Event-Triggered Islanding in Inverter-Based Grids

Ioannis Zografopoulos, Graduate Student Member, IEEE, Charalambos Konstantinou, Senior Member, IEEE

The authors are with the  Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia.
E-mail<EMAIL_ADDRESS>
The decentralization of modern power systems challenges the hierarchical structure of the electric grid and requires the implementation of automated schemes that can overcome adverse conditions. This work proposes an adaptive isolation methodology that can segregate a grid topology in autonomous islands that maintain stable and economic operation in the presence of deliberate (e.g., cyberattacks) or unintentional abnormal events. The adaptive isolation logic is event-triggered to avoid false positives, improve detection accuracy, and reduce computational overheads. A measurement-based stable kernel representation (SKR) triggering mechanism inspects distributed generation controllers for abnormal behavior. The SKR notifies a machine learning (ML) ensemble classifier that detects whether the system behavior is within acceptable operational conditions. The event-triggered adaptive isolation framework is evaluated using IEEE RTS-24 bus system. Simulation results demonstrate that the proposed framework detects anomalous behavior in real-time and identifies stable partitions minimizing operating costs faster than traditional islanding detection techniques.

Attacks, detection, distributed generation, event-triggered, islanding, isolation, resilience.

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# Group delay time of fermions in graphene through tilted potential barrier

Youssef Fattasse  Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Miloud
Mekkaoui  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Jellal
<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Canadian
Quantum Research Center, 204-3002 32 Ave Vernon, BC V1T 2L7, Canada  Abdelhadi
Bahaoui  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco

###### Abstract

The group delay time of Dirac fermions subjected to a tilting barrier
potential along the $x$-axis is investigated in graphene. We start by finding
the eigenspinor solution of the Dirac equation and then relating it to
incident, reflected, and transmitted beam waves. This relationship allows us
to compute the group delay time in transmission and reflection by obtaining
the corresponding phase shifts. We discovered that the barrier width, incident
energy, and incident angle can all be used to modify the group delay time, and
that the particles travel through the barrier at the Fermi velocity $v_{F}$.
Our findings also show that the transmission group delay might be controlled,
and that gate voltage control could be useful in graphene-based tilting
barriers.

###### pacs:

72.80.Vp, 73.21.-b, 71.10.Pm, 03.65.Pm
Keywords: Graphene, tilted barrier, transmission, group delay time.

##  I Introduction

The ability to change particle behavior by adjusting the gate bias voltage has
prompted a lot of interest in quantum tunneling [1, 2, 3]. Electrical
conduction in gapless graphene cannot be turned off using the control voltages
required for conventional transistor operation [4]. This problem can be solved
in graphene by establishing an energy gap in the energy spectrum of particles.
The desired gap is a measurement of the threshold voltage and the on-off ratio
of the field effect transistors [5, 6]. However, significant progress has been
made in understanding quantum phenomena in graphene systems, with the group
delay time being one of the most important parameters connected to the dynamic
aspect of the tunneling process [7, 8]. By studying the behavior of wave
packets, Hartman demonstrated that the group delay may be described in terms
of the derivative of the phase shift with respect to the energy [7]. The
Hartman effect states that the effective group velocity of a particle can
become superluminal for sufficiently large barriers [9, 8].

We previously investigated the quantum tunneling for Dirac fermions in
graphene scattering by a linear vector potential [14, 15]. In [14], it was
demonstrated that the infinite mass boundary condition discretizes the
transverse momentum. As a result, an effective massive 1D Dirac equation is
derived in which the quantized transverse momentum behaves as an effective
mass. In [15], the Goos-Hänshen shifts were studied using the solutions of the
energy spectrum of graphene in a linear barrier potential. The procedure
begins with the transmission and reflection probabilities being used to
determine the appropriate phase shifts. According to a numerical analysis,
incident energy, barrier height, and width have a significant impact on the
Goos-Hänshen shifts, which can change positively or negatively under
particular conditions.

We investigate the group delay time in transmission and reflection for Dirac
fermions in graphene subjected to a tilting barrier potential based on our
prior work [14, 15]. We illustrate how to calculate the group delay time as a
function of different physical parameters based on phase shifts and GH shifts
in transmission using the energy spectrum solution. We propose a numerical
investigation under various conditions to provide a better understanding of
our findings. In particular, the tilting barrier is shown to be able to manage
the group delay time.

The following is a breakdown of the current paper’s structure. We define our
problem in section II, write out the corresponding Hamiltonian and energy
spectrum solutions for different regions. We then calculate the transmission
and reflection probabilities from which the phase shifts are determined. As a
result, we determine the group delay time in terms of the physical parameters
that define our system using traditional definitions in section III. We
numerically examine and highlight the basic aspects of the group delay time in
section IV. In the concluding section, we summarize our findings.

##  II Energy spectrum

As seen schematically in Fig. 1, massless Dirac fermions in graphene are
scattered by tilted barrier potentials $V_{0}$ and $V_{1}$. The current system
is divided into three zones designated by the numbers $j=1,2,$ and 3, each of
which has a distinct potential.

Figure 1: (color online) Configuration of a tilted barrier potential of width
$d$, height $V_{0}$ and $V_{1}$ applied to an intermediate region of graphene.

The following Dirac-like Hamiltonian can be used to describe the current
system

$H=v_{F}{\bm{\sigma}}\cdot\textbf{p}-\left(\beta
x-V_{0}\right)\Theta(xd-x^{2}){\mathbb{I}}_{2}$ (1)

with the Heaviside step function $\Theta$, the Fermi velocity ${v_{F}\approx
10^{6}m/s}$, the Pauli matrices ${{\bm{\sigma}}=(\sigma_{x},\sigma_{y})}$,
$\textbf{p}=-i\hbar(\partial_{x},\partial_{y})$, the $2\times 2$ unit matrix
${\mathbb{I}}_{2}$, $\beta=\frac{V_{0}-V_{1}}{d}$. The spinor $\Psi(x,y)$ at
energy $E$ has a time-independent Dirac equation, which is given by

$\left[v_{F}{\bm{\sigma}}\cdot\textbf{p}-\left(\beta
x-V_{0}\right)\Theta(xd-x^{2}){\mathbb{I}}_{2}\right]\Phi(x,y)=E\Phi(x,y).$
(2)

The system is considered to have a finite width $W$. The spinor meets the
infinite mass boundary condition at the interfaces $y=0$ and $y=W$ along the
$y$-direction [17, 16]. As a result, the transverse momentum $k_{y}$ is
quantized

$k_{y}=\frac{\pi}{W}\left(n+\frac{1}{2}\right),\qquad n=0,1,2\cdots.$ (3)

The spinor is then obtained by separating the variables and, therefore, we
write
$\Psi_{j}(x,y)=\left(\varphi_{j}^{+}(x),\varphi_{j}^{-}(x)\right)^{\dagger}e^{ik_{y}y}$.
In region 1 ($x<0$), we determine the two components of the eigenspinor after
solving the eigenvalue equation

$\displaystyle\Psi_{1}(x,y)=\left(\begin{array}[]{c}{1}\\\ {z_{1}}\\\
\end{array}\right)e^{i(k_{1}x+k_{y}y)}+r\left(\begin{array}[]{c}{1}\\\
{-z_{1}^{-1}}\\\ \end{array}\right)e^{i(-k_{1}x+k_{y}y)},\qquad
z_{1}=s_{1}e^{i\phi}$ (8)

where $r$ is the reflection coefficient and $\phi=\tan^{-1}(k_{y}/k_{1})$ is
the incident angle, while the sign function
$s_{1}={\mbox{sign}}{\left(E\right)}$ indicates the conduction and valence
bands. The corresponding dispersion relation is straightforward to obtain

$\displaystyle E=s_{1}\hbar v_{F}\sqrt{k_{1}^{2}+k_{y}^{2}}.$ (9)

In region 2 ($0<x<d$), the parabolic cylinder function can be used to express
the general solution, with the two components being [14]

$\displaystyle\Xi^{+}=b_{1}D_{\eta-1}\left(\Lambda\right)+b_{2}D_{-\eta}\left(-\Lambda^{*}\right)$
(10)
$\displaystyle\Xi^{-}=-\frac{b_{1}}{k_{y}}\sqrt{2\varrho}e^{-i\pi/4}D_{\eta-1}\left(\Lambda\right)-\frac{b_{2}}{k_{y}}\left[2(\epsilon-\varrho
x)D_{-\eta}\left(-\Lambda^{*}\right)+\sqrt{2\varrho}e^{i\pi/4}D_{-\eta+1}\left(-\Lambda^{*}\right)\right]$
(11)

and we have defined $\eta=\frac{ik_{y}^{2}}{2\varrho}$,
$\Lambda(x)=\sqrt{\frac{2}{\varrho}}e^{i\pi/4}\left(-\varrho
x+\epsilon\right)$, $\varrho=\frac{\beta}{\hbar v_{F}}$,
$\epsilon=\frac{E-V_{0}}{\hbar v_{F}}$, with $b_{1}$ and $b_{2}$ are two
constants. The following are the components of the spinor solution of the
Dirac equation (2) in region 2

$\displaystyle\varphi^{+}(x)=\Xi^{+}+i\Xi^{-},\qquad\varphi^{-}(x)=\Xi^{+}-i\Xi^{-}$
(12)

which results in the spinor

$\displaystyle\Psi_{2}(x,y)$ $\displaystyle=$ $\displaystyle
a_{1}\left(\begin{array}[]{c}\chi^{+}(x)\\\ \chi^{-}(x)\\\
\end{array}\right)e^{ik_{y}y}+a_{2}\left(\begin{array}[]{c}\xi^{+}(x)\\\
\xi^{-}(x)\\\ \end{array}\right)e^{ik_{y}y}$ (17)

where the functions $\chi^{\pm}(x)$ and $\xi^{\pm}(x)$ are written as follows:

$\displaystyle\chi^{\pm}(x)$ $\displaystyle=$ $\displaystyle
D_{\eta-1}\left(\Lambda\right)\mp\frac{1}{k_{y}}\sqrt{2\varrho}e^{i\pi/4}D_{\eta}\left(\Lambda\right)$
(18) $\displaystyle\xi^{\pm}(x)$ $\displaystyle=$
$\displaystyle\pm\frac{1}{k_{y}}\sqrt{2\varrho}e^{-i\pi/4}D_{-\eta+1}\left(-\Lambda^{*}\right)\pm\frac{1}{k_{y}}\left(-2i\epsilon_{0}\pm
k_{y}+2i\varrho x\right)D_{-\eta}\left(-\Lambda^{*}\right)$ (19)

$a_{1}$ and $a_{2}$ are two constants.

The following spinor is found in region 3 $(x>d)$:

$\Psi_{3}(x,y)=t\left(\begin{array}[]{c}{1}\\\ {z_{1}}\\\
\end{array}\right)e^{i(k_{1}x+k_{y}y)}$ (20)

propagating with the same wave vector $k_{1}$ as in region 1. The transmission
$t$ and reflection $r$ coefficients associated with phase shifts will be
computed using the previous solutions.

##  III Group delay time

Let us begin by defining the following abbreviations in order to determine the
group delay time. This is about the eigenspinors’ components

$\chi^{\pm}(0)=\chi_{0}^{\pm},\qquad\chi^{\pm}(d)=\chi_{d}^{\pm},\qquad\xi^{\pm}(0)=\xi_{0}^{\pm},\qquad\xi^{\pm}(d)=\xi_{d}^{\pm}.$
(21)

Because the spinors must be consistent at all interfaces, we get a set of
equations stated in terms of transfer matrices $M_{jj+1}$ between different
regions. Then, over the entire tilted barrier, the full transfer matrix can be
expressed as

$\left(\begin{array}[]{c}1\\\ r\\\
\end{array}\right)=M\left(\begin{array}[]{c}t\\\ 0\\\
\end{array}\right)=\prod_{j=1}^{4}M_{jj+1}\left(\begin{array}[]{c}t\\\ 0\\\
\end{array}\right)$ (22)

where $M_{12}$, $M_{23}$ are transfer matrices that connect the $j$-th region
wavefunction to the $(j+1)$-th region wavefunction. Explicitly, we have

$\displaystyle M_{12}=\left(\begin{array}[]{cc}1&1\\\ z_{1}&-z^{\ast}_{1}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}\chi_{0}^{+}&\xi_{0}^{+}\\\
\chi_{0}^{-}&\xi_{0}^{-}\\\ \end{array}\right),\quad
M_{23}=\left(\begin{array}[]{cc}\chi_{d}^{+}&\xi_{d}^{+}\\\
\chi_{d}^{-}&\xi_{d}^{-}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}e^{\textbf{\emph{i}}k_{1}d}&e^{-\textbf{\emph{i}}k_{1}d}\\\
z_{1}e^{\textbf{\emph{i}}k_{1}d}&-z_{1}^{\ast}e^{-\textbf{\emph{i}}k_{1}d}\\\
\end{array}\right)$ (31)

and then

$\displaystyle M=\left(\begin{array}[]{cc}m_{11}&m_{12}\\\ m_{21}&m_{22}\\\
\end{array}\right).$ (34)

As a result, the transmission and reflection coefficients are given by

$\displaystyle t=\frac{1}{|m_{11}|}e^{i\varphi_{t}},\qquad
r=\left|\frac{m_{21}}{m_{11}}\right|e^{i\varphi_{r}}$ (35)

where

$\displaystyle\varphi_{t}=\arctan\left(i\frac{t^{\ast}-t}{t+t^{\ast}}\right),\qquad\varphi_{r}=\arctan\left(i\frac{r^{\ast}-r}{r+r^{\ast}}\right)$
(36)

are the phase shifts of the transmission and reflection amplitudes,
respectively. After some lengthy but straightforward algebra, we obtain the
following coefficients:

$\displaystyle
t=\frac{e^{-ik_{1}d}\left(1+z_{1}^{2}\right)\left(\xi^{+}_{d}\chi^{-}_{d}-\xi^{-}_{d}\chi^{+}_{d}\right)}{\vartheta},\qquad
r=\frac{\delta}{\vartheta}$ (37)

as well as having defined

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\chi^{+}_{0}\xi^{-}_{d}+z_{1}\left(\chi^{-}_{d}\xi^{-}_{0}+\chi^{+}_{0}\xi^{+}_{d}-\chi^{+}_{d}\xi^{-}_{0}+\chi^{+}_{0}\xi^{-}_{d}\right)-z_{1}^{2}\left(\chi^{+}_{d}\xi^{+}_{0}-\chi^{+}_{0}\xi^{+}_{d}-\chi^{-}_{d}\xi^{+}_{0}\right)$
(38) $\displaystyle\vartheta$ $\displaystyle=$
$\displaystyle\left(\xi^{+}_{0}+z_{1}\xi^{-}_{0}\right)\left(\chi^{-}_{d}-z_{1}\chi^{+}_{d}\right)-\left(\chi^{+}_{0}+z_{1}\chi^{-}_{0}\right)\left(\xi^{-}_{d}-z_{1}\xi^{+}_{d}\right).$
(39)

The transmission $T=\frac{J_{t}}{J_{i}}$ and reflection
$R=\frac{J_{r}}{J_{i}}$ probabilities are calculated using the current of
densities $J_{i}$, $J_{r}$, and $J_{t}$ representing the incident, reflected
and transmitted waves, respectively. We get the current density from the
Hamiltonian

$\displaystyle J=e\upsilon_{F}\psi^{+}\sigma_{x}\psi$ (40)

leading to the probabilities

$\displaystyle T=\left|t\right|^{2},\qquad R=\left|r\right|^{2}.$ (41)

A transverse wave vector $ky=k_{y_{0}}$ and an incident angle
$\phi(k_{y_{0}})\in[0,\frac{\pi}{2}],$ indicated by the subscript 0, are then
used to analyze the GH shifts and group delay time. A temporal-spatial wave
packet, which is the weighed superposition of plane wave spinors, can be used
to describe a finite pulsed electron beam. As a result, the incident,
reflected at $x=0$, and transmitted wave packets at $x=d$ wave functions can
be expressed as a double Fourier integral over $k_{y}$ [18, 19]

$\displaystyle\psi_{i}(x,t)=\iint f(k_{y},\omega)e^{-i(k_{y}y-\omega
t)}dk_{y}d\omega$ (42) $\displaystyle\Psi_{r}(x,t)=\iint
rf(k_{y},\omega)e^{-i(k_{y}y-\omega t)}dk_{y}d\omega$ (43)
$\displaystyle\Psi_{t}(x,t)=\iint tf(k_{y},\omega)e^{-i(k_{y}y-\omega
t)}dk_{y}d\omega$ (44)

where the involved spinors are solutions of Dirac equation (2). Here,
$f(k_{y},\omega)=w_{y}e^{-w_{y}^{2}(k_{y}-\omega)^{2}}$ is the Gaussian
angular spectral distribution, with $\omega=E/\hbar$ and $w_{y}$ is the half
beam width at waist [20]. The total phases of reflected and transmitted waves
at $x=0,d$, respectively, are given by

$\displaystyle\Phi_{\nu}=\varphi_{\nu}+k_{y}-\omega t,\qquad\nu=t,r.$ (45)

Using the stationary phase approximation, we may get analytical equations for
group delay and lateral GH shift by assuming that the distribution
$f(k_{y},\omega)$ is a smooth and steeply peaked function around the center
energy/wavevector [21, 22]. GH shifts are obtained from

$S_{\nu}=-\frac{\partial\varphi_{\nu}}{\partial.k_{y}}.$ (46)

For retaining the nice shape throughout propagation, the equation of motion is
determined using the constraint $\partial\Phi_{\nu}/\partial\omega=0$. This
provides the group delay time

$\displaystyle\tau_{\nu}$
$\displaystyle=\frac{\partial\varphi_{\nu}}{\partial\omega}+\frac{\partial
k_{y}}{\partial\omega}S_{\nu}$ (47)
$\displaystyle=\tau_{\nu}^{s}+\tau_{\nu}^{\varphi}$ (48)

where $\tau_{\nu}^{\varphi}$ represents the time derivative of phase shifts

$\tau_{t}^{\varphi}=\hbar\frac{\partial\varphi_{t}}{\partial
E}+\frac{\hbar}{2}\frac{\partial\phi}{\partial
E},\qquad\tau_{r}^{\varphi}=\hbar\frac{\partial\varphi_{r}}{\partial E}$ (49)

and the second $\tau_{\nu}^{s}$ results from the contribution of $S_{\nu}$

$\tau_{t}^{s}=\frac{\sin\phi}{\upsilon_{F}}S_{t},\qquad\tau_{r}^{s}=\frac{\sin\phi}{\upsilon_{F}}S_{r}.$
(50)

We shall proceed with numerical analysis after obtaining closed form equations
of the group delay in various energy domains.

##  IV Numerical analysis

We compute the group delay time in transmission for electrons passing through
a tilting barrier under various incident angle $\phi$, potential height
$V_{0}$ and width $d$, and incident energy $E$. Dimensionless group delay time
$\tau_{t}/\tau_{0}$, which results in transversal time
$\tau_{0}=\frac{d}{v_{F}}\cos\phi$, is convenient for our task. The key
findings of this study are depicted in the following seven figures, each with
a distinct set of physical parameters.

In Fig. 2, the group delay in transmission $\tau_{t}/\tau_{0}$ is shown versus
the incident angle $\phi$ by choosing different values of the remaining
physical parameters. It is evident that at normal incidence, i.e. $\phi=0$,
the particles propagate through the barrier with the Fermi velocity $v_{F}$ (
$\tau_{t}/\tau_{0}=1$), but that as $\phi$ increases, $\tau_{t}/\tau_{0}$
begins to progressively increase until it reaches a maximum, after which it
decays exponentially. It approaches zero when $\phi=90^{\circ}$, since the
wave vector inside the barrier becomes imaginary, and the wave function in the
barrier region becomes an evanescent wave. The behavior of $\tau_{t}/\tau_{0}$
is affected by incident energy $E$, barrier width $d$, and height $V_{1}$, as
it drops as $E$ and $d$ increase, but increases as $V_{1}$ grows. This means
that the linear potential can affect the group delay time by modulating it.

(a)  (b)

(c)  (d)

Figure 2: (color online) The group delay time in transmission
$\tau_{t}/\tau_{0}$ as a function of the incident angle $\phi$ for $V_{0}=80$
meV, $d=80$ nm. (a): $V_{1}=0$ meV and (b): $V_{1}=20$ meV with $E=20$ meV
(blue line) $E=25$ meV (red line), $E=30$ meV (green line). (c): $V_{1}=50$
meV, $E=30$ meV (blue line), $E=35$ meV (red line), $E=40$ meV (green line).
(d): $V_{1}=50$ meV, $E=30$ meV, $d=40$ nm (green line), $d=60$ nm (red line),
$d=80$ nm (blue line).

In Fig. 3, we plot the group delay time $\tau_{t}/\tau_{0}$ as a function of
the barrier widths $d$ for the barrier heights (a): $V_{1}=0$ meV, (b):
$V_{1}=20$ meV, (c): $V_{1}=50$ meV and (d): $V_{1}=V_{0}$. The other
computation parameters being $V_{0}=80$ meV, $\phi=30^{\circ}$ and for
different values of $E=20$ meV, $E=25$ meV and $E=30$ meV. The possibility of
modulating $\tau_{t}/\tau_{0}$ by changing the height of the potential barrier
through different applied gate voltages is also present in the barrier tilting
structure. Particularly interesting is that if $V_{0}$ is kept constant and
$V_{1}$ is modified, $\tau_{t}/\tau_{0}$ shows identical behavior in Fig. 3a
for $V_{1}=0$ meV and Fig. 3b for $V_{1}=20$ meV. But if $V_{1}$ is increased
from $V_{1}=50$ meV up to $V_{1}=V_{0}$, the oscillations in
$\tau_{t}/\tau_{0}$ are again observed as illustrated in Fig. 3c and Fig. 3d
such that their number increases and also their peak value decreases. On the
other hand, we observe $\tau_{t}/\tau_{0}$ increasing with the increase of
incident energy $E$. Therefore the incident energy modifies the period and
amplitude of the oscillations by increasing them. As the barrier width $d$ is
increased, the peaks show a discernible spread due to the Fabry-Pérot
enhancement. The particles travel back inside the tilting barrier due to
Fabry-Pérot resonances between the barrier edges, which explains
$\tau_{t}/\tau_{0}$.

(a)

(b)

(c)

(d)

Figure 3: (color online) The group delay time in transmission
$\tau_{t}/\tau_{0}$ as a function of the barrier width $d$ for $V_{0}=80$ meV,
$\phi=30^{\circ}$, $E=20$ meV (blue line), $E=25$ meV (red line), $E=30$ meV
(green line). (a): $V_{1}=0$ meV, (b): $V_{1}=20$ meV, (c): $V_{1}=50$ meV and
(d): $V_{1}=V_{0}=80$ meV.

(a)

(b)

(c)

(d)

Figure 4: (color online) The group delay time in transmission
$\tau_{t}/\tau_{0}$ as a function of the barrier width $d$ for $V_{0}=80$ meV,
$E=30$ meV, $\phi=5^{\circ}$ (red line), $\phi=10^{\circ}$, (blue
line),$\phi=15^{\circ}$ (green line). (a): $V_{1}=0$ meV, (b): $V_{1}=20$ meV,
(c): $V_{1}=50$ meV, (d): $V_{1}\approx V_{0}=80$ meV.

Fig. 4 shows the group delay time $\tau_{t}/\tau_{0}$ as a function of the
barrier width $d$ for different values of the incident angle $\phi=5^{\circ}$
(red line), $\phi=10^{\circ}$ (blue line), $\phi=15^{\circ}$ (green line),
$E=30$ meV and we choose the remaining parameters as in Fig. 3. With the Fermi
velocity $v_{F}$, the particles pass through the barrier, but when $d$
increases the group delay time in transmission $\tau_{t}/\tau_{0}$ starts to
oscillates with peak increasing. Notice that for $V_{1}=50$ meV and
$V_{1}\approx V_{0}=80$ meV as presented, respectively, Fig. 4c and Fig. 4d,
$\tau_{t}/\tau_{0}$ oscillates twice and the peak value decreases compared to
Fig. 4a and Fig. 4b, where $\tau_{0}$ is the time it would take a particle to
travel the distance $d$ if the barrier did not exist. Here we observe that the
particles propagate through the barrier with the Fermi velocity $v_{F}$
($\tau_{t}/\tau_{0}=1$). When $d$ increases one sees that $\tau_{t}/\tau_{0}$
begins to oscillate and the associated amplitude increases with the increase
of the incident angle $\phi$, while the peaks did not get influenced and they
are still in the same positions. When $v_{F}$ is equivalent to the speed of
light $c$ in optics, the group delay time $\tau_{t}$ may be smaller than
$\tau_{0}$ ($\tau_{t}/\tau_{0}<1$), indicating superluminality. This phenomena
faster than light is relevant for the Hartman effect in the tunneling process.
On the other hand, we discover that particles propagate past the barrier at
speeds greater than the Fermi velocity $v_{F}$.

(a)

(b)

Figure 5: (color online) The group delay time $\tau_{t}/\tau_{0}$ (blue line)
and transmission probability $T$ (red line) as a function of the incident
energy $E$ at two incident angles (a): $\phi=15^{\circ}$ and (b):
$\phi=30^{\circ}$, with $V_{0}=60$ meV, $V_{1}=20$ meV, $d=80$ nm.

In Fig. 5 we plot the group delay time in transmission $\tau_{t}/\tau_{0}$
(blue line) and transmission probability $T$ (red line) as a function of the
incident energy $E$ for two values of the incident angle
$\phi=15^{\circ},30^{\circ}$ in Fig. 5a and Fig. 5b with $V_{0}=60$ meV,
$V_{1}=20$ meV and $d=80$ nm. It can be seen that when $E$ increases, both
quantities exhibit closed behavior and they can be modulated by modifying the
incident angle. Additionally, we see oscillating behavior in the group delay
time and transmission probability. This is due to the overlapping of the
reflected and incident waves, which causes self-interference delay.
Furthermore, we see a peak in the $\tau_{t}/\tau_{0}$ behavior, which
increases in tandem with the increase in the incident angle. As shown in Fig.
5a, when $E<V_{0}-V_{1}$, $T$ decays exponentially due to the wave function in
the region of the barrier becoming an evanescent wave, and goes down to a
minimum, then starts increasing again. However, $T$ approaches zero before
starting to increase again when $E>V_{0}-V_{1}$ as depicted in Fig. 5b.

(a)

(b)

Figure 6: (color online) The group delay time $\tau_{t}/\tau_{0}$ (blue line)
and transmission probability $T$ (red line) as a function of the barrier
height $V_{0}$ at incident angles (a): $\phi=15^{\circ}$ and (b):
$\phi=30^{\circ}$, with $E=50$ meV, $V_{1}=0$ meV, and $d=80$ nm.

Fig. 6 presents the group delay time $\tau_{t}/\tau_{0}$ and the transmission
probability $T$ as a function of the barrier height $V_{0}$ for $d=80$ nm,
$E=50$ meV, and $V_{1}=0$ meV, with Fig. 6a and Fig. 6b illustrate the case of
$\phi=15^{\circ}$ and $\phi=30^{\circ}$, respectively. For small values of
$V_{0}$, one sees that the particles propagate through the tilted barrier at
the Fermi velocity $v_{F}$, which they can transmit perfectly, $T=1$. When
$V_{0}$ is increased, $\tau_{t}/\tau_{0}$ and $T$ begin to rapidly fall toward
a constant value that is independent of $V_{0}$. We observe that the incident
angle has an effect on $\tau_{t}/\tau_{0}$ and $T$’s behaviors since they
decrease as it increases. For $\phi=30^{\circ}$, $T$ stabilizes at zero
regardless of the value of $V_{0}\geq 80$ meV, resulting in a total
reflection.

(a)

(b)

Figure 7: (color online) (a): The group delay time in transmission $\tau_{t}$
as a function of the barrier width $d$ for $V_{0}=40$ meV, $E=40$ meV,
$\phi=30^{\circ}$, with two values $V_{1}=20$ meV (red line) and $V_{1}\approx
V_{0}=40$ meV (green line). (b): $\tau_{t}$ as a function of the incident
energy $E$ for $V_{0}=40$ meV, $V_{1}=20$ meV, $\phi=30$, with three values
$d=90$ nm (blue line), $d=80$ nm (red line), $d=70$nm (green line).

Fig. 7a depicts the group delay time in transmission $\tau_{t}$ as a function
of the barrier width $d$ for tilting barrier ($V_{0}=40$ meV, $V_{1}=20$ meV)
as well as barrier square $V_{1}=V_{0}=40$ meV with $E=40$ meV,
$\phi=30^{\circ}$. One notices that $\tau_{t}$ rapidly grows as $d$ increases,
eventually stabilizing at a maximum. We get the same behavior shape with
$V_{1}=V_{0}$, but with a decline. As a result, Fig. 7a shows that the
Hartmann effect exists at $V_{1}=V_{0}=40$ meV, because $\tau_{t}$ saturates
at a constant when the barrier width $d$ is increased [23]. Fig. 7b
illustrates $\tau_{t}$ as a function of incident energy $E$ for for $V_{0}=40$
meV, $V_{1}=20$ meV, $\phi=30^{\circ}$, demonstrating the impact of barrier
widths $d=90$ nm (blue line), $d=80$ nm (red line) and $d=70$ nm (green line).
We can see that $\tau_{t}$ oscillates as $E$ increases, and that its peak
increases as $d$ increases. When $E$ grows, $\tau_{t}$ tends to settle at a
certain value.

##  V Conclusion

We investigated the group delay time in transmission $\tau_{t}/\tau_{0}$ for
Dirac fermions in graphene scattered along the $x$-axis by a linear barrier
potential. The group delay time has been demonstrated to oscillate in response
to several physical parameters such as barrier width $d$, incident angle
$\phi$, incident energy $E$, and two barrier heights ($V_{0},V_{1}$). When the
barrier width $d$ becomes large enough, our theoretical investigation supports
the existence of group delay time saturation. It also proves that quantum
interference has a significant impact on particle tunneling in graphene via a
tilted barrier.

Also we demonstrated that the physical parameters that characterize our system
can be used to modify the behavior of $\tau_{t}/\tau_{0}$. Furthermore, we
discovered that the group delay time in transmission equals unity at certain
critical values of incidence energy, incident angle, and barrier width, i.e.
$\tau_{t}/\tau_{0}=1$, implying that particles travel across the barrier with
the Fermi velocity $v_{F}$. Finally, we expect that all of the findings will
be valuable not only for the theoretical research of the tunneling effect but
also for graphene’s technological applications.

## Acknowledgment

## References

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#  Why stars inflate to and deflate from red giant dimensions, II:
replies to critics

Alvio Renzini1
1INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5,
I-35122 Padova, Italy E-mail<EMAIL_ADDRESS>
(Accepted 2023 January 10. Received 2022 December 16)

###### Abstract

In a 1992 paper of ours the role of opacity-driven thermal instabilities in
shaping the course of stellar evolution was amply illustrated. This included
the classical issue of “why stars become red giants” as well as the subsequent
formation of extended “Cepheids” blue loops during the helium burning phases.
Our explanation of these evolutionary phenomena has been occasionally
dismissed with just a few words in refereed or not refereed publications. In a
most recent case, the fact that, through the years, I did not reply to these
criticisms is interpreted as evidence that they were well founded. In this
paper it is made clear that this is not at all the case, the leading role of
such instabilities is instead reaffirmed and the criticisms are shown to be
insubstantial.

###### keywords:

stars: evolution – stars: interiors – Hertzsprung-Russell and colour-magnitude
diagrams

††pagerange: Why stars inflate to and deflate from red giant dimensions, II:
replies to critics–Why stars inflate to and deflate from red giant dimensions,
II:  replies to critics††pubyear: 2002

##  1 Introduction

The founding fathers of stellar evolution, Subrahmanyan Chandrasekhar, Martin
Schwarzschild, Alan Sandage and Fred Hoyle, set up the stage, made the early
discoveries and introduced the language of stellar evolution. But left a
problem unsolved, which is usually formulated as a question: “why stars become
red giants?”222footnotetext: The reason to use italics fonts for this verb
will be made clear in the course of the paper. Then, in the early ’sixties,
adequate computers became available and, thanks to scientists such as Rudolf
Kippenhahn, Pierre Demarque and Icko Iben, full realistic stellar evolutionary
sequences began to be produced. Computers had no difficulty whatsoever to make
red giants, and to account, for instance, for the so-called Hertzsprung gap,
i.e., the paucity of stars in between the main sequence (MS), spectral types O
to F, and the red giants branch (RGB), spectral types K and M. Models swept
quickly through to the gap and found a rest only as red giants. Problem
solved? Yes and no. The clear success of the models satisfied almost
everybody, and the question Why? became kind of academic and few continued to
pay attention to it. A common wisdom was that it is not easy to predict,
without computers, what the solutions are of a system of four, non-linear,
differential equations.

A first hint came to me when realising what was happening in the evolution of
a $3\,\hbox{$M_{\odot}$}$ stars in one of the classical Iben’s papers (,
1965). In its Figure 6, one can see that some 10 Myr after hydrogen exhaustion
in the core, the surface luminosity of the star was starting to drop while the
nuclear luminosity (provided by the hydrogen burning shell) was still
increasing for another $\sim 2$ full Myr, and only then it started to drop as
well. So, something was happening to the envelope that the core did not know
yet. The envelope had become incapable to transmit outwards and radiate away
all the energy being generated in the interior, i.e., the envelope had become
thermally unstable! This incapacity was clearly causing radiative energy being
trapped in the envelope, which in turn was causing its runaway expansion.
Thus, a thermal instability of the stellar envelopes was first proposed as the
driving of the stars to red giant dimensions (, 1984, 1984) This was the
beginning of the story.

##  2 Runaway inflations and deflations of stellar envelopes

This explanation was more effectively formulated in a paper with the same
title of the present one (, 1992), along with the presentation of stellar
evolutionary sequences and toy envelope models illustrating what physically
happens to the stars.333footnotetext: Most of these calculations were made
with Iben’s stellar evolution code that Icko kindly provided to us. I could
find no better way for summarising the paper than use its very abstract, which
is reproduced here:

“We demonstrate that a unique physical process is responsible for the runaway
expansion of stars to red giant dimension, as well as for their subsequent
recollapse leading to the so-called blue loops. In response from an increasing
luminosity from the core, the stellar envelope expands keeping its thermal
equilibrium444footnotetext: A star is said to be in thermal equilibrium when
its surface luminosity remains close to the nuclear energy generated in the
interior. insofar the envelope thermal conductivity increases. However,
expansion implies local cooling, ion recombination, and thus increasing
opacity, in such a way that a time comes when further expansion causes a drop
in thermal conductivity in the envelope. As the luminosity transferred outward
and radiated away from the surface drops, thermal equilibrium is broken and an
increasing fraction of the core luminosity is trapped in the envelope, causing
further expansion and further drop of the thermal conductivity: the resulting
runaway inflation of the envelope brings the star into the red giant region of
the H-R diagram, and thermal equilibrium is not restored until convection
penetrates inwards and the whole envelope becomes convective. The reverse
process is responsible for the formation of the blue loops. During the early
helium-burning phase, the core luminosity decreases and the star descends
along the Hayashi track. By contraction the envelope heats up, the heavy
elements ionize and the opacity drops. As the inner part of the envelope
returns to radiative equilibrium, the envelope departs again from thermal
equilibrium, since by contraction the temperature increases, the heavy ions
ionize, the opacity drops, the thermal conductivity increases, and so do the
radiative energy losses. Thus the envelope catastrophically deflates inside
the potential well of the star. We present detailed analyses of these runaway
inflations and deflations, and apply those concepts to achieve a deeper
physical understanding of several major features of stellar evolution, etc…”

So, from the beginning ours was not just an explanation of why stars become
giants, but also of why some stars retreat from a red giant size and
experience extended blue loops during their core-helium burning phase, another
unexplained issue at that time. (1971) had indeed realized that what they
called a “secular instability” was developing “during” the loops, but could
not identify its physical origin. Our 1992 paper has been often criticized,
but only for the red giant issue, while the loops part was ignored. This was
somehow disappointing, because ours ambition had been to unify the two
problems as a manifestation of a single process: a thermal instability arising
in stellar envelopes, where opacity plays a leading role. For many years I
left unanswered the criticisms, hoping that time would have settled the issue,
but this did not happen. Once every few years a paper appears cursorily
dismissing our explanation and venturing into attempts to find the real one,
but never claiming success. The latest case is a paper by (2022) that has
recently appeared on the ArXiv, where one reads: “The discussions about why
stars become red giants have sometimes turned into heated debates (1997, 1997,
2005, 2005), while some other times authors have ignored criticism, and
continue to develop ideas (1992, 1994) that had already been seriously
questioned by other researchers (1989, 1993)”. So, our explanation was
dismissed on the basis that others had already claimed that it is bogus and
the fact that we did not further react is interpreted as an implicit admission
that the criticism was well founded. So, this paper is meant to make clear
that this is not at all the case. I still believe that the explanations in our
1992 paper are correct and describe why (some) stars become red giants, and
others don’t.

##  3 The critics

In this section the criticisms made by the mentioned authors are quoted and
comment. Let’s start from (1989). In his paper Achim Weiss examines the
criterion for thermal stability of stellar envelopes that I had proposed in
1984 and he found some cases in which apparently it would not apply. In
particular, an approximate rendition for the criterion was found inaccurate.
This last point was admitted in our 1992 paper and the other criticisms where
answered. But in the end Weiss concluded: “I agree with Renzini’s discussion
on the differences in redward evolution (Renzini 1984) of stars of different
masses, different metal content and his conclusion about the connection with
stellar opacities. Also, this paper confirms that the expansion is a pure
envelope phenomenon that has to be initiated somehow (e.g. by a rapid
contraction of the core and the presence of a burning shell). It might be
acceptable that the final expansion in some stars after having reached a
maximum luminosity is indeed a thermal runaway effect connected with some
specific opacity features.” So, rather than demolish it, Weiss confirmed the
proposed physical origin of the inflation to red giants.

Sugimoto & Fujimoto (2000) confine their criticism to the following sentences:
“Renzini et al. (1992) and Renzini & Ritossa (1994) have argued that thermal
instability in the envelope, which results in the deep penetration of surface
convection, is the cause of the expansion to a red giant. It is true that the
red giant branch is separated from the branch of stable blue giants without
surface convection by the well-known Hertzsprung gap, as first noted by (1971)
and (1972) and also seen from Figure 7. A star does undergo thermal
instability in the envelope when crossing the gap. This is not the cause of
the large expansion, however, but is a result of potentially extraordinarily
large radii, as already noted by many authors (1995, 1989, 1991, 1993)”. So,
Sugimoto and Fujimoto concur that a thermal instability develops in the
envelope, however it would not be the thermal instability that drives the
expansion but rather the instability would be driven by the willingness of the
star to assume a stable giant configuration that terrestrial mathematicians
had proved to exist. I don’t know how to otherwise interpret this sentence.

According to Fujimoto & Iben (1991): “Renzini (1984) discusses the thermal
instability in the envelope as a cause of the red giant structure.” This
sentence does not properly represent what said in my 1984 paper, where the
thermal instability is seen the cause of the expansion to the red giant
structure, rather than of the existence of static red giant solution to the
four equations. Then Fujimoto and Iben go on: “… during the evolution to a red
giant, a thermal instability in the envelope occurs and the surface convective
zone extends deeply into the interior… however, the envelope instability is
just an episode which occurs when the surface temperature grows low enough in
the course of expansion, forcing the helium ionization zone inward (in the
case of helium stars) or forcing the hydrogen and helium ionization zones
inward (in the case of hydrogen stars), and causing the effective polytropic
index to increase. This instability is not itself responsible for the
development of a red giant structure (see also Weiss 1989).” However, this is
not what Weiss concluded, as reported above, if by “development” one means
expansion of a dwarf to become a giant. If instead they meant existence of a
red giant structure solution to the equations, then the same comment to the
point of view of Sugimoto & Fujimoto applies, as above. Incidentally, in our
1992 paper it is shown that through the expansion the polytropic index in the
envelope does not change much at all (see Figure 9 there).

Iben (1993) had four objections to our 1992 paper, partly following the lines
of Fujimoto & Iben (1991). Contrary to what stated by Miller Bortolami (2022),
we did not leave unanswered the four objections, but in (1994) we demonstrated
all the four points to be invalid, so there is no need to expand here and the
interested reader may look directly at Iben’s views and our replies.

##  4 STATIC STELLAR MODELS

In the course of their evolution, stars either expand or contract all the
time. Why are they doing so? On the MS and beyond, what drives stellar
evolution is nuclear burning in the deep interior, whereas the envelope has a
passive role; it has just to adjust to the changing conditions in the core.
However, details of the core internal structure are irrelevant, as there is no
underground communication between the envelope and the core. What matters to
the envelope are just three core quantities, namely, its mass, radius and
luminosity, which set the the boundary conditions at the base of the envelope.
Of the three, by far the the dominant one for the driving of envelope changes
is luminosity. For example, take a star during its core hydrogen-burning
phase: in the conversion of hydrogen into helium three particles disappear,
pressure would drop (because $P=nkT$), so the core by loosing pressure support
is forced to contract. In doing so it heats up thus being able to maintain its
hydrostatic equilibrium. As core density and temperature increase, so does the
nuclear luminosity, and this is the trigger to envelope expansion. Suppose the
envelope does not change at all while the nuclear luminosity increases. The
envelope would receive from its bottom more energy than it is able to emit
from the surface, hence its total energy (thermal plus gravitational) would
increase: it expands. In general, when the core luminosity increases the star
expands, when it decreases the star contracts (at least insofar the envelope
is close to thermal equilibrium). Anyway, during these changes deviations from
hydrostatic equilibrium are tiny and indeed we speak of quasi-static evolving
configurations.

Several papers devoted to the issue “why stars become red giant?” deal instead
with fully static configurations, i.e., having set to zero the gravitational
energy generation (the so called $\epsilon_{\rm g}$). This is equivalent to
freeze evolution and explore instead potential stellar structures in full
thermal equilibrium that may or may not be realized in Nature. As such, these
studies may answer the question “what is the structure of red giants”, but
having switched off evolution they are by construction incapable of answering
the other question, i.e., how such structure becomes established in real
stars, and in stellar evolutionary sequence alike? The key point is that the
word become implies evolution, not static, unevolving states. The transition
to red giant can only be understood in a full evolutionary context.

Most of the papers mentioned in the previous section deal instead with static
models, and thus are intrinsically unable to answer the usual question. Fairly
enough, (1989) entitled his paper “Why red giant are giant”. This is indeed a
question that static models can answer. But intrinsically cannot account for
evolutionary transitions. Not only static models were used, but a specific
subset of them (e.g., 1995, 1991, 2005, 2005). Perhaps with the ambition to
succeed where the founding fathers have failed, they tried to use their same
old tools, specifically the (in)famous homology invariants $U$ and $V$. For
what matters here, suffice to say that $U$ and $V$ are built using only two of
the four equations, namely those for hydrostatic equilibrium and mass
conservation. No energy generation. No luminosity, i.e., no energy transfer
through the star. Hence, in such approach the drivers of evolution are
expunged altogether. I dare to say that virtually nothing about stellar
evolution can be understood by cruising the $UV$ plane. Well, in absence of
better tools the founding fathers got something out of it. Not much, but
something. However, after $\sim\\!60$ years of full stellar evolution through
computers the regression to the rudimentary tools of the fathers sounds
inexplicable to me. Apparently, there was a kind of reluctance to look at what
in fact happens inside stars from computer outputs, as if knowledge achieved
in that way was lacking some sort of sublimity. Perhaps these feelings were
best expressed by (2005), when saying “The end result is that the post-main-
sequence developments of all stars -low- mass, intermediate-mass and high-
mass- as they expand to become giants, are finally seen to be example of one
underpinning fact: that dense cores with surrounding shells naturally follow
hydrogen exhaustion. While this has been known all along from oft-repeated
computer calculations, we now know why analytically. That matters to true
theorists.” (p. 150).

##  5 A chicken and egg problem?

(2022)555footnotetext: It has been thanks to this paper that I discovered
Faulkner’s article of 2005, as it never appeared on ArXiv. fully endorses
Faulkner’s (2005) criticism of our physical explanation of the dwarf-to-giant
transition, so I feel obliged to comment on such criticism. I cut and paste
here all his points relative to us, and comment them.

“For the most part, such stars [massive post-main-sequence stars] are out of
thermal equilibrium, but not thermally unstable as Renzini has claimed, alone
or in concert, in a series of papers, e.g. Renzini et al. (1992).” (p. 190) In
his whole article there is no attempt at elucidating what would be the
difference between being out of thermal equilibrium and being thermally
unstable, specifically in stars expanding to red giants. In the early hydrogen
shell-burning phase stars can be in or very close to thermal equilibrium for
quite some time (see the run of nuclear and surface luminosity in Iben’s
Figure 6), until suddenly they start to depart precipitously and increasingly
from thermal equilibrium. What is this if not a thermal instability?

A few pages later, he states: “We now see that it is most definitively not an
envelope thermal instability that drives the expansion of the entire star, as
Renzini, either alone or with a succession of (sometimes absconding)
colleagues has long asserted. To the absolute contrary, the envelopes are sent
out of equilibrium by an expansion imperative dictated by developments in the
deepest and densest interiors of the stars.” (p. 196).

Then Faulkner goes on: “We can now contrast this with Renzini’s proposed
explanation for the expansion of stars to the red-giant branch, an explanation
he advanced particularly for intermediate mass stars. Renzini suggested that a
star of intermediate mass starts to expand from the main sequence - a fact
taken as a given - recombination occurring in its cooler regions increasing
the opacity there, leading to absorption of luminous energy from below. That
in turn expands the star’s outermost layers still further, leading to more
cooling and yet more absorption, as the hypothesized recombination wave sweeps
inward through the star. A thermal instability develops from the outside
inwards, for which the luminosity dips are advanced as evidence of this
absorption of flux in the outermost regions. He has furthermore claimed that
this opacity effect is naturally smaller when there are fewer heavy elements
in a star, and that this is why such stars do not expand as much. In this
explanation, once the initial expansion from the main sequence has made the
outermost layers cool enough, the natural behavior of those layers takes over
and promotes the major, fast transition to the red-giant branch itself. Thus
the behavior of the outermost layers leads the star to the giant branch.”

This rendition of our interpretation of the phenomenon of the evolution to red
giants (for intermediate mass stars) is fair enough, though Faulkner doesn’t
spare us a couple of pricks666footnotetext: Does anybody doubt that stars
expand during their MS phase? Or, are we the first to claim that metals
contribute to stellar opacity? That ion recombination takes place was not
an“hypothesis” but it is a fact that can be easily verified on stellar
evolutionary sequences..

But then Faulkner continues: “I have shown above, instead, that the behavior
of the central regions is the main driver. Far from leading the rest of the
star along, the envelope regions are lagging behind where they would have been
had there been time for them to reach either a complete new equilibrium
mandated by the change in central conditions (in particular the increasing
central condensation) or some ‘moving target’ analogue.” (p. 199-200)

Here Faulkner is half right, hence half wrong. It is obviously true that the
ultimate driver of stellar evolution are the nuclear transformations taking
place in the core (including the shell). It is the ensuing increase in
luminosity emanating from the core that drives the initial expansion of the
envelope, during the first times after hydrogen shell ignition. But once the
thermal instability suddenly erupts, it is the envelope that takes the lead.
As mentioned in Section I, it is the luminosity radiated by the envelope that
starts dropping, while the shell (nuclear) luminosity still keeps increasing
for a while. Then, with some delay, also the nuclear luminosity starts
dropping fast, which is due to the feedback effect from the envelope. As
documented in our 1992 paper, with the runaway expansion of the envelope, its
weight on the shell drops, i.e., when the accelerated expansion started from
the surface sweeps inside all the way to reach the upper part of the shell,
then density and temperature in the upper shell drop and so does the
generation of nuclear energy.777footnotetext: This is what (1965) called the
“shell narrowing phase”, see Iben’s Figure 6. In this phase the envelope is
clearly leading and the core (the shell) is lagging, to the extent that the
envelope expansion almost succeeds in switching off the shell.

Faulkner’s argument seems instead to consist in that the core would be
changing too rapidly for the envelope to follow, and then the envelope would
lag behind. Hence, there would be no genuine thermal instability in the
envelope. But this not what happens, especially in intermediate-mass stars:
Past shell ignition there is a long period during which the core shrinks
slowly, nuclear and surface luminosities are almost identical, and yet the
thermal instability suddenly erupts (see again Iben’s Figure 6). Moreover, in
our 1992 paper we already countered this option: in Figure 4 there we showed
indeed that the thermal instability erupts, no matter how slowly the core
luminosity increases. In other words, it is not that the core would be
evolving too rapidly for the envelope to adjust, it is instead that the
envelope becomes incapable of transferring outwards all the energy being
provided by the core. However, in stars more massive than $\sim
10\,\hbox{$M_{\odot}$}$, after hydrogen exhaustion the core contracts very
rapidly, out of thermal equilibrium, shell ignition is violent and there is no
approach the thermal equilibrium in the envelope which instead is immediately
pushed into runaway expansion. Still, the recombination wave starts from near
the surface adding the usual thermal instability to an already complex
structure which is out of thermal equilibrium everywhere, from the core,
through the shell and then in the envelope.

So, whether it is the core or the envelope that comes first, cannot be reduced
to a chicken and egg problem. During most of the evolution of a star it is the
stellar core that leads and the envelope follows, struggling to adjust. But
when envelope thermal instabilities develop, and they do, it is the envelope
itself that drives further changes, including those deep in the shell, and
does so with the short, thermal timescale. The relative roles of core and
envelope in determining the transition to red giants where clearly stated in
the conclusions of our 1992 paper, where it was said: “We have demonstrated
that stars become red giants in response to the increasing luminosity being
provided by the core, and that the runway expansion -when it takes place- is
triggered by the thermal conductivity of the envelope reaching a maximum and
then decreasing. The decrease of thermal conductivity is caused by the opacity
increase promoted by the recombination of heavy ions in the envelope, as the
envelope itself expands and cools.” (Point 1 of the concluding section).
Ultimately, it is the core that drives the envelope to the edge of its
catastrophic thermal instability, which however first erupts near the surface
and then drills through the star until it reaches to expand even the burning
shell itself. All this can be easily verified, suffice to look, and with
humility pay attention, at what happens inside models in stellar evolutionary
sequences.

##  6 Stars that do not become red giants

The statement above “… the post-main-sequence developments of all stars -low-
mass, intermediate-mass and high-mass- as they expand to become giants …” (,
2005) is not correct. Not all stars become red giants during their post-MS
phase. Section 5.3 of our 1992 paper was dedicated to the effect of
metallicity on the evolution to red giant configurations. If metal opacity is
the culprit, one expects a big effect of metal abundance on the phenomenon.
And indeed the effect is big. Even before 1992, it was known that
intermediate- and high-mass stars of low (or zero) metallicity do NOT become
red giants during their post-MS phase, but fail to incur in the thermal
instability and ignite helium in the core as blue giants. In this respect, as
evidence we quoted (1977), (1984) and (1985), see also (1993). These stars
were developing a large central concentration, a steep density and molecular
weight gradient at the edge of their hydrogen-exhausted core, ignited the
shell, reached and surpassed the Schönberg-Chandrasekhar limit, just like
their more metal rich stars, but failed to become red giants. Perhaps it is
worth repeating here that “We emphasize that in no other proposed explanation
of why stars become red giants one has ever attempted to answer the question
‘why very metal poor (intermediate-mass) stars do NOT become red giants?’, a
question which instead finds its most natural answer in the frame of our
physical interpretation” (, 1992). And this remains fully valid even to these
days, thirty years later, in particular for those articles that have dismissed
our demonstration as reported in the previous sections.

Figure 1: Evolutionary sequences for the indicated masses, from the pre-MS all
the way to past helium exhaustion at the center and helium shell burning. In
blue are the tracks for standard opacities. and metallicity $Z=0.001$. In red
are the sequences have assumed that the only source of opacities is pure
electron (Thomson) scattering. Courtesy of Santi Cassisi.

Just to illustrate this further, Figure 1 shows several stellar evolutionary
sequences, as described in the caption. Let us consider first the (blue)
tracks computed with standard opacities, and relatively low metallicity,
$Z=0.001$. In the $9\,\hbox{$M_{\odot}$}$ sequence helium is ignited at the
center while the star is still a blue giant, with an effective temperature of
$\sim 13,000$ K, when it begins slowly contracting and spends all the core
helium-burning phase in the blue. It is not before central helium exhaustion
and helium-shell ignition (corresponding to the prominent loop in the track)
that the fast excursion to the red begins, with the onset of the thermal
instability signalled by the luminosity drop past the loop. In practice, it is
only after central helium exhaustion that the luminosity grows high enough to
trigger the envelope instability, whereas past hydrogen exhaustion the
luminosity had not reached such threshold, hence failed to trigger the
instability. This illustrates the point made before that what matters to the
envelope is only the luminosity at its base, irrespective of the structure
inside the outer(most) shell, rather than unspecified “developments in the
deepest and densest interiors of the stars”, as advocated by Faulkner. In
other words, there is no subterranean communication between the core and the
envelope.

The case of the $7\,\hbox{$M_{\odot}$}$ track is quite similar, with the main
difference being that helium ignites while the star has reached the slightly
cooler temperature of $\sim 9,000$ K. In the case of the
$5\,\hbox{$M_{\odot}$}$ star, instead, the thermal instability clearly erupts
and the star inflates to red giant dimension, though an RGB phase is promptly
aborted by helium ignition. The $3\,\hbox{$M_{\odot}$}$ sequence is much
similar to the case of more metal rich stars, with inflation to the RGB and
deflation from it that gives raise to the (first) blue loop. (A hint for the
second blue loop is barely visible shortly after the arrival on the RGB.) The
$2\,\hbox{$M_{\odot}$}$ star develops the thermal instability and its helium
core becomes electron degenerate, which allows for the extended rise in
luminosity terminated by the helium flash. The extended RGB is even more
prominent in the $1\,\hbox{$M_{\odot}$}$ case, where no thermal instability
erupts (see next section).

Turning to the red sequences, for which the opacity has been artificially
restricted to pure electron scattering, one can notice the following. First,
all tracks are systematically brighter and hotter compared to the those with
standard opacity, as expected. Past central hydrogen exhaustion, the stellar
luminosity steadily increases, no thermal instability sets in and helium is
ignited while the star is still at a high effective temperature, that
decreases with decreasing stellar mass. Fast excursions to low effective
temperatures start only after helium-shell ignition, corresponding the
(second) blue loops. Moreover, no envelope convection sets in (opacity is too
low) and tracks run to very low effective temperatures. Being now globally
hotter, the $2\,\hbox{$M_{\odot}$}$ star does not develop an electron-
degenerate helium core, and helium is ignited under non-degenerate conditions.
This pure Thompson-scattering experiment shows that stellar models can be
constructed whereby these models reach large dimensions even in absence of a
thermal runway instability, suffice for them to become bright enough. Yet,
this happens only after helium exhaustion in the core, but not after central
hydrogen exhaustion. However, this is not what real stars do, as their opacity
is not limited to electron scattering. This experiment shows that (on the
computer) a giant configuration can always be achieved, provided that enough
luminosity is pumped into the envelope. Incidentally, it also shows that a
proper RGB is produced only if using realistic opacities.

##  7 Stars that become red giants without ever breaking their thermal
equilibrium

One objection heard frequently was: but low mass stars do become red giants
and yet they do not develop a thermal instability. True! Suffice to look at
the colour-magnitude diagrams of galactic globular clusters, where main
sequence, turnoff and subgiant branch join smoothly to the RGB. Most recently,
(2022) states: “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.” True, for low-mass stars, but not true for intermediate-mass
ones, just more massive than, say $\sim 1.1-1.2\,\hbox{$M_{\odot}$}$, which
instead do become red giants as a result of the thermal instability.

This point was addressed in our 1992 paper (in Section 5.2) where we said: “In
general, the violence of the runaway decreases with decreasing mass…It does so
to the extent that for masses below $\sim 1\,\hbox{$M_{\odot}$}$ (the actual
value depends on metallicity) the drop in surface luminosity … vanishes, and
so does the runaway expansion itself. Low mass stars become red giant without
ever departing from TE… This follows from the fact that low-mass stars begin
their evolution already close to the Hayashi line and their envelope becomes
convective before the thermal instability has a chance to take place: the
early establishment of convection suppresses the thermal instability of the
envelope.”

So, why do they become red giants? The reason was already mentioned in Section
3, when saying: “In general, when the core luminosity increases the star
expands, when it decreases the star contracts.” And the core luminosity does
indeed steadily increase and does so by about three orders of magnitude (see
Figure 1): as the shell around the electron-degenerate helium core burns
through, the core mass increases and core radius decreases (this is what
happens to degenerate cores and white dwarfs alike). Hence, gravity, pressure,
density and temperature in the shell all increase and so does the nuclear
energy generated there. To accommodate the increasing luminosity from the
core, and radiate it away, the envelope is forced to expand. Contrary to the
case of stars experiencing the thermal instability, the core now strongly
holds the lead all the way through –from the MS to the RGB tip– and the
envelope lags behind. Again, the key factor for the expansion is the
luminosity and its steady increase, something that is pathetically absent in
the $UV$ homology invariants.

##  8 Summary and Conclusions

In this paper we have quoted in full the specific criticisms that have been
moved to our answer to the question of “why stars become red giants?” It can
be appreciated that these dismissals are embarrassingly shallow, being
typically limited to just a few, sometimes even cryptic words. Several authors
admit that a thermal instability develops in the envelope, but just state that
it would not be responsible for the runaway expansion. They don’t explain why
this would not be the case nor what other physical process would drive the
expansion instead. Another author admits that stars sweeping to become red
giant are indeed in thermal imbalance but this would be not caused by a
thermal instability of the envelope. Without guessing what other physical
process would trigger the thermal imbalance itself. (1997), at the beginning
of his one page article says: “Thermal imbalance or even thermal instabilities
may in some cases describe how, but they do not tell us why stars expand to
giant dimensions.” As far as physics is concerned, I don’t see much difference
between the how and the why, and Faulkner did not try to explain what the
difference would be. In any event, I am satisfied for having proposed how the
expansion takes place, and if others believe that a different, whimsical why
exists, and I don’t think it does, then I am happy to pass the hand to
$UV$-plane explorers. Ultimately, distinguishing evolutionary phases in
thermal equilibrium from those that are thermally unstable is critical to
properly understand stellar evolution and so the shapes of evolutionary
sequences in the HR diagram.

## Acknowledgments

I would like to thank Santi Cassisi for his critical reading of the manuscript
and for constructive comments. I also thank him for having computed on my
request the evolutionary tracks shown in Figure 1.

## Data Availability

No new data were generated or analyzed in support of this research.

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[0000-0002-7093-7355]A. Renzini