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Stellar Structure: TCD 2006: 5.1
5-7 constitutive physics
Stellar Structure: TCD 2006: 5.2
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) 2.35
Stellar Structure: TCD 2006: 5.3
5 equation of state
Stellar Structure: TCD 2006: 5.4
5 equation of statePremise: 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: 5.5
5.1 density of states
A g a s c o n s i s t s o f a n e n s e m b l e o f p a r t i c l e s . C o n s i d e r a b o x o f v o l u m e V . E a c h p a r t i c l e b e h a v e s l i k e a w a v e w i t h m o m e n t u m p = h / . T h e n u m b e r o f w a v e s - o r q u a n t u m s t a t e s - w i t h m o m e n t u m b e t w e e n p a n d p + d p i s
dph
Vgfdppn ip 3
)( 5 . 2
w h e r e g i i s a p a r t i t i o n f u n c t i o n a n d f ( p ) i s a n o c c u p a t i o n p r o b a b i l i t y .
T h e t o t a l n u m b e r o f p a r t i c l e s i s t h e n
p
dpppnN 24 5 . 3
a n d t h e i n t e r n a l e n e r g y o f t h e g a s i s
p p dpppnE 24
5 . 4
Stellar Structure: TCD 2006: 5.6
5.2 pressure
… .
p p dpppnE 24
5 . 4
F r o m k in e t i c t h e o r y , t h e p r e s s u r e i s g iv e n b y
p
dppppvnP 243
1 5 . 5
H e n c e , f o r n o n - r e la t i v i s t i c ( N R ) p a r t i c le s w i t h p = p 2 / 2 m = p v / 2 .
EP3
2
5 . 6
a n d f o r u l t r a - r e la t i v i s t i c p a r t i c le s ( U R ) w i t h p = p c a n d v = c ,
EP3
1
5 . 7
Stellar Structure: TCD 2006: 5.7
5.3 classical ideal gas
A g a s i s c l a s s i c a l w h e n t h e o c c u p a t i o n p r o b a b i l i t y f ( p ) ,
1/exp
1
kTff
pFB
f o r b o t h b o s o n s a n d f e r m i o n s ( e l e c t r o n s ) . T h i s i s e q u i v a l e n t t o t h e a v e r a g e
s e p a r a t i o n o f p a r t i c l e s b e i n g l a r g e c o m p a r e d w i t h t h e i r d e B r o g l i e
w a v e l e n g t h . I f s o
nkTkTV
NP 5 . 8
( s e e 1 9 9 6 n o t e s f o r p r o o f )
C o m p a r i n g 5 . 8 w i t h 5 . 6 a n d 5 . 7 , w e f i n d t h e a v e r a g e k i n e t i c e n e r g y p e r
p a r t i c l e
( N R ) < p > = ( 3 / 2 ) k T ,
( U R ) < p > = 3 k T . 5 . 9
Stellar Structure: TCD 2006: 5.8
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
= 2.410-12 m (for the electron)
Stellar Structure: TCD 2006: 5.9
5.4 degenerate electron gas
Q u a n t u m e f f e c t s d o m i n a t e w h e n n > > n Q , o r k T < < h 2 n 2 / 3 / 2 m H .
A q u a n t u m g a s i s a c o l d g a s : “ c o l d n e s s ” i s s e t b y d e n s i t y , n o t t e m p e r a t u r e .
A c o l d g a s i s d e g e n e r a t e b e c a u s e t h e p a r t i c l e s o c c u p y t h e l o w e s t p o s s i b l e
e n e r g y s t a t e s a n d b e c a u s e , f o r e l e c t r o n s , t h e s e s t a t e s a r e f i l l e d a n d
e l e c t r o n s o b e y t h e P a u l i - e x c l u s i o n p r i n c i p a l . T h e e n e r g y o f t h e m o s t
e n e r g e t i c e l e c t r o n s i n a c o l d e l e c t r o n g a s , F , i s t h e F e r m i - e n e r g y .
T h e z e r o - t e m p e r a t u r e l i m i t i s k n o w n a s t h e F e r m i - D i r a c d i s t r i b u t i o n .
W e s t a t e t h a t f o r a N R d e g e n e r a t e e l e c t r o n g a s :
P = K N R n 5 / 3 , w h e r e
3/22
8
3
5
m
hK NR 5 . 1 0 a
S i m i l a r l y , f o r a n U R d e g e n e r a t e e l e c t r o n g a s :
P = K U R n 4 / 3 , w h e r e
3/1
8
3
4
hc
K UR 5 . 1 0 b
( s e e 1 9 9 6 n o t e s f o r p r o o f )
Stellar Structure: TCD 2006: 5.10
Fermi-Dirac distribution function
Stellar Structure: TCD 2006: 5.11
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 = 85k4 / 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: 5.12
5.6 total pressure
T o t a l p r e s s u r e i s o f t e n g i v e n i n t e r m s o f t h e s u m o f p r e s s u r e s , i n c l u d i n g t h e
i o n a n d e l e c t r o n p r e s s u r e s :
P t = P g + P r = P i + P e + P r 5 . 1 4
R e c a l l t h a t i n t e r n a l t e m p e r a t u r e o f a s t a r g r a v i t a t i o n a l P . E . ,
h e n c e T I ~ M / R . M e a n w h i l e p a r t i c l e d e n s i t y n ~ M / R 3 .
T h u s i f t h e p a r t i c l e s f o r m a c l a s s i c a l g a s ,
23
333431M
RM
RM
nn
T
kTnkTn
aT
P
P
ie
I
IiIe
I
g
r
5 . 1 5
T h u s f o r i n c r e a s i n g m a s s , r a d i a t i o n p r e s s u r e b e c o m e s i n c r e a s i n g l y
i m p o r t a n t , a n d u l t i m a t e l y ( M ~ 5 0 M ) c a u s e s t h e s t a r t o b e c o m e u n s t a b l e .
Stellar Structure: TCD 2006: 5.13
EoS regimes
Stellar Structure: TCD 2006: 5.14
6 stellar opacity
Stellar Structure: TCD 2006: 5.15
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: 5.16
photon-electron interactions
n = 3
n = 2
0
h = 3-2
h = 2
ion
h = ion+½mv2
½mv2
h = ½m(v22-v1
2)
Excitation energy
bound-bound bound-free free-free
Stellar Structure: TCD 2006: 5.17
6.1 photons + ions
bound-bound
photon frequency 12 interacts with atom containing energy levels 1 and 2, where h12 = 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() 6.2
Stellar Structure: TCD 2006: 5.18
6.2 photons + electrons
free-free
photon frequency ff interacts with free electron which can occupy states with energy ½mevn
2 .
if hff = ½me (v22-v1
2), photon absorbed with probability aff()
total absorption coefficient obtained by averaging over electron velocities (v):
ff() = ions v aff() Nions ne(v) dv 6.3
normally assume Maxwellian velocity distribution ne(v): <v>=(kT/me)
e-
v1
v2
Stellar Structure: TCD 2006: 5.19
6.3 electron scattering
An elastic collision between two particles e.g. photon and electron
If 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=8e4/3c4m2
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 6.4
Most important in fully ionized stellar cores.
e-
Stellar Structure: TCD 2006: 5.20
6.4 total absorption coefficient
T h e t o t a l m o n o c h r o m a t i c a b s o r p t i o n c o e f f i c i e n t i s g i v e n b y t h e s u m :
( ) = b b ( ) +
b f ( ) + f f ( ) +
e s 6 . 5
S t e l l a r o p a c i t y c a l c u l a t i o n s m u s t c o n s i d e r a l l a t o m s a n d i o n s . F o r s t e l l a r s t r u c t u r e , b e s t t o u s e a w e i g h t e d a v e r a g e o v e r a l l f r e q u e n c i e s . W e u s e t h e P l a n c k f u n c t i o n t o m a x i m i s e t h e o p a c i t y c o n t r i b u t i o n w h e r e t h e f l u x i s l i k e l y t o b e s t r o n g e s t :
00
1
ddT
dBd
dT
dB 6 . 6
i s t h e R o s s e l a n d m e a n a b s o r p t i o n c o e f f i c i e n t . .
Stellar Structure: TCD 2006: 5.21
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 6.7b) intermediate T: = 0 T–3.5 (Kramer’s law) 6.8c) high T: = es 6.9where 0 = 4.34 1025 g/t Z (1+X)
Fig 6-5 compares approximations with tabulated opacities. electron conductionIn 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: 5.22
tabulated opacities
Stellar Structure: TCD 2006: 5.23
approximate opacities
Stellar Structure: TCD 2006: 5.24
Course Information
Website: star.arm.ac.uk/~csj/teaching/
Contact: [email protected]
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: 5.25
7 nucleosynthesis
Stellar Structure: TCD 2006: 5.26
7 nucleosynthesis
nuclear reactions
nuclear energy production
nuclear reaction networks - hydrogenpp chains
CN+ cycle
nuclear reaction networks - helium and beyond3 and -capture reactions
others
stable nuclides
synthesis of the elements
neutrinos
Stellar Structure: TCD 2006: 5.27
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 / 40 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: 5.28
7.1 nuclear interaction
N a t u r e p r o v i d e s f o u r f o r c e s w h i c h c o n t r o l t h e i n t e r a c t i o n b e t w e e n 2 p r o t o n s
f o r c e s o u r c e r a n g e n u c l e a r r e a c t i o n s
g r a v i t a t i o n a l m a s s 1 / r 2 n o
e l e c t r o s t a t i c c h a r g e 1 / r 2 y e s
w e a k n u c l e a r b a r y o n - l e p t o n 1 / r w : w > > 2 s o m e
s t r o n g n u c l e a r b a r y o n - b a r y o n 1 / r s : w > > 2 y e s
T h e c o m b i n e d p o t e n t i a l i s i l l u s t r a t e d i n F i g 7 - 1 .
S i n c e E k i n < < E p o tm a x , c l a s s i c a l p h y s i c s s t a t e s t h a t t w o p r o t o n s c a n n o t
a p p r o a c h o n e a n o t h e r t o w i t h i n a s e p a r a t i o n r 1 . H o w e v e r , q u a n t u m m e c h a n i c s d e s c r i b e s t h e p r o t o n a s a w a v e - f u n c t i o n g i v e n b y t h e s o l u t i o n o f t h e S c h r ö d i n g e r e q u a t i o n
02
22
2
potkin EE
m
r 7 . 3
Stellar Structure: TCD 2006: 5.29
p-p potential and wavefunction
Stellar Structure: TCD 2006: 5.30
barrier penetration
02
22
2
potkin EE
m
r 7 . 3
F o r r > r 1 a n d r < r 2 , ( E k i n - E p o t ) i s p o s i t i v e a n d i s r e a l :
r > r 1 ~ s i n k r k = 2 m / h 2 ( E k i n - E p o t )
r 2 < r < r 1 ~ e - k r 7 . 4
r < r 2 ~ s i n k r
r e p r e s e n t s t h e b a r r i e r p e n e t r a t i o n p r o b a b i l i t y . T h e r e i s a f i n i t e p r o b a b i l i t y o f t h e p r o t o n ‘ t u n n e l l i n g ’ t o r 2 a n d c o m b i n i n g w i t h t h e t a r g e t p r o t o n . S e e w a v e f u n c t i o n i n F i g . 7 - 1 ( b o t t o m ) .
T u n n e l l i n g a l s o a l l o w s a l p h a - a n d b e t a - d e c a y p r o c e s s e s t o o c c u r , w h e r e b y a p a r t i c l e c a n e s c a p e f r o m t h e p o t e n t i a l w e l l i n t h e a t o m i c n u c l e u s i f i t h a s s u f f i c i e n t k i n e t i c e n e r g y .
Stellar Structure: TCD 2006: 5.31
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: 5.32
7.2 nuclear energy production
The rest mass energy of protons, neutrons, atomic nuclei, etc is given by
E = mc2 7.5Atomic 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 7.6For any nuclear reaction we are interested ina) the reaction rate:
rij=ninj < v > / 7.7where = 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: 5.33
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 proton-proton reaction
deuterium, positron and neutrino
n(, -’)p neutron decay
electron, antineutrino and proton
Stellar Structure: TCD 2006: 5.34
pp chains
Stellar Structure: TCD 2006: 5.35
pp chains
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 7.9
1H (p, + ) 2H 2H (p, ) 3He
3He (3He, 2p) 4He 3He (, ) 7Be
7Be (–, ) 7Li 7Be (p, ) 8B
7Li (p, ) 4He 8B 8Be* + + +
8Be* 2 4He
PP I PP II PP III
+ 13.05 MeV + 25.7 MeV + 19.1 MeV
85% 15% 0.02%
Stellar Structure: TCD 2006: 5.36
CN cycles
The CN cycle is a “catalytic” process.
CN = 0 XH XN14 T13
7.10
Stellar Structure: TCD 2006: 5.37
CN cycles12C (p, ) 13N
13N 13C + + +
13C (p, ) 14N
14N (p, ) 15O 14N (p, ) 15O
15O 15N + + + 15O 15N + + +
15N (p, ) 12C 15N (p, ) 16O 15N (p, ) 16O
16O (p, ) 17F 16O (p, ) 17F 16O (p, ) 17F
17F 17O + + + 17F 17O + + + 17F 17O + + +
17O (p, ) 14N 17O (p, ) 18F 17O (p, ) 18F
18F 18O + + + 18F 18O + + +
18O (p, ) 15N 18O (p, ) 19F
19F (p, ) 16O
CN(12C destroyed)
CNO (16O destroyed)
NO OF
Stellar Structure: TCD 2006: 5.38
relative rates
Stellar Structure: TCD 2006: 5.39
7.4 reaction networks - helium burning
3 = 0 XHe3
2 T40
7.11
4He (, ) 8Be – 22 kEV8Be (, ) 12C* – 282 kEV12C* 12C + 2 + .66 MeV
12C (, ) 16O + 0.16 MeV16O (, ) 20Ne + 4.73 MeV20Ne (, ) 24Mg + ...
Stellar Structure: TCD 2006: 5.40
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: 5.41
7.5 the stable nuclides
Stellar Structure: TCD 2006: 5.42
How are the elements made?
starssupernovae
– >> 10,000,000 K
– helium-burning– carbon-burning– neutron capture decay– fission
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
How are the elements made ...?
Stellar Structure: TCD 2006: 5.46
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: 5.47
7.6 Can we really see inside the stars?
Stellar Structure: TCD 2006: 5.48
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, ~10-442 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: 5.49
Measuring neutrinos
Helioseismology says solar models are right.
Neutrinos must have mass!
Stellar Structure: TCD 2006: 5.50
Stellar Structure: TCD 2006: 5.51
Trinity College +
Armagh Observatory
Final Year Astronomy ProjectsProjects 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 high-performance 60 cpu computer cluster. Assistance with accommodation.
More information: Contact [email protected]