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Nucleosynthesis Flowchart
BB
1,2H
3,4He
7Li
Intergalactic medium
Galaxy formation inflow
Gal. winds, stripping, mergers
Interstellar medium
Star formation
Small
stars
D, Li
?
Winds, PN, Novae
He, 7Li, C, N
D, Li, Be, B
Middling
stars
Big
stars
WD
NS
BH
Explosive
r-process
SN
Spallation
6Li, Be, B
Cosmic
rays
Lecture 9: Supernova Rates
Star-Formation Efficiency, Yield
How many supernovae per year for each galaxy type ?
Use power-law IMF, Salpeter slope -7/3 = -2.33
N(M ) µ M
logN(M)
-7/3 20 M.
Limits of validity,
not well known
0.1M.
slope = -7/3 = -2.33
log M
0.1 M
8 M
20 M
Supernova limit
“Universal” IMF (Kroupa 2002)
N(M ) µ M a
-a
a ~ - 7/3 M > 1 M
- 4/3 0.1 - 1 M
- 1/3 M < 0.1 M
M42 M35 Pleiades local
MW MC
GC local
log( M / M)
log( M / M )
Integrating a Power-Law IMF
Number of stars :
N=
ò N(M) dM = A ò M B dM =
A
M B +1 ( if B ¹ -1)
B +1
Fraction of stars with M > 8 M ( for B = -7/3 )
number of SNe
fN º
=
number of stars
ò
ò
20
8
20
0.1
M B dM
M B dM
A
Most stars at
B+1 20
-4/3 20
M
M
low-mass end!
0.018 - 0.063
8
B
+1
8
fN =
=
=
20
20
-4/3
A
B+1
0.018 - 21.544
M
M
0.1
0.1
B +1
500 stars --> 1 supernova!
Þ
f N = 0.2%
SN Mass Fraction
Supernovae are rare, but each is very massive.
What fraction of the mass goes into SNe?
fM
ò
=
ò
=
Þ
20
8
20
0.1
M
M
M ´ M -7/3 dM
M ´ M -7/3 dM
-1/3 20
8
-1/3 20
0.37-0.50
=
0.37 - 2.15
0.1
f M = 7.2%
Most of mass is in
low-mass stars.
“Typical” SN Mass
Median mass:
ò
ò
M SN
1
=
2
Þ
M ´ M -7/3 dM
-1/3
M
SN - 0.50
8
=
20
0.37 - 0.50
M ´ M -7/3 dM
8
M SN = 12.2 M .
Mean mass:
20
1
-1/3
-7/3
M
M
´
M
dM
ò
8
8
-1/3
M =
=
20
-7/3
1
-4/3 20
M
dM
ò8
M
8
-4/3
4 ´ (20-1/3 - 8-1/3 ) 4 ´ (0.37 - 0.50)
=
=
= 12 M .
-4/3
-4/3
20 - 8
0.018 - 0.062
20
SN Rates vs Galaxy Type
Spiral Galaxy: SFR: ~ 8 M yr-1.
Þ
SN rate:
7.2% have M > 8 M .
(8 M yr-1) x 0.072 ~ 0.6 M yr-1 go into SNe
0.6 M . yr -1 1
~
yr -1 (fewer seen due to dust)
12.2 M .
20
Irregular Galaxy: ~10x this rate during bursts (1 SN per 2 yr)!
No SNe between bursts.
SN Rates: Ellipticals
t* = 1 Gyr
e-folding time
t = 10 Gyr
age
a = 0.95
efficiency
M0 = 1011 M total mass = initial gas mass
Gas consumption:
MG (t) = M0 e-t / t* = M0 - a MS (t)
Star formation:
M S (t) =
M0
a
( 1- e )
m(t) = e-t / t
*
1
gas
1
S(t) = (1- e-t/t* ) / a
a
-t / t*
stars
-t / t*
10 M . ) e
(
dM S
M
e
-3
-1
0
˙
º MS =
=
=
5
´10
M
yr
.
dt
a t*
(0.95) (10 9 yr)
-10
11
SN rate:
(0.072) (5 ´10 -3 M. yr -1)
12.2 M .
t*
» 3 ´10 -5 yr -1
3 SN per 105 yr.
t
t*
Negligible!
t
What Star Formation Efficiency a
and Yields of H, He and Metals ?
X = 0.75
Y = 0.25
Z = 0.00
MG = M0
MS = 0
MG = 0
MS = M0
KABOOM!
KABOOM!
MG = (1-a) M0
MS = a M0
a=?
X=?
Y=?
Z=?
Estimates for efficiency a , yield in X, Y, Z
Assume:
1. Type-II SNe enrich the ISM.
(Neglect: Type-I SNe, stellar winds, PNe, ....)
2. Closed Box Model:
(Neglect: Infall from the IGM, outflow to the IGM)
3. SN 1987A is typical Type-II SN.
Better models include these effects.
What do we know about SN 1987A?
SN 1987A
23 Feb 1987 in LMC
Brightest SN since 1604!
First SN detected in neutrinos.
Visible (14 --> 4.2 mag) to naked eye,
in southern sky.
Progenitor star visible:
~20 Msun blue supergiant.
3- ring structure (pre-SN wind)
UV flash reached inner ring in 80 d.
Fastest ejecta reached inner ring in ~6 yr.
Fast ejection velocity v~c/30~11,000 km/s.
Slower (metal-enriched) ejecta asymmetric.
SN 1987A
23 Feb 1987 in LMC
Brightest SN since 1604!
2010
First SN detected in neutrinos.
Visible (14 --> 4.2 mag) to naked eye,
in southern sky.
Progenitor star visible:
~20 Msun blue supergiant.
3- ring structure (pre-SN wind)
Shockwave reaches inner ring 2003.
2003
Star Formation Efficiency
Use SN 1987A to calculate a and yield.
SN 1987A: progenitor star mass = 20 M
remnant neutron star mass = 1.6 M
mass returned to the ISM = 18.4 M
From IMF, 7.2% of MS is in stars with M > 8 M
 = Fraction of MS returned to ISM:
b=
mass returned to gas
18.4
= 0.072 ´
» 6.6%
mass turned into stars
20
Star Formation Efficiency
a = fraction of MS retained in stars:
a =1- b = 93%
SN 1987A Lightcurve
Powered by radioactive decay of r-process nuclei.
Use to measure metal abundances in ejected gas.
56Ni
=> 56 Co
56Co => 56 Fe
6d half-life
78d half-life
X, Y, Z of ejecta from SN1987A
Initial mass
~ 20 M
NS mass
~ 1.6 M
Mass ejected ~ 18.4 M
Þ
in H
9.0 M
He
7.0 M
Z
2.4 M
9
X=
» 0.49
18.4
2.4
Z=
» 0.13
18.4
}
= 18.4 M
7
Y=
» 0.38
18.4
Q1: What changes to the particle content of the
expanding Universe occur at the epochs of:
• Annihilation:
– pair soup -> quark soup (109 photons/quark)
• Baryogenesis:
– quarks bound (by strong force) into baryons.
– UUD = proton DDU = neutron
• Nucleosynthesis:
– Atomic nuclei: 75% H, 25% He, traces of Li, Be
• Recombination:
– Neutral atoms form as free electrons recombine
– photons fly free
Q2: Given present-day density parameters
WM = 0.3 for matter and WR = 5x10-5 for radiation,
at what redshift z were the energy densities equal ?
volume R3
N particles of mass m
photon wavelengths stretch:
1
l µRµ
1+ z
e M = r M c 2 = WM ( rcrit c 2 ) (1+ z )
3
rM =
Nb m
3
µ
(1+
z)
R3
Ng h n
4
-4
e R = WR ( rcrit c ) (1+ z )
eR =
µ
R
µ
1+
z
( )
3
R
e M WM 1
WM
0.3
1=
=
Þ 1+ z =
=
= 6000
-5
e R WR 1+ z
WR 5´10
2
4
Q3 a) Evaluate the neutron/proton ratio in
thermodynamic equilibrium at high and low T.
æ Dm c 2 ö
N n æ mn ö
= çç ÷÷ exp ç ÷
N p è mp ø
è kT ø
3/2
mn = m p + Dm = 1.0014 m p
T ®¥
æm ö
Nn
3/2
n
® çç ÷÷ exp ( 0 ) = (1.0014) » 1
N p è mp ø
T ®0
N n æ mn ö
® çç ÷÷ exp ( -¥) = 0
N p è mp ø
3/2
3/2
b) Evaluate the n/p ratio and Yp if mn = mp.
mn
=1
mp
mn =m p
Þ Dm = 0
Nn = N p
Þ 100% He Yp =1
Nn
3/2
® (1) exp ( 0) =1
Np
Q4 Alien’s CMB-meter reads 5.1K and 4.9K in the
fore and aft directions. Evaluate the velocity.
V DT 0.1K
=
=
c T
5K
ÞV =
c
= 6000 km/s
50
Are humans present on Earth at this time?
T = 5K T0 = 2.7K
lµR Þ T µ
1
R
mater dominated expansion: R µ t 2/3
3/2
3/2
æ
ö
æ
ö
æ
ö
t
R
T
2.7K
time :
=ç ÷ = ç 0 ÷ = ç
÷ = 0.40
èT ø
è 5K ø
t0 è R0 ø
1
now: t0 ~
~ 13´10 9 yr Age of Sun: ~5´10 9 yr
H0
3/2
look-back time: t0 - t = 0.6t0 ~ 8 ´10 9 yr ( Before Sun was born! )
Cosmological Models
Assume a Universe filled with uniform density fluid.
[ OK on large scales > 100 Mpc ]
Density:
Energy density:
e = r c2
11
3
H
1.4
´10
Msun
-26
-3
0
Critical density: r º
» 10 kg m »
c
8p G
(Mpc) 3
2
3 components:
1. Radiation
2. Matter
3. “Dark Energy”
Total
{
“Dark Matter”
baryons
WB ~ 0.04
Only ~4% is matter
as we know it!
Energy Density of expanding box
volume R 3
N particles
particle mass m
momentum p
2
p
energy E = hn = m 2c 4 + p 2c 2 = m c 2 +
+ ...
2m
Cold Matter: ( m > 0, p << mc )
E » m c 2 = const
N m c2
-3
eM »
µ
R
R3
Radiation: ( m = 0 )
Hot Matter: ( m > 0, p >> mc )
l µ R (wavelengths stretch) :
E =hn =
eR =
hc
l
N hn
-4
µ
R
R3
µ R-1
3 Eras: radiation…matter…vacuum
radiation :
rR µ R
matter :
r M µ R-3
r L = const
vacuum :
R
1
aº
=
R0
1+ z
r=
r R,0
a
4
+
r M ,0
a
3
log r
-4
rR
rM
z = redshift
+ rL
rR = rM at a ~ 10
log R
e
log R
t
-4
t ~ 10 yr
4
rM = r L at a ~ 0.7 t ~ 1010 yr
rL
t
2/3
1/ 2
log t
t
• Q1: Given the density parameters W=0.3 for
matter and W=0.7 for Dark Energy, evaluate
the redshift z at which the energy densities of
matter and Dark Energy are equal.
 W= crit ~ R-3
 W= crit ~ R0
1 + z = R0 / R
    when W   z   W
• 1+z = ( W / W )1/3 = ( / )1/3 = 1.326
•
z = 0.326
• Q2: What changes to the particle content of the
expanding Universe occur at the following epochs:
• Annihilation: particles and anti-particles
annihilate, producing photons. Small excess of
particles (~1 per 109 photons)
• Baryogenesis: free quarks confined by strong
force in (colourless) groups of 3 producing
neutrons (ddu) and protons (uud).
• Nucleosynthesis: protons and neutrons bind to
form 2D, then 4He. Yp set by p/n ratio, as virtually
all n go into 4He leaving residual p as H.
• Recombination: H and He nuclei capture free
electrons. Universe now transparent to photons.
• Q3: If the neutron decay time were 1 s,
rather than 900s, what primordial helium
abundance Yp would emerge from Big Bang
Nucleosynthesis?
• n(t) = n(0) exp(- t /  )
• p(t) = p(0)+(n(0)-n(t))
• t~300s
  = 900s => 1s
• Yp = 2n/(p+n) => 0 since virtually all
neutrons decay.
• Q4: Name and describe three effects that give rise to
anisotropy in the Cosmic Microwave Background,
indicating which are most important on angular
scales of 10, 1 and 0.1 degrees.
• 10o Sachs-Wolf effect - photons last scattered from
higher-density regions lose energy climbing out of
the potential well.
• 1o Doppler effect - velocity of gas on last-scattering
surface shifts photon wavelengths.
• 0.1o Sunyaev-Zeldovich effect - re-ionised gas (e.g.
X-ray gas in galaxy clusters) scatters CMB photons
passing thru, changing photon direction and energy.
In dimensionless form
MG (t)
= fraction of M0 in gas
m(t) º
M0
M S (t) = fraction of M0 that has been
S(t) º
turned into stars
M0
m(t)
MG = M 0 - a M S
1
m = 1- a S
slope = - a
OK, since some gas is recycled.
1
S¥
S(t)