Download hot CNO cycle

Lecture 9
Hydrogen Burning Nucleosynthesis,
Classical Novae, and X-Ray Bursts
Once the relevant nuclear physics is known in terms of the
necessary rate factors,  = NA<sv>, the composition can be solved
from the coupled set of rate equations:
dYI
  YI Y j   jk ( I ) 
dt
j ,k

k , j ,L  Lk  I  j
YL Yk  kj ( L)
The rather complicated looking restriction on the second summation
simply reflects the necessary conservation conditions for the
generic forward reaction, I(j,k)L and its reverse, L(k,j)I.
k and j are typically n, p, a, or g.
In the special case of weak interactions one substitutes for Yj
or Yk, the inverse mean lifetime against the weak interaction,
 (I or L) = 1/(I or L), where  can be beta-decay, positron decay
or electron capture. The mean lifetime is the half-life divided
by ln 2 = 0.693...
Aside on implicit solution of rate equations
dY1
  Y1 1  Y2 2
dt
dY2
 Y1 1 Y2 2
dt
Y1
  Y1  Y1 1  Y2  Y2 2
t
Y2
 Y1  Y1 1  Y2  Y2 2
t
1

Y1   1   Y2  2    Y1 1  Y2 2
 t

1

Y1  1   Y2   2   Y1 1  Y2 2
 t

note limits:
(Y11 Y2 2 )
Y2   Y1 
1 /  t  1  2 
An example is the differential equation for the abundance
of 14 N in the CNO cycle:
dY (14 N )
  Y (14 N )Y p  pg (14 N )  Y (13 C)Y p  pg (13 C)
dt
If the reactions creating and destroying 14 N were in steady state,
i.e., balanced one another then one would have
dY (14 N )
 0   Y (14 N )Y p  pg (14 N )  Y (13 C)Y p  pg (13 C)
dt
in which case
pg (13 C)
Y(14 N )
=
13
Y ( C)
 pg (14 N )
The nuclei would have abundances inversely proportional to their
destruction rate.
Similarly, restricting our attention for now to just the
main loop of the CNO cycle, once steady state is achieved
12
Y (13C)  pg ( C)

12
Y ( C)  pg (13 C)
 pg (14 N )
Y (15 N )

14
Y( N)
 pg (15 N )   pa (15 N )
and by repeated application
 pg (14 N )
Y (12C)

14
Y ( N )  pg (12 C)
but this would only be true if
dYi
 0
dt
for every nucleus, i, in the cycle.
etc.
How long does that take for a pair of nuclei?
The time to reach steady state (not the same thing as
equilibrium) between two nuclei connected by a single
reaction is approximately the reciprocal of the
destruction rate for the more fragile nucleus.
Eg. for
12
C,
13
C
 t  time to reach steady state =
The larger
term initially
dY (13 C)
 Y (13 C) Yp  pg (13 C)  Y(12 C) Yp  pg (12 C)
dt
13
12
13

Y

(
C)

Y

(
C)
dY ( C)
p pg
p pg
13
12
 t  Y( C)
 Y ( C)
13
dt
Yp  pg ( C)
Yp  pg (13 C)
 Y (13 C)  Yinitial (13 C)  Ysteadystate (13 C)
1
Yp  pg (13 C)
 = 1 to 10 would be more appropriate for massive stars where T is this high,
so the real time scale should be about 10 times greater. Also lengthened by convection.
Yp 100
At T6  30

Y 
Y 
Y 
Y 
Y 

( N )   1.4 10 sec
( O)   2.9 10 sec
( N )   4.2  10 sec
( O)   1.6  10 sec
( C)   5.6  10 sec
Yp  pg (13 C)
p
p
p
p
p
15
pa
17
1
1
1
 1.5108 sec
pg
16
pg
pg
12
1
N  14 N
!5
11
17
1
1
C 13 C
6
pa
14
12
10
O  16O
one cycle of the main CNO cycle
O  14 N
12
16
8
12
C 14 N
Steady state after several times these time scales.
Yp 100
At T6  20

Y 
Y 
Y 
Y 
Y 

( N )   44 yr
( O)   2.9 10 yr
( N )   1.2 10 yr
( O)   8.8 10 yr
( C)   8200 yr
Yp  pg (13 C)
p
p
p
p
p
15
pa
17
1
 2400 yr
1
1
8
pa
14
pg
16
pg
pg
12
1
1
1
12
C 13 C
!5
N  14 N
17
O  16O
6
7
one cycle of the nain CNO cycle
16
O  14 N
12
C 14 N
Provided steady state has been achieved the abundance
ratios are just given by the ’s. After the operation of the CNO
cycle, some nuclei may achieve super-solar ratios in the
stellar envelope.
More recent measurements of
17O(p,a) suggest that it is not.
Hydrogen Burning Nucleosynthesis Summary
• 12C
• 13C
• 14N
• 17O
• 15N
- destroyed, turned into 13C if incomplete cycle, 14N
otherwise
- produced by incomplete CN cycle. Probably made in low
mass stars and ejected into the ISM by red giant winds
and plaetary nebulae
- product of the CNO cycle. At comparatively low T,
12C -> 14N; at higher T and over longer time scales
16O -> 14N. Mostly made in low mass stars and ejected
by red giant winds and planetary nebulae. However, some
part from high mass stars, especially at high Z and if
the He core peenetrates the H-envelope in low Z stars.
- complicated. Used to be considered a massive star product
from the CNO bicycle. Now new rate measurements suggest that
it may need to be relegated to classical novae
- certainly not made in the classical CNO cycle in stable stars.
• 18O
• 23Na
• 26Al
- made in helium burning in massive stars by
14N (ag 18F (e+  )18O
- partly a product of the Ne-Na cycle in hydrogen
burning, but mostly made by carbon burning
- gamma-ray line emitter. Partly made in hydrogen burning
by Mg - Al cycle. Mostly made in carbon and neon burning.
Suppose keep raising the temperature of the CNO cycle. How fast
can it go?
• As 14N(p,g)15O goes faster and faster there comes a point
where the decays of 14O and 15O cannot keep up with it.
1/2 (14O) = 70.64 s against positron emission.
1/2(15O) = 122. 24 s.
• Material then accumulates in 14O, 15O - more than in
14N.
The lifetimes of these two radioactive nuclei give
the energy generation that now becomes insensitve to
temperature and density.
 nuc  5.9  1015 Z erg g-1 s-1
• As the temperature and density continue to rise, other reactions
become possible.
15
O(a , g )19 Ne( p, g )20 Na( p, g )21 Mg ....
-Limited or “Hot” CNO cycle
(p,g)
14O
(e+)
Slowest rates are weak decays of 14O and 15O.
Hot
CNO
cycle
Cold
CNO
cycle
but 13N decays in 10 min
Classical Novae
• Distinct from “dwarf novae” which are probably accretion
disk instabilities
• Thermonuclear explosions on accreting white dwarfs. Unlike supernovae,
they recur, though generally on long (>1000 year) time scales.
• Rise in optical brightness by > 9 magnitudes
• Significant brightness change thereafter in < 1000 days
• Evidence for mass outflow from 100’s to 5000 km s-1
• Anomalous (non-solar) abundances of elements from carbon to sulfur
• Typically the luminosity rises rapidly to the Eddington
luminosity for one solar mass (~1038 erg s-1) and stays there
for days (fast nova) to months (slow nova)
• In Andromeda (and probably the Milky Way) about 40
per year. In the LMC a few per year.
• Evidence for membership in a close binary –
0.06 days
2.0 days
(GQ-Mus 1983)
(GK Per 1901)
see Warner, Physics of Classical Novae,
IAU Colloq 122, 24 (1990)
Discovery Aug 29, 1975
Magnitude 3.0
V1500 Cygni
A “fast” nova
Nova Cygni 1992
The brightest nova since 1975.
Visible to the unaided eye. Photo at
left is from HST in 1994. Discovered
Feb. 19, 1992. Spectrum showed
evidence for ejection of large amounts
of neon, oxygen, and magnesium,
Peak magnitude 4.4;
3.2 kpc
A “neon” nova - ejecta rich in
Ne, Mg, O, N
Ejecta ~ 2 x 10-4 solar masses
H burning ceased after 2 years
(uv continuum sudden drop)
Fast nova – rise is very steep and the principal
display lasts only a few days. Falls > 3 mag
within 110 days
Slow nova – the decline by 3 magnitudes takes
at least 100 days. There is frequently a decline
and recovery at about 100 days associated
with dust formation.
Very slow nova – display lasts for years.
Effect of embedded companion star?
Recurrent novae – observed
to recur on human time scales.
Some of these are accretion
disk instabilities
Red dwarf stars are very
low mass main sequence
stars
An earth mass or so is ejected at speeds of 100s to 1000s of
km/s. Years later the ejected shells are still visible. The next page
shows imgaes from a ground-based optical survey between 1993 and
1995 at the William Hershel Telescope and the Anglo-Australian
Telescope.
Nova Persei (1901)
GK Per
Nova Hercules (1934)
DQ - Her
Nova Cygni (1975)
V1500 Cygni
Nova Pictoris (1927)
RR Pic
Nova Serpentis (1970)
FH Ser
http://www.jb.man.ac.uk/~tob/novae/
Models
A white dwarf composedof either C and O or O, Mg, and Ne
accretes hydrogen rich material from a companion star at a
rate of 10-91 M e / yr
As the matter piles up, it becomes dense and hot. It is heated
at its base chiefly by gravitational compression, though the
temperature of the white dwarf itself may also play a role.
Ignition occurs at a critical pressure of
2  1019 dyne cm -2 (Truran and Livio 1986);
basically this is the condition that Tbase ~107 K
This implies a certain critical mass since
4ΉPign R WD 4
M ign 
~ 10-5 -10-4 M e
G M WD
dP GM

;
4
dm 4 r
dm  4 r 2  dr
i.e.,
where we have used:
1/ 2
R WD
2/ 3
2/ 3






M
M
8.5108 1.286  WD 
 0.777  WD   cm

 Me 
 Me  


Approximately,
R  M -1/3
Eggleton (1982) as quoted in Politano et al (1990)
This gives a critical mass that decreases rapidly (as M-7/3) with
mass. Since the recurrence interval is this critical mass divided
by the accretion rate, bursts on high mass white dwarfs occur
more frequently
Nomoto (1982)
The mass of the accreted hydrogen envelope at the time the hydrogen ignites is
a function of the white dwarf mass and accretion rate. Bigger dwarfs and higher
accretion rates have smaller critical masses for surface runaways.
Truran and Livio (1986)
using Iben (1982) – lower limits
especially for high masses
Mass WD
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.35
Interval
(105 yr)
12.9
7.3
4.2
2.4
1.2
0.64
0.28
0.09
0.04
M ~108 M e / yr
Even though the average mass
white dwarf is 0.6 – 0.7 solar masses
the most often observed novae have
masses around 1.14 solar masses.
These would be white dwarfs
composed of Ne, O, and Mg. It
is estimated that ~ 1/3 of novae,
by number, occur on NeOMg WDs
even though they are quite rare.
see also Ritter et al, ApJ,
376, 177, (1991)
Politano et al (1990) in Physics of Classical
Novae
For typical values the density at ignition is somewhat degenerate:
M WD 1.0 M
RWD  5500 km
Accreted layer R  150 km
4 R 4 Pcrit
M 
 7 105 M
GM
M
-3
~
3000
g
cm
4 R 2 R
(hydrostatic eq.)
or if one assumed ideal gas
G M WD M crit
4
7
~
~10
for
T
=10
K
4
4 R WD kT N A
from integrating hydrostatic equilibrium assuming T const
NAk T d
GMWD

 dm 4 RWD 4
Laccrete 
GMWD M&
~1034 erg s-1
R
is radiated away
M& ~ 109 M e yr 1
(L WD alone ~10-2 Le )
Heating of base by compression:
dV
dV
~ Pbase
dt
dt
dr
~ Pbase 4 R 2
dt
Lcompression ~ P
Thickness ~ 100 km ~
 M 
or could use Pbase 


 base 
which is also about 1032 erg s-1
M
4 R 2 
M
~ 70,000yr
(7  10-5 M e ; 109 M e yr 1 )
M&
dr r
~
~ 5 106 cm s-1
(100 km in 7 x 104 years)
dt t
L ~ (2  1019 )(4 )(5108 )2 (5106 )
t ~
 31032 erg s-1
& and comparable to L but small compared to L
Note  M
WD
accrete
Though partially degenerate and dominated by beta-limited
CNO burning at first, the nova instability is
basically an example of the thin shell instability.
P  constant because radius at base is constant
T goes up; density goes down.
dP GM
GM M


P

 constant
4
4
dm 4 r
4 RWD
so long as the region where most of the mass is
concentrated remains << RWD
For the beta-limited CNO cycle
 nuc  5.9  1015 Z erg g -1 s-1
Z ~ 0.01 - 0.1
for M = 10-5 M e ; Z = 0.01
L   nuc M ~ 1042 erg s-1
So the initial flash is quite super-Eddington at the bottom. The layer convects.
As the accreted layer becomes convective thoughout, an adiabatic gradient
is established throughout, expansion occurs and T stops rising and in fact
begins to decline.
Basically the limiting condition is that the temperature stays
high enough to provide an Eddington luminosity to the layer
until it is all ejected.
The binding energy per gm of nova material is
GM
~
 2 1017 erg gm -1
R
This is considerably less than the energy released by
burning a gram of hydrogen to helium, (6 x 1018 erg gm-1)
so most of the hydrogen is ejected unburned.
However, for a violent outburst, it is not adequate to use just the
CNO in the accreted matter. Mixing with the substrate must occur
and this enriches the runaway with additional catalyst for
CNO burning
Energy budget for 3  10 -5 M e
G M M
~ 10 46 erg
R
M v2 ~ 10 45 erg (v ~ 1000 km/s)
38
-1
7
45 - 46
L
dt
~
few

10
erg
s

10
s
=
10
erg

So the integrated kinetic energies, potential energy, and light
output are all comparable. A part of this energy may come from
a “common envelope” effect with the companion star.
Nucleosynthesis in Novae
Basically 15N and 17O
The mass fraction of both in the ejecta is ~0.01,
so crudely …
M nova (15 O) ~  0.01  3 105   30  1010  ~ 105 M
Woosley (1986)
X Pop I  17 O  ~ 105 / 3  1010 ~ 3  106  the solar mass fraction
approximate Pop I
material in the Galaxy
within solar orbit
of 15 N and 17O in the sun.
In the sun, the mass fractions of 15 N and 17O
are 4.4 x 10-6 and 3.9 x 10 -6 respectively.
The half lives of 15O and 17 F are about the same.
Novae also make interesting amounts of 22Na
and 26Al for gamm-ray astronomy
Typical temperatures reached in hydrogen burning
in classical novae are in the range 1.5 - 3.0 x 108 K,
sufficient that burning is primarily by the beta-limited
CNO cycle. It would take temperatures of about
3.5 x 108 K to break out of the CNO cycle and produce
heavier elements by the rp-process.
This is not ruled out for the more massive novae.
E.g., 1.35 Msun model reached 356 million K.
Livio and Truran, ApJ, 425, 797, (1994)
Politano et al, ApJ, 448, 807, (1995)
Typical heavy element mass fractions in novae are typically
>10% showing strong evidence for mixing with the substrate
during or prior to the explosion. E.g. QU Vul was 76 and 168
times solar in neon at 7.6 and 19.4 yr after explosion
Gehrz et al, ApJ, 672, 1167, (2008)
Some issues
•
Burning is not violent enough to give fast
novae unless the accreted layer is significantly
enriched with CNO prior to or early during the
runaway. Also nucleosynthesis strongly suggests mixing.
Shear mixing during accretion
Convective “undershoot” during burst
• Relation to Type Ia supernovae. How to grow MWD?
• How hot do they get?
Over time, matter is removed from the white dwarf,
not added and this poses a problem to making Type
Ia supernovae by this route.
The rp-Process
Wallace and Woosley, ApJS,
45, 389 (1981)
Aside - T to produce heavy elements is reduced if there
is a lot of Ne and Mg already present as in novae on NeOMg
white dwarfs.
Al (13)
Mg (12)
Na (11)
Ne (10)
Burst Ignition:
15 16
13 14
F (9)
O (8)
N (7)
C (6)
B (5)
Be (4)
Li (3)
He (2)
3 4 5
H (1)
n (0)
2
0 1
11 12
9 10
8
7
6
Prior to ignition
~0.20 GK Ignition
: hot CNO cycle
: 3a
: Hot CNO cycle II
~ 0.68 GK breakout 1: 15O(ag)
~0.77 GK breakout 2: 18Ne(a,p)
(~50 ms after breakout 1)
Leads to rp process and
main energy production
The rp-Process
Wallace and Woosley, ApJS, 45, 389, (1981)
a)
15O(ag)19Ne
b)
19Ne(p,g)20Na
c)
rp-process limited by
weak interactions, not
15O(ag)19Ne.
d)
e) (a,p) reactions start
to bridge waiting points
comparable
to Hburning lifetime
appoximately
equal to 19Ne positron decay
Endpoint: Limiting factor I – SnSbTe Cycle
The Sn-Sb-Te cycle
Known ground state
a emitter
(g,a)
105Te
106Te
107Te
108Te
104Sb
105Sb
106
107Sb
Sb
Sb (51)
Sn (50)
In (49)
Cd (48)
Ag (47)
Pd (46)
Rh (45)
Ru (44)
(p,g)
103Sn
102In
104Sn
103In
105Sn
104In
106Sn

5758
56
5455
Rb (37)
Kr (36)
Br (35)
Se (34)
As (33)
Ge (32)
Ga (31)
Zn (30)
Cu (29)
37383940
Ni (28)
Co (27)
33343536
Fe (26)
Mn (25)
3132
Cr (24)
V (23)
2930
Ti (22)
Sc (21)
25262728
Ca (20)
K (19)
2324
Ar (18)
Cl (17)
2122
S (16)
P (15)
17181920
Si (14)
Al (13)
1516
Mg (12)
Na (11)
14
Ne (10)
F (9)
11 1213
O (8)
N (7)
9 10
C (6)
B (5)
7 8
Be (4)
Li (3)
He (2)
5 6
H (1)
3 4
0 1 2
59
Tc (43)
Mo (42)
Nb (41)
Zr (40)
Y (39)
Sr (38)
105In
(Schatz et al. PRL 86(2001)3471)
Xe (54)
I (53)
Te (52)
53
5152
4950
45464748
424344
41
Principal Application(s)
•
Type I X-ray bursts on accreting neutron stars
• Unusually violent novae using Mg or Ne as starting point
• Neutrino-driven wind. Early on in a supernova
explosion proton-capture in a region with Ye > 0.50
may produce many “proton-rich” nuclei above the
iron group (part of the “p-process”)
Type I X-Ray Bursts
(e.g., Strohmayer & Bildsten 2003)
• Burst rise times < 1 s to 10 s
• Burst duration 10’s of seconds to minutes
• Occur in low mass x-ray binaries
• Persistent luminosity from 0.2 Eddington to < 0.01
Eddington
• Spectrum softens as burst proceeds. Spectrum
thermal. A cooling blackbody
• Lpeak = 3.8 x 1038 erg s-1. Evidence for radius expansion
above that. T initially 3 keV, decreases to 0.5 keV,
then gets hotter again.
Burst energy
thermonuclear
Persistent flux
gravitational energy
(much more energy)
• Of 13 known luminous globular cluster x-ray
sources, 12 show x-ray bursts. Over 50 total X-ray
bursters are known.
• Distances 4 – 12 kpc. Until 2005, none outside
our galaxy. Now two discovered in M31 (Pietsch and
Haberl, A&A, 430, L45 (2005).
• Some “superbursts” observed lasting for several hours
•
Low B-field < 108-9 gauss
• Rapid rotation (at break up?)
• Very little mass lost (based upon models)
(Woosley & Taam 1976)
Woosley et al, ApJS, 151,
175 (2004)
Reaction network
red triangles have experimentally determined masses.
The rest are theoretical more or less
from He burning
  8.9 10 5 g cm
M 1.75 10 9 Me yr -1
T = 2.7  10 8 K
Z = 0.05 Ze
Total nuclear energy generation at this stage is 3.4 x 1035 erg s-1.
The time is one minute before the burst.
Just before the burst starts, most of the layer is convective. The total
power is 8 x 1037 erg s-1, but only 1.3 x 1035 erg s-1 is escaping from
the surface - small compared with the accretion luminosity.
Woosley et al (2004)
Z=Z / 20
on a longer
time scale
Times offset by 41,700 s of accretion at 1.75 x 10-9 solar masses/yr
at the base
9.07 108 K
1.44 106 gm cm
-3
begining of
second burst
maximum T developed in the burst about 1.5 x 109 K
Fourteen consecutive flashes.
The first is a start up transient.
M = 1.75 109 M yr -1
Z  Z / 20
Embarrassingly good
agreement!
Model A3
GS 1826-24
Heger, Cumming, Gallaoway
and Woosley (2007, ApJ, 671, 141)