Lecture 9

Hydrogen Burning Nucleosynthesis,Classical Novae, and X-Ray Bursts

, , ,

( ) ( )II j jk L k kj

j k k j L L k I j

dYY Y I Y Y L

dtρ λ ρ λ

∋ + = +

=− +∑ ∑

Once the relevant nuclear physics is known in terms of the necessary rate factors, λ= NA<v>, the composition can be solvedfrom the coupled set of rate equations:

The rather complicated looking restriction on the second summationsimply 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, , or .

In the special case of weak interactions one substitutes for Yjρλor Ykρλ, the inverse mean lifetime against the weak interaction,λ(I or L) = 1/(I or L), where can be beta-decay, positron decayor electron capture. The mean lifetime is the half-life divided by ln 2 = 0.693...

Aside on implicit solution of rate equations

dY1

dt=−Y1λ1 + Y2 λ2

dY2

dt= Y1λ1 −Y2 λ2

δY1δt

=− Y1 +δY1( )λ1 + Y2 +δY2( )λ2

δY2

δt= Y1 +δY1( )λ1 −Y2 +δY2( )λ2

δY11δt

+λ1⎛⎝⎜

⎞⎠⎟+ δY2 −λ2( ) =−Y1λ1 + Y2 λ2

δY1 −λ1( ) + δY21δt

+λ2⎛⎝⎜

⎞⎠⎟= Y1λ1 −Y2 λ2

δY2 =−δY1 =(Y1λ1−Y2λ2 )

1 / δt + λ1 + λ2( )note limits:

An example is the differential equation for the abundance

of 14N in the CNO cycle:

dY (14 N )

dt=−Y(14 N)Ypρλp (

14 N) + Y(13C)Ypρλp (13C)

Ifthereactionscreatinganddestroying14Nwereinsteadystate,i.e.,balancedoneanotherthenonewouldhave

dY(14 N)

dt=0 =−Y(14 N)Ypρλp (

14 N) + Y(13C)Ypρλp (13C)

inwhichcase

Y(14N)Y(13C)

=λp (

13C)λp (

14 N)

Thenucleiwouldhaveabundancesinverselyproportionaltotheirdestructionrate.

Similarly, restricting our attention for now to just the

main loop of the CNO cycle, once steady state is achieved

Y (13C)

Y (12 C)=λp (

12C)

λp (13C)

Y(15 N)Y(14 N)

=λp (

14 N)

λp (15 N) + λp (

15 N)

andbyrepeatedapplication

Y(12C)Y(14 N)

=λp (

14 N)

λp (12C)

etc.

butthiswouldonlybetrueif

dYi

dt= 0

foreverynucleus,i,inthecycle.

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 12C, 13C δt ≡timetoreachsteadystate=1

ρYpλp (13C)

dY(13C)dt

=−Y(13C)ρYpλp (13C) + Y(12C)ρYpλp (

12C)

dY(13C)dt

δt =−Y(13C)ρYpλp (

13C)

ρYpλp (13C)

+ Y(12C)ρYpλp (

12C)

ρYpλp (13C)

δY(13C)=−Yinitial (13C) + Ysteadystate(

13C)

The largerterm initially

ρ = 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.

At T6=30 ρYp =100

ρYpλp (13C)( )

−1 =1.5×108 sec 12C↔13 C

ρYpλp (15 N)( )

−1=1.4×106 sec!5N↔ 14N

ρYpλp (17O)( )

−1=2.9×1011 sec 17O↔ 16O

ρYpλp (14 N)( )

−1=4.2 ×1010sec onecycleofthemainCNOcycle

ρYpλp (16O)( )

−1=1.6 ×1012sec 16O↔ 14N

ρYpλp (12C)( )

−1=5.6 ×108sec12C↔14 N

Steady state after several times these time scales.

At T6=20 ρYp =100

ρYpλp (13C)( )

−1 = 2400 yr 12C↔13 C

ρYpλp (15 N)( )

−1= 44yr!5N↔ 14N

ρYpλp (17O)( )

−1=2.9×108 yr 17O↔ 16O

ρYpλp (14 N)( )

−1=1.2 ×106 yr onecycleofthenainCNOcycle

ρYpλp (16O)( )

−1=8.8 ×107 yr 16O↔ 14N

ρYpλp (12C)( )

−1=8200yr12C↔14 N

Provided steady state has been achieved the abundanceratios are just given by the λ’s. After the operation of the CNOcycle, some nuclei may achieve super-solar ratios in the stellar envelope.

More recent measurements of17O(p,) suggest that it is not.

Hydrogen Burning Nucleosynthesis Summary

• 12C - destroyed, turned into 13C if incomplete cycle, 14N otherwise

• 13C - produced by incomplete CN cycle. Probably made in low mass stars and ejected into the ISM by red giant winds and plaetary nebulae

• 14N - 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.

• 17O - 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

• 15N - certainly not made in the classical CNO cycle in stable stars.

• 18O - made in helium burning in massive stars by 14N (18F (e+ )18O

• 23Na - partly a product of the Ne-Na cycle in hydrogen burning, but mostly made by carbon burning

• 26Al - 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 fastcan it go?

• As 14N(p,)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.

• As the temperature and density continue to rise, other reactions become possible.

εnuc = 5.9 × 1015 Z erg g-1 s-1

15O(,)19 Ne(p,)20 Na(p,)21Mg ....

14O(p,)(e+)

Slowest rates are weak decays of 14O and 15O.

-Limited or “Hot” CNO cycle

ColdCNO cycle

Hot 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 (GQ-Mus 1983) 2.0 days (GK Per 1901) see Warner, Physics of Classical Novae, IAU Colloq 122, 24 (1990)

V1500 Cygni

Discovery Aug 29, 1975Magnitude 3.0

A “fast” nova

Nova Cygni 1992

The brightest nova since 1975.Visible to the unaided eye. Photo atleft is from HST in 1994. DiscoveredFeb. 19, 1992. Spectrum showedevidence for ejection of large amountsof neon, oxygen, and magnesium,

Peak magnitude 4.4; 3.2 kpcA “neon” nova - ejecta rich in Ne, Mg, O, NEjecta ~ 2 x 10-4 solar masses

H burning ceased after 2 years (uv continuum sudden drop)

Fast nova – rise is very steep and the principaldisplay lasts only a few days. Falls > 3 magwithin 110 days

Slow nova – the decline by 3 magnitudes takesat 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.

Recurrent novae – observedto recur on human time scales.Some of these are accretion disk instabilities

Effect of embedded companion star?

Red dwarf stars are very low mass main sequencestars

An earth mass or so is ejected at speeds of 100s to 1000s ofkm/s. Years later the ejected shells are still visible. The next page shows imgaes from a ground-based optical survey between 1993 and1995 at the William Hershel Telescope and the Anglo-AustralianTelescope.

Nova Cygni (1975) V1500 Cygni

Nova Serpentis (1970) FH Ser

Nova Pictoris (1927) RR Pic

Nova Hercules (1934) DQ - Her

Nova Persei (1901) GK Per

http://www.jb.man.ac.uk/~tob/novae/

i.e.,dP

dm=

GM4πr4

;

dm=4πr2 ρdr

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 Me / yr

Asthematterpilesup,itbecomesdenseandhot.Itisheated

atitsbasechieflybygravitationalcompression,thoughthe

temperatureofthewhitedwarfitselfmayalsoplayarole.

Ignitionoccursatacriticalpressureof

2×1019 dynecm-2 (TruranandLivio1986);

basicallythisistheconditionthatTbase~107K

This implies a certain critical mass since

ΔM ign≈4πPign

G

RWD4

MWD

~10-5 -10-4 Me

-1/3

Approximately,

R M∝

RWD

≈8.5×108 1.286MWD

Me

⎛

⎝⎜

⎞

⎠⎟

−2/3

−0.777MWD

Me

⎛

⎝⎜

⎞

⎠⎟

2/3⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

1/ 2

cm

Eggleton(1982)asquotedinPolitanoetal(1990)

This gives a critical mass that decreases rapidly (as M-7/3) withmass. Since the recurrence interval is this critical mass dividedby the accretion rate, bursts on high mass white dwarfs occurmore frequently

where we have used:

The mass of the accreted hydrogen envelope at the time the hydrogen ignites isa function of the white dwarf mass and accretion rate. Bigger dwarfs and higheraccretion rates have smaller critical masses for surface runaways.

Nomoto (1982)

0.60 12.90.70 7.30.80 4.20.90 2.41.00 1.21.10 0.641.20 0.281.30 0.091.35 0.04

Even though the average mass white dwarf is 0.6 – 0.7 solar massesthe most often observed novae havemasses 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 WDseven though they are quite rare.

Mass WD Interval (105 yr)

Politano et al (1990) in Physics of Classical Novae

see also Ritter et al, ApJ, 376, 177, (1991)

Truran and Livio (1986) using Iben (1982) – lower limitsespecially for high masses

&M ~10−8 Me / yr

For typical values the density at ignition is somewhat degenerate:

WDM 1.0 M

5500 km

Accreted layer R 150 kmWDR

=

≈Δ ≈

e

45

-32

47 10

~ 3000 g cm4

critR PM M

GMM

R R

π

ρπ

−Δ ≈ = ×

ΔΔ

e

:

or if one assumed ideal gas

ρ ~GMWDμΔMcrit

4πRWD4 kTNA

~104 forT=107 K

fromintegratinghydrostaticequilibriumassumingT≈const

NAkTμ

dρdm

=GMWD

4πRWD4

(hydrostatic eq.)

Laccrete

=GMWD

&MR

~1034 ergs-1 &M ~10−9 Me yr−1

isradiatedaway(LWD alone~10-2 Le )

Heatingofbasebycompression:

Lcompression~PdVdt

~Pbase

dVdt

~Pbase4πR2 drdt

Thickness~100km~ΔM

4πR2ρ

Δt~ΔM&M

~70,000yr(7×10-5 Me ; 10−9 Me yr−1)

drdt

~ΔrΔt

~5×10−6 cms-1 (100kmin7x104 years)

L ~(2×1019 )(4π )(5×108 )2(5×10−6 )

=3×1032 ergs-1

Note∝ &MandcomparabletoLWDbutsmallcomparedtoLaccrete

base

32 -1

or could use

which is also about 10 erg s

base

MP

ρ⎛ ⎞⎜ ⎟⎝ ⎠

&

Though partially degenerate and dominated by beta-limitedCNO burning at first, the nova instability isbasically an example of the thin shell instability.

constant because radius at base is constant

T goes up; density goes down.

P≈

dP

dm=

GM4πr4

⇒ P≈GMΔM4πRWD

4=constant

solongastheregionwheremostofthemassisconcentratedremains<<RWD

For the beta-limited CNO cycle

εnuc =5.9 ×1015 Z ergg-1 s-1 Z~0.01-0.1

forM=10-5 Me ;Z=0.01

L =εnuc M ~1042 ergs-1

Sotheinitialflashisquitesuper-Eddingtonatthebottom.Thelayerconvects.Astheaccretedlayerbecomesconvectivethoughout,anadiabaticgradientisestablishedthroughout,expansionoccursandTstopsrisingandinfactbeginstodecline.

Basically the limiting condition is that the temperature stayshigh enough to provide an Eddington luminosity to the layeruntil it is all ejected.

The binding energy per gm of nova material is

~GM

R≈2×1017 erggm-1

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 ×10-5Me

GMΔM

R~1046erg

ΔMv2 ~1045erg(v~1000km/s)

Ldt~few×1038ergs-1 ×107s=1045-46erg∫

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 …

( )( )( )( )( )

15 5 10 5

17 5 10 6Pop I

15 17

( O) ~ 0.01 3 10 30 10 ~ 10

X O ~ 10 / 3 10 ~ 3 10 the solar mass fraction

of N and O in the sun.

novaM M−

−

×

× × ≈

e

Novae also make interesting amounts of 22Naand 26Al for gamm-ray astronomy

approximate Pop I material in the Galaxywithin solar orbit

Woosley (1986)

In the sun, the mass fractions of 15N and 17O

are 4.4 x 10-6 and 3.9 x 10-6 respectively.

The half lives of 15O and 17F are about the same.

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-limitedCNO cycle. It would take temperatures of about 3.5 x 108 K to break out of the CNO cycle and produceheavier 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 substrateduring or prior to the explosion. E.g. QU Vul was 76 and 168times 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.

• Relation to Type Ia supernovae. How to grow MWD?

• How hot do they get?

Shear mixing during accretion

Convective “undershoot” during burst

Over time, matter is removed from the white dwarf,not added and this poses a problem to making TypeIa supernovae by this route.

The rp-Process

Wallace and Woosley, ApJS, 45, 389 (1981)

Aside - T to produce heavy elements is reduced if thereis a lot of Ne and Mg already present as in novae on NeOMgwhite dwarfs.

0 1

2

3 4 5

6

7

8

9 10

11 12

13 14

15 16

17 18 19 20

21 22

23 24

25 26 27 28

29 30

31 32

33 34 35 36

37 38 39 40

41 42 43 44

45 46

47 48 49 50

51 52 53 54

n (0)

H (1)

He (2)

Li (3)

Be (4)

B (5)

C (6)

N (7)

O (8)

F (9)

Ne (10)

Na (11)

Mg (12)

Al (13)

Si (14)

P (15)

S (16)

Cl (17)

Ar (18)

K (19)

Ca (20)

Sc (21)

Ti (22)

V (23)

Cr (24)

Mn (25)

Fe (26)

Co (27)

Ni (28)

Cu (29)

Zn (30)

Ga (31)

Ge (32)

As (33)

Se (34)

Br (35)

Kr (36)

Rb (37)

Sr (38)

Y (39)

Zr (40)

Nb (41)

Mo (42)

Tc (43)

Ru (44)

Rh (45)

Pd (46)

Ag (47)

Cd (48)

In (49)

Sn (50)

Burst Ignition:

Prior to ignition : hot CNO cycle~0.20 GK Ignition : 3

: Hot CNO cycle II

~ 0.68 GK breakout 1: 15O()

~0.77 GK breakout 2: 18Ne(,p)

(~50 ms after breakout 1)Leads to rp process and main energy production

The rp-Process

a) 15O()19Ne comparable to Hburning lifetime

b) 19Ne(p,)20Na appoximatelyequal to 19Ne positron decay

c) rp-process limited byweak interactions, not15O()19Ne.

d) e) (,p) reactions startto bridge waiting points

Wallace and Woosley, ApJS, 45, 389, (1981)

0 1 2

3 45 6

7 8

9 10

111213

14

1516

17181920

2122

2324

25262728

2930

3132

33343536

3738394041

424344

45464748

49505152

535455

56

5758

59

H (1)He (2)Li (3)

Be (4) B (5) C (6) N (7)

O (8) F (9)

Ne (10)Na (11)

Mg (12)Al (13)Si (14) P (15)

S (16)Cl (17)

Ar (18) K (19)

Ca (20)Sc (21)

Ti (22) V (23)

Cr (24)Mn (25)

Fe (26)Co (27)

Ni (28)Cu (29)

Zn (30)Ga (31)

Ge (32)As (33)

Se (34)Br (35)Kr (36)Rb (37)

Sr (38) Y (39)

Zr (40)Nb (41)

Mo (42)Tc (43)

Ru (44)Rh (45)Pd (46)Ag (47)

Cd (48)In (49)

Sn (50)Sb (51)

Te (52) I (53)

Xe (54)

The Sn-Sb-Te cycle

104Sb 105Sb 106 107Sb

103Sn 104Sn 105Sn 106Sn

105Te 106Te 107Te 108Te

102In 103In 104In 105In

(,a)

Sb

+

(p, )

Known ground state emitter

Endpoint: Limiting factor I – SnSbTe Cycle

(Schatz et al. PRL 86(2001)3471)

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”)

• 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.

Type I X-Ray Bursts(e.g., Strohmayer & Bildsten 2003)

Burst energythermonuclear

Persistent fluxgravitational 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

Total nuclear energy generation at this stage is 3.4 x 1035 erg s-1.

The time is one minute before the burst.

ρ =8.9×105 g cm

T = 2.7 × 108 K

&M =1.75×10−9 Me yr-1

Z=0.05Ze

from He burning

Just before the burst starts, most of the layer is convective. The totalpower 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.

Times offset by 41,700 s of accretion at 1.75 x 10-9 solar masses/yr

Z=Z / 20e

Woosley et al (2004)

on a longer time scale

8

6 -3

at the base

9.07 10 K

1.44 10 gm cm

×

× begining ofsecond burst

maximum T developed in the burst about 1.5 x 109 K

Fourteen consecutive flashes.The first is a start up transient.

9 -1M = 1.75 10 M yr

/ 20Z Z

−×

=e

e

&

GS 1826-24

Heger, Cumming, Gallaowayand Woosley (2007, ApJ, 671, 141)

Model A3

Embarrassingly goodagreement!