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<!DOCTYPE html>
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<head>
<!-- 2025-02-06 Thu 09:04 -->
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<title>400A - Qualitative evolution</title>
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<p>
<b>Materials</b>: Onno Pols' lecture notes Chapter 9, 10, 11, Kippenhahn's
book Chapters 22-24, 26-33, Hansen, Kawaler, Trimble book, Chapter 2,
<a href="https://ui.adsabs.harvard.edu/abs/2022MNRAS.512.4116F/abstract">Farrell et al. 2022</a>.
</p>
<div id="outline-container-org7bdf6eb" class="outline-2">
<h2 id="org7bdf6eb"><a href="#org7bdf6eb">Pre-main sequence and Hayashi track</a></h2>
<div class="outline-text-2" id="text-org7bdf6eb">
<p>
We have developed a set of 4 non-linear, coupled ODEs to describe the
<i>structure</i> and <i>evolution</i> of stars (with physical approximations
built-in), with an EOS for closure conditions plus a set of equations
that describe the changes in composition due to nuclear processing and
mixing (see for example <a href="./notes-lecture-neutrinos.html#orgaf3db0c">this summary</a>).
</p>
<p>
We have also worked out the ordering of timescales of the problem:
</p>
<div class="latex" id="org7628f76">
\begin{equation}
\tau_\mathrm{nuc} \gg \tau_\mathrm{KH} \gg \tau_\mathrm{free\ fall} \ \ ,
\end{equation}
</div>
<p>
which implies that for a star in hydrostatic equilibrium and LTE at
every location throughout the star (i.e., global gravothermal
equilibrium), the equations describing the (slow) <i>evolution</i> of the
composition decouple from the 4 ODEs+EOS that describe the <i>structure</i> of
the star.
</p>
<p>
The <code>MESA-web</code> stellar models at start at the <i>top right</i> of the
Herzsprung-Russell diagram (so high L low T<sub>eff</sub>) as a uniform sphere of
gas with set composition that is in hydrostatic but not <i>necessarily</i>
thermal equilibrium <i>yet</i>.
</p>
<p>
<b>N.B.:</b> there may also be short initial numerical transients lasting a
few timesteps and irrelevantly short simulated physical time.
</p>
<p>
<b>N.B.:</b> the expression <i>top right</i> above is relative, in its future
evolution the mode may evolve at even higher L later on (but not lower
T<sub>eff</sub>!)
</p>
<p>
These models thus evolve on a thermal timescale ∼τ<sub>KH</sub> losing energy (L
goes down) and contracting – picture lines of constant R=(L/(4πσ
T<sub>eff</sub><sup>4)</sup>))<sup>1/2</sup>! This <i>is</i> the gravothermal KH contraction which leads to
the increase in average ⟨ T ⟩ by the virial theorem, and it goes on
until either:
</p>
<ol class="org-ol">
<li><b>degeneracy pressure stops this process by changing the EOS, tapping
into quantum mechanical effects to provide extra pressure</b>: this
should not happen to any of your models, but it is how you form a
<b>Brown Dwarfs</b> (BD) and a planets. The distinction between these is
whether some minor nuclear burning, energetically unimportant but
still affecting the composition happens before the object sets on
gravothermal equilibrium: BD experience deuterium burning, but
deuterium is a loosely bound nucleus and its burning doesn't release
a lot of energy.</li>
<li><b>an energy source (namely nuclear burning) stops the collapse</b>. This
is what happens to your stars, where H burning (as the lightest and
most energy-releasing fuel) starts. The point where L=L<sub>nuc</sub> because
of H core burning is referred to as <b>Zero Age Main Sequence</b> (ZAMS),
since the "main sequence" of the color magnitude diagram containing
most stars is made of stars during their longest phase of evolution,
hydrogen core burning.</li>
</ol>
<p>
From the ignition of H in the core onwards, for timescales τ ≪ τ<sub>nuc</sub>,
once the composition is decided (i.e., for fixed mass fractions {X<sub>i</sub>})
the structure is determined by those 4 ODEs+EOS. This means that the
<i>initial</i> structure of the star is <i>not</i> sensitive to the (complex,
not-yet-fully understood) star formation process, and the ZAMS
structure is fully determined by the composition!
</p>
</div>
<div id="outline-container-org9239e2d" class="outline-3">
<h3 id="org9239e2d"><a href="#org9239e2d">Pre-main sequence stellar structure</a></h3>
<div class="outline-text-3" id="text-org9239e2d">
<p>
If the gravothermal collapse of the mass of gas considered ends with
the ignition of nuclear fuel (option 2. above), that is, if we make a
star (recall ⟨ T ⟩ ∝ μ M/R, so the mass determines whether we get
here), then the phase from the beginning of the simulation until ZAMS
is called <i>pre-main-sequence</i> (pre-MS).
</p>
<p>
During this phase the models start from the top-right of the diagram
at high L, large R (⇒ low ⟨ T ⟩), and progressively contract and heat
up. This is an idealization: often as stars are doing that, they still
accrete mass, in the most massive stars (which have shorter KH
timescales and pre-MS phases), accretion may not be over until even
after ZAMS! See for instance the recent review by <a href="https://ui.adsabs.harvard.edu/abs/2023ASPC..534..275O/abstract">Offner et al. 2023</a>.
</p>
<p>
<b>N.B.:</b> this accretion drives "outflows" in young stellar objects, which
can be "dusty" and thus opaque, making the pre-main sequence evolution
<i>embedded</i> and hard to directly observe.
</p>
<p>
The ⟨ T ⟩ is initially low (the gas that is collapsing is coming from
the interstellar medium), and increasing. Since ∇ = ∂ ln(T)/∂ ln(P) ∝ κ L and κ
generally increases at low T, initially the stars have steep
temperature gradient and are <i>fully convective</i>! This homogenizes their
composition (but deuterium burning during the pre-MS can change this),
and also determines that ∇≅∇<sub>ad</sub> to very good approximation.
</p>
<p>
Since almost arbitrarily high energy flux can be carried by
(efficient) convection maintaining an adiabatic gradient, this means
the luminosity of a fully convective star does not depend much on its
structure (unlike a stably stratified <i>radiative</i> star where the
temperature gradient of the <i>structure</i> is determined by the need of
carrying the flux out).
</p>
<p>
This leads to the evolution of these stars along an almost vertical
initial path on the HR diagram, the so-called <i>Hayashi track</i> (see also
Sec. 9.1.1. of Onno Pols' lecture notes for an analytic approximation)
after <a href="https://en.wikipedia.org/wiki/Chushiro_Hayashi">Chushiro Hayashi</a>. The Hayashi track corresponds to solutions for
fully <a href="./notes-lecture-convection.html">convective</a> stars. Wiggles around this vertical line are due to
recombination and partial ionization zones leading to deviations of
the temperature gradient ∇ from adiabatic and burning of light
elements (<sup>2</sup>H,<sup>7</sup>Li) that releases some energy.
</p>
<p>
<b>N.B.:</b> The energy release per nucleon of the burning of light elements
is very low and does not manage to alt the thermal-timescale
quasi-static collapse.
</p>
<p>
<b>The Hayashi line effectively determines a right, low-T<sub>eff</sub> boundary
on the HRD for stars in hydrostatic equilibrium</b>: if a star were to be
colder, it would have a steeper-than-adiabatic gradient somewhere,
which would imply a higher convective flux (cf. <a href="notes-lecture-convection.html#orgbc73ac8">convective energy
flux</a>)) and thus increase the luminosity of the star, moving the star
upwards back onto the Hayashi track.
</p>
<p>
<b>N.B.:</b> for these cool temperatures, we already know that the opacity is
dominated by H<sup>-</sup>, molecules, and dust, and we have approximate powerlaw
scalings with T<sub>eff</sub> for analytic considerations, but <code>MESA-web</code> uses
tabulated values (cf. <a href="./notes-lecture-kappa.html">opacity lecture</a> and references therein).
</p>
<p>
The location in T<sub>eff</sub> of the Hayashi track is dependent on the mass M
of the star: more massive stars are hotter since the very beginning.
This can be analytically derived imposing ∇=∇<sub>ad</sub> and solving the
remaining 3 ODEs assuming some form for κ≡κ(T,ρ) at the photosphere:
effectively the outer boundary condition and atmospheric physics
determines this.
</p>
<p>
Stars to the left, hotter side of the Hayashi track instead must <i>not
be /fully convective</i> and have some radiative layers (recombination and
light-elements burning chaging κ and μ)!
</p>
</div>
</div>
</div>
<div id="outline-container-orgb4324da" class="outline-2">
<h2 id="orgb4324da"><a href="#orgb4324da">Main sequence</a></h2>
<div class="outline-text-2" id="text-orgb4324da">
<p>
As the gravothermal collapse continues and ⟨ T ⟩ increases, at some
point, if we are making a star, by <i>definition</i> nuclear burning turns on
(option 2. above). This is when the central temperature (which at this
stage is the highest temperature in the star), is sufficient to obtain
enough tunneling through the Coulomb barriers.
</p>
<p>
Because it is abundant, and its burning releases a lot of energy per
nucleon (∼ 6.5MeV/nucleon) because it produces the double-magic
nucleus \(^{4}\mathrm{He} \equiv \alpha\) (neutrons <i>and</i> protons fill their nuclear "shells",
by analogy with electron shells in atomic physics), hydrogen is the
first fuel to ignite, see also <a href="./notes-lecture-nuclear-burning.html">nuclear burning lecture</a>.
</p>
</div>
<div id="outline-container-org26890a4" class="outline-3">
<h3 id="org26890a4"><a href="#org26890a4">Structure during the main sequence</a></h3>
<div class="outline-text-3" id="text-org26890a4">
<p>
As we discussed in the <a href="./notes-lecture-nuclear-cycles.html">nuclear reaction cycles lecture</a>, hydrogen
burning can occur in two different ways: pp-cycle and CN-NO bi-cycle.
</p>
<p>
Looking at <code>MESA-web</code> models, we can see that the pp-cycle is sufficient
to achieve the equilibrium condition L<sub>nuc</sub>=∫ dm ε<sub>nuc</sub>
≡ L in low mass stars (<b>N.B.:</b> L∝ M<sup>x</sup> with x≥1). This is because the
pp-cycle has lower Coulomb barriers (shallower relation between ε<sub>nuc</sub>
and T) but a higher normalization (cf. <a href="notes-lecture-nuclear-cycles.html#org1ad4443">pp → CNO transition</a>).
</p>
<ul class="org-ul">
<li><b>Very low M main sequence ⇒ fully convective</b></li>
</ul>
<p>
For the lowest-mass stars, T<sub>eff</sub> remains cold and the opacity remains
high: they burn through the pp cycle, but remain <i>convective</i> throughout
the main sequence. In this case, <i>all</i> of the stellar material is
available to burn, there is no core/envelope structure at all! These
stars however have (relatively speaking) very low L, thus they evolve
very slowly. All these stars in the Universe are still on the
main-sequence! This is the case of the 0.3M<sub>☉</sub> star you computed for
a homework, which has an approximately polytropic EOS because it is
fully convective, thus has ∇=∇<sub>ad</sub> ⇒ P∝ρ<sup>Γ<sub>1</sub></sup>.
</p>
<ul class="org-ul">
<li><b>Low M main sequence ⇒ radiative core, convective envelope</b></li>
</ul>
<p>
Moving slightly higher in mass, meaning also to higher T<sub>eff</sub>, a
radiative core appears. the burning is very concentrated in the
innermost region, but they are cool enough to have high κ at the
surface, and thus retain a convective <i>envelope</i>:
</p>
<p>
<b>N.B.:</b> we are seeing that the cooler T<sub>eff</sub> is the deeper the convective
envelope! Increasing T<sub>eff</sub> the convective layer disappear in the
deepest layers. This can be shown analytically (see Onno Pols' lecture
notes sec. 7.2.3).
</p>
<ul class="org-ul">
<li><b>High M main sequence ⇒ convective core, radiative envelope</b></li>
</ul>
<p>
Increasing M ⇔ T<sub>eff</sub> further, the equilibrium condition L=L<sub>nuc</sub> cannot
be satisfied anymore with the pp-chain, and the CN-NO bi-cycle kicks
in. Because of its higher Coulomb barriers, it has a steeper
temperature dependence: the energy release is even more concentrated,
implying that ∇ in the core is very steep (recall ∇∝ κ L ∝ κ L<sub>nuc</sub>),
thus <i>the core becomes convective</i>. This means that convective mixing
makes a larger mass of hydrogen available to the very central burning
zone. At the same time, higher M ⇒ higher T<sub>eff</sub> and the envelope
becomes radiative.
</p>
<figure id="org4221cdb">
<img src="./images/conv_ZAMS.png" alt="conv_ZAMS.png" width="100%">
<figcaption><span class="figure-number">Figure 1: </span>The "initial" gravothermal equilibrium structure of a star is determined only by mass M and composition. The figure (Fig. 9.8 in Onno Pols' notes, modified from Kippenhahn & Weigert) shows in gray the region in mass coordinate y=m/M that are convective as a function of the total mass M=∫ dm for Z=0.02 models. Red lines indicate where 50 and 90 % of the luminosity L is generated (the "burning region") and the blue dashed lines show r(m)=0.25M and r(m)=0.5M.</figcaption>
</figure>
<p>
<b>N.B.:</b> The threshold initial masses dividing the three regimes above are
somewhat uncertain and dependent on input physics and modeling
assumptions.
</p>
<ul class="org-ul">
<li><b>Q</b>: for your <code>MESA-web</code> models, what is the highest mass with a
radiative main sequence core, and the lowest with convective main
sequence core?</li>
</ul>
</div>
</div>
<div id="outline-container-org1aef0be" class="outline-3">
<h3 id="org1aef0be"><a href="#org1aef0be">Evolution during the main sequence</a></h3>
<div class="outline-text-3" id="text-org1aef0be">
<p>
During the main sequence L steadily increases on τ∼τ<sub>nuc</sub>. This is
because the conversion of hydrogen into helium decreases X (and
increases Y), which enter in two key quantities, mean molecular weight
and electron scattering opacity:
</p>
<div class="latex" id="orgdaac5fa">
\begin{equation}\label{eq:microphysics_XY}
\mu \simeq \frac{1}{2X+\frac{3}{4}Y+\frac{Z}{2}} \ \ , \\
\kappa_\mathrm{es} = 0.2(1+X) \ \ \mathrm{cm^{2}\ g^{-1}} \ \ \ .
\end{equation}
</div>
<p>
Assuming a star to be in gravothermal equilibrium and assuming
radiative energy transport (which we have just seen is not verified
everywhere by <code>MESA-web</code> models!), we know that:
</p>
<div class="latex" id="orgf7ae275">
\begin{equation}\label{eq:L_scaling}
L\propto \frac{\mu^{4} M^{3}}{\kappa} \ \ ,
\end{equation}
</div>
<p>
This scaling relation is approximate and does not exactly hold if a
star is not fully radiative (which we have already seen is not
accurate!), but it tells that:
</p>
<ul class="org-ul">
<li>the higher κ, that is, the harder it is for photons to get out, the
lower the luminosity</li>
<li>the higher the mass, the higher the luminosity (⇒ the higher the
nuclear burning rate for a given fuel!), and since the mass exponent
is larger than 1, this implies that <i>more massive stars have shorter
lifetimes w.r.t. lower mass stars</i>. They do have more fuel available
(∝ M), but they burn through it at a higher rate (∝ M<sup>3</sup>)! In fact
single-star lifetimes of stars that burn all the way to iron is only
∼10-50Myr (M<sub>ZAMS</sub>≥7.5M<sub>☉</sub>, with the exact lower limit depending
on Z, rotation, binary interactions, cf. for example <a href="https://ui.adsabs.harvard.edu/abs/2017PASA...34...56D/abstract">Doherty et al.
2017</a> and <a href="https://ui.adsabs.harvard.edu/abs/2017ApJ...850..197P/abstract">Poelarends et al. 2017</a>), compared to > 10<sup>9</sup> years for
M<sub>ZAMS</sub>≤2M<sub>☉</sub>.</li>
</ul>
<figure id="org4b9e3d5">
<img src="./images/stellar_lifetimes.png" alt="stellar_lifetimes.png" width="100%">
<figcaption><span class="figure-number">Figure 2: </span>Stellar lifetime as a function of initial masses from <a href="https://ui.adsabs.harvard.edu/abs/2017A%26A...601A..29Z/abstract">Zapartas et al. 2017</a>. <code>MESA</code> and <code>GENEC</code> models are shown, focusing on masses that result in a final core-collapse event. The bottom panel shows the deviations between the analytic fit and the numerical models.</figcaption>
</figure>
<ul class="org-ul">
<li>the higher the mean molecular weight μ (= number of particles per
baryonic mass), the higher the luminosity.</li>
</ul>
<p>
Using Eq. \ref{eq:L_scaling} we can infer that the high power of μ
drives the luminosity evolution of the stars during the main sequence:
because hydrogen is converted into helium (X → Y), the <b>mass-weighted
average ⟨ μ ⟩ = ∫ dm μ(m)/∫ dm increases and thus L increases</b>.
</p>
<p>
<b>N.B.:</b> massive and low mass stars however have a very different
morphology of the main sequence. For stars with radiative cores
(burning through the pp-chain, M≤1.2M<sub>☉</sub>), L increases, R varies
little, thus since L=4π R<sup>2</sup>σ T<sub>eff</sub><sup>4</sup> in equilibrium, we also see a
slight increase in temperature of the star during the main sequence.
Conversely, massive stars with convective cores (burning through the
CNO cycle, M≥1.2M<sub>☉</sub>) increase in radius and actually become <i>cooler</i>
as they evolve during the main sequence. One can derive (see Onno
Pols' notes chapter 7) analytic R(M) relations assuming a specific
scaling for the energy generation to qualitatively explain this. In
reality, the details of the core evolution (influenced by uncertain
processes such as convective boundary mixing) and envelope (influenced
by wind uncertainties) matter for the details.
</p>
<p>
<b>N.B.:</b> The relative role of μ and κ is slightly sensitive to
metallicity too (because at lower Z the approximation κ≅κ<sub>es</sub> is
progressively better since fewer bound-bound and bound-free
transitions are available, see also <a href="https://ui.adsabs.harvard.edu/abs/2022MNRAS.516.5816X/abstract">Xin et al. 2022</a>). The opacity κ is
dominant in determining the L and R at ZAMS for Z≅0.02, but the change
in μ is determining their <i>evolution</i> along the main sequence.
</p>
<ul class="org-ul">
<li><b>Q</b>: based on the scaling in Eq. \ref{eq:L_scaling}, how does the
luminosity of two identical stars differing only in Z compare? Which
star has the highest L? (<b>Hint</b>: you can compute more <code>MESA-web</code> models
of your mass varying Z to check your answer!)</li>
</ul>
<p>
Looking at the Kippenhahn diagrams and composition diagrams from
<code>MESA-web</code> we can also see what the model does in the core (something
not <i>directly</i> accessible to observations - if not through neutrinos).
For low mass stars with radiative cores and high ρ<sub>center</sub> (something
you can derive from the virial theorem + hydrostatic equilibrium +
EOS), partial degeneracy already plays a role in sustaining the
structure during the main sequence, and as the central burning region
converts hydrogen into helium, the helium core becomes hot and
degenerate - thus sustaining itself against gravitational collapse
with the quantum effects due to the Fermi-Dirac statistics of
electrons.
</p>
<p>
Conversely, high mass stars have a convective core: convective mixing
connects the innermost burning region with a larger fuel reservoir.
The progressive burning of hydrogen changes the center opacity (well
approximated by electron scattering only in the hot, fully ionized
interior) κ≅κ<sub>es</sub>=0.2(1+X) cm<sup>2</sup> g<sup>-1</sup>. Specifically, as X decreases, so
does κ, and since ∇ = ∂ ln(T)/∂ ln(ρ) ∝ κ L, the temperature gradient
becomes "less steep", meaning there is less need for convection:
<i>during the main sequence of massive stars, the convective core
receeds in mass coordinate</i>.
</p>
<figure id="org74c7bf2">
<img src="./images/1Msun_X_M.png" alt="1Msun_X_M.png" width="100%">
<figcaption><span class="figure-number">Figure 3: </span>Hydrogen mass fraction X as a function of mass coordinate m for a single, non-rotating, 1M<sub>☉</sub>, Z=0.02 <code>MESA</code> model across its main sequence evolution. The color go from dark (∼ ZAMS) to light (∼ TAMS).</figcaption>
</figure>
<figure id="org3c29f44">
<img src="./images/20Msun_H_profile.png" alt="20Msun_H_profile.png" width="100%">
<figcaption><span class="figure-number">Figure 4: </span>Hydrogen mass fraction X as a function of mass coordinate m for a single, non-rotating, 20M<sub>☉</sub>, Z=0.001 <code>MESA</code> model across its main sequence evolution. The color go from dark (∼ ZAMS) to light (∼ TAMS), and as time passes the core recedes because of the change in κ.</figcaption>
</figure>
</div>
</div>
</div>
<div id="outline-container-orgdee94fc" class="outline-2">
<h2 id="orgdee94fc"><a href="#orgdee94fc">End of the main sequence</a></h2>
<div class="outline-text-2" id="text-orgdee94fc">
</div>
<div id="outline-container-org8f713c2" class="outline-4">
<h4 id="org8f713c2"><a href="#org8f713c2">"Low" mass stars with radiative cores</a></h4>
<div class="outline-text-4" id="text-org8f713c2">
<p>
Very low mass stars smoothly evolve off the main sequence: if you look
at the T(ρ) diagram in the movie produced by <code>MESA-web</code>, from the
outlines of the track you can see where the nuclear burning moves.
</p>
<figure id="org2424ed6">
<img src="./images/1Msun_TAMS.png" alt="1Msun_TAMS.png" width="100%">
<figcaption><span class="figure-number">Figure 5: </span>Screenshot of a <code>MESA-web</code> calculation of a 1M<sub>☉</sub> star shortly after the main sequence. The HRD (bottom left) shows a smooth end of the main sequence, and the Kippenhahn diagram and T(ρ) tracks (middle) show that all the burning is in a shell surrouding the inert He core. The bottom right panel shows that the inner region as a flattening T profile because of conduction efficiently transporting energy and erasing the dT/dr.</figcaption>
</figure>
<p>
Since these are stars that were burning radiatively (the fully
convective ones have not yet finished their main sequence even if they
had been burning since the birth of the Universe!), they have just
outside the region hot enough for hydrogen burning fresh fuel
available that has not been mixed in the burning region. Therefore,
<b>hydrogen ignites in a shell</b> around the now H-depleted, He-rich core.
</p>
<p>
Because of the gap in T to bridge the Coulomb barriers for
hydrogen-burning and 3α, Helium core burning does <i>not</i> ignite
immediately: the Helium core sits inert, contracts, degeneracy
pressure starts to matter and conduction becomes important, leading to
an almost <i>isothermal</i> He core sitting below the H shell.
</p>
<p>
The morphology of the end of the main sequence for low mass stars with
radiative cores is <i>smooth</i>: the core contracts, the shell above it
contracts and it is immediately hot enough to burn. The temperature of
the shell is determined by the <i>contraction</i> of the inert He core,
rather than by the energy generation by nuclear physics. Therefore,
the shell is typically becoming hot enough to burn through the CNO
cycle even for a low mass star.
</p>
</div>
</div>
<div id="outline-container-org363300f" class="outline-4">
<h4 id="org363300f"><a href="#org363300f">"High" mass stars with convective cores</a></h4>
<div class="outline-text-4" id="text-org363300f">
<p>
Increasing the mass above the threshold for activating the CN-NO
bi-cycle (somewhere ∼1.1-1.3M<sub>☉</sub> depending on assumptions), the
morphology of the end of the main sequence changes.
</p>
<figure id="orgaff3b66">
<img src="./images/30Msun_TAMS.png" alt="30Msun_TAMS.png" width="100%">
<figcaption><span class="figure-number">Figure 6: </span>Screenshot of a <code>MESA-web</code> calculation of a 30M<sub>☉</sub> star shortly after the main sequence. The HRD (bottom left) shows the "Henyey hook" feature, the Kippenhahn diagran and T(ρ) track shows that there is an off-center H-burning shell but the He in the core ignites promptly too. The core is not degenerate, but convective again, and mantains a nearly adiabatic temperature gradient.</figcaption>
</figure>
<p>
In this case, during the main sequence the <i>burning</i> is even more
centralized in mass and radius coordinate than for lower-mass
pp-chain-sustained stars, but that drives <i>convection</i>. Therefore,
convective mixing refuels the burning region from a larger reservoir,
and when the fuel runs out, it means that there is a gap in the star
between where T is hot enough for nuclear reactions and where viable
fuel is. This causes an "overall contraction phase", also known as
"Henyey hook", where the star, out of energy sources resumes its
gravothermal collapse and shrinks in radius.
</p>
<p>
This process increases the temperature profile until the H-rich fuel
left at the edge of the convective core ignites in a shell. However,
the He core below, whose mass is set by the extent of convection
(+convective boundary mixing) during the main sequence, is too big to
be sustained by electron degeneracy pressure and too hot to be
degenerate (recall that ⟨ T ⟩ ∝ μ M/R): below the shell the
contraction continues until He also promptly ignites through the 3α
reaction, driving core convection!
</p>
</div>
</div>
</div>
<div id="outline-container-orga42d04a" class="outline-2">
<h2 id="orga42d04a"><a href="#orga42d04a">H-shell and He burning</a></h2>
<div class="outline-text-2" id="text-orga42d04a">
<p>
"[The post main sequence acts as a] <i>sort of magnifying glass, also
revealing relentlessly the faults of calculations of earlier phases</i>" -
Kippenhahn.
</p>
</div>
<div id="outline-container-org6a41f98" class="outline-3">
<h3 id="org6a41f98"><a href="#org6a41f98">Low mass star "flashes"</a></h3>
<div class="outline-text-3" id="text-org6a41f98">
<p>
For low mass stars the He core is sufficiently small to be
electron-degeneracy supported, and there is H-rich fuel available
right outside the region that was burning during the main sequence:
after exhausting H in their core, they smoothly transition to a
H-shell burning/He core degenerate phase. During this phase the core
contracts and the envelope expands dramatically: the star appears as a
red giant (RG)!
</p>
<p>
<b>N.B.:</b> during this phase the He core is degenerate and <i>conduction</i> by
electrons efficiently transports energy making the whole core
approximately isothermal. This leads to the Schonberg-Chandrasekhar
maximum mass that it can have.
</p>
<p>
The microphysical reason for this expansion is not perfectly
understood (and roughly once per decade a new tentative partial
explanation is put forward). Nevertheless, we are confident that this
does occur as we can see it happening across stellar populations. One
partial explanation often invoked is the so called "mirror principle":
when there is a shell source of energy, as the inner region contracts
the outer regions expand (and viceversa). This "mirror principle" can
be understood in terms of the virial theorem in its most complete form
(including the \(\ddot{I}\) term dependent on the moment of inertia):
since the core contracts (decreasing the moment of inertia), the
envelope needs to expand to compensate (increasing the moment of
inertia). Another way to justify this semi-empirical "mirror
principle" is to keep the shell energy generation constant (see Onno
Pols' lecture notes, chapter 10).
</p>
<p>
The H-shell ignites wherever there is available fuel, its lower
boundary temperature thus is determined by the structure of the
contracting core, which typically exceeds the T threshold for the CNO
cycle: even stars that burn through the pp-chain on the main sequence
will do the CNO cycle later! The shell energy release also determines
the structure of the envelope above: once the star is <i>not homogeneous</i>
anymore, the simple gravothermal collapse due to the virial theorem
complicates!
</p>
<p>
This also implies that it is the core structure which determines the
properties of the shell, which determines the envelope properties
(namely the luminosity): in fact we observe tight correlations between
the core mass and the luminosity of the star.
</p>
<p>
As the evolution proceeds, the shell "climbs up in mass coordinate"
(though its radius may stay constant or decrease even as the
underlying inert He core contracts). The T<sub>eff</sub> decreases and the
convective envelope deepens (T<sub>eff</sub> drops, T<sub>shell</sub> is set by the core
contraction and locked by nuclear reactions, thus ∇ steepens), this
can reach the inner most layers (partially enriched in He, especially
\(^{3}\mathrm{He}\), and possibly \(^{14}\mathrm{N}\) if the star experienced
some CN cycle), leading to the "first dredge up": material from the
inner layers above the H-shell is mixed outwards by convection and
becomes visible in the stellar atmosphere.
</p>
<p>
As the shell moves upwards by consuming H fuel (and dumping He ashes
onto the core), it will encounter a layer mixed by convection in the
first dredge up. The outward mixing of nuclearly processed material
also corresponds to inward mixing of H-rich envelope material: the
shell thus reaches a region that is <i>more fuel rich</i> than before! This
makes the shell briefly exceed the L<sub>nuc</sub> = L condition, the
overproduction of energy pushes the envelope to higher L, lower T<sub>eff</sub>,
and lowers the ρ in the shell, causing a decrease of L<sub>nuc</sub>. This
process ultimately results in stars crossing a certain luminosity
threshold 3 times: observationally this produces a cumulation of stars
at a certain luminosity or in other words a "bump" in the luminosity
distribution.
</p>
<p>
<b>N.B.:</b> for massive stars, discussed below, the "first dredge up" may
not occur as described here, but the H-shell will also move outwards
towards more H-rich fuel causing a 3× crossing of a certain
luminosity.
</p>
</div>
<div id="outline-container-orga850d69" class="outline-4">
<h4 id="orga850d69"><a href="#orga850d69">He flash</a></h4>
<div class="outline-text-4" id="text-orga850d69">
<iframe width="560" height="315" src="https://www.youtube.com/embed/2_Km4RTdkPw?si=ZkacE_zcP7g67kIN" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<p>
Above is a <code>pgstar</code> movie of the He flash(es) in a 1M<sub>☉</sub> star computed
with <code>MESA</code> by <a href="https://www.stellarphysics.org/">M. Cantiello</a>. Note the panels are <i>different</i> than in the
<code>MESA-web</code> configuration, and the HRD does <i>not</i> show the pre-main
sequence.
</p>
<p>
As the H burning shell adds nuclear ashes to the underlying inert He
core, until it reaches a mass that cannot be sustained by degeneracy
pressure anymore, and He ignites. This typically occurs for M<sub>He</sub>≅0.45M<sub>☉</sub>.
</p>
<p>
This ignition however happens in a degenerate environment where P does
<i>not</i> depend on T! Therefore the energy released by the burning of He
initially does not increase dP/dr and does not cause an expansion of
the core, instead it all remains as internal energy, raising the
temperature and increasing the nuclear burning rate: this situation
(which presents itself any time there is a nuclear ignition in a
degenerate environment) is clearly unstable and leads to the so called
"Helium flash". Burning rises T until P transitions from being mostly
due to electron degeneracy to being ideal gas again: this causes an
abrupt change in pressure and a temporarily <i>dynamical</i> phase of the
evolution!
</p>
<p>
Because this requires a specific He core mass, and the He core mass
before the flashes is determining the total luminosity of the red
giant, this means that pre-flash there is a "standardizable" maximum
luminosity of red giants, the so called "tip of the red giant branch",
which is nowadays used as an alternative method to measure distances
for cosmological applications.
</p>
<p>
The occurrence of neutrino cooling in the core can cause the burning
during the He flash to be initially off-center. Moreover, the star can
react to the flash by (finally) expanding the core and decreasing the
burning rate, and on a span of a few thermal timescale, minor
secondary flash can occur as the core re-collapses, until He core
burning finally stabilizes, lifting degeneracy and causing core
convection.
</p>
</div>
</div>
<div id="outline-container-org549e582" class="outline-4">
<h4 id="org549e582"><a href="#org549e582">Red clump and Horizontal branch</a></h4>
<div class="outline-text-4" id="text-org549e582">
<p>
During He core burning, low mass stars have a convective core burning
thought the 3α (and later \(^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}\)), surrounded
by an inert He layer, and a H-burning shell wherever H becomes
available. Above the H-burning shell, if there is a substantial H-rich
envelope, it will be convective: these stars are close to the Hayashi
track (by radius they are mostly convective), but on the hotter side
(because of the existing radiative layers).
</p>
<p>
Since the He flash occurs as soon as the He core mass reaches a
sufficient mass, all these stars have similar luminosities, and form
the so-called "red clump" on the HR diagram, a noticeable feature in
cluster and galaxy populations that can also be used for distance and
age estimates (see also for example <a href="https://www.annualreviews.org/content/journals/10.1146/annurev-astro-081915-023354">Girardi 2016</a>). Since the mass of
the He core at ignition for low mass stars is set by the He flash at
∼0.45M<sub>☉</sub>, the lower mass stars will have less envelope at this
point (more has been processed into He to reach the threshold mass for
the flash): from the red clump a there is a continuous almost
horizontal line (they all have roughly the same luminosity set by the
core mass) of stars in the HR diagram for low mass core-He burning
stars whose coolest end is the red clump.
</p>
<p>
(continuing reading about the evolution of low mass <a href="#orgf9df2b6">here</a>)
</p>
</div>
</div>
</div>
<div id="outline-container-org2b7b367" class="outline-3">
<h3 id="org2b7b367"><a href="#org2b7b367">High mass stars and "Hertzsprung gap"</a></h3>
<div class="outline-text-3" id="text-org2b7b367">
<p>
Stars with masses sufficiently high for the core to be convective
during the hydrogen core burning main sequence (M≥1.2M<sub>☉{}</sub> roughly,
depending on assumptions) will <i>not</i> have a phase of evolution with an
inert, isothermal He core: the core is too big for degeneracy pressure
to sustain it and after the main sequence it continues contracting
until the 3α reaction activates and He burns. The prompt post "Henyey
hook" appearance of two nuclear energy sources (He core and H shell)
drives the star towards the cool side of the HR diagram very quickly
(∼ τ<sub>KH</sub>), becoming red supergiants (RSG)
</p>
<p>
Thus, in the HRD of a coeval stellar population, there will be many
stars on the main sequence (τ∼τ<sub>nuc,H</sub>) and close to the Hayashi track
as RSG (τ∼τ<sub>nuc,He</sub>), but very few in between: this is often referred to
as the "Hertzsprung gap". <b>N.B.:</b> the scarcity of stars in the gap is
only due to the timescales of evolution, it is not a forbidden region
of the HRD.
</p>
<p>
Some stars may experience "blue loops" as their H-shell climbs upward
in mass coordinate and encounters layers with more H (see for example
<a href="https://ui.adsabs.harvard.edu/abs/2015MNRAS.447.2951W/abstract">Walmswell et al. 2015</a>). The occurrence of these is very sensitive to
numerical approximations and make solid predictions hard, but their
physical nature in some cases is supported by observations. Depending
on metallicity, some stars may even spend most of their He core
burning time in a blue loop appearing hotter than a typical RSG.
</p>
</div>
<div id="outline-container-org474029c" class="outline-4">
<h4 id="org474029c"><a href="#org474029c">Mass loss and single-star evolution path to Wolf-Rayet</a></h4>
<div class="outline-text-4" id="text-org474029c">
<p>
As M increases (and consequently even more so L), mass loss becomes a
progressively more important ingredient for the evolution of stars.
</p>
<p>
Stars can lose mass through:
</p>
<ul class="org-ul">
<li>stellar winds (pressure driven for low mass stars, radiation driven
for high mass stars)</li>
<li>eruptive events (e.g., "luminous blue variable eruptions")</li>
<li>binary interactions</li>
</ul>
<p>
All of these can directly or indirectly impact the internal structure
of the star, and its appearance. Very massive stars may have such high
mass loss rates that they lose their entire H-rich envelope already
during the main sequence (becoming WNh stars). Moving to lower masses,
they may evolve red-ward on the HR diagram (which increases the
opacity κ and thus presumably the wind mass-loss rate, although this
is highly debated presently, see <a href="https://ui.adsabs.harvard.edu/abs/2014ARA%26A..52..487S/abstract">Smith 2014</a>, <a href="https://ui.adsabs.harvard.edu/abs/2017A%26A...603A.118R/abstract">Renzo et al. 2017</a>, <a href="https://ui.adsabs.harvard.edu/abs/2020MNRAS.492.5994B/abstract">Beasor
et al. 2020</a>, <a href="https://ui.adsabs.harvard.edu/abs/2024A%26A...681A..17D/abstract">Decin et al. 2024</a>), and then shed their H-rich envelope.
</p>
<p>
A star which has lost its envelope will "reveal" its He core, and if
luminous enough, this will drive a thick wind that can enshroud the
star and hide it below a "pseudo-photosphere". These winds can be so
dense that collisional excitation produces emission lines, making the
stars appear as WR (see e.g., <a href="https://ui.adsabs.harvard.edu/abs/2024arXiv241004436S/abstract">Shenar 2024</a>).
</p>
<p>
<b>N.B.:</b> stripped low and intermediate mass stars not luminous enough to
drive WR-like outflows that produce emission lines are predicted and
observed and require binary interactions to form, see <a href="https://ui.adsabs.harvard.edu/abs/2023Sci...382.1287D/abstract">Drout et al.
2023</a>.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgff22c22" class="outline-2">
<h2 id="orgff22c22"><a href="#orgff22c22">Late evolution</a></h2>
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<h3 id="orgf9df2b6"><a href="#orgf9df2b6">Low mass stars: AGB thermal pulses and WD cooling</a></h3>
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<p>
After the end of He core burning, the <i>vast</i> majority of stars (∼ 98% of
all stars integrating over the birth-mass distribution for M<sub>ZAMS</sub>≤
7.5M<sub>☉</sub>) is left with a carbon/oxygen rich degenerate core which is
not massive enough to ignite further nuclear burning, and electron
degeneracy sustains it. These stars however still need to lose their
H-rich extended envelope and He-rich shell (which remain temporarily
sustained by nuclear burning in shells) before they can finally rest
as white dwarfs entirely sustained by degeneracy pressure. This
process is relatively fast and involves copious episodic stellar
outflows which are still an active topic of research.
</p>
<p>
A star in this phase is referred to as an "Asymptotic Giant Branch"
(AGB) star: for most of its life the He layer is inert (no nuclear
burning) and (partially) degenerate too, and it grows in mass because
of the ashes of the overlaying H-burning shell, which sustains the
H-rich envelope above it.
</p>
<p>
As the He layer grows in mass, it temporarily ignites: this energy
release causes a <i>flash</i> (similar to He ignition in low mass stars in
the first place), and expands the He layer, pushing outward the inner
boundary of the H-shell, often until its density becomes too low for
H-burning. Thus, as a consequence of the He shell flash, matter is
pushed out and cools (possibly forming dust and increasing κ and thus
the mass loss from the star), the H shell shuts off, but the He shell
too does. This is because the flash was not hydrostatic self-regulated
burning! The outer layers then re-collapse on a thermal timescale and
as they contract, the H-burning shell ignites first (it's easier to
burn H than He!), returning to the initial situation, but with a
little less mass. This process of "thermal AGB pulses" ultimately will
lead to the loss of all the H and He, leaving a "bare" CO core
exposed, with only a very thin H/He atmosphere.
</p>
<p>
<b>N.B.:</b> ignition in a (partially) degenerate environment causes an
abrupt increase in T and thus P from the ideal gas EOS, but the
environment was supported by a T-independent degeneracy pressure: this
leads to a discontinuity in time of the pressure and thus a <i>dynamical</i>
event, referred to as a Flash. This can also occur in the core of
massive stars!
</p>
<figure id="org5514363">
<img src="./images/Trho_TPAGB.png" alt="Trho_TPAGB.png" width="100%">
<figcaption><span class="figure-number">Figure 7: </span>T(ρ) diagram of a 1M<sub>☉</sub> <code>MESA-web</code> model during an AGB thermal pulse. Note the temperature inversion in the core, the presence of 2 burning shells in this snapshot (the He shell marked by the orange outline and the H shell marked by the yellow outline).</figcaption>
</figure>
<p>
<b>N.B.:</b> because of ν cooling, in AGB stars the center cools faster than
the layers above it: this can lead to a "temperature inversion". At
the boundary between this evolutionary end and the end of massive
stars, the so-called "super-AGB" stars will ignite C off-center, but
the ignition of carbon will then move inwards (in a ∼ meter thin
shell) until it reaches the center, lifting the electron degeneracy by
releasing nuclear energy, and allowing the star to evolve past C core
burning.
</p>
<p>
After losing their envelopes to thermal pulses (possibly accompanied
by a late enhancement of their stellar winds), low mass stars rapidly
move from the top right (high L low T<sub>eff</sub>) corner of the HR diagram to
the lower left (low L high T<sub>eff</sub>) corner becoming white dwarfs (WD):
this process of contraction occurs on a thermal timescale. In WDs the
gravothermal collapse stops because of the electron degeneracy
pressure: the degeneracy decoupled their structure (which can be
approximated assuming k<sub>B</sub>T≪ ε<sub>Fermi</sub> ⇒ T≅ 0)
and their radiative properties.
</p>
<p>
These sit in the bottom left corner of the HR diagram: they actually
have <i>hotter</i> surface temperatures compared to a main sequence star!
This high T means they do radiate and lose energy, but because of the
small radius (R≅0.01R<sub>☉</sub>≅ 1000km) they have a low luminosity
L=4π R<sup>2</sup> σ T<sub>eff</sub>: their radiative cooling is very slow (timescale
of billions of years). The WD will just "slide down slowly" on a
cooling track. The WD cooling sequence provides a <i>clock</i> for stellar
populations!