are generated as cosmic rays collide with atomic nuclei in the atmosphere. SuperKamiokande detects more neutrinos from the atmosphere above them than from the atmosphere
on the other side of the Earth, which is explained by that some of the latter electron neutrinos
changes to another kind of neutrino during their flight. Later on the Canadian SNO experiment
used heavy water, which is sensitive to all kinds of neutrinos, to detect all kinds of neutrinos from
the Sun.
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The red giant phase
When the core runs out of hydrogen, the hydrogen-burning will continue in a shell surrounding
the core. This inert core will then become isothermal, but there is an upper limit to the core mass
Mc , beyond which the core cannot produce enough pressure to balance the weight of the rest of
the star. This is called the Schönberg-Chandrasekhar mass. Assume that the core has a radius Rc ,
a volume Vc , a mean molecular weight µc , and a temperature Tc . From the virial theorem we get
Z Vc
1 GMc2
.
(23)
P dV = Ps Vc + α
3 Rc
0
We can calculate the pressure integral for an isothermal gas
Z Vc
Z
R
R
P dV =
Tc ρdV =
Tc Mc .
µ
µ
c
c
0
3
(24)
Using Vc = 4πRc3 /3 we get
Ps (Rc ) =
αG Mc2
3 RTc Mc
−
.
4π µc Rc3
4π Rc4
(25)
We notice that this pressure is 0 for
R0 =
αG Mc µc
,
3R Tc
(26)
and consequently the core would collapse if Rc < R0 . The pressure has a maximum at
R1 =
4αG Mc µc
,
9R Tc
(27)
and the maximal pressure is
Ps,max (Mc ) = constant
Tc4
.
Mc2 µ4c
(28)
This pressure must balance the pressure of the surrounding envelope, Penv . If we assume the core
to be a point mass we find that
GM 2
.
(29)
Penv >
8πR4
The stability criterion is therefore that Ps,max should exceed this value. We can use a homology
relation for the envelope to calculate
Tc ∝
µenv G M
,
R R
(30)
and then we get
Mc
≤ constant
M
µenv
µc
2
.
(31)
Schönberg and Chandrasekhar determined the constant to be 0.37, and if the envelope has a solar
composition and the core consists of helium we have that µenv = 0.6 and µc = 1, which gives us
Mc /M ≤ 0.13. As the core mass exceeds this value the core contracts rapidly.
For stars larger than 2M the core at the end of the main-sequence phase exceeds the SchönbergChandrasekhar limit, so it collapses until it establishes a temperature gradient that allows it to
establish hydrostatic equilibrium. The temperature in the hydrogen-burning envelope increases in
response to the increase of the core temperature, which results in that since the hydrogen burning
is due to the CNO cycle that we get L > Lnuc . For this reason the envelope must expand and
the star becomes a red giant. The transition time from the main sequence to the red giant phase
is so short that this part of the HR-diagram is empty and is called the Hertzsprung gap. The
helium core increases in mass as the shell hydrogen burning progresses, and consequently the core
contracts and heats up, which accelerates the hydrogen burning, and leads to an increase of the
stellar luminosity. Since the energy flux is increasing at the same time as the shell is cool, and
thus has a high opacity, the envelope will become convective from the top of the hydrogen burning
shell to the surface. This will lead to dredge-up”, the convection is transporting up the products
of the nuclear burning to the surface, which will then change the composition of the surface layer.
The deep convective zone means that the star is approaching the Hayashi limit.
The electrons in the core will become degenerate for
Ps,max (Mc ) ≤ K1
3Mc
4πRc3
5/3
,
(32)
which is fulfilled for the helium core, when the mass is smaller than 2M . The core contraction is
then slow and gradual. Like in the more massive stars the temperature increases and the burning
rate increases so that the envelope expands and the luminosity increases. Thus these stars will
gradually evolve towards the red giants. When the core has grown to 0.5M the conditions will be
such that helium burning starts in the core, but this helium burning is unstable since the core is
4
degenerate. We get a helium flash at which the core luminosity increases to 1011 L . This helium
flash will not be reflected in the stellar luminosity though, since the energy is absorbed by the
envelope, and the helium flash will end as the temperature becomes large enough to break the
degeneracy.
There are also some low-mass red giants that lose so much mass that they cannot reach helium
ignition. The core will then eventually become a helium white dwarf.
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