5-7 constitutive physics
Stellar Structure: TCD 2006: 5.1
constitutive relations
In addition, , ,  refer to energy generation, density and energy transport,
the last depending on  and , the convective efficiency and opacity. These
quantities describe the physics of the stellar material and may be
expressed in terms of the state variables (P and T) and of the composition
of the stellar material (X,Y,Z or Xi). These constitutive relations are required
to close the system of ode’s:
 = (P,T,Xi)
(equation of state)
2.31
 = (,T,Xi)
(nuclear energy generation rate)
2.32
 =  (,T,Xi)
(opacity)
2.33
 =  (,T,Xi)
(convective efficiency)
2.34
 = (,T,,,Xi) (energy transport)
Stellar Structure: TCD 2006: 5.2
2.35
5 equation of state
Stellar Structure: TCD 2006: 5.3
5 equation of state
Premise: matter inside stars consists of an almost perfect gas.
The gas is ionized (plasma),
 allows greater compression (10-15 m cf. 10-10 m for a neutral gas -- why ?).
Particles in thermodynamic equilibrium with radiation,
intensity governed by Planck’s law.
Particles may be non-classical and non-relativistic,
effects of quantum mechanics and special relativity must be considered.
The properties of the gas are often referred to as state variables.
Macroscopic properties of a gas described completely by three quantities, e.g.
P – Pressure
P = –(dE/dV)S,N
T – Temperature
T = (dE/dS)V,N
 – chemical potential
 = (dE/dN)S,V.
The first law of thermodynamics changes in the internal energy dE to changes in
entropy dS, volume dV and the number of particles dN
dE = TdS – PdV + dN
5.1
The chemical potential describes roughly how the number density can change
without affecting other quantities, for example if ionization state were to change.
Stellar Structure: TCD 2006: 5.4
5.1 density of states
A gas consists of an ensemble of particles. Consider a box of volume V.
Each particle behaves like a wave with momentum p=h/. The number of
waves - or quantum states - with momentum between p and p+dp is
V
n( p )dp  f  p g i 3 dp
h
5.2
where gi is a partition function and f(p) is an occupation probability.
The total number of particles is then
N   n  p 4p 2 dp
p
5.3
and the internal energy of the gas is
E    p n  p 4p 2 dp
p
Stellar Structure: TCD 2006: 5.5
5.4
5.2 pressure
….
E    p n  p 4p 2 dp
p
5.4
From kinetic theory, the pressure is given by
P
1
2


pvn
p
4

p
dp

p
3
5.5
Hence, for non-relativistic (NR) particles with p= p2/2m = pv/2.
P
2
E
3
5.6
and for ultra-relativistic particles (UR) with p=pc and v=c,
1
P E
3
5.7
Stellar Structure: TCD 2006: 5.6
5.3 classical ideal gas
A gas is classical when the occupation probability f(p),
fB  fF 
1
 1
exp  p    / kT 
for both bosons and fermions (electrons). This is equivalent to the average
separation of particles being large compared with their de Broglie
wavelength. If so
N
P  kT  nkT
V
5.8
(see 1996 notes for proof)
Comparing 5.8 with 5.6 and 5.7, we find the average kinetic energy per
particle
(NR) <p>=(3/2)kT,
(UR) <p>=3kT.
5.9
Stellar Structure: TCD 2006: 5.7
reminder
de Broglie wavelength:
a particle of mass m moving with a velocity v will under suitable
conditions exhibit the characteristics of a wave with wavelength
B = h / mv = h / p
Compton wavelength:
when the particle is moving relativistically, the de Broglie
wavelength may be written
C = h / mc
= 2.410-12 m (for the electron)
Stellar Structure: TCD 2006: 5.8
5.4 degenerate electron gas
Quantum effects dominate when n>>nQ, or kT << h2n2/3/2mH.
A quantum gas is a cold gas: “coldness” is set by density, not temperature.
A cold gas is degenerate because the particles occupy the lowest possible
energy states and because, for electrons, these states are filled and
electrons obey the Pauli-exclusion principal. The energy of the most
energetic electrons in a cold electron gas, F, is the Fermi-energy.
The zero-temperature limit is known as the Fermi-Dirac distribution.
We state that for a NR degenerate electron gas:
P = KNR n5/3, where K NR
h2  3 

5m  8 
2/3
5.10a
Similarly, for an UR degenerate electron gas:
P = KUR n4/3, where K UR
hc  3 

4  8 
1/ 3
5.10b
(see 1996 notes for proof)
Stellar Structure: TCD 2006: 5.9
Fermi-Dirac distribution function
Stellar Structure: TCD 2006: 5.10
5.5 photons
Thermal radiation may be characterized as a photon gas, (zero-mass
bosons with zero chemical potential). The photon number and energy
density may be written
n = bT3,
b = 2.03 107 K-3 m-3
5.11
The internal energy density is
U=aT4 , a = 85k4 / 15(hc)3 = 7.565 10-16 J K-4 m-3
5.12
and the pressure due to this radiation is
Pr = U/3 = aT4/3
5.13
Stellar Structure: TCD 2006: 5.11
5.6 total pressure
Total pressure is often given in terms of the sum of pressures, including the
ion and electron pressures:
Pt = Pg + Pr = Pi + Pe + Pr
5.14
Recall that internal temperature of a star  gravitational P.E.,
hence TI ~ M/R. Meanwhile particle density n ~ M/R3.
Thus if the particles form a classical gas,
Pr
1 3 aTI4
TI3
M 3 R3
2




M
Pg ne kTI  ni kTI ne  ni
M R3
5.15
Thus for increasing mass, radiation pressure becomes increasingly
important, and ultimately (M~50M) causes the star to become unstable.
Stellar Structure: TCD 2006: 5.12
EoS regimes
Stellar Structure: TCD 2006: 5.13
6 stellar opacity
Stellar Structure: TCD 2006: 5.14
6 stellar opacity
The ability of stellar material to absorb heat
The inverse of conductivity
Interaction of photons with atoms:
i. bound-bound absorption
ii. bound-free absorption
iii. free-free absorption
iv. electron scattering
Rosseland Mean Opacity
Thermal conduction
Stellar Structure: TCD 2006: 5.15
photon-electron interactions
bound-bound
bound-free
free-free
h = ½m(v22-v12)
½mv2
ion
n=3
h = 3-2
n=2
Excitation
energy
h = 2
h = ion+½mv2
0
Stellar Structure: TCD 2006: 5.16
6.1 photons + ions
bound-bound
photon frequency 12 interacts with atom containing energy levels
1 and 2, where h12 = 2-1.
photon absorbed with transition probability a12() = B12.
multiply by occupation numbers N1 and sum over all transitions
between all levels in all ions
bb() = ions 1 N1 2 a12()
6.1
bound-free
photon frequency  interacts with atom of ionisation energy I
containing energy levels n.
if  > I-n, photon absorbed with probability abf(n, )
multiply by occupation numbers Nn and sum over all levels in all
ions
bf() = ions n Nn abf()
Stellar Structure: TCD 2006: 5.17
6.2
6.2 photons + electrons
free-free
photon frequency ff interacts with free electron
which can occupy states with energy ½mevn2 .

eif hff = ½me (v22-v12), photon absorbed with
probability aff()
total absorption coefficient obtained by averaging over electron
velocities (v):
ff() = ions v aff() Nions ne(v) dv
normally assume Maxwellian velocity distribution ne(v):
<v>=(kT/me)
Stellar Structure: TCD 2006: 5.18
6.3
v2
v1
6.3 electron scattering
An elastic collision between two particles
e.g. photon and electron

eIf h << mc2,
scatterer (m) not moved and photon  not altered.
Scattering independent of frequency,
but depends on density and degree of ionisation (ne).
Absorption coefficient per electron: e=8e4/3c4m2
Absorption coefficient per unit mass: es = e ne / 
For a fully ionized mixture of H, He, …
es = e mp2(1+X)/6 = 0.20 (1+X) cm2 g-1
Most important in fully ionized stellar cores.
Stellar Structure: TCD 2006: 5.19
6.4
6.4 total absorption coefficient
The total monochromatic absorption coefficient is given by the sum:
() = bb() + bf() +ff() + es
6.5
Stellar opacity calculations must consider all atoms and ions.
For stellar structure, best to use a weighted average over all
frequencies.
We use the Planck function to maximise the opacity contribution
where the flux is likely to be strongest:
1



0
dB d
dT  


0
dB
d
dT
 is the Rosseland mean absorption coefficient.
.
Stellar Structure: TCD 2006: 5.20
6.6
numerical and approximate values
A vast number of data contribute to the Rosseland mean. Since the
total opacity is an harmonic mean, the opacity must be
recalculated for every chemical mixture; thus =(,T,Xi). Hence,
detailed tables of precalculated opacities are usually used, e.g.
Fig 6-4.
However, it is often useful to use approximations in specific ranges
of T, in order to construct simple stellar models. For example:
a) low T:
 = 0 0.5 T4
6.7
b) intermediate T:
 = 0  T–3.5 (Kramer’s law)
6.8
c) high T:
 = es
6.9
where
0 = 4.34  1025 g/t Z (1+X)
Fig 6-5 compares approximations with tabulated opacities.
electron conduction
In very dense stellar material, the mean free path of the photon
becomes so small that it is no longer the most efficient carrier of
energy. Thermal conduction by electrons becomes the dominant
transport mechanism,
Stellar Structure: TCD 2006: 5.21
tabulated opacities
Stellar Structure: TCD 2006: 5.22
approximate opacities
Stellar Structure: TCD 2006: 5.23
Course Information
Website: star.arm.ac.uk/~csj/teaching/
Contact: csj@arm.ac.uk
Lectures 1 - 4: slides online
Problem Sheet 1: solutions online
Problem Sheet 2: issued
Lectures 5 - 6: slides online Feb 27
Lectures 7 - 8: Mar 2
Tutorial: Mar 9
Stellar Structure: TCD 2006: 5.24
7 nucleosynthesis
Stellar Structure: TCD 2006: 5.25
7 nucleosynthesis
nuclear reactions
nuclear energy production
nuclear reaction networks - hydrogen
pp chains
CN+ cycle
nuclear reaction networks - helium and beyond
3 and -capture reactions
others
stable nuclides
synthesis of the elements
neutrinos
Stellar Structure: TCD 2006: 5.26
the alchemists’ stone
Atoms have masses which are integral multiples of the mass of the
hydrogen atom. Therefore, given a suitable mechanism, all
atoms could be created from the fusion of hydrogen.
Problem: electrostatic force implies a strong repulsion between
atomic nuclei, which all carry positive electric charge.
For 2 protons, separated by 2 proton radii (~10-15 m), e-s P.E.:
Epot = e2 / 40 r ~ 3 10-13 J
7.1
Average K.E. of a proton at 107 K is
Ekin = 3/2 kT ~ 2 10-16 J
7.2
Not enough!
Eddington argued that interiors of stars were likely sites for
synthesis of elements. Antagonists pointed out the energetics
were against it. Eddington’s rejoinder was “We do not argue with
the critic who urges that stars are not hot enough for this
process; we tell him to go and find a hotter place.”
Stellar Structure: TCD 2006: 5.27
7.1 nuclear interaction
Nature provides four forces which control the interaction between 2 protons
force
source
range
nuclear reactions
gravitational
mass
1/r2
no
electrostatic
charge
1/r2
yes
weak nuclear
baryon-lepton
1/rw: w>>2
some
1/rs: w>>2
yes
strong nuclear baryon-baryon
The combined potential is illustrated in Fig 7-1.
Since Ekin<<Epotmax, classical physics states that two protons cannot
approach one another to within a separation r1. However, quantum
mechanics describes the proton as a wave-function  given by the solution
of the Schrödinger equation
 2 2m
 2 Ekin  E pot   0
2
r

Stellar Structure: TCD 2006: 5.28
7.3
p-p potential and wavefunction
Stellar Structure: TCD 2006: 5.29
barrier penetration
 2 2m
 2 E kin  E pot   0
2
r

7.3
For r>r1 and r<r2, (Ekin-Epot) is positive and  is real:
r>r1
 ~ sin kr
r2<r<r1
 ~ e-kr
r<r2
 ~  sin kr
k = 2m/h2 (Ekin-Epot)
7.4
 represents the barrier penetration probability.
There is a finite probability of the proton ‘tunnelling’ to r2 and combining
with the target proton. See wavefunction in Fig. 7-1 (bottom).
Tunnelling also allows alpha- and beta-decay processes to occur, whereby
a particle can escape from the potential well in the atomic nucleus if it has
sufficient kinetic energy.
Stellar Structure: TCD 2006: 5.30
reaction cross-section < v >
The cross-section < v > for a fusion reaction is represented by the
product of the particle energy distribution and the tunnelling
probability
Stellar Structure: TCD 2006: 5.31
7.2 nuclear energy production
The rest mass energy of protons, neutrons, atomic nuclei, etc is
given by
E = mc2
7.5
Atomic nuclei consist of Z protons and N neutrons.
The total rest mass energy of a nucleus is always less than the rest
mass energy of the constituent particles.
The deficit represents the binding energy of the nucleus
Q(Z,N) = [Zmp+Nmn-m(Z,N)] c2
7.6
For any nuclear reaction we are interested in
a) the reaction rate:
rij=ninj < v > / 
7.7
where  = tunnelling probability, v = the particle velocity distribution,
and ninj / the densities of interacting particles.
b) the energy released:
ij = rij Qij
7.8
where Qij = energy released per reaction.
Stellar Structure: TCD 2006: 5.32
7.3 reaction networks - H burning
Some notation
Shorthand to describe nuclear reactions
i1(i2,o3)o4
i: input particles, i1 is the principle
o: output particles, o4 is the principle
Examples:
1H(p,+)2H
n(, -’)p
proton-proton reaction
 deuterium, positron and neutrino
neutron decay
 electron, antineutrino and proton
Stellar Structure: TCD 2006: 5.33
pp chains
Stellar Structure: TCD 2006: 5.34
pp chains
PP I
1H
PP II
PP III
(p, + ) 2H
2H
(p, ) 3He
3He (3He,
2p) 4He
3He
(, ) 7Be
7Be
(–, ) 7Li
7Li
(p, ) 4He
7Be
8B
(p, ) 8B
 8Be* + + + 
8Be*
 2 4He
+ 13.05 MeV
+ 25.7 MeV
+ 19.1 MeV
85%
15%
0.02%
1: Chain only operates as fast as slowest reaction: cycle = rslowest Qcycle
2: Branching ratio depends on relative cross-sections
3: Energy yields depend on how much energy removed by neutrinos
pp = 0 XH2  T4
Stellar Structure: TCD 2006: 5.35
7.9
CN cycles
The CN cycle is a “catalytic” process.
CN = 0 XH XN14  T13
Stellar Structure: TCD 2006: 5.36
7.10
CN cycles
12C
13N
(p, ) 13N
 13C + + + 
13C
(p, ) 14N
14N
(p, ) 15O
15O
 15N + + + 
15N
(p, ) 12C
14N
15O
(p, ) 15O
 15N + + + 
15N
(p, ) 16O
15N
(p, ) 16O
16O
(p, ) 17F
16O
(p, ) 17F
17F
 17O + + + 
17O
(p, ) 14N
17F
 17O + + + 
17O
18F
 18O + + + 
18O
(12C
CN
destroyed)
(16O
CNO
destroyed)
(p, ) 18F
(p, ) 15N
NO
Stellar Structure: TCD 2006: 5.37
16O
17F
 17O + + + 
17O
18F
(p, ) 17F
(p, ) 18F
 18O + + + 
18O
(p, ) 19F
19F
(p, ) 16O
OF
relative rates
Stellar Structure: TCD 2006: 5.38
7.4 reaction networks - helium burning
3 = 0 XHe3 2 T40
7.11
4He
(, ) 8Be
8Be (, ) 12C*
12C*  12C + 2
– 22 kEV
– 282 kEV
+ .66 MeV
(, ) 16O
16O (, ) 20Ne
20Ne (, ) 24Mg
+ 0.16 MeV
+ 4.73 MeV
12C
+ ...
Stellar Structure: TCD 2006: 5.39
other reactions
T9~0.5-1:
12C + 12C 23Na + p + 2.24 MeV (56%)
12C + 12C 20Ne +  + 4.62 MeV (44%)
T9>1:
16O + 16O 31P + p + 7.68 MeV (61%)
16O + 16O 28Si +  + 9.59 MeV (21%)
16O + 16O 31Si + n + 1.5 MeV (18%)
T9>3:
28Si “burning”
Stellar Structure: TCD 2006: 5.40
7.5 the stable nuclides
Stellar Structure: TCD 2006: 5.41
How are the elements made?
stars
supernovae
–
–
–
–
–
–
Stellar Structure: TCD 2006: 5.42
>> 10,000,000 K
helium-burning
carbon-burning
neutron capture
 decay
fission
How are the elements made ...?
Stellar Structure: TCD 2006: 5.43
How are the elements made ...?
Stellar Structure: TCD 2006: 5.44
How are the elements made ...?
Stellar Structure: TCD 2006: 5.45
nucleosynthesis of elements
Burbidge, Burbidge, Fowler &
Hoyle (1956)
Ann. Rev. Mod. Phys. 29, 547
Synthesis of the elements in the
stars
=> Nobel prize for Physics (1983)
Stellar Structure: TCD 2006: 5.46
7.6 Can we really see inside the stars?
Stellar Structure: TCD 2006: 5.47
neutrinos
Neutrinos produced as electron (or positron) decay/capture
products in nearly all nuclear reaction networks. Neutrinos
remove energy because interaction cross-section is very small.
Typically, the neutrino capture cross section, ~10-442 cm2 where
 is the neutrino energy. The mean free path is ~1020-2/ cm.
For ordinary stars,  is very large, but in supernovae cores,
~25 m can be obtained.
Normal neutrino losses are modest. Measurements of solar
neutrino flux used to test models of stellar structure.
‘Neutrino luminosity’ can be crucial during some stages of evolution
- they can lead to a negative flux gradient! Particularly severe in
stellar collapse when large numbers of neutrinos can be
created. In supernovae, neutrino flux is comparable with the
photon flux.
In addition to nuclear decays/captures, neutrinos also be produced
in other ways which become important in late stages in stellar
evolution.
Stellar Structure: TCD 2006: 5.48
Measuring
neutrinos
Helioseismology
says solar models
are right.
 Neutrinos
must have mass!
Stellar Structure: TCD 2006: 5.49
Stellar Structure: TCD 2006: 5.50
Trinity College
+
Armagh Observatory
Final Year Astronomy Projects
Projects in Stellar Physics
One or more projects offered at Armagh Observatory for Autumn Term 2006.
There will be a mark requirement. Possible topics include following areas:
Stellar Spectroscopy - nucleosynthesis in action - abundances in evolved stars
Stellar Evolution - theoretical models of horizontal-branch stars
Stellar Atmospheres - opacity in chemically peculiar stars - preparing for GAIA
Friendly student community (10+ graduate students). Dedicated highperformance 60 cpu computer cluster. Assistance with accommodation.
More information: Contact csj@arm.ac.uk
Stellar Structure: TCD 2006: 5.51