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<p>
<b>Materials</b>: chapter 2 of Onno Pols' lecture notes, chapters 1 &amp; 2 of
Kippenhahn's book.
</p>

<div id="outline-container-org10f06e0" class="outline-2">
<h2 id="org10f06e0"><a href="#org10f06e0">The first two equations of stellar structure</a></h2>
<div class="outline-text-2" id="text-org10f06e0">
<p>
So far we have mostly been discussing astronomical facts on stars
(e.g., use of populations to understand phenomena too slow to observe,
the CMD/HRD diagrams, various types of binaries and how to use
observables to derive masses), albeit anticipating some physical
modeling (e.g., some basics of black-body radiation to introduce
T<sub>eff</sub>). In this lecture, we will move on to proper astro <b>physics</b> of
stars: we want to derive a set of equations to model the parts of the
stars that we cannot see, the interior structure of stars and how they
evolve in time. This will take a few lectures to construct, and it
will require several approximations - I will try to highlight these
when we encounter them.
</p>

<p>
These equations express the conservation of mass, momentum, and
energy, plus equations that describe the changes in chemical
composition, and an equation that relates thermodynamic quantities to
each other. These are typically expressed in a specific form for
stellar evolution applications, to make the problem of determining the
<i>internal structure</i> of a star and how it evolves tractable.
</p>

<p>
Note that deriving these equations will take us away from directly
observable phenomena, and push us into theoretical modeling based on
physical understanding that needs to be <i>indirectly</i> validated on
observations.
</p>

<p>
Before we get into arguments specific to stars, let's review briefly
hydrodynamical coordinate systems that we will be using to describe
the star.
</p>
</div>

<div id="outline-container-org3e45872" class="outline-3">
<h3 id="org3e45872"><a href="#org3e45872">Lagrangian and Eulerian coordinates</a></h3>
<div class="outline-text-3" id="text-org3e45872">
<p>
In hydrodynamics one is typically interested in describing fields of
variables that take one value for each point in the fluid (e.g.,
density, velocity, pressure, temperature, etc..). Let's call a generic
field &Psi; (=&rho;, v, P, T, etc&#x2026;) and discuss the two ways we can use to
described it.
</p>
</div>

<div id="outline-container-org69da340" class="outline-4">
<h4 id="org69da340"><a href="#org69da340">Eulerian description</a></h4>
<div class="outline-text-4" id="text-org69da340">
<p>
In a Eulerian description the fields are described as a function of
position in space (and time): &Psi; &equiv; &Psi;(x, y, z, t) in Cartesian
coordinates or &Psi;(r, &theta;, &varphi;, t) in spherical-polar coordinates (of
course, one could use any other spatial coordinate system as well, the
point is the field is expressed as a function of a fixed location in
space-time).
</p>

<p>
This is like looking at the flow from on top of a bridge at a fixed
position, describing the variables at that specific position.
</p>


<figure id="orgbf8e004">
<img src="./images/Eulerian-bridge.jpg" alt="https://www.flinilckr.com/photos/frixan/114822407/" width="75%">

<figcaption><span class="figure-number">Figure 1: </span>Eulerian description is like putting many bridges over a fluid and describing the properties as a function of what goes on just below the bridge.</figcaption>
</figure>

<p>
For a star assuming spherical symmetry (as we already did to relate
the stellar flux with the luminosity and obtain L=4&pi; R<sup>2</sup>&sigma; T<sub>eff</sub><sup>4</sup>),
we would have no dependence on the angular variables &part;<sub>&theta;</sub> &Psi;
&equiv; &part;<sub>&varphi;</sub>&Psi; &equiv; 0 and we could express &Psi;&equiv;&Psi;(r, t).
</p>
</div>
</div>

<div id="outline-container-org25c228f" class="outline-4">
<h4 id="org25c228f"><a href="#org25c228f">Lagrangian description</a></h4>
<div class="outline-text-4" id="text-org25c228f">
<p>
In a Lagrangian description the fields are described as a function of
the fluid element, following the fluid element itself: &Psi; &equiv;&Psi;(fluid
element, t).
</p>

<p>
This is like hopping on a boat that moves with the fluid, and
describing the properties of the fluid as the boat moves with it.
</p>


<figure id="org3e8e01e">
<img src="./images/Lagrangian_kayak.jpg" alt="https://www.snowaddiction.org/2015/10/the-zen-of-kayaking-i-photograph-the-fjords-of-norway-from-the-kayak-seat.html?m=1" width="75%">

<figcaption><span class="figure-number">Figure 2: </span>Lagrangian description is like following the fluid on a boat and describe what goes on as a function of the fluid particle being followed.</figcaption>
</figure>

<p>
In a Lagrangian system of coordinates, the total time derivative can
be written using the "chain rule" as:
</p>

<div class="latex" id="orgda23a82">
\begin{equation}
 \frac{d}{dt} = \frac{\partial}{\partial t} + \frac{\partial x}{\partial t}\frac{\partial}{\partial x} + \frac{\partial y}{\partial t}\frac{\partial}{\partial y} +\frac{\partial z}{\partial t}\frac{\partial}{\partial z} \equiv \frac{\partial}{\partial t} + v\cdot\nabla \ \,
\end{equation}

</div>

<p>
where the second term on the r.h.s. is the "advective" term, due to
the motion of the fluid element.
</p>

<p>
Because of the extremely large range of spatial scales in stellar
evolution (e.g., in the Sun from the center at r&cong;0 to the outer radius
r&cong; 10<sup>11</sup> cm, and in its future as a red giant 100-1000&times; larger - even
using R<sub>☉</sub> as unit there are &sim;3-5 orders of magnitude of dynamic
range without even considering neutron stars and black holes!),
compared to the limited range in mass (in the appropriate units 0-1
M<sub>☉</sub> for the Sun!), typically a Lagrangian description is preferred.
This means, for example we can chose as independent coordinate in the
star not the radius r, but instead the amount of mass enclosed by a
given radius m&equiv; m(r), thus express &Psi;(r,t) =&Psi;(r(m), t)&rarr; &Psi;(m(r), t).
</p>

<p>
Derivatives can then be expressed with the chain rule:
</p>
<div class="latex" id="orgcb45cce">
\begin{equation}
 \frac{\partial}{\partial m} = \frac{\partial r}{\partial m} \frac{\partial}{\partial r} \ \ .
\end{equation}

</div>

<p>
This is what is typically done in stellar evolution, and it
works because the m(r) function is invertible, and it is in fact
monotonically increasing: as r increases, the amount of stellar mass
enclosed can also only increase!
</p>

<p>
<b>N.B.</b>: For some applications, Eulerian and Lagrangian descriptions can
also be "combined", for example with "moving mesh" numerical
approaches, see for example <a href="https://ui.adsabs.harvard.edu/abs/2020ApJS..248...32W/abstract">AREPO hydrodynamics code</a>, <a href="https://ui.adsabs.harvard.edu/abs/2016ApJS..226....2D/abstract">DISCO code</a>.
</p>
</div>
</div>
</div>

<div id="outline-container-orgfa68081" class="outline-3">
<h3 id="orgfa68081"><a href="#orgfa68081">Spherical symmetry and one dimensional approximation</a></h3>
<div class="outline-text-3" id="text-orgfa68081">
<p>
We have defined a star to be a <i>self-gravitating mass of gas that can
produce energy through internal sources</i>. Thus, from the
<i>self-gravitating</i> and the <i>gas</i> parts of this definition, at
zeroth-order, the dynamical elements (i.e., the forces!) that
determine the structure of a star are:
</p>

<ul class="org-ul">
<li>Gravity: the weight of the gas forming the star itself</li>
<li>Pressure: the thermal pressure of the gas, sometimes with
contribution from the radiation and degeneracy</li>
</ul>

<p>
Gravity is a central force that depends only on the radius (&prop; r<sup>-2</sup>),
and pressure is isotropic. Therefore at zeroth order, we expect stars
to be well approximated as <i>spheres</i>. This mathematically means that we
can express all the variables that characterize the structure of a
star as a function of a single variable, for example the radius from
the center of the star. This allows for the calculation of
</p>

<ul class="org-ul">
<li><b>Q</b>: can you think of cases where a star  may not be spherical?</li>
</ul>
</div>
</div>

<div id="outline-container-orgda10f25" class="outline-3">
<h3 id="orgda10f25"><a href="#orgda10f25">Mass conservation</a></h3>
<div class="outline-text-3" id="text-orgda10f25">
<p>
Let's consider the amount of mass in a parcel of stellar gas. This will
depend on the local gas density \(\rho(r, t)\) (or equivalently in the
Lagrangian formalism \(\rho(m, t)\)!) and the amount of volume in the shell
</p>

<div class="latex" id="org293528e">
\begin{equation}
dm = \rho dAdr
\end{equation}

</div>

<p>
where dA is the element base area, and dr its radial thickness. We can
integrate over the base to get the parcel to be a spherical shell
</p>

<div class="latex" id="org99f30a0">
\begin{equation}
\int dA = 4\pi r^{2}
\end{equation}

</div>

<p>
where r is the radius of the shell, therefore
</p>

<div class="latex" id="orgf1eb0bf">
\begin{equation}
dm = 4\pi \rho r^{2} dr \ \ .
\end{equation}

</div>

<p>
In principle gas could also flow in/out of the shell at a rate
determined by the inflow/outflow velocity such that in a time interval
dt an amount \(-\rho v dA dt\) flows out (for \(v>0\), the quantity is negative)
or in (\(v<0\)). Again integrating over \(dA\):
</p>

<div class="latex" id="org9cb56a8">
\begin{equation}
\label{eq:mass_continuity}
dm = 4\pi \rho r^{2} dr - 4\pi r^{2} \rho v dt \ \ .
\end{equation}

</div>

<p>
This is the complete mass continuity equation in spherical symmetry.
From this complete form we can take the partial derivatives w.r.t. \(r\)
(at fixed \(t\)) and \(t\) (at fixed \(r\)):
</p>

<div class="latex" id="org4b2a158">
\begin{equation}\label{eq:mass_continuity_rt}
 \frac{\partial m}{\partial r} = 4\pi r^{2} \rho \ \ , \\
 \frac{\partial m}{\partial t} = - 4\pi r^{2} \rho v \ \ .
\end{equation}

</div>

<p>
We can also derive the first equation above w.r.t. \(t\) and the second one
w.r.t \(r\), and demand the two forms are the same. Since \(r\) and \(t\) are the
independent variables here (i.e., \(\partial r/\partial t = 0\)) we obtain:
</p>

<div class="latex" id="org535b863">
\begin{equation}
\frac{\partial \rho}{\partial t} = - \frac{1}{r^{2}}\frac{\partial (r^{2}\rho v)}{\partial r} \Leftrightarrow \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho v) = 0 \ \ ,
\end{equation}

</div>
<p>
with \(\partial_{\theta }\equiv \partial_{\phi }\equiv 0\) for the last one, that is
the typical form of the mass continuity equation in spherical
symmetry.
</p>

<p>
To turn these equations in the more typical form for stellar
structure, just take the first equation in \ref{eq:mass_continuity_rt}
and express it with \(m\) as independent variable:
</p>

<div class="latex" id="org6b0de5e">
\begin{equation}\label{eq:mass_conservation}
\frac{\partial r}{\partial m} = \frac{1}{4\pi r^{2} \rho} \ \ ,
\end{equation}

</div>

<p>
where the partial derivatives become total derivatives in a static
situation (where by definition \(\partial_{t} = 0\), which is also why we don't
typically focus on the second equation in
\ref{eq:mass_continuity_rt} - by the end of this lecture we will be
able to discuss whether this is an acceptable approximation). This is
the first stellar structure equation that expresses mass conservation,
and it depends on a yet unknown variable, the gas density \(\rho\).
</p>
</div>
</div>

<div id="outline-container-org916196c" class="outline-3">
<h3 id="org916196c"><a href="#org916196c">Momentum conservation and hydrostatic equilibrium</a></h3>
<div class="outline-text-3" id="text-org916196c">
<p>
Consider the equation of motion of a parcel of stellar gas, \(F = dp/dt
= ma\) (for constant \(m\)), or often more conveniently in fluid dynamics,
work per unit volume with \(f = dF/dV\) and thus \(f=\rho a\) with \(\rho = dm/dV\) and
\(dV=dAdr \Rightarrow V=\int dAdr\) the volume. Let's start by writing down explicitly
the forces that we think are important for an <b>isolated</b>, <b>non-rotating</b>,
<b>non-magnetic</b> star.
</p>
</div>

<div id="outline-container-orga55a5ac" class="outline-4">
<h4 id="orga55a5ac"><a href="#orga55a5ac">Gravity</a></h4>
<div class="outline-text-4" id="text-orga55a5ac">
<p>
Since by definition a star is a self-gravitating body (<b>N.B.:</b> so is a
planet, that's not the whole definition of a star!), we want to
include the gravitational force on the l.h.s. of our f=&rho; a equation.
This can be obtained as the gradient of the gravitational potential &Phi;
which is a solution of the Poisson equation:
</p>

<div class="latex" id="org38c91ce">
\begin{equation}
\nabla^{2} \Phi = 4\pi G\rho \Rightarrow \frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial \Phi}{\partial r} \right) = 4\pi G\rho \ \ ,
\end{equation}

</div>

<p>
where the second form assumes already spherical symmetry. Note how
this equation does not make the problem worse: we have a new variable
\(\Phi\) but the r.h.s. only depends on the density \(\rho\) which is already
appearing in Eq. \ref{eq:mass_conservation}.
</p>

<p>
We can introduce the gravitational acceleration \(g = - \nabla\Phi\), which in
spherical symmetry only has a non-zero radial component &rArr; \(g = - d\Phi/dr\)
which from Newton's theory of gravity we know to be
</p>

<div class="latex" id="org1f40b2a">
\begin{equation}
- \nabla \Phi = g \equiv g(m(r))= \frac{Gm(r)}{r^{2}} \ \ ,
\end{equation}

</div>

<p>
where \(m\equiv m(r)\) is the mass enclosed within a certain radius \(r\), which we
already encountered. Thanks to the spherical symmetry assumption, we
don't even need to really solve Poisson's equation to make a stellar
model! The gravitational force acting on a spherical shell of mass \(dm
= 4\pi r^{2}\rho dr\) is thus just \(-gdm = -Gmdm/r^{2}\), or per unit volume \(f_\mathrm{grav}
= -g\rho = -Gm\rho/r^{2}\), where the minus sign is to explicitly indicate that
this force points towards the center of the star.
</p>
</div>
</div>

<div id="outline-container-org056e5c9" class="outline-4">
<h4 id="org056e5c9"><a href="#org056e5c9">Pressure gradient</a></h4>
<div class="outline-text-4" id="text-org056e5c9">
<p>
The other contribution we need to include in our \(f = \rho a\) equation is
from the pressure. We could already use dimensional analysis to guess
in what form pressure can enter the l.h.s. of the equation:
</p>

<div class="latex" id="org493d7e7">
\begin{equation}
[P] = [\mathrm{force}]/[\mathrm{area}] \Rightarrow [P]/[\mathrm{length}] = [\mathrm{force}]/[\mathrm{volume}] \equiv [f]
\end{equation}

</div>

<p>
This suggests that the pressure divided an appropriate length scale
has the right dimension to enter the force per unit volume f. This in
turn suggests that maybe what we need is the pressure <i>gradient</i>!
</p>


<figure id="orgc7c3751">
<img src="./images/HSE-sketch.png" alt="HSE-sketch.png" width="50%">

<figcaption><span class="figure-number">Figure 3: </span>Sketch of the force balance for an internal layer of a spherically symmetric star. Modified from Onno Pols' lecture notes Fig. 2.1</figcaption>
</figure>


<p>
Let's now have a slightly more formal look. Because of
spherical symmetry, the pressure in the horizontal direction (which in
stellar context always means in the plane orthogonal to the radial
direction) is perfectly balanced, and the pressure only depends on the
radius \(P\equiv P(r) (\equiv P(r(m)) \equiv P(m))\).
</p>

<p>
The net force per unit area on each side of a spherical shell of gas
of thickness dr is \(P(r)\) at the inner boundary and \(P(r+dr)\) at the outer
boundary. Therefore, \(dF_\mathrm{press} = P(r)dA - P(r+dr)dA \simeq dP/dr dA\) where we
used \(P(r+dr)\simeq P(r)+(dP/dr)dr\). Now using \(dm = \rho drdA\) and dividing by \(dV
= drdA\) we finally obtain \(f_\mathrm{press} = - dP/dr\).
</p>
</div>
</div>

<div id="outline-container-org581613d" class="outline-4">
<h4 id="org581613d"><a href="#org581613d">Combining the two</a></h4>
<div class="outline-text-4" id="text-org581613d">
<p>
We have now an explicit form for the two most important forces in a
(isolated, non-rotating, non-magnetic) star \(f = f_{grav} + f_{pres} = -g\rho -
dP/dr \equiv \rho a\).
</p>

<p>
Since stars don't change that much on short timescales (we will see
exceptions later, and define relevant timescales too), we can assume
that overall the acceleration a of each parcel of gas is zero in most
cases, that is \(a=0\). <i>Stars are generally in hydrostatic equilibrium</i>. In
this case the conservation of momentum becomes
</p>

<div class="latex" id="org226864e">
\begin{equation}
\frac{dP}{dr} = -g\rho = -\frac{Gm}{r^{2}}\rho \ \ ,
\end{equation}

</div>

<p>
or changing to have \(m\) has the independent variable, to have a
Lagrangian treatment:
</p>

<div class="latex" id="org2f1a083">
\begin{equation}
\frac{dP}{dr} = \frac{dP}{dm}\frac{dm}{dr} = \frac{dP}{dm}4\pi r^{2}\rho
\end{equation}

</div>

<p>
and thus
</p>

<div class="latex" id="org301ebcd">
\begin{equation}\label{eq:HSE}
\frac{dP}{dm} = -\frac{Gm}{4\pi r^{4}} \ \ ,
\end{equation}

</div>

<p>
which is the second stellar structure equation that expresses the fact
that the gravitational pull of the stellar gas is compensated by the
pressure gradient inside the star. This also means that it is the
gravity of the star that imposes the pressure stratification of the
star and ultimately its structure: the pressure in each layer is just
what is needed to support the weight of the layer(s) above. And finally,
the fact that \(dP/dm<0\), that is the pressure decreases as the enclose
mass increases, or equivalently, the pressure increases towards the
center (smaller radii, smaller amount of enclosed mass) makes sense,
if the gradient has to compensate the gravitational pull.
</p>

<p>
<b>All of stellar evolution can be though as gas re-arranging itself to
fight against gravity, delaying gravitational collapse.</b>
</p>

<p>
<b>N.B.:</b> The hydrostatic equilibrium equation can also be obtained
starting from the Navier-Stokes equation assuming no viscosity (the
microscopic viscosity is generally negligible in stars).
</p>

<p>
<b>N.B.:</b> We have implicitly assumed that the star we model is
sufficiently far away from anything else that there are no external
forces. This may not hold in a binary system, for which in the
Euler equation there will be other terms, such as the gravity of the
binary companion, and tidal forces arising from its presence. While
these are important, they often affect most directly only the outer
layers of a star (that can be significantly tidally distorted), and
can maybe be neglected further in the interior.
</p>

<p>
<b>N.B.:</b> Similarly, we have neglected rotation, which also breaks the
spherical symmetry by adding in the reference frame co-rotating with
the star non-inertial forces (centrifugal, Euler, and Coriolis).
However, the centrifugal foce depends on the distance from the
rotation axis (r sin(&theta;)) and thus mostly impacts the outer layers of
the stars and is less critical in the inner regions (though not at all
always negligible!). The Euler force depends on d&omega;/dt with &omega; rotation
rate, so it is typically negligible on evolutionary timescales for the
star. The Coriolis force depends on &omega; &times; v, so it does not affect
static gas (but it does have important effects if there are
velocities, think for examples hurricanes in the Earth atmosphere).
Moreover, rotation can interplay with many hydrodynamical and secular
instabilities affecting the stellar gas in ways that are only roughly
approximated in models. All these complications introduced by rotation
and how to model them in stellar evolution are still active
research topics (see for example <a href="https://ui.adsabs.harvard.edu/abs/2000ARA%26A..38..143M/abstract">Maeder &amp; Meynet 2000</a> and <a href="https://ui.adsabs.harvard.edu/abs/2000ApJ...528..368H/abstract">Heger et al.
2000</a>).
</p>

<p>
<b>N.B.:</b> Finally, we have neglected also the impact of magnetic fields on
the stellar gas. We know that stellar magnetism exists from
observational phenomena such as stellar flares, seeing Zeeman
splitting in stellar spectra, etc. We should also expect magnetic
fields theoretically, because stars are giant balls of ionized gas.
However, the global dynamical impact of magnetic field should be small
in most cases, given the success of stellar evolution in explaining
many observations neglecting them. Stellar magnetism (and it's
important interplay with rotation) is also an active field of research
both observationally and theoretically.
</p>

<p>
Equations Eq. \ref{eq:mass_conservation} and \ref{eq:HSE} are two
differential equations, that under the assumption of spherical
symmetry are ordinary differential equations (\(\partial_{r} \rightarrow d/dr\)),
for the function \(m\equiv m(r)\) that depend on \(P\), \(\rho\). We thus have three
variables \((m, P, \rho)\) and two equations: we cannot yet solve for the
structure of a star. We will close the system of equations (meaning,
obtain as many equations as variables, so we can solve for the stellar
structure) later in the course.
</p>
</div>
</div>

<div id="outline-container-org2757f7e" class="outline-4">
<h4 id="org2757f7e"><a href="#org2757f7e">Estimate for the central pressure</a></h4>
<div class="outline-text-4" id="text-org2757f7e">
<p>
A first estimate for the central pressure can be obtained substituting
the local gradient with the difference from surface to the core across
the entire mass of the star \(dP/dm \rightarrow (P_{surface} - P_{center})/M \simeq
-P_{center}/M\), where we also use \(P\) increases inwards and thus it is
legitimate to expect \(P_{center}\gg P_{surface}\). Then, on the l.h.s. of Eq.
\ref{eq:HSE}, we should take as estimates some fraction of the total
mass M and radius R. For the sake of simplicity, let's take the
fraction to be 1 and drop the 4&pi;:
</p>

<div class="latex" id="orga6fe5ee">
\begin{equation}
P_\mathrm{center} = \frac{GM^{2}}{R^{4}}\ \ ,
\end{equation}

</div>

<p>
Plugging in the numbers for the Sun this gives \(P_{center}\simeq 10^{16}
dyne cm^{-2}\simeq 10^{10}\) atmospheres. Although this a is very imprecise
estimate, it already gives the idea that the pressure in the center of
the Sun must be extremely high. See Onno Pols chapter 2 for more
precise estimates and lower bounds.
</p>
</div>
</div>
</div>

<div id="outline-container-org5cb62bc" class="outline-3">
<h3 id="org5cb62bc"><a href="#org5cb62bc">Dynamical timescale estimates</a></h3>
<div class="outline-text-3" id="text-org5cb62bc">
<p>
Let's say that the star was not in hydrostatic equilibrium, but still
spherically symmetric. Returning to the general form for the momentum
conservation \(f = \rho a \equiv \rho \partial^{2}r/\partial t^{2}\) we have
</p>

<div class="latex" id="org5f6d74a">
\begin{equation}\label{eq:dyn}
\rho \frac{\partial^{2} r}{\partial t^{2}} = -\frac{dP}{dr} -\frac{Gm}{r^{2}}\rho \ \ ,
\end{equation}

</div>

<p>
where since P decreases outwards, \(dP/dr<0\), so the first term on the
l.h.s. pushes outwards (positive radial acceleration), while gravity
pulls inward, as one would expect.
</p>

<p>
Normally, for a star, we expect these two terms to balance each other,
but what happens if we turn one off?
</p>
</div>

<div id="outline-container-org5134097" class="outline-4">
<h4 id="org5134097"><a href="#org5134097">Explosion timescale</a></h4>
<div class="outline-text-4" id="text-org5134097">
<p>
Let's turn off gravity, setting \(g = - Gm/r^{2 }\rightarrow 0\)! To
estimate how long it takes for the pressure gradient to push the gas
out to a radius comparable to the radius of the star we can do the
following rough substitution in the dynamical equation above:
</p>
<ul class="org-ul">
<li>\(\partial^{2} r \rightarrow R\) (outer radius of the star)</li>
<li>\(\partial t^{2} \rightarrow \tau_mathrm{expl}^{2}\) (what we want to estimate)</li>
<li>\(dP/dr\rightarrow P_\mathrm{avg}/R\) with \(P_\mathrm{avg}\) some averaged pressure in the star</li>
<li>\(\rho \rightarrow \rho_\mathrm{avg}\) some averaged density of the star</li>
</ul>
<p>
and we obtain:
</p>

<div class="latex" id="org7e4c6ba">
\begin{equation}
\tau_\mathrm{expl} \simeq \frac{R}{\sqrt{\frac{P_{avg}}{\rho_{avg}}_{}}} \simeq \frac{R}{c_{s}}\ \ .
\end{equation}

</div>

<p>
where, if we interpret \(P\) and \(\rho\) as some average values throughout the
star the sound speed \(c_{s}^{2} = P/\rho\) appears! The timescale for the
internal pressure of the stellar gas to "blow up" the star is of the
order of the sound-crossing time.
</p>
</div>
</div>

<div id="outline-container-orgccbd6ce" class="outline-4">
<h4 id="orgccbd6ce"><a href="#orgccbd6ce">Free fall timescale</a></h4>
<div class="outline-text-4" id="text-orgccbd6ce">
<p>
Almost by definition, this is how the star would collapse if there
were no forces other than gravity, so let's turn off the pressure
gradient \(dP/dr\rightarrow0\). Then, as above:
</p>
<ul class="org-ul">
<li>\(\partial^{2} r \rightarrow R\) (outer radius of the star)</li>
<li>\(\partial t^{2} \rightarrow \tau_{ff}^{2}\) (what we want to estimate)</li>
<li>\(m \rightarrow M\) (total mass)</li>
</ul>
<p>
we get:
</p>

<div class="latex" id="org54cc847">
\begin{equation}
\tau_\mathrm{ff} \simeq \sqrt{\frac{R^{3}}{GM}} \simeq \sqrt{\frac{1}{G\rho_\mathrm{avg}}}\ \ ,
\end{equation}

</div>
<p>
with \(\rho_\mathrm{avg} = 3M/(4\pi R^{3})\) average density of the star. Note that here we
have been very loose with the &pi; factors and averages.
</p>

<ul class="org-ul">
<li><b>Q</b>: you all have estimated the Sun's mean density, calculate the Sun
free fall time now. Does the Sun vary on this timescale? Do you
think this justifies our assumption of hydrostatic equilibrium?</li>
<li><b>Q</b>: are stars in hydrostatic equilibrium? How do we know
observationally?</li>
</ul>
</div>
</div>
</div>

<div id="outline-container-orgf56001b" class="outline-3">
<h3 id="orgf56001b"><a href="#orgf56001b">Introduction to <code>MESA_web</code></a></h3>
<div class="outline-text-3" id="text-orgf56001b">
<p>
We will discuss in detail stellar evolution codes, numerical
strategies for solving the stellar structure equations, and what goes
on in MESA/MESA-web. For now I just want to introduce this tool and
show you how you can obtain numerical stellar models.
</p>

<ul class="org-ul">
<li><a href="http://user.astro.wisc.edu/~townsend/static.php?ref=mesa-web-input">Description of Input</a></li>
<li><a href="http://user.astro.wisc.edu/~townsend/static.php?ref=mesa-web-submit">Submission website</a></li>
<li>Example output:
<ol class="org-ol">
<li>Download the zip file from the email you receive when the
calculation is done</li>
<li>Unzip the file, the content has a <code>*.mp4</code> video with the evolution
of some quantities (depending on the star you asked, it may be
very short), an <code>input.txt</code> file that reminds you of what you put
into <code>MESA-web</code>, the  <code>trimmed_history.data</code>  and a few
<code>profile*.data</code>, and a <code>profiles.index</code> that contains a map of which
<code>profile*.data</code> maps to which "model number" (i.e., timestep of the
code).</li>
<li><p>
You can inspect the <code>txt, list</code>, and <code>*.data</code> files using your text
editor.
</p>

<p>
The <code>trimmed_history.data</code> contains in each column global variables
of the star (e.g., surface luminosity, outer radius, etc.) and
each row correspond to a specific timestep. This is what you can
use to plot, for example, an Herzsprung-Russell diagram using the
columns <code>log_L</code> and <code>log_Teff</code>.
</p>

<p>
The <code>profile*.data</code> files contain each a snapshot of the internal
structure of the star you simulated at fixed time, so each column
corresponds to a quantity that takes different values at
different locations in the star (e.g., Lagrangian mass
coordinate, density, pressure, opacity). Each row corresponds to
a "mesh point", that is a discretized spatial coordinate (we will
see later what the full set of equations is and how codes like
MESA solve them).
</p>

<p>
Refer to the <a href="http://user.astro.wisc.edu/~townsend/static.php?ref=mesa-web-output">MESA-web output page</a> for a full description of the
output.
</p></li>
<li><i>If</i> you want you can use the python module <a href="http://user.astro.wisc.edu/~townsend/resource/tools/mesa-web/mesa_web.py"><code>mesa_web.py</code></a> provided by
<code>MESA-web</code> to read the output in the <code>*.data</code>, but remember these are
just plain text, so you can also write your own.</li>
</ol></li>
</ul>
</div>
</div>
</div>

<div id="outline-container-orgfd6903c" class="outline-2">
<h2 id="orgfd6903c"><a href="#orgfd6903c">Homework</a></h2>
<div class="outline-text-2" id="text-orgfd6903c">
<ul class="org-ul">
<li>Calculate the Keplerian period of a point mass orbiting at the
surface of a star of mass M and radius R and compare it to the free
fall timescale of the star.</li>
<li>Calculate the free fall timescale for the Sun, for a Red Supergiant
with M=10M<sub>☉</sub> and R=1000R<sub>☉</sub> and a White Dwarf with M=1M<sub>☉</sub>
and R=1000 km, and a Neutron star with M=1M<sub>☉</sub> and radius R=10km.
Compare also their average densities.</li>
<li>Skim <a href="https://ui.adsabs.harvard.edu/abs/2023arXiv230915930F/abstract">MESA-web paper by Fields et al. 2022</a>.</li>
<li>Using <a href="http://user.astro.wisc.edu/~townsend/static.php?ref=mesa-web-submit">MESA-web</a> make a 1 M<sub>☉</sub> star until age 4.5&times;10<sup>9</sup> years (a
very rough model of the Sun as it is today!). <b>Hint:</b> look a the
<code>Custom Stopping condition</code> menu for possible definitions of "end of
the simulation". Plot m(r), make sure
to label your axes properly (including units!). Are there other
variables with a qualitatively similar behavior that one could use
as independent coordinate for the stellar structure? Try to make
other plots to find some, and explain what is the mathematical
property that allows to use m(r) and or any other variable you found
as a coordinate.</li>
<li>With the model above, check the central pressure of the star (you
can also plot P(m) and P(r), or look at the final frame in the
movie made by <code>MESA-web</code> for you) and compare it with the estimate
above and the one provided in Onno Pols' lecture notes.</li>
<li>Check also the outer luminosity: is it the value you expected?</li>
</ul>
</div>
</div>
</div>
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