Stellar Structure: TCD 2006: 5.1 5-7 constitutive physics

Stellar Structure: TCD 2006: 5.1

5-7 constitutive physics


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


5 equation of state


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.


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


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


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


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)


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 )


Fermi-Dirac distribution function


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


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 .


EoS regimes


6 stellar opacity


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


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


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


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


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-


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 . .


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,


tabulated opacities


approximate opacities


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


7 nucleosynthesis


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


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.”


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


p-p potential and wavefunction


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 .


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


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 .


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


pp chains


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%


CN cycles

The CN cycle is a “catalytic” process.

CN = 0 XH XN14 T13

7.10


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


relative rates


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 + ...


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”


7.5 the stable nuclides


How are the elements made?

starssupernovae

– >> 10,000,000 K

– helium-burning– carbon-burning– neutron capture decay– fission


How are the elements made ...?






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)


7.6 Can we really see inside the stars?


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.


Measuring neutrinos

Helioseismology says solar models are right.

Neutrinos must have mass!



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]

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Stellar Structure: TCD 2006: 5.1 5-7 constitutive physics