# A Red Giants’ Toy Story

Marcelo M. Miller Bertolami Instituto de Astrofísica de La Plata, Consejo
Nacional de Investigaciones Científicas y Técnicas
Avenida Centenario (Paseo del Bosque) S/N, B1900FWA La Plata, Argentina.
Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata
Avenida Centenario (Paseo del Bosque) S/N, B1900FWA La Plata, Argentina.

(Received February 29, 2021; Revised February 30, 2022; Accepted February 31,
2023)

###### Abstract

In spite of the spectacular progress accomplished by stellar evolution theory,
some simple questions remain unanswered. One of these questions is “Why do
stars become red giants?”. Here we present a relatively simple analytical
answer to this question. We validate our analysis by constructing a
quantitative toy model of a red giant and comparing its predictions to full
stellar evolutionary models.

We find that the envelope forces the value of $\nabla=d\ln T/d\ln P$ at, and
above, the burning shell into a very narrow range of possible values. Together
with the fact that the stellar material at the burning shell both provides and
transports most of the stellar luminosity, this leads to tight relations
between the thermodynamic variables at the burning shell and the mass and
radius of the core—$T_{s}(M_{c},R_{s})$, $P_{s}(M_{c},R_{s})$, and
$\rho_{s}(M_{c},R_{s})$. When complemented by typical mass-radius relations of
the helium cores, this implies that for all stellar masses the evolution of
the core dictates the values of $T_{s}$, $P_{s}$ and $\rho_{s}$. We show that
for all stellar masses evolution leads to an increase in the pressure and
density contrasts between the shell and the core, forcing a huge expansion of
the layers on top of the burning shell.

Besides explaining why stars become red giants our analysis also offers a
mathematical demonstration of the so-called shell homology relations, and
provides simple quantitative answers to some properties of low-mass red
giants.

Stellar structures — Stellar Evolution — Giant Branch — Stellar Interiors

††journal: ApJ††software: LPCODE: Miller Bertolami (2016)

##  1 Introduction

About a century from its humble beginnings (Eddington, 1926), stellar
evolution theory has developed into a full-fledged predictive theory
(Kippenhahn et al., 2012) whose results are routinely used as inputs for other
fields of astrophysics (Greggio & Renzini, 2011; Cassisi & Salaris, 2013). Its
predictions have been confirmed by a variety of different observational tests,
and are continuously checked by countless numerical simulations by means of
specifically tailored numerical codes (Lebreton et al., 2008; Silva Aguirre et
al., 2020). In spite of the spectacular progress accomplished, some simple
questions remain unanswered. One of these questions is “Why do stars become
red giants?”. Although the existence of a red giant solution is known since
the early days of automatic numerical computations (Öpik, 1938; Hayashi, 1947,
1949; Hoyle & Schwarzschild, 1955), no simple explanation of the reason for
the existence of such solution has been provided. The absence of a universally
accepted answer for this question can be clearly appreciated by reading the
related chapters in textbooks of stellar evolution. When trying to explain the
evolution after the end of core-hydrogen (H) burning, different authors choose
very different paths. Some authors choose to describe extensively the details
of numerical models (Iben, 2013), others choose to establish an ad hoc
principle that applies exclusively to stellar evolution (e.g. the so-called
“mirror principle of radial motions” Kippenhahn et al., 2012), others try to
convince themselves that not all physical processes can be understood in
simple step-by-step terms and that we should accept the raw output of
numerical simulations (Prialnik, 2009), while others openly admit that we lack
a definitive explanation of the precise physical reason(s) that drive the
expansion (Hansen et al., 2004; Salaris & Cassisi, 2005).

The lack of an accepted answer does not mean that no answers have been
proposed but just that proposed answers have never attained wide popularity.
And for good reasons. Proposed answers encompass a wide variety of ideas, too
vast and complex for this introduction. With no intention of being exhaustive,
but to highlight both the complexity of the problem and its many possible
angles, let us mention a few of them. As mentioned by Eggleton & Faulkner
(1981), one of the most ubiquitous myths among nonspecialists is the idea that
the gravitational energy released by the contracting core is absorbed by the
envelope causing it to expand, ignoring the fact that most of the energy comes
from the burning shell and the envelope has no way to tell were the energy is
coming from. Similarly ubiquitous is the statement that the increase in the
stellar radius is a consequence of the increase in the energy output of the
burning shell due to the heating of the core, also ignoring the fact that in
low-mass stars the core is mostly heated by the surrounding envelope (and not
the other way around), and that there is no obvious reason why the burning
shell would not settle at a lower temperature, which would imply a huge
decrease in the energy output of the burning shell. While it might be expected
that nonspecialists find it hard to pin down the reasons why stars become red
giants, it is much more interesting that experts in stellar structure have
also failed to agree on the actual causes of this transformation. While
studying this problem some authors have highlighted the importance of the
gravitational field generated by the compact core by numerically solving
steady-state solutions (thermal equilibrium, $dS/dt=0$) (Höppner & Weigert,
1973; Eggleton & Faulkner, 1981; Weiss, 1983). Others have focused on the
chemical gradients developed due to nuclear reactions at the burning shell
(Hoyle & Lyttleton, 1949). Some authors have highlighted the key importance of
the degree of central condensation of the core and its consequences on the
envelope (Eggleton & Cannon, 1991; Faulkner, 2005). While all the works
mentioned before have been focused on the study of the consequences of
different core properties in steady-state solutions, some other authors have
concluded that giantness is a consequence of thermal instabilities in the
envelope due to the high luminosity released by the burning shell (Renzini,
1984; Applegate, 1988; Renzini et al., 1992; Renzini & Ritossa, 1994). Also in
this connection, some authors have pointed to the development of some
gravothermal catastrophe of the core and the entropy gradient above it
(Fujimoto & Iben, 1991). The methods adopted to try to answer this question
have also been extremely diverse. For example, some authors have tried to
answer the question by constructing simplified ad hoc polytropic models
(Eggleton & Cannon, 1991; Frost & Lattanzio, 1992; Eggleton et al., 1998),
some have tried to answer this question by means of dimensional analysis
(Bhaskar & Nigam, 1991), while others have tried to answer the question by a
detailed evaluation of numerical solutions on the $U-V$ plane (Fujimoto &
Iben, 1991; Sugimoto & Fujimoto, 2000). While some authors have highlighted
the importance of some critical curves and singular points in the $U-V$ plane
(Yahil & van den Horn, 1985; Sugimoto & Fujimoto, 2000), others have
disparaged these explanations as mere descriptions of the results from
numerical models (Iben, 1993). Proposed answers also encompass a wide
diversity in complexity, from very simple (Hauptmann et al., 2000) to
extremely complex (Whitworth, 1989). The discussions about why stars become
red giants have sometimes turned into heated debates (Sugimoto, 1997;
Faulkner, 1997; Sugimoto & Fujimoto, 2000; Faulkner, 2005), while some other
times authors have ignored criticism and continue to develop ideas (Renzini et
al., 1992; Renzini & Ritossa, 1994) that had already been seriously questioned
by other researchers (Weiss, 1989; Iben, 1993). The reader interested in the
subtleties and shortcomings of the different ideas is referred to the appendix
of Sugimoto & Fujimoto (2000), Appendix 10.B of Faulkner (2005), the
introductions to Applegate (1988) and Stancliffe et al. (2009), and Section
3.3 of Bhaskar & Nigam (1991).

Both Whitworth (1989) and Iben (1993) concluded that a simple explanation of
why stars become red giants is not possible. Iben (1993) issued a warning
against attempts to find simple explanations, claiming that simple
explanations might be forcefully misleading. Although we are aware of these
warnings, this paper is an attempt to shed some light on a possible simple
explanation on the question of why do stars become red giants. We understand
this to be a worthwhile project because toy models and dimensional analysis
play a key role in how our minds approach the understanding of physical
problems. We believe that toy models are key to extrapolate the results of a
set of numerical simulations to all _similar_ stars and to other _analogous_
physical problems. In fact, in order to define what the words _similar_ and
_analogous_ in the previous sentence mean, a toy model description of the
problem is a necessity. For this reason we have searched for a toy model
description of red giants that is both accurate and convincing. In
constructing our toy model we have kept in mind the warnings by Iben (1993)
regarding “convoluted arguments” and “circular explanations”. We have forced
ourselves to avoid the use of particular properties observed in detailed
stellar models in the construction of our toy model, and kept the line of
reasoning as clear as possible.

In our opinion, numerical experiments with steady state solutions (thermal
equilibrium) like those performed by Höppner & Weigert (1973), Weiss (1983),
and Iben (1993), together with the fact that low-mass red giants evolve in a
nuclear timescale and develop the most extreme case of giantness, clearly show
that thermal instabilities in the envelope are not what pushes stars into red
giant dimensions. In fact, as pointed out by Faulkner (1997), even when they
expand in thermal timescales envelopes lag behind, rather than lead, as can be
tested by shutting off chemical changes in a stellar evolution code and
letting the envelope reach its final steady-state solution consistent with the
structure of the core. One of the most insightful presentation to date was
given by Faulkner (2005) who describes giantness as a structural property of
thermal equilibrium solutions consisting of a compact core, the presence of a
burning shell, and a massive envelope that imposes a ”mass storage problem” to
the star. The fact that it is a structural property of thermal equilibrium
solutions does not imply that a star out of thermal equilibrium cannot be in a
giant configuration, but just that in those cases we should understand them as
evolving toward the corresponding structure in thermal equilibrium in a
Kelvin-Helmholtz timescale. In his article, Faulkner (2005) proceeds first by
defining a set of principles of stellar structure and the meaning of a dense
core. Then, assuming a simple model with a discontinuity at the burning shell,
proceeds to derive an “asymptotic theory” of low-mass red giants. For this, he
first motivates that the temperature at the discontinuity follows a very
specific relation (his equation 10.2) with the help from polytropic envelope
integrations (performed in his Appendix 10.A.2). From these expressions, and
using some numerical results of polytropic envelope integrations, he argues
that massive envelopes have forcefully a very small value of his parameter
$\tau$, which effectively means that temperature at the burning shell has a
tight dependence on the mass and radius of the core. Using this result, he
then derives two ”theorems” for how the luminosity of the star and the density
of the shell relate to the mass and radius of the core—his equations 10.7 and
10.8; the latter only valid for a very specific temperature dependence of the
energy generation rate. Once this is done he proceeds to extend his results to
a unified treatment of main-sequence stars and red giants, horizontal branch
stars, and finally, to giant stars with luminous cores. However, we understand
the work by Faulkner (2005) to be problematic due to a florid but
mathematically obscure presentation, and the reliance on assumptions difficult
to justify. Among these, his derivation of his equations 10.5 and 10.18 stick
out, as their derivation require that $P_{\rm rad}/P_{\rm gas}$ is constant
throughout the burning shell, but nonzero, imposing a relation between $T^{4}$
and $P$ which disappears when radiation pressure is negligible111Which implies
that $1-\beta$ is close to zero. Oddly enough radiation pressure is negligible
in the conditions relevant for red giants, and the appearance of the $1-\beta$
factor is rather disturbing.. Another feature that makes his equation 10.5
suspicious is the fact that the energy generation from the burning shell is
proportional to the cube of the radius of the burning shell, while in a thin
shell approximation the expected dependence would be with the square of the
radius. More worrying is the fact that, within his asymptotic theory, the
core-mass-luminosity relation (his equation 10.7) is obtained without any
mentioning regarding the development of convection in the envelope. This
result is a consequence of the assumption, in his asymptotic theory, that
$\tau=0$, which is not rigorously demonstrated and only based on the
discussion of polytropic envelopes on the $U-V$ plane. In fact, it has been
known since the work of Sandage & Schwarzschild (1952) that without the
inclusion of an outer convective region, post main sequence models expand
without increasing their luminosity (see for example Fig. 3 in Sandage &
Schwarzschild 1952 and Fig. 2 of Stancliffe et al. 2009). This issues,
together with his assumption of a very specific temperature dependence for the
CNO-Cycle ($\eta\simeq 15$, his equations 10.8 and 10.10), his use of the
$U-V$ plane (in his Appendix 10.A.2), and his emphasis on polytropic envelopes
(particularly the $n=3$ polytrope), makes his final conclusions difficult to
accept. In spite of these mathematical shortcomings, we believe his perceptive
physical description of the red giant structure is worth exploring.

The aim of this paper is to present a compelling toy model of red giants that
improves our insight of how these stars work. Note in passing that the key
feature of toy models is not precision but the accurate description of the
main processes involved222“All models are wrong but some are useful” Box
(1979)., albeit distilled to its most simple elements. This is the ideal we
pursue through the paper. The paper is structured as follows; first on Section
2 we analyze the burning shell and its surroundings. This sets the key
constraints imposed by the burning shell and the envelope on the thermodynamic
variables. Moreover, this section offers the first proof we are aware of for
the underlying hypothesis of the so-called shell homology relations; i.e. that
temperature ($T_{s}$), pressure ($P_{s}$), density ($\rho_{s}$) at the
location of the burning shell only depend on the radius ($R_{s}$) and mass
($M_{c}$) of the core. In addition, these sections will clarify the problems
with some assumptions behind Faulkner’s asymptotic theory and highlight the
role of convection in the outer layers. Then on Section 3 we check the
accuracy of the derived expressions by constructing a toy model of a full low-
mass red giant and compare its predictions with those of detailed stellar
evolution computations. Interestingly this toy model helps us understand why
all low-mass stars develop the Helium flash at nearly the same core mass, why
this mass is about a half a solar mass and why the burning shell in low mass
stars remains at an almost constant location through the evolution. After this
validation, on Section 4 we make a dimensional analysis of red giant stars of
all masses. Dimensional analysis of the resulting expressions shows why the
presence of a burning shell makes homologous contraction of the whole star
impossible and forces the formation of bright red giants. A simple description
of the formation of red giants at all masses is presented on Section 5. We
conclude the paper by summarizing our results and the physical insights gained
from the present model. We hope that the last two sections will convince our
most numerically minded readers of why simple models can improve our
understanding of the inner workings of stars, even after so many decades of
numerical simulations.

##  2 The burning shell and its surroundings

One of the defining features of a red giant star is the presence of a very
localized burning shell. Fig. 1 shows a schematic description of the region of
the star around the burning shell333Note that our treatment of the burning
shell is rather different from that of Faulkner (2005). In that work the
burning shell was adopted as a discontinuous transition making no distinction
between the values of the variable below, at and above the burning shell. In
view that some of these variables can have significant changes (e.g. $L_{c}\ll
L_{s}$) we think that approximation is not justified.. We will indicate by
$T_{-}$, $P_{-}$, $\rho_{-}$ and $r_{-}$ the structure variables at the bottom
boundary of the burning shell, and by $T_{+}$, $P_{+}$, $\rho_{+}$, and
$r_{+}$ those at the upper boundary of the burning shell. Energy generation
happens between $r_{-}$ and $r_{+}$ where the luminosity $l$ increases from
the luminosity of the core ($L_{c}$) to the luminosity of the envelope
($L_{c}+L_{s}$). $R_{s}$ indicates the location of the middle of the burning
shell, where the thermodynamic variables are $T_{s}$, $P_{s}$, $\rho_{s}$.
Note that for a burning shell to have an impact on the structure of the star
its luminosity must be at least similar to that of the core ($L_{s}\gtrsim
L_{c}$). In most interesting cases in fact $L_{s}$ will be $L_{s}\gg L_{c}$.
Due to the extreme sensitivity of nuclear reaction rates with the temperature
(e.g. $\epsilon_{\rm CNO}\simeq\epsilon_{0}\rho^{t}T^{\nu}$, with $t=1$, and
$\nu\simeq 23\textendash 10$ for $T\simeq 10^{7}\textendash 10^{8}$K), a small
decrease in the temperature is enough to produce a huge drop in the energy
generation rate. A small decrease of $\delta T/T\simeq 1/\nu$ leads the energy
release to drop almost to zero $\delta\epsilon/\epsilon\simeq\nu\,\delta
T/T\simeq 1$. As a consequence the energy released by the burning shell is
concentrated in a thin region of $\sim 2\delta T$ around the characteristic
temperature of the shell $T_{s}$. This will allow us to work with first-order
approximations around the shell temperature $f(T_{s}+dT)\simeq
f(T_{s})+\frac{df}{dT}(T_{s})\,dT$ and consider the burning shell to be thin
for most computations (i.e. $r_{-}\simeq R_{s}\simeq r_{+}$).

Figure 1: Schematic description of the evolution of the luminosity $l(r)$
around the burning shell for a core with generic luminosity $L_{c}$. Note that
$r_{-}$, and $r_{+}$ indicate the position of the inner and upper boundaries
of the burning shell, and $L_{s}$ indicates the total energy per unit time
generated in the burning shell.

Due to the presence of the H-burning shell, the stellar plasma at the shell
cannot be degenerate444Otherwise a flash would develop, injecting huge amounts
of energy in that layer, and lifting degeneracy.. Then, as the material on the
burning shell is not degenerate, and $T$, $\rho$, and $P$ are continuous,
there must exist regions, immediately below and above the burning shell, where
the material behaves as a classical ideal gas
($P=\Re\,\rho\,T/\mu$)555Strictly speaking as a classical ideal gas plus
radiation, but radiation pressure always plays a minor role at the burning
shell. In fact, we know from observations that most giant stars are far from
the Eddington luminosities of their cores..

###  2.1 Above the burning shell

As a first step on our way to understanding the physics of red giants, we will
look at how the outer boundary conditions establish some constraints on the
thermodynamical quantities immediately above the burning shell. In particular,
we will see that the outer boundary conditions strongly restrict the possible
values of the temperature gradient $\nabla={\mathrm{d}}\ln T/{\mathrm{d}}\ln
P$.

Under the assumption that the envelope is in a steady state (thermal
equilibrium, $dS/dt\simeq 0$, $\int_{0}^{M_{\star}}|\epsilon_{g}|dm\ll
L_{\star}$), the thermodynamic variables in the envelope above the burning
shell must fulfill the hydrostatic equilibrium equation,

$\frac{dP}{dm}=-\frac{Gm}{4\pi r^{4}},$ (1)

the connection between the local radius and the lagrangian mass coordinate,

$\frac{dr}{dm}=-\frac{1}{4\pi r^{2}\rho},$ (2)

and the heat transport equation,

$\frac{dT}{dm}=-\frac{GmT}{4\pi r^{4}P}\nabla.$ (3)

In the last equation $\nabla$ is given by

$\nabla=\nabla_{\rm rad}=\frac{3}{16\pi acG}\frac{\kappa L_{\star}P}{mT^{4}},$
(4)

when $\nabla_{\rm rad}<\nabla_{\rm ad}$ and energy is transported by
radiation, or $\nabla=\nabla_{\rm ad}$ when convection develops
(Schwarzschild, 1906). All quantities in eqs. 2, 3, and 4 have been defined as
in the classic textbook by Kippenhahn et al. (2012), and
$L_{\star}=L_{s}+L_{c}$ is the luminosity of the star.

Given that we want to understand the general properties imposed by the
envelope on the burning shell for a core of arbitrary mass ($M_{c}$) and
radius ($R_{s}$) it is convenient to use adimensional variables. Using the
mean pressure of the core $\bar{P_{c}}$, the mean density of the core
$\bar{\rho_{c}}$ and a characteristic temperature $T_{c}$666Defined as the
temperature of a classical ideal gas with the mean hydrostatic pressure
$\bar{P_{c}}$ and mean density $\bar{\rho_{c}}$ of the core.,

$\bar{P_{c}}=\frac{G{M_{c}}^{2}}{8\pi{R_{s}}^{4}},\ \ \ \ \ \
\bar{\rho_{c}}=\frac{3}{4\pi}\frac{M_{c}}{{R_{s}}^{3}},\ \ \ \ \ \
T_{c}=\frac{\mu GM_{c}}{6\Re R_{s}}$ (5)

we define $q=m/M_{c}$, $x=r/R_{s}$, $y=P/\bar{P_{c}}$, $z=\rho/\bar{\rho_{c}}$
and $t=T/T_{c}$. Assuming that the gas is a classical ideal gas and that the
opacity can be approximated by a power law
$\kappa=\kappa_{0}P^{a}T^{b}$($a\geq 0$, $b\leq 0$ under normal conditions),
eqs. 1, 2, 3, 4 can be written as

$\frac{dy}{dq}=\frac{-2q}{x^{4}},\ \ \ \ \ \ \frac{dx}{dq}=\frac{t}{3x^{2}y},\
\ \ \ \ \ \frac{dt}{dq}=\mathbb{C}\frac{y^{a}t^{b-3}}{x^{4}},$ (6)

and

$\nabla=\min\left(\mathbb{C}\frac{y^{a+1}t^{b-4}}{-2q},0.4\right).$ (7)

The meaning of the constant $\mathbb{C}$ ($\mathbb{C}<0$) can be made clear by
defining a characteristic luminosity777The reader is warned not to confuse
$L_{0}$, which is a reference luminosity for a core of given mass and radius,
with the luminosity of the core $L_{c}$ to be defined in the next sections.
$L_{0}$ for a core of mass $M_{c}$ and radius $R_{s}$ as

$L_{0}=\frac{64\pi^{2}ac}{3}\frac{{T_{c}}^{4}{R_{s}}^{4}}{M_{c}\kappa_{c}},$
(8)

where $\kappa_{c}=\kappa(\bar{P_{c}},T_{c})$. With this definition it becomes
clear that $\mathbb{C}$ is an adimensional version of the luminosity of the
star and $\mathbb{C}=-L_{\star}/L_{0}$.

Due to hydrostatic equilibrium pressure at the burning shell is necessarily
lower than the mean pressure at the core, and even more so at the top boundary
of the burning shell ($y_{+}<1$, and much lower than unity for dense cores).
On the other hand, as temperature does not change significantly across the
burning shell it is usually $t_{+}\lesssim 1$. Note that as $T_{c}$
corresponds to the temperature required by a classical ideal gas at the mean
density of the core to match the hydrostatic pressure, for degenerate cores
$T_{c}$ is actually larger than the real temperature of the core and thus of
the shell, and then $t_{+}<1$. We can get a clear idea of the relevant values
of $\mathbb{C}$ by replacing the quantities in eq. 8 with those of a typical
envelope ($\mu\simeq 1.176$, $\kappa\simeq 0.34$cm2g-1) and obtain,

$L_{0}=6.77\times
10^{35}\left(\frac{M_{c}}{M_{\odot}}\right)^{3}\hbox{erg/s}=177\left(\frac{M_{c}}{M_{\odot}}\right)^{3}L_{\odot}$
(9)

We see from eq. 9 that for typical values of core masses in red giants,
$L_{0}$ is lower than the typical luminosities of the same stars on the main
sequence. Physically interesting values of $\mathbb{C}$ correspond to
$\mathbb{C}\gtrsim 1$ and even $\mathbb{C}\gg 1$ in the case of low-mass
stars.

Using eqs. 6 we can integrate the envelopes from the upper boundary of the
burning shell ($y_{+},t_{+}$) outward. Note that one of the key assumptions
here is that the envelope is massive enough so that we can integrate outward
in $q$ without reaching the photosphere (up to 30% of the core mass in Fig.
2). Fig. 2 shows the result of these envelope integrations for
$y_{+}\in(10^{-4},0.5)$, $\log|\mathbb{C}|\in(-1,5)$ and for a Kramers’
opacity law ($a=1$ and $b=-4.5$). Similar results are obtained for a classical
Thomson electron scattering opacity ($a=0$ and $b=0$) and other values of
$t_{+}\sim 1$ (see Appendix D).

Figure 2: Envelope integration for different values of $\mathbb{C}$ and
$y_{+}$, for $t_{+}=0.75$ and for a Kramers’ opacity.

As it is clear from Fig. 2 two main families of solutions are possible.
Solutions in black correspond to solutions in which the radiative gradient
increases as we move outward from the burning shell. In most cases these
envelopes become convective ($\nabla=0.4$). Conversely, solutions in orange
correspond to those cases in which the radiative gradient drops extremely fast
as we move outward (in fact more than exponentially as can be seen in Fig. 2).
As shown in Fig. 2 these solutions quickly become isothermal at a temperature
close to that of the shell. This second family of solutions does not
correspond to physical stellar envelopes as they cannot satisfy photospheric
boundary conditions. Using Eddington’s approximation we know that $d\log
T/d\tau|_{\rm ph}\simeq 3/16$ and from a zero-order integration for the
pressure $d\log P/d\tau|_{\rm ph}\simeq 3/2$, and consequently the temperature
gradient near a stellar photosphere must be $\nabla_{\rm ph}\simeq 1/8$. Such
condition cannot be satisfied by the orange solutions that stay radiative
throughout the whole integration as $\nabla_{\rm rad}$ decreases
monotonically, and for which $\nabla_{\rm rad}\ll 1$ very close to the core.
Interestingly, the watershed between both families of solutions corresponds to
those envelopes that start at the burning shell with a temperature gradient of
$\nabla_{+}=(a+1)/(4-b)$ (shown as a blue horizontal line in Fig. 2.

To better understand this behavior it is instructive to look at the derivative
of the temperature gradient. Using eqs. 6 and 7, it is easy to show that

$\frac{d\nabla_{\rm rad}}{d\log q}=\nabla_{\rm
rad}\left[\left((a+1)+(b-4)\nabla_{\rm rad}\right)\frac{d\log y}{d\log
q}-1\right],$ (10)

where due to hydrostatic equilibrium ${d\log y}/{d\log q}<0$. It is now clear
that if $\nabla_{\rm rad}\leq(a+1)/(4-b)$ at some point, then eq. 10 behaves
locally as $\frac{d\nabla_{\rm rad}}{d\log q}=\mathbb{A}\times\nabla_{\rm
rad}$ with $\mathbb{A}<0$ and $\nabla_{\rm rad}$ drops exponentially. Then
solutions with $\nabla_{\rm rad}\leq(a+1)/(4-b)$ already at the burning shell
are not physical. Physically meaningful solutions are then confined to values
of $\nabla>(a+1)/(4-b)$. On the other hand, due to convection, we know that
the temperature gradient cannot exceed the adiabatic value. Then, we have
shown that outer envelope imposes a very strict constraint on the value of the
temperature gradient in the envelope, and immediately above the burning shell
—$(a+1)/(4-b)<\nabla<0.4$.

Interestingly, as it can be guessed from Fig. 2, physical solutions with
$\nabla_{+}$ near the adiabatic value will correspond to envelopes that turn
convective very quickly (deep convection). We see that, unless we are in the
case of deep convection, with the bottom of the convective envelope closer
than $q\simeq 1.1$, the temperature gradient stays close to the critical limit
$\nabla^{\rm lim}=(a+1)/(4-b)$. Those solutions with values of $\nabla_{+}$
near $(a+1)/(4-b)$ will show relatively massive radiative regions on top of
the burning shell and can even stay radiative until they reach the
photosphere888It is worth noting that solutions with $\nabla_{+}>(a+1)/(4-b)$
are not forced to increase monotonically and can start to decrease depending
on the local value of $\frac{d\log y}{d\log q}$. In fact, we know from
integrations of photospheric conditions immediately below the photosphere
$\nabla_{\rm rad}\simeq\tau/(\tau+2/3)$, where $\tau$ is Rosseland optical
depth.. In addition, one shared property of all physical solutions is that
they show a significant decrease in temperature outside the burning shell and
a very strong decrease in pressure, in particular for those solutions that
start with $y_{+}\ll 1$. In Appendix C we show that if the convective zone
reaches all the way to the burning shell, then detailed models predict a
completely different behavior and a much smaller radius.

Most importantly, for typical opacity laws we have $\nabla^{\rm lim}=0.25$
(Thomson electron scattering, $a=b=0$) and $\nabla^{\rm lim}=0.2353$ (Kramers’
opacity, $a=1$, $b=-4.5$). This means that the value of $\nabla$ above the
burning shell varies less than a factor of two throughout the envelope. This
result will be very useful to make simple estimations in the next sections.

We end this section noting that, although these conclusions are strictly valid
for an opacity that follows a power law, the conclusions are quite general. In
fact, for any opacity law we can locally approximate it by a power law
($\kappa\simeq\kappa_{0}P^{a^{\prime}+1}T^{b^{\prime}-4}$). Due to the faster-
than-exponential nature of eq. 10 if $\nabla$ drops, in some region, below the
local critical value then the envelope solutions will quickly become
unphysical. As a consequence, in envelopes with real opacity laws $\nabla$
needs to stay between the local critical value $\nabla^{\rm
lim}=(a^{\prime}+1)/(4-b^{\prime})$ and the adiabatic value $\nabla_{\rm ad}$.
In particular, for real envelopes near the burning shell we will have in most
cases $\nabla_{+}\simeq(a+1)/(4-b)$ and only for deep convection
$\nabla_{+}\simeq\nabla_{\rm ad}$.

###  2.2 Temperature gradient inside the burning shell

Having shown in Section 2.1 that $(a+1)/(4-b)\lesssim\nabla_{+}\lesssim 0.4$
we now turn to analyze how $\nabla$ changes inside the burning shell where
$l(r)$ is not constant anymore (see Fig. 1).

From eqs. 1 and 3 we see that inside the burning shell we have

$\frac{dT}{dP}=\frac{3}{16\pi\,a\,c\,G}\frac{\kappa l}{m\,T^{3}}.$ (11)

Using again that the opacity inside the shell follows
$\kappa=\kappa_{0}P^{a}T^{b}$ we can write

$\frac{T^{4-b}}{P^{a+1}}\nabla\frac{1}{l}=\mathbb{K},$ (12)

where $\mathbb{K}>0$ is a constant.

Evaluating the left-hand side of eq. 12 both at the upper boundary and the
center of the shell, we obtain

$\nabla_{+}=\left(\frac{T_{s}}{T_{+}}\right)^{4-b}\left(\frac{P_{+}}{P_{s}}\right)^{a+1}\nabla_{s}\frac{L_{c}+L_{s}}{L_{c}+L_{s}/2}.$
(13)

We know from the high sensitivity of the energy generation to temperature that
temperature changes only slightly inside the burning shell $T_{s}-T_{+}=\Delta
T=T_{s}/\nu$. Let us call $x=\Delta P/P_{s}$ ($P_{+}=P_{s}(1-x)$), then
$dP/P=(\nabla_{s})^{-1}dT/T$ and we see that $x=(\nabla_{s}\nu)^{-1}$. Note
that, as $\Delta T/T_{s}\ll 1$ and $(a+1)/(4-b)\lesssim\nabla_{+}\lesssim 0.4$
then also $x\ll 1$ for typical values of the opacity. Then eq. 13 can be
written as

$\nabla_{+}=\left(1+\frac{1}{\nu}\right)^{4-b}\left(1-x\right)^{a+1}\frac{1}{x\nu}\frac{L_{c}+L_{s}}{L_{c}+L_{s}/2}.$
(14)

Defining $F$ as the factor due to the luminosity of the core,
$F=(L_{c}+L_{s})/(L_{c}+L_{s}/2)$ ($F=2$ for inert cores), we can write at
first order in $1/\nu$

$\displaystyle\nabla_{+}=$
$\displaystyle\left\\{1+\left(\frac{4-b}{\nu}\right)\right.$ (15)
$\displaystyle\times$
$\displaystyle\left.\left[1-\left(\frac{a+1}{4-b}\right)\frac{1}{\nabla_{s}}\right]+\mathcal{O}(1/\nu)^{2}\right\\}F\nabla_{s}.$

We see here that the dominant factor for $\nu\gg 1$ is due to the change in
the luminosity throughout the shell ($F$) and that at the dominant order the
value of $\nabla_{s}$ is

$\nabla_{s}\simeq\frac{\nabla_{+}}{F}=\left[\frac{1/2+L_{c}/L_{s}}{1+L_{c}/L_{s}}\right]\,\nabla_{+}$
(16)

We see that the temperature gradient is changed by a relative large factor
${F}^{-1}$ in the small region of the burning shell. In particular, for an
inert core ($F=2$, $L_{c}\ll L_{s}$) and the value of $\nabla_{s}$ becomes
half its value at the upper boundary. Like $\nabla_{+}$, $\nabla_{s}$ is also
tightly constrained by the outer envelope.

If we now estimate $\nabla_{-}$, we see that the temperature gradient at the
bottom of the burning shell is

$\nabla_{-}\simeq\left[\frac{L_{c}}{L_{s}+L_{c}}\right]\nabla_{+},$ (17)

where it becomes clear that in the case of the inert core $L_{c}\ll L_{s}$ the
material becomes isothermal at the lower boundary of the burning shell, as we
know from numerical models of low-mass stars (see Appendix A).

Knowing the value of $\nabla_{s}$ we can make some useful estimations of how
pressure and density change across the burning shell. Using that at the
burning shell $dP/P\simeq(\nabla_{s})^{-1}dT/T$, and using the equation of
state of an ideal gas we get

$\displaystyle P_{+}\simeq$ $\displaystyle
P_{-}\exp\left[\frac{-2F}{\nu\nabla_{+}}\right],$ (18)
$\displaystyle\rho_{+}\simeq$ $\displaystyle\rho_{-}\frac{\mu_{\rm
env}}{\mu_{\rm
core}}\exp\left[\frac{-2}{\nu}\left(\frac{F}{\nabla_{+}}-1\right)\right],$

where $\mu_{\rm env}$ and $\mu_{\rm core}$ are the mean molecular weights of
the envelope and core, respectively.

###  2.3 The upper mantle and the drop in $P$, $\rho$ and $T$ above the
burning shell

We now turn to analyze how the radiative region above the burning shell (from
now on, the upper mantle) imposes some tight constraints on the
thermodynamical quantities at the upper boundary of the burning shell, and
consequently on the burning shell itself.

Let us first note that due to hydrostatic equilibrium the drop in pressure and
temperature immediately above the burning shell is

$\left.\frac{dP}{P}\right|_{+}=\frac{\bar{P_{c}}}{P_{+}}\frac{dm}{M_{c}},\ \ \
\left.\frac{dT}{T}\right|_{+}=\frac{\bar{P_{c}}}{P_{+}}\nabla_{+}\frac{dm}{M_{c}}.$
(19)

This implies that, as the core becomes more compact and the pressure contrast
between the upper boundary of the shell and the core increases by orders of
magnitude ($\bar{P_{c}}\gg P_{+}$), the drop in pressure and temperature above
the shell will become large even in a region of almost negligible mass
($\Delta m/M_{c}=(m-M_{c})/M_{c}\ll 1$). In the following we will show that,
when this happens, very tight constraints on the burning shell can be derived.
Then we will show that these constraints are approximately valid even at the
very early stages of shell burning (i.e. immediately after the end of the main
sequence).

One of the key results from Section 2.1 is the fact that $\nabla$ is strictly
constrained above the burning shell to values $(a+1)/(4-b)<\nabla<0.4$ which
in practice prevents $\nabla_{+}$ from varying more than a factor two in that
region. If we now restrict ourselves to a region of negligible mass above the
burning shell ($\Delta m/M_{c}\ll 1$) we can approximate

$\frac{dT}{dr}\simeq-\frac{-\mu_{\rm
env}GM_{c}\bar{\nabla}}{\Re}\frac{1}{r^{2}},$ (20)

where $\bar{\nabla}$ is typical mean value of $\nabla$ that fulfills
$(a+1)/(4-b)<\bar{\nabla}\leq 0.4$. Integrating this expression downward from
a point ($r=R_{0}$) where our approximations are still valid we can obtain

$T\simeq\frac{G\,M_{c}\mu_{\rm
env}}{r\,\Re}\bar{\nabla}\frac{(1-r/R_{0})}{(1-T_{0}/T)}$ (21)

We see that in those cases where $T$, $\rho$ and $P$ drop significantly in a
massless region above the burning shell, this is when the core becomes compact
in the sense that $\bar{P_{c}}\gg P_{+}$, then we can move outwards to a point
$R_{o}\gg r$ and $T_{0}\ll T$ and the factors on the right-hand side of eq. 21
become close to unity. We will see below, however, that even under not very
extreme conditions, the factors on the right are of order one.

In particular, eq. 21 sets a tight relation for the temperature immediately
above the burning shell,

$T_{+}\simeq\frac{G\,M_{c}\mu_{\rm
env}}{R_{s}\,\Re}\bar{\nabla}\frac{(1-R_{s}/R_{0})}{(1-T_{0}/T_{+})}$ (22)

Again, here the adimensional factor $(1-R_{s}/R_{\rm 0})$ will approach unity
fast, as the density contrast between the shell and the core increases, as
when $\rho_{+}\ll\rho_{c}$ it is possible to integrate outward to large values
of $R_{0}$ keeping the approximation $m\simeq M_{c}$. It will be shown later
that even in the early stages of shell burning these adimensional factors are
all close to unity. It is then useful to write all the factors close to one
as999It is worth noting that, while eq. 23 has similarities with equation 10.2
in Faulkner (2005) (when $\tau=0$) they rely on very different justifications.
While the assumption of $\tau\simeq 0$ in Faulkner (2005) is based on the
analysis of polytropic envelope integrations in the U-V plane with no
discussion about the role of convection, eq. 23 is based on the analysis done
in Section 2.1. One of the key ingredients in the derivation of eq. 23 is the
fact that the development of convection together with outer boundary
conditions strongly constrain the value of $\nabla$. This highlights the role
of convection in the formation of bright red giants and explains why the early
models that neglected the existence of convection (Sandage & Schwarzschild,
1952) did not form luminous red giants.

$T_{+}\simeq\frac{G\,M_{c}\mu_{\rm env}}{R_{s}\,\Re}\bar{\nabla}\zeta,$ (23)

where $\zeta$ will be very close to one101010It must be emphasized that the
argument presented in the next sections does not require $\zeta=1$ but only
that $\zeta$ is of order one, even if within a factor of a few. The important
feature is that $\bar{\nabla}\zeta$ is tightly constrained, and eq. 23 sets an
effective constraint between $T_{+}$, $M_{c}$ and $R_{s}$. In fact the case of
$\zeta=1$ corresponds to the case when boundary terms can be neglected and
$\rho$, $T$ and $P$ can be assumed to have a simple power-law dependence on
$r$, leading to a polytropic relation among them (see Chapter §19.2 in
MacDonald, 2015). and $(a+1)/(4-b)<\bar{\nabla}\leq 0.4$. We will see below
that when the density and pressure drop by orders of magnitude in a massless
region above the burning shell, then $\bar{\nabla}\simeq(a+1)/(4-b)$.

It is useful to note that, when the factors on the right in eq. 21 can be
neglected we also have that

$\frac{d\rho}{dr}\simeq-\frac{\rho}{r}(1-\bar{\nabla})\frac{(1-T_{0}/T)}{(1-r/R_{0})}$
(24)

To have a more quantitative idea of the implications in eqs. 21, 23, and 24 we
can assume that energy is transported by radiation in the region immediately
above the burning shell. Note that if energy is transported by convection the
situation is straightforward as then $\nabla$ is completely constant and
$\nabla=0.4$. Again, if we restrict to a region where $\Delta m/M_{c}\ll 1$,
we can replace eq. 1 into eq. 3 and obtain

$dT=\frac{3L_{\star}}{16\pi\,a\,c\,G}\frac{\kappa}{M_{c}\,T^{3}}\,dP.$ (25)

Writing the opacity dependence close to the burning shell as
$\kappa=\kappa_{0}P^{a}T^{b}$ ($a\geq 0$, $b\leq 0$ under normal conditions),
eq. 25 can be integrated in the standard way (Kippenhahn et al., 2012) from a
point ($r=R_{0}$) where our approximation is still valid down to the burning
shell

$(T_{\rm 0}^{4-b}-T^{4-b})=\frac{4-b}{a+1}\frac{3\kappa_{0}\,L_{\star}}{16\pi
acGM_{c}}(P_{\rm 0}^{a+1}-P^{a+1}).$ (26)

We can see now that the value of $\nabla$ inside the upper mantle is

$\nabla=\frac{(4-b)}{(a+1)}\frac{[1-(T_{\rm 0}/T)^{4-b}]}{[1-(P_{\rm
0}/P)^{a+1}]}.$ (27)

Under these conditions, the temperature and density at the upper boundary of
the burning shell must fulfill

$T_{+}^{4-b}=\frac{\Pi}{\Theta}\frac{4-b}{a+1}\frac{3\kappa_{0}\,L_{\star}}{16\pi
acGM_{c}}P_{+}^{a+1},$ (28)

with $\Theta=(1-(T_{\rm 0}/T_{+})^{4-b})$ and $\Pi=(1-(P_{\rm
0}/P_{+})^{a+1})$. As before, as soon as $P_{c}\gg P_{s}$ pressure and
temperature drop quickly above the burning shell and $\Theta/\Pi\simeq 1$. The
value of $\nabla$ at the upper boundary of the burning shell is then

$\nabla_{+}=\left.\frac{d\ln T}{d\ln
P}\right|_{+}=\frac{a+1}{4-b}\frac{\Theta}{\Pi}.$ (29)

Then, once $\Theta\simeq 1$ and $\Pi\simeq 1$ we see that typical values of
$\bar{\nabla}\simeq\nabla_{+}\simeq(a+1)/(4-b)$ are 4/17 for a Kramers’
opacity and 1/4 for Compton scattering.

Interestingly, the envelope integrations discussed in Section 2.1 show that
this also applies to massive radiative envelopes for which we cannot
approximate $m\simeq M_{c}$ through all the integration. This is because
$\nabla$, due to eq. 10, needs to stay close to $\nabla^{\rm lim}=(a+1)/(4-b)$
to avoid a runaway increase in $\nabla$, which would turn the envelope
convective, but also larger than this critical value to avoid a runaway drop
in $\nabla$, which would make it impossible to satisfy photospheric boundary
conditions.

As discussed above, when $\rho_{+}\ll\rho_{c}$ and $P_{+}\ll P_{c}$ we can
strictly demonstrate that $\Theta\simeq 1$, $\Pi\simeq 1$, $\zeta\simeq 1$,
$T_{0}\ll T_{+}$ and $R_{s}\ll R_{0}$ and all previous expressions are very
simple. In Appendix E we show that these approximations are still acceptable
in the early stages of the burning shell, when the pressure and density
contrast are not that large. Readers not inclined to these rough estimations
can skip this Appendix and wait for Section 3.2, where the predictions of
these approximation will be compared, and validated, against full stellar
evolution models.

###  2.4 Energy release at the burning shell

A second relation between temperature gradient, temperature, density, and
shell luminosity can be derived from the fact that $L_{s}$ is the consequence
of the energy generated in the shell itself. From the energy conservation
equation

$\frac{dl}{dm}=\epsilon,$ (30)

we can estimate that $L_{s}\simeq\langle\epsilon\rangle\Delta
m=\langle\epsilon\rangle\rho_{s}4\pi{R_{c}}^{2}\Delta r$. Noting that almost
all energy is released in a region where $\delta T/T\simeq 2/\nu$ ($T_{s}/\nu$
corresponds to half a burning shell), we can use that $\Delta
r\simeq-(dT/dr)^{-1}\,2\,T_{s}/\nu$

$L_{s}=\langle\epsilon\rangle
4\pi{R_{c}}^{2}\rho_{s}\frac{-T_{s}\,2}{\nu}\left(\frac{dT}{dr}\right)^{-1}.$
(31)

Given that energy transport at the burning shell happens through radiation, we
can estimate that the temperature gradient at the peak of the H-burning shell,
i.e. in the middle of the burning shell where $l_{\rm mid-
shell}=L_{c}+L_{s}/2=(L_{s}/2)(1+2\,L_{c}/L_{s})$, also fulfills

$\left.\frac{dT}{dr}\right|_{\rm shell}\approx\frac{-3}{16\pi
ac}\frac{\kappa_{s}\rho_{s}}{{R_{c}}^{2}{T_{s}}^{3}}(L_{s}/2+L_{c}).$ (32)

Replacing $L_{s}$ from eq. 32 into eq. 31, and noting that the mean energy
release is about half its value at the peak111111This is exact for a linear
luminosity increase in the shell.,
$\langle\epsilon\rangle\approx\epsilon(\rho_{s},T_{s})/2\approx\epsilon_{0}/2{\rho_{s}}^{t}{T_{s}}^{\nu}$,
and replacing that in eq. 3 we have

$\displaystyle\left(\left.\frac{dT}{dr}\right|_{\rm mid-
shell}\right)^{2}\simeq$
$\displaystyle\frac{3}{8ac\nu}\left(\frac{\rho_{s}}{T_{s}}\right)^{2}$ (33)
$\displaystyle\kappa(\rho_{s},T_{s})\epsilon(\rho_{s},T_{s})\left(1+\frac{2L_{c}}{L_{s}}\right)$

or

$\displaystyle\left(\left.\frac{dT}{dr}\right|_{\rm mid-
shell}\right)^{2}\simeq$
$\displaystyle\frac{3}{8ac}\frac{\kappa_{0}\epsilon_{0}\Re^{a}}{\mu^{a}}$ (34)
$\displaystyle\frac{{\rho_{s}}^{2+s+t}\,{T_{s}}^{a+b+\nu-2}}{\nu}\left(1+\frac{2L_{c}}{L_{s}}\right).$

###  2.5 Constraints set by the burning shell on the thermodynamic variables

Eqs. 33 and 34 tell us how the temperature gradient has to be, for the total
energy of the shell to be released in a region where temperature drops by
$\sim 2T_{s}/\nu$. In addition to eqs. 33 and 34, due to the radiative
transport and hydrostatic equilibrium equations we know that at location
$r=R_{s}$ the temperature gradient must be such that it transports the local
luminosity of the star at that point. This is

$\displaystyle\left.\frac{dT}{dr}\right|_{\rm mid-shell}=$
$\displaystyle\nabla_{s}\frac{T_{s}}{P_{s}}\left(\left.\frac{dP}{dT}\right|_{\rm
mid-shell}\right)$ (35) $\displaystyle=$
$\displaystyle\frac{\nabla_{+}}{F}\times\left(\frac{-GM_{c}\mu_{s}}{{R_{s}}^{2}\Re}\right)$

Equating the gradients $dT/dt({\rm mid-shell})$ in eqs. 33 and 35 we find that
the existence of the burning shell forces

$\displaystyle\left(\frac{-GM_{c}\mu_{s}}{{R_{s}}^{2}\Re}\right)^{2}\frac{\left(1+\frac{2L_{c}}{L_{s}}\right)}{\left(1+\frac{L_{c}}{L_{s}}\right)^{2}}\times(\nabla_{+})^{2}=$
(36)
$\displaystyle\frac{3}{2ac\nu}\left(\frac{\rho_{s}}{T_{s}}\right)^{2}\kappa(\rho_{s},T_{s})\epsilon(\rho_{s},T_{s})$

where we have replaced the factor $1/F$ by its dependence on the luminosity to
emphasize the behavior of the equation in the case of luminous shells
$L_{c}/L_{s}\ll 1$. Eq. 36 is the key equation to understand the behavior of
the stellar structure in the presence of a burning shell121212Eq. 36 is
closely related to equation 10.5 in Faulkner (2005). However, here the weird
dependence on the cube of the radius of the burning shell has been replaced by
the more reasonable ${R_{s}}^{2}$, and the weird appearance of the radiation
density constant has been removed. In addition, the explicit appearance of
$\nabla_{+}$ highlights the role of the constraints imposed by the envelope. .
We have seen that $(a+1)/(4-b)<\nabla_{+}<\nabla_{\rm ad}$ and, as soon as
their core becomes dense $\nabla_{+}\simeq(a+1)/(4-b)$ (eq. 29). The first
thing to note is that in the case of a dim shell $L_{s}\ll L_{c}$ this
equation becomes meaningless (in fact) as it only implies that
$\epsilon_{s}\simeq 0$, which is a trivial result as nothing should happen in
that situation. As soon as the energy generated in the burning shell becomes
relevant ($L_{s}\sim L_{c}$) eq. 36 starts to put constraints on the values of
$T_{s}$ and $\rho_{s}$ as a function of the mass and radius of the core
($M_{C}$, $R_{s}$). We see that the key consequence of the development of a
burning shell is the emergence of an additional constraint (eq. 36), between
$\rho_{s}$, $T_{s}$ and $R_{s}$ and $M_{c}$.

To fully appreciate the power of eq. 36 we need to complement it with eq. 23,
a constraint that is always present in stellar envelopes. Noting that
$T_{s}=T_{+}(1+1/\nu)$ we have

$T_{s}\simeq\frac{G\,M_{c}\mu_{\rm env}}{R_{s}\,\Re}(1+1/\nu)\nabla_{+}.$ (37)

where we have approximated $T_{0}\ll T_{s}$ and $R_{s}\ll R_{0}$, which
according to our previous discussion are good approximations under rather
general situations. This equation implies an additional constraint on $T_{s}$,
and together with eq. 36 it implies that the values of $T_{s}$, $\rho_{s}$ and
$P_{s}$ are completely defined by the values of $M_{c}$ and $R_{s}$. In fact
eqs. 36, and 37 demonstrate the key hypothesis in the derivations of “shell
homology relations” (see Chapter §33.2 in Kippenhahn et al., 2012), i.e. that
$T_{s}(M_{c},R_{s})$, $\rho_{s}(M_{c},R_{s})$, and $P_{s}(M_{c},R_{s})$.

An alternative version of eq. 36 can be obtained by using eq. 37 on the left-
hand side of eq. 36 to obtain

$\displaystyle\left(\frac{\mu_{s}}{\mu_{e}}\right)^{2}\frac{\left(1+\frac{2L_{c}}{L_{s}}\right)}{\left(1+\frac{L_{c}}{L_{s}}\right)^{2}}(1+1/\nu)^{2}=$
(38)
$\displaystyle\frac{3}{2ac\nu}\left(\frac{{\rho_{s}}^{2}{R_{s}}^{2}}{{T_{s}}^{4}}\right)\kappa(\rho_{s},T_{s})\epsilon(\rho_{s},T_{s}).$

To good approximation we have

$\left(\frac{\mu_{s}}{\mu_{e}}\right)^{2}\simeq\frac{3}{2ac\nu}\left(\frac{{\rho_{s}}^{2}{R_{s}}^{2}}{{T_{s}}^{4}}\right)\kappa(\rho_{s},T_{s})\epsilon(\rho_{s},T_{s}).$
(39)

In the next section we will test and validate all the expressions derived here
by building a toy model of a low-mass red giant and comparing it to the
predictions of detailed full stellar models. Before this, let us summarize
what we have found so far. In Section 2.1 we have shown that the existence of
an envelope of nonnegligible mass, together with the photospheric boundary
conditions, imply that the temperature above the burning shell must follow
$(a+1)/(4-b)\lesssim\nabla_{+}\lesssim 0.4$, where $(a+1)/(4-b)\simeq
0.23\textendash 0.25$. Then, on Section 2.2 we have seen that the high
temperature sensitivity of nuclear reactions and hydrostatic equilibrium can
be used to show that the temperature gradient inside the burning shell is
$\nabla_{s}\simeq\nabla_{+}/F$, where $F\simeq 2$ when the shell is much more
luminous than the core ($L_{s}\gg L_{c}$). As shown in Section 2.3 this
implies, among other things, that the temperature at the shell follows
$T_{s}\propto M_{c}/R_{s}$. Finally, the fact that the luminosity generated at
the burning shell must also be transported through the burning shell leads to
a tight constraint of $\rho_{s}$ and $T_{s}$ as a function of $M_{c}$ and
$R_{s}$. Together these last two results demonstrate the validity of the key
hypothesis of the so-called shell homology relations (see Chapter §33.2 in
Kippenhahn et al., 2012), i.e. that we can write $T_{s}(M_{c},R_{s})$,
$\rho_{s}(M_{c},R_{s})$, and $P_{s}(M_{c},R_{s})$.

##  3 The validation: A toy model for a low-mass star

Figure 3: Main structural parts of a low-mass red giant.

To better understand the equations derived in Section 2 and to check their
accuracy, we will construct here a toy model of a low-mass red giant (Fig. 3).
Fortunately, the degenerate core of low-mass red giants can be approximated by
a self-gravitating sphere sustained by the pressure of cold nonrelativistic
electrons.

A cold core supported by fully degenerate nonrelativistic electrons has a very
tight mass-radius relationship that corresponds to an $n=3/2$ polytrope. In
cgs units, this relationship is

$R_{\rm dc}=1.12\times 10^{20}{M_{\rm dc}}^{-1/3}.$ (40)

This tight constraint indicates that the geometrical size of the core of low-
mass stars only evolves due to the increase in their mass. This increase in
$M_{c}$ is provided by the steady burning of H at the burning shell.
Consequently the radius of the degenerate core will evolve on a nuclear
timescale. Then, due to the degeneracy of the He core, gravothermal energy
release is to a first approximation negligible in comparison with the
intensity of the H-burning shell; i.e. mathematically we can say that
$L_{s}\gg L_{c}$. This fact, together with the high thermal conductivity
provided by degenerate electrons will allow us to approximate the degenerate
core as an isothermal, thermally relaxed ($\int_{0}^{M_{c}}\epsilon_{g}dm=0$)
structure with $L_{c}\ll L_{s}$.

One feature, usually neglected when discussing low-mass giants, and which is
key to understand the $M_{c}(R_{s})$ relation of low-mass red giants, is the
existence of a very light mantle of nondegenerate ideal gas between the
burning shell and the degenerate core. The existence of such an inner mantle
can be easily motivated as follows. Due to the presence of the H-burning shell
the stellar plasma at the shell cannot be degenerate131313Otherwise a flash
would develop, injecting huge amounts of energy in that layer, and lifting
degeneracy.. Then, as the material on the burning shell is not degenerate, and
$T$, $\rho$, and $P$ are continuous, there must exist regions immediately
below the burning shell where the material behaves as a classical ideal gas
($P=\Re\,\rho\,T/\mu$). We will show that it plays a key role in the
$M_{c}(R_{s})$ relation of low-mass red giants even when its mass ($\Delta m$)
is negligible (in the sense that $\Delta m\ll M_{c}$).

In low-mass stars after the end of core H burning the contraction of the tiny
He core left is stopped by degeneracy, and the temperatures around the
H-exhausted core reach $T_{s}\sim 10^{7}$K. Note that this happens before the
development of a giant structure. From this point onward, the star can be
described as composed of an almost pure inert He core, initially of less that
a tenth of a solar mass and increasing with time, and a H-rich envelope that
harbors the rest of the mass of the star, both separated by the presence of a
H-burning shell. In addition the He core harbors a He mantle where the
material behaves as a classical ideal gas. Such a configuration is depicted in
Fig. 3.

###  3.1 The inner mantle

As evolution proceeds in a nuclear timescale we can assume at first order that
$L_{c}=0$. The temperature profile of the inner mantle is then given by eq.
17, which for $L_{c}=0$ corresponds to $\nabla_{-}=0$. We see that the
presence of the H-burning shell forces the existence of an isothermal mantle
with $T\simeq T_{-}=T_{s}(1+1/\nu)$ in between the H-burning shell and the
degenerate core141414This mantle is clearly seen in detailed stellar models;
see Appendix A.. This mantle is in hydrostatic equilibrium and has to fulfill
eq. 1. As long as the density does not increase significantly the isothermal
mantle can be considered as an ideal classical gas. The mean molecular weight
of the inner mantle corresponds to that of pure and fully ionized He
($\mu_{c}\simeq 4/3$). In this region the density must follow

$\frac{d\rho}{dr}=-\frac{Gm\mu_{c}}{\Re\,r^{2}T_{-}}\rho,$ (41)

and if we restrict ourselves to the outermost regions of the core, where
$\Delta m(r)\ll M_{c}$, the equation becomes

$\frac{d\rho}{\rho}=-\frac{GM_{c}\mu_{c}}{\Re\,T_{-}}\frac{dr}{r^{2}}.$ (42)

This equation can be integrated inward from the burning shell $r=R_{s}$, and
will be a good description of the structure as long as the electrons remain
nondegenerate and the mass of the region is negligible in comparison to that
of the core ($\Delta m(r)\ll M_{c}$). As $\rho_{s}$ is, in most cases, much
lower than the mean density of the core ($\rho_{s}\ll\bar{\rho_{c}}$), then
eq. 42 can be integrated quite far from the burning shell ($|r-R_{s}|\lesssim
R_{s}$) and still fulfill $\Delta m(r)\ll M_{c}$. Integrating the density
inward we find that in the (almost) massless mantle below the burning shell,
the density follows

$\rho(r)=\rho_{-}e^{\frac{G\mu_{c}M_{c}}{\Re
T_{-}}\left(\frac{R_{s}-r}{R_{s}\,r}\right)}.$ (43)

Eq. 43 is valid as long as $\Delta m(r)\ll M_{c}$, and when
$\rho_{s}\ll\bar{\rho_{c}}$ this equation will be valid far from the burning
shell $|r-R_{s}|\lesssim R_{s}$.

A cold core supported by fully degenerate nonrelativistic electrons is
described by eq. 40. However, due to eq. 43 the density drops exponentially
between the degenerate core and the burning shell. The point at which the core
stops to behave as a degenerate electron gas corresponds to the point at which
the pressure from the nondegenerate ion gas starts to be more important than
that of the degenerate electrons. The density at the boundary of the
degenerate core ($\rho_{\rm Bdc}$) can be estimated by setting
$P_{e^{-}}=P_{\rm ions}$,151515Note that it is reasonable to assume that
electrons are degenerate in this estimation as the density at which electrons
stop to behave as a degenerate gas corresponds to $\epsilon_{\rm Fermi}\approx
kT$, which corresponds to  $\rho(\epsilon_{\rm Fermi}\approx kT)\approx
1.2\times 10^{-8}\,T^{3/2}$ which is lower than the density in eq. 44.

$\rho_{\rm Bdc}\approx 1.7\times 10^{-8}\,{T_{s}}^{3/2}[cgs].$ (44)

Where we have used that in the absence of neutrino cooling the degenerate core
behaves isothermally due to the high heat transport efficiency of degenerate
electrons. Note that for typical H-burning temperatures the density at the
boundary of the degenerate core $\rho_{\rm Bdc}\ll\langle\rho_{\rm
dc}\rangle=3M_{\rm dc}/(4\pi{R_{\rm dc}}^{3})$, which means that $\rho_{\rm
Bdc}$ is almost zero in comparison with $\langle\rho_{\rm dc}\rangle$ and its
location can be approximated by radius of the $n=3/2$ polytrope (eq. 40). The
condition $\rho_{\rm Bdc}\ll\langle\rho_{\rm dc}\rangle$ also implies that eq.
43 is valid for values of $|r-R_{s}|\gtrsim R_{s}$, and the inner mantle can
be treated as massless; $M_{\rm dc}\simeq M_{c}$. In particular, if we now
integrate the inner mantle (eq. 43) down to the location of the upper boundary
of the degenerate core, we get

$1.7\times 10^{-8}\,{T_{s}}^{3/2}\approx\rho_{\rm
Bdc}\approx\rho_{-}e^{\frac{G\mu_{c}M_{c}}{\Re T_{s}}\left(\frac{R_{s}-R_{\rm
dc}}{R_{s}\,R_{\rm dc}}\right)}.$ (45)

Eq. 45 gives a new relation between $\rho_{s}$, $T_{s}$ and $M_{c}$ and
$R_{s}$. This relation must be fulfilled together with those coming from eqs.
36, and 37. Note in passing that each of these equations represents a relation
imposed by the three distinctive regions of our star: the core (eq. 45), the
burning shell (eq. 36), and the envelope (eqs. 29 and 37).

###  3.2 A complete toy

Before continuing with our analysis we will write eq. 45 in a more favorable
way. Replacing eq. 37 into equation 45, and using
$\nabla_{+}\simeq(a+1)/(4-b)$ we can write

$\displaystyle\rho_{s}\simeq$ $\displaystyle 5.33\times
10^{5}\frac{\mu_{s}}{\mu_{c}}\exp\left(-\frac{1}{\nu}\frac{(7-2b-a)}{(a+1)}\right)\left(1+1/\nu\right)^{3/2}$
(46)
$\displaystyle{T_{9}}^{3/2}\,\exp\left(\frac{(4-b)}{(a+1)}\frac{1}{(1+2/\nu)}\frac{\mu_{c}}{\mu_{\rm
env}}\left(1-\frac{R_{\rm dc}}{R_{s}}\right)\right).$

At the densities and temperatures typical of the burning shell and the upper
mantle, the opacity is dominated by electron scattering161616While the core is
below $M_{c}\lesssim 0.23M_{\odot}$ and the shell temperature is close to
$\simeq 3\times 10^{7}$K, Thomson scattering is not the only relevant source
of opacity, and other sources, mostly bound-free opacities are also important.
It is only after $M_{c}\simeq 0.25M_{\odot}$ that Thomson scattering becomes
dominant at the shell temperature. In Appendix B we show that qualitative
behavior is unaltered when a Kramers’ bound-free opacity law is adopted., and
can be approximated as

$\kappa_{e}=0.2(1+X)\,\,\hbox{\rm cm${}^{2}$/g}.$ (47)

At the burning shell (hydrogen mass fraction $X\simeq 0.35$) we have
$\kappa_{s}\simeq 0.27$ cm2/g. The energy generation due to CNO burning in
that temperature range, $T_{s}/10^{9}=T_{9}\in(0.001,0.1)$, is well
approximated by

$\epsilon_{\rm CNO}\simeq 8.24\times 10^{25}X_{\rm
CNO}X\rho_{s}{T_{9}}^{-2/3}\exp\left({-15.231T^{-1/3}}\right).$ (48)

Where for a typical composition we have $X_{\rm CNO}\simeq 0.01$. The choice
of Thomson scattering and CNO burning corresponds to $a=b=0$ and $t=1$ in the
expressions of $\epsilon$ and $\kappa$, as discussed in Section 2. The
temperature dependence $\nu$ of the nuclear generation rate is,

$\nu=\frac{d\ln\epsilon_{\rm CNO}}{d\ln
T_{9}}\simeq\left(-\frac{2}{3}+5.077\,{T_{9}}^{-1/3}\right).$ (49)

This expression gives $\nu\simeq 23$, 13 and 10 for $T_{9}\simeq 0.01$, 0.05,
and 0.1 respectively, as it should.

In these conditions, eq. 39 becomes

$\left(\frac{\mu_{s}}{\mu_{\rm env}}\right)^{2}\simeq 5.099\times
10^{-10}\frac{{\rho_{s}}^{3}{R_{s}}^{2}{T_{9}}^{-14/3}}{\nu}\exp\left({-15.231\,{T_{9}}^{-1/3}}\right).$
(50)

Replacing eq. 46 into eq. 50 we have, for the case of Thomson scattering

$\displaystyle 1.2953\times 10^{-8}$
$\displaystyle\frac{\mu_{c}^{3}}{\mu_{s}\mu_{\rm
env}^{2}}\nu\exp(21/\nu)\left(1+1/\nu\right)^{-9/2}$ (51) $\displaystyle=$
$\displaystyle{T_{9}}^{-1/6}{R_{s}}^{2}\exp\left[\frac{12}{(1+2/\nu)}\frac{\mu_{c}}{\mu_{\rm
env}}\left(1-\frac{R_{s}}{R_{\rm dc}}\right)\right.$ $\displaystyle-$
$\displaystyle\left.15.231{T_{9}}^{-1/3}\right]$

Now we can rewrite eq. 51 and eq. 37 in a more convenient way by properly
normalizing $R_{s}$, leaving an explicit dependence on the mass of the core.
These equations result in

$\frac{R_{s}}{R_{\rm dc}}\simeq\frac{\mu_{\rm
env}}{\mu_{c}}\frac{0.6}{T_{9}}\left(\frac{M_{\rm
dc}}{M_{\odot}}\right)^{4/3},$ (52)

and

$\displaystyle 1.633\times 10^{-26}$
$\displaystyle\frac{\mu_{c}^{3}}{\mu_{s}\mu_{\rm
env}^{2}}\nu\exp(21/\nu)\left(1+1/\nu\right)^{-9/2}$ (53) $\displaystyle=$
$\displaystyle{T_{9}}^{-1/6}\left[\frac{R_{s}}{R_{\rm
dc}}\right]^{2}\left[\frac{M_{c}}{M_{\odot}}\right]^{-2/3}$
$\displaystyle\times$
$\displaystyle\exp\left[\frac{12}{(1+2/\nu)}\frac{\mu_{c}}{\mu_{\rm
env}}\left(1-\frac{R_{\rm s}}{R_{\rm dc}}\right)\right.$ $\displaystyle-$
$\displaystyle\left.15.231{T_{9}}^{-1/3}\right]$

Typical values for the mean molecular weight are $\mu_{c}/\mu_{s}\simeq 1.58$,
$\mu_{c}/\mu_{\rm env}\simeq 2.167$. The actual value of $T_{9}(M_{c})$ and
$R_{s}(M_{c})$ is not strongly affected by factors of order 1 on the left-hand
side of eq. 53 and is dominated by the exponents in $T_{9}$, $R_{s}$ and the
exponential function. In particular, we see that the exponential contains a
factor $\mu_{c}/\mu_{\rm env}$ which explains the well-known fact that the
temperature of the burning shell, and consequently the luminosity of the
shell, is affected by the molecular weight contrast between the envelope and
the core. This is the reason why some previous works have found that molecular
weight contrast between the envelope and the core helps the development of the
giant structure (Stancliffe et al., 2009). In addition, these expressions show
us that estimating the impact of the molecular weight on the luminosity of the
shell by means of shell homology relations where one considers the changes in
$\mu$ and $R_{c}$ as independent is not completely right (see Chapter §33.2 in
Kippenhahn et al., 2012). In a full stellar model altering the molecular
weight of the envelope will have an impact on the radius of the burning shell,
which also alters the temperature of the shell.

By replacing $T_{9}(R_{s})$ or $R_{s}(T_{9})$ from eq. 52 into eq. 53 we
obtain the solutions for $T_{9}(M_{c})$ and $R_{s}(M_{c})$. These solutions
are shown in Fig. 4, and can be very well approximated by

Figure 4: Values of $R_{s}$ and $T_{9}$ for different core masses that fulfill
eqs.52 and eq. 53 and the corresponding equations for Kramers’ bound-free
opacities; see Appendix B.

$\displaystyle T_{9}$ $\displaystyle=$ $\displaystyle
0.0071+0.12\left(\frac{M_{c}}{M_{\odot}}\right)^{1.13}$
$\displaystyle\frac{R_{s}}{R_{\rm dc}}$ $\displaystyle=$ $\displaystyle
2.181+0.589\,\ln\left(\frac{M_{c}}{M_{\odot}}-0.008\right).$ (54)

Similar expressions for a Kramers’ opacity are derived in Appendix B. Fig. 4
shows the predicted evolution of the temperature and radius of the H-burning
shell as the mass of the core increases as a consequence of nuclear burning in
our toy model. The resulting density of the burning shell is shown in Fig. 5

Figure 5: Dependence of the density of the shell as a function of the mass of
the core compared with other key densities of our toy model. Thick lines
indicate the relations obtained for Thomson scattering, while thin lines
depict the same relations for a Kramers’ bound-free opacity.

We see that the burning shell settles near the degenerate core but moves away
from it as the shell becomes hotter. As a consequence the burning shell
settles at an almost constant value of $R_{s}\simeq 0.026$ (middle and lower
panels in Fig. 4). The fact that our toy model is able to reproduce the radius
of the burning shell predicted by the full stellar evolution model to less
than 10% is a remarkable confirmation of the validity of our assumptions. We
now see that the constancy in the radius of the burning shell is the
consequence of the existence of an ideal gas mantle on top of a degenerate
core. As the mass of the core increases, it forces a drop in the density of
the burning shell. As the degenerate part of the helium core contracts
(following $R_{\rm dc}\propto M_{c}^{-1/3}$) the density of the isothermal
mantle between the degenerate core and the burning shell drops (so that
density in this region can go from $\rho_{\rm Bdc}$ to $\rho_{s}$). This
stretches the isothermal mantle of ideal gas, counterbalancing the mild
contraction of the degenerate part of the core, and leading to an almost
constant radius of the burning shell.

In spite of all simplifications in the treatment of the core and the fact that
the envelope is only required to be massive enough and fulfill outer boundary
conditions, the model gives a very good quantitative agreement with the
temperature and radius of the shell derived from the detailed computation of
the full set of stellar evolution equations by means of a state-of-the-art
stellar evolution code (LPCODE, Miller Bertolami, 2016). This is particularly
true once the thin shell approximation adopted throughout this paper becomes
good and the density contrast between the shell and the core is high, but the
agreement is still good even at the beginning of the shell burning stage as
$R_{s}$ in the simple model deviates by less than a factor two from the real
value of $R_{s}$ (see Fig. 4). This level of agreement is an astonishing
confirmation of the accuracy of eqs. 29, 37 and 39 and the approximations
$\Theta\simeq 1$, $\Pi\simeq 1$, $\zeta\simeq 1$ and $R_{s}\ll R_{0}$ on which
it is based. As we mentioned at the beginning of this article, toy models are
not made to be precise but to be an accurate description of the main
mechanisms of a given process, so we find this level of precision remarkable.

Besides the confirmation of our key equations, our toy model offers some very
interesting insight on the structure of low-mass red giants. The model shows
why the existence of a mantle of negligible mass above the degenerate core
increases its radius, so that, as the core increases its mass and the radius
of the degenerate regions decreases as ${M_{c}}^{-1/3}$, the radius of the
burning shell stays at a constant value of $R_{s}\simeq
0.027...0.03R_{\odot}$. In addition, the model helps to understand why all
low-mass stars develop a He flash when the core mass is about half a solar
mass irrespective of initial mass, metallicity, or mass of the envelope. We
see that, for a star with a completely isothermal and thermally relaxed core,
the H-burning shell (and consequently the core) reaches He-burning
temperatures when the mass of the core is $M_{c}\simeq 0.75M_{\odot}$,
independent of the metallicity and the mass of the envelope. This sets an
upper limit for the mass of the core at the moment of the He-core flash. The
fact that the He flash happens before the core reaches this limiting mass is
due to gravothermal heat release (see discussion in Appendix A). As the core
increases its mass and the H-burning shell heats up, the timescale of nuclear
reactions becomes shorter and comparable with the thermal (Kelvin-Helmholtz)
time scale of the the core. This additional heating of the core anticipates
the development of the He-flash. However, as this additional heating is a
function of the rate at which mass is added to the core ($\dot{M_{c}}$), and
$\dot{M_{c}}$ is solely determined by $T_{s}$, the mass of the core at which
the helium flash will happen will be almost constant and irrespective of
initial mass, metallicity, and mass of the envelope.

##  4 Dimensional analysis

Now that we have confirmed the accuracy of the equations derived in Section 2
we can turn to analyze their implications for the global structure of stars.

From eq. 37 we know that $T_{s}\propto M_{c}/R_{s}$, replacing now this into
eq. 36 we find that

$P_{s}=\mathcal{K}\,{R_{s}}^{(\nu-t-8+b)/(a+t+2)}{M_{c}}^{(6-\nu+t-b)/(a+t+2)},$
(55)

where $\mathcal{K}$ is a constant that can be computed from eq. 36. Due to
hydrostatic equilibrium we know that the mean pressure in the core is

$\bar{P_{c}}\approx G\,{M_{c}}^{2}/8\pi\,{R_{s}}^{4}.$ (56)

Eqs. 55 and 56 show that the action of the burning shell destroys the
possibility of homologous contraction for these stars as it gives

$\frac{P_{s}}{\bar{P_{c}}}=\mathcal{K^{\prime}}{R_{s}}^{\frac{\nu+3t+b+4a}{a+t+2}}{M_{c}}^{\frac{2-\nu-
t-b-2a}{a+t+2}}.$ (57)

In fact as $\nu$ is the dominant term in both exponents, eq. 58 tells us that,
as soon as $R_{s}$ contracts or $M_{c}$ increases by a small amount, pressure
at the burning shell drops by orders of magnitude. To quantify this point lets
focus in the most common situation in both low- and high-mass red giants,
Thomson scattering ($a=b=0$) with CNO burning ($t=1$ and $\nu\simeq 23...13$).
Under those conditions, eq. 58 gives

$\frac{P_{s}}{\bar{P_{c}}}=\mathcal{K^{\prime}}{R_{s}}^{1+\nu/3}{M_{c}}^{\frac{1-\nu}{3}}.$
(58)

For typical values of $\nu$, $(1+\nu/3)\simeq 8.5...5.3$, then a small
decrease in the radius of $\sim 10$% leads to a $\sim 50...80$% drop in the
ratio between the pressure at the shell and that at the core. This is the
situation in massive and intermediate-mass stars (IM&M), where the location of
the shell decreases by at least a factor of two, between the end of core H
burning and the beginning of He burning, while the mass of the core remains
constant, leading to a huge drop of more than two orders of magnitude in
$P_{s}/\bar{P_{c}}$. Note in passing that for IM&M stars $\dot{r}$ does not
change sign at the maximum of energy release, as stated by Kippenhahn et al.
(2012), but immediately above the burning shell in the region where the
density drops many orders of magnitude. On the contrary, in low-mass stars, as
we have seen in the previous section the interplay between the inner mantle
and the degenerate part of the core renders the position of the shell almost
constant (i.e. $\dot{r}=0$) while the mass of the core increases.

Using eq. 37 and the equation of state we can also obtain

$\rho_{s}=\mathcal{K^{\prime\prime}}\,{R_{s}}^{(\nu-6+b+a)/(a+t+2)}{M_{c}}^{(4-\nu-
a-b)/(a+t+2)}.$ (59)

A knowledgeable reader will recognize the exponents in eqs. 55, 58 and 59 as
those provided by shell homology relations. This should come as no surprise as
in Section 2.5 we mathematically demonstrated that to this level of
approximation the key hypothesis of shell homology relations is rigorously
valid (i.e. that $T_{s}(M_{c},R_{s})$, $\rho_{s}(M_{c},R_{s})$ and
$P_{s}(M_{c},R_{s})$). Consequently, any dimensional analysis based on our
relations should cast the same result. The main difference between eqs. 55 and
59 and those derived by shell homology relations is that here we are able to
compute (if required) the numerical values of the proportionality constants
$\mathcal{K}$,$\mathcal{K^{\prime}}$, and $\mathcal{K^{\prime\prime}}$.

##  5 A red giants’ toy story

We are now in condition to qualitatively understand the development of red
giant structures after the main sequence. The existence of an envelope of
nonnegligible mass forces the mean value of $\nabla\simeq 0.23\textendash 0.4$
above the burning shell, which leads to a tight dependence of the shell
temperature on the mass and radius of the core $T_{s}\propto M_{c}/R_{s}$ (eq.
23). Additionally, the switching-on of the burning shell and the fact that the
heat is both generated and transported at the burning shell leads to a tight
relationship between $\rho_{s}$, $T_{s}$, $M_{c}$ and $R_{s}$ (eq. 36).
Together, eqs. 23 and 36 imply that $\rho_{s}(M_{c},R_{s})$. For a given mass-
radius relation of the core, $M_{c}(R_{s})$ this implies that the shell
density will evolve as a function of $M_{c}$ or $R_{s}$. Below we will analyze
the typical mass-radius relations for low-mass stars and IM&M stars.
Interestingly we will see that these two cases correspond quite well with
evolutions at constant $R_{s}$ or $M_{c}$, respectively.

The situation is simpler in more massive stars, where the core is an
undifferentiated contracting sphere with the same equation of state, while it
is more complicated for low-mass stars where the core can be divided into a
degenerate core and a surrounding mantle that behaves as an ideal classical
gas. In the latter case we will make use of the simple $R_{s}(M_{c})$ relation
derived in Section 3.

###  5.1 Low-mass stars

As soon as H burning is exhausted in the core the layers around it start to
contract and heat up as a consequence of the negative gravothermal specific
heat of typical of nondegenerate stellar material171717In stars with masses
between $1.1M_{\odot}\lesssim M_{\star}\lesssim 2M_{\odot}$ the former
convective core first contracts in a Kelvin-Helmholtz timescale until the
layers surrounding the He core reach H-burning temperatures. On the contrary,
in lower-mass stars $M_{\star}\lesssim 1.1M_{\odot}$ the transition from core
H burning to shell H burning happens continuously., leading to the ignition of
a H-burning shell around the exhausted core.

The formation of a degenerate core surrounded by a hot mantle (Section 3.2)
leads to stringent mass-radius relation for the core. Together with the
constraints coming from the massive envelope and the burning shell (eqs. 23
and 36) this forces the radius of the burning shell to settle at a very
specific constant location and at specific values of temperature and density,
all of which are solely defined by the mass of the core below the burning
shell (eq. 54 and Figs. 4 and 5). As $M_{c}$ only increases due to the H
consumption at the burning shell, $T_{s}(M_{c})$,
$\rho_{s}(M_{c})$,$P_{s}(M_{c})$,$R_{s}(M_{c})$ will evolve on a nuclear
timescale. For low-mass red giants (eq. 54) $R_{s}$ is forced to remain almost
constant, and we get $P_{s}/\bar{P_{c}}\propto{M_{c}}^{(2-\nu-
t-b-2a)/(a+t+2)}$ and
$\rho_{s}/\bar{\rho_{c}}\propto{M_{c}}^{(2-\nu-2a-b-t)/(a+t+2)}$. At first the
switching-on of the H-burning shell does not lead to a particularly increase
in the geometrical size of the star, because for low core masses
$M_{c}\lesssim 0.12M_{\odot}$ the density of the shell does not differ so much
from that of the core, which means that similar amounts of mass are harbored
in similar volumes both in the core and around the burning shell (Fig. 5).
However as the mass of the core increases due to nuclear burning, the density
of the shell and the density of the core evolve in opposite directions and
soon much larger volumes are needed to harbor masses around the burning shell.
In our toy homogeneous model already at $M_{c}\approx 0.12M_{\odot}$ the
density of the shell is about one order of magnitude lower than the the mean
density of the core $\bar{\rho_{\rm c}}$. The drop in density of the shell
leads to a drop in the density of the envelope leading to an increase in size.
This situation is made extreme by the enforcement of hydrostatic equilibrium
on the upper mantle as we will show in the next paragraph.

Eq. 24 is valid as long as $m(r)\simeq M_{c}$. Let us define a point $R_{\rm
BAP}$ which sets the boundary of the validity for the massless approximation,
i.e. $(m(R_{\rm BAP})-M_{c})=\mathcal{Q}\,M_{c}$ with $\mathcal{Q}\ll 1$. For
example, with $\mathcal{Q}=0.1$ our estimations of $T$, $\rho$ in that mantle
will be wrong by less than 10%. Within that upper mantle density drops as

$\rho(r)\approx\rho_{s}\left(\frac{r}{R_{s}}\right)^{-\delta}$ (60)

with $\delta=(3-b-a)/(a+1)$. For the usual situation of Thomson Scattering
$\delta=3$. Integrating the mass up to $R_{\rm BAP}$ we have

$\Delta M=\mathcal{Q}\,M_{c}=\rho_{s}\int_{R_{s}}^{R_{\rm BAP}}4\pi
r^{2}\left(\frac{r}{R_{s}}\right)^{-\delta}r^{2}dr.$ (61)

Solving the integral and using the definition of $\bar{\rho_{c}}$ we find
that, for $\delta=3$

$R_{\rm
BAP}=R_{s}\exp\left(\frac{\mathcal{Q}}{3}\frac{\bar{\rho_{c}}}{\rho_{s}}\right),$
(62)

which is a remarkable result. Eq. 62 shows that as soon as the density
contrast between the core and the shell increases the radius of the _massless_
upper mantle on top of the shell increases exponentially. This is a very well-
known feature of numerical models. The situation is even more extreme for
bound-free opacities $\delta=3.5$ and density drops faster.

The direct consequence of this drop in the density contrast is that the
boundary of the massless mantle is moved far away from the core ($R_{\rm
BAP}\gg R_{s}$), and the density of the material drops orders of magnitude to

$\rho_{\rm
BAP}=\rho_{s}\exp\left(\frac{\mathcal{Q}}{3}\frac{\bar{\rho_{c}}}{\rho_{s}}\right)^{-\delta}.$
(63)

while the pressure drops as

$P_{\rm
BAP}=P_{s}\exp\left(\frac{\mathcal{Q}}{3}\frac{\bar{\rho_{c}}}{\rho_{s}}\right)^{-(1+\delta)}.$
(64)

As pointed out by Faulkner (2005), the star is then left with the daunting
task of harboring the whole massive envelope with a very low density and under
a very low gravitational potential (due to the large increase in $R_{\rm
BAP}$). A simple estimation from the hydrostatic equilibrium equation (setting
$P_{\rm surface}\approx 0$) gives

$\bar{R_{\rm env}}\simeq\frac{(G/4\pi)^{1/4}(M_{c}\,M_{\rm env})}{P_{\rm
BAP}^{1/4}},$ (65)

and the exponential drop in $P_{\rm BAP}$ leads to a large increase in the
value of the typical radius $\bar{R_{\rm env}}$ of the mass shells in the
massive envelope. Note that this qualitative conclusion is valid regardless of
whether the massive envelope is radiative or convective181818It is affected,
however, by very deep convection so that convection reaches down to the
burning shell itself. In fact, very deep convection is known to reduce the
size of red giant stars; see Appendix C.

Note that while $\rho_{s}$ stays within one order of magnitude of
$\bar{\rho_{c}}$ the exponent $(\mathcal{Q}\bar{\rho_{c}})/(\rho_{s}3)$ is
close to unity and nothing happens. In our toy model (Fig. 5) this corresponds
to $M_{c}\lesssim 0.1M_{\odot}$ (depending on the exact value of $\mathcal{Q}$
adopted). We see that our simple model predicts that the change in the
behavior of the radius of low-mass red giants should appear once $M_{c}\gtrsim
0.1M_{\odot}$, but not before. Fig. 6 shows the behavior of the stellar radius
as a function of the mass of the core for 1 $M_{\odot}$ and 1.8 $M_{\odot}$
full evolutionary models computed with LPCODE. It is clear from Fig. 6 that
there is a change in the behavior of the radius as a function of the mass of
the core at $M_{c}\sim 0.125M_{\odot}$. This is particularly easy to see in
the case of the $1M_{\odot}$, in which H burned in the core radiatively and
thus had a smooth transition to shell burning. Stellar models with cores
smaller than this threshold change their radii by less than a factor of 2 even
with an increase in the mass of the core of more than one order of magnitude.

Figure 6: Evolution of the radius of a $1M_{\odot}$ model ($Z=0.02$) as a
function of the mass of the He core.

###  5.2 Intermediate-mass and massive stars (IM&M)

The situation in IM&M stars is fortunately much simpler that that of low-mass
stars. This is because their core is not differentiated and can be modeled as
a contracting sphere of ideal gas. For stars in this mass range the mass of
the exhausted core after the main sequence is larger191919For stars below
$\simeq 3M_{\odot}$ this limit is reached later after enough fresh He has been
added to the core by the burning shell, see Salaris & Cassisi (2005). than the
Schönberg-Chandrasekhar limit, and the exhausted core contracts on a thermal
timescale. As this timescale is shorter than the nuclear timescale at which
the burning shell adds mass to the core, the evolution will proceed at almost
constant $M_{c}$. As soon as core H burning is finished and as temperatures
around the burning shell reach $T\simeq 10^{7}$K we can estimate the mean
pressure in the core to be

$\bar{P_{c}}\approx\frac{G{M_{c}}^{2}}{8\pi{R_{s}}^{4}}+P_{s},$ (66)

and the mean density of the core is

$\bar{\rho_{c}}\approx\frac{3{M_{c}}}{4\pi{R_{s}}^{3}}.$ (67)

As the equation of state in the contracting cores of massive and intermediate-
mass stars corresponds to that of a classical ideal gas, we know that typical
temperatures in the core are

$\bar{T_{c}}\approx\frac{\mu\bar{P_{c}}}{\Re\bar{\rho_{c}}}.$ (68)

The evolution of those cores is ruled by the speed at which they lose energy
from their surface, as $L_{c}\neq 0$. The Virial theorem for such structures
tell us that

$2E_{i}+E_{g}=4\pi{R_{s}}^{3}P_{s}$ (69)

Using the expression for the internal energy of an ideal monoatomic gas and
for the gravitational energy we find that

$k^{\prime}\frac{3M_{c}\Re\bar{T_{c}}}{\mu_{c}}-\frac{kG{M_{c}}^{2}}{R_{c}}=4\pi{R_{s}}^{3}P_{s}$
(70)

where $k^{\prime}$ and $k$ are form factors of the order of unity that depend
on the details of the mass distribution of the core. In the most
representative case of Thomson scattering, $P_{s}$ can be replaced as a
function of $M_{c}$ and $R_{s}$ to give:

$k^{\prime}\frac{3\Re\bar{T_{c}}}{\mu_{c}}=\frac{kG{M_{c}}}{R_{c}}+4\pi
k^{\prime\prime}{R_{s}}^{\nu/3}M^{(4-\nu)/3}$ (71)

The time evolution of the core of massive stars is given by
$-d(E_{i}+E_{g})/dt=L_{c}$, and consequently $L_{c}=dE_{i}/dt=-(dE_{g}/dt)/2$.
As heat is radiated away from the core ($L_{c}>0$) at constant
mass202020Because contraction happens on timescales much shorter than those of
nuclear reactions., the temperature will rise and the radius will shrink. Due
to the large value of $\nu$, the surface terms in eqs. 71 and 66 become
quickly irrelevant. As contraction proceeds at constant mass we see that

$\displaystyle\frac{\rho_{s}}{\bar{\rho_{c}}}\propto{R_{s}}^{1+\nu/3}$
$\displaystyle\frac{P_{s}}{\bar{P_{c}}}\propto{R_{s}}^{1+\nu/3}$
$\displaystyle\frac{T_{s}}{\bar{T_{c}}}\propto{R_{s}}^{0}=1$ (72) (73)

Due to the large exponents we see that the contrasts in the density and
pressure increase very rapidly with moderate contractions of the core. Again,
as in the case of low-mass stars, the development of a large density contrast
has enormous consequences for the hydrostatic equilibrium of the upper mantle
on top of the burning shell. In fact, the derivations of eqs. 62, 63, 64, and
65 done in Section 5.1 are independent from the specific characteristics of
the core and are valid also when the stellar core corresponds to an ideal
classical gas. A small contraction of the core leads to an increase in the
density contrast of the star and, as before, this will lead to a further drop
in the pressure and density ($P_{\rm BAP}$, $\rho_{\rm BAP}$) and an increase
in the radius ($R_{\rm BAP}$) at the bottom of the massive envelope. The
envelope of mass $M_{\rm env}$ has to be accommodated in a very low
gravitational potential and with very low densities, which causes a huge
increase in the typical radius ($\bar{R_{\rm env}}$) of its layers.

Note that although the shell remains at an almost constant location in the
case of low-mass stars this is not true for IM&M stars (see Fig. 7). Due to
the different central temperatures at the end of the main sequence
($T/10^{7}=$ 3.6, 4.3, and 5.8 at the end of the main sequence for the 3,5 and
10 $M_{\odot}$ models), different degrees of core contraction are required
before He-burning temperatures are attained ($T_{c}\sim M_{c}/R_{s}$). For
example, in the sequences shown in Fig. 7 between the end of the main sequence
and the beginning of He burning, the core contracts by
$R_{s}/R_{s}^{0}\simeq$2.5, 2.2 and 1.9 for the 3, 5 and 10 $M_{\odot}$
sequences. These changes in $R_{s}$ correspond to increases in the density of
$\bar{\rho_{c}}/\bar{\rho_{c}}^{0}\simeq 15.6$ 10.6, and 6.8.

Figure 7: Evolution of the core mass $M_{c}$, shell radius $R_{s}$ and stellar
radius of three full evolutionary sequences with initial masses $3\
M_{\odot}$, $5\ M_{\odot}$, and $10\ M_{\odot}$ ($Z=0.02$).

##  6 Conclusions and final remarks

In this work we have revisited the long standing question of “why stars become
red giants” by means of a detailed analysis of the stellar structure
equations. Contrary to the suggestions by Iben (1993) and Whitworth (1989) we
find that a simple analytic answer is possible. Our analysis is inspired by
the physical insight offered by Faulkner (2005) but it is based on a more
solid logical and mathematical derivation, offering deeper insight into the
structure of red giants. Most importantly, our analysis done in Section 2.1
highlights the role played by convection in the development of (bright) red
giants. Contrary to the conclusion of Faulkner (2005), that convection only
acts to make stars smaller (see item $(e)$ in Appendix 10.A.2 of Faulkner
2005), we find that the development of convection is a key ingredient. It is
the development of convection what sets the upper boundary to the value of
$\nabla_{+}$, leading to the existence of the $T_{s}(M_{c},R_{s})$ relation.
This fact explains why earlier models in which convection was neglected did
not become luminous red giants—see for example Fig. 3 in Sandage &
Schwarzschild (1952) but also Fig. 2 of Stancliffe et al. (2009). The fact
that Faulkner (2005) is able to derive a core mass-luminosity relation (his
equation 10.7) with no mentioning of the role of convection highlights the
problem with the mathematical derivations in that work, and in particular with
the analysis done in their Appendix 10.A.2.

Our findings can be summarized as follows. The existence of an envelope of
nonnegligible mass forces $\nabla$ to have a very narrow range of possible
values ($0.23\lesssim\nabla\leq 0.4$, for typical opacity laws), which leads
to a tight dependence of the shell temperature on the mass and radius of the
core $T_{s}\propto M_{c}/R_{s}$ (eq. 23). Additionally, the switching-on of
the burning shell and the fact that the heat is both generated and transported
at the burning shell leads to a tight relationship between $\rho_{s}$,
$T_{s}$, $M_{c}$, and $R_{s}$ (eq. 36). Together, eqs. 23 and 36 imply that
$\rho_{s}(M_{c},R_{s})$ (and consequently $P_{s}(M_{c},R_{s})$). In a full
stellar structure, these constraints are then complemented by mass radius
relations for the core $M_{c}(R_{s})$. The addition of this restriction leads
to unique values for $T_{s}$, $\rho_{s}$ and $P_{s}$ as a function of either
$R_{s}$ or $M_{c}$. This forces a very different evolution for $\rho_{s}$ and
$P_{s}$ as a function of either $R_{s}$ or $M_{c}$ in comparison to their core
counterparts $\bar{P_{c}}$ and $\bar{\rho_{c}}$, making homologous contraction
impossible.

Although the $M_{c}(R_{s})$ relation of the cores is different in low-mass
stars and in intermediate-mass and massive (IM&M) stars, the consequences are
the same. In low-mass stars the combination of a degenerate massive inner
region and a massless hot mantle below the burning shell leads to an almost
constant value of $R_{s}$ while the mass of the core increases as nuclear
burning consumes the H envelope. On the contrary, on IM&M stars the Kelvin-
Helmholtz contraction of the classical ideal gas in the core leads to the
decrease in the radius of the core but at almost constant mass. In both low-
mass and IM&M stars, however, evolution leads to an increase in the mean
pressure $\bar{P_{c}}$ and density $\bar{\rho_{c}}$ of the core. Most
importantly, evolution in both low-mass and IM&M stars leads to an increase in
the pressure and density contrasts between the shell and the core
($\bar{P_{c}}/P_{s}$ and $\bar{\rho_{c}}/\rho_{s}$). This increase in
$\bar{\rho_{c}}/\rho_{s}$ leads to a huge expansion of a mantle on top of the
burning shell, where both the pressure and density drop by orders of
magnitude. The massive envelope on top of this mantle is then forced to very
low densities and low gravitational potentials, leading to very large stellar
radii. The storage of a massive envelope at low densities leads to the
formation of a red giant.

Besides finding a toy-model description for the formation of red giants some
additional findings of the present work are worth mentioning. First, the
$T_{s}(M_{c},R_{s})$, $\rho_{s}(M_{c},R_{s})$, and $P_{s}(M_{c},R_{s})$
relations derived from the analysis of the burning shell (and the lower
envelope) are, to the best of our knowledge, the first mathematical
demonstration of this key hypothesis behind shell homology relations
(Kippenhahn et al., 2012). Second, while it is known since the work of Hayashi
(1947) that stellar models become giants even when convection is suppressed,
our simple model shows that convection plays a key role in the formation of
the actual luminous red giants that we observe. Next, it is clear from the
current presentation why a molecular weight gradient helps the development of
a giant structure, although it is neither a sufficient nor a necessary
condition (Stancliffe et al., 2009). The development of a weight gradient
between the core and the shell helps increase the density contrast
$\bar{\rho_{c}}/\rho_{s}$ between the core and the envelope, but it is not the
only way to attain a large density contrast. In addition, our toy model shows
that it is not completely correct to estimate changes in the luminosity of the
shell due to changes in the molecular weight without including the feedback of
this change on the radius of the shell (e.g. Kippenhahn et al., 2012).
Finally, our simple toy model of a low-mass red giant offers the first simple
explanation of why all low-mass stars develop a He-core flash at basically the
same core mass, and why this mass is of the order of half a solar mass. In a
star with an inert degenerate core the temperature of the shell solely depends
on the mass of the core. In particular, this temperature reaches He-burning
luminosities at $M_{c}\simeq 0.75M_{\odot}$, independently of the initial mass
or metallicity. This sets a clear maximum value for the mass of the He core at
the He flash. The helium flash develops well before this point because, for
the high temperatures of the burning shell, the nuclear timescales became
comparable to the timescale for gravitational contraction. As consequence,
gravitational contraction provides an additional heating source, leading to an
advance of the He flash. Interestingly, as this advance is a consequence of
the rate of growth of the degenerate core, and that rate ($dM_{c}/dt\propto
L_{s}$) solely depends on the mass of the core, the helium flash is advanced
by the same amount in all low-mass stars.

From a pedagogical perspective, we believe that having a simple physical model
to explain how stars become luminous red giants will improve our teaching of
stellar evolution, and also our interpretation of results from detailed
numerical models. In future works we will apply the model presented here to
the explanation of other properties of stellar models like the red giant bump
(Christensen-Dalsgaard, 2015; Hekker et al., 2020) or the critical mass at
which stars depart from the red giant branch (Soker, 2008).

In closing let us mention that, given the past history of this topic, we do
not expect this paper to end the discussion about the subject. We hope,
however, that the ideas presented in this paper might offer a different
perspective for future discussions.

M3B warmly thanks Jørgen Christensen-Dalsgaard, Achim Weiß, Tiara Battich,
Santi Cassisi, Leandro G. Althaus, Alfred Gautschy, and the two anonymous
referees for their comments and criticism on different versions of the
manuscript that helped to improve and correct the paper. Jørgen Christensen-
Dalsgaard and Achim Weiß are particularly acknowledged, the former for his
very detailed feedback and encouragement, and the latter for his patience with
the author during the many discussions about this topic in the last 15 years.
Tiara Battich is also acknowledged for her continuous support during the
realization of this work. M3B is partially supported by PIP 2971 from CONICET
and PICT 2020-03316 from Agencia I+D+i. The author also thanks the Max Planck
Institute for Astrophysics without whose library this research would have been
impossible. The author also thanks I. Newton for the idea of what to do during
a pandemic lockdown. Finally, M3B is especially grateful to the editor (Dr.
Steven Kawaler) for his patience and positive approach to the peer review
process. This work is dedicated to Mabel A. Bertolami who taught the author
the values of hard work and perseverance, which were key for the successful
realization of this work.

##  Appendix A The inner and upper mantle in detailed numerical models

In this section we show the main structural properties of the core, shell, and
upper mantle in a detailed numerical simulation of a $1M_{\odot}$, $Z=0.02$
star on the red giant branch. We do this for the sake of completeness and also
to show the reasons behind the identification of the main parts of a low-mass
red giant (Fig. 3). Fig. 8 shows the temperature and density structure of a
detailed $1M_{\odot}$ model as it evolves on the red giant branch.

The existence of the inner isothermal mantle is apparent in all models right
below the burning shell (upper panel of Fig. 8). As expected from the
discussion in our toy model, in this region the density drops exponentially
with increasing radius, from the typical values in the degenerate core down to
the density of the burning shell (lower panel of Fig. 8). Note that, although
extended in radius, the upper mantle region shown in Fig. 8 harbors a
negligible amount of mass, due to its low densities in comparison to the core.
This is clearly appreciated in Fig. 9 where the properties of the models are
shown as a function of the lagrangian coordinate $m(r)=\int_{0}^{r}4\pi
r^{\prime 2}\rho dr^{\prime}$. In this figure it is clear that, as soon as the
star develops a relatively dense core with a significant mass ($M_{c}\gtrsim
0.2M_{\odot}$), the density drops orders of magnitude in a region of
negligible mass above the burning shell (our so-called upper mantle), as a
consequence of eq. 35.

A selected sample of the properties of the models at different stages of the
evolution is shown in Table 1.

Figure 8: Density and temperature profiles of a $1M_{\odot}$ model ($Z=0.02$) for different core masses. Black dots indicate the location of the peak of the H-burning shell of each model. Figure 9: Density profiles of a $1M_{\odot}$ model ($Z=0.02$) as a function of the lagrangian coordinate $m(r)$. Black dots indicate the location of the peak of the H-burning shell of each model. $M_{c}$ | $R_{c}$ | $T_{s}$ | $\rho_{s}$ | $R_{\star}$ | $\log L/L_{\odot}$
---|---|---|---|---|---
$[M_{\odot}]$ | [$10^{9}$ cm] | [$10^{7}$ K] | [g/cm3] | [$R_{\odot}$] |
0.0317 | 3.30 | 1.88 | 314‘ | 1.286 | 0.256
0.0489 | 3.65 | 1.90 | 318 | 1.367 | 0.295
0.0788 | 3.86 | 1.97 | 332 | 1.519 | 0.355
0.0992 | 3.83 | 2.01 | 332 | 1.646 | 0.380
0.1247 | 3.23 | 2.12 | 358 | 1.916 | 0.378
0.1492 | 2.64 | 2.27 | 330 | 2.543 | 0.528
0.1752 | 2.39 | 2.44 | 247 | 3.722 | 0.826
0.1999 | 2.28 | 2.63 | 188 | 5.574 | 1.133
0.2501 | 2.05 | 2.97 | 132 | 10.51 | 1.593
0.3000 | 2.04 | 3.48 | 70.9 | 26.17 | 2.222
0.3500 | 2.02 | 3.99 | 48.7 | 53.87 | 2.698
0.4000 | 1.99 | 4.45 | 35.8 | 96.72 | 3.059
0.4500 | 1.95 | 4.87 | 28.2 | 156.5 | 3.336
Table 1: Properties of the shell and the star for selected snapshots in the
evolution of the $1M_{\odot}$ ($Z=0.02$) sequence.

A clear distinction between our toy model and the actual solution of the
detailed models is the fact that the cores of red giants are clearly not
isothermal for two different reasons. On the one hand the material is not
completely degenerate ($T=0$) as it is still too hot for that approximation in
the outermost regions of the degenerate core, and on the other hand the
existence of neutrino emission leads to a decrease in the temperature of the
core close to the center. Fig 10 shows a full evolutionary model near the He-
core flash in comparison to the corresponding structure when gravothermal
energy release is relaxed to zero (for the same chemical structure). Clearly,
in thermal equilibrium the degenerate core attains a temperature almost equal
to that of the burning shell, with a slight decrease near the center of the
core, due to neutrino energy losses. If neutrino losses are also removed, then
the core becomes completely isothermal (Fig. 10). Fig. 10 shows that the He
flash happens for a lower core mass than it would were it the case that
nuclear reactions had a much larger timescale than the Kelvin-Helmholtz
timescale of the core.

Figure 10: Temperature stratification of the $1M_{\odot}$ full evolutionary
model (FEM) near the He-flash. Green and cyan lines indicate the corresponding
temperature stratification when the same chemical structure is fixed and the
gravothermal energy release is relaxed to zero $\epsilon_{g}=0$ (green), and
when in addition to setting $\epsilon_{g}=0$ neutrino energy losses are also
set to zero.

##  Appendix B Relations for a Kramers’ bound-free opacity law

In Section 2 we derived the equations for the upper mantle under the
assumption that opacity near the burning shell is dominated by classical
electron scattering. While this is true once $M_{c}\gtrsim 0.23M_{\odot}$ (for
a $1M_{\odot}$ star), below that core mass (and the consequent shell
temperature) other sources of opacity, mostly bound-free opacities, are
important. To explore the impact of this assumption we rederive the equations
used in Sections 2 and 5 when Kramers’ bound-free opacity is adopted.

A very rough approximation to bound-free opacities in stellar interiors is
provided by Hansen et al. (2004), which for a typical composition of the
envelope ($X=0.7$, $Z=0.02$) is (in cgs)

$\kappa_{\rm b-f}=4.3\times 10^{-8}\rho{T_{9}}^{-7/2}.$ (B1)

With this new opacity law we have $a=1$ and $b=-4.5$, and eqs. 28, 33 can be
derived in exactly the same way as before. Using these equations and the
opacity law in eq. B1 we can now derive $R_{s}(M_{c})$ and $T_{9}(R_{s})$
relation similar to eqs. 52 and 53. These new relations are

$\frac{R_{s}}{R_{\rm dc}}\simeq\frac{\mu_{\rm
env}}{\mu_{c}}\frac{0.56}{T_{9}}\left(\frac{M_{\rm
dc}}{M_{\odot}}\right)^{4/3},$ (B2)

and

$\displaystyle 1.90\times$ $\displaystyle
10^{-25}\frac{\mu_{c}^{4}}{\mu_{s}^{2}\mu_{\rm
env}^{2}}\nu\exp(30/\nu)\left(1+1/\nu\right)^{-6}=$ (B3)
$\displaystyle{T_{9}}^{-13/6}\left[\frac{R_{s}}{R_{\rm
dc}}\right]^{2}\left[\frac{M_{c}}{M_{\odot}}\right]^{-2/3}$
$\displaystyle\exp\left[\frac{17}{(1+2/\nu)}\frac{\mu_{c}}{\mu_{\rm
env}}\left(1-\frac{R_{\rm s}}{R_{\rm dc}}\right)-15.231{T_{9}}^{-1/3}\right]$

The resulting $T_{9}(M_{c})$ $R_{s}(M_{c})$ are shown in Fig. 4 where it is
clear that they share the same key features described by eqs. 52 and 53, the
burning shell (and consequently the isothermal core below) attains He-burning
temperatures at $M_{c}\simeq 0.7M_{\odot}$ and the H-burning shell settles at
a constant radius of $R_{s}\simeq 0.028R_{\odot}$, very close to that
predicted by full stellar evolution models.

The main difference introduced by a Kramers’ opacity law is to increase the
temperature of the shell as compared to that predicted by Thomson scattering
for the more massive cores. Note, however, that as soon as the temperature of
the shell increases to $T\gtrsim 3\times 10^{7}$ K, electron scattering
becomes the main opacity source.

For the sake of completeness let us mention that the solutions shown in Fig. 4
can be very well approximated by

$\displaystyle T_{9}$ $\displaystyle=$ $\displaystyle
0.0037+0.136\left(\frac{M_{c}}{M_{\odot}}\right)^{1.16}$
$\displaystyle\frac{R_{s}}{R_{\rm dc}}$ $\displaystyle=$ $\displaystyle
1.884+0.357\,\ln\left(\frac{M_{c}}{M_{\odot}}-0.039\right).$ (B4)

##  Appendix C Very deep convection

Key equations derived in the paper rely on the radiative nature of the burning
shell. Here we analyze how expressions presented in the paper are affected by
convection. Let us call $L_{s}$ and $L_{c}$ the luminosities of the shell and
core (where $L_{s}\gtrsim L_{c}$). Let us assume now that, for some reason,
convection reaches deep into the burning shell to a point where the luminosity
is $L_{\chi}=L_{s}\,\chi$ with $0<\chi<1$. At the convective boundary we have
that $\nabla_{\rm ad}=\nabla_{\rm rad}$, and, by equating the temperature
gradients we have

$\left.\frac{dT}{dr}\right|_{\chi}=-0.4\frac{GM_{c}\mu}{\Re{R_{s}}^{2}},$ (C1)

and from the energy generation and energy transport equations we can write

$\left(\left.\frac{dT}{dr}\right|_{\chi}\right)^{2}\simeq\frac{3}{8\pi
ac}\frac{{\rho_{s}}^{2}}{{T_{s}}^{2}}\frac{\kappa_{s}\epsilon_{s}}{\nu}\left(\frac{L_{c}}{L_{s}}+\chi\right)\mathcal{D},$
(C2)

where $1<\mathcal{D}<2$ if convection has not reached the location of the peak
of the burning shell and $1=\mathcal{D}$ if it has. Together eqs. C1 and C2
give

$\left(0.4\frac{GM_{c}\mu}{\Re{R_{s}}^{2}}\right)^{2}\frac{1}{\left(\frac{L_{c}}{L_{s}}+\chi\right)}\approx\frac{3}{8\pi
ac}\frac{{\rho_{s}}^{2}}{{T_{s}}^{2}}\frac{\kappa_{s}\epsilon_{s}}{\nu}\mathcal{D}.$
(C3)

Although the equations remain formally unchanged, it is clear that the
deepening of convection inside the burning shell will have a strong impact. As
convection deepens, then $\chi\longrightarrow 0$ and the left- and right-hand
sides of eq. C3 are decoupled. Without this constraint all the arguments that
allowed us to conclude that a red giant will be formed cannot be stated. In
fact, if somehow a steady burning shell would develop in a completely
convective star, that object would not develop a giant structure, as those
stars are $n=3/2$ polytropes and, as such, cannot develop a density contrast
between the envelope and the core. It is clear that convection inside the
burning shell will conspire with the formation of a giant structure. Moreover,
numerical experiments show that, as soon as convective transport penetrates
inside the main regions of the burning shell, the radius of the star will
decrease. Fig. 11 shows a numerical experiment in which the bottom of the
adiabatic temperature is forced down into the burning shell itself. Note that
these are static structures in which the gravothermal heat term $\epsilon_{g}$
is forced to zero throughout the star. We see that, as long as the adiabatic
gradient stays far from the burning shell, the luminosity of the star remains
unchanged because both the upper mantle and the burning shell can be
considered radiative so the temperature of the burning shell and luminosity
are determined by the equations derived in this paper. The radius of the star,
on the other hand, is reduced as a consequence of the less steep density
gradient imposed by adiabatic convection. However, as soon as convection
reaches down to the burning shell itself luminosity is strongly affected, and
the radius of the star drops suddenly. The more convection reaches into the
burning shell, the smaller the final radius of the star. We find that,
although relatively deep convection remains the main picture of the paper, as
long as convection does not reach the bottom of the burning shell, stars with
deep convective zones in the sense of this section (i.e. those in which
convection reaches down to the burning shell itself) display much smaller
radius.

Figure 11: Impact of very deep convection on the luminosity and radius of the
star.

##  Appendix D Envelope Integrations

Figs. 12, 13, 14, and 15 show envelope integrations similar to those displayed
by Fig. 2 but for different conditions at the burning shell.

Figure 12: Envelope integration for different values of $\mathbb{C}$ and
$y_{+}$, for $t_{+}=1$ and for a Kramers’ opacity.

Figure 13: Envelope integration for different values of $\mathbb{C}$ and
$y_{+}$, for $t_{+}=0.5$ and for a Kramers’ opacity.

Figure 14: Envelope integration for different values of $\mathbb{C}$ and
$y_{+}$, for $t_{+}=1.5$ and for a Kramers’ opacity.

Figure 15: Envelope integration for different values of $\mathbb{C}$ and
$y_{+}$, for $t_{+}=0.75$ and for a Thomson scattering-like opacity.

##  Appendix E Early evolution of the burning shell

As discussed in Section 2.3, when $\rho_{+}\ll\rho_{c}$ and
$P_{+}\ll\bar{P_{c}}$ we can strictly demonstrate that $\Theta\simeq 1$,
$\Pi\simeq 1$, $\zeta\simeq 1$, $T_{0}\ll T_{+}$ and $R_{s}\ll R_{0}$ and all
previous expressions are very simple. We show below that these approximations
are still acceptable even on the very early stages of shell burning. From the
hydrostatic equilibrium equation and using the definitions in eq. 5 we can see
that the drop in pressure, temperature, and density immediately above the
burning shell follows

$\displaystyle\left.\frac{dP}{P}\right|_{+}\simeq-6\left(\frac{T_{c}}{T_{+}}\right)\frac{R_{s}\,dr}{r^{2}},$
(E1)
$\displaystyle\left.\frac{dT}{T}\right|_{+}\simeq-6\nabla_{+}\left(\frac{T_{c}}{T_{+}}\right)\frac{R_{s}\,dr}{r^{2}},$
$\displaystyle\left.\frac{d\rho}{\rho}\right|_{+}\simeq-6\left(1-\nabla_{+}\right)\left(\frac{T_{c}}{T_{+}}\right)\frac{R_{s}\,dr}{r^{2}}.$

From eqs. 18 we see that once the shell becomes active $F\simeq 2$ (eq. 15) in
the early stages of H-shell burning ($\nu\simeq 23$ at low temperatures of the
CNO cycle), we have for $\nabla_{+}\simeq 0.25$ very roughly that
$\rho_{+}\sim\rho_{-}/2$ and $P_{+}\sim P_{-}/2$. Using the very conservative
approximation that at the lower boundary of the burning shell density and
pressure are half their values at the core, we then have
$\rho_{+}\sim\rho_{c}/4$ and $P_{+}\sim\bar{P_{c}}/4$ (in fact, in low-mass
giants it will be much smaller than this due to the presence of a near
isothermal inner mantle; see Section 3.1). From eqs. E1 and as long as
$r\simeq R_{s}$ we can make the very rough approximation that
$d\rho/\rho|_{+}\simeq-18/4dr/r$ immediately above the burning shell. From the
previous estimation we get that $\rho(r)\simeq\rho_{+}(r/R_{s})^{-18/4}$.
Integrating the upper mantle

$\mathcal{Q}M_{c}=\int_{R_{s}}^{R_{\rm 0}}\rho(r)4\pi r^{2}dr$ (E2)

where $\mathcal{Q}$ is the fraction of the mass of the core we consider
acceptable for $m(r)\simeq M_{c}$. Integrating out to $\mathcal{Q}=0.2$ so
that the approximation $m\simeq M_{c}$ is accurate to $\sim 20$%, we get that
$R_{\rm 0}\simeq 1.4\,R_{s}$, or the width of the upper mantle to be $\Delta
R/R_{s}\simeq 0.4$. From this estimation we see that, already at the very
early stages of H-shell burning the factor $(1-R_{s}/R_{\rm 0})$ in eq. 23 is
a factor three lower than unity but not negligible. From this value of $\Delta
R/R_{s}$ and the estimations of $dP/P$ and $dT/T$ at the upper boundary we get
$T_{\rm 0}\sim T_{+}\exp(-6\nabla_{+})(R_{0}/R_{s})\sim 0.31T_{+}$ and $P_{\rm
0}\sim\exp(-6)(R_{0}/R_{s})\sim 0.0035P_{+}$. Then the factors $\Theta$ and
$\Pi$ in eq. 28 are both close to one even at this early stage and considering
these conservative assumptions regarding the link between the values at the
lower boundary ($\rho_{-}$, $P_{-}$) and their mean values at the core.

The importance of the estimations from the previous paragraphs cannot be
downplayed as they tell us that already from the very early stages of H-shell
burning the factors $\Pi$, $\Theta$, and $(1-R_{\rm 0}/R_{s})$ in eqs. 28 and
23 are of order unity, and consequently, these equations are actual
restrictions for the values of $\bar{P_{c}}$, $T_{c}$ and $R_{c}$. These
restrictions play a key role in the increase in the density and pressure
contrast ($\bar{\rho_{c}}/\rho_{s}$ and $\bar{P_{c}}/P_{s}$) between the shell
and the core, so it is important to show that such relations are valid (even
if only approximately) before the values of $\bar{\rho_{c}}/\rho_{s}$ and
$\bar{P_{c}}/P_{s}$ become huge. Once $\bar{\rho_{c}}/\rho_{s}$ and
$\bar{P_{c}}/P_{s}$ become large, the approximation $\Theta=1$, $\Pi=1$
becomes very precise. It is then reasonable to assume for most practical
estimations that $\Theta\simeq 1$, $\Pi\simeq 1$,$\zeta\simeq 1$ and $R_{s}\ll
R_{0}$.

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