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<p>
<b>Materials:</b> Mihalas <i>Stellar atmospheres</i> book, Chapter 3. de Koter's
<a href="https://staff.fnwi.uva.nl/a.dekoter/ARTv1.103.00.pdf">lecture notes on radiative transfer and stellar atmospheres</a>, Shore's
<i>Astrophysical Hydrodynamics</i> book, Lamers &amp; Cassinelli 1994 book.
</p>

<div id="outline-container-org4b30cf6" class="outline-2">
<h2 id="org4b30cf6"><a href="#org4b30cf6">Stellar atmospheres</a></h2>
<div class="outline-text-2" id="text-org4b30cf6">
<p>
So far in this course we have mostly dealt with the <i>interior</i> of stars
and derived equation to determine the stellar <i>structure</i> and what
drives their <i>evolution</i> (which is typically "slow", \(\tau_\mathrm{nuc} \gg \tau_\mathrm{KH}
\gg \tau_\mathrm{ff}\)).
</p>

<p>
The picture we have derived is that gravity drives a contraction of
the (hot) stellar gas, which, because it has a finite temperature,
shines. This loss of internal energy translates into a loss of
gravitational potential because of the virial theorem relating the
two, and consequently the temperature in the center increases. This
continues until nuclear burning ignites to compensate for the energy
losses: <i>stars don't shine because they burn, stars burn <span class="underline">because</span> they
shine</i>.
</p>

<p>
The nuclear burning can be seen just as an attempt by the star to
<i>delay</i> its gravitational collapse: as nuclear fuel is depleted the star
is forced to evolve (until either gravity "wins" or the EOS deviates
from the classical ideal gas, and quantum mechanical effects provide
sufficient pressure that win over gravity).
</p>

<p>
However, many of the assumptions we have made to derive these
equations and the picture of stellar evolution described above <i>do not
apply to the stellar atmosphere</i>, which is the layer that produces the
<i>detectable</i> photons from stars! In this lecture we will consider in
more details the physics of the stellar atmosphere which are crucial
to obtain empirical evidence to test the picture of stellar evolution
we now can build, and to provide the outer boundary conditions needed
to calculate a stellar model.
</p>

<ul class="org-ul">
<li><b>Q</b>: which assumptions that hold in the stellar interior don't apply
to the atmosphere?</li>
</ul>
</div>

<div id="outline-container-org3a59498" class="outline-3">
<h3 id="org3a59498"><a href="#org3a59498">Definition</a></h3>
<div class="outline-text-3" id="text-org3a59498">
<p>
The <i>stellar atmosphere</i> is by definition the layer where the radiation
field is <i>not</i> isotropic, but the photon flux has a net <i>radial</i>
component.
</p>

<p>
<b>N.B.:</b> This does <i>not</i> mean that all the photons move radially! Just that
on average there are more photons moving in the positive radial
direction than the negative radial direction, but photons can still
move in all direction!
</p>

<p>
This means that <i>by definition</i> the stellar atmosphere is <i>not</i> a black
body, and it is <i>partially</i> transparent to photons (resulting in a
positive radial component of the photon flux). We <i>cannot</i> use the
diffusion approximation to treat radiative transport here, since the
mean free path of photons \(\ell_{\gamma}\) are becoming comparable to the
thickness of the atmosphere (\(\rho\) decreases moving outward!).
</p>

<p>
To make models of stellar <i>spectra</i> and determine the outer boundary
conditions we need to consider how radiation from the bottom flows
through the atmospheric layer and consequently how this layer
stratifies, determining the outer boundary pressure and temperature.
</p>

<p>
<b>N.B.:</b> real stars are even more complicated: in the photosphere
radiative and (magneto)-hydrodynamical processes can launch stellar
winds that remove mass from the star and connect smoothly the stellar
material with the interstellar material!
</p>

<p>
By definition, the bottom layer of the atmosphere is the <i>photosphere</i>,
that is this fictional surface that has T<sub>eff</sub> such that a black body of
this temperature would emits the same energy per unit time as the star
does. This is from where the bulk of the stellar radiation comes from
<i>by definition</i> (recall the <a href="notes-lecture-CMD-HRD.html#org68cd301">discussion on spectral types!</a>).
</p>

<p>
<b>N.B.:</b> \(dL/dm = \varepsilon_\mathrm{nuc} - \varepsilon_{\nu} + \varepsilon_\mathrm{grav} \equiv 0\) in the outer
layers of the star where \(T\) is not high enough for nuclear burning, \(\rho\)
is not high enough for neutrino emission, and we assume gravothermal
equilibrium, so \(\varepsilon_\mathrm{grav} \propto ds/dt = 0\), so \(L\) is constant.
</p>
</div>
</div>
</div>

<div id="outline-container-orga3a3a99" class="outline-2">
<h2 id="orga3a3a99"><a href="#orga3a3a99">Basics of radiative transfer</a></h2>
<div class="outline-text-2" id="text-orga3a3a99">
<p>
The problem we need to consider in the stellar atmosphere is how
radiation coming from the <i>photosphere</i> flows through the overlaying
gas, and how this impact the observable radiation itself, and the
structure of these gas layers (thus determining the pressure at the
base of the atmosphere, which is the outer boundary condition for the
interior!).
</p>

<p>
<b>N.B.:</b> a useful thing for this problem is to "think like a photon", and
imagine the physics it will encounter during its trip towards the
detector!
</p>

<p>
Since by definition the radiation coming out from the photosphere is a
black body, it is isotropic, so if you think of the atmosphere as a
slab of gas with a thickness \(dr\), the radiation illuminating it from
below may come at an angle \(\theta\) w.r.t. the slab.
</p>

<p>
We can define the specific intensity (per unit frequency or
wavelength) \(I_{\nu}\) as the amount of energy flowing through a surface
element \(dA\) in a time interval \(dt\) and coming within a solid angle
\(d\Omega\) around the direction \(\mathbf{n}\) with frequency in the range
between \(\nu\) and \(\nu+d\nu\):
</p>

<div class="latex" id="orgc2e99a2">
\begin{equation}
I_{\nu} \equiv I_{\nu}(\theta) = \frac{dI}{d\nu} = \frac{dE_{\nu}}{d\nu dt dA d\Omega} \mathbf{n} \ \ ,
\end{equation}

</div>

<p>
which has the dimensions of [E]/([L<sup>2</sup>][t][&nu;][solid angle]). This would
be constant as photons propagate along a path of length ds along the
direction \(\mathbf{n}\), however there can be processes that add
photons:
</p>
<ul class="org-ul">
<li>scattering from another direction onto the direction of interest;</li>
<li>emission processes;</li>
</ul>
<p>
and process that can remove photons:
</p>
<ul class="org-ul">
<li>scattering from the direction of propagation onto another direction;</li>
<li>absorption processes.</li>
</ul>
<p>
Moreover, \(I_{\nu}\) itself may in general be time dependent (although
this is not the case for stellar atmosphere), so we can write down the
equation of radiative transfer as
</p>

<div class="latex" id="orgf03ac97">
\begin{equation}\label{eq:radTrans}
\frac{dI_{\nu}}{ds} = \frac{1}{c}\frac{\partial I_{\nu}}{\partial t} + \mathbf{n}\cdot\nabla I_{\nu} = -\kappa_{\nu}\rho I_{\nu} + j_{\nu}\rho \ \ ,
\end{equation}

</div>

<p>
where the l.h.s. expresses the total change in specific intensity
along the direction \(\mathbf{n}\) due to the intrinsic time-dependence
(\(\partial_{t}\)) of \(I_{\nu}\) and the spatial dependence along the direction we are
considering (\(\mathbf{n}\cdot\nabla\)), and the r.h.s. expresses the loss of
radiation intensity due to scattering <i>and</i> absorption processes, which
depends on \(\kappa_{\nu}\) (the specific opacity we have already encountered) and
is proportional to \(I_{\nu}\) itself (you can't lose photons you don't
have!), and the addition of radiation intensity from emission
processes and scattering along the line of sight which depends on the
emission coefficient \(j_{\nu}\).
</p>

<p>
<b>N.B.:</b> dimensional analysis reveals that each side has the units of
[\(I_{\nu}\)]/[L], this equation describes how the intensity changes along
its path. The fact that photons propagate at the speed of light c make
the leftmost factor of \(1/c\) appear: \(d/ds = c\partial_{t} + \mathbf{n}\cdot\nabla\). The
density \(\rho\) on the l.h.s. expresses that the more matter there is (per
unit volume), the more likely there will be absorption and emission.
</p>

<p>
The specific intensity at the bottom of the atmosphere is related to
the photospheric emission by:
</p>

<div class="latex" id="orgd82264c">
\begin{equation}\label{eq:flux_BB}
F \equiv \int_{0}^{+\infty} d\nu F_{\nu} \equiv \sigma T_\mathrm{eff}^{4} =  \int_{0}^{+\infty} d\nu \int d \Omega \cos(\theta) I_{\nu} \ \ ,
\end{equation}

</div>

<p>
that is the black body flux \(F\) is obtained by integrating the specific
intensity over the solid angles. Note the factor \(\cos(\theta)\) that arises
because \(I_{\nu}\) is a vector and we only want the component normal to the
surface element \(dA\).
</p>

<p>
This last expression is going to be useful to connect the physics in
the atmosphere with the interior, since we <i>define</i> the photosphere to
have a flux \(\sigma T_\mathrm{eff}^{4}\).
</p>

<p>
<b>N.B.:</b> Because of Eq. \ref{eq:flux_BB} the photosphere flux acts as
inner boundary condition for the problem of the radiative transfer
through the atmosphere. We have already seen that it acts as outer
boundary condition for the stellar interior.
</p>
</div>

<div id="outline-container-org81db0e1" class="outline-3">
<h3 id="org81db0e1"><a href="#org81db0e1">Simple solutions of the steady state radiative transfer equation</a></h3>
<div class="outline-text-3" id="text-org81db0e1">
</div>
<div id="outline-container-orgffd5b09" class="outline-4">
<h4 id="orgffd5b09"><a href="#orgffd5b09">Steady state without emission</a></h4>
<div class="outline-text-4" id="text-orgffd5b09">
<p>
In absence of an explicit time dependence (\(\partial_{t} I_{\nu} =0\)) and emission
processes (\(j_{\nu}=0\)), this equation is easily solved calling s the
length element along the direction \(\mathbf{n}\) so that \(\mathbf{n}\cdot\nabla
\equiv d/ds\), and the solution becomes:
</p>

<div class="latex" id="orgc3d1b5b">
\begin{equation}
I_{\nu} = I_{\nu,0} e^{-\kappa_{\nu}\rho s} =  I_{\nu,0} e^{-\tau_{\nu}} \ \ ,
\end{equation}

</div>
<p>
where we introduce the definition of specific optical depth \(d\tau_{\nu} =
\kappa_{\nu}\rho ds\). This variable is useful because it gives the
scale-length of the problem as depending on \(\kappa_{\nu}\rho =
1/\ell_{\gamma,\nu}\) with \(\ell_{\gamma,\nu}\) the mean free path for a photon
of frequency between \(\nu\) and \(\nu+d\nu\). Effectively, this allows us to
use \(\tau_{\nu}\) as the independent coordinate for the propagation of photons
of frequency between \(\nu\) and \(\nu+d\nu\).
</p>
</div>
</div>

<div id="outline-container-org87a17d9" class="outline-4">
<h4 id="org87a17d9"><a href="#org87a17d9">Steady state with emission and absorption canceling each other</a></h4>
<div class="outline-text-4" id="text-org87a17d9">
<p>
With the definition of \(d\tau_{\nu}\), we can re-write Eq. \ref{eq:radTrans}
(still assuming no explicit time dependence, \(\partial_{t}I_{\nu} = 0\)) as:
</p>

<div class="latex" id="orgb33dc17">
\begin{equation}\label{eq:rad_trans_tau}
\frac{dI_{\nu}}{d\tau_{\nu}} = \frac{j_{\nu}}{\kappa_{\nu}} - I_{\nu} \equiv S_{\nu} - I_{\nu} \ \ ,
\end{equation}

</div>

<p>
where in the last step we define the source function
\(S_{\nu}=j_{\nu}/\kappa_{\nu}\). In thermal equilibrium and at high optical
depth, for instance in the interior region of a star,
\(dI_{\nu}/d\tau_{\nu}=0\) and \(I_{\nu} = B(\nu,T)\) is the black body function
for the intensity, and this equation states \(S_{\nu} = I_{\nu} \equiv B(\nu,
T)\).
</p>

<p>
This effectively is a statement that at thermal equilibrium, the
emission processes, the absorption processes, and scattering in and
out of the direction of interest all cancel each other out.
</p>
</div>
</div>
</div>

<div id="outline-container-org7588199" class="outline-3">
<h3 id="org7588199"><a href="#org7588199">Eddington atmosphere</a></h3>
<div class="outline-text-3" id="text-org7588199">
<p>
The simplest stellar atmosphere model that allows to define
non-trivial outer boundary conditions is the so called "Eddington gray
atmosphere", which provides an analytic \(T(\tau)\) relation in the
atmosphere that can be smoothly attached to the stellar interior where
\(T \equiv T_\mathrm{eff}\) and used to calculate the pressure needed at such
boundary to have hydrostatic equilibrium. In other words, the
Eddington gray atmosphere allows one to define (non-trivial) outer
boundary conditions for the stellar interior problem.
</p>

<p>
Let's start with the assumption of a <i>plane parallel atmosphere</i>, that
is we neglect the <i>curvature</i> of the stellar atmosphere, which is
acceptable if its radius is much larger than the length scale of
interest at any point in it. This assumption reduces the problem to a
one-dimensional problem along the vertical direction, and \(ds =
-dz/cos(\theta)\) for the element of length along a generic photon path \(ds\),
and rewrite the steady state (\(\partial_{t} = 0\)) radiative transfer equation as:
</p>

<div class="latex" id="orga7966ca">
\begin{equation}
\cos(\theta)\frac{d I_{\nu}}{d\tau_{\nu}} = - (S_{\nu}-I_{\nu}) \ \ .
\end{equation}

</div>

<p>
<b>N.B.:</b> we define ds and \(dz\) to be antiparallel (introducing a minus
sign), because we want \(d\tau\) to be positive moving inwards toward
negative \(z\).
</p>

<p>
The second approximation of the Eddington atmosphere is that we assume
a "gray" radiative transfer, meaning the opacity is <i>independent of
frequency</i> \(\kappa_{\nu}\rightarrow\kappa\), thus \(\tau_{\nu}\rightarrow\tau\).
We also neglect the frequency dependence of the source term \(S_{\nu}\).
With these hypotheses we can now integrate this in \(d\nu\) from 0 to +&infin;
and obtain:
</p>

<div class="latex" id="org6a6aae9">
\begin{equation}\label{eq:gray_eq}
\cos(\theta) \frac{dI}{d\tau} = - (S-I) \ \ ,
\end{equation}

</div>
<p>
which can be solved analytically (multiply by \(e^{-\tau/cos(\theta)}\),
rewrite the l.h.s. as a total derivative and integrate in \(d\tau\))
getting
</p>

<div class="latex" id="orgc0a4d90">
\begin{equation}
I(\tau,\theta) = \frac{\exp(\tau/\cos(\theta))}{\cos(\theta)} \int_{\tau}^{+\infty} S\exp(-\tau/\cos(\theta))d\tau \ \ ,
\end{equation}

</div>
<p>
where the r.h.s. is integrated from a certain optical depth \(\tau\)
outwards. We can recover the \(\nu\) dependence of \(S \rightarrow S(\tau)\) as an
optical depth dependence in this integral.
</p>

<p>
We can also define the radiation energy density \(u\), the total flux
\(F\), and the radiation pressure as moments of the intensity \(I(\tau,\theta)\)
w.r.t. \(\cos(\theta)\) (since \(\theta\) always appears in a cosine, it is usual to
change variable to \(\cos(\theta)=\mu\) in radiative transfer calculations):
</p>

<div class="latex" id="orga7717c5">
\begin{equation}\label{eq:momenta_rad}
u \equiv u(\tau) = \frac{2\pi}{c} \int_{-1}^{1} I(\tau, \theta)d\cos(\theta) \ \ ,\\
F \equiv F(\tau) = 2\pi\int_{-1}^{1} I(\tau, \theta)\cos(\theta)d\cos(\theta) \ \ , \\
P_\mathrm{rad} \equiv P_\mathrm{rad}(\tau) = \frac{2\pi}{c}\int_{-1}^{1} I(\tau, \theta)\cos^{2}(\theta)d\cos(\theta) \ \ .
\end{equation}

</div>

<p>
We can also define the average specific intensity as \(J(\tau) = (4\pi)^{-1}\int
I(\tau)d\Omega \equiv 0.5\int_{-1}^{+1} I(\tau)dcos(\theta)\), so that \(J=c u/4\pi\). and dividing Eq.
\ref{eq:gray_eq} by two and integrating between -1 and 1 in \(\cos(\theta)\) we
have
</p>

<div class="latex" id="org5cc9db8">
\begin{equation}\label{eq:J_S}
\frac{1}{4\pi}\frac{d F}{d\tau} = J-S \ \ .
\end{equation}

</div>

<p>
Now the total radiative gray flux in the atmosphere has to be
constant, \(dF/d\tau = 0\): there is radiative equilibrium and what goes in
must come out! So this equations tells us \(J=S\).
</p>

<p>
We can also take Eq. \ref{eq:gray_eq} and multiply it by \(\cos^{2}(\theta)\)
and integrate between -1 and 1 in \(\cos(\theta)\) to obtain:
</p>

<div class="latex" id="org002760c">
\begin{equation}\label{eq:sol_S}
\frac{dP_\mathrm{rad}}{d\tau} = \frac{F}{c} \ \ .
\end{equation}

</div>

<p>
The r.h.s. is constant, so this can be integrated to give
\(P_\mathrm{rad} = F\tau/c + \mathrm{constant}\). One more hypothesis of
the Eddington approximation is to <i>assume</i> that the gas is radiation
pressure dominated \(P_\mathrm{rad} \gg P_\mathrm{gas}\) (this was to
allow him to proceed further): then we also know from thermodynamics
that \(P_\mathrm{rad}=u/3 \equiv 4\pi J/3c\) (using the definition of \(J\) and
its relation with the radiation energy density \(u\)). Putting all these
findings together:
</p>

<div class="latex" id="orgbb69020">
\begin{equation}
S = J = \frac{3 P c}{4\pi} = \frac{3F}{4\pi}\left(\tau + \mathrm{constant}\right) \ \ ,
\end{equation}

</div>
<p>
that is we have an expression for the source function!
Substituting for S in the solution for I we get:
</p>
<div class="latex" id="org476c740">
\begin{equation}
I(\tau, \cos(\theta)) = \frac{3F}{4\pi}\frac{\exp(\tau/\cos(\theta))}{\cos(\theta)}\int_{\tau}^{+\infty} \left(\tau+\mathrm{constant}\right) \exp\left(-\frac{\tau}{\cos(\theta)}\right)d\tau \Rightarrow I(0,\cos(\theta)) = \frac{3F}{4\pi}(\cos(\theta)+\mathrm{constant}) \ \ .
\end{equation}

</div>

<p>
To determine the constant of integration, we can use the second
Eq. \ref{eq:momenta_rad} which defines F using the solution for
I(&tau;=0,cos(&theta;)) in the integral:
</p>

<div class="latex" id="org071dc2d">
\begin{equation}
F = 2\pi\int_{-1}^{1}I\cos(\theta)d\cos(\theta) = \frac{3F}{2}\int_{-1}^{1}\left(\cos^{2}(\theta)+\mathrm{constant}\cos(\theta)\right)d\cos(\theta) = \frac{3F}{2}(\frac{1}{3}+\frac{\mathrm{constant}}{2})\\
 \Rightarrow \mathrm{constant} = \frac{2}{3} \ \ .
\end{equation}

</div>
<p>
We obtained this constant imposing the flux to come from &tau;=0 &rArr; &kappa; = 0,
so from the layer after which there is nothing impeding the photons
anymore (<b>N.B.:</b> the only other option is &rho;=0, so there is nothing, or
ds=0, so the photons have not moved!). With this constant, we
completely specified the source function S &equiv; S(&tau;) and we can obtain I&equiv;
I(&tau;) and use it to calculate the pressure!
</p>
</div>

<div id="outline-container-org18620cc" class="outline-4">
<h4 id="org18620cc"><a href="#org18620cc">Outer boundary conditions of the stellar problem: T<sub>eff</sub> and P</a></h4>
<div class="outline-text-4" id="text-org18620cc">
<p>
From Eq. \ref{eq:J_S} and \ref{eq:sol_S} we now have:
</p>

<div class="latex" id="org7736580">
\begin{equation}
J = S =  \frac{3F}{4\pi}\left(\tau+\frac{2}{3}\right)  \ \ ,
\end{equation}

</div>

<p>
but also, assuming that the atmosphere is also in LTE (including
radiation!), \(J=S=B(\nu,T)=\sigma T^{4}/\pi\), so using that \(F=\sigma T_\mathrm{eff}^{4}\) we
obtain:
</p>

<div class="latex" id="org425c31a">
\begin{equation}
T^{4} = \frac{3}{4}T_\mathrm{eff}^{4}(\tau+\frac{2}{3}) \ \ ,
\end{equation}

</div>

<p>
which is the Eddington \(T(\tau)\) relation which connects the effective
temperature of the black body to the outer temperature \(T(\tau)\) under the
approximations for the atmosphere:
</p>
<ol class="org-ol">
<li>plane parallel</li>
<li>gray (i.e., independent on frequency &nu;)</li>
<li>radiation dominated</li>
<li>Local thermal equilibrium.</li>
</ol>


<p>
<b>N.B.:</b> In the stellar atmosphere, \(T\) is a steep function of \(\tau\) in this
approximation!
</p>

<p>
<b>N.B.:</b> in this approach the photosphere correspond to \(\tau=2/3\), this
factor comes from imposing \(T=T\mathrm{eff}\) in the radiation
dominated, gray, plane parallel atmosphere. This number is a direct
consequence of these specific approximations, and it makes sense that
it is of order &sim;1: the black body radiation from the stellar interior
comes from the region where &tau; goes from \(\tau\gg1\) (where \(I_{\nu} = S_{\nu}\) and
we have a black body distribution for radiation) to \(\tau\le1\) (where the
optical depth is low and we cannot assume black body) to . Once again,
it is important to remember that the photosphere is an idealization,
and nothing that special occurs at \(\tau=2/3\), it's just a convenient
location where we can stitch the Eddington gray atmospheric model to
the interior model.
</p>


<p>
Finally, to find the outer boundary pressure, we need to integrate
downward from \(\tau=0\) to \(\tau(T=T_\mathrm{eff})\simeq2/3\) the hydrostatic
equilibrium equation. We typically assume that the atmosphere is in
hydrostatic equilibrium, however <i>can</i> be a big assumption, depending on
the star and whether it loses mass and whether the interaction between
radiation and the gas drives non-trivial dynamics. Furthermore, we
usually assume that the gravity is constant, or in other words, we
neglect the atmosphere's "self-gravity" since the bulk of the mass is
inside its inner boundary. One can just assume that \(\kappa\) is constant
throughout the atmosphere, an oversimplification that allows for an
analytic calculation which yields:
</p>

<div class="latex" id="orgfb2381f">
\begin{equation}
P(\tau) = \frac{GM}{R^{2}\kappa}\tau \Rightarrow P(\tau=2/3) = \frac{2}{3}\frac{GM}{R^{2}\kappa}\ \ ,
\end{equation}

</div>
<p>
where \(M\) is the total mass of the star, \(R\) is the radius such that
\(L/(4\pi R^{2}) = \sigma T_\mathrm{eff}^{4}\), \(\kappa\) is the opacity assumed constant
in the atmosphere, and we <i>define</i> the bottom of the atmosphere at
\(\tau=2/3\) because of the Eddington \(T(\tau)\) relation. Alternatively, one
could use tabulated values of \(\kappa\) and a \(T(\tau)\) to perform the
integral.
</p>

<p>
Together with \(T=T_\mathrm{eff}\), we now have specified the outer
boundary conditions fixing \(T\) and \(P\) at \(\tau=2/3\) and completely
determined the mathematical problem of the structure and evolution of
a single, non-rotating, non-magnetic star of known total (initial)
mass \(M\) and composition.
</p>

<p>
<b>N.B.:</b> While Eddington atmosphere are the simplest non-trivial case, it
is still on approximations which can (and sometimes are) relaxed in
stellar evolution modeling: this can move the outer boundary in \(\tau\)
location too by changing some of the assumptions of the Eddington gray
atmosphere!
</p>

<p>
<b>N.B.:</b> A "classic" generalization of this atmospheric model is the
generic class of gray atmospheres where the constant of integration is
<i>not</i> a constant, but a function of \(\tau\) itself.
</p>
</div>
</div>
</div>
</div>

<div id="outline-container-org93aabbf" class="outline-2">
<h2 id="org93aabbf"><a href="#org93aabbf">Saha equation</a></h2>
<div class="outline-text-2" id="text-org93aabbf">
<p>
Let's also assume that LTE holds in the stellar atmosphere, therefore,
the rate at which atoms are ionized \(I\) matches the rate at which there
are recombinations \(R\) (principle of detailed balance). Therefore:
</p>

<div class="latex" id="org14f5b98">
\begin{equation}
 n_{e} n_{+} R = n_{0} I \Rightarrow \frac{n_{e} n_{+}}{n_{0}} = \frac{I}{R}\ \ ,
\end{equation}

</div>
<p>
where \(n_{e}\), \(n_{+}\), and \(n_{0}\) are the number densities of electrons,
positive ions, and neutral atoms respectively (so we are imposing a
balance per unit volume). But that must also be equal to the ratio of
available states to all these particles, which in the limit of ideal
gas we can calculate using Maxwell-Boltzmann statistics! The momentum
terms of the ions and neutral atoms cancel each other in the ratio
(neglecting the small mass difference between these 2), and we are
left with
</p>
<div class="latex" id="org064732a">
\begin{equation}
\frac{n_{e} n_{+}}{n_{0}} = 2\frac{(2\pi m_{e} k_{B}T)^{3/2}}{h^{3}} \exp\left(-\frac{\chi}{k_{B}T}\right) \ \ ,
\end{equation}

</div>
<p>
where the first term comes from the momentum phase space of the
electron (with 2 factor for its spin) and the exponential depends on
the ionization potential \(\chi\). This is the so called Saha equation named
after <a href="https://en.wikipedia.org/wiki/Meghnad_Saha">Megnhad Saha</a>, which under the assumption of LTE (sometimes
questionable in stellar atmospheres) allows to calculate the free
electron and ion densities.
</p>

<p>
<b>N.B.:</b> The exponential factor comes from the Maxwell-Boltzmann
statistical distribution of \(dn_{0}\), \(dn_{e}\), and \(dn_{+}\)!
</p>

<p>
For any ion/atom for which we can calculate (or empirically determine
in a lab) the ionization potential \(\chi\), or more in general the
difference in their energy levels, we can write a similar equation!
Thus once the temperature \(T\) of a gas is specified this allows us to
predict what the photons filtering through the atmosphere will
encounter, and thus what we expect will be "removed" from the
distribution of photons coming out of the photosphere and the
resulting spectrum of the stars.
</p>

<p>
<b>N.B.:</b> This equation also allows us to determine the number of free
electrons and thus the chemical potential in the partial ionization
zones of the stars!
</p>
</div>
</div>

<div id="outline-container-org105f6dd" class="outline-2">
<h2 id="org105f6dd"><a href="#org105f6dd">Spectral line formation</a></h2>
<div class="outline-text-2" id="text-org105f6dd">
<p>
Lines form because the black body spectrum coming from the photosphere
(by definition) filters through the overlaying <i>atmospheres</i> where atomic
radiative processes (mainly bound-bound and bound-free transitions)
can <i>remove</i> some photons from the spectrum.
</p>

<p>
To predict the spectrum of a star, one needs to know the temperature,
density, and velocity structure of the atmosphere (to be able to
calculate the Doppler shifts!), whether it is in LTE (so electron
populations are described by the Saha equation above) or non-LTE
effects need to be accounted for (e.g., for maser lines), and solve
the radiative transfer equation.
</p>

<p>
In some cases, the velocity structure depends on the radiation itself
making this process extremely complicated, or more precisely, in the
momentum equation of the gas, a radiative acceleration term dependent
on the velocity (because of the Doppler-dependence of &kappa;<sub>&nu;</sub>) appears,
making the dynamics of the radiation+gas highly non-linear. This is,
for example, the case of radiatively driven stellar winds from massive
stars (see for instance the book by Lamers &amp; Cassinelli 1994 or the
review <a href="https://ui.adsabs.harvard.edu/abs/2014ARA%26A..52..487S/abstract">Smith 2014</a>).
</p>
</div>

<div id="outline-container-org0acdc74" class="outline-3">
<h3 id="org0acdc74"><a href="#org0acdc74">Broadening mechanisms</a></h3>
<div class="outline-text-3" id="text-org0acdc74">
<p>
While treating in detail all these processes would require an entire
course on its own (see for example <a href="https://staff.fnwi.uva.nl/a.dekoter/ARTv1.103.00.pdf">the notes by Prof. de Koter</a>), we
can give a brief qualitative description of some key effects here.
</p>

<p>
While considering these remember that for virtually all stars (except
the Sun), the projected disk on the sky is <i>unresolved</i> (the size of the
point-spread function of the telescope is bigger than the size of the
stellar disk projected on the sky): in an observed spectrum you see
all the surface at the same time!
</p>
</div>

<div id="outline-container-orgaa0c281" class="outline-4">
<h4 id="orgaa0c281"><a href="#orgaa0c281">Intrinsic width of lines</a></h4>
<div class="outline-text-4" id="text-orgaa0c281">
<p>
Because of the uncertainty principle, an electron in an ion allowing
for a bound-bound or bound-free transition is not perfectly localized.
A consequence of this is that the spectral lines formed by one
particular ion in a particular energy state is not an infinitely sharp
delta function \(\delta(\nu_{0})\) centered at \(\nu_{0}=\Delta E/h\), but instead it is a
Lorentzian profile with an intrinsic width.
</p>
</div>
</div>

<div id="outline-container-orga21cd07" class="outline-4">
<h4 id="orga21cd07"><a href="#orga21cd07">Rotational broadening</a></h4>
<div class="outline-text-4" id="text-orga21cd07">
<p>
If the star is rotating, some parts of the disk will be moving away
from the observer (at a velocity \(v_\mathrm{rot} \times sin(i)\) with \(i\)
inclination angle to the line of sight), and some parts will be moving
towards the observer (unless \(i=0\), i.e. the star is seen rotation
pole on, as seems to be the case for the North Star!).
</p>

<p>
This will introduce a Doppler shift from each part of the disk: this
<i>rotational broadening</i> is usually described by a Gaussian, that needs
to be convolved in frequency space with the intrinsic Lorentzian
distribution coming from the QM of the transition.
</p>

<p>
The <i>convolution</i> of a Lorentzian and a Gaussian gives a Voigt profile
after <a href="https://en.wikipedia.org/wiki/Woldemar_Voigt">Woldermar Voigt</a>.
</p>
</div>
</div>

<div id="outline-container-orgdf798b3" class="outline-4">
<h4 id="orgdf798b3"><a href="#orgdf798b3">Pressure broadening</a></h4>
<div class="outline-text-4" id="text-orgdf798b3">
<p>
In a star, even in the relatively low \(\rho\) atmosphere, ions/atoms
interacting with radiation are <i>not</i> in isolation! The presence of
external forces (due to other ions/atoms, or global magnetic field,
etc.) can modify the energy levels of each atom's Hamiltonian, and
thus the central frequency \(\nu_{0}\) <i>and</i> the width of specific bound-bound
transition. Collectively this is referred to as "pressure broadening".
</p>

<p>
As a concrete example, Zeeman splitting of the degenerate (in absence
of magnetic field) \(\ell=1\), \(m=0,\pm1\) triplet can result in small (non
resolved) shift in frequency that are observed as a broadened line.
</p>
</div>
</div>

<div id="outline-container-org92e5523" class="outline-4">
<h4 id="org92e5523"><a href="#org92e5523">P Cygni profiles</a></h4>
<div class="outline-text-4" id="text-org92e5523">
<p>
If the atmosphere is "moving", for example because there is a wind
outflow, a particular shape of the spectral lines will form. This is
called after the first star in which this was detected a "P Cygni"
line.
</p>

<p>
The wind moving toward the observer will absorb radiation like any
atmosphere, but because of its motion the absorption will be moved to
shorter wavelengths (blue-shifted). Viceversa, the wind moving in
directions away from the observer will have electrons de-exciting and
thus photon <i>emission</i> (if the de-excitation is radiative and not
collisional), which will be redshifted to longer wavelengths, causing
a specific shape of the line:
</p>


<figure id="orgfc280ab">
<img src="./images/P_Cygni.png" alt="P_Cygni.png">

</figure>
</div>
</div>
</div>


<div id="outline-container-org1b6049a" class="outline-3">
<h3 id="org1b6049a"><a href="#org1b6049a">Emission lines</a></h3>
<div class="outline-text-3" id="text-org1b6049a">
<p>
Some stars not only show <i>absorption</i> lines (i.e., "lack" of photons at
certain wavelengths compared to the underlying black body spectrum
produced at the photosphere), but also <i>emission lines</i>.
</p>

<p>
The P Cygni profiles mentioned above are in a sense an "intermediate"
behavior between these two regimes.
</p>
</div>

<div id="outline-container-org653b454" class="outline-4">
<h4 id="org653b454"><a href="#org653b454">Be stars</a></h4>
<div class="outline-text-4" id="text-org653b454">
<p>
These are stars of spectral class B (recall the <a href="./note-lecture-CMD-HRD.html">lecture on CMD/HRD</a>),
so fairly hot and massive, which show <i>emission</i> lines, typically H&alpha;. A
star is classified as Be if it is a B-type star that ever showed H&alpha; in
emission, even though these can be intermittent and disappear: long
term spectroscopic followup, including the crucial contribution of
amateur observers is important to understand the spectral behavior and
thus the nature of these objects (see <a href="http://basebe.obspm.fr/basebe/">BeSS catalog</a> containing
professionally taken and amateur spectra of many bright Be stars!).
</p>

<p>
These stars are interpreted as being <i>fast rotating</i>
(\(\omega\ge0.7\omega_\mathrm{crit}\)) which shed a "decretion disk": the emission
lines are not from the star directly, but from the disk of the star! A
clear indication of the presence of the disk is the "double peaked"
morphology of the H&alpha; emission:
</p>


<figure id="orge2209c2">
<img src="./images/Alcyone_Halpha.png" alt="Alcyone_Halpha.png" width="100%">

<figcaption><span class="figure-number">Figure 1: </span>Spectrum of Alcyone (&eta; Tau) on March 18<sup>th</sup> 2019 centered around H&alpha; (&lambda;&sim;6562 Angstrom) showing the typical double peaked emission suggesting the presence of a disk, obtained by the amateur astronomer <a href="https://www.astronomie.be/erik.bryssinck/aboutme.html">E. Bryssinck</a>.</figcaption>
</figure>


<ul class="org-ul">
<li><b>Q</b>: can you infer why the double peaked morphology suggests a disk?</li>
</ul>

<p>
The formation path of these stars is still being actively
investigated, but the fact that none are found with main sequence
binary companions and many are found instead with a neutron star
companion (periodically plunging through the disk producing X-rays
making the system a Be-X-ray binary!) suggest that these may be
accretor stars spun up by binary interactions (cf. <a href="https://ui.adsabs.harvard.edu/abs/1994A%26A...288..475P/abstract">Pols &amp; Marinus
1994</a>, <a href="https://ui.adsabs.harvard.edu/abs/2020A%26A...641A..42B/abstract">Bodensteiner et al. 2020</a>, <a href="https://ui.adsabs.harvard.edu/abs/2020MNRAS.498.4705V/abstract">Vinciguerra et al. 2020</a>), although
single star evolutionary pathways also exist (e.g., <a href="https://ui.adsabs.harvard.edu/abs/1998A%26A...329..551L/abstract">Langer 1998</a>), see
also the review by <a href="https://ui.adsabs.harvard.edu/abs/2013A%26ARv..21...69R/abstract">Rivinus et al. 2013</a>.
</p>
</div>
</div>

<div id="outline-container-org888277f" class="outline-4">
<h4 id="org888277f"><a href="#org888277f">B[e] stars</a></h4>
<div class="outline-text-4" id="text-org888277f">
<p>
These are also B-type stars showing emission lines, but <i>forbidden</i>
emission lines, that is radiative transitions where the angular
momentum of the electron changes which are exponentially disfavored.
These can only occur in low-density environments: if the density was
high, the atoms/ions would much rather de-excite collisionally than
with a radiative transition with \(\Delta \ell >0\).
</p>

<p>
Thus, the presence of a forbidden line (indicated by the squared
brackets) suggests a very low density environment surrounding these
stars. They tend to be brighter than Be stars (presumably, more
massive!), and whether there is an evolutionary relation between the
two classes is presently unclear.
</p>
</div>
</div>

<div id="outline-container-org9bd0b9f" class="outline-4">
<h4 id="org9bd0b9f"><a href="#org9bd0b9f">Wolf-Rayet stars</a></h4>
<div class="outline-text-4" id="text-org9bd0b9f">
<p>
Wolf-Rayet stars are a spectroscopic class <i>defined</i> by the presence of
emission lines and the deficiency (but not necessarily total lack) of
hydrogen (see also review by <a href="https://ui.adsabs.harvard.edu/abs/2024arXiv241004436S/abstract">Shenar 2024</a>).
</p>

<p>
They are further subdivided in classes based on the dominant lines
visible (WNh if there is still significant amount of hydrogen, WN is
it's nitorgen, WC for carbon, WO for oxygen). These are massive stars
which have somehow shed a large portion of their H-rich envelopes
(either because of winds of binary interactions) and are bright enough
to drive strong outflows that are so dense that they are optically
thick (remember \(\tau(r)=\int_{0}^{r }\kappa\rho dr'\)). In these dense winds
collisional excitation of atoms/ions is possible followed by radiative
de-excitation producing an <i>excess</i> of photons at the specific frequency
of the atom/ion and transition considered, resulting in the emission
line.
</p>
</div>
</div>
</div>
</div>
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