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Lecture 7
Evolution of Massive
Stars on the Main Sequence
and During Helium Burning Basics
Key Physics and Issues
• Nuclear Physics
• Equation of state
• Opacity
• Mass loss
• Convection
• Rotation (magnetic fields)
• Binary membership
• Explosion physics
• Evolution in HR diagram
• Nucleosynthesis
• Surface abundances
• Presupernova structure
• Supernova properties
• Remnant properties
• Rotation and B-field
of pulsars
Massive Stars
Generalities:
Because of the general tendency of the interior temperature of
stars to increase with mass, stars of just over one solar mass are
chiefly powered by the CNO cycle(s) rather than the pp cycle(s).
This, plus the increasing fraction of pressure due to radiation,
makes their cores convective. The opacity is dominantly due to
electron scattering. Despite their convective cores, the overall
main sequence structure can be crudely represented as an
n = 3 polytrope.
Following eq. 2- 286 through 2-289 in Clayton it is
easy to show that for
Pgas   Ptotal 
N A kT 

1 4
aT = (1-  )Ptotal
3
Prad =
that
1/ 3
Ptotal
 N k  4 3  1    
=  A  
 




a



4/3
That is, a star that has constant throughout its mass
will be an n = 3 polytrope
It is further not difficult to show that a sufficient
condition for  = constant in a radiative region
is that
L(r ) (r )
 (1 -  ) = constant
M (r )
That is, if the energy generation per unit mass interior
to r times the local opacity is a constant,  will be a
constant and the polytrope will have index 3.
This comes from combining the equation for radiative
equilibrium with the equation for hydrostatic equilibrium
and the definition of 
On the other hand, convective regions (dS = 0), will
have n between 1.5 and 3 depending upon whether
ideal gas dominates the entropy or radiation does.
Ideal gas (constant composition):
P = const    T     2/3 =  5/3
since
 =
T3/2

 constant
n+1
3
 n=
n
2
Radiation dominated gas:
P=
since
 =
T3

1 4
aT   4/3
3
 constant
n+1
 n=3
n
Half way through hydrogen burning
For “normal” mass stars, ideal gas entropy
always dominates on the main sequence, but
for very massive stars Srad and Sideal can become
comparable.
So massive stars are typically a hybrid
polytrope with their convective cores having
3 > n > 1.5 and radiative envelopes with n
approximately 3.
Overall n = 3 is not bad.
Then
1- 

2
2
 M 

 18 M e 
 =   (1+Z i ) Yi 
Clayton 2-307
1
 0.84 for 35% H, 65% He
For very large M,   M -1/2 ; for 18 M e   0.80
1    fraction of the pressure
from radiation
inner ~5 Msun is
convective
Near the surface
the density declines
precipitously making
radiation pressure
more important.
L
(1   ) decreases as M(r) 
M
because L is centrally concentrated,
so  increases with M(r)
inner ~8 Msun
convective
Convection plus entropy from ideal gas
implies n = 1.5
d ln  1

d ln P 
  5 / 3 for ideal gas
at constant entropy
40% of the mass
Most of the mass and
volume.
d ln  1

d ln P 
  4 / 3 for standard model
(with  = const) in radiative
regions
For the n=3 polytrope
2/3
 M 
6
Tc  4.6  10 K  
1/c 3

 Me 
For stars on the main sequence and half way through
hydrogen burning,   0.84 and, unless the star is very massive,
  0.8 - 0.9. Better values are given in Fig 2-19 of Clayton.  declines
slowly with mass
The density is not predicted from first principles, but detailed models
 10 M e 
--3
(following page) give c  10 
gm
cm
,So

 M 
 M 
7
Tc  3.9 10  

 10 M e 
1/ 3
K
M
All evaluated in
actual models at
a core H mass
fraction of 0.30
for stars of solar
metallicity.
 0.8
9
12
15
20
25
40
60(57)
85(78)
120(99)
Tc/107

3.27
3.45
3.58
3.74
3.85
4.07
4.24
4.35
4.45
9.16
6.84
5.58
4.40
3.73
2.72
2.17
1.85
1.61
C
L/1037
2.8
7.0
13
29
50
140
290
510
810
L  M2.5
c decreases with mass as a general consequence of the fact that
Tc3
c
 M 2  3  3 and H burning happens at a relatively constant
temperature. Until about 40 M e , the density decreases roughly
as M -1 . After that it decreases more slowly. Recall   M -1/2 for
very large masses
Homology
n ~ 18
  const if eleectron scattering
From homology relations on the previous page and taking
= constant (electron scattering), an ideal gas equation of state,
and n = 18, one obtains:
L
Tc
 4M 3
o
M
 1/ 3 M 4 / 21
4 / 21

M
if composition constant
1/ 21
( o o )
R
where  was defined in lecture 1.
If radiation pressure dominates, as it begins to for very large
values of mass,
1/ 42
L
M
o
T
M
( o o )1/ 21
hence

M
L
Overall L ~ M2.5
M 2 (lower end) to a constant (high end)
M 100M
Competition between the p-p
chain and the CNO Cycle
The Primary CNO Cycle
In a low mass star
The slowest reaction is 14N(p,)15O. For temperatures near 2 x 107 K.
 nuc  T
n
n=
 -2
3
1/ 3
 22

7
1


14  1
  4.248 

0.020




14  1
 60.0
n =18
(More on nucleosynthesis later)
CNO tri-cycle
CN cycle (99.9%)
O Extension 1 (0.1%)
Ne(10)
F(9)
O Extension 2
O Extension 3
O(8)
N(7)
C(6)
3
4
5
6
7
8
9
neutron number
All initial abundances within a cycle serve as catalysts and accumulate at largest 
Extended cycles introduce outside material into CN cycle (Oxygen, …)
Mainly of interest for nucleosynthesis
In general, the rates for these reactions proceed through known
resonances whose properties are all reasonably well known.
There was a major revision of the rate for 14 N(p, )15O in
2001 by Bertone et al., Phys. Rev. Lettr., 87, 152501.
The new rate is about half as large as the old one,
so the main sequence lifetime of massive stars is longer
(but definitely not linear in the reciprocal rate). Mainly
affected globular cluster ages (0.7 to 1 Gy increase in
lifetime due to the importance of the CNO cycle at the
end of the MS life and during thick H shell burning).
Equation of state
Well defined if tedious to calculate up to the point
of iron core collapse.
Ions - ideal gas - P =

NA k T

1 4
Radiation P =
aT
3
Electrons - the hard part - can have arbitrary relativity and degeneracy
(solve Fermi integrals). At high T must include
electron-positron pairs.
Beyond 1011 g cm -3 - neutrino trapping, nuclear force, nuclear excited
states, complex composition, etc.
Opacity
In the interior on the main sequence and within the helium
core for later burning stages, electron capture dominates.
In its simplest form:
e 
ne Th


 N A Ye  Th
 Ye(N A  Th )

8  e2 
 Th  
3  me c 2 
2
 e  0.40 Ye cm 2 gm -1
Recall that for 75% H, 25% He, Ye  0.875, so  e  0.35
For He and heavier elements  e  0.20.
There are correction terms that must be applied to es especially at
high temperature and density
1) The electron-scattering cross section and Thomson cross section
differ at high energy. The actual cross section is smaller.
Klein-Nishina
 KN
  2h  26  h  2

  Th 1  
 
 ...
2 
2 
  me c  5  me c 

h  1020 Hz
2) Degeneracy – at high density the phase space for the scattered electron
is less. This decreases the scattering cross section.
3) Incomplete ionization – especially as the star explodes as a supernova.
Use the Saha equation.
4) Electron positron pairs may increase  at high temperature.
Effects 1) and 2) are discussed by
Chin, ApJ, 142, 1481 (1965)
Flowers & Itoh, ApJ, 206, 218, (1976)
Buchler and Yueh, ApJ, 210, 440, (1976)
Itoh et al, ApJ, 382, 636 (1991) and references therein
Electron conduction is not very important in massive stars but is
important in white dwarfs and therefore the precursors to Type Ia
supernovae
Itoh et al, ApJ, 285, 758, (1984) and references therein
For radiative opacities other than es, in particular bf and bb,
Iglesias and Rogers, ApJ, 464, 943 (1996)
Rogers, Swenson, and Iglesias, ApJ, 456, 902 (1996)
see Clayton p 186 for a definition of terms.
“f” means a continuum state is involved
Note centrally concentrated
nuclear energy generation.
convective
During hydrogen
burning
Convection
All stellar evolution calculations to date, except for brief snapshots,
have been done in one-dimensional codes.
In these convection is universally represented using some variation
of mixing length theory.
Caveats and concerns:
• The treatment must be time dependent
•
Convective overshoot and undershoot
•
Semiconvection (next lecture)
• Convection in parallel with other mixing processes,
especially rotation
• Convection in situations where evolutionary time scales are
not very much longer than the convective turnover time.
Kuhlen, Woosley, and Glatzmaier
exploried the physics of stellar convection
using 3D anelastic hydrodynamics.
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
The model shown is a 15 solar mass star
half way through hydrogen burning. For now
the models are not rotating, but the code
includes rotation and B-fields. (Previously
used to simulate the Earth’s dynamo).
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Convective structure
Note growth of the
convective core with M
from Kippenhahn and Wiegert
The adiabatic condition can be written in terms of
the temperature as
 dT
dP
 2
0
P 1  2 T
This defines  2
(see Clayton p 118)
For an ideal gas  2  5 / 3, but if radiation is
included the expression is more complicated
Convective instability is favored by a large fraction of radiation
pressure, i.e., a small value of 

1  T  dP 
 dT 

1


 


  convection
 dr star   2  P  dr 
32  24   3 2
4
5
2 



(Clayton 2-129)
2
2
24  18  3
3
3
 1 
 1 
For  =1, 1-   0.4; for  = 0, 1-   0.25
 2 
 2 
 1 
for  = 0.8 1-  = 0.294
 2 
So even a 20% decrease in  causes a substantial decrease in the
critical temperature gradient necessary for convection.
 decreases with increasing mass. Hence more massive stars
are convective to a greater extent.
 (1  )1/ 2 
M
18.0 

Me
  2 2 

M 
M
9
12
15
20
25
40
60(57)
85(78)
120(99)
All evaluated at
a core H mass
fraction of 0.30
for stars of solar
metallicity.
Tc/107

3.27
3.45
3.58
3.74
3.85
4.07
4.24
4.35
4.45
9.16
6.84
5.58
4.40
3.73
2.72
2.17
1.85
1.61
L/1037
C
2.8
7.0
13
29
50
140
290
510
810
Q
conv core
0.26
0.30
0.34
0.39
0.43
0.53
0.60
0.66
0.75
Several things to note:
a) Increasing size of convective core
12 - 15 M
b)
d ln Tc
d ln M
0.17
20 - 25 M
0.13
60 - 85 M
0.10
4
1
 0.19 ( =1) to
 0.048 ( =0)
21
21
c) Polytropic index varies from >1.5 to 3.0 as M increases. n = 3 not bad overall
Homology gives
d)
C M 2  3
TC
3
 constant, though n does vary significantly; in fact,
c M
Tc 3
 constant
First 3 Burning Stages in the Life of a Massive Star
0
blue = energy generation
purple = energy loss
green = convection
Surface convection zone
H-burn
He-burn
The convective core shrinks during hydrogen burning
During hydrogen burning the mean atomic weight is increasing from
near 1 to about 4. The ideal gas entropy is thus decreasing. As the
central entropy decreases compared with the outer layers of the star
it is increasingly difficult to convect through most of the star’s mass.
For an ideal gas plus radiation:
(see Clayton p. 123)
1
4
  for pure hydrogen;
for pure helium
2
3
change in entropy during He
burning is small
The convective core grows during helium burning.
During helium burning, the convective core grows largely because the
mass of the helium core itself grows. This has two effects:
a) As the mass of the core grows so does its luminosity, while the radius
of the convective core stays nearly the same (density goes up). For
a 15 solar mass star:
He mass fraction
Radius conv core
Lum conv core
Lum star
3.0 M
1
0.87 x 1010 cm
3.2 x 1037 erg s-1
2.16 x 1038 erg s-1
3.6 M
0.5
1.04 x 1010 cm
6.8 x 1037 erg s-1
2.44 x 1038 erg s-1
The rest of the luminosity is coming from the H shell..
blue = energy generation
purple = energy loss
green = convection
Surface convection zone
H-burn
He-burn
b) As the mass of the helium core rises its decreases.
T3

M 
2
3
1/ 3aT 4 1  
 N A kT

M 2 3
This decrease in  favors convection.
The entropy during helium burning also continues to decrease,
and this would have a tendency to diminish convection, but the
 and L effects dominate and the helium burning convective core
grows until near the end when it shrinks both due to the decreasing
central energy generation.
This growth of the helium core can have two
interesting consequences:
• Addition of helium to the helium convection
zone at late time increases the O/C ratio made
by helium burning
• In very massive stars with low metallicity the
helium convective core can grow so much that
it encroaches on the hydrogen shell with major
consequences for stellar structure and
nucleosynthesis.
Convective Overshoot Mixing
Initially the entropy is nearly flat in a zero age main sequence star
so just where convection stops is a bit ambiguous. As burning proceeds
though and the entropy decreases in the center, the convective extent
becomes more precisely defined. Still one expects some overshoot
mixing.
A widely adopted prescription is to continue arbitrarily the
convective mixing beyond its mathematical boundary by some fraction,
a, of the pressure scale height. Maeder uses 20%. Stothers and Chin
(ApJ, 381, L67), based on the width of the main sequence, argue
that a is less than about 20%. Doom, Chiosi, and many European
groups use larger values. Woosley and Weaver use much less. Nomoto
uses none.
Overshoot mixing:
Overshoot mixing has many effects. Among them:
• Larger helium cores
• Higher luminosities after leaving the main sequence
• Broader main sequence
• Longer lifetimes
• Decrease of critical mass for non-degenerate C-ignition.
Values as low as 5 solar masses have been suggested but
are considered unrealistic.
Overshoot mixing:
DeMarque et al, ApJ, 426, 165, (1994) – modeling main sequence
widths in clusters suggests a = 0.23
Woo and Demarque, AJ, 122, 1602 (2001) – empirically for low mass
stars, overshoot is < 15% of the core radius. Core radius
a better discriminant than pressure scale height.
Brumme, Clune, and Toomre, ApJ, 570, 825, (2002) – numerical 3D
simulations. Overshoot may go a significant fraction of
a pressure scale height, but does not quickly establish an
adiabatic gradient in the region.
Differential rotation complicates things and may have some of
the same effects as overshoot.
See also http:/www.lcse.umn.edu/MOVIES
METALLICITY
Metallicity affects the evolution in four distinct ways:
• Mass loss
• Energy generation (by CNO cycle)
• Opacity
• Initial H/He abundance
lower main sequence:
 decreases if Z decreases L
Tc
 o 16 /13  o1/13
  o o 
2 /15
For example, 1 M at half hydrogen depletion
Z = 0.02
Z = 0.001
log Tc
7.202
7.238
L
L
2.0 L
Because of the higher luminosity, the lifetime of the lower
metallicity star is shorter (it burns about the same fraction of its mass).
Upper main sequence:
The luminosities and ages are very nearly the same because the
opacity is, to first order, independent of the metallicity. The
central temperature is a little higher at low metallicity because of the
decreased abundance of 14N to catalyze the CNO cycle.
For example in a 20 solar mass star at XH = 0.3
log Tc
Z  0.02
7.573
Z  0.001
7.647
log L / L
4.867
4.872
Qcc
0.390
0.373
M /M
19.60
19.92 (mass loss)
Schaller et al. (1992)
M
d ln  MS
d ln M
M
d ln  MS
d ln M
1-2
-2.9
2-4
4-9
-2.6
-2.0
12 - 15 20 - 25 85 - 120
-1.4
-1.2
-0.3
There is a slight difference in the lifetime on
the upper main sequence though because of the
different initial helium abundances. Schaller et al.
used Z = 0.001, Y = 0.243, X=0.756 and Z = 0.02,
Y = 0.30, X = 0.68.
So for the higher metallicity there is less hydrogen to
burn.
But there is also an opposing effect, namely mass loss.
For higher metallicity the mass loss is greater and the
star has a lower effective mass and lives longer.
Both effects are small unless the mass is very large.
For helium burning, there is no effect around 10 solar mases,
but the higher masses have a longer lifetime with higher
metallicity because mass loss decreases the mass.
For lower masses, there is a significant metallicity dependence
for the helium burning lifetime. The reason is not clear.
Perhaps the more active H-burning shell in the solar
metallicity case reduces the pressure on the helium core.
For 2 solar masses half way through helium burning
log Tc
Z = 0.001
Z = 0.02
8.089
8.074
a small difference but the helium burning rate goes as
as a high power of T at these temperatures. The above
numbers are more than enough to explain the difference
in lifetime.
Zero and low metallicity stars may end their lives as
compact blue giants – depending upon semiconvection
and rotationally induced mixing
For example, Z = 0, presupernova, full semiconvection
a)
20 solar masses
R = 7.8 x 1011 cm Teff = 41,000 K
b) 25 solar masses
R=1.07 x 1012cm Teff = 35,000 K
Z = 0.0001 ZO
a)
25 solar masses, little semiconvection
R = 2.9 x 1012 cm Teff = 20,000 K
b) 25 solar masses, full semiconvection
R = 5.2 x 1013 cm Teff= 4800 K
Caveat: Primary 14N production
Caution - rotationally induced mixing changes these results
Very massive stars
As radiation pressure becomes an increasingly dominant part of
the pressure,  decreases in very massive stars.
This implies that the luminosity approaches Eddington.
E. g. in a 100 solar mass star L = 8.1 x 1039 erg s-1,
or LEdd/1.8. But  still is 0.55.
In regions where the star is supported entirely by radiation pressure
and the radiation is diffusing the luminosity is the Eddington
luminosity.
The stellar structure equation for the radiative temperature gradient is
dT
3  L(r)

dr
4ac T 3 4 r 2
Massive stars are always
radiative near their surfaces,
implies
even though they can become
c
2 d 1
4
L(r) 
4 r
aT 
convective throughout most

dr  3

of their mass.
if the pressure is given by
Prad = (1- ) Ptot
where  = Pideal /Ptot , then
4 r 2 c d
L(r) 
(1   ) Ptot
 dr
which if  is a constant gives (with hydrostatic equilibrium)
GM (r) 
4 GMc

(1


)
.
2


r
If  =0, this is the Eddington luminosity
L(r) 
4 r 2 c
(1   )
This suggests that massive stars, as  approaches 0, will
approach the Eddington luminosity with L proportional to M.
In fact, except for a thin region near their surfaces, such stars
will be entirely convective and will have a total binding energy
that approaches zero as approaches zero. But the calculation applies
to those surface layers which must stay bound.
Completely convective stars with a luminosity proportional
to mass have a constant lifetime, which is in fact the shortest
lifetime a (main sequence) star can have.
LEdd 
 M   0.34 
1.47 10 erg s 



M





qnuc  4.8 1018 erg/g
4 GMc
38
-1
 MS  qnuc M / LEdd  2.1 million years
(exception supermassive stars over 105 solar masses – post-Newtonian
gravity renders unstable on the main sequence)
Similarly there is a lower bound for helium
burning. The argument is the same except one
uses the q-value for helium burning to carbon and
oxygen.
One gets 7.3 x 1017 erg g-1 from burning 100% He to
50% each C and O.
Thus the minimum (Eddington) lifetime for helium
burning is about 300,000 years.
Limit
Limit
Since  ~ 4/3, such stars are loosely bound (total energy much
less than gravitational or internal energy) and are subject to
large amplitude pulsations. These can be driven by either
opacity instabilities (the mechanism) or nuclear burning
instabilities (the  mechanism).  is less than 0.5 for such
stars on the main sequence, but ideal gas entropy still dominates.
For solar metallicity it has long been recognized that such stars
(say over 100 solar masses) would pulse violently on the main
sequence and probably lose most of their mass before dying.
Ledoux, ApJ, 94, 537, (1941)
Schwarzschild & Harm, ApJ, 129, 637, (1959)
Appenzeller, A&A, 5, 355, (1970)
Appenzeller, A&A, 9, 216, (1970)
Talbot, ApJ, 163, 17, (1971)
Talbot, ApJ, 165, 121, (1971)
Papaloizou, MNRAS, 162, 143, (1973)
Papaloizou, MNRAS, 162, 169, (1973)
Mass lost
~35 solar
masses
100 M e (main sequence mass) star when the central hydrogen mass fraction hits 0.01.
The luminosity of the star is 6.18  10 39 erg s-1 (about 0.1% as bright as a Type II supernova)
L
100Me at hydrogen depletion

Eddington luminosity for 65 solar masses
LEdd 
4 GMc

 0.34 
 9.6 1039 erg s-1 
  
So about model is ~2/3 Eddington luminosity for electron
scattering opacity near the surface.
As the star contracts and ignites helium burning its luminosity
rises to 8 x 1039 erg s-1 and the mass continues to decrease
by mass loss. Super-Eddington mass ejection?
L excess R
-1
M:
: 0.001M e y 1 L 38 R13 M100
GM
L excess depends on  which is at least as large
as electron scattering.
Luminous blue
variable stars?
Structural adiabatic exponent is nearly 4/3
Radial pulsations and an upper limit
1941, ApJ, 94, 537
(1941)
Also see Eddington (1927, MNRAS, 87,
539)
Upper mass limit: theoretical predictions
Stothers & Simon (1970)
Upper mass limit: theoretical predictions
Ledoux (1941)
radial pulsation, e- opacity,
H
100 M
Schwarzchild & Härm (1959)
radial pulsation, e- opacity,
H and He, evolution
65-95 M
Stothers & Simon (1970)
radial pulsation, e- and atomic
Larson & Starrfield (1971)
pressure in HII region
50-60 M
Cox & Tabor (1976)
e- and atomic opacity
Los Alamos
80-100 M
Klapp et al. (1987)
e- and atomic opacity
Los Alamos
440 M
Stothers (1992)
e- and atomic opacity
Rogers-Iglesias
120-150 M
80-120 M
Upper mass limit: observation
R136
Feitzinger et al. (1980)
250-1000 M
Eta Car
various
120-150 M
R136a1
Massey & Hunter
(1998)
136-155 M
Pistol Star
Figer et al. (1998)
140-180 M
Damineli et al. (2000)
~70+? M
LBV 1806-20
Eikenberry et al. (2004)
150-1000 M
LBV 1806-20
Figer et al. (2004)
130 (binary?)
M
HDE 269810
Walborn et al. (2004)
150 M
WR20a
Bonanos et al. (2004)
Rauw et al. (2004)
82+83 M
Eta Car
R122 in LMC
Calculations suggested that strong non-linear pulsations
would grow, steepening into shock in the outer layers and
driving copious mass loss until the star became low
enough in mass that the instability would be relieved.
But what about at low metallicity?
Ezer and Cameron, Ap&SS, 14, 399 (1971) pointed out that Z = 0
stars would not burn by the pp-cycle but by a high temperature
CNO cycle using catalysts produced in the star itself,
Z ~ 10-9 to 10-7.
Maeder, A&A, 92, 101, (1980) suggested that low metallicity might
raise Mupper to 200 solar masses.
Surprisingly though the first stability analysis was not performed for
massive Pop III stars until Baraffe, Heger, and Woosley, ApJ, 550, 890,
(2001).
Baraffe et al found that a) above Z ~ 10-4 solar, the  instability
dominated and below that it was negligible and b) that the -instability
was also suppressed for metallicities as low as 10-7 solar.
Reasons
No heavy elements to make lines and grains
High temperature of H-burning made the reactions less
temperature sensitive.
They concluded that stars up to about 500 solar masses would
live their (main sequence) lives without much mass loss.
In the same time frame, several studies suggested that the IMF for
Pop III may have been significantly skewed to heavier masses.
Abel, Bryan, and Norman, ApJ, 540, 39, (2000)
Larson, astroph 9912539
Nakamura and Umemura, ApJ, 569, 549,(2002)
So - very massive stars might have formed quite early in the universe,
might have lived their lives in only a few million years and might have
died while still in possession of nearly their initial mass.
Or given the discussion of the Eddington limit, they might not
As we shall see later the supernovae resulting from such large
stars and their nucleosynthesis is special.
These stars may also play an important role in reionizing the
universe (even if only 0.01% of the matter forms into such stars).
8 Mev/nucleon
13eV/nucleon
106
There are really three important distinct issues: What masses were
the early stars born with, what were there typical masses on the
main sequence, and with what mass did they die?
CAVEAT!