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<p>
<b>Materials</b>: Onno Pols' lecture notes Chapter 7,8, Tauris &amp; van den
Heuvel book chapter 4.
</p>

<div id="outline-container-org7b2f5f9" class="outline-2">
<h2 id="org7b2f5f9"><a href="#org7b2f5f9">Mass transfer in binary systems</a></h2>
<div class="outline-text-2" id="text-org7b2f5f9">
<p>
We have already seen in the <a href="./notes-lecture-BIN.html#org2c7b98c">second lecture</a> that, in the Universe,
binary systems composed of two stars orbiting each other are common,
the higher the mass of the stars the more so (see for example <a href="https://ui.adsabs.harvard.edu/abs/2023ASPC..534..275O/abstract">Offner
et al. 2023</a>). Historically, this fact has been leveraged to design
empirical <i>dynamical mass measurement</i> which rely only on Kepler's laws
to obtain masses.
</p>

<p>
However, often enough, stars in binaries are so close to each other
than as they evolve, they will "touch" each other and exchange mass.
In today's lecture we will discuss in more detail how being in a
binary can lead to dramatic consequences for the evolution of <i>both</i>
stars in the system, starting with the Roche model of the potential.
</p>

<ul class="org-ul">
<li><b>Q</b>: We will see this in more detail in the coming in class activity,
but you can already answer based on your 1M<sub>☉</sub> stellar models:
what happens to the stellar radius as a typical star evolves?</li>
<li><b>Q</b>: Which stars evolve faster: higher or lower mass?</li>
</ul>
</div>

<div id="outline-container-org9cd0d9e" class="outline-3">
<h3 id="org9cd0d9e"><a href="#org9cd0d9e">Roche model</a></h3>
<div class="outline-text-3" id="text-org9cd0d9e">
<p>
To describe the gas of the stars in a binary system subject to the
gravity of both stars is effectively a <i>three body problem</i>, hard to
solve even before we fold in anything else but gravity.
</p>

<p>
A useful approximation often used is the <a href="https://en.wikipedia.org/wiki/%C3%89douard_Roche">Roche</a> model of the
gravitational potential. This is based on the following key
assumptions:
</p>

<ol class="org-ol">
<li>e=0: the mutual orbit of M<sub>1</sub> and M<sub>2</sub> is circular;</li>
<li>M<sub>1</sub> and M<sub>2</sub> are treated as point masses;</li>
<li>M<sub>1</sub> &ge; M<sub>2</sub> &Gt; m: the masses of the two stars are much larger than
the test mass m we consider to map the potential;</li>
</ol>

<p>
These defined the <i>restricted circular three body problem</i>. Of these
assumptions the most restrictive is perhaps the first: binaries can be
eccentric, and updates to this model to account for that are actively
being developed (e.g., <a href="https://ui.adsabs.harvard.edu/abs/2009ApJ...702.1387S/abstract">Sepinsky 2009</a>). The second assumption is not
critical, since the density increases steeply moving inwards in the
star: <i>most of the mass is in the center</i>. The last assumption is
typically easily verified as long as one considers small parcels of
gas.
</p>

<p>
With these three assumptions, one can put themselves in a <b>frame
centered in the binary center of mass and co-rotating with the binary</b>
at frequency n=2&pi;/P. Since it is a rotating frame, it is non-inertial,
but the gravitational potential of the two stars can be written as:
</p>

<div class="latex" id="org955617f">
\begin{equation}
\Phi(\mathbf{r}) = -\frac{GM_{1}}{\lvert \mathbf{r}-\mathbf{r}_{1}\rvert}-\frac{GM_{2}}{\lvert \mathbf{r}-\mathbf{r}_{2}\rvert}-\frac{1}{2}(\mathbf{n}\times\mathbf{r})^{2} \ \ ,
\end{equation}

</div>
<p>
Where r<sub>i</sub> is the distance to M<sub>i</sub> (considered a point-mass), so the first
two terms represent the gravity of each star, respectively, and the
last term represents the centrifugal force.
</p>

<p>
Taking the gradient, the gravitational force from the combined masses
of the stars in the binary is F=-&nabla; &Phi;(\mathbf{r}). But because we are
in non-inertial frame, there is also another term to add to the
momentum equation which cannot be expressed as a potential, because it
depends on the velocity of the test mass m: the Coriolis force &prop;
2n&times; v. Per unit volume the momentum conservation is thus
</p>

<div class="latex" id="orgc221b23">
\begin{equation}
\partial_{t} v + (v\cdot\nabla )v = -\nabla \Phi-\frac{\nabla P}{\rho}-2\mathbf{n}\times \mathbf{v} \ \ .
\end{equation}

</div>

<p>
Finally, one could also relax the hypothesis 2. above and include a
tidal term due to the finite size of the stars in the potential (this
again works because most of the mass of the star is in it's core,
saving the Roche model, and the tides are a relatively small
perturbation on top of this):
</p>


<figure id="org0ebb275">
<img src="./images/RochePotential.jpeg" alt="RochePotential.jpeg" width="100%">

<figcaption><span class="figure-number">Figure 1: </span>Visualization of the Roche potential (grid) and the equipotential surfaces intersecting the orbital plane for a binary with q=M<sub>2</sub>/M<sub>1</sub>=0.5. The three Lagrangian points L<sub>1</sub>, L<sub>2</sub>, and L<sub>3</sub> are marked. From <a href="https://en.wikipedia.org/wiki/Roche_lobe#/media/File:RochePotential.jpg">wikipedia</a>.</figcaption>
</figure>

<p>
On the line connecting the two binaries, this potential has three
zeros, meaning points where there is no net
(gravitational+centrifugal) force. All three are <i>saddle points</i> and
they are called Lagrangian points:
</p>
<ul class="org-ul">
<li>L<sub>1</sub> sits in between the two stars, and it is the point where the
gravitational pull from each star cancel each other</li>
<li>L<sub>2</sub> is the point behind star 2 (many telescopes, including for
example <i>JWST</i> or <i>Gaia</i> sit close to the L<sub>2</sub> point of the Earth-Sun
binary system)</li>
<li>L<sub>3</sub> is the point behind star 1.</li>
</ul>

<p>
<b>N.B.:</b> The position of the center of mass depends on the mass
distribution (here the masses are assumed to be point like), while the
position of the Lagrangian points depends on the dynamics (gravity and
centrifugal forces)
</p>

<p>
The equipotential shapes are close to spheres for r<sub>1</sub> and/or r<sub>2</sub> &Lt;
r<sub>L<sub>1</sub></sub>, then they get progressively elongated along the axis connecting
the stars, and flattened by the centrifugal potential.
</p>

<p>
For short period binaries, P&lt;10 days (though the exact value depends
on many things, especially mass M<sub>1</sub>, mass ratio q=M<sub>2</sub>/M<sub>1</sub>, metallicity,
etc.), we observe that the stars co-rotate with the orbit, that is
there is complete <i>tidal synchronization</i>. In that case, in the
co-rotating frame, the gas of both stars is still, \(\mathbf{v}=0\), and
the momentum conservation reduces to
</p>

<div class="latex" id="orgff86730">
\begin{equation}
- \rho \nabla\Phi = \nabla P \ \ ,
\end{equation}

</div>
<p>
that is equipotential surfaces are also isobaric surfaces (&nabla;&Phi; = 0 &hArr; &nabla;
P=0), and since &rho; &sim; dP/d&Phi; has to remain finite, it is also constant on
these surfaces!
</p>

<p>
Therefore, for a tidally synchronized binary the stellar gas will be
shaped like equipotential surfaces in the Roche potential, producing
non-axisymmetric, tear-drop-shaped stars.
</p>

<p>
For larger periods, tidal synchronization may not occur (tides are a
steep function of R/a with R stellar radius and a orbital separation),
and in that case the rotational bulge may lag behind the orbit, adding
an extra deformation.
</p>

<p>
On top of the necessary hypothesis (e=0, M<sub>1 </sub>&ge; M<sub>2</sub> &Gt; m, point masses
and in co-rotation) to derive the Roche approximation, it is worth
noting that for r&Gt; r<sub>L<sub>2</sub></sub> the Roche model becomes less and less
physically meaningful. By construction, it is based on the assumption
of co-rotation with the binary, but a parcel of gas significantly
outside L<sub>2</sub> will not maintain this co-rotation, an it is more likely to
only "see" the inner binary as a point mass with total mass M<sub>1</sub>+M<sub>2</sub>.
</p>


<figure id="org1af626c">
<img src="./images/potential_Roche.png" alt="potential_Roche.png" width="100%">

<figcaption><span class="figure-number">Figure 2: </span>cross section of the potential on a plane perpendicular to the orbit and passing through the three Lagrangian points. The dotted line outside of L<sub>2</sub> mark where co-rotation is not achievable and the Roche approximations don't hold anymore. The dark gray binary with both stars within their Roche lobes is a <i>detached</i> binary, the intermediate gray one, where M<sub>1</sub> fills its Roche lobe is a <i>semidetached</i> binary, and the lightest gray where both stars fill their lobes is a contact binary. This is Figure 6.1 of Onno Pols' lecture notes.</figcaption>
</figure>
</div>


<div id="outline-container-org0945767" class="outline-4">
<h4 id="org0945767"><a href="#org0945767">Spherical approximation</a></h4>
<div class="outline-text-4" id="text-org0945767">
<p>
The Roche potential is <i>clearly</i> not spherically symmetric: how can we
use this in stellar evolution simulations?! In reality, thanks to the
fact that in depth, the potential is very closed to spherically
symmetric, and most of the deformation is in a large by radius by
extremely small in mass region of the star (nevertheless important for
atmospheric effects and observable predictions!). So for the stellar
<i>interior</i> calculation, the spherical symmetry is still acceptable for
most of the mass domain and as long as the stars are detached (see
below).
</p>

<p>
Each Roche lobe can then be interpreted as the <i>sphere of gravitational
influence</i> of each star.
</p>

<p>
What we need to know then is the <i>volume</i> of each Roche lobe, and we can
then <i>define</i> a sphere that has the same volume and compare the volume
of the stellar gas to the volume of such sphere. There are multiple
formulae to fit the Roche volume as a function of the binary
parameters (e.g., <a href="https://ui.adsabs.harvard.edu/abs/1971ARA%26A...9..183P/abstract">Paczsynki 1971</a>), but probably the most common one is
<a href="https://ui.adsabs.harvard.edu/abs/1983ApJ...268..368E/abstract">Eggleton 1983</a>'s formula, which provides a fit accurate to &sim;1% and
continuous across a large range of mass ratios:
</p>

<div class="latex" id="org2786b30">
\begin{equation}
R_{RL,i}= a \frac{0.49 q_{i}^{2/3}}{0.6q_{i}^{2/3}+\ln(1+q_{i}^{2/3})} \ \ ,
\end{equation}

</div>
<p>
where a is the orbital separation, q<sub>i</sub> = M<sub>i</sub>/M<sub>j</sub>, and 4&pi; R<sub>RL,i</sub><sup>3</sup>/3 &sim; Roche
volume of star i.
</p>
</div>
</div>
</div>
</div>

<div id="outline-container-org225146b" class="outline-2">
<h2 id="org225146b"><a href="#org225146b">Type of binary interactions</a></h2>
<div class="outline-text-2" id="text-org225146b">
<p>
As we will see in the next in class activity, stars tend to grow
bigger as the evolve: even a binary system that stars as detached may
come into contact as the stars evolve, triggering the onset of mass
transfer between the stars in the binary system. In fact, gas reaching
L<sub>1</sub> will be equally bound to either stars, and it can easily be
perturbed to fall into the other star (see also review by <a href="https://ui.adsabs.harvard.edu/abs/2024ARA%26A..62...21M/abstract">Marchant &amp;
Bodensteiner 2024</a>).
</p>

<p>
Depending on the stability of the <i>orbital response</i> to the
transfer of mass, we distinguish two cases: <i>stable Roche lobe overflow</i>
or <i>dynamically unstable common envelope</i>.
</p>
</div>

<div id="outline-container-org755c15e" class="outline-3">
<h3 id="org755c15e"><a href="#org755c15e">Dynamically stable mass-transfer</a></h3>
<div class="outline-text-3" id="text-org755c15e">
<p>
If Roche lobe overflow does not hit a runaway response of the orbit
(causing more and more overflow), then it is dynamically stable.
</p>

<p>
This will occur in the majority (50-70%) of massive stars, and it
can have important consequences for <i>both</i> stars that will be modified
by this interaction (see below).
</p>
</div>
</div>

<div id="outline-container-org5e87be0" class="outline-3">
<h3 id="org5e87be0"><a href="#org5e87be0">Dynamically unstable common envelope</a></h3>
<div class="outline-text-3" id="text-org5e87be0">
<p>
Sometimes, the orbit and or the evolution of the stars respond to mass
transfer increasing the amount of overflow, leading to an unstable
situation: in this case the system enters in contact first, and then
ultimately in a <i>common envelope event</i> (see reviews by <a href="https://ui.adsabs.harvard.edu/abs/2013A%26ARv..21...59I/abstract">Ivanova et al.
2013</a>, <a href="https://ui.adsabs.harvard.edu/abs/2023LRCA....9....2R/abstract">Ropke &amp; de Marco 2023</a> and Ivanova et al. 2020s book)
</p>

<p>
During a common envelope the gas of the envelope of both stars fills
equipotentials beyond L<sub>2</sub>, cannot maintain co-rotation, and thus start
exerting a friction on the orbit of the two cores (or core and star)
inside this shared envelope.
</p>

<p>
This results in an inspiral that can end either with:
</p>
<ul class="org-ul">
<li>a stellar merger</li>
<li>the successful ejection of the shared envelope (interrupting the
drag), and the formation of a tight period binary</li>
</ul>

<p>
Common envelope evolution, since it's theorization in the 1970s by
Paczynski, Webbink, Taam, and Ostriker, remains one of the biggest
open questions in stellar physics that impacts the formation of all
compact binaries (cataclysmic variables made of a main sequence plus a
white dwarf, binary white dwarfs, gravitational wave mergers, etc.).
</p>
</div>
</div>


<div id="outline-container-org894a855" class="outline-3">
<h3 id="org894a855"><a href="#org894a855">Nomenclature: Case A, B, and C</a></h3>
<div class="outline-text-3" id="text-org894a855">
<p>
Depending on when mass transfer starts, we can have three different
categories (defined by <a href="https://ui.adsabs.harvard.edu/abs/1967ZA.....66...58K/abstract">Kippenhahn et al. 1967</a> and <a href="https://ui.adsabs.harvard.edu/abs/1970A%26A.....7..150L/abstract">Lauterborn 1970</a>):
</p>
<ul class="org-ul">
<li>case A RLOF: donor is burning hydrogen in its core, thus it occurs
in the tightest (smaller separation a) binaries when the stars are
still relatively small in radius.</li>
<li>case B RLOF: the donor has a helium core (possibly inert and
sustained by electron degeneracy or burning), typically this is a
faster mass, although this may also depend on the metallicity and
its impact on where on the HR diagram Helium ignites (see e.g.,
<a href="https://ui.adsabs.harvard.edu/abs/2022A%26A...662A..56K/abstract">Klencki et al. 2022</a>). Since stars typically expand after they run
out of hydrogen-rich fuel in the core, this mode of mass transfer
is the most common overall.</li>
<li>case C RLOF: for low mass stars, this is typically defined after
He ignition, for high mass stars, it is typically defined after
core He depletion.</li>
</ul>

<p>
Because of the different L and &phi; factors for each nuclear burning
phase (<a href="./notes-lecture-nuclear-burning.html#orgda5a33e">see lecture on nuclear burning for notation</a>), the timescales
for mass transfer and therefore responses of both stars and the orbit
to mass transfer can be very different for these three, leading to
divergent evolution of the binaries and the outcome: case A typically
has a short thermal timescale phase, followed by a much longer nuclear
timescale phase. Case B and C tend to be faster (thermal timescale or
shorter), though this may also be metallicity dependent .
</p>
</div>
</div>
</div>

<div id="outline-container-org8820411" class="outline-2">
<h2 id="org8820411"><a href="#org8820411">Orbital evolution during mass transfer</a></h2>
<div class="outline-text-2" id="text-org8820411">
<p>
Because of the evolution of the stellar radius R(t), many stellar
binaries will evolve from an initially <i>detached</i> state (i.e., the mass
of each star is contained within their respective Roche lobes), to a
<i>semi-detached</i> phase (when one star's material fills and even exceeds
its Roche lobe, causing matter to be pulled away by the companion's
gravity), and a fraction even into <i>over-contact</i> (<b>both</b> stars filling
their Roche lobes).
</p>

<p>
During a phase of mass transfer the structure of <i>both</i> stars changes,
but also the orbit! The details of this are arguably the biggest
uncertainty in binary stellar evolution:
</p>
<ul class="org-ul">
<li>in most cases this process is <i>fast</i> (&prop; thermal timescale),
making direct observations rare (with the exception of case A RLOF
on nuclear timescales)</li>
<li>where, how much, and how mass is transferred and lost from the
system determines the torques that the orbit feels and thus it's
angular momentum and energy evolution</li>
<li>a variety of physical processes (from magnetic fields, jets, etc.)
can intervene in the mass transfer, and with different degrees of
importance on the process depending on the masses and mass-ratios
of the system, and the evolutionary stage: a general recipe valid
for <i>all</i> binaries is not known and may not even exist because of the
wide range of physical regimes in which mass transfer could occur!</li>
</ul>

<p>
To study the evolution of the orbit, it is useful to consider the
total orbital angular momentum of the binary:
</p>

<div class="latex" id="org17c53fe">
\begin{equation}
J_\mathrm{tot} = J_\mathrm{orb}+J_{1}+J_{2} \ \ ,
\end{equation}

</div>
<p>
with J<sub>i</sub> = I<sub>i</sub> &omega;<sub>i</sub> spin angular momentum of the two stars, and J<sub>orb</sub>
orbital angular momentum. A simple order of magnitude calculation can
show that
</p>
<div class="latex" id="orgb30b13c">
\begin{equation}\label{eq:J_ordering}
J_{i} \simeq M_{i} R_{i}^{2} \omega_{i} \ll J_\mathrm{orb} \simeq \frac{M_{1}M_{2}}{M_{1}+M_{2}} a^{2} n \ \ ,
\end{equation}

</div>
<p>
where a&Gt;{R<sub>1</sub>, R<sub>2</sub>} is the semimajor axis of the orbit, much larger than
the radii of the stars, and &omega;<sub>i</sub> &le; n is the spin frequency of the
rotation of the stars which is typically lower than the orbital
frequency n = 2&pi;/P and at best equal when tides can synchronize
rotation and revolution (e.g., in the Moon+Earth binary!). Note
however that there are known exception to this (see for example
<a href="https://ui.adsabs.harvard.edu/abs/2024A%26A...684A..35B/abstract">Britavski et al. 2024</a>), but they don't invalidate the ordering of Eq.
\ref{eq:J_ordering} because of the squared dependence on the lever arm
length of the angular momentum:
<i>the spin angular momentum of both stars in a binary is typically small
compared to the orbital angular momentum</i>.
</p>

<p>
For the orbital term, using Kepler's 3<sup>rd</sup> law and a<sub>1</sub>M<sub>1</sub>
= a<sub>2</sub>M<sub>2</sub> = a M<sub>1</sub>M<sub>2</sub>/(M<sub>1</sub>+M<sub>2</sub>) we can write:
</p>
<div class="latex" id="orgceff91b">
\begin{equation}
J_\mathrm{orb} = M_{1} a_{1}^{2} v_{1} + M_{2} a_{2}^{2} v_{2} = \sqrt{\frac{GM_{1}^{2}M_{2}^{2} a(1-e^{2})}{M_{1}+M_{2}}} \ \ .
\end{equation}

</div>
<p>
and thus taking a logarithmic derivative obtain:
</p>
<div class="latex" id="org181c45e">
\begin{equation}\label{eq:am_balance}
\frac{\dot{a}}{a} = 2 \frac{\dot{J_\mathrm{orb}}}{J_\mathrm{orb}}-2\frac{\dot{M_{1}}}{M_{1}}-2\frac{\dot{M_{2}}}{M_{2}}+\frac{\dot{M_{1}}+\dot{M_{2}}}{M_{1}+M_{2}} +2\frac{e\dot{e}}{1-e^{2}} \ \ ,
\end{equation}

</div>
<p>
which allows us to calculate the change in orbital separation a if we
know the mass change rates <i>and</i> the orbital angular momentum losses
from the system <i>and</i> eccentricity changes. In other words, we need to
know the forces and torques exerted by the gas being transferred/lost
onto the binary.
</p>

<p>
For the eccentricity, a common (but questionable and questioned)
approach is to <i>assume</i> that tides pre-mass transfer will circularize
the binary (e=0). While this seems plausible since pre-mass transfer
the donor star is by definition almost as large as it can be, allowing
for tides to torque it, in reality post-mass transfer eccentric
systems are known (see e.g., <a href="https://ui.adsabs.harvard.edu/abs/2009MNRAS.400L..20E/abstract">Eldridge 2009</a>): it is possible that the
timescale for circularization \(e/\dot{e}\) is comparable or longer than
the thermal timescale of the donor star (e.g., for case B Roche lobe
overflow), making it impossible for tides to circularize the system
before/during mass transfer.
</p>

<p>
The orbital angular momentum losses can occur do to a variety of
phenomena:
</p>
<ul class="org-ul">
<li><b>magnetic braking</b>: if the stars are magnetic (e.g., because of the
presence of a convective envelope), expelled gas can remain magnetically
tethered to them, providing a long lever arm to that gas to extract
angular momentum</li>
<li><b>tidal LS coupling</b>: if the system is small enough that tides matter,
they can redistribute angular momentum between the spins and orbit.
In extreme mass ratio binaries with q=M<sub>2</sub>/M<sub>1</sub> &Lt; 1/3 this can result
in unstable situations where the orbit doesn't have enough angular
momentum to allow for tidal synchronization, leading to a runaway
extraction of angular momentum from the orbit and ultimately a
merger (this is called a Darwin instability, after the nephew of the
more famous Charles Darwin of the evolutionary theory)</li>
<li><b>gravitational wave emission</b>: this term is important for very long
lived and compact binaries, and can ultimately lead to mergers of
compact stellar remnants (white dwarfs, neutron stars, and black
holes)</li>
<li><b>mass loss</b>: where and how mass may be lost by the system (e.g., if
the accretor star does not accept all the mass) can also eat up
some orbital angular momentum, and the rate depends on poorly
understdood details.</li>
</ul>

<p>
The last term is usually the dominant one, but a general solution
would require to treat all of the effects above.
</p>

<p>
Let's now consider some cases where we can solve the equation for the
orbital evolution.
</p>
</div>

<div id="outline-container-org02510eb" class="outline-3">
<h3 id="org02510eb"><a href="#org02510eb">Stellar wind mass loss</a></h3>
<div class="outline-text-3" id="text-org02510eb">
<p>
Consider for simplicity a circular binary (e=0) where the only mass
lost is due to stellar winds (no mass exchange).
</p>

<p>
The winds have velocity v<sub>wind,i </sub>&ge; v<sub>esc,i</sub> &sim;(2GM<sub>i</sub>/R<sub>i</sub>)<sup>1/2</sup> for each
star, while the orbital velocity of the binary is set by Kepler's 3<sup>rd</sup>
law, and each individual star has
</p>
<div class="latex" id="orgf8d8623">
\begin{equation}
v_{1} = \frac{M_{2}}{M_{1}+M_{2}}\sqrt{\frac{G(M_{1}+M_{2})}{a}} \ \ ,\\
v_{2} = \frac{M_{1}}{M_{1}+M_{2}}\sqrt{\frac{G(M_{1}+M_{2})}{a}} \ \ ,\\
\end{equation}

</div>
<p>
where the square root term is v<sub>orb</sub> (i.e., the velocity around the
center of mass of the reduced-mass point-mass), and the prefactors
come from momentum balance in the center of mass frame of the system.
</p>

<p>
Thus, taking the ratio, we see that:
</p>
<div class="latex" id="orgb4185ab">
\begin{equation}
\frac{v_\mathrm{wind, 1}}{v_\mathrm{orb,1}} \geq \frac{v_\mathrm{esc, 1}}{v_\mathrm{orb, 1}} = \sqrt{2\frac{M_{1}(M_{1}+M_{2})}{M_{2}^{2}}\frac{a}{R_{1}}} \gg 1 \ \ , \\
\frac{v_\mathrm{wind, 2}}{v_\mathrm{orb,2}} \geq \frac{v_\mathrm{esc, 2}}{v_\mathrm{orb, 2}} = \sqrt{2\frac{M_{2}(M_{1}+M_{2})}{M_{1}^{2}}\frac{a}{R_{2}}} \gg 1 \ \ , \\
\end{equation}

</div>
<p>
where the last comes from a &Gt; R<sub>i</sub>. This means that to a good
approximation one can assume winds from the stars to leave the system
instantaneously without exerting any torque on the binary.
In this case, the material lost to winds will carry the specific
angular momentum (per unit mass) of the wind-losing star moving on its
orbit:
</p>
<div class="latex" id="org3b7933c">
\begin{equation}
dJ_\mathrm{orb} = \frac{J_\mathrm{orb, 1}}{M_{1}}dM_{1} + \frac{J_\mathrm{orb, 2}}{M_{2}}dM_{2} \equiv \frac{M_{1} J_\mathrm{orb}}{M_{2} (M_{1}+M_{2})}dM_{1} + \frac{M_{2} J_\mathrm{orb}}{M_{1}(M_{1}+M_{2})}dM_{2} \ \ ,
\end{equation}

</div>
<p>
where in the last we used J<sub>orb, i</sub> M<sub>i</sub> = J<sub>orb</sub> M<sub>1</sub>M<sub>2</sub>/(M<sub>1</sub>+M<sub>2</sub>) (angular
momentum conservation in the center of mass frame) and dM<sub>i</sub>&lt;0 in both
cases. The Eq. \ref{eq:am_balance} reduces to
</p>
<div class="latex" id="org2a00f77">
\begin{equation}
\frac{\dot{a}}{a} = - \frac{\dot{M_{1}}+\dot{M_{2}}}{M_{1}+M_{2}} \ \ ,
\end{equation}

</div>
<p>
which since dM<sub>i</sub>&lt;0 means that in the approximation of <i>fast</i> stellar wind
mass loss (w.r.t. to the orbital velocity) and assuming the wind to
take the specific orbital angular momentum of the mass-losing star
around its orbit and no accretion, then <i>the binary widens</i> and the
relative rate at which is widens is equal to the relative rate at
which the binary loses mass! Note that the orbital angular momentum is
decreasing, and yet the binary widens. This is sometimes referred to
as "Jeans mode".
</p>
</div>
</div>

<div id="outline-container-orga788675" class="outline-3">
<h3 id="orga788675"><a href="#orga788675">Conservative mass transfer evolution</a></h3>
<div class="outline-text-3" id="text-orga788675">
<p>
Another case that allows for analytic consideration is the case of
<i>conservative</i> mass transfer, that is
</p>
<div class="latex" id="org979487d">
\begin{equation}
\dot{M_{1}} = - \dot{M_{2}} \ \ \mathrm{and}\ \  \dot{J_\mathrm{orb}}=0 \ \,
\end{equation}

</div>
<p>
that is all the mass lost by star 1 is accreted by star 2, no spills
from the system occurs and thus no orbital angular momentum can be
lost.
</p>

<p>
In this case Eq. \ref{eq:am_balance} becomes:
</p>
<div class="latex" id="orgf168d91">
\begin{equation}
\frac{\dot{a}}{a} = 2\left(\frac{M_{1}}{M_{2}}-1\right)\frac{\dot{M_{1}}}{M_{1}} \ \ ,
\end{equation}

</div>
<p>
which tells us that since dM<sub>1</sub>&lt;0 when star 1 is the one filling its
Roche lobe and donating the mass, the orbit initially shrinks, until
the condition M<sub>2</sub>=M<sub>1</sub> is reached because of mass transfer, after which
further mass loss results in <i>widening</i> of the orbit.
</p>

<p>
Effectively, what we have found is that conservation of angular
momentum imposes that the orbit initially shrinks and then widens once
the mass ratio flips from q=M<sub>2</sub>/M<sub>1</sub>&lt;1 to q&gt;1.
</p>
</div>
</div>

<div id="outline-container-orgfb3cb61" class="outline-3">
<h3 id="orgfb3cb61"><a href="#orgfb3cb61">General case</a></h3>
<div class="outline-text-3" id="text-orgfb3cb61">
<p>
In general, we can assume that a fraction &beta;<sub>RLOF</sub> of the mass lost by
the donor star ends up on the accretor star (thus a fraction 1-&beta;<sub>RLOF</sub>
is instead lost from the system), and that the non-accreted material
takes away a specific angular momentum per unit mass &gamma;<sub>RLOF</sub>. With this
parametrization of the uncertain mass transfer efficiency (represented
by the poorly known parameter &beta;<sub>ROLF</sub>) and angular momentum losses
(represented by &gamma;<sub>RLOF</sub>) we can re-write Eq. \ref{eq:am_balance} for a
circular system (e=0) as:
</p>
<div class="latex" id="org707ea75">
\begin{equation}
\frac{\dot{a}}{a} =-2\frac{\dot{M_{1}}}{M_{1}}\left[1-\beta_{RLOF}\frac{M_{1}}{M_{2}}-(1-\beta_{RLOF})\left(\gamma_{RLOF}+\frac{1}{2}\right)\frac{M_{1}}{M_{1}+M_{2}}\right] \ \ ,
\end{equation}

</div>
<p>
where in general &beta;<sub>RLOF</sub> and &gamma;<sub>RLOF</sub> are going to be unknown functions of
the masses and separation (or period), but they can be analytically
specified for some physically relevant cases (alternative but
equivalent parametrizations exist, see for example <a href="https://ui.adsabs.harvard.edu/abs/1997A%26A...327..620S/abstract">Sobermann et al.
1997</a>).
</p>

<p>
These parameter may also depend on the formation of an accretion disk
around the initially less massive star (as opposed to direct impact of
the stream of matter with the star), and possibly of a circumbinary
disk (see for example <a href="https://ui.adsabs.harvard.edu/abs/1975ApJ...198..383L/abstract">Lubow &amp; Shu 1975</a>).
</p>
</div>
</div>
</div>

<div id="outline-container-orge48a0e2" class="outline-2">
<h2 id="orge48a0e2"><a href="#orge48a0e2">Modifications to the structure of stars because of mass transfer</a></h2>
<div class="outline-text-2" id="text-orge48a0e2">
<iframe width="600" height="400" src="https://www.youtube.com/embed/Zn1E23saY-A?si=w0m_yOw7DRnEqX7s" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

<p>
The artist impression in the video above shows a <i>conservative</i> (no
spill over), dynamically stable mass transfer between a 20M<sub>☉</sub> and a
15M<sub>☉</sub> star producing a direct impact of the L<sub>1</sub> stream with the
accretor. Although this is an artist impression (as you can tell from
the fact that the stars are resolved), it illustrates several
important modifications that will occur in <i>both</i> stars as they go
through mass transfer.
</p>
</div>

<div id="outline-container-org3fdb702" class="outline-3">
<h3 id="org3fdb702"><a href="#org3fdb702">Donor: loss of the envelope, revealing the core</a></h3>
<div class="outline-text-3" id="text-org3fdb702">
<p>
The donor will lose (most of) its hydrogen rich envelope, becoming a
low opacity, high temperature, small in radius, helium enriched star.
This is something that only very massive stars may do if single (and
how massive they need to be is a matter of debate because of wind
uncertainties).
</p>

<p>
This means that post-mass transfer, the donor star is the <i>exposed</i> core
of the star, which allows a unique opportunity to directly see (with
photons!) the interior of a star.
</p>
</div>
</div>

<div id="outline-container-org44367cf" class="outline-3">
<h3 id="org44367cf"><a href="#org44367cf">Accretor: Spin up, pollution, rejuvenation</a></h3>
<div class="outline-text-3" id="text-org44367cf">
<p>
The mass transferred comes with angular momentum and will easily spin
up the accretor star, making it fast rotating up to the point where at
the equator centrifugal force and radiation pressure balance the
gravitational pull. This is what is invoked to explain Be stars in
Be-X-ray binaries for example.
</p>

<p>
Secondly, the matter from the donor may be chemically enriched by
nuclear processes in <i>the donor's core</i>. For example, if the star
transfers matter from the outermost layers of the core, it may be
CNO-processed! This can change the chemical composition in the
accretor structure, and thus its spectrum.
</p>

<p>
Finally, for massive stars, there is a <i>rejuvenation</i> effect. Since they
burn through the CN-NO cycle, &epsilon;<sub>CNO</sub>&prop; T<sup>16</sup> releases a lot of energy
in a small hot-enough volume, leading to a steep temperature gradient
and thus convection. As the mass of the accretor increases because of
accretion itself, &lang; T &rang; &prop; M by the virial theorem means that the
average (and central) temperature have to increase to sustain the
increased mass. This leads to a steepening of the temperature gradient
in the core, and drives further <i>convection</i>. This "extra" convection
takes Hydrogen of the accretor that would <i>not</i> have burned had the star
been single, and mixes it in the burning region itself, elongating the
lifetime of the star.
</p>
</div>
</div>
</div>

<div id="outline-container-orgad95e91" class="outline-2">
<h2 id="orgad95e91"><a href="#orgad95e91">Mass transfer rate calculation</a></h2>
<div class="outline-text-2" id="text-orgad95e91">
<p>
Say we have a binary that is starting to transfer mass through the L<sub>1</sub>
Lagrangian point: how can we decide the mass transfer <i>rate</i> per unit
time? This is a crucial question, since the response of both stars to
mass transfer will depend on the \(dM_{i}/dt\) of both stars (or in other
words on how the mass-change timescale \(M_{i}/\dot{M_{i}}\) compares to the
<i>local</i> timescales of the stellar surfaces), and thus whether it remains
dynamically stable or devolves into a common envelope!
</p>

<p>
The classic approach is to <i>postulate</i> that the mass transfer rate is a
function of the amount of overflow in radius. If the donor is star 1:
</p>

<div class="latex" id="org8118f7a">
\begin{equation}
\dot{M_{1}} \equiv \dot{M_{1}}(\Delta R_{1}/R_{RL,1}) \equiv \dot{M_{1}}(\frac{R_{1}-R_{RL,1}}{R_{RL,1}}) \ \ .
\end{equation}

</div>

<p>
By dimensional analysis we can infer that the mass loss rate by the
donor is related to the gas density and velocity at L<sub>1</sub> (the point
where it becomes gravitationally unbound) times the cross section of
the nozzle of gas across the plane perpendicular to the axis
connecting the two stars and going through L<sub>1</sub>
</p>

<div class="latex" id="org7b6a16e">
\begin{equation}
\dot{M_{1}} = \rho_{L_{1}} v_{L_{1}} A_{L_{1}} \ \ .
\end{equation}

</div>

<p>
<b>N.B.:</b> this is reminiscent of the time-dependent mass-conservation term
we obtained during the <a href="./notes-lecture-HSE.html">lecture on hydrostatic equilibrium</a>, not by
accident!
</p>

<p>
Each term needs to be estimate based on thermo- and hydro-dynamical
considerations on the flow of the gas, but typically v<sub>L<sub>1</sub></sub>&ge;
v<sub>esc,1</sub>&cong; c<sub>sound,1</sub> because the gas has to leave the gravitational
potential well of star 1 and the last step is a consequence of the
Virial theorem, &rho;<sub>L<sub>1</sub></sub> can be estimated using an EOS and mass
continuity from the stellar surface to L<sub>1</sub>, and to estimate A<sub>L<sub>1</sub></sub> we
consider that the Roche potential has a saddle point in L<sub>1</sub>, therefore
we can do a Taylor approximation along the y-direction the plane of
interest and get:
</p>

<div class="latex" id="org8a4847d">
\begin{equation}
\Delta \Phi \simeq \frac{\partial \Phi}{\partial y}\rvert_{L_{1}} y + \frac{1}{2}\frac{\partial^{2} \Phi}{\partial y^{2}}\rvert_{L_{1}}y^{2} \ \ ,
\end{equation}

</div>
<p>
where &part;<sub>x</sub>&Phi;(L<sub>1</sub>)=&part;<sub>y</sub>&Phi;(L<sub>1</sub>)0, and using the Roche potential plus Kepler's 3<sup>rd</sup>
law this gives:
</p>
<div class="latex" id="org815ec86">
\begin{equation}
\Delta \Phi \simeq \frac{1}{2}{n^{2}}y^{2} \ \ ,
\end{equation}

</div>
<p>
with n= 2&pi;/P the orbital frequency. A<sub>L<sub>1</sub></sub>&cong; y<sup>2</sup> for y determined
by how much outside of its Roche lobe the donor star is. This can be
related by looking at a point far from L<sub>1</sub>'s direction where the
gravity of the companion does not distort the equipotential surfaces
too much, resulting in &Phi; &sim; GM<sub>1</sub>/(R<sub>1</sub>), and thus:
</p>
<div class="latex" id="orgd9ababa">
\begin{equation}
\Delta \Phi \simeq \frac{GM_{1}}{R_{1}}\frac{\Delta R}{R} \ \ ,
\end{equation}

</div>
<p>
and putting things together:
</p>
<div class="latex" id="orgddad763">
\begin{equation}
A_{L_{1}} \simeq y^{2}(\Delta R) = \frac{2GM_{1}}{n^{2}R_{1}}\frac{\Delta R}{R_{1}} \ \ ,
\end{equation}

</div>

<p>
More detailed calculations use the Bernoulli principle applied to the
gas streaming from the outer layers of the donor star to the L<sub>1</sub> point,
and differ in whether the inner point is in the stellar atmosphere
(which is optically thin, likely non-isothermal) or inside the
photosphere (optically thick), see for instance <a href="https://ui.adsabs.harvard.edu/abs/1988A%26A...202...93R/abstract">Ritter 1988</a>, <a href="https://ui.adsabs.harvard.edu/abs/1990A%26A...236..385K/abstract">Kolb &amp;
Ritter 1990</a>, <a href="https://ui.adsabs.harvard.edu/abs/2021A%26A...650A.107M/abstract">Marchant et al. 2021</a>, <a href="https://ui.adsabs.harvard.edu/abs/2023MNRAS.524..471C/abstract">Cehula &amp; Pejcha 2023</a>, <a href="https://ui.adsabs.harvard.edu/abs/2024ApJ...971...64I/abstract">Ivanova et
al. 2024</a>.
</p>

<p>
Once the mass loss rate from the donor is determined, several
processes occurring during the travel of the gas in between the stars
(does it form a disk or not? does the gas have time to cool and change
its entropy?) and when it reaches the accretor (what is the shear and
entropy contrast with the accretor's outer layers? Is the composition
the same?) will determine the mass accretion rate of the accretor, and
enter in the parameters &beta;<sub>RLOF</sub>, and &gamma;<sub>RLOF</sub> we introduced above while
determining the orbital evolution in the general case.
</p>

<p>
Note that a lot of the physics at play here may happen while the stars
are <i>not</i> in gravothermal equilibrium, something that is neglected in
<i>rapid</i> binary population synthesis calculations based on pre-computed
single star models (where the single stars were always in gravothermal
equilibrium themselves), cf. <a href="./notes-lecture-neutrinos.html#org2f5a84b">notes on codes</a>.
</p>


<ul class="org-ul">
<li>if there is matter flowing through L<sub>1</sub>, can it be convective at that
location (<b>Hint</b>: think of what force drives the instability)?</li>
</ul>
</div>
</div>

<div id="outline-container-orgc7fe4b5" class="outline-2">
<h2 id="orgc7fe4b5"><a href="#orgc7fe4b5">Homework</a></h2>
<div class="outline-text-2" id="text-orgc7fe4b5">
<ul class="org-ul">
<li>Assuming that star 1 in a binary is filling its Roche lobe and
transferring mass, find a relation between it's average density and
the orbital parameters P and q=M<sub>2</sub>/M<sub>1</sub>. Since these are (in some
cases) observationally measurable, this relation gives a way to
measure the average density of stars during mass transfer!</li>
</ul>
</div>
</div>
</div>
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