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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2025-02-06 Thu 09:04 -->
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<title>400A - CCSNe and compact object formation</title>
<meta name="author" content="Mathieu Renzo" />
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<p>
<b>Materials</b>: <a href="https://ui.adsabs.harvard.edu/abs/2012ARNPS..62..407J">Janka 2012</a>, <a href="https://ui.adsabs.harvard.edu/abs/2024OJAp....7E..31S/abstract">Soker 2023</a>, <a href="https://ui.adsabs.harvard.edu/abs/2022MNRAS.510.4689V">Burrows et al. 2023</a>, see also <a href="https://www.as.arizona.edu/~mrenzo/materials/cores_of_massive_stars.pdf">my
guest lecture notes</a> and references therein.
</p>
<p>
<b>Disclaimer:</b> Here, we discuss the collapse and possible explosion of
massive stars, that is "core collapse supernovae" (CCSNe). These are
<i>not</i> the only kind of supernova transient that can be observed, and our
focus here excludes SNIa, pair-instability supernovae, luminous red
novae, tidal disruption events, etc. (see also <a href="./projects.html">projects</a>).
</p>
<div id="outline-container-org3fb77c5" class="outline-2">
<h2 id="org3fb77c5"><a href="#org3fb77c5">Gravity's victory over nuclear energy release</a></h2>
<div class="outline-text-2" id="text-org3fb77c5">
<p>
"Massive stars" can be defined as those that will end their life
collapsing either into a neutron star (NS) or a black hole (BH). This
requires them to consume (at least partially) the oxygen-rich core,
that is, for the widest range of initial masses, to build a
silicon-rich core and subsequently iron-rich core that is too massive
to be sustained by the electron degeneracy pressure (M<sub>Fe</sub> ≥
M<sub>Chandrasekhar</sub>). This core is therefore doomed to collapse under the
influence of its own self-gravity, and this can result in a successful
supernova explosion. For single massive stars, this happens for
initial mass M<sub>ZAMS</sub> ≥ 7-10M<sub>☉</sub>, depending on their metallicity and
rotation rate (<a href="https://ui.adsabs.harvard.edu/abs/2017PASA...34...56D">Doherty et al. 2017</a>). The presence of companions, which
is the rule rather than the exception (<a href="http://adsabs.harvard.edu/abs/2012Sci...337..444S">Sana et al. 2012</a>), can also
change this threshold (<a href="http://adsabs.harvard.edu/abs/2017A%26A...601A..29Z">Zapartas et al. 2017</a>, <a href="https://ui.adsabs.harvard.edu/abs/2017ApJ...850..197P/abstract">Poelarends et al. 2017</a>).
</p>
<p>
<b>N.B.:</b> At the lower mass end, super-AGB supernova progenitors may reach
the onset of core collapse because of electron-captures in the oxygen,
neon, magnesium degenerate core producing a so-called "electron
capture SN" (cf. <a href="in-class-evol-wrap-up.html#org07b2e4b">sAGB evolution</a>) instead of because electron-captures
in the iron core. In reality, al CCSNe <i>are</i> electron-capture SNe
because electron captures always play a role in accelerating the
collapse (see below).
</p>
<p>
<b>N.B.:</b> In other subfields, for example asteroseismology or exoplanets,
"massive" may just mean "with a convective main sequence core",
meaning a much lower threshold M<sub>ZAMS</sub> ≥ 1.2-1.3M<sub>☉</sub>.
</p>
<p>
At the very end of the evolution, the degenerate iron core is too
massive to be sustained by the electron degeneracy pressure, and
therefore it collapses under its own gravity, reaches (super–)nuclear
density (ρ≥10<sup>14</sup> g cm<sup>-3</sup>), bounces due to the repulsive core of the
nuclear force (cf. <a href="./notes-lecture-nuclear-burning.html">lecture on microphysics of nuclear reactions</a>) and
this triggers a shock-wave.
</p>
<p>
<b>N.B.:</b> The collapse of the exhausted core takes a <i>dynamical</i> timescale,
which for the densities reached at the end of the nuclear life of the
star (ρ≥10<sup>10</sup> g cm<sup>-3</sup>) is extremely short, τ<sub>free fall</sub>∼0.1 sec ≤
blink of an eye!. Therefore, while the inner core collapses, the outer
layer barely notice that something is happening: the collapse, bounce
and reversal of the velocity therefore has to trigger a shock
(discontinuity in the velocity of the gas)!
</p>
<p>
The shock wave launched is thought to explode the star, unbinding the
stellar envelope. However, historically (hydrodynamical) simulations
have had limited success in producing explosions. In most cases, the
shock just stops or reverts while still deep in the star. Therefore, a
``shock-revival'' mechanism (usually neutrino heating behind the shock
aided by asymmetries in the flow) is needed to push it further and
make it unbind the stellar envelope.
</p>
<p>
The dynamics of the collapse, and therefore the success or failure of
the explosion, depend on the structure (temperature, density, etc.)
and details of the composition of the core (see also <a href="https://ui.adsabs.harvard.edu/abs/2014ApJ...783...10S/abstract">Sukhbold &
Woosley 2014</a>, <a href="https://ui.adsabs.harvard.edu/abs/2016ApJS..227...22F/abstract">Farmer et al. 2016</a>, <a href="https://ui.adsabs.harvard.edu/abs/2024RNAAS...8..152R/abstract">Renzo et al. 2024</a>, <a href="https://ui.adsabs.harvard.edu/abs/2024arXiv240902058L/abstract">Laplace et al.
2024</a>). But the details of the stellar structure depend on the late
burning stages (namely, <a href="./notes-lecture-nuclear-cycles.html#orgb831943">Silicon burning</a>) of the star, and on the
mixing processes during these phases.
</p>
<p>
Several attempts to define a (small set of) parameter(s) describing
the pre-collapse core structure that could predict the outcome of the
simulation of a SN explosion (success or failure, NS or BH remnant)
have been made (e.g., <a href="https://ui.adsabs.harvard.edu/abs/2011ApJ...730...70O/abstract">O'connor & Ott 2011</a>, <a href="https://ui.adsabs.harvard.edu/abs/2016ApJ...818..124E/abstract">Ertl et al. 2016</a>, <a href="http://adsabs.harvard.edu/abs/2019arXiv190201340C">Couch et
al. 2019</a>). Because the late burning phases and ultimately the
formation of the collapsing core itself is a multi-physics,
multi-scale, and likely stochastic problem, any attempt to define such
a parameter is necessarily an attempt to average and (over?) simplify
the structure (<a href="https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.5304M/abstract">Mueller 2019</a>).
</p>
<p>
The thermal state and composition of the iron core are of crucial
importance for the collapse itself: a slight variation in one of these
cause significant differences in the density profile of the star and
can in principle influence the outcome of the evolution (i.e.
successful explosion, or not). As an example, the amount of free
electrons in the core, which is quantified by \(Y_e=1/\mu_{e}\):
</p>
<div class="latex" id="org9ba0b33">
\begin{equation}
Y_e=\ \sum_i \frac{Z_i}{A_i}X_i \ \ ,
\end{equation}
</div>
<p>
enters quadratically in the effective Chandrasekhar
mass,
</p>
<div class="latex" id="org81772e1">
\begin{equation}\label{eq:Mcha}
M_\mathrm{Fe} \geq M_\mathrm{Ch}^\mathrm{eff} \simeq (5.80 M_☉) Y_e^2\left[1 +
\left(\frac{s_e}{\pi Y_e}\right)^2\right]%1.44M_☉(2 Y_e)^2 \ \ ,
\end{equation}
</div>
<p>
that is the maximum mass that can be sustained by electron degeneracy
pressure, where s<sub>e</sub> is the electrons specific entropy and the term
dependent on it is a "thermal correction" to account for the fact that
not the entire core is fully degenerate and with k<sub>B</sub>T≪ε<sub>Fermi</sub>.
</p>
</div>
</div>
<div id="outline-container-orgfc14771" class="outline-2">
<h2 id="orgfc14771"><a href="#orgfc14771">Physics of core collapse</a></h2>
<div class="outline-text-2" id="text-orgfc14771">
</div>
<div id="outline-container-org09e5cc1" class="outline-3">
<h3 id="org09e5cc1"><a href="#org09e5cc1">Core Bounce</a></h3>
<div class="outline-text-3" id="text-org09e5cc1">
<p>
Since the nuclei of the iron group are the most tightly bound (cf.
<a href="./notes-lecture-nuclear-burning.html">lecture on microphysics of nuclear reactions</a>), the fusion of two of
them would require energy input greater than the energy released.
Therefore, inside the iron core, fusion reactions cannot compensate
the energy loss of the star, and the core is doomed to collapse. The
conventional definition for the onset of collapse is
</p>
<div class="latex" id="org6eef88b">
\begin{equation}\label{eq:onset_cc}
\mathrm{max}\{ |v| \} \geq 10^3 \ \mathrm{km \ s^{-1}} \ \ ,
\end{equation}
</div>
<p>
where v is the radial infall velocity. This is the very latest point
to which a stellar evolution model can be computed with a stellar
evolution code (e.g., <code>MESA</code> and/or <code>MESA-web</code>), but it is just an
arbitrary threshold set by Eq. \ref{eq:onset_cc} is motivated by the
fact that, at this point, the star is a few tenths of seconds (roughly
a dynamical timescale) away from ``core bounce'' (see below). The
central density is so high (\(\rho \gtrsim 10^{10} \mathrm{ \ g \ cm^{-3}}\))
that stellar evolution codes usually cannot properly simulate the
physics needed (e.g., the high density regions require a different
equation of state - EOS, hydrostatic equilibrium does not hold any
longer, neutrinos start to be trapped because of the higher density
and neutrino opacity). However, this is a purely technical threshold,
while in nature the evolution of such a star is continuous during
collapse.
</p>
<p>
During collapse, electron capture reactions, e.g.,
</p>
<div class="latex" id="org9b3ceed">
\begin{equation}
\label{eq:ecap}
p+e^-\rightarrow n+\nu_e \ \ , \ \ ^AZ+e^-\rightarrow ^A(Z-1)+ \nu_e \ \ ,
\end{equation}
</div>
<p>
decrease Y<sub>e</sub>, and diminishing \(M_\mathrm{Ch}^\mathrm{eff}\) (see Eq.
\ref{eq:Mcha}) accelerating the collapse further. Together with
positron capture reactions, electron capture reactions form the
so-called <a href="./notes-lecture-neutrinos.html#org2af9c50">URCA process</a>, responsible for the lion's share of the
cooling (provided by neutrinos) during the collapse phase.
</p>
<p>
Moreover, matter becomes so hot that iron photodisintegration can
occur too, causing internal energy losses that also accelerate the
collapse.
</p>
<p>
As the infall velocity progressively increases, the core divides into
two separate parts <a href="http://adsabs.harvard.edu/abs/2002RvMP...74.1015W">Woosley et al. 2002</a>:
</p>
<ul class="org-ul">
<li><p>
<b>Inner Core</b>: in sonic contact and collapsing self-similarly (i.e.,
the infall velocity |v| ∝ r). Its mass is given by:
</p>
<div class="latex" id="org7fdf2f9">
\begin{equation}
\label{eq:InnerCoreMass} M_\mathrm{i.c.}=\int_{|v(r)|\leq c_s(r)}4\pi\rho(r) r^2 dr \ \ ,
\end{equation}
</div>
<p>
where c<sub>s</sub> ≡ c<sub>s</sub>(r) is the local sound speed, and the integral can
be evaluated analytically assuming <a href="./notes-lecture-EOS2.html#org69ba73e">ultra-relativistic electron
degeneracy</a> dominates the pressure. The value of \(M_\mathrm{i.c.}\) at
core bounce is almost independent of the stellar progenitor and it
is about \(1\,M_☉\), <a href="http://adsabs.harvard.edu/abs/1980ApJ...238..991G">Goldreich & Weber 1980</a>, <a href="http://adsabs.harvard.edu/abs/1982ASIC...90...53Y">Yahil & Lattimer 1982</a>.
</p></li>
<li><b>Outer Core</b>: in supersonic collapse, because at lower density the
sound speed c<sub>s</sub> decreases, so no information about the inner core
can reach into the outer core.</li>
</ul>
<p>
The collapse goes on until the central density is so high (ρ<sub>c</sub>∼10<sup>14</sup>
g cm<sup>-3</sup>) that the repulsive core of the nuclear force becomes relevant.
This repulsive contribution causes a sudden stiffening of the EOS, and
triggers the so-called core bounce, which is conventionally defined by
an arbitrary threshold on the specific entropy at the edge of the
inner core: s=3 (in units of the Boltzmann constant k<sub>B</sub> times the
Avogadro number N<sub>A</sub>). The inner core overshoots the equilibrium density
of the stiffened EOS, stops collapsing and reverses its radial
velocity. This launches a shock wave at the edge of the inner core. It
is thought that this shock wave at least in some cases, successfully
disrupts the star, producing a SN. However, in most cases, the shock
needs to be ``revived'' by some mechanism (see below).
</p>
<p>
<b>N.B.:</b> The collapse and bounce is analogous to a ball bouncing off a
wall which momentarily compresses a bit and momentum changes sign in
the direction perpendicular to the wall progressively across the ball
volume. In this analogy, for a collapsing star, the wall is the inner
core reaching super-nuclear densities.
</p>
<figure id="org8ab5dd4">
<img src="./images/v_profile_core_bounce.png" alt="v_profile_core_bounce.png" width="100%">
<figcaption><span class="figure-number">Figure 1: </span>Velocity profile for the core of an initially 15 M<sub>☉</sub> star at the onset of core-collapse (left panel) at core bounce (right panel). Note the linear behavior of the infall in the inner core on the left panel. Note also the scales on the two panels: at the onset of core collapse, the infall velocity is still subsonic and directed inward (v<0) everywhere. The data in the right panel are obtained using the open-source code GR1D (<a href="http://adsabs.harvard.edu/abs/2010CQGra..27k4103O">O'connor & Ott 2010</a>), with the data at the onset of core collapse (left panel) as input. The energy source to drive the explosion is the gravitational binding energy released by the collapse of ∼1.4 M<sub>☉</sub> of Fe-rich material with radius of almost ∼1000 km to a proto-NS with radius of ∼10 km. This figure is modified from <a href="https://etd.adm.unipi.it/theses/available/etd-05062015-125630/unrestricted/Thesis_colored_10052015.pdf">Renzo 2015</a>.</figcaption>
</figure>
<p>
The collapse of the core releases gravitational binding energy, which
ultimately is the energy source of the possible SN explosion:
</p>
<div class="latex" id="org2f91935">
\begin{equation}
\label{eq:3}
\Delta E_\mathrm{bind} \simeq \frac{G M_\mathrm{Fe}^2}{R_{\rm NS}} - \frac{G
M_\mathrm{Fe}^2}{R_{\rm Fe}} \sim 10^{53}\,\mathrm{erg} \ \ ,
\end{equation}
</div>
<p>
This largely exceeds the total binding energy of the stellar envelope,
and roughly speaking 100 times more that the typical kinetic energy
of a SN explosions (1 Bethe ≡ 1 f.o.e. ≡ 10<sup>51</sup> erg). Note that the
energy radiated away by a SN is typically a small fraction of the
kinetic energy (\(\int L_\mathrm{SN}(t)\,dt \simeq10^{49}\,\mathrm{erg}\)).
Thus, the CCSN explosion problem is about how this energy can be
harvested by the shock to explode the star.
</p>
<figure id="org711810f">
<img src="./images/shockR.png" alt="shockR.png" width="100%">
<figcaption><span class="figure-number">Figure 2: </span>Evolution of the minimum, maximum (solid lines), and average shock radius (dashed lines) for the explosion of an 18 M<sub>☉</sub> stellar model. Red curves are computed in 1D: in spherical symmetry the shock stalls and its radius stays roughly constant. This is Fig.~5 of <a href="http://adsabs.harvard.edu/abs/2016PASA...33...48M">Muller 2016</a>.</figcaption>
</figure>
<p>
As the shock wave propagates in the outer core, it loses energy by
<i>heating and photodisintegrating the infalling material</i>, and overcoming
the <i>ram pressure</i> (P<sub>ram </sub>≅ ρ v<sup>2</sup>) of this same material.
Neutrinos are emitted from the cooling proto-NS and absorbed in a
layer, called "gain layer" behind the shock, who is still in a region
of the star so dense that neutrinos are <i>not</i> free to stream out. All
the neutrinos that are not absorbed in the gain region contribute to
decreasing the total energy of the material behind the shock. The
energy loss through these mechanisms leads to a stalled shock (the
shock radius remains roughly constant for ∼0.1 millisec) in most
simulations available to date. An uncertain ``shock revival
mechanism'' must act to revive the shock and restart its radial
expansion allowing it to unbind the stellar envelope and produce a SN
explosion.
</p>
<p>
In the <i>neutrino-driven paradigm</i> (for reviews see <a href="https://ui.adsabs.harvard.edu/abs/2012ARNPS..62..407J">Janka 2012</a>, <a href="https://ui.adsabs.harvard.edu/abs/2022MNRAS.510.4689V">Vartanyan
et al. 2022</a>), the shock is pushed by the large neutrino emission from
the hot proto-NS formed in the inner part of the bouncing inner core.
These neutrinos are mostly produced by electron captures to
``neutronize'' the Fe core (in the innermost ``cooling-region'' where
the proto-NS is), cf. Eq. \ref{eq:ecap} and from <a href="./notes-lecture-neutrinos.html#org1c43654">cooling processes</a>. A
small fraction of these neutrinos will interact in a region behind the
shock (the so-called ``gain-region''). This is because the stellar
plasma has now densities and temperatures higher than during core Si
burning, and the cross section for neutrino absorption is not
negligible anymore.
</p>
<figure id="orgded7c43">
<img src="./images/87A.jpeg" alt="87A.jpeg" width="100%">
<figcaption><span class="figure-number">Figure 3: </span>Strong empirical evidence for the crucial involvement of neutrinos in CCSNe came from the direct detection of ∼12 high energy neutrinos coincident with SN1987A which went off in the LMC, the latest supernova within ∼50kpc (see <a href="https://ui.adsabs.harvard.edu/abs/1989ARA%26A..27..629A/abstract">Arnett 1989</a> review).</figcaption>
</figure>
<p>
Note, however, that other explosion mechanism relying less heavily on
the neutrino flux have been proposed. For instance, accretion on the
forming compact object in the inner core might trigger energetic jets
that might help pushing the shock, and this might be the dominant
explosion mechanism for long gamma ray bursts and SNIc showing broad
emission lines (i.e., very large ejecta velocities).
</p>
<figure id="orgaf29124">
<img src="./images/sketch_mechanism.png" alt="sketch_mechanism.png" width="100%">
<figcaption><span class="figure-number">Figure 4: </span>Sketch of the key ingredients for a successful explosion in the neutrino driven paradigm, from <a href="https://ui.adsabs.harvard.edu/%5C#abs/2017IAUS..329...17M">Muller 2017</a></figcaption>
</figure>
</div>
</div>
<div id="outline-container-org0adf9d0" class="outline-3">
<h3 id="org0adf9d0"><a href="#org0adf9d0">Shock revival mechanisms</a></h3>
<div class="outline-text-3" id="text-org0adf9d0">
<p>
As mentioned above, historically, numerical simulations of
core-collapse SNe would always find stalling and ultimately receding
shocks, that is failed explosions. This lead to the realization that
the <i>asymmetries</i> in the flow are a key ingredient to achieve a
successful explosion.
</p>
<p>
Several sources of asymmetry (both local and global) exist in the
collapsing core of a massive star:
</p>
<ul class="org-ul">
<li>The neutrino heat the bottom of the gain region, driving convection
(a steep temperature and entropy gradient can develop because of the
neutrino heating);</li>
<li>Convection implies the presence of turbulent flow and an associated
turbulent pressure (P<sub>turb</sub>≅ ρ v<sub>turb</sub><sup>2</sup>) that can help pushing
the shock (<a href="https://ui.adsabs.harvard.edu/%5C#abs/2018ApJ...865...81O">O'Connor et al. 2018</a>),</li>
<li>Standing accretion Shock Instability (SASI, <a href="http://adsabs.harvard.edu/abs/2003ApJ...584..971B">Blondin et al. 2003</a>, see
also embedded video below): when the shock stalls its surface is
perturbed by the infalling material. These perturbation (e.g., in
terms of the local velocity) are advected downwards by convection
and amplified which leads to a sloshing motion of the shock
(although recently it was suggested that the growth time of this
instability may be too long for it to play a role in successful
explosions, where other mechanism revive the shock faster than SASI
can develop, see <a href="https://ui.adsabs.harvard.edu/abs/2023ApJ...957...68B">Burrows et al. 2023</a>.) <b>N.B.:</b> there is a mathematical
analogy between the problem of the propagation of a perturbed shock
with convection behind it and the draining of shallow water, and the
same instability can develop in a table top experiment, see
<a href="https://www.youtube.com/watch?v=5fcsSA31rkE">for example this experiment</a>.</li>
<li>Lepton Emission Self-Sustained Emission (LESA, <a href="https://ui.adsabs.harvard.edu/%5C#abs/2014ApJ...792...96T">Tamborra et al 2014</a>),
found in 3D simulations where the neutrino emission is roughly
dipolar.</li>
</ul>
<p>
The overall effect of asymmetries is to (i) increase the amount of
time spent by matter in the gain region, where the energy of the
neutrinos can be harvested and used to push the shock (ii) provide
extra pressure terms (e.g., due to turbulence). The combination of
these two should result, at least in some cases, in successful
explosions (since we do observe SNe and NS!).
</p>
<p>
The first axisymmetric (i.e., 2D) simulations showed some successful
explosion, but it was soon realized that the symmetry imposed
artificially in these calculation was changing the turbulent cascade:
instead of dripping towards smaller scale and be dissipated at the
viscous scale, energy is pumped to larger scales in 2D, which
artificially helps the explosion.
</p>
<figure id="org1925bec">
<img src="./images/2v3dCCSN.png" alt="2v3dCCSN.png" width="100%">
<figcaption><span class="figure-number">Figure 5: </span>Visualization of the shock morphology in 3D (left) and 2D(right): clearly the artificial imposition of a symmetry by running at lower dimensionality changes the dynamics of the explosion.</figcaption>
</figure>
<p>
As of early 2019, there is an emerging picture from the 3D
core-collapse SN simulations of different research groups (<a href="http://adsabs.harvard.edu/abs/2018ApJ...855L...3O">Ott et al.
2018</a>, <a href="http://adsabs.harvard.edu/abs/2018arXiv180101293K">Kuroda et al. 2018</a>): <i>not only the asymmetries</i> during the first
millisecond after core-bounce are necessary for successful explosions,
but also <i>the pre-collapse core-structure and in particular the Si/O
interface is crucial</i>.
</p>
<p>
The most massive stellar cores, for which the Si/O interface is at a
large Lagrangian mass coordinate, develop strong neutrino driven
convection, which together with the contribution of the turbulent
pressure drives a successful explosion with significant fallback and
result in the formation of a BH.
</p>
<p>
<b>N.B.:</b> BHs may form if this explosion mechanism fails (possibly with
very little more than the disappearance of the star as a transient,
see for instance <a href="https://ui.adsabs.harvard.edu/abs/2015MNRAS.450.3289G/abstract">Gerke et al. 2015</a>, <a href="https://ui.adsabs.harvard.edu/abs/2021MNRAS.508.1156B/abstract">Basinger et al. 2021</a>, <a href="https://ui.adsabs.harvard.edu/abs/2024arXiv241007055T/abstract">Tsuna et al.
2024</a>), <i>or</i> with an explosions and a visible transient anyways (see for
instance <a href="https://ui.adsabs.harvard.edu/abs/2013ApJ...769..109L/abstract">Lovegrove & Woosley 2013</a>, <a href="https://ui.adsabs.harvard.edu/abs/2019MNRAS.485L..83Q/abstract">Quartaet et al. 2019</a>, <a href="https://ui.adsabs.harvard.edu/abs/2022MNRAS.511..176A/abstract">Antoni et al.
2022</a>) – it is possible both occur in nature depending on the mass of
the progenitor (and BH) considered and this is highly debated
presently.
</p>
<p>
Intermediate mass cores show a steep density gradient at the Si/O
interface: if the shock can reach this interface, it will
significantly accelerate outwards (because of mass continuity and the
drop in the impinging ram pressure). The neutrino driven convection
and turbulent pressure combined with the density drop result in
successful explosions with neutron star remnant. Smaller cores show
strong SASI oscillations of the shock and delayed explosions, likely
also resulting in NS remnants. However, note that the landscape on
explosion physics in the literature is itself very dynamic, with
multiple groups working on this problem and disagreeing on the
details.
</p>
<p>
Example of a 3D neutrino radiation hydrodynamics simulation (3D ν-RHD)
of a SASI-driven explosion:
</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/4SPq9_-h0Bs?si=-Ljb_roZT__rprtf" 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>
Example of a 3D ν-RHD simulation dominated by neutrino convection:
</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/5fcsSA31rkE?si=BNgv4wtP0vdJs4lL" 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>
<b>N.B.:</b> A SN shock successful in exploding the star needs to harvest ∼1%
of the gravitational binding energy released by the collapse of the
core (mostly in the form of neutrinos), and this requires
non-spherically-symmetric physical ingredients.
</p>
</div>
</div>
<div id="outline-container-org446297e" class="outline-3">
<h3 id="org446297e"><a href="#org446297e">Neutron star kicks</a></h3>
<div class="outline-text-3" id="text-org446297e">
<p>
The current understanding of core-collapse dynamics suggests that
asymmetries are likely the key to the success of the explosion. This
is further confirmed by the large velocities at which we observe some
single NSs moving. The proper motion of radio pulsars can correspond
to velocities in excess of ∼1000 km s<sup>-1</sup>, which is much higher than the
maximum velocity at which we observe O and B type stars moving through
the galaxy (≤10km s<sup>-1</sup> for the bulk of the population).
</p>
<figure id="orgbda796f">
<img src="./images/guitar_nebula.png" alt="guitar_nebula.png" width="100%">
<figcaption><span class="figure-number">Figure 6: </span>Guitar nebula formed by the bow shock of a NS travelling at ∼1000km s<sup>-1</sup> ≫ v<sub>OB</sub> ∼ 10 km s<sup>-1</sup>. From <a href="https://ui.adsabs.harvard.edu/abs/2004ApJ...600L..51C/abstract">Chatterjee & Cordes 2003</a>.</figcaption>
</figure>
<p>
These are explained invoking an energy and momentum re-distribution
between the forming compact object and the SN ejecta allowed by the
asymmetries. One possible source of asymmetry is if the neutrino flux
from the cooling region is itself non-spherical, however, since the
proto-NS that occupies most of the volume of the cooling region is
convective with a convective turnover timescale faster than the
explosion, this explaination is currently disfavored. Another
possibility is that hydrodynamical instabilities lead to aspherical
flows.
</p>
<figure id="orgbe19c8f">
<img src="./images/sketch_perturbations.png" alt="sketch_perturbations.png" width="100%">
<figcaption><span class="figure-number">Figure 7: </span>Schematic of the development of a large asymmetry in a 3D CCSN simulation resulting in a large natal kick. From \cite{muller:17}.</figcaption>
</figure>
<p>
Whether the SN shock achieves a runaway radial growth (successful
explosion,) or it stalls and reverts (failed explosion, likely to
result in BH formation) is typically decided within the first few 100s
of millisec after core-bounce. However, until few seconds after
core-bounce the deeper layers of the ejecta and the proto-NS are still
dynamically connected, and interact through gravity (see, e.g, <a href="http://adsabs.harvard.edu/abs/2013MNRAS.434.1355J">Janka
2013</a>, <a href="https://ui.adsabs.harvard.edu/abs/2023arXiv231112109B">Burrows et al. 2023b</a>). If a clump of ejecta is more dense
because of asymmetries in the flow (a situation routinely realized in
3D ab-initio simulations of the core-collapse process), it can
gravitationally pull the newly born compact object in its direction
and accelerate it.
</p>
<p>
A key prediction of this ``tug-boat'' model is that the SN shock is
faster in the direction opposite to the one in which the compact
object is accelerated. A faster shock is more efficient at
photodisintegrating nuclei, producing light particles that can be
accreted by the surviving nuclei: this model predicts stronger
explosive nucleosynthesis in the direction opposite to the compact
object! This prediction seems to be consistent with observations of
supernova remnant for which we can find the associated NS,
<a href="http://adsabs.harvard.edu/abs/2017ApJ...844...84H">Holland-Ashford et al. 2017</a>, <a href="http://adsabs.harvard.edu/abs/2018ApJ...856...18K">Katsuda et al. 2018</a>. Note however, that
recent simulations from <a href="https://ui.adsabs.harvard.edu/abs/2023arXiv231112109B">Burrows et al. 2023b</a> disagree with this
picture, and produce sizable natal kicks just through momentum
conservation in asymmetric ejecta.
</p>
<p>
If a SN occurs in a binary system (a situation which should <i>not</i> be
uncommon, given that <a href="./notes-lecture-BIN.html">massive star progenitors are preferentially born
in binary or higher multiplicity systems</a>), because of the SN kick, the
kinetic energy of the compact object is greatly increased, and this
can lead to an increase of the total (orbital) energy of a putative
binary system:
</p>
<div class="latex" id="org23cf258">
\begin{equation}
\label{eq:1}
E_\mathrm{orb} =
\frac{1}{2}M_1v_1^2+\frac{1}{2}M_2v_2^2-\frac{GM_1M_2}{a}
\stackrel{\mathrm{SN}}{\rightarrow}
\frac{1}{2}(M_1-M_\mathrm{ej})(v_1+v_\mathrm{kick})^2+\frac{1}{2}M_2v_2^2-\frac{GM_1M_2}{a}
> 0
\end{equation}
</div>
<p>
where we implicitly assume an instantaneous explosion (compared to the
orbital period of the binary), which leaves the gravitational
interaction term unchanged. If E<sub>orb</sub> becomes positive, the binary
system is unbound. The SN kick thus breaks most massive binary systems
by giving a large velocity to the compact object, but without
modifying significantly the instantaneous velocity of the companion
star.
</p>
<p>
<b>N.B.:</b> Do not confuse v<sub>2</sub> with the orbital velocity \(v_\mathrm{orb} =
\sqrt{\frac{G(M_1+M_2)}{a}}\) which represents the velocity of the
point of reduced mass M<sub>1</sub>M<sub>2</sub>/(M<sub>1</sub>+M<sub>2</sub>) orbiting around the center of mass
of the binary, and not the velocity of a physical object!}
</p>
<div class="latex" id="orgeb73ce8">
\begin{equation}
\label{eq:2}
v_2 = \frac{M_1}{M_1+M_2}v_\mathrm{orb}\equiv\frac{M_1}{M_1+M_2}\sqrt{\frac{G(M_1+M_2)}{a}} \ \ .
\end{equation}
</div>
<p>
The companion star is thus shot out of the binary with its
pre-explosion v<sub>2</sub>, and if v<sub>2</sub>>30 km s<sup>-1</sup> it becomes a runaway star <a href="https://ui.adsabs.harvard.edu/abs/2019A&A...624A..66R">Renzo
et al. 2019</a>. This however tends to happen rarely, since mass transfer
during the binary evolution tends to increase the separation a,
decrease the mass M<sub>1</sub>, and increase the mass M<sub>2</sub>.
</p>
<p>
Note typically SN ejecta achieve velocities of ∼10<sup>4</sup> km s<sup>-1</sup> ≫ v<sub>orb</sub>,
thus we can neglect the orbital motion of the binary during the SN
(although see also <a href="http://adsabs.harvard.edu/abs/2017ApJ...846L..15B">Batta et al. 2017</a>): effectively this corresponds to
an instantaneous loss of mass from the exploding star, which is not
the center of mass of the binary. This off-center mass loss (from the
point of view of the binary) is also referred to as or "Blaauw kick"
and can modify the orbit, and in extreme case where
\(M_\mathrm{ejecta}\geq (M_1+M_2)/2\) it can change it from an
circle/ellipse to a parabola/hyperbole - so unbind the binary <a href="http://adsabs.harvard.edu/abs/1961BAN....15..265B">Blaauw
1961</a> (see also homework on virial theorem!). Typically in massive
binary evolution, the exploding star loses its H-rich envelope to the
companion long before its SN explosion, therefore M<sub>ej</sub> is rarely
sufficiently large to unbind the binary without a natal kick due to
asymmetries.
</p>
<p>
From the observation of the pulsars proper motions we know that at
least some NS receive such large kicks, however there is still debate
on whether SN resulting in the formation of a BH can also provide
significant kicks, and consequently whether most BHs remain bound to
their stellar companion, or whether their formation breaks the
binaries in which they form. In at least a handful of cases, it can be
shown that large mass BHs were formed with negligible natal kicks,
<a href="https://ui.adsabs.harvard.edu/abs/2023arXiv231001509V">Vigna-gomez et al. 2024</a>, although observations of the kinematic
properties of X-ray binaries hosting BHs accreting mass from a
companion seem to require moderate kicks (see e.g., <a href="https://ui.adsabs.harvard.edu/abs/2019MNRAS.489.3116A/abstract">Atri et al. 2019</a>).
</p>
<p>
The observation of double pulsars <a href="http://adsabs.harvard.edu/abs/1975ApJ...195L..51H">Hulse & Taylor 1975</a> also raises the
question of whether some successful SN explosion resulting in NS
formation might also lead to systematically smaller kicks, allowing
for a binary to survive two consequent explosions. The idea is that
ultra-stripped SNe (i.e. the explosion of a star that has lost a very
substantial amount of mass in multiple binary mass transfer episodes),
and/or electron-capture SNe from collapsing ONeMg cores, or yet
core-collapse SN of small Fe cores would more easily result in a
successful explosion (no shock stalling, and no time to develop
significant asymmetries during a shock stalling phase) with
consequently small kicks.
</p>
</div>
</div>
</div>
<div id="outline-container-org4d185d9" class="outline-2">
<h2 id="org4d185d9"><a href="#org4d185d9">The supernova zoo</a></h2>
<div class="outline-text-2" id="text-org4d185d9">
<figure id="org95fd1e3">
<img src="./images/SN_taxonomy.png" alt="SN_taxonomy.png" width="100%">
<figcaption><span class="figure-number">Figure 8: </span>Schematic representation of the SN taxonomy, based on spectral features and light curve shape. The dot-dashed line indicates the possible connection between SN events detected in late stages and classified as type Ib/Ic and SNe of type IIb. This figure is from <a href="https://www.as.arizona.edu/%5Csimmrenzo/materials/Thesis/Renzo_MSc_thesis.pdf">Renzo 2015</a> and inspired by Fig.~2 in <a href="https://ui.adsabs.harvard.edu/abs/2001ASSL..264..199C/abstract">Cappellaro et al. 2001</a>.</figcaption>
</figure>
<p>
The classification of supernovae (SNe), as many things in astronomy,
is rooted in history and was designed before the development of a
physical understanding of what is observed.
</p>
<p>
The "modern" version is based on a combination of <i>photometric</i> and
<i>spectroscopic</i> criteria which allow to classify transients observed in
the sky.
</p>
<p>
<b>N.B.:</b> the criteria are purely phenomenology-base and empirical!
</p>
<p>
The term "nova" in latin means "new" (omitting "star"), and in the
time of visual observations it was used for any new source appearing
in the sky. Keeping track of "changes" in the sky was especially
developed in Eastern Asian astronomy (and astrology), while it
received less attention in Western cultures where the dominant
Aristotelian philosophy postulated some "perfection" of the
"superlunar" world (and if it's perfect, why would it "change"?)
</p>
<p>
The term "supernova" was introduced by <a href="https://en.wikipedia.org/wiki/Walter_Baade">W. Baade</a> and <a href="https://en.wikipedia.org/wiki/Fritz_Zwicky">F. Zwicky</a> in 1931
to identify "new sources in the sky" with a particularly high
luminosity.
</p>
<p>
The first step to classify transients is to look at Hydrogen lines in
the spectrum, and in absence of Hydrogen lines, whether there are
Silicon lines. No hydrogen and silicon is called a type Ia SN which is
theoretically associated to the thermonuclear obliteration of a white
dwarf (see corresponding project for more information, the rest of
this lecture focuses on massive stars explosions).
</p>
<p>
All other SN types are associated instead to the death of massive
stars: in absence of hydrogen lines and presence of helium lines we
speak of type Ib (so the progenitor star had no more H-rich envelope
at death), absence of both hydrogen and helium we have a type Ic
(progenitor stripped of H and He – though a small amount of He may be
"hidden"). There is then a "transitional class" of type IIb SNe which
show H only very early on. Collectively type Ib/Ic/IIb are often
referred to as "stripped envelope SNe".
</p>
<p>
SNe showing hydrogen in their spectrum are called type II SNe.
Type II SNe are more common than stripped envelope ones, and they are
further classified base on the light curve:
</p>
<ul class="org-ul">
<li>type IIP showing a plateau (when the photosphere is locked by H recombination)</li>
<li>type IIL showing a linear decay in log(L) vs. time</li>
</ul>
<p>
Moreover, finer distinction and sub-classes exist, for example SNIIn,
Ibn, Icn which are SN with the conditions mentioned above also showing
narrow (∼1000km s<sup>-1</sup>) <b>emission</b> lines.
</p>
<p>
SNe are also said to be "super-luminous" if their absolute bolometric
magnitude exceeds M=-21 at peak.
</p>
</div>
<div id="outline-container-org4c6e412" class="outline-3">
<h3 id="org4c6e412"><a href="#org4c6e412">Supernova rates</a></h3>
<div class="outline-text-3" id="text-org4c6e412">
<p>
The CCSN rate in Milky-way galaxies (with star formation rate of
approximately few M<sub>☉</sub> yr<sup>-1</sup>) is about ∼ 1 per century. We are
currently awaiting for one!
</p>
<p>
The figure below shows a breakdown per SN type:
</p>
<figure id="org426b738">
<img src="./images/smith11.png" alt="smith11.png" width="50%">
<figcaption><span class="figure-number">Figure 9: </span><a href="https://academic.oup.com/mnras/article-lookup/doi/10.1111/j.1365-2966.2011.17229.x">Smith et al. 2011</a> break down of CCSN by type in a volume limited sample.</figcaption>
</figure>
</div>
</div>
</div>
<div id="outline-container-org3b4b309" class="outline-2">
<h2 id="org3b4b309"><a href="#org3b4b309">Light curves</a></h2>
<div class="outline-text-2" id="text-org3b4b309">
<p>
Once a SN is detected (typically from ground-based observatories), it
can be followed up to measure a light curve and spectra. This is
necessary to understand the type of transient identified (is it
"really" a supernova or something else?), classify it (what spectral
type and light curve morphology does it have?), and understand the
physical mechanism driving the explosion and the progenitor star that
caused it.
</p>
<p>
<b>N.B.:</b> Presently, uncertainties in progenitor structure and evolution
are dominant in this problem space!
</p>
</div>
<div id="outline-container-orgafbe21c" class="outline-3">
<h3 id="orgafbe21c"><a href="#orgafbe21c">Shock breakout</a></h3>
<div class="outline-text-3" id="text-orgafbe21c">
<p>
If a successful shock is launched by the collapse and bounce of the
core, it will propagate as a radiation mediated shock throughout the
star, effectively "whipping the star" and injecting throughout energy.
</p>
<p>
This shock propagation <i>starts</i> within hundreds of millisecond after the
initiation of the collapse, and lasts a timescale of the order of the
dynamical timescale τ<sub>free fall</sub><sub>nil</sub>. As we know, this is a function of the
average density ⟨ ρ ⟩, which depends on the extent of the envelope of
the exploding star, from τ<sub>free fall</sub>∼ minutes-hours if the
progenitor star has lost its envelope (e.g., because of binary
interactions or strong stellar winds) to ∼10days for a very extended
RSG. While the burst of neutrinos pushing the shock occurs within the
first hundreds of millisecond from deep within the core, for a
dynamical timescale no electromagnetic phenomena is detectable, since
everything is occurring within the optically thick photosphere.
</p>
<p>
Once the shock approaches the surface of the star with low optical
depth, photons from the heated material just behind the shock are less
and less impeded by matter, and they can thus leak ahead of the shock!
Specifically, if τ≤ c/v<sub>shock</sub> the photons will be able to stream ahead
of the shock and leak out, providing the first electromagnetic signal
of a SN, the so called "shock breakout".
</p>
<p>
The duration of the shock breakout constrains:
</p>
<ul class="org-ul">
<li>the light-travel time across the stellar surface R<sub>photo</sub>/c (thus the
radius of the progenitor star)</li>
<li>the presence of surface inhomogeneities (e.g., because of
convection) smears out the signal, with the less dense (⇒ lower
optical depth τ) parts of the stars allowing an earlier shock
breakout compared to the denser and optically thicker parts (cf.
<a href="https://ui.adsabs.harvard.edu/abs/2022ApJ...933..164G/abstract">Goldberg et al. 2021</a>).</li>
</ul>
<p>
By the end of shock breakout, the shock has deposited energy in the
entire volume of the star above the inner core that bounced. This
excess energy (coming from the pre-explosion gravitational binding
energy, released mostly in the form of neutrinos partially harvested
by the shock) unbinds the outer layers – and whatever remains bound
behind the shock and/or falls back on the central compact object will
determine the compact object mass.
</p>
</div>
</div>
<div id="outline-container-orge3ddad5" class="outline-3">
<h3 id="orge3ddad5"><a href="#orge3ddad5">Adiabatic expansion and radioactive energy input</a></h3>
<div class="outline-text-3" id="text-orge3ddad5">
<p>
The stellar gas energized by the shock will start an adiabatic
expansion and cooling. This leads to the SN light curve "rise time",
as R increases L increases too: early observations of the SN light
curve constrain the pre-explosion radius of stars (although
complications due to the presence of circumstellar material make this
a "dirty" observational signal from an astrophysics perspective).
</p>
<p>
While propagating in the densest region of the star (roughly anywhere
with ρ≥10<sup>6</sup> g cm<sup>-3</sup>, that is within the carbon-oxygen core), the shock
also photodisintegrates the infalling material, which after the shock,
if ρ is sufficiently high re-combines itself into new nuclei. This
explosive nucleosynthesis caused by the shock produces a large amounts
of radioactive material, especially \(^{56}\mathrm{Ni}\), which is an
unstable nucleus that initiates the following decay cascade:
</p>
<div class="latex" id="org964d634">
\begin{equation}
^{56}\mathrm{Ni} \rightarrow e^{+} + \nu_{e} + ^{56}\mathrm{CO} \\
^{56}\mathrm{CO} \rightarrow e^{+} + \nu_{e} + ^{56}\mathrm{Fe} \ \ ,
\end{equation}
</div>
<p>
where the first β<sup>+</sup> decay has a half-life of ∼6 days and the second
∼77days. The positrons released immediately will annihilate with an
electron producing γ rays (E<sub>γ</sub>∼ 2 × m<sub>e</sub>c<sup>2</sup> ∼ 1MeV). The
stellar material in the process of being ejected is opaque to these γ
rays, and thus they deposit their energy as thermal energy in the gas,
providing a long-lasting energy source.
</p>
<p>
For a typical CCSN, the amount of \(^{56}\mathrm{Ni}\) synthesized is of
the order of few × 0.01M<sub>☉</sub>. This radioactive energy source is
crucial to explain the long term light curves of these explosions.
</p>
<p>
The adiabatic expansion phase reaches a maximum when the energy
injected by the decay of this material has had enough time to diffuse
through the ejecta: the width of the post-shock-breakout peak
constrains the amount of mass in the ejecta, and its luminosity the
amount of radioactive material produced.
</p>
<p>