2 | Classnotes 3 | Classnotes
4 | Classnotes 5 | Classnotes
6 | Classnotes 7 | Classnotes
9 | Classnotes
10 | Classnotes 11 | Classnotes 12 | Classnotes
13 | Classnotes 14 | Classnotes 15 | Classnotes
Evolution of stars beyond the main sequence
depends, of course, on the initial mass of the star.
- M > 8M-- Star evolves to
an iron core and explodes as a Type II supernova.
Remnant is either a neutron star or a black hole.
- M >0.8M but < 8M
-- Star evolves to a red giant and sheds its outer
- envelope to form a planetary nebula with
the core remaining to cool as a white dwarf.
- M < 0.8M -- Evolution is
too slow for any star to have yet consumed its supply of hydrogen.
The following piece on 'Evolution of low-mass
stars' is adapted from "The Physical Universe"
by F. H. Shu.
Ascending the Giant Branch
To fix ideas, let us first discuss a low-mass
star like our Sun. After hydrogen has been exhausted in the core,
heat continues to leak out. Since there is no more nuclear energy
generation in the core to make up the deficit, the core must
contract gravitationally, as much as Kelvin or Helmholtz originally
envisioned in the nineteenth century. As the core contracts,
it heats itself up as well as the layers just above it. At the
new higher temperatures, hydrogen can begin to burn in a shell
just outside the hydrogen-exhausted core (Fig. 8.3).
||The helium core
itself still has no nuclear energy generation, and as it continues
to lose heat to the cooler overlying layers, it must continue
to contract. This contraction is abetted as the surrounding hydrogen-burning
shell drops more and more helium "ash" onto the core.
The shrinkage of the core accompanied by the addition of more
mass makes the gravity at the border of the core stronger and
stronger. But the pressure in the shell equals the weight of
a column of material of unit area above it. This pressure must
thereore try to increase to counterbalance the increasing gravity
of the core. The pressure of the ordinary gas in the shell can
be increased, in accordance to the perfect-gas law, either by
raising the density or by raising the temperature. In fact, both
occur, and both increase the rate of hydrogen burning in the
|Figure 8.3 The structure
of a star (a) on the main sequence and (b) as it begins to leave
the main sequence because of core-hydrogen exhaustion.
However, not all the high luminosity generated
in the shell finds its way to the surface. As long as the envelope
remains radiative, the luminosity that it can carry is
limited by the photon diffusion rate. The latter is nearly fixed
for a star of a given mass. The difference between the luminosity
generated in the shell source and that leaving the surface goes
into heating up the intermediate layers, causing them to expand.
This expansion increases the total radius R; given a nearly
constant value for the surface luminosity L, thre must
be a decrease of the effective temperature Te,
in accordance with the relation The immediate post-main
sequence evolution of a radiative star therefore moves the star's
position more-or-less horizontally to the right in the H-R diagram,
turning the dwarf star into a subgiant. The cooling and
expanding surface layers cause the star to turn red in
Figure 8.4 Ascent of a
low-mass star to the red-giant branch. (adapted from Icko Iben,
Ann. Rev. Astr. Ap., 5, 1967, 571.)
As the star expands, however, the effective
temperature cannot continue to fall to arbitrarily low values.
Sooner or late, the tracs of low mass stars travel almost vertically
upwards, turning the red subgiant into a red giant (Fig.
8.4). The accompanying increase in the amount of shell luminosity
which makes its way to the surface is too much for radiative
diffusion to carry outward stably, and the entire envelope of
the red giant becomes convective (Fig. 8.5).
|Figure 8.5 The structure
of a red giant. The left figure shows the entire star from core
to photosphere. The right figure shows an enlarged picture of
the region near the core. Notice that the core, which may contain
about half the total mass of a low-mass star at this point, occupies
only one ten-billionth of the total value.
Meanwhile, the core continues to contract,
and in a low-mass star, the free electrons become so tightly
packed that they become degenerate. If we could artificially
peel off the overlying layers of a red giant at this point, the
core would essentially be a low-mass (about 0.4M) helium
white dwarf.. The very large border gravities associated with
this "white dwarf" cause the hydrogen in the shell
source to burn furiously, sending the star quickly up the red-giant
branch. At the tip of the red giant branch (in Fig. 8.4), the
temperatures in the core rise to about 108 K, which
is high enough to ignite helium and "burn" it in to
carbon via the triple-alpha process.
The Helium Flash and Descent to the
The ignition of helium in the core of a
low-mass star, however, occurs under degenerate conditions; so
it lacks the safety-valve feature which characterizes core-hydrogen
burning on the main sequence (see Classnotes 21 -- also Seeds
Box 10-- will explain what is meant by "degenerate"
conditions.) There, a slight increase in the core tempearture
will lead to a pressure increase that decreases the temperature.
Here, the pressure increases primarily because of the degeneracy
effects, not because of thermal motions; so an increase of the
core temperature leads to an overproduction of nuclear energy
without a compensating pressure increase and a compensating expansion.
Hence, an increase of the temperature, once helium has been ignited,
tends to lead to a runaway production of nuclear energy:
Higher temperatures --> more nuclear
energy --> even higher temperatures, etc.
Therefore, helium burning turns on in low-mass
stars with a "flash". So much energy is released in
the "flash" that the core's temperature rises enough
to remove the degeneracy. Normal thermal pressure then dominates
over electron-degeneracy pressure, and the core expands. This
expansion lowers the border gravity of the core, which weakens
the hydrogen-shell source. Thus, although the star now has two
nuclear sources -- helium burning in the core and hydrogen burnin
in a shell -- the prodigious shell source is now so weakened
that the star actually produces less luminosity than before.
The lowered total luminosity is too little to keep the star in
its distended red-giant state, and the star both shrinks in size
and becomes intrinsically dimmer (Fig. 8.6).
After the helium flash is completed, the
core contains an ordinary (i.e., nondegenerate) helium plasma
which is stably fusing helium into carbon. Surrounding this core
is a hydrogen-burning shell, whose strength depends on the mass
of teh overlying envelope. This state of core-helium burning
and shell-hydrogen burning is called the horizontal branch.
The exact location of a horizontal branch in the theoretical
H-R diagram depends not only on its initial mass and chemical
composition of the main sequence, but also on the amount of envelope
mass lost by the star as it ascended the red-giant branch. This
mass loss can be expected because, with the low gravity at its
distended surface, a red giant finds it even more difficult to
retain coronal surface than the Sun. Theoretical reasoning alone
cannot yet predict quantitatively the amount of mass loss to
be expected, but observations of red-giant stars show substantial
mass loss. For a group of stars which start on the lower main
sequence with similar initial masses and chemical compositions,
the stars which lose more mass on the red-giant branch end up
on the horizontal branch with smaller envelope masses and, therefore,
weaker shell sources in addition to the core source. In particular,
a horizontal branch star which had lost all its envelope mass
would be a chemically homogeneous helium star burning helium
into carbon at its center. Such a state is often called the "helium
main sequence" by analogy with the usual (hydrogen) main
sequence. Even a relatively low mass (say, 0.5M) helium
star appears quite blue because of its moderately large luminosity
and moderately small radius.
Descent of a low-mass star with poor heavy-element abundances
(Population II star) from the tip of the red-giant branch to
the horizontal branch. Track A corresponds to a star which suffered
a relatively large loss of mass during the red-giant stage of
stellar evolution. Track B corresponds to a star which suffered
relatively little loss of mass.(Adapted from Icko Iben, Ann.
Rev. Astr. Ap., 5, 1967, 571.)
|Most horizontal-branch stars,
however, have a finite envelope mass on top of such a helium-burning
core, and the hydrogen-burning shell associated with the weight
of this envelope keeps the envelope relatively distended. Thus,
true horizontal-branch stars tend to have slightly higher luminositites
than a "helium-main-sequence" star of the same core
mass, as well as apppreciably lower effective temperatures. If
we started with a group of stars, therefore, with similar initial
masses and chemical compositions, and if this group suffered
varying amounts of envelope-mass loss ascending the red-giant
branch, we should expect them to end up on the horizontal branch
with nearly the same luminosities but with different effective
temperatures. In other words, such a group would occupy a horizontal
locus in the H-R diagram, and this feature gives these stars
Ascending the Asymptotic Giant Branch
What happens when helium in the core of
a horizontal-branch star is exhausted (by burning into carbon
and oxygen)? Well, obviously, the core must contract, which increases
the pressure and temperature of the overlying layers. Thus, helium
ignites in a shell just outside the core, and hydrogen burns
in a shell outside of that.
||The star is now
in a double-shell-burning stage. The mass of the inert carbon-oxygen
core continues to increase, an it continues to contract just
as the helium core did when the star ascended the red-giant branch
again, and the double-shell-source phase is also known as the
asymptotic giant branch (Fig. 8.8). Eventually, the shrinkage
of the core again causes the free electrons to become degenerate.
If the overlying envelope were now to be stripped away (say,
by mass loss), the core would be a hot carbon-oxygen white dwarf.
Now, however, the mass of the degenerate core is larger than
before because of the additional "ash" in the core,
and the radius of the incipient white dwarf is smaller thn its
helium counterpart at the tip of the red-giant branch. Thus,
the gravity of any overlying shell sources would be correspondingly
larger, forcing them to generate higher luminosities yet. Stars
at the end of the double-shell-burning phase may become red super-
giants. At such tremendous rates of expenditures of energy,
the star cannot live much longer.
|Figure 8.8 The structure
of an asymptotic giant. The figure on the left shows the entire
star from the core to photosphere. The figure on the right shows
an enlarged picture of the region near the core.
What happens to stars at these very late stages of stellar evolution
is theoretically quite uncertain; several complications make
detailed computation difficult. One of these is the onset of
"thermal relaxation oscillations" when the helium-shell
source becomes spatially very thin. The origin of this instability
is very different from the "helium flash" discussed
earlier. There, an initial overproduction of nuclear energy leads
to a runaway because of the degeneracy of the nuclear-burning
region. Here, an initial overproduction of nuclear energy also
leads to a thermal runaway, but for an entirely different reason.
Here, the nuclear burning region is nondegenerate, but it is
a spatially thin shell. Thus, with the input of excess nuclear
energy, the layer can and will expand. But the expansion of a
thin shell does little to relieve the weight of the overlying
material; this material is only lifted a little. Thus, the weight
hardly changes, and therefore the pressure that the thin shell
has to maintain to offset this weight also hardly changes. Meanwhile,
the tempreature has increased, and if the rate of nuclear-energy
generation is sufficiently sensitive to temperature changes --
as the triple-alpha reaction is -- then it will also increase
further before the excess heat has a chance to diffuse away.
Thus, a thermal runaway ensues. The runaway is checked only after
considerable expansion of the layer and the appearance of convection
to carry away the excess heat. But a basic problem remains. After
the runaway is checked, and when the star tries to adopt the
"natural" double-shell-burning configuration appropriate
for this stage of its evolution (i.e., when it tries to "relax"
back to the "natural" stage of equilibrium), it finds
itself in the same difficulty. Thus, the star undergoes a series
of "thermal relaxation oscillations" which consist
of one or more sharp pulses of extra energy generation followed
by relatively long periods of quiet evolution (Fig. 8.9). Each
thermal runaway is followed by the development of a convection
zone which extends from the helium-burning shell almost to the
inner convection zone may actually connect to the out convective
envelope through the hydrogen-burning shell (see Fig. 8.8). If
this happens, the products of helium burning, carbon and oxygen,
as well as s-process material, may be brought to the surface
of the asymptotic red giant.
|Figure 8.9 Thermal relaxation
oscillations associated with helium-shell flashs. (Adapted from
M. Schwarzschild and R. Harm, ApJ, 150, 1967, 961.)
Planetary Nebulae and White Dwarfs
Another complication is that considerable
mass loss may occur on the asymptotic giant branch. Observations
suggest that asymptotic branch stars beyond the red-giant tip
lose mass very rapidly. Among the many promising mechanism which
have been considered is the suggestion that small specks of dust
may form in cool atmospheres and be driven out subsequently by
the radiation pressure of the star. Quantitative calculations,
unfortunately, are difficult. Observations indicate that stars
that originally had less than about six solar masses seem to
lose so much mass during such high-luminosity stages that they
become (perhaps periodically) planetary nebulae illuminated by
a hot central core. This hot central core is presumably an incipient
white dwarf, with mass necessarily below the Chandresakhar limit
1.4M. From the central-star stage of planetary nebulae,
the exposed core burns out its hydrogen and helium shells, loses
its extended envelope, and descends the H-R diagram to enter
the region occupied by white dwarfs proper. Figure 8.10 summarizes
the complete evolution of a low-mass star from the main sequence
to carbon-oxygen white dwarf. The final approach to a white dwarf
from an asymptotic giant star is shown in dashed lines, to emphasize
that the theory is incomplete for these late stages of stellar
a good look at Figure 8.10. You may be staring at the future
of the Sun. When the Sun swells up to become a red giant for
the first time, it will occupy about 30° in the sky. What a sunset
that would make! When the sun swells up for the second time (assuming
it does not lose so much mass that it becomes a helium white
dwarf), it may well engulf the Earth. Any inhabitants, of course,
would have long since been roasted. Be consoled at least that
the Sun itself will ultimately be able to rest in peace as a
senescent white dwarf.
|Figure 8.10 The complete
evolution of a low-mass star from the main sequence to a white
dwarf. The track from the asymptotic giant branch to the white
dwarf (via a planetary nebula is uncertain and is shown as a
Syllabus | Classnotes 2 | Classnotes
3 | Classnotes 4 | Classnotes
5 | Classnotes 6 | Classnotes
7 | Classnotes 8
9 | Classnotes
10 | Classnotes 11 | Classnotes 12 | Classnotes
13 | Classnotes 14 | Classnotes 15 | Classnotes