Modelling SN Type II: microphysics From Woosley et al. (2002) Woosley Lectures
Dec 21, 2015
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
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.
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.
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
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.
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)
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)