Modelling SN Type II: microphysics From Woosley et al. (2002) Woosley Lectures

Modelling SN Type II: microphysics

From Woosley et al. (2002)

Woosley Lectures

Solar-system composition

The s and r processes

Solar-system isotopic composition

Arnett: Supernovae and nucleosynthesis

Solar-system isotopic composition-2

-8

-6

-10

-4



A little on the equation of state

Actually the dimensionsof Y are Mole/gm and NA has dimensions particles per Mole.

0.5 2

0 ConvectiondS

dr

Tgram

T

S = k log W

ZentralfriedhofVienna, Austria

34 4

2

34 4

2 3

3

3

rad

Eg., just the radiation part

1/

4 1 1

3

4 1

3

44

34

3

4So S

3

V

T dS d P dV

aTT dS dT aT d aT dV

aTdT aT dV aT dV

dS aT V dT aT dV

d aT V

aT

For radiation:

41

3AN k T

P aT

3/ 2 3 3/ 2( / ) /( / )T T T

for ideal gas plus radiation

dividing by NA k makes s dimensionless

Cox and Guili (24.76b)

expressionCox and GuiliPrinciples of Stellar StructureSecond editionA. Weiss et alCambridge Scientific Publishers

ReifFundamentals of Statisticaland Thermal PhysicsMcGraw Hill

Note: here has a different definition

( 10.20)

where , the chemical potential is defined by

hence

/ ( )

/

/

/

e

e e e

e ee

e

e

A

AA

e

e

CG

S

kT

T S E PV

V P n T

P

Ps S N k

n

N

S

k

N

Y

T

T

n

N

Y

For an ideal gas

3 1

2 AP N kT

i.e., non-relativistic, non- degenerate

For an ideal gas

2

CG (24.134b). ., eYi e

4/3

23 3

3 3 2

2 44 4

3 3 2 4

2

2

2

2

2 2

2 2( )

For >>1 (great degeneracy)

8 1 1

3

8 1 2 71

3 4 15

21

1

/

14

1

3

21e

e e e

e e

e

e

e

e

e e F

e e

n kTc h

P kTc h

P

n kT

u P

Su P n

k

TV

n

n kT n k

P

T

2e e

eA

S Y

N

T

sk

The entropy of most massivestars is predominantly dueto electrons and ions. Radiation is ~10%correction.

Implication: The Chandrasekhar mass will be relevant to the late evolution of the core

Iben (1985; Ql. J. RAS 26, 1)

3T

Woosley et al (2002; RMP 74, 1015)

Burning Stages in the Life of a Massive Star

0


11,000

Stellar Neutrino Energy Losses

(see Clayton p. 259ff, especially 272ff)

and and in

comparable amounts

22 2

2 2

2 2 4 2 2 2 2 4

49 -3

2

45 22

21

3

1.41 10 erg cm

21.42 1 10 cm

W e

e

e e

W

e

G c m E

v m c

E m c p c m c

G

c E

v m c

1) Pair annihilation

2910% (especially 0.5)e

e e

kT m c T

e e radiation

Want energy loss per cm3 per second. Integrate over thermal distributionof e+ and e- velocities. These have, in general, a Fermi-Dirac distribution.

-

3 2

2

2

9 2

-

1 ( 1)

exp( ) 1

5.93/ c/m energy

= Chemical potential/kT

(determined by the condition that

n (matter) =

e e

e

e

e

e

P n n vE

m c W W dWn

W

m c ET W

kT m c

n n

A N )eY

Fermi Integral

Clayton (Sect. 3.6) and Lang in Astrophysical Formulae give some approximations (not corrected for neutral currents)

18 3 -3 -19 9

2e

15 9 -3 -19

15

9

( ) 4.9 10 exp ( 11.86 / ) erg cm s

2m /

( ) 4.2 10 erg cm s

(better is 3.2 10 )

Note origin of T :

If n is relativistic

NDNR P T T

c kT

NDR P T

3

2 2

, n (like radiation)

< v> E ( )

energy carried per reaction ~ kT

T

kT

6 2 9P T T T T

n n v E

9 3T

v cancels v-1 in

T9 < 2

More frequently we use the energy loss rate per gram per second

-1 -1 erg gm sP

-1

In the non-degenerate limit from pair annihilation

declines as .

In degenerate situations, the filling of phase space

suppresses the creation of electron-positron pairs

and the loss rate plummets.

Beaudet et al. (1967; ApJ 150, 979)

e-

e-

W-

2) Photoneutrino process: (Clayton p. 280)

Analogue of Compton scattering with the outgoing photonreplaced by a neutrino pair. The electron absorbs the extramomentum. This process is only of marginal significance in stellar evolution – a little during helium and carbon burning.

e e

When non-degenerate and non-relativistic Pphoto is proportional to the density (because it dependson the electron abundance)and ,photo is independent of the density. At high density, degeneracy blocks the phase space for the outgoing electron.


3) Plasma Neutrino Process: (Clayton 275ff)

plasma

This process is important at high densities where the

plasma frequency is high and can be comparable

to or greater than kT. This limits its applicability to

essentially white dwarfs, and to a le

sser extent, the evolved

cores of massive stars. It is favored in degenerate environments.

A photon of any energy in a vacuum cannot decay into e+ ande- because such a decay would not simultaneously satisfy the conservation of energy and momentum (e.g., a photon that hadenergy just equal to 2 electron masses, h = 2 mec2, would also havemomentum h/c = 2mec, but the electron and positron that are created,at threshold, would have no kinetic energy, hence no momentum.Such a decay is only allowed when the photon couples to matter that can absorb the excess momentum.

The common case is a -ray passing of over 1.02 MeV passing neara nucleus, but the photon can also acquire an effective mass by propagating through a plasma.

plasmon e e

Plasma frequency

Plasmon dynamics

An electromagnetic wave propagating through a plasma has an excess energy above that implied by its momentum. This excess is available for decay

A”plasmon” is a quantized collective charge oscillation in an ionized gas. For our purposes it behaves like a photon with restmass. The frequency of these oscillations is given by the plasmafrequency:

24 1/ 2

p

222

p

2

1/2

1/22/3

1/2

4ND 5.6 10

4D 1 3

/2

ee

e

ee

e e

F e

n en

m

n en

m m c

m c

suppression for degeneracy

increases with density

Plasmon dynamics

6 3221 -3 -1

2

7.5 3/ 2221 -3 -1

2

)

7.4 10 erg cm s

)

3.3 10 exp( / ) erg cm s

p

p eplasma

e

p

p eplasma p

e

a kT

m cP

m c kT

b kT

m cP kT

m c kT

For moderate values of temperature and density, raising the densityimplies more energy in the plasmon and raising the temperature excitesmore plasmons. Hence the loss rate increases with temperature and density.

p

However, once the density becomes so high that,

for a given temperature , raising the density

still further freezes out the oscillations. The thermal plasma

no longer has enough energy to exite t

kT

hem. The loss rate

plummets exponentially.

This is a relevant temperaturefor Type Ia supernovae


A

4) Neutrino bremsstrahlung - of minor significance

in Type Ia supernova ignition

+ Ae Z Z

Festa and Ruderman (1969)Itoh et al (1996)

5) Neutral current excited state decay – not very important. maybe assists in white dwarf cooling. Crawford et al. ApJ, 206, 208 (1976)

Neutrino loss mechanisms

Itoh et al. (1989; ApJ 339, 354)

= 10 T93

net nuclear energy generation (burning + neutrino losses)

net nuclear energy loss (burning + neutrino losses)

convection semiconvectiontotal mass of star(reduces by mass loss)ra

dia

tiv

e e

nv

elo

pe

(blu

e g

ian

t)

convective envelope (red giant)

H b

urn

ing

He

bu

rnin

g

C b

urn

ing

(rad

iati

ve)

C s

hel

lb

urn

ing

Ne O

burning

C shell burning

OO O O shell burning

Si

Si

erg/g/sec

erg/g/sec

-1014 -107 -100

100 107 1014

Heger (2002)


= 10 T93

Burning Stages in the Life of a Massive Star

0


11,000


Helium burning


25 M¯, end of helium burning, wind and matter outside mass cut (including fall-back)


Nuclear production, end of He burning

16O § 0.3 dex

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Modelling SN Type II: microphysics From Woosley et al. (2002) Woosley Lectures