You should begin making your way through Binney & Merrifield's Chapter 5. We will be hotting most of the topics in that Chapter 6 this semester.
(Very) Brief Review of Stellar Structure and Evolution
Fundamentals of Stellar Interiors and Evolution
Theory of stellar structure and evolution obviously important
to understanding stellar populations and evolution of clusters
Subject of an entire other graduate course.
Can only broadly summarize important aspects here (more "what"
Lives of stars governed by the basic principle by which larger structures, like
star clusters and galaxies, also live:
Let heat flow outward so that inner regions may become denser and, therefore,
Basic elements of standard stellar structure theory:
Hydrostatic equilibrium (first eqn. below): In order for a star
to be static, must have a pressure gradient to counteract gravity.
Physical parameters of gas sphere described/connected by
set of differential equations:
The first relates the decrease in pressure with
radius, P(r), to the local weight as:
The function M(r) represents the mass
interior to r:
The conservation of energy then requires that
the energy emerging each second from the sphere of
radius r is related to the power generated
per gram of matter, ε(r)
The thermal gradient is a by-product of energy
transport mechanisms. If the material is in radiative
equilibrium, the temperature gradient is related to the
However, if the temperature gradient required for
equilibrium is excessive and cannot be accommodated
by radiative transfer (the "Schwarzschild criterion"),
the matter becomes convectively
unstable, and the thermal gradient is given by:
Here γ is the "adiabatic gamma" and
is the ratio of the specific heat (heat required
to raise temperature of unit mass by 1K) at constant pressure
to the specific heat at constant volume. It is related
to the equation of state (and is 5/3 for an ideal gas).
μ is the mean molecular weight compared to that
of hydrogen, mH .
These equations require information on the physical
properties of the matter from which the star is made,
the constitutive relations or equations of state:
The pressure equation of state:
P = P( ρ, T, X),
where X is the chemical composition.
- Under many circumstances, ideal gas law reasonable approximation.
- Things get more complex in dense matter, high temperature situations.
Opacity. Cannot necessarily be described analytically:
κ = κ( ρ, T, X)
Energy generation rate:
ε = ε( ρ, T, X)
Two important nuclear reactions take place in main sequence
stars to "burn" hydrogen into helium (with net effect to turn 4 protons
into a He nucleus):
p-p chain (e.g., dominant in Sun):
(Note: The reactions on the right are the reason that Be and Li are heavily
depleted in stars, and why deep convection acts to destroy the "surface" abundances
of these elements in stellar atmospheres.)
In hotter stellar cores, the CNO cycle dominates:
This is only one (but the "dominant one") of the nuclear
reaction chain endings in the CNO cycle that produce a 12C
nucleus and a 4He nucleus.
The rate of these reactions is proportional to
ρ2 and increases rapidly with T.
Boundary conditions that define limits of integration.
At r = 0, M(r) --> 0, L(r) --> 0
At surface of star, roughly
(T, P, ρ) --> 0
If you specify mass, radius, composition and luminosity at the surface
of a star, apply boundary conditions and begin integrating inwards, a successful model will
also satisfy the internal boundary condition.
Cannot pick arbitrary combinations of radius and luminosity
after mass and composition specified.
This is a statement of the Vogt-Russell Theorem, which is
The mass and chemical composition of a star uniquely determine
its radius, luminosity, internal structure and subsequent
Since the dependence of a star's evolution on mass and composition
is a consequence of the change in composition due to nuclear
reactions, the Vogt-Russell Theorem states more simply that:
The internal structure of a star is uniquely determined
by its age, mass, and chemical composition.
(The Vogt-Russell Theorem is a "first order" theorem.
It ignores second order effects like magnetic fields, stellar rotation, mass loss,
The volution of stars, especially as witnessed in the observational CMD,
can be summarized by the
fact that as stars become hotter and denser in their cores with time, progressively
heavier nuclei are formed (H --> He --> C,N,O ... up to Fe depending on mass of star).
The energy released in the creation of heavier nuclei temporarily
sustains the requirement for the outward heat flux while stalling further
compression of the interior.
These active periods of successful nuclear burning and structural
stasis are interrupted by brief periods at the point of fuel
exhaustion, where the star rearranges itself by
further concentration of its interior as needed to
achieve stasis again.
Places in the CMD of a stellar populations
where stars "pile up" correspond to
places where the "stable", fuel burning periods place stars (mini-"main
sequences"), whereas the CMD is less well populated by stars in the
rapid, structural reorganization phases.
The above is one statement of the fuel consumption theorem
we will see later: The number of stars at each burning stage
varies roughly as the time it takes to exhaust the nuclear fuel burnt
at that stage.
Through the Vogt-Russell theorem we can theoretically predict
where stars should be in the "theoretical HR diagram" of
effective temperature and bolometric luminosity.
(But recall our previous discussions about connecting this theoretical plane to the observational
For example: Solutions for "Zero-Age'', chemically homogeneous, H-burning
Main Sequence stars, the ZAMS, give an arrangement that is a
From Binney & Merrifield. The dashed lines show regions where variable stars
are found, including the luminosity limit to
non-variable stars specified by de Jager (1984). Stars beyond this limit
tend to have large mass loss rates.
Individual stars spend most of their lives in this limited part of
the HR diagram, slowly using up hydrogen in the core and drifting away
from the ZAMS.
This is the longest period of a star's life because it is a period of
fuel economy: Even though the star releases more nuclear energy after the MS,
the star is also more luminous and more quickly exhausts fuel.
The rate of hydrogen core exhaustion is a function of mass.
More massive stars use up their hydrogen faster, causing them to evolve off
the MS faster.
This results in the classical notion of stellar populations "burning down
the MS like a candle wick".
From VandenBerg (198x).
As chemical composition in core slowly changes, star's interior
conditions slowly change and, at first, there is slow evolution from ZAMS;
then evolution suddenly speeds up.
Overall evolution of stellar luminosity from MS to RGB tip:
From Binney & Merrifield.
Initial phase of slow evolution corresponds to increase in
mean molecular weight, μ = < m > / mH, in core as He made.
By ideal gas law, P = ρ kT/ μ mH ,
unless there is an increase in ρ or T to compensate, there
will be an insufficient pressure to support the layers above;
thus the stellar core contracts to increase P.
This releases gravitational potential energy, half of which is radiated
away (by the virial theorem) and half put into thermal energy (i.e.,
temperature) of gas.
Because of the increase in interior temperature, there is an increase in
the rate of nuclear reactions and the region undergoing them; as a
result, the star becomes slowly more luminous.
The Sun is in this state now.
Eventually H exhausted in core, and star gets to point 3 in the
"Figure 13.4" below.
Transfer to shell H burning with no fusion in core.
After point 3, the shell luminosity actually exceeds the
previous core production rate, and some of this energy goes into
Meanwhile, lack of energy production in the now isothermal
He core means it must shrink,
at the same time that it is being made more massive by the addition
of more He produced in shell.
This is the subgiant phase.
Expansion of envelope drops its T and opacity increases.
Convection must take over energy transport.
Eventually, convection decreases H- opacity, allowing more energy to escape,
while core shrinks more and releases more energy.
Thus star increases luminosity in red giant phase.
Subsequent evolution reflects structural changes that reflect
chemical composition changes.
Evolution of 5 solar mass star:
Above images taken from Carroll & Ostlie.
Stars of all masses go through similar evolution, though,
as mass decreases, the rate of evolution is slower, and the
number of left-right excursions decreases.
From Binney & Merrifield.
Uncertainties in the Models
The above process is responsible for the major sequences in the typical color-magnitude (or HR) diagram of
a stellar population.
However, there are several details that lead to important variations in the
character of the observed stellar spectrum and position on the CMD.
Generally each of these is poorly understood at present -- at least the
theories about them are presently more rudimentary.
These create uncertainties in the true structures of stars, and these
uncertainties accumulate as models evolve (i.e., once an uncertainty in
the theory is introduced at a point in evolution, all subsequent phases of
evolution are uncertain in the theory).
Generally these effects are not predicted, but used as free parameters:
Treatment of convection is one of the largest sources of uncertainty
in stellar evolution theory.
This high-resolution image reveals the multitude of
convective cells, each about the size of Texas, that cover
most of the solar surface. In these cells, hot, buoyant gas
from the interior rises to the surface where it expands and
cools. The cooler, denser gas slides towards the edges and
eventually sinks down into the cooler, darker, network of lanes.
The large dark spot is a sunspot, a point of particularly
strong magnetic field.
Just now beginning to undertake complex 3-D hydrodynamical
simulations of convection, as seen
(Description of the convective model is
More often, convection is generally regarded in the context of
mixing length theory.
Generally understood to be "too simplistic'',
nevertheless, makes it easy to incorporate the theory.
In mixing length theory,
matter travels in rising and falling cells
that travel one mixing length, l, before
dissolving into ambient medium.
Modelers describe the mixing length in terms
of a normalization with respect to one scaleheight,
H (distance over which pressure drops by 1/e),
producing a dimensionless parameter:
= l / H
Stellar evolution theory is sensitive to ,
which determines the radius over which chemical gradients are
Convection zones are often bounded by regions
where there is a sudden change in chemical composition:
For example, the lower boundary might be where fusion is occurring
(so fuel levels and emissivity different above and below
burning region and requiring different energy transport mechanism
--- see Figure 13.5 above).
The upper boundary would mark the extent over which the internal
layers --- including their nucleosynthetic yields --- are mixed and
homogenized, creating a difference in composition from layers above.
But, in truth the boundaries are not sharp:
because of turbulence, convection can encroach on -- or
overshoot -- into normally non-convective regions.
Convective overshoot is too hard to predict, and the size
of the convective overshoot region, β (also
normalized by H), is typically adopted as a
free parameter (along with ), that
one fits to get better match to observations.
Typically play with α and β to seek best fit of
theory to observations of Sun and other well-studied stars, but not
clear how well these apply to other stars.
The effects of overshoot in the core,
increasing the mixed core mass, are
to modify the evolutionary tracks of stars
(affecting the isochrone shape) and to lengthen the evolutionary
lifetimes of stars (affecting age determinations and luminosity
functions near the main-sequence turnoff).
Effect of convective core overshoot on stellar evolution for
solar-metallicity stars of 1.5 to 1.0 M. Overshoot tracks have a
redder MS hook due to the lengthened H coreburning lifetime.
From Woo & Demarque (2001, AJ, 122, 1602).
Affects SED dating of young and intermediate-age
galaxies observed at large redshifts.
Affects the age dating of star clusters.
For example, start with the normal problem of isochrone
fitting to young open clusters, which are often sparse and
therefore have poorly represented SGBs and blue
MSTO hooks, where there is a potential
degeneracy between age, distance modulus and reddening:
The blue hook of the main sequence can actually be fit
with quality photometry of an open cluster. This example is
the open cluster NGC 2420 by
Demarque et al. (1994, ApJ, 426, 165). Note how the required
distance modulus and extinction, E(B-V) for each model depends critically on how well
the blue hook is matched by different ages: If one decreases the age
of the population, the intrinsic MSTO becomes brighter and bluer
and this requires you to invoke a large distance modulus and reddening
to get a match (proceed from lower right to upper left in the
above sequence). This is one
example of the distance-age uncertainty problem we will address
in more detail soon.
Now mix in the additional uncertainty regarding
the degree of convective overshoot (here β is
given by Dmix):
A comparison of non-convective against convective overshoot models
for the cluster NGC 3680 by Kozhurina-Platais et al. (1997, AJ, 113, 1045).
Note the effects of the assumed overshoot on the age determination even when the distance modulus and reddening are fixed (as is done in this case).
Cosmic Helium Abundance
Most He in Universe was created in the Big Bang.
Hence, even the oldest, most metal poor stars should
contain substantial He.
ASIDE: We defined earlier the "[element/H]'' nomenclature
for chemical abundances. This is a definition by number.
It is often useful to give fractional abundances of
elements by weight.
We define parameters (X,Y,Z) which are the fractional
abundances by weight of hydrogen, helium, and everything
For example, the Sun is usually take to have
(X,Y,Z) = (0.70, 0.28, 0.02).
One finds (cf. Bertelli et al. 1994, A&AS, 106, 275):
logZ = 0.977[Fe/H] - 1.699
We very much would like to know the cosmic Y value,
since it tells us a minimum Y for abundances in all stars,
and it also is an important constraint on Big Bang nucleosynthesis.
How do we measure Y?
Recall that He lines are only expressed in the
Because of "dredge-up'' (see below), evolved stars
are "contaminated'' by nucleosynthetic yield from their
own cores, so these are not appropriate subjects for this
Want to inspect most metal poor sources because these
will be the least pre-enriched by Y, Z from
Have to look for hot, unevolved "Pop III'' sources.
For example, look at most metal poor, hot blue, young
stars one can find. Generally in
other galaxies (necessarily nearby
Or look for emission/absorption lines of 4He
spectra of metal-poor, "primeval'' galaxies;
e.g., Izotov & Thuan Blue Compact Dwarf Galaxies (BCDs).
Can't really accomplish this, so generally
extrapolate Y(Z) to point where Z = 0.
Analysis of extragalactic HII regions by Pagel et al. (1992,
MNRAS, 255, 325).
Click here to see similar,
more recent work by
Izotov & Thuan (1998, ApJ, 500, 188).
Or, hyperfine line of 3He+ in HII regions
in M.W.: Bania, Rood, et al. find no variation with metallicity,
suggesting that this isotope not created or destroyed
in stars in large amounts --> primordial value observed.
Izotov & Thuan (1998) find cosmical
Y = 0.2443 ± 0.0015, slightly higher than previous values
of 0.228 ± 0.005 from Pagel et al. (1992), Balges et al.
From our standpoint, the effect of lowering Y at fixed
Z raises (brightens) the ZAMS level:
The helium content affects the mean molecular weight
in the entire star, which in the core affects nuclear burning
(as above), and in the envelope affects opacity.
(Mbol / Y)Z,Teff ~ 3.
Hence, for Z = 0.02, for Y going from
0.28 to 0.23 raises ZAMS by about 0.15 mag.
From Binney & Merrifield.
Y is ~0.23 in subdwarfs, and ~0.28 in solar
A more modern analysis of the helium contant
by Bertelli et al. (2009) .
Notice also that the shape of the subgiant branch is
altered as Y changes.
Knowing Y is thus important for getting distances
right (e.g., shifting ZAMS level for MS-fitting technique), and this
(as well as changes in the subgiant shape)
effects the age determinations for clusters
(as we shall see).
In thermal equilibrium, nuclear burning establishes a natural
stratification of chemical composition with heavier elements
closer to the core.
But stars made of pre-enriched gas are initially homogeneously mixed.
Gravitational settling sorts the primordial
elements with time.
But rate of settling is uncertain.
most of the helium in a star is primordial and initially homogeneously
Downward helium diffusion increases the mean molecular
weight of the core (and, by the way,
decreases the mean molecular weight in the
envelope --> slight expansion), which accelerates the
exhaustion of hydrogen in the core --> evolution off of the
Thus the rate of helium diffusion shortens the MS
lifetime of stars of a given mass.
This can be seen in the lower (fainter) MSTO magnitudes for
higher diffusion the same age populations in the
isochrones shown in Figures 5 and 7 below.
Effect of He diffusion on evolutionary tracks, from
Proffitt & VandenBerg (1991, ApJS, 77, 473).
Effect of He diffusion on isochrones, from
Proffitt & VandenBerg (1991, ApJS, 77, 473). In left figure, note the
use of both the theoretical (left panel) and observational (right panel)
planes! Right panel stresses the difference in predicted ages one
would get from the MSTO magnitude with and without diffusion for
In the onion-skin model of stellar evolution, nuclear burning creates
a star with elements in layers, and some layers may be undergoing
different fusion processes.
The varying relative
energy contributions from the different shells can create
oscillations in star that induce complex convection patterns.
A suddenly deepened convection zone can dredge-up
heavier elements from lower layers to the star's surface.
affects the He abundance observed in stars -->
age determinations in clusters.
creation of carbon stars out of
Changes spectral line distribution of star --> bolometric
Note reverse process, elements from surface brought down
to hotter temperatures.
In particular, surface
lithium brought to burning temperatures
and becomes depleted.
Three kinds of dredge-up are usually discussed:
"First Dredge-Up'' -- first ascent up red giant branch, brings
up ashes from hydrogen burning (namely byproducts of ancillary
reactions to the CNO cycle, which overproduces 14N,
and leads to 13C and 17O being brought
See dramatic decrease in 12C / 13C, and
16O / 17O ratios in spectrum.
Also see a decrease in [C/N].
It has recently been shown that the degree to which [C/N] is altered by first dredge-up
is mass-dependent, which means that the observed [C/N] in giants is therefore age-dependent (because the ZAMS mass of a star that is currently on the RGB will depend on the star's age).
This has recently developed into a powerful new tool to measure the ages of
red giant stars just from spectroscopy!!
Calibrated by asteroseismology (using Kepler + APOGEE data), this plot shows how the combination of [C/N] and [m/H]
can be used to infer the age of a first-ascent RGB star. From Martig et al. (2016, MNRAS, 456, 3655).
An age map of the Milky Way using [C/N] abundances in red giants as measured by APOGEE.
From Ness et al. (2016, ApJ, 823, 114).
Leaves in place a discontinuity
in the star at the depth of the dredging.
For low-mass stars, this discontinuity is erased on the RGB when the
H burning shell goes through it.
For >2.5 solar mass stars, the discontinuity is left in place (star
flashes to HB before shell gets there -- see Figure 13.5 above [NOT the Figure 13.5 below!]).
Evolution of a solar mass (left) and a 5 solar mass
(right) star, with phases of dredge-up indicated. From
Carroll & Ostlie's "An Introduction to Modern
Astrophysics, 2nd Ed.".
"Second Dredge-Up'' -- second ascent, asymptotic giant branch,
brings up ashes from hydrogen burning.
Convective layers widen again on way up AGB.
For low mass star, doesn't get as deep as first dredge-up --
For high mass (>4 solar masses) star,
the second convection goes deeper and brings up
new stuff -- second dredge-up.
"Third Dredge-Up'' -- process by which ashes from helium burning
nucleosynthesis (including s-process elements)
are transported from the interior to the surface of
low- and intermediate-mass stars (M < 7 solar masses)
during the AGB phase --> carbon star
Happens due to periodic thermal instability "pulses" developing in He-burning shell
of AGB star leading to readjustments of the star structure (but not completely
Makes S, C, S/C class stars (stars enriched with ZrO, carbon molecules, and both,
Spectra of a carbon star and a carbon/S type star from the
Digital Atlas of Spectra of Cool Stars by Turnshek et al.
Also see stars with greatly enhanced s-process element abundances.
These stars are important because their enriched envelopes, when ejected,
increase the presence of these elements in the ISM.
Stellar Winds/Mass Loss
The boundary condition above that density is zero at
"surface" of star is not true for all stages of stellar evolution.
Plasma flows from most stars via stellar winds into
the interstellar medium.
Extremely hard to predict.
Becomes most important in later stages of evolution (see
the Carroll & Ostlie Figures 13.4 and 13.5 just above).
For example, critical aspect of HB formation, where amount of
mass lost determines the effective temperature of the
Red Giant Branch, Mass Loss and the Horizontal Branch
The subgiant branch is occupied by stars in hydrogen shell burning,
but not yet fully convective envelopes.
An important phase in the life of < 8 solar mass stars is the
red giant branch.
The vertical RGB, where a star moves up the Hayashi
track, is a shell hydrogen burning star that has a fully convective envelope.
(The Hayashi track in the HR diagram is a nearly vertical downward path followed by
contracing pre-main-sequence stars,
or upward vertical path for expanding post-main-sequence, red giant stars; the vertical track is a result
of the star not being out of hydrostatic equilibrium and heavily convective.)
When a star with mass less than about 2 solar masses
evolves to the tip of the RGB, it undergoes a helium flash,
when the triple alpha nuclear fusion reaction is initiated.
Stars more massive than this ignite He more quiescently because, unlike
low mass stars, their cores are not electron degenerate.
These higher mass stars go blueward, like low mass stars, but do
not become significantly fainter (see blue excursions in "Figure 5.2" above).
This explosive phenomenon for lower mass stars causes an almost immediate loss of mass.
The resulting reconfiguration of the star is not easy to model, and the
evolution to the Zero Aged Horizontal Branch is not possible to
The details of the transition are therefore handled
Once on the ZAHBs, evolution slows again, and stellar structure models
can be implemented again.
The final luminosity of HB stars is almost independent of mass, because
it depends on the mass of the He core, and only stars that have a He core mass
of about 0.45 solar masses ignite helium in a "flash''.
If all low mass stars evolved identically, they would lose the same amount of
mass and end up at exactly the same point on the ZAHBs.
But the observed horizontal branches of clusters
are extended in length.
The M15 CMD we saw earlier, from Yanny et al. (1994, AJ, 107, 1745).
Since all stars near the tip of the RGB have the same mass,
and since stellar evolution models show that there is little
movement (especially left-right) in the CMD during core helium
burning, the observed extended HB lengths imply
variable mass loss, ΔM = MRG - MHB
for stars at the point of the helium
To explain the observed structures of the HB,
Rood (1973) modeled the distribution function of HB masses
as a Gaussian as follows:
where Mc is the helium core mass.
The limits ensure that stars
do not gain mass nor lose part of their helium cores, and <MHB> is the
mean mass of stars arriving on the HB and σM is the dispersion.
By fitting this mass loss model to the data, one typically finds:
RG - <HB> ~ 0.2 ☉
Each observed HB is therefore a mass sequence:
The total mass varies along the HB.
The mass of the He core is roughly constant
One finds that the types of HB formed for different
Z results in different looking ZAHBs, with the spread in
mass loss creating spreads in colors.
The above figure shows that the primary effect driving the
appearance of an HB for a population is the abundance.
As we have seen above, metal-rich, Population I stars have a
short red clump.
The HB becomes bluer for more metal-poor
This trend is shown in the plot below:
Taken from Binney & Merrifield.
We can quantitatively describe the morphology of an HB
using the statistic shown in the above figure:
Nvar is the number of RR Lyrae variables
(as we shall see, and is shown in Figure 5.1 above,
the HB is intersected by the so-called "instability strip"
and this intersection point is the location of the RR Lyrae
Nblue is the number of HB stars blueward
of the RR Lyrae stars, and
Nred is the number of HB stars redward
of the RR Lyrae stars.
When this HB morphological index is negative, the HB is more
red than blue, and when positive more blue than red.
(Note that an earlier version of the above morphology index
excluded the Nvar term in the denominator.)
The difference in colors with fractional mass loss has
to do with the amount of envelope that has been peeled off:
more mass loss means there is less envelope to puff out (star
becomes "less giant -like" in structure) and you see hotter
layers of the star.
Note, Binney & Merrifield say "As for the MS and SGB, most
of the trend is caused by line blanketing in metal-rich stars",
but this is actually not the dominant effect.
The ad hoc approach of an analytical description of the HB,
like Rood's, is ultimately unsatisfying, because it is only an empirical
description and does not explain:
...why some stars lose more mass than others.
...why some clusters of same metallicity have
...why some populations have bimodal HBs.
Three clusters of similar metallicity ([Fe/H] ~ -1.2) but
one each having a blue, a bimodal,
and a red horizontal branch. From Bellazzini et al. (2001, AJ, 122, 2569).
The open circles are RR Lyrae stars, and triangles are probably non-members.
Students should be able to figure out why the open circles appear
so much less confined to the HB in this figure.
While it is obvious that metallicity plays a primary role in driving
the degree of mass loss to the HB,
clearly there is a second parameter driving mass loss
variations that we do not yet fully understand.
Among the proposed second parameters have been:
in clusters, cluster central density and concentration
(though this doesn't explain dSphs, some that
show an internal second parameter spread)
magnetic field (affects mass loss)
-capture element abundances
oxygen abundances affecting mass loss rates
primordial He abundance
dredged-up He abundance
rotation (delays He flash so there is more time to
lose mass on RGB, may also influence
stellar wind strength)
differences in the number of swallowed planets, brown
dwarfs in RGB phase
(swallowed planets can spin-up RGB stars -- see Soker & Harpaz, 2000,
MNRAS, 317, 861).
A "tourist's map'' of the effects of different proposed
second parameters on the HB color, from Fusi Pecci et al. (1996,
in Formation of the Halo ... Inside and Out, ASP Conf. Ser. 92,
eds. H. Morrison & A. Sarajedini).
There is no consensus on this:
Age is the most often cited second parameter, but debated
(e.g., Sarajedini et al. 1997 versus Stetson et al. 1998).
But a thorough analysis of the second parameter effect using the largest
set of HST photometry on globulars (Dotter et al. 2010, ApJ, 708, 698)
reveals the strongest correlation of the second parameter to cluster
There could be several second parameters...
Dotter et al. (2010) suggest that the cluster central density may
be a viable third parameter.
All-in-all it is very likely that age is
one of the secondary parameters --- if not THE secondary parameter ---
driving horizontal branch morphology, for reasons
I will give when we return to this topic when we discuss clusters,
If age is the second parameter, the morphology of a population's
HB could then provide us with a useful, but easy (though coarse), chronometer (a very desirable
Stars on the HB slowly brighten above the ZAHB in the CMD as they evolve.
Therefore, in a population of stars, the HB is thickened on account
of the presence of stars of different HB ages.
Once a star exhausts He in the core and transitions to He shell burning, the
star evolves quickly in a second red giant-like
stage, called the asymptotic giant branch.
Name because the AGB evolutionary path parallel to, but just blueward, of
RGB, but asymptotes to RGB near the tip.
Just as with the RGB, the transition to AGB luminosity occurs quickly
for low mass stars.
As seen in the Figure above and in the figure below (which was already
the ratio in luminosity between the ZAHB and the tip of the AGB
is much less for higher mass stars, which quickly transition back and forth
across the HR diagram for successive core fuel ignitions.
As a star moves up the AGB, it loses mass more quickly.
Many stars lose most of their mass to the ISM before they die, with much
of it during post-HB phases.
The complicated He shell and H shell burning creates oscillations in
the AGB star, and a complex convection pattern is developed --> second/third dredge-up.
If the convection brings up carbon nuclei through the He burning shell
and into the envelope, you get a carbon star.
Carbon stars are extremely luminous, and easily identifiable by
extremely red colors compared to almost any other type of star. Their brightness
in the red/IR and their ready identification by color
makes them extremely useful "tracer populations" in Galactic research.
Mass loss on the AGB creates OH/IR stars.
Luminous, very red stars with strong emission at 1612 MHz from
masing OH radicals.
Particularly bright in mid-infrared from emitted dust shells.
Both come from matter ejected from star.
Both properties make OH/IR stars easy to identify, thus
they are useful population tracers.
OH/IR stars are the extremes of a type of pulsational variable
found at the top of the AGB, called Mira variables or
long period variables (more on that in a bit).
Post-AGB Evolution:Low Mass Stars
After reaching the top of the AGB, a lower mass star sheds almost all of its remaining
As with the HB/mass loss situation, loss of the outer envelope exposes very hot
interior and the star becomes suddenly extremely blue.
Intense ionizing radiation from the star may cause fluorescence of the
ejected envelope --> planetary nebula.
An approximately half-solar mass degenerate core is all that remains and the
star cools down the white dwarf sequence.
Most of the support of the WD is supported by electron-degeneracy
pressure, so the star's radius is nearly constant as it cools.
Hence, the white dwarf cooling line in the CM diagram is given
approximately by L = constant * Teff4, or
Mbol = -10 log Teff + constant
The theory of cooling of such a star is actually reasonably well
understood, and so white dwarf cooling ages have recently become a very
useful chronometer (for populations where such faint stars can be seen!).
White Dwarf sequence in the globular cluster M4 by Richer
et al. (1997, ApJ, 484, 741).
A theoretical cooling sequence for C-core WDs is shown on the
right, as is an age scale. Note that the end of the WD sequence
is not yet discerned in this very deep imaging, but can be seen
in the even deeper HST imaging study by
Hansen et al. (2002, ApJ, 574, L155) which shows M4 to have an
age of at least 12.7 ± 0.7 Gyr (2 sigma error).
High Mass Stars
For stars with initial masses larger than about 8 solar masses, the carbon created in helium
burning can ignite, and nuclear burning proceeds in stages (onion shell model) until iron
is created in the core.
Each ignition of a new fuel instigates a blue-red excursion across the CMD.
Iron is the most tightly bound nucleus, so its creation signals the end of exothermic nuclear
The stellar core must then contract, which drives up the Fermi energy of the electrons
in the core and heats the core.
A catastrophic collapse ensues when the electrons get driven into the nuclei and
these nuclei are shattered into alpha particles by energetic photons.
Both processes remove ambient photon and electron degeneracy pressure and further
collapse occurs, which speeds up the reactions and the increased loss of pressure -->
Runaway core implosion releases energy that may lead to supernova and formation of neutron
star or black hole in the center of the star.
elements are created (though may not be the primary source of r-process elements overall,
compared to neutron star-neutron star mergers) and the ISM is enriched -- subject of another lecture.
Summary of (Major Sequences in CMDs)=(Phases in the Evolution of a Star)
Table from Binney & Merrifield.
From Carroll & Ostlie.
Simple Numerical Relations
Though the above evolutionary description comes from large-scale
computer simulations, there are some simple rule of thumb, order of magnitude
approximations useful to remember.
ZAMS Mass-Luminosity Relation
On the ZAMS, the luminosity of a star goes roughly as
A more careful assessment yields:
The luminosity of HB stars is roughly 50L☉
(independent of mass).
This corresponds to an absolute visual magnitude of roughly +0.5.
This rough rule of thumb is useful for judging immediately
the approximate distance of a globular cluster from looking at the
apparent V magnitude of the HB.
Note that since bluish HB stars are A type stars, their color
is near enough to 0 that the rule of thumb holds roughly for any
filter being given on the ordinate of the CMD.
In reality, there is a metallicity trend to the HB absolute magnitude.
This dependence is still debated (and is an important issue surrounding
the distance scale problem -- i.e., using the HB as a standard candle), but
one group finds:
MV(HB) = 0.17[Fe/H] + 0.82
(according to Lee et al. 1990, for Y = 0.23).
Main Sequence Lifetime
The Main Sequence lifetime of a star is fixed by the length of time that its luminosity can be
supported by thermonuclear conversion of H to He.
The energy released from the fusion of 4 protons into a He nucleus releases 26.7 MeV, of which
25 MeV is injected into the burning layer and the rest lost in neutrino flux.
Converting units yields the total energy per burned mass
Δ as E = 0.0067
From evolution models, the MS lifetime of a star ends when about 1/10 of a star's
mass has been converted from H to He. Thus, one can calculate the lifetime in years
using the formula given.
The solar MS lifetime is about 10 Gyr, so that we can normalize to solar units
Horizontal Branch Lifetime
A similar argument can be used to obtain the HB lifetime.
The energy released in He conversion to C and O is E = 0.00072
The HB lifetime is the time to burn the ~0.45
As we saw above, the luminosity is 50L☉.
And for supergiants:
Energy released for H --> Fe is E = 0.0085
Amount converted is ~1.4 ☉.
For luminosity L = L3 X
More accurate estimates for these lifetimes are obtained from evolutionary models, and
are given in Binney & Merrifield. Relative Number of Stars in Different Evolutionary Phases for Simple Stellar Population
The post-MS phases of stellar evolution are typically much shorter than the MS lifetime
for that same star.
Thus (apart from the White Dwarf sequence), every evolved star in a Simple Stellar
Population comes from a predecessor MS star of about the same mass = (the mass of stars at
Under these circumstances, the relative number of stars, Ni ,
in any particular evolutionary phase, i , is given by the relative amount of time
τi a star of mass (MSTO) spends in that phase.
It can also be shown (HW problem) that the contribution of phase i to the integrated light of a
Simple Stellar System is proportional to the amount of fuel a star burns when it is in that phase.