ASTR 5610, Majewski [SPRING 2020]. Lecture Notes

## ASTR 5610 (Majewski) Lecture Notes

### STELLAR EVOLUTION AND THE COLOR-MAGNITUDE DIAGRAM

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 and galaxies.

• Subject of an entire other graduate course.
• Can only broadly summarize important aspects here (more "what" than "why").
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, even hotter.

Basic elements of standard stellar structure theory:

• Spherical symmetry.
• 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) as:

• 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 luminosity as:

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 evolution.

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.

Vogt-Russell Theorem

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 color-magnitude plane...)
• For example: Solutions for "Zero-Age'', chemically homogeneous, H-burning Main Sequence stars, the ZAMS, give an arrangement that is a mass sequence.
• ##### 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 expanding envelope.

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:

• Convective Overshoot
• 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. From http://www.amnh.org/education/resources/rfl/web/sunscapes/sunscapes.xml.html.
• Just now beginning to undertake complex 3-D hydrodynamical simulations of convection, as seen here.

(Description of the convective model is here).

• 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 flattened out.
• 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 else.

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 hottest stars.
• 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 problem.
• Want to inspect most metal poor sources because these will be the least pre-enriched by Y, Z from earlier populations/supernovae.
• 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 to resolve).

Or look for emission/absorption lines of 4He in integrated 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. (1993).
• 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 metallicity stars.
• ##### 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).

• Helium Diffusion
• 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.
• Specifically, most of the helium in a star is primordial and initially homogeneously mixed.

• 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 main sequence.
• 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 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 different metallicities.

• Dredge-Up
• 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.

For example:

• affects the He abundance observed in stars --> age determinations in clusters.

• creation of carbon stars out of M giants.
• Changes spectral line distribution of star --> bolometric corrections, colors.
• 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 to surface).
• 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!!

##### 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 -- no difference.

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 formation.
• 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 understood).

Makes S, C, S/C class stars (stars enriched with ZrO, carbon molecules, and both, respectively).

##### 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 resulting star.

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 follow.
• The details of the transition are therefore handled semi-empirically.
• 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 flash.
• 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 however.

• 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 populations.
• 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:

HB morphology = (Nblue - Nred ) / (Nblue + Nvar + Nred )
where

• 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 variables),

• 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 different HBs.
• ...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:

• age
• 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 age.

• 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, coming up.
• 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 tool!).

Post-HB Evolution

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 presented above), 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 hydrogen.

• 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 fusion.

• 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 collapse.
Runaway core implosion releases energy that may lead to supernova and formation of neutron star or black hole in the center of the star.
• r-process 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.
Boom.

Summary of (Major Sequences in CMDs)=(Phases in the Evolution of a Star)
##### 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 3.5.

• A more careful assessment yields:

ZAHB Magnitude

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).

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 Δ c2.
• 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 and obtain:

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 Δ c2
• The HB lifetime is the time to burn the ~0.45 helium core.
• As we saw above, the luminosity is 50L. Thus:

And for supergiants:

• Energy released for H --> Fe is E = 0.0085 Δ c2
• Amount converted is ~1.4 .
• For luminosity L = L3 X 103L, and:
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 the MSTO).
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.

All material copyright © 2003, 2006, 2008, 2010, 2012, 2014, 2016, 2018, 202, 2020 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 and Astronomy 5610 at the University of Virginia.