[a,b]Vadim Guzey

# UPCs as probes of partonic structure – exclusive and inclusive processes

###### Abstract

Ultraperipheral collisions (UPCs) at the LHC and RHIC provide important new
information on the partonic structure of the proton and nuclei and small-$x$
dynamics in QCD. We review phenomenological applications of the collinear
factorization at leading and next-to-leading orders of perturbative QCD and
the dipole model to coherent and incoherent $J/\psi$ photoproduction in Pb-Pb
UPCs at the LHC emphasizing the strong leading twist gluon nuclear shadowing,
the role of quark-antiquark-gluon dipoles, and a possible onset the gluon
saturation in nuclei. We also discuss inclusive and diffractive dijet
photoproduction in UPCs, which give complementary constraints on nuclear
parton distributions and the pattern of factorization breaking in diffraction.

##  1 Ultraperipheral collisions as a proxy for photon-nucleus collider

Ultraperipheral collisions (UPCs) constitute an important part of the nuclear
physics program at the Large Hadron Collider (LHC) and Relativistic Heavy Ion
Collider (RHIC). In UPCs, colliding ions pass each other at large impact
parameters of the order of tens of fermi. In this case, the strong hadron-
hadron interaction is suppressed and the reaction is mediated by quasi-real
photons in the Weizsäcker-Williams equivalent photon approximation. The flux
of these photons is enhanced by $Z^{2}$ ($Z$ is the ion electric charge) and
the maximal photon energy is proportional to the ion Lorentz factor
$\gamma_{L}$. This makes UPCs a proxy for a high-energy photon-nucleus
collider reaching at the LHC well into the TeV-range. Thus, UPCs give an
opportunity to address open questions of the partonic structure of protons and
nuclei and the strong interaction dynamics in small-$x$ QCD [1, 2, 3, 4].

One of the most frequently studied UPC process is photoproduction of light
($\rho)$ and heavy ($J/\psi$, $\Upsilon$) vector mesons $V$. Since both
colliding nuclei can serve as a source of photons and as a target, the UPC
cross section is given by a sum of two terms,

$\frac{d\sigma^{AA\to
VAA^{\prime}}}{dydp_{T}}=\left[kN_{\gamma/A}(k)\frac{d\sigma^{\gamma A\to
VA^{\prime}}}{dp_{T}}\right]_{k=k^{+}}+\left[kN_{\gamma/A}(k)\frac{d\sigma^{\gamma
A\to VA^{\prime}}}{dp_{T}}\right]_{k=k^{-}}\,,$ (1)

where $y$ and $p_{T}$ are the vector meson rapidity and transverse momentum,
respectively; $kN_{\gamma/A}(k)$ is the photon flux known from QED with an
additional input to suppress the strong interaction at small impact
parameters. Equation (1) leads to a two-fold ambiguity in the photon energy
$k$ as a function of $y$, $k^{\pm}=(M_{V}/2)e^{\pm y}$, where $M_{V}$ is the
vector meson mass. The underlying photon-nucleus cross section
$d\sigma^{\gamma A\to VA^{\prime}}/dp_{T}$ can be either coherent
($A^{\prime}=A$, target intact) or incoherent ($A^{\prime}\neq A$, target
breaks up). The two cases contain complementary information on small-$x$
dynamics and are experimentally distinguished by measuring the $p_{T}$
distribution and comparing it to predictions of the STARlight Monte Carlo [5].

Both coherent and incoherent scattering can be accompanied by mutual
electromagnetic excitation of the colliding ions with their subsequent de-
excitation and emission of forward neutrons [6, 7]. Measurements of UPCs in
conjunction with the detection of forward neutrons in any two channels (0n0n,
0nXn, XnXn) allow one to circumvent the photon-energy ambiguity and thus to
access very low values of $x$ [8, 9]. In the case of coherent $J/\psi$
photoproduction in Pb-Pb UPCs at 5.02 TeV, this was recently demonstrated by
CMS [10] and ALICE [11] collaborations at the LHC.

##  2 Exclusive $J/\psi$ photoproduction in collinear factorization of pQCD

Studies of exclusive $J/\psi$ photoproduction in UPCs are motivated by the
result that in the leading double logarithmic approximation of perturbative
QCD (pQCD) and the static limit for the charmonium wave function, the cross
section of this process is proportional to the small-$x$ gluon density of the
target squared [12],

$\frac{d\sigma^{\gamma p\to J/\psi p}(t=0)}{dt}=\frac{12\pi^{3}}{\alpha_{\rm
e.m.}}\frac{\Gamma_{V}M_{V}^{3}}{(4m_{c}^{2})^{4}}\left[\alpha_{s}(Q_{\rm
eff}^{2})xg(x,Q_{\rm eff}^{2})\right]^{2}C(Q^{2}=0)\,,$ (2)

where $\Gamma_{V}$ is the $J/\psi\to l^{+}l^{-}$ leptonic decay width,
$x=M_{J/\psi}^{2}/W_{\gamma p}^{2}$ and $Q_{\rm eff}={\cal O}(m_{c})$ with
$W_{\gamma p}$ being the photon-nucleon center-of-mass energy and $m_{c}$ the
charm quark mass. The factor of $C(Q^{2}=0)$ accounts for effects beyond the
non-relativistic approximation for the $J/\psi$ vertex.

Applying Eq. (2) to nuclear targets, one can express the $p_{T}$-integrated
cross section of coherent $J/\psi$ photoproduction in the following form,

$\sigma^{\gamma A\to J/\psi A}(W_{\gamma p})=\frac{d\sigma^{\gamma p\to J/\psi
p}(t=0)}{dt}\left[\frac{xg_{A}(x,Q_{\rm eff}^{2})}{Axg_{p}(x,Q_{\rm
eff}^{2})}\right]^{2}\int_{|t_{\rm min}|}^{\infty}dt|F_{A}(t)|^{2}\,,$ (3)

where $xg_{A}/(Axg_{p})$ is ratio of the gluon density in the nucleus to that
in the free proton, and $F_{A}(t)$ is the nucleus form factor. To quantify the
magnitude of nuclear modifications, it is convenient to introduce the nuclear
suppression factor $S_{Pb}(x)$ [13, 14],

$S_{Pb}(x)=\left[\frac{\sigma^{\gamma A\to J/\psi A}(W_{\gamma
p})}{\sigma_{\rm IA}^{\gamma A\to J/\psi A}(W_{\gamma
p})}\right]^{1/2}=\frac{xg_{A}(x,Q_{\rm eff}^{2})}{Axg_{p}(x,Q_{\rm
eff}^{2})}=R_{g}(x,Q_{\rm eff}^{2})\,,$ (4)

where $\sigma_{\rm IA}^{\gamma A\to J/\psi A}$ is the impulse approximation
(IA) for the cross section in Eq. (3), where one neglects nuclear
modifications of $xg_{A}$. The values of $S_{Pb}(x)$ extracted from ALICE and
CMS measurements give direct evidence of the large gluon nuclear shadowing
[13, 14],

$R_{g}(x=6\times 10^{-4}-10^{-3},Q_{\rm eff}^{2}=3\ {\rm GeV}^{2})\approx
0.6\,.$ (5)

It confirmed predictions of the leading twist approximation (LTA) to nuclear
shadowing [15] and the EPS09 nuclear parton distribution functions (nPDFs)
[16].

One can go beyond the approximation of Eq. (2) and use the formalism of
collinear factorization for hard exclusive processes at next-to-leading order
(NLO) accuracy. In this framework, the amplitude of exclusive $J/\psi$
photoproduction is expressed in terms of generalized parton distributions
(GPDs) and NLO gluon and quark coefficient functions $T_{g}$ and $T_{q}$ [17].
The nuclear photoproduction amplitude is given by the following convolution
[18, 19],

${\cal M}^{\gamma A\to J/\psi A}(t)\propto\sqrt{\langle
O_{1}\rangle_{J/\psi}}\int^{1}_{-1}dx\left[T_{g}(x,\xi)F_{A}^{g}(x,\xi,t)+T_{g}(x,\xi)F_{A}^{q,S}(x,\xi,t)\right]\,,$
(6)

where $\langle O_{1}\rangle_{J/\psi}$ is the non-relativistic QCD (NRQCD)
matrix element related to $\Gamma_{V}$, and $F_{A}^{g}$ and $F_{A}^{q,S}$ are
the nuclear gluon and quark singlet GPDs. While only the gluons contribute at
leading order (LO), there is both gluon and quark contributions at NLO.

GPDs interpolate between usual PDFs, distribution amplitudes and elastic form
factors and, as a result, depend on the two momentum fractions $x$ and
$\xi\approx M_{J/\psi}^{2}/(2W_{\gamma p}^{2})$, the momentum transfer $t$ and
the factorization scale $\mu_{F}$ (implicit). However, at small $\xi$, GPDs
can be expressed in terms of usual PDFs because the $\mu_{F}$ evolution washes
out information on the possible $\xi$ dependence of GPDs at the input scale
[20, 21]. Therefore, with a good accuracy, one can use the following model for
the gluon nuclear GPD (the quark case is similar),

$F_{A}^{g}(x,\xi,t,\mu_{F})=xg_{A}(x,\mu_{F})F_{A}(t)\,.$ (7)

Figure 1: NLO pQCD predictions for $d\sigma^{AA\to AAJ/\psi}/dy$ as a function
of the $J/\psi$ rapidity $y$ in Pb-Pb UPCs at 5.02 TeV for EPPS21, nNNPDF3.0,
and nCTEQ15WZSIH nPDFs. For comparison, the Run 2 LHC data are also shown.
From [19].

Figure 1 shows NLO pQCD predictions for the rapidity dependence of the cross
section of coherent $J/\psi$ photoproduction in Pb-Pb UPCs at 5.02 TeV, which
are compared to the Run 2 LHC data. In the calculation [19], three state-of-
the art nPDFs (EPPS21, nCTEQ15WZHIS, and nNNPDF3.0) have been used. The shaded
bands showing the propagation of the nPDF uncertainties are much larger than
the experimental errors, which implies that these data have the potential to
improve the determination of nPDFs.

It is well known that NLO pQCD calculations for this process are characterized
by a very strong $\mu_{F}$ dependence, which originates from the double
logarithmic terms in the NLO coefficient functions enhanced by $\log(1/\xi)$
at small $\xi$. One can nevertheless find an optimal scale providing a
reasonable description of both Run 1 and 2 UPC data (these values are shown in
the figure).

A surprising result of the NLO pQCD analysis [18, 19] is the dominance of the
quark contribution at central rapidities $y$ because of the strong
cancellations between the LO and NLO gluon contributions. It challenges the
perturbative stability of the NLO calculations and the interpretation of the
UPC data in terms of the gluon nuclear shadowing. This can be partially
mitigated by considering the ratio of UPC cross sections for two different
targets, for instance, for oxygen and lead, where the $\mu_{F}$ dependence is
dramatically reduced and becomes comparable to nPDF uncertainties [19].

At the same time, NLO pQCD predictions for $\Upsilon$ photoproduction are
under better theoretical control than those for $J/\psi$: NLO corrections are
moderate, GPD modeling benefits from the longer evolution in $\mu_{F}$, and
relativistic corrections are expected to be small. This was demonstrated in
the case of $\Upsilon$ photoproduction in Pb-Pb UPCs at the LHC [22], which
was shown to be dominated by the gluon contribution and which hence can be
used to study the $\mu_{F}$ dependence of the gluon nuclear shadowing.

It was argued that perturbative stability of the NLO predictions for heavy
vector meson photoproduction can be improved by a special choice of the
factorization scale $\mu_{F}=m_{c}$ and the so-called $Q_{0}$-subtraction
eliminating certain double counting between NLO and LO terms [23, 24]. In the
case of the proton target, it was shown to restore the gluon dominance leading
to new constraints on the gluon distribution at small $x$ [25, 26].

##  3 Exclusive $J/\psi$ photoproduction in dipole picture

In the dipole picture of high-energy scattering, the photon is viewed as a
superposition of long-lived quark-antiquark, quark-antiquark-gluon, etc.
dipoles, which elastically scatter on the target nucleons. This leads to a
high-energy factorization for the nuclear scattering amplitude, which is
expressed in terms of the overlap of the photon and $J/\psi$ wave functions
and the eikonal form for the dipole cross section $\sigma(r_{T})$,

${\cal M}^{\gamma A\to J/\psi A}=2\int d^{2}{\bf r}_{T}\int\frac{dz}{4\pi}\int
d^{2}{\bf
b}_{T}[\Psi^{\ast}_{J/\psi}\Psi_{\gamma}]\left(1-e^{-\frac{1}{2}\sigma(r_{T})T_{A}(b_{T})}\right)\,,$
(8)

where $r_{T}$ is the dipole transverse size, $b_{T}$ is the impact parameter,
$z$ is the light-cone momentum fraction shared by the charm quark and
antiquark, and $T_{A}(b_{T})$ is the nuclear optical density. The application
of this implementation of the dipole model to coherent $J/\psi$
photoproduction in Pb-Pb UPCs at 2.76 TeV overestimates the data at $y\approx
0$ [27, 28] because nuclear shadowing resulting from scattering of small
dipoles with $r_{T}\sim 0.3$ fm on target nucleons is weak. It is an example
of the general observation that successive scattering of small dipoles on
nucleons of a nuclear target corresponds to a higher-twist effect [29].

To cure this, one needs to include higher Fock states, in particular, the
quark-antiquark-gluon dipoles. The improved, more complete dipole model
predictions agree well with the LHC data on coherent $J/\psi$ photoproduction
in Pb-Pb UPCs [30, 31]. Note that the inclusion higher Fock states allows one
to model inelastic (leading-twist) contribution to nuclear shadowing as
required by the Glauber-Gribov theory of nuclear shadowing [32].

Instead of building up the nuclear scattering amplitude as the Glauber series,
see Eq. (8), one can implement nuclear geometry in the initial condition of
the Balitsky-Kovchegov (BK) equation. This can be interpreted as an onset of
the gluon saturation in nuclei, but not necessarily in the nucleons. The
resulting approach provides a good description of the LHC data on coherent
$J/\psi$ photoproduction in Pb-Pb UPCs [33].

To summarize, none of the available theoretical approaches describe the
$y$-dependence of coherent $J/\psi$ photoproduction in Pb-Pb UPCs at 5.02 TeV
[34, 35]. The suppression at $y\approx 0$ can be interpreted as the strong
leading twist gluon nuclear shadowing in the collinear framework or as an
indication of the importance of higher Fock states and the gluon saturation in
the dipole picture. Large values of $y$ correspond to small shadowing, where
predictions of different models generally converge, but there they are at the
border of their applicability. In this respect, recent measurements of UPCs
with neutron emission [10, 11] provide important complementary information on
nuclear shadowing in a wide range of $x$, $10^{-5}<x<0.03$.

Turning to incoherent $J/\psi$ photoproduction in UPCs, one notices that it
requires a new sub-nucleon scale, which can be realized in the form of “hot
gluon spots” and geometric fluctuations of the proton. Implementation of these
ideas in the dipole picture provides a fair description of the ALICE data [36,
37]. The competing explanation is based on the leading twist approximation
(LTA) to nuclear shadowing [38]. Note that the two approaches provide
compatible predictions for the $t$-dependence of the $J/\psi$ photo-nuclear
cross section, which are characterized by a shift toward smaller $|t|$ in the
coherent case [39, 40] and flattening for large $|t|$ in the incoherent case.

##  4 Inclusive and diffractive dijet photoproduction in Pb-Pb UPCs

Complementary information on the partonic structure of nuclei and small-$x$
QCD dynamics can be obtained in dijet photoproduction in UPCs. In the
framework of collinear factorization and NLO pQCD, the cross section of
inclusive dijet photoproduction in Pb-Pb UPCs reads [41]

$d\sigma^{AA\to A+{\rm 2jets}+X}=\sum_{a,b}\int dy\int dx_{\gamma}\int
dx_{A}f_{\gamma/A}(y)f_{a/\gamma}(x_{\gamma},\mu^{2})f_{b/A}(x_{A},\mu^{2}){\hat{\sigma}}_{ab\to{\rm
jets}}\,,$ (9)

where $f_{\gamma/A}(y)$ is the photon flux, $f_{a/\gamma}(x_{\gamma})$ are
photon PDFs in the resolved-photon case, $f_{b/A}(x_{A})$ are nuclear PDFs,
and ${\hat{\sigma}}_{ab\to{\rm jets}}$ is the cross section to produce jets in
hard scattering of partons $a$ and $b$ (in the direct photon case, $a$ stands
for the photon). The distributions in Eq. (9) depend on the corresponding
momentum fraction and the factorization scale $\mu$.

Figure 2 shows predictions of Eq. (9) with the implementation and input
specified in [41] and compares them to the preliminary ATLAS data at 5.02 TeV
[42]. One can see from the figure that NLO pQCD describes well the shape and
normalization of the data. This cross section is sensitive to nuclear
modifications of nPDFs at the level of $10-20$% and can be used to reduce
their uncertainties by approximately a factor of 2 [43]. Note that this
process can also be used to look for nonlinear effects within the color glass
condenstate (CGC) framework [44].

Figure 2: NLO pQCD predictions for the cross section of inclusive dijet
photoproduction in Pb-Pb UPC as a function of $x_{A}$ in different bins of the
dijet transverse momentum $H_{T}$ vs. the ATLAS data. From [41].

By requiring that the target nucleus stays intact, one study diffractive dijet
photoproduction in UPCs. In NLO pQCD, the $k^{+}$ contribution [see Eq. (1)]
to the cross section of diffractive dijet photoproduction in Pb-Pb UPCs has
the following form [45],

$\displaystyle d\sigma^{AA\to A+{\rm 2jets}+X+A(+)}$ $\displaystyle=$
$\displaystyle\sum_{a,b}\int dy\int dx_{\gamma}\int dx_{P}\int dz_{P}$ (10)
$\displaystyle\times$ $\displaystyle
f_{\gamma/A}(y)f_{a/\gamma}(x_{\gamma},\mu^{2})f_{b/A}^{D(3)}(x_{P},z_{P},\mu^{2}){\hat{\sigma}}_{ab\to{\rm
jets}}\,,$

where $x_{P}$ and $z_{P}$ refer to the Pomeron-in-nucleus and parton-in-
Pomeron momentum fractions. Such measurements will probe the novel nuclear
diffractive PDFs $f_{b/A}^{D(3)}$ at small $x$ [15] and may also shed some
light on the mechanism of factorization breaking in diffractive scattering
[45].

##  5 Summary

There is continuing interest in using UPCs at the LHC and RHIC to obtain new
constraints on the partonic structure of the proton and nuclei and the QCD
dynamics at small $x$. The available data challenge both collinear
factorization and dipole model frameworks. Large nuclear suppression of
coherent $J/\psi$ photoproduction in Pb-Pb UPCs at the LHC can be explained by
the strong gluon/quark shadowing at small $x$, the contribution of quark-
antiquark-gluon dipoles, or an onset of saturation. It will be important to
test these prediction in $\Upsilon$ photoproduction in Pb-Pb UPCs.

While the framework of collinear factorization and NLO pQCD provides a fair
description of coherent $J/\psi$ photoproduction in Pb-Pb UPCs at the LHC
within large factorization scale and nPDF uncertainties, there remain
outstanding challenges related to NRQCD corrections to the $J/\psi$ vertex and
small-$x$ resummation of the NLO coefficient functions. One way to circumvent
these issues is to consider the ratio of UPC cross sections for different
targets, where the theoretical uncertainties are significantly reduced. Recent
progress in calculations of exclusive vector meson prodiction at NLO in the
dipole picture should help to clarify interpretation of the UPC data.

## Acknowledgments

This research was funded by the Academy of Finland project 330448, the Center
of Excellence in Quark Matter of the Academy of Finland (projects 346325 and
346326), and the European Research Council project ERC-2018-ADG-835105
YoctoLHC.

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Resilience of Snowball Earth to Stochastic Events

Guillaume Chaverot1,2,3, Andrea Zorzi4, Xuesong Ding5Now at Bureau of Economic Geology, The University of Texas at Austin, Austin, TX, US, Jonathan Itcovitz6,7, Bowen Fan8, Siddharth Bhatnagar1,3,9, Aoshuang Ji10, Robert J. Graham8, Tushar Mittal10

1Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
2Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
3Life in the Universe Center, Geneva, Switzerland
4Department of Earth and Planetary Sciences, Stanford University, Stanford CA, US
5Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, US
6Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, UK
7Department of Civil and Environmental Engineering, Imperial College London, London, UK
8Department of the Geophysical Sciences, University of Chicago, Chicago, IL, USA
9Group of Applied Physics and Institute for Environmental Sciences, University of Geneva, Bd. Carl-Vogt 66, CH-1205 Geneva, Switzerland
10Department of Geosciences, Pennsylvania State University, University Park, PA, US

Guillaume<EMAIL_ADDRESS>
* We use an Energy Balance Model (EBM) to simulate the response of Snowball climate after stochastic events.

* Impact simulations and estimates of ash dispersion are used to inform initial conditions.

* Earth's Snowball states seem resilient to termination by stochastic events within our modeling framework.

Earth went through at least two periods of global glaciation (i.e., “Snowball Earth” states) during the Neoproterozoic, the shortest of which (the Marinoan) may not have lasted sufficiently long for its termination to be explained by the gradual volcanic build-up of greenhouse gases in the atmosphere. Large asteroid impacts and supervolcanic eruptions have been suggested as stochastic geological events that could cause a sudden end to global glaciation via a runaway melting process. Here, we employ an energy balance climate model to simulate the evolution of Snowball Earth's surface temperature after such events.
We find that even a large impactor (diameters of $d \sim 100\,\mathrm{km}$) and the supervolcanic Toba eruption ($74\,\mathrm{kyr}$ ago), are insufficient to terminate a Snowball state unless background CO$_2$ has already been driven to high levels by long-term outgassing. We suggest, according to our modeling framework, that Earth's Snowball states would have been resilient to termination by stochastic events.

§ PLAIN LANGUAGE SUMMARY

The terminations of Earth's longest periods of global glaciation are commonly understood to have occurred due to the gradual build-up of greenhouse gases in the atmosphere from volcanism. However, the sudden ends of Earth's shorter global glaciation periods likely cannot be explained by the same mechanisms. Large asteroid impacts and supervolcanic eruptions have been suggested as geophysical phenomena that could cause abrupt ends to global glaciation periods. Here, we model the evolution of the planet's surface temperature in the aftermath of such events. Impacts and eruptions open up gaps in the global ice sheet, and also partially cover the ice in far-spreading dust and ash, both of which increase the amount of solar radiation that is absorbed by the planet comparing to the highly reflective surface of ice and snow. Greater absorption of radiation leads to higher surface temperatures, which increases ice melting, and generates a feedback loop that can melt the entire planet surface.
However, we find that the scales of impact or eruption required to produce global melting are too great to have likely occurred at the times of Earth's global glaciations. Other mechanisms must, therefore, be explored to explain Earth's short glaciation periods.

§ INTRODUCTION

The Snowball Earth hypothesis <cit.> proposes global surface coverage of Earth in a thick ice layer. In contrast, the most recent glacial episode, the Last Glacial Period, witnessed only partial ice coverage extending down from the poles but not reaching equatorial regions <cit.>e.g.,>batchelor2019. Snowball events have occurred on at least two occasions during the Neoproterozoic era ($1000-538.8 \,\mathrm{Ma}$). In particular, there is good evidence for a snowball event during the Cryogenian period ($720-635 \,\mathrm{Ma}$), with other potential Snowball states also being suggested <cit.>e.g.,>chumakov2009baykonurian. Evidence for Snowball Earth's existence comes in many forms, including paleomagnetic data in Australia <cit.>, Australian cap carbonates <cit.>, and extreme carbon isotope excursions in Norway and Greenland <cit.>. Geologic measurements have been complemented by modeling efforts <cit.>e.g.,>Caldeira1992 to understand both the origins and demise of the Snowball state.

Under Earth's current insolation, a Snowball state is one of three possible stable climate regimes for Earth, alongside a temperate state and a hot steam atmosphere state <cit.>.
However, as shown in brunetti_co-existing_2019, the number and nature of these stable climate regimes can vary depending on feedbacks considered within the model.
The Snowball state is stable against moderate variations in insolation through the high albedo of ice, which acts to prevent increases in global temperature <cit.>. This creates a bi-stability between the temperate Earth and Snowball Earth through the difference of albedo between ice and liquid water <cit.>. It is, therefore, impossible to deglaciate Snowball Earth under such conditions without other climate forcings.
One such forcing that is commonly invoked is the build-up of atmospheric CO2 <cit.>. Magmatic volatile emissions from volcanism, coupled with the absence of CO2 sinks of Earth's temperate-state carbon cycle (e.g., silicate weathering being prevented by the covering of ice) leads to a strong greenhouse effect that warms the planet, triggering the Snowball termination <cit.>. However, the time required to accumulate sufficient CO2 to trigger melting <cit.>$4-30 \,\mathrm{Myr}$, >Hoffman1998 is too long to account for the short Marinoan glaciation <cit.>. Suggested alternatives include impact events <cit.> and massive volcanic eruptions <cit.>, which we hereafter refer to as stochastic events due to the nature of their occurrences. However, quantitative estimates of climate after stochastic events remain relatively unexplored.

Impacts of large asteroids can generate global climate effects <cit.>e.g.,>Brugger2017,Pierazzo1998,Artemieva2017,turbet_environmental_2020. Simulations of such impacts have estimated the masses of water vapor injected into the atmosphere by these events, but the climatic implications have not been investigated  <cit.> or have been inconclusive due to uncertainties on the radiative effect of water vapor <cit.>. Nonetheless, the temporal coincidence between the youngest Paleoproterozoic glacial deposits and the Yarrabubba event has sparked interest in the role of impacts in climate evolution <cit.>. Further, given their large kinetic energy, impacts can melt the ice table covering the planet surface and vaporize rocks from the underlying crust. Therefore, among other effects, impacts can (1) increase the planet's surface temperature by hundreds of degrees for large distances from the impact site, and (2) deposit ejecta material onto the icy surface far from the impact site, affecting the planet's surface albedo.

Supervolcanic eruptions are also suggested candidates to escape the Snowball Earth state. Such events are defined by either their magnitude ($M\geq8$, proportional to the erupted mass) or their ejecta volume <cit.>$1,000 \,\ \mathrm{km^3}$,>desilva2022, and are thousands of times larger than typical volcanic eruptions <cit.>e.g.,>Hansen1978. The largest known super-eruption is Toba ($M=9.1$), which is recorded in the 74,000-year-old Youngest Toba Tuff <cit.>YTT,>Rose1990. The ash ejected by the event covered $\sim$ 40 million $\mathrm{km^2}$ of Earth's surface at a depth greater than 5 mm <cit.>. Toba-like super-eruptions are expected to happen $1-2$ times per million years <cit.>. Supervolcanic eruptions have significant global climate effects <cit.>e.g.,>Hansen1978. Large volumes of volcanic gases are injected into the atmosphere, such as (1) CO2, which acts as a classical greenhouse gas, (2) SO2, which acts as a solar-reflecting coolant, and (3) H2O, which can act as a greenhouse gas, but can also form radiative-active clouds. Further, eruptions deposit ash and dust on the icy surface, reducing the albedo in comparison to clean ice and hence increasing the amount of sunlight absorbed by the planet's surface. Although often cited as a possible cause of the onset of global glaciation <cit.>, there is no detailed exploration of the viability of this scenario.

In this study, we quantify the heating effects of stochastic events on a globally glaciated Earth, testing the potential of such events to generate a runaway melting process that would lead to the termination of the Snowball state. Through impact simulations and ash-/ejecta-dispersal modeling (Sections <ref> & <ref>), we quantify the changes in albedo and surface temperature produced by such events. Using a 1-D Energy Balance Model (EBM; Section <ref>), we then analyze the radiative perturbation caused by such temperature and albedo changes, following the evolution of the post-event surface temperature profile and ice coverage. By varying the scale of our stochastic events (e.g., impactor size, magnitude of supervolcanic eruption), we assess the capabilities of such events to trigger global deglaciation (Section <ref>). We discuss the likelihood of occurrence of our event scales during the Neoproterozoic, and thus the likelihood of deglaciation by stochastic events, in Section <ref>.

§ METHOD

§.§ Climate modeling

budyko1969 and sellers1969's independent seminal works on the Earth's climate established the usefulness and robustness of 1-D EBMs. In a latitudinal EBM, the prognostic surface temperature is zonally averaged, thus leaving only the meridional dimension.
This gives a remarkably good first-order estimate to solving the climate of a rocky planet <cit.>. The 1D-nature of the planet is further exemplified by parameterizing the meridional atmospheric heat transport as temperature-driven diffusion.
Because bi-stability between the temperate Earth and Snowball Earth is a radiative balance process that exists through the difference of albedo between ice and liquid water <cit.>, we use an EBM focused on radiative balance <cit.> to model the evolution of snowball deglaciation. The corresponding temperate and snowball temperature profiles are shown in Supplementary Figure S2. As 1-D EBMs are not longitudinally defined, we developed a method to account for spatially-located events, correcting input calculations from impact simulations and estimations of ash-dispersion from eruptions (see Supplementary Text S4).

For our simulations, we assume a planet following a circular orbit at 1 AU around a Sun-like star, with a modern Earth solar constant. We fix the planetary obliquity to 0° and also assume zero eccentricity. Land and oceans are uniformly distributed across the planet such that each latitude cell consists of a 70:30 modern Earth ocean-to-land ratio. Heat capacity and albedo parameterizations between ice and liquid water are smooth transition functions taken from williams_habitable_1997 and spiegel_habitable_2008 (see Supplementary Text S4). To constrain our study within a simple framework, we neglect the effects of aerosols and ice-melting latent heat: potential caveats from our assumptions are discussed in Section <ref>. However, the neglected processes (e.g., dust enrichment of the stratosphere due to impacts, SO2 production of volcanism) tend to cool the atmosphere, making deglaciation more difficult to achieve. Even in these conditions, a significant warming of the planet is hardly reachable as shown in Section <ref>. Nonetheless, we account for variable relative humidity (RH), cloud forcing <cit.> and the potential addition of CO2 released by volcanic eruptions. We also evaluate the influence of the location of stochastic events by modeling eruptions and impacts at the equator and at 40° N. The effect of impacts is modeled by modifying the initial temperature distribution as per hydrocode simulations (see Section <ref>). Eruptions do not modify the temperature distribution, but affect the surface albedo due to ash-dispersion.

§.§ Impact simulations

We use the shock physics code iSALE <cit.> to model asteroidal impacts on a Snowball Earth state and produce 1-D thermal profiles for the surface of the planet in the aftermath of the impact. We additionally analyze the spread of the impact ejecta curtain in order to estimate the reduction in surface albedo associated with covering the ice with ejecta material.

We initialize simulations based on estimates of Snowball Earth's surface state <cit.>. A $5\;\mathrm{km}$ thick global ice sheet thus sits at the surface of the target planet, with a rocky crust, mantle, and iron core sitting below. An average surface temperature of $227\;\mathrm{K}$ is prescribed for the pre-impact temperature profile, matching the steady-state temperature found by the EBM in the absence of a stochastic climate forcing event. Impactors consist entirely of dunite, matching closely the thermodynamic behaviors of the chondritic material that are expected from such objects <cit.>. We run simulations for impactors with diameters between $40-100 \;\mathrm{km}$ to assess the effects of impact scale. The impact velocity is constant for all impacts at $1.5 \times$ mutual escape velocity ($v_\text{esc}$), representing the most likely impact velocity <cit.>, and we consider only head-on collisions, which is an approximation accounting for the small mass of the impactors relative to the Earth (more informations about the impact setup are provided in Supplementary Text S2).

Simulations are run until the motion in the mantle region excavated by the impact settles out (i.e., the mantle has rebounded and subsequent oscillatory motions have terminated). A 1-D surface temperature profile is then extracted, and used as the initial perturbation of the EBM. Because we use a 1-D latitudinal EBM, the initial temperature at the impact site results from the average of the high-temperature impact site longitudes and the low-temperature unperturbed longitudes (see Supplementary Text S4 for more details). As a result, the initial temperature in the EBM (zonal mean temperature, $200-350 \,\mathrm{K}$) is lower than the temperature at the impact site (local temperature, $>1000 \,\mathrm{K}$).
We find that this approach slightly facilitates deglaciation by marginally overestimating the surface albedo ($<1 \,\%$).

An albedo profile is similarly determined, accounting for the exposure of crustal material after melting of the ice layer and the deposition of material from the impact ejecta curtain on top the ice table outside of the impact crater.
We assume that ejecta reaches distances up to $5 \times$ the crater radius <cit.>e.g.,>RICHARDSON2005, corresponding to distances of $2250 \;\mathrm{km}$, $3850 \;\mathrm{km}$, $3900 \;\mathrm{km}$ and $5000 \;\mathrm{km}$ from the impact site for the $40 \;\mathrm{km}$, $60 \;\mathrm{km}$, $80 \;\mathrm{km}$ and $100 \;\mathrm{km}$ impactors, respectively. The albedo for ejecta-covered areas is set to $0.2$, similar to that of cryoconite <cit.>. A step-like transition is then assumed between dark ejecta and ice albedo ($0.7$). Finally, we re-scale the temperature increase with the surface area that is affected by the impact and the size of latitudinal band, assuming the impact energy is conserved during the zonal transport. The temperature rise is submitted as the input to the EBM.

§.§ Supervolcanic eruptions

We implement the formalism of pyle1989thickness to model the variation of volcanic ash thickness ($d$) with distance from the eruption center. The exponential relationship between $d$ and the isopach area $A_\mathrm{iso}$ is given by:

\begin{equation}
d = d_0 e^{-k A_\mathrm{iso}^{1/2}}
\label{eq:Ash Spread}
\end{equation}

where $d_0$ is the ash maximum thickness (units of $\mathrm{m}$) and $k$ is the rate of ash thinning with isopach area, also in $\mathrm{m}$ <cit.>. We find $k$ using ($d$, $A_\mathrm{iso}$) values of volcanic ash fall from Toba, reconstructed through several tens of thickness measurements of the YTT tephra deposit <cit.> and digitized with QGIS <cit.>.

The ash deposition region is divided into fully- ($d >0.014 \,\mathrm{mm}$) and partially-covered ($0.002<d<0.014 \,\mathrm{mm}$), the latter known as dusty ice <cit.>. Assuming an elliptical geometry for ash-dispersal, the ash deposition area is converted in latitude and longitude. For Toba, the fully-covered area extends for $\pm 43^\circ$ in latitude and $\pm 86^\circ$ in longitude; the dusty ice area spans a wider range between $\pm 50^\circ$ in latitude and $\pm 100^\circ$ in longitude. At visible wavelengths, coarse dust has an albedo of $0.18$ <cit.>. Thus, assuming a clean ice albedo of $0.7$, we assign the dusty ice albedo as changing linearly between clean ice and dust (Supplementary Figure S3). Such values are consistent with measurements of bare ice mean albedo <cit.>. Fine dust correspond to higher albedo values, thus making deglaciation harder <cit.>.

Temporal evolution of surface temperature after a Toba-like eruption (panel A-D) and impacts of varying size (panel E-H). For the eruptions, each panel corresponds to a different set of initial conditions, highlighting the differences that such effects produce under the same climate forcing event. For the impacts, all panels show the same cloud-free atmosphere with $400\;\mathrm{ppm}$ CO2 of (A). The global-mean temperature (solid lines), minimum and maximum temperatures (dashed lines), and the melting temperature of the surface ice (dotted lines) are all shown.

§ RESULTS

The EBM is able to provide the temporal evolution of surface temperature for up to several years after the moment of climate forcing. We find, however, that temperatures tend to converge within around two years (e.g., Figure <ref>). Geospatial variation in temperature, due to the regional nature of the stochastic events, can be well represented by recording global temperature minima, maxima, and mean averages (Figure <ref>).

We first analyzed the effects of a Toba-like eruption (Figure <ref>A-D) for a cloud-free atmosphere similar to the present-day Earth atmosphere, with variable water vapor, $1 \;\mathrm{bar}$ of nitrogen and $400 \;\mathrm{ppm}$ of CO2, and with the eruption happening at the equator (Figure <ref>A). Additional simulations then included cloud radiative effects <cit.>$14 \,\mathrm{W \ m^{-2}}$,>abbot_resolved_2014 and increased atmospheric CO2 at $800 \,\mathrm{ppm}$ (Figure <ref>B) in order to test the robustness of the result to initial atmospheric conditions, and to account for greenhouse gases released by the eruption. Finally, we repeated these set simulations with the climate forcing occurring at $40^\circ \,\mathrm{N}$ (Figure <ref>C). In none of these simulations does a Toba-like eruption change surface temperatures sufficiently to escape the Snowball state. In the best-case scenario for deglaciation (i.e., accounting for the cloud-warming effect and with $800 \,\mathrm{ppm}$ of CO2), the maximum equatorial temperature is only $255 \,\mathrm{K}$ after relaxation from forcing. Other effects were further tested, but were found to contribute minorly to results within realistic bounds, including: the addition of humidity due to volcanic moisture, weaker atmospheric circulation, different ice albedo, and greater effective stellar insolation (see Supplementary Text S5 and Figure S1).

In the case of our impactors, the largest ($d = 100 \;\mathrm{km}$) generates a transient water belt (note this may not be a true water belt but rather a function of the EBM's spatial structure), with maximum extents of $4.5^\circ \;\mathrm{N}$ and $4.5^\circ \;\mathrm{S}$, and lasting for around two months (Figure <ref>H). During this time, the surface temperature of the planet decreases due to thermal emission. Once the freezing point is reached, the change of albedo induces a rapid runaway re-glaciation of the transient water belt. We thus find that even the $100 \;\mathrm{km}$ impactor cannot melt a surface large enough to break the ice albedo feedback, which prevents the Snowball Earth deglaciation. The lower resultant temperatures and reduced geospatial domain of our smaller impactors ($d = 40-80 \;\mathrm{km}$) results in no production of a water belt and indeed limited melting of the planet's surface (Figure <ref>E-G), with the same result of no deglaciation.

Impactors larger than $100 \;\mathrm{km}$ will melt more ice and will generate greater temperature perturbations. However, such impacts would also melt part of the planet crust and/or mantle, for which there is little evidence in the geologic record, and such massive impactors are highly unlikely by the time of the Neoproterozoic (Section <ref>). A simple analytical calculation tends to show that the energy required to deglaciate could only be delivered by an impactor that is larger than 100 km (Text S1).

The equilibrium spatial temperature profiles (i.e., spatially resolved profiles for the last times shown in Figure <ref>A-H) can provide further insight into the climate forcing effects of our stochastic events (Figure <ref>).
Without any stochastic forcing, we observe the usual latitudinal distribution of surface temperatures, with a maximum of $\sim205 \;\mathrm{K}$ at the equator and a minimum of $\sim 190 \;\mathrm{K}$ at the poles.

For a Toba-like eruption, we find that deposition of the volcanic ash leads to greatest temperature rises near the deposition site itself (Figure <ref>A). Ash deposition affects the surface albedo on a limited area, locally increasing the absorbed flux.
For this reason, considering the eruption at $40^\circ \,\mathrm{N}$ (Figure <ref>A, red line) centers the peak of the temperature profile to this location, and lower the maximum temperature of about $5 \;\mathrm{K}$ due to a weaker incoming flux at this latitude.
Additionally, for both cases, heat diffusion (atmospheric circulation) allows a global increase of temperature.
By accounting for the radiative effect of the clouds and by assuming $800 \;\mathrm{ppm}$ of CO2, the extra warming is around $10 \;\mathrm{K}$, independent of latitude and of the eruption location.
We note that the results shown here do not account for the ash injected into the stratosphere, which reduces the bond albedo of the planet and thus induces global cooling <cit.>e.g.,>abbot_mudball_2010. We discuss such processes, as well as others that we have chosen to neglect in the presented results, and their effect on our conclusions in Section <ref>.

For impact events, the converged temperature profiles are warmer than the non-perturbed simulation (Figure <ref>B), due to ejecta affecting the surface albedo. The rise in temperature is a function of the area covered by ejecta: the greater the area covered, the more the albedo of the surface is affected. However, the maximum temperatures for all impact scenarios are colder than those found in scenarios forced by the supervolcanic eruptions. The injection of heat that is provided by the impactors is a relatively minor effect in comparison to the change in surface albedo of the ice that is brought about by the ejecta curtain. Evidence for this can also be found in the temporal evolution of temperatures after the impacts (Figure <ref>E-H). In the four cases, the large radiative imbalance at the impact site dissipates heat rapidly relative to the timescale of global temperature changes. Therefore, the colder converged temperatures of our impacts are due to the diminished ejecta coverage that they produce in comparison to the coverage of the supervolcanic ash (see Supplementary Figure S3). Interestingly, for $80 \;\mathrm{km}$ and $100 \;\mathrm{km}$ impactors, we find a cold patch in the converged temperature profile at the impact site. We determine this to be due to the transient water belt that forms after the impact: when the planet cools, this liquid water freezes at the impact site as clean ice with greater albedo than the ejecta-covered ice around the crater.

Latitudinal temperature profiles after (A) Toba-like eruptions, both at the equator (blue lines) and at $40^\circ \;\mathrm{N}$ (red lines), and with both the cloud-free atmosphere with $400\;\mathrm{ppm}$ CO2 (solid lines) and an atmosphere including cloud radiative effects and $800\;\mathrm{ppm}$ CO2, and (B) impacts with impact diameters of $40 \;\mathrm{km}$ (green), $60 \;\mathrm{km}$ (blue), $80 \;\mathrm{km}$ (orange), and $100 \;\mathrm{km}$ (red), all striking at the equator and under cloud-free atmosphere with $400\;\mathrm{ppm}$ CO2 (solid lines). The red dashed line in panel B corresponds to a $100 \;\mathrm{km}$ impactor under a cloudy atmosphere, assuming $400\;\mathrm{ppm}$ of CO2. The initial temperature profile (i.e., before climate forcing stochastic events) is shown for both types of events (dotted lines).

§ DISCUSSION

§.§ Limitations of the climate model

H2O and CO2 are efficient greenhouse gases and should facilitate the warming and deglaciation of the ice sheet. However, we find that variation of the gas partial pressures, within sensible bounds for the climate of a Neoproterozoic Snowball Earth, cannot push our stochastic events into regimes where they cause global deglaciation (see Supplementary Text S5).

There are a variety of effects that are not included in the EBM that could be considered as important. We neglect the formation of stratospheric aerosols, which would act to cool the planet <cit.>. In order to minimize the degrees of freedom in the model, we do not account for ice thickness or the latent heat released by ice melting, which could be important in accounting for the time-scales over which the water belt induced by the $100$-$\mathrm{km}$ impactor freezes. In much the same way, oceanic circulation will affect the creation of ice in the water belt <cit.>e.g.,>yang_transition_2019, brunetti_co-existing_2019. However, a commonality between these effects is that their inclusion would serve only to increase the challenge in producing global deglaciation, which our models already suggest is not achievable given the scales of events considered. We thus suggest that our simplified EBM is able to point out possible directions for further works using more complex models.

§.§ Assumptions on impacts and large eruptions

Our simulations suggest that impacts larger than $100 \;\mathrm{km}$ are necessary, albeit not sufficient, to initiate deglaciation on the planet. However, dynamical models indicate that, outside of the early Solar System, such impacts are expected to occur less than once every billion years <cit.>. Moreover, geologic evidence for such event should be present, since a large projectile should have globally spread an ejecta layer analogous to the Chicxulub impact event <cit.>. So far, no impact signature caused by a $\sim 100$-$\mathrm{km}$ impactor has been found on Earth <cit.>; as per allen2022, the largest impact crater found is caused by a $20-25 \,\mathrm{km}$ impactor.

The results we present are based on a single impact scenario, comprising of a fixed stratigraphy and an asteroidal projectile hitting the surface at $17 \;\mathrm{km \,s^{-1}}$. We chose an ice thickness of $5 \;\mathrm{km}$ following Erickson2020 as a reasonable upper limit for the actual size of glaciers covering the planet. In reality, the ice thickness will depend on the location, as suggested by the geologic record <cit.>e.g.,>McMechan2000 and climate modeling <cit.>, ranging from meters to $5 \;\mathrm{km}$. Variation in ice thickness will produce a mechanically different response to impact. However, given the scale of impactor that our models find would be necessary to cause global deglaciation (i.e., $d > 100 \;\mathrm{km}$), the ice thickness is relatively shallow in comparison, and we thus expect that the consequences of this effect would be minor. Further, we did not consider impactors of alternative compositions, such as icy bodies, in our models. While short-period comets <cit.>impact velocity of $30 \;\mathrm{km \,s^{-1}}$, >CHYBA1991 have kinetic energies similar to that of the asteroidal impacts we consider, long-period comets <cit.>$50 \;\mathrm{km \,s^{-1}}$, >CHYBA1991 might change the impact mechanics, but are less likely to hit the Earth <cit.>. Such events would leave behind a distinct geochemical signal from a rocky impactor analogue that, while not yet discovered, should similarly perhaps not be excluded for now. Further modeling of such stochastic events could be entertained by future works, although will likely find challenges of scale similar to our work (i.e., requiring extremely unlikely comet sizes to cause deglaciation). Finally, we assumed zero porosity as a worst case scenario: for a given size, non-porous material ensures the maximum kinetic energy available to melt ice and vaporize target rocks. Similar to the climate choices made, therefore, relaxing this assumption only serves to make deglaciation more challenging.

In modeling volcanic eruptions, results are sensitive to the assumed dust distribution and the absence of lava warming and water vapor. Based on Icelandic dust and ash <cit.>, a thickness of $15 \;\mathrm{mm}$ is considered as the threshold for ice to melt due to the decrease in albedo. A thicker layer would thermally insulate the ice below, preventing melting. Increasing such a threshold would expand the dust-covered area subject to melting, favoring deglaciation. Furthermore, the local temperature increase due to erupting lava is disregarded, since its effect is negligible when averaged longitudinally. Similarly, we do not consider a long-term (i.e., multi-year) eruption, which would force a higher temperature around the volcano for a longer time. Additionally, as seen for the Hunga Tonga-Hunga Ha'apai eruption, a substantial amount (up to $10^{11} \;\mathrm{kg}$) of water vapor can be injected into the stratosphere where it can have multi-year long radiative effects <cit.>net warming of the surface, >Millan2022. Finally, aerosols ejected up to the stratosphere could increase snowfall, thus covering low albedo dusty ice.

§.§ Possible shortening of snowball events

In this work, we consider a low CO2 pressure as the initial climate state. However, during a snowball event the carbon cycle is altered due to the ice shield preventing CO2 (partial) dissolution into the ocean and restricted silicate weathering on land. On modern Earth, CO2 outgassing is about 47 bar/Gyr <cit.> through a combination of continental and submarine volcanism. Although there is no consensus so far on the exact volcanic outgassing flux for snowball Earth, multiple studies suggest that the value was much lower than present day.
fischer_agu_2020 estimate the arc production around 5.7 bar/Gyr while dutkiewicz_duration_2024 argue that the mid-ocean ridge outgassing was even lower.
We found that an extra radiative forcing of 50 W/m$^2$ (in addition to 14 W/m$^2$ of forcing induced by the clouds) is required, coupled with the effect of a 100 km impactor, to deglaciate the planet. That is corresponding approximately to 0.08 bar of CO2 according to wolf_evaluating_2018. In comparison, without any stochastic event the required forcing to deglaciate is equal to 90 W/m$^2$ (in addition to 14 W/m$^2$ of forcing induced by the clouds).
Using modern day Earth outgassing from fischer_agu_2020 as boundary values, it takes between 1.7 Myr and 14 Myr to reach a CO2 pressure for which a 100-km impactor could induce a deglaciation, in our framework. An intermediate outgassing value of 25 bar/Gyr gives approximately 3.2 Myr to reach the required CO2 pressure.

These estimates should be considered as lower limits because we neglect the potential sinks of CO2 existing on snowball Earth, as well as cooling processes of impacts (e.g., increase of the bond albedo due to aerosols). However, this timescale is shorter than the duration time of the Marinoan and Sturtian glaciations. Although a large impact during this period seems highly unlikely, the possibility of deglaciating an extra-solar planet in this way cannot be ruled out.

§ CONCLUSIONS

Earth's periods of global glaciation are traditionally conceived to have ended through a gradual build-up greenhouse gases in the atmosphere (e.g., CO2). However, the shortest of these so-called Snowball Earth states were geologically short and stochastic events such as impacts and supervolcanic eruptions has been suggested as plausible deglaciation precesses. Here, we present modeling on the response of Snowball Earth to such events under differing scenarios. We use impact simulations and ash spreading models to estimate the initial effect of each stochastic event, and follow these up by evolving Earth's surface environment through time using a 1-D EBM.

In our modeling framework, surface albedo of the planet is the dominant driving force of long-term temperature change in the aftermath of stochastic events. For impacts, this takes the form of the ejecta curtain stemming from the impact site. For eruptions, volcanic ash is transported over a wide range of latitude and longitudes. We thus observe that eruptions produce a greater change in Earth's surface albedo and hence a greater temperature response for typical scale events. However, even an impactor radius of $100 \;\mathrm{km}$ is not sufficient to induce deglaciation of Snowball Earth. The warming at the impact site melts ice locally in the short-term, but due to a large radiative local imbalance, ice reforms in less than a year, returning the planet to the snowball state. Even if larger impactors could induce deglaciation, they are dynamically unlikely to have occurred on Earth at the time of its snowball episodes. Similarly, no recorded supervolcanic eruption could have single-handedly deglaciated Earth. The surface temperature increase due to dust deposition from the largest recorded event, Toba, does not reach the melting temperature of ice anywhere on the planet.

Our model thus places constraints on the magnitude of stochastic events that are incapable of triggering Snowball Earth termination. However, these constraints should be refined with more comprehensive modeling. For impacts, 3-D modeling can better resolve the ejecta distribution through accounting for oblique impacts. Impact simulations should also be able to account for the ice erosion effect due to the tsunami created upon impact, which would increase the low-albedo area. For the climate response, 3-D GCM modeling is required to include longitudinal and vertical resolutions, as well as to provide better representations of clouds, rain/snow, atmospheric dynamics, ocean dynamics, and distribution of aerosols ejected by the impactor <cit.>e.g.,>abbot_jormungand_2011,ashkenazy_dynamics_2013,turbet_environmental_2020 or released by the volcano <cit.>. For instance, even 2-D models could show more complex transitions between two steady states if specific eigenmodes are triggered by an impact happening at the right location <cit.>.

Lastly, in this work, we treated impacts and supervolcanoes as independent and instantaneous events all starting from the same initial state. Future analyses should investigate some of the prolonged effects associated with our stochastic events (e.g., multi-year lava eruptions). Additionally, the combined effect of impacts and eruptions should be considered, with impact-induced volcanism having been suggested as a potential supplement for the K-T extinction <cit.>e.g.,>Renne2015, although still debated <cit.>e.g.,>Bhandari1995. Multiple such events would likely stack their temperature responses, although unlikely to be in a linear manner, and could thus instigate a deglaciation of Snowball Earth.

§ OPEN RESEARCH

At present, the iSALE code <cit.> is not fully open-source. It is distributed via a private GitHub repository on a case-by-case basis to academic users in the impact community, strictly for non-commercial use.

The temperature profile after impact, used to perform climate simulations can be found from chaverot_resilience_2024.

The project was made possible by the 2022 Cooperative Institute for Dynamic Earth Research (CIDER) workshop in UC Berkeley, funded by the NSF. This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. GC acknowledges the financial support of the SNSF (grant number: 200021_197176, 200020_215760 and P500PT_217840). AZ acknowledges the financial support of Stanford University. JI was supported by the UK Science and Technology Facilities Council (STFC) grant number ST/T505985/1. RG acknowledges funding by the National Aeronautics and Space Administration (NASA) through a contract with ORAU.

We gratefully acknowledge the developers of iSALE-2D, including Gareth Collins, Kai Wünnemann, Dirk Elbeshausen, Tom Davison, Boris Ivanov and Jay Melosh.

We thank the two anonymous reviewers for the time they spent providing useful comments that helped to improve this manuscript.
Authors gratefully acknowledge R. T. Pierrhumbert for useful discussions during the 2022 CIDER workshop.

Authors declare that they have no competing interests.

Author contributions

Conceptualization: GC, BF

Methodology: AZ, GC, SB, XD, BF, JI, AJ, TM

Investigation: GC

Project administration: AZ, GC

Writing – original draft: AZ, GC

Writing – review & editing: AZ, GC, XD, BF, SB, JI, TM, RG

Data Curation: AZ, GC, JI

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