No dilute core produced in simulations of giant impacts on to Jupiter

MNRAS 542, 947–959 (2025) https://doi.org/10.1093/mnras/staf1105
Advance Access publication 2025 July 4
No dilute core produced in simulations of giant impacts on to Jupiter
T. D. Sandnes ,1‹
V. R. Eke ,1
J. A. Kegerreis ,2,3,4
R. J. Massey 1
and L. F. A. Teodoro 5,6
1Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
2Department of Earth Science and Engineering, Imperial College London , London SW7 2BP, UK.
3SETI Institute, 339 Bernardo Avenue, Suite 200, Mountain View, CA 94043, USA
4NASA Ames Research Center, MS 245-3, Moffett Field, CA 94035, USA
5Faculty of Mathematics and Natural Sciences, University of Oslo, Sem Sælands vei 24, 0371 Oslo, Norway
6School of Physics and Astronomy, University of Glasgow, G12 8QQ Scotland, UK
Accepted 2025 July 4. Received 2025 July 4; in original form 2024 December 8
ABSTRACT
A giant impact has been proposed as a possible formation mechanism for Jupiter’s dilute core – the planet’s inferred internal
structure in which the transition between its core of heavy elements and its predominantly hydrogen–helium envelope is gradual
rather than a discrete interface. A past simulation suggested that a head-on impact of a 10 M⊕ planet into an almost fully formed,
differentiated Jupiter could lead to a post-impact planet with a smooth compositional gradient and a central heavy-element
fraction as low as Z ≈ 0.5. Here, we present simulations of giant impacts on to Jupiter using improved numerical methods to
reassess the feasibility of this scenario. We use the REMIX smoothed particle hydrodynamics (SPH) formulation, which has
been newly developed to improve the treatment of mixing in SPH simulations. We note that, as in previous works, chemical
mixing is not included in these models and that incorporating such processes at sub-particle scales could improve numerical
convergence. We perform giant impact simulations with varying speeds, angles, pre-impact planet structures, and equations of
state. In all of our simulations, heavy elements re-settle over short time-scales to form a differentiated core, even in cases where
the core is initially disrupted into a transiently mixed state. A dilute core is not produced in any of our simulations. These results,
combined with recent observations that indicate Saturn also has a dilute core, suggest that such structures are produced as part
of the extended formation and evolution of giant planets, rather than through extreme, low-likelihood giant impacts.
Key words: methods: numerical – planets and satellites: gaseous planets – planets and satellites: individual: Jupiter – planets
and satellites: interiors.
1 INTRODUCTION
Measurements of Jupiter’s gravitational moments by the Juno space-
craft have led to models of the planet’s interior that suggest the exis-
tence of a dilute core: an extended compositional gradient between
Jupiter’s central core of heavy elements and its hydrogen–helium
envelope (Nettelmann 2017; Wahl et al. 2017; Vazan, Helled &
Guillot 2018; Debras & Chabrier 2019; Miguel et al. 2022; Militzer
et al. 2022; Howard et al. 2023; Militzer & Hubbard 2024). This
is inconsistent with traditional giant planet formation models that
predict a differentiated internal structure (Müller, Helled & Cumming
2020). With ring seismology suggesting that Saturn also has a dilute
core (Mankovich & Fuller 2021), understanding the processes that
govern the formation of such compositional gradients would provide
key insights into the evolution of giant planets and planetary systems.
Several mechanisms have been proposed to explain Jupiter’s dilute
core (Helled et al. 2022). An extended planetesimal-dominated
accretion phase could lead to the dilute core being in place prior
to runaway gas accretion (Venturini & Helled 2020; Stevenson et al.
2022). Alternatively, convective processes could gradually erode a
E-mail: thomas.d.sandnes@durham.ac.uk
differentiated core until it reaches a mixed state (Moll et al. 2017). Liu
et al. (2019; hereafter L19) proposed a giant impact as an alternative
mechanism.
The head-on impact simulation of L19 presented the disruption
of a differentiated core by a 10 M⊕ impactor into a well-mixed,
diluted state with a heavy-element fraction in the centre of the planet
of Z 0.5. This extreme giant impact would deliver approximately
half of the planet’s heavy elements in a single event. The simulations
of L19 with a larger impact parameter or a smaller impactor mass
did not produce a dilute core. The hydrodynamic simulations of
L19 were carried out using the adaptive mesh code FLASH (Fryxell
et al. 2000). By separately modelling the subsequent thermodynamic
evolution, L19 found that this compositional gradient could persist
for Gyr time-scales until the present day. However, overmixing in
regions of large bulk motion relative to the stationary grid points is
a typical shortcoming of Eulerian methods (Robertson et al. 2010;
Springel 2010). This spurious diffusion arises from the advection
terms necessary in this non-Lagrangian method. Additionally, the
accuracy of the treatment of self-gravity is sensitive to choices made
in the multipole approximation of the gravitational potential (Couch,
Graziani Flocke 2013), and L19 used idealized equations of state
(EoS) that do not capture the complexities of metallic hydrogen
within Jupiter’s deep interior (Chabrier, Mazevet Soubiran 2019).
© The Author(s) 2025.
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium,
provided the original work is properly cited.
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948 T. D. Sandnes et al.
MNRAS 542, 947–959 (2025)
Therefore, further investigation is warranted of the impact scenario
using fundamentally different modelling approaches, to assess the
potential sensitivity of dilute core production to the specifics of the
numerical methods employed.
Lagrangian hydrodynamic methods, where Galilean invariance is
maintained, do not experience the artificial mixing from advection
through grid points that is observed in methods that use a stationary
mesh, since interpolation points move with the fluid velocity. In
particular, smoothed particle hydrodynamics (SPH) is widely used
for simulations of giant impacts since it: inherently tracks the
evolution of fluid element trajectories and thermodynamics; is able
to deal with vacuum regions and evolving free surfaces efficiently;
offers geometry-independent adaptive resolution; and couples el-
egantly with gravity solvers (Gingold Monaghan 1977; Lucy
1977). In traditional SPH (tSPH) formulations, however, mixing at
density discontinuities is typically suppressed by spurious surface
tension-like effects (Agertz et al. 2007). These artificial effects are
considerable – and challenging to remedy – at boundaries between
dissimilar, stiff materials, for which more significantly erroneous
estimates of fluid pressure suppress mixing more strongly (Ruiz-
Bonilla et al. 2021). Therefore, a more advanced SPH construction
that addresses these known sources of error is needed to utilize
the benefits of the SPH formulation to reliably investigate dilute
core formation in giant impact simulations like these where material
mixing is the key physical mechanism of interest.
REMIX is an advanced SPH scheme designed to directly address
the sources of numerical error that suppress mixing in SPH sim-
ulations (Sandnes et al. 2025). The REMIX scheme incorporates
a range of novel and recently developed improvements to tSPH
formulations, and its construction is generalized to address sources of
error independent of material type or EoS. It demonstrates significant
improvements in the treatment of both mixing and instability growth,
including in simulations with materials and conditions representative
of those in giant impact simulations (Sandnes et al. 2025). REMIX
is integrated into the open-source, state-of-the-art SWIFT code,1
whose computational efficiency enables simulations of planetary
giant impacts to be performed at high resolutions (e.g. Kegerreis
et al. 2022).
Here, we use REMIX SPH to investigate whether Jupiter’s dilute
core could be formed by a giant impact. First, in Section 2 we describe
the methods used to perform simulations and to construct initial con-
ditions. In Section 3, we test REMIX in fluid instability simulations
under conditions representative of Jupiter’s deep interior. Then, in
Section 4 we use REMIX to model giant impacts on to Jupiter. We
carry out simulations of: head-on impacts (Section 4.1); isolated
planets with a pre-constructed dilute core to assess the potential
stability of dilute-core structures in our simulations (Section 4.2);
impacts at a range of impact speeds and angles (Section 4.3); and
impacts with pre-impact planet structures and EoS set up to closely
follow and compare with those of L19 (Section 4.4). We discuss our
results in Section 5 and summarize our findings in Section 6.
2 METHODS
2.1 REMIX smoothed particle hydrodynamics
REMIX is an SPH formulation designed to address key sources of
error that suppress mixing and instability growth in tSPH simulations,
particularly at density discontinuities. By adopting a generalized,
1SWIFT (Schaller et al. 2024) is publicly available at www.swiftsim.com.
material-independent approach, REMIX is able not only to improve
the treatment of contact discontinuities within a single material
but also to handle well the more challenging case of interfaces
between dissimilar, stiff materials. Like tSPH, REMIX inherently
conserves mass, energy, and momentum; is constructed from a basis
of thermodynamic consistency; and is fully Lagrangian, ensuring
Galilean invariance. REMIX has been extensively tested with stan-
dard hydrodynamics and giant impact-relevant test scenarios, with
full details presented in Sandnes et al. (2025). Here, we summarize
the primary features of REMIX and set up some additional test
scenarios tailored directly to a Jupiter core-mixing context.
In tSPH formulations used for applications in astrophysics, the
fluid density at the positions of particles is estimated by kernel
interpolation using an extended, Gaussian-like kernel function (Price
2012). The standard SPH density estimate will smooth the density
field on kernel length-scales. In regions where the density varies
smoothly, this will be a minor effect. However, in sharply varying
regions, and in particular at discontinuities in the underlying field,
the reconstructed density field will inevitably be smoothed.
The effect of this kernel smoothing can be clearly seen at interfaces
between different material layers in the pre-impact planets used for
our giant impact simulations. Pre-impact planetary equilibrium pro-
files and the corresponding SPH particle placements are calculated
using the publicly available WOMA2
and SEAGEN3
codes. Prior to
impact simulations, additional adiabatic ‘settling’ simulations are
performed to allow particles to rearrange themselves towards an
equilibrium configuration. In these simulations, particle entropies
are fixed to their initial value to enforce adiabatic evolution. Settling
simulations are carried out separately for each planet and are run
for a simulation time of 5000 s. REMIX reduces the errors that
traditionally make calculations of particle accelerations sensitive to
the local particle configuration. Therefore, the amount of particle
motion in settling simulations is reduced.
The radial density and pressure profiles of a proto-Jupiter planet,
to be used in our impact simulations, is shown in Fig. 1 from settling
simulations using tSPH and REMIX. With tSPH the density field
is smoothed by kernel interpolation, leading to diverging pressures
at the core–envelope boundary that act as an artificial barrier to
mixing across the interface. With REMIX, the density discontinuity
stays sharp and the pressure remains continuous across the material
boundary. To address kernel smoothing error, REMIX uses a differ-
ential form of the density estimate by which particle densities are
evolved with time rather than recalculated from the instantaneous
distribution of particle masses. All density discontinuities, including
those at material boundaries and free surfaces, are not erroneously
smoothed.
The smoothing error introduced by using an extended kernel
function combines with the error introduced by the discretisation
of the underlying fluid into a finite set of particles (Price 2012;
Spreng et al. 2020). To deal with discretisation error and mitigate
the accumulation of error over time in evolved densities and internal
energies, REMIX uses: linear-order reproducing kernels (Frontiere,
Raskin Owen 2017) that adapt to treat free surfaces as vacuum
boundaries; a choice of free functions in the SPH equations of motion
2The WOMA code (Ruiz-Bonilla et al. 2021) for producing spherical and
spinning planetary profiles and initial conditions is publicly available with
documentation and examples at github.com/srbonilla/WoMa, and the PYTHON
module WOMA can be installed directly with pip.
3SEAGEN (Kegerreis et al. 2019) is publicly available at github.com/jkeger/
seagen, or as part of WOMA.
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MNRAS 542, 947–959 (2025)
Figure 1. Radial profiles of density (a, b) and pressure (c, d) for a two-layer
proto-Jupiter settling simulation at time t = 5000 s. Columns show profiles
from simulations using a tSPH formulation (a, c) and REMIX SPH (b, d).
Individual particles are coloured by material type: blue for the central core of
ice and orange for the hydrogen–helium envelope.
that limits discretisation error (Read, Hayfield Agertz 2010);
a kernel normalizing term in the density evolution calculations;
advanced formulations of artificial viscosity as well as artificial
diffusion of internal energy and density between materials of the
same type.
All simulations presented here use particles of equal mass
across the simulation, a case that was specifically considered in
the validation of REMIX (Sandnes et al. 2025), and we employ
the Wendland C2
kernel with η = 1.487 to construct linear-order
reproducing kernels (Wendland 1995; Dehnen Aly 2012). Particles
have approximately 100 neighbours within their kernel. REMIX
has been developed with computational efficiency in mind and
therefore, as demonstrated by the simulations presented here, can
be used in simulations at state-of-the-art resolutions for giant impact
simulations.
2.2 Equations of state
The EoS characterize the thermodynamic behaviour of a material. In
the SPH simulations presented here, the EoS are used to calculate
pressures and sound speeds from densities and internal energies.
These quantities are then used both to directly evolve the simulated
fluid and in calculations of time-step durations.
Models of Jupiter’s internal structure are sensitive to uncertainties
in the hydrogen–helium EoS used to calculate the planet’s envelope
profiles (Miguel, Guillot Fayon 2016; Mazevet, Licari Soubiran
2022; Howard et al. 2023), with much work ongoing to create EoS
that accurately reproduce the behaviour of hydrogen–helium at the
extreme densities and pressures in the interiors of giant planets
(Saumon, Chabrier van Horn 1995; Militzer Hubbard 2013;
Chabrier et al. 2019). For simulations of giant impacts on to Jupiter
and hydrodynamic tests using Jupiter-like materials, we use the
Chabrier Debras (2021) hydrogen–helium EoS (hereafter CD21
H–He), with a helium mass fraction of Y = 0.245 (Chabrier et al.
2019). For simulations of impacts on to Jupiter aiming to reproduce
directly the initial conditions of L19, we use an ideal gas with
adiabatic index γ = 2.
For heavy elements, we use the AQUA EoS (Haldemann et al.
2020) to represent ice and the ANEOS forsterite EoS (Stewart et al.
2020) for rocky material. For direct L19 comparison simulations, we
use Tillotson ice and granite (Melosh 1989).
2.3 Impact initial conditions
For the majority of our simulations, we use a differentiated two-
layer proto-Jupiter with a heavy-element core of ice and a H–He
envelope, and a single-layer ice impactor. For simulations set up to
most closely match the initial conditions of the simulations of L19,
we use three-layer pre-impact planets with layers of rock, ice, and gas
for both target and impactor. The choice to focus on impacts between
planets with a reduced number of layers is made to further reduce
any potential barriers to mixing. In all of our simulations, we follow
L19’s scenario and use an impactor with a total mass of 10 M⊕ and
a proto-Jupiter of total mass 308 M⊕, with core mass of ∼10 M⊕,
where M⊕ = 5.972 × 1024
kg. The total mass of the system is
therefore the present-day mass of Jupiter, MJ = 1.898 × 1027
kg.
Some simulations with three-layer planets have a slightly more
massive core of 11.6 M⊕ to give equilibrium profiles that more
closely match those of L19, with different EoS, although we find
that changes to the initial profiles do not significantly affect the
evolution of the impact. H–He layers are chosen to be adiabatic
with surface temperatures (defined by where P = 1 bar) of 165 K
for proto-Jupiters and 500 K for impactors. For two-layer proto-
Jupiters and single-layer impactors, the ice layer is also chosen to be
adiabatic with an impactor surface temperature of 200 K. For three-
layer planets, the temperature–density relation of heavy-element
layers is chosen somewhat arbitrarily, to attempt to match the radii
of material interfaces of the simulations of L19: all are isothermal
except for the impactor ice layer which has T ∼
√
ρ.
All impact simulations are performed in 3D. They are set up
1 h prior to impact as detailed in Kegerreis et al. (2025, appendix
B.2), defined as the planets’ individual centres of mass reaching
the summed distance of their initial radii, such that the shapes of
the planets are allowed to realistically distort under tidal forces.
At Jupiter’s orbital distance from the Sun, we expect the peculiar
velocity of the impactor to be small compared with the mutual escape
speed, vesc = 54 km s−1
, and therefore we simulate most impacts with
an impact velocity of v = vesc. Note that L19 simulate impacts with
v = 46 km s−1
at the point of impact. Our impact parameter space
exploration includes speeds as low as v = 40.5 km s−1
to test the
potential implications of this choice.
We carry out a suite of simulations to systematically probe the
effect of impact speed (v = 0.75, 1.0, 1.5 vesc); impact angle (with
impact parameter b = 0.0, 0.2, 0.4, 0.6); and numerical resolution
(with particle number N = 105
–108
in logarithmic steps of 100.5
).
These simulations are based on our fiducial simulation that uses
planets with a reduced number of layers, is head-on, is at the mutual
escape velocity, and has resolution 107
. We also carry out simulations
to replicate the impact of L19 even more closely, with three-layer
bodies using the EoS used in their simulations, as well as with the
more sophisticated EoS detailed above.
2.4 Measures of material mixing
Parametrizing material mixing will enable us to quantitatively de-
scribe the degree to which core material may be diluted throughout
the impact. We measure the state of mixing in our impact simulations
using two parameters: the local heavy-element mass fraction, Z̄, and
the total mass of mixed material across the simulation, Mmix. These
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quantities describe the local and global state of material mixing
respectively. In our simulations, mixing is treated at the particle
scale and not below. Therefore, the material of each particle remains
fixed for the duration of the simulation. To estimate mixing, we
therefore calculate these quantities as weighted estimates based on
the localized distributions of particle material types.
We use kernel interpolation to estimate the local heavy-element
mass fraction. The quantity Z̄ is calculated based on weighted
contributions from nearby SPH particles. This parameter describes
the fraction of local mass that is represented by heavy-element SPH
particles, such that Z̄ = 0 in regions where no local particles are
heavy elements and Z̄ = 1 where they all are. We estimate Z̄ at the
positions of particles with the standard SPH approach:
Z̄i ≡

j ζj mj Wij Vj

j mj Wij Vj
. (1)
Here subscripts denote quantities either sampled at the position
of, or associated with, a particle i or its neighbouring particles j.
Sums are approximations of integrals over discrete volume elements
Vj = mj /ρj , where mj and ρj are particle masses and densities.
The kernel function Wij ≡ W(rij , hi) contributes weighting based
on the particle separation rij ≡ ri − rj and is characterized by the
smoothing length hi. The parameter ζj takes the value 1 if particle
j’s material represents heavy elements, and is 0 otherwise. We use
the spherically symmetric Wendland C2
kernel function for these
calculations (Wendland 1995). We use this rather than the linear-
order reproducing kernels used in REMIX, since it provides a more
simple and method-independent measure of the mixing.
To estimate the total mass of mixed material in our simulations
we first define what constitutes a mixed state. Since each particle
retains its material for the duration of the simulation, we determine
that a particle with neighbours of different material-types only – with
no neighbours of its own type – is in a maximally mixed state. For
a particle i we estimate the local mass fraction of particle i’s own
material, similarly to Z̄, by
w̄i ≡

j κij mj Wij Vj

j mj Wij Vj
, (2)
where κij = 1 for particle pairs of the same material and κij = 0
otherwise. We note that, unlike Z̄, the value of w̄ will never reach 0
because of the contribution of i itself in this calculation. We estimate
that the contribution to mi from materials different from that of i to
be mmix, i ≡ (1 − w̄i) mi. The total mixed mass in the simulation is
then given by
Mmix ≡

i
mmix, i =

i
(1 − w̄i) mi , (3)
where we sum over all simulation particles.
We note that both Z̄ and Mmix will be spatially smoothed on
the scale of the smoothing length, since they are calculated by
interpolation using an extended kernel. Therefore material near sharp
material interfaces will be measured as mixed even if particles of
different materials have not crossed the interface.
3 FLUID INSTABILITIES AND MIXING
Before running the primary impact simulations, we first test
REMIX in simulations of Kelvin–Helmholtz instabilities (KHI) and
Rayleigh–Taylor instabilities (RTI) with materials and conditions
representative of material interfaces in giant impacts on to Jupiter.
Although no converged reference solutions exist for these scenarios,
demonstrating that REMIX alleviates the purely numerical known
issues of tSPH at the material interface will verify that the material-
independent improvements of REMIX are effective in this regime,
where core-material and metallic hydrogen have been predicted to
be miscible (Wilson Militzer 2011, 2012). We carry out fluid
instability simulations in 3D and with particles of equal mass across
the simulation to validate our hydrodynamic treatment for our impact
simulations, as done in Sandnes et al. (2025) for similar but not
Jupiter-specific tests.
3.1 Kelvin–Helmholtz instability
The KHI arises as perturbations at shearing fluid interfaces grow
to form spiralling vortices (Chandrasekhar 1961). We examine
the growth of the KHI between layers of ice and H–He at con-
ditions representative of Jupiter’s deep interior. In our simula-
tions these materials are treated as inviscid fluids and so, since
the growth of the instability is predominantly inertial, we ex-
pect a qualitatively similar evolution to analogous, well-studied
ideal gas simulations (Price 2008; Robertson et al. 2010; Mc-
Nally, Lyra Passy 2012; Frontiere et al. 2017; Rosswog 2020).
We characterize the growth of a mode of wavelength λ by the
time-scale
τKH =
(ρ1 + ρ2) λ
√
ρ1ρ2 |v1 − v2|
, (4)
where ρ1 and ρ2 are the densities in regions separated by the shearing
interface and |v1 − v2| is their relative speed (Price 2008).
Initial conditions are constructed similarly to those of Sandnes
et al. (2025). H–He particles are initialized in a 3D cubic lattice
in a periodic domain with 128 × 128 × 18 particles in the x, y, z
directions. The size of the simulation domain in x and y is 1 RJ,
where the radius of Jupiter is RJ = 69.9 × 103
km, and particle
masses are chosen to give a density of ρ1 = 3.5 g cm−3
. A region
occupying the central half of the domain in y and spanning the
full domain in x and z is replaced by a region of higher density
ice at ρ2 = 8.43 g cm−3
. These densities are chosen to correspond
to the densities at the core–envelope interface in the pre-impact
proto-Jupiter, as plotted in Fig. 1. Since we use particles of equal
mass across the simulation, the cubic lattice of ice particles is
initialized with a smaller grid-spacing. The particle configurations
in both regions are constructed to maintain their grid-spacing across
boundaries of the periodic domain and for the two regions to be
separated by the mean of the two grid-spacings at both interfaces. The
two regions are initialized with relative speeds of v1 = −10−4
RJ s−1
and v2 = 10−4
RJ s−1
. A mode of wavelength λ = 0.5 RJ and of form
vy = 0.01|v1 − v2| sin (2πx/λ) seeds the instability. Initial internal
energies are set such that the regions are in pressure equilibrium
with P (ρ, u) = 3.2 × 1012
Pa. We note that the spurious smooth-
ing of the density discontinuity in tSPH means that, unlike with
REMIX, simulations with tSPH are not truly initialized in pressure
equilibrium.
The evolution of the KHI with these initial conditions, from
simulations using tSPH and REMIX, is shown in Fig. 2. The growth
of the instability is clearly and strongly suppressed with tSPH:
the characteristic spirals of the KHI do not form and particles
are prevented from crossing the density discontinuity by spurious
surface tension-like effects. REMIX directly addresses the sources
of error that lead to these effects and so allows the instability
to grow, and particles of different materials are able to intermix.
The instability grows over a similar time-scale, scaled by τKH,
to the analogous KHI simulations with an ideal gas, and also
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Figure 2. KHI growth with materials and conditions representative of a pre-impact Jupiter’s core–envelope interface. Snapshots show two times from
simulations using a tSPH formulation and REMIX. Individual particles are plotted on a grey background and coloured by their material type and
density. Particles at all z are plotted, so the grey background is visible in regions that have maintained their grid alignment in z from the initial
conditions.
Figure 3. RTI for Jupiter-like materials and conditions, plotted as in Fig. 2. Snapshots show three times for simulations using a tSPH formulation and REMIX.
The regions of fixed boundary particles at the top and bottom of the simulations have been cropped from the figure; their positions and densities do not
change.
those between Earth-like materials presented in Sandnes et al.
(2025).
3.2 Rayleigh–Taylor instability
The RTI occurs due to the displacement of a high-density fluid
by a low-density fluid (Chandrasekhar 1961). We consider a
gravity-driven case in which a region of dense ice sits above
a region of H–He, initially in approximate hydrostatic equilib-
rium other than a small velocity seed perturbation. As in the
KHI, spurious surface tension-like effects at the density discon-
tinuity strongly suppress the growth of this instability in tSPH
simulations.
Initial conditions are constructed similarly to those of Sandnes
et al. (2025). Particles are placed in a periodic simulation domain
in two cubic lattices. The domain has dimensions of 0.5 RJ, 1 RJ
in the x and y directions, with a thin 3.5 × 10−3
RJ domain size in
the z dimension. The low density H–He region has 256 × 256 × 18
particles with density ρ1 = 3.5 g cm−3
and occupies the bottom half
of the domain. The upper ice region is constructed to satisfy similar
grid-spacing constraints as in the KHI simulation, with density
ρ2 = 8.43 g cm−3
. Particles in the top and bottom 0.05 RJ of the
domain are fixed in place throughout the course of the simulation.
Initial internal energies are set to satisfy hydrostatic equilibrium for a
constant gravitational acceleration g = −31.4 m s−2
, and an interface
pressure of P0 = 3.2 × 1012
Pa, representative of the gravitational
acceleration and pressure at the core–envelope boundary in the proto-
Jupiter used for our fiducial giant impact simulations. Particles are
initially at rest, other than an initial velocity perturbation that seeds
the instability,
vy(x, y) = δy (1 + cos [8π (x + 0.25)]) (1 + cos [5π (y − 0.5)])
(5)
in the region 0.3 RJ y 0.7 RJ and vy = 0 otherwise. We use a
perturbation amplitude of δy = 0.025 RJ s−1
.
The evolution of the RTI with these initial conditions is shown
in Fig. 3, for simulations using tSPH and REMIX. In the tSPH
simulation, the RTI plumes grow slowly and material is prevented
from crossing the interfaces. This is in contrast with the REMIX
RTI, where we observe the unimpeded growth of both the primary
and secondary instabilities. This leads to mixing across a range of
length-scales as particles are not artificially prevented from crossing
the interface and instabilities grow to drive turbulent mixing.
The results of these KHI and RTI simulations demonstrate that
REMIX does not suppress mixing and fluid instability growth in
conditions representative of the giant impact simulations presented
in the following section.
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Figure 4. Snapshots from the fiducial, head-on impact on to Jupiter carried out using REMIX SPH. Individual SPH particles are plotted in cutaways from 3D
simulations and are coloured by material-type and density. An animation of this impact is available at www.icc.dur.ac.uk/giant impacts/jupiter remix 1e7.mp4
and a 3D-rendered animation of the equivalent 108-particle resolution impact is available at https://youtu.be/xkpZSNlrWTg.
4 GIANT IMPACTS
Here, we investigate dilute-core formation in simulations of: head-on
impacts (Section 4.1); isolated planets with a pre-constructed dilute
core to test whether our simulation methods would in principle be
able to produce a dilute core that is stable for the duration of our
impact simulations (Section 4.2); impacts at a range of speeds and
angles (Section 4.3); and impact simulations with initial conditions
set up to closely replicate those of L19, including an alternative
version with more sophisticated EoS (Section 4.4).
4.1 Fiducial scenario
As a basis for investigations of impact configuration and numerical
resolution in later sections, we consider a fiducial scenario of
the head-on impact between a 10 M⊕ impactor and a 308 M⊕
proto-Jupiter with a 10 M⊕ core, at the mutual escape speed
of the two bodies. We choose to focus primarily on impacts
with a two-layer proto-Jupiter with only layers of ice and H–
He and a single-layer, ice impactor. We do this to deliberately
reduce both the number of density discontinuities and the size
of the core–envelope density contrasts in the initial conditions, to
remove barriers to forming a dilute core in our simulations. We
use the more advanced CD21 H–He and AQUA EoS for these
simulations.
We simulate the fiducial scenario using both REMIX and tSPH. In
the REMIX simulation, ice particles can mix freely into the envelope,
as seen in Fig. 4. The core reaches a temporarily somewhat-mixed
state, however, heavy elements rapidly settle under gravity to re-form
a differentiated core over short time-scales of ∼10 h. The snapshots
at 43 h show a later time where post-impact bulk-material oscillations
have dissipated. No dilute core is produced, even with the improved
treatment of mixing in the REMIX scheme. In the tSPH simulation,
spurious surface tension-like effects are strong, suppressing mixing
of ice and H–He particles. Heavy elements remain in a largely
cohesive mass throughout the simulation, which settles to form a
core with a discrete interface between the two different materials. A
figure showing snapshots from the tSPH simulation is presented in
Appendix A.
We additionally perform REMIX simulations of this impact with
resolutions N = 105
–108
SPH particles in logarithmic steps of 100.5
.
Although higher resolution allows turbulence to be resolved at lower
length-scales, therefore extending the time materials take to separate,
all of these simulations produce an undiluted core over the short time-
scales simulated, as shown in Appendix A.
Although the constituent equations of both REMIX and tSPH
conserve energy, the kick-drift-kick algorithm with individual parti-
cle time-step sizes introduces slight variations in the system’s total
energy during the simulation. For tSPH and REMIX, these variations
remain within 0.056 per cent and 0.073 per cent of the initial total
energy, respectively.
The evolution of the local heavy-element fraction (equation 1) of
the REMIX impact is shown in Fig. 5, for a thin cylinder aligned along
the direction of the impact. At early times, Z̄ reaches intermediate
values as material mixes due to the erosion of the impactor as it
travels through the envelope, as seen in Fig. 5(a), and due to the
disruption of the core by the impact, the immediate aftermath of
which is shown in Fig. 5(b). At 10 h, the core, not positioned at
the centre of mass of the planet due to the oscillations, consists
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Figure 5. Profiles of localized heavy-element mass fraction, Z̄, sampled in a thin, 0.05 RJ radius cylinder along the axis of head-on impact, x, from the fiducial
REMIX simulation. Panels correspond to times: (a) immediately prior to core-disruption by the impactor; (b) when heavy elements are mixed with envelope
material; (c) when the core has settled to form a discrete boundary, although still mixed with some envelope material; (d) when core and envelope material
have largely separated. Black, solid lines show the median particle value in 100 bins along the plotted region. Grey shading spans percentiles such that the
enclosed regions correspond to 68 per cent of particles in each bin. The upper blue and lower orange dashed lines show heavy-element fractions of pure ice
and hydrogen–helium, respectively. The x-axis is centred at the centre of mass of the system; deviations of the core position from x = 0 are due to post-impact
oscillations.
Figure 6. Evolution of the total mass of mixed material, Mmix, in simulations
of head-on impacts on to Jupiter. The black, solid line shows results from a
REMIX simulation and the pink, dashed line from a tSPH simulation. The
total mass of heavy elements is 20 M⊕ in these simulations.
largely of heavy elements, and the core–envelope interface is already
sharp. By 12 h the core is close to consisting purely of heavy
elements.
The evolution of the total mixed mass (equation 3) in these
simulations is shown in Fig. 6. There is considerably more mixing
with REMIX than with tSPH. In the REMIX simulation, mixing
peaks at a time 2.5 h after impact, at which time the core- and
impactor-material particles have been maximally disrupted and
mixed with the H–He envelope. After this time, the mass of mixed
material falls as the system settles under gravity and materials
separate, with material being largely separated by ∼10 h. We carry
out simulations until later times to allow the large, dynamical
oscillations to dissipate, although for the majority of this time the
boundary of the core is already sharp and oscillations only affect its
shape.
4.2 Stability of a pre-constructed dilute core
We now address the possibility that material separation in the
REMIX impact arises from numerical errors that would prevent
these hydrodynamic methods from sustaining a non-transient dilute
core over these time-scales no matter the scenario. To test this, we
construct a planet with initial profiles of heavy-element fraction
and density that match the post-impact planet produced in the
simulation of L19 that produced a dilute core (from their fig. 2
a and the initial frame of their supplementary information video
5). These simulations are performed to assess whether a dilute-
core structure can persist for the runtime of our impact simula-
tions, rather than to infer the stability of the specific dilute core
produced by the giant impact of L19. Therefore, some evolution
in the planet’s radial profiles is not a concern, and is expected
due to differences between the simulations, such as the number of
materials and the treatment of mixed materials in the simulation
methods.
The initial planet is constructed by placing ∼107
particles in a
configuration that corresponds to the desired density profile. Each
particle’s material is set probabilistically based on the heavy-element
fraction profile. For instance, at a radius where the heavy-element
fraction is Z = 0.3, a particle will have a 30 per cent chance of
being assigned the AQUA ice EoS and a 70 per cent chance of the
CD21 H–He EoS. Specific internal energies are then chosen such
that their pressures satisfy hydrostatic equilibrium, with a pressure
of 107
Pa at the vacuum boundary, as this was found to give a
relatively stable vacuum interface. Note that here, in keeping with
the giant impact simulations of the previous section, we represent
heavy elements only by ice, rather than by ice and rock, as done in
the simulations of L19. Therefore, the materials and thermodynamics
of our initial planet are not directly equivalent to L19’s post-impact
planet; we are primarily focused on the comparative stability of a
dilute-core structure rather than a specific planetary profile. Although
these simulations are of a planet in isolation here, unlike the settling
simulations we perform prior to impacts, we do not fix particle
entropies. This approach more directly addresses whether a non-
transient dilute core could, in principle, be produced in our impact
simulations.
The radial profiles of density and heavy-element fraction at five
points in time in these simulations, carried out using REMIX and
tSPH, are shown in Fig. 7. The times plotted are chosen to show
the stability of a dilute core for the time-scale of the duration of
our planetary impact simulations, as for an impact, the core rapidly
settles to a differentiated state already by ∼10 h. With REMIX the
core remains dilute with a smooth interface. Although the profiles
themselves evolve in time in both cases, the profiles evolve less
substantially between t = 20–40 h than in the first 20 h. Further
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Figure 7. Evolution of radial profiles of heavy-element fraction (a, b) and
density (c, d) in simulations of a Jupiter-like planet with a pre-constructed
dilute core. Simulations are carried out with REMIX (a, c) and tSPH (b, d)
and profiles are plotted at five times to show the stability of a dilute core over
the time-scales of our giant impact simulations. The plotted profiles show the
mean values of the quantities within 50 radial shells.
advancements in simulation methodology – particularly in capturing
mixing processes below the particle scale – might improve the
stability of the compositional gradients. At t = 40 h, the planet has
a central heavy-element fraction of Z = 0.83 and the dilute-core
structure extends out to 0.4–0.5 RJ. With tSPH, materials separate
within the first 10 h to form an undiluted core, as in the impact
simulations using either method.
4.3 Impact speed and angle
Although no stable dilute core is produced in the head-on scenario
of Section 4.1, this result could perhaps be sensitive to the speed
and angle of the impact. The core of the planet is more likely
to be disrupted in head-on or low-angle impacts, but impactor
material might mix into the envelope more effectively by erosion in
higher-angle impacts. Additionally, one might speculate that higher
impact speeds may act to increase the initial material mixing. Or,
conversely, perhaps lower speeds could lead to post-impact heavy-
element distributions and internal energy profiles that are more stable
to convection that may otherwise facilitate demixing of materials.
Therefore, investigating a wide range of impact speeds and angles
will enable us to examine the sensitivity of dilute core production to
the impact configuration.
For this parameter study, we use the same initial planetary
bodies as in the fiducial scenario, although we run simulations with
all additional combinations of four impact parameters and three
impact speeds, listed in Section 2.3. The choices of v = 0.75 vesc
and v = 1.5 vesc represent extreme scenarios to probe the sensi-
tivity of our results to large changes in the impact kinematics.
All of these simulations were performed using REMIX with 107
particles.
Snapshots from impact simulations with different speeds and
angles are shown in Fig. 8. Although these examples constitute a
small selection of the impacts simulated, they specifically correspond
to speeds and angles in which the core is significantly disrupted.
During the impacts, heavy elements mix into the envelope through
Figure 8. Snapshots from REMIX simulations of giant impacts on to Jupiter
for different impact parameters, b, and speeds, v. Impact velocities are scaled
to the mutual escape speed of the two bodies. Times correspond to core-
disruption and long after impact, when heavy elements have settled to form
an undiluted core. Particles are coloured by their material-type and density.
both the erosion of the impactor and the disruption of the core, in
particular for low impact angles. In head-on impacts energy is more
effectively transferred to the core and so the post-impact core-density
in these impacts is lower than for off-axis impacts. However, for all
impact configurations, heavy elements settle over short time-scales
to form an undiluted core with a discrete boundary to the H–He
envelope.
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4.4 Direct Liu et al. (2019) comparison
Here we simulate impacts that are set up to directly follow the
impact of L19, which they found to produce a well-mixed dilute
core. By closely matching their initial conditions and the EoS
used, any remaining differences in simulation outcomes should be
primarily due to the numerical methods used in the simulations. We
additionally simulate an equivalent impact using more advanced EoS
for both core and envelope materials. This will allow us to investigate
the sensitivity of our results to the EoS used.
In the simulations of L19, planetary profiles are initially con-
structed using the SESAME EoS (Lyon 1978) and then swapped for
Tillotson and ideal gas EoS for the impact simulations, replacing
the initial particle internal energies to recover the SESAME pressure
and density profiles (S.-F. Liu personal communication, 2020). For
our subset of comparison simulations set up to match those of L19,
we therefore carry out a similar process by swapping the EoS from
profiles calculated using ANEOS forsterite, AQUA, and CD21 H–He
to Tillotson EoS and ideal gas with γ = 2. We verify that our proto-
Jupiter profile closely matches the one used in L19’s simulations, so
the difference between using these EoS or SESAME to construct the
pre-swapped profiles is minor. We find that planets with a swapped-
in ideal gas envelope are not stable for the duration of settling
simulations. Therefore, impact simulations that use these EoS are
run without prior settling simulations. As in previous simulations, the
proto-Jupiter has a total mass of 308 M⊕ and the impactor has a mass
of 10 M⊕.
Snapshots from impacts of both the direct L19 comparison
and the equivalent simulation with improved, more sophisticated
EoS are shown in Fig. 9. Although there are small differences
in dynamics during the course of these impacts, they each fol-
low a similar evolution, both to each other and to all previ-
ous impacts simulated here. The core is disrupted and material
temporarily mixes into the envelope, but heavy elements set-
tle to form a discrete core–envelope interface over short time-
scales. Some rock and ice core materials remain mixed with each
other at later times, however they are not diluted by envelope
material.
5 DISCUSSION
None of the giant impact simulations presented here produced a dilute
core. The heavy-element mass fraction profiles of the post-impact
planets from our simulations are plotted in Fig. 10. No dilute core was
produced in our simulations (1) spanning a range of impact speeds
and angles; (2) at different resolutions; and (3) between planets
with different compositions. We carried out simulations both under
conditions set up to directly mirror those of L19 and in conditions
set up to facilitate mixing and remove potential barriers to it, in an
attempt to offer the best chance of dilute core production. The red line
in Fig. 10(c) shows the profile of post-impact heavy-element fraction
from the simulation of L19 that produced a well-mixed dilute core
with a central heavy-element fraction of Z 0.5. The categorical
difference in our results and the simulation of L19 are likely due to
differences in the simulation methodologies used.
Our primary suite of simulations were performed using REMIX
SPH, an advanced SPH formulation that was developed specifically
to improve the treatment of material mixing in SPH simulations.
As a Lagrangian method where interpolation points move with the
fluid velocity, we do not encounter the issues in regions of large bulk
motion that are known to affect grid-based codes (e.g. figs 33 and
36 of Springel 2010). In this aspect, the methods used by L19 face
potential limitations in the application of studying mixing in regions
where there is large advection through the stationary grid points. The
well-established overmixing in grid-based codes in regions of large
bulk motion through the grid may be an explanation for their results:
in their simulations, the core rapidly mixes into the envelope as it is
accelerated by the impactor.
Although we demonstrate significant improvements in the treat-
ment of mixing in our simulations compared with simulations
carried out with a tSPH scheme, we have not considered mixing
of material below the length-scale of SPH particles (Greif et al.
2009) or the chemical reactions that might affect the evolution of
materials as they mix. The extension of the simulation methods
to include these potentially important mechanisms may also help
address the resolution dependence of mixing observed in these highly
turbulent scenarios (Appendix A). Further progress in quantifying
the discrepancy between simulated scenarios and their physical
analogues – which remains challenging due to the chaotic nature
of impact dynamics and the lack of observational constraints in
this regime – would help to better assess the sensitivity of large-
scale simulation outcomes to both physical processes and numerical
uncertainties. Future work should focus on developing improved
numerical methods to model mixing and demixing processes oc-
curring below the resolution scale of SPH particles. Different
simulation approaches should be compared in isolated test scenarios,
like those we present in Section 3, to further analyse the demix-
ing processes observed during the simulated impacts, providing
a physically motivated framework for quantifying the differences
in the material separation mechanisms with different numerical
methods.
However, as shown in Section 4.2, a dilute core structure can be
sustained in our REMIX simulations with limited material separation
over the time-scale of the impact simulations. This suggests that our
impact simulations – despite lacking sub-particle scale mixing and
chemistry – could in principle produce a dilute core if a more stable
configuration were to be reached. Therefore, the absence of a dilute
core in our impact simulations appears due to the giant impacts’
inability to disrupt the core to a more stable diluted state.
In addition to the hydrodynamic methods, there are differences
in the approaches taken in the calculation of self-gravity. In our
simulations, we employ the fast multipole method (Greengard
Rokhlin 1987; Cheng, Greengard Rokhlin 1999), which partitions
the simulation domain into a hierarchical tree of spatial cells.
Gravitational interactions between nearby particles are calculated
directly, while interactions over larger distances are approximated by
multipole expansions of particle groups within cells (Schaller et al.
2024). In contrast, L19 estimate the full gravitational potential using
a single multipole expansion centred on the system’s centre of mass
(Liu et al. 2015). Choosing the centre of mass as the expansion centre,
rather than alternatives like the location of peak density (Sellwood
1987) or a ‘square-density-weighted mean location’ (Couch et al.
2013), can introduce errors in gravitational force estimates, particu-
larly when the centre of mass deviates significantly from the peak-
density location. During the giant impact simulations by L19, the
disruption of the core of heavy elements coincides with a significant
shift of this high-density region from the centre of mass, which is
primarily set by the much larger mass of the envelope. Therefore it is
not clear whether error in the calculation of gravity may also play a
role in the rapid mixing of material during core disruption, observed
in their simulation.
Separately from this discussion of the numerical methods, it should
be noted again how extreme and specific the dilute core-producing
impact simulation of L19 is: the impact conditions require the head-
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Figure 9. Snapshots from REMIX simulations of head-on impacts on to Jupiter set up to most closely follow the initial conditions of the impact of L19 that
produced a dilute core (top) and the equivalent scenario with more advanced EoS (bottom). Particles are coloured by their material-type and density.
Figure 10. Radial profiles of heavy-element mass fraction at 43 h after impact, from simulations (a) at different impact speeds and angles; (b) with different
numerical resolution; (c) set up to closely follow the initial conditions of L19, including the equivalent profile from their simulation that produced a dilute core.
Heavy-element fraction is measured by the ratio of heavy-element SPH particle mass to total mass in 300 radial shells.
on impact of a 10 M⊕ on to an almost fully formed Jupiter that
has accreted almost all of its final envelope mass yet only half its
heavy elements. This, combined with the inference of a dilute core
in Saturn (Mankovich Fuller 2021) in addition to Jupiter, might
suggest that it is more likely that dilute cores are produced as part
of the extended processes that underlay the formation and evolution
of giant planets, rather than through low-likelihood stochastic events
(Helled Stevenson 2024).
Recent models of giant planet formation indicate that composition
gradients naturally arise during the formation process (Helled
Stevenson 2017; Lozovsky et al. 2017; Stevenson et al. 2022) and that
an extended period of planetesimal accretion could deliver sufficient
energy to delay runaway gas accretion (Venturini Helled 2020).
Since the material delivered by runaway gas accretion is less rich in
heavy elements, offsetting this phase could allow the compositional
gradients that constitute the dilute core to extend further from the
planet’s centre. Alternatively, under certain conditions, thermal con-
vection after giant planets have formed could lead to the convective
mixing of core and envelope material (Moll et al. 2017). These
formation pathways are perhaps more promising than a single, low-
probability giant impact, which our results suggest could be unable
to produce a dilute core even under the extreme impact conditions
considered here.
6 CONCLUSIONS
We have presented results from REMIX SPH simulations of giant
impacts on to Jupiter to investigate the feasibility of this as the
process by which the planet’s dilute core was formed. We varied
impact speed, angle, numerical resolution, the number of layers in the
pre-impact planets, and the EoS used to represent proto-Jupiter and
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impactor materials. The impact dynamics in all simulations followed
the same trend: initial disruption and partial mixing, followed by
settling and re-formation of an undiluted heavy-element core on
∼10 h time-scales. However, our simulations do not account for
mixing below the resolution scale of SPH particles or the effects
of chemical reactions, which could, in principle, influence the large-
scale outcomes of the simulations. The first of these may also be
required for numerical convergence of these highly turbulent impact
scenarios.
Our results contrast with the simulation of Liu et al. (2019) that
produced a highly dilute core with a central heavy-element fraction
of Z 0.5 and a smooth transition to the envelope. Their result
is potentially an artefact of numerical issues, such as the well-
established overmixing in grid-based codes in regions of large bulk
motion through the grid.
The REMIX SPH scheme was specifically designed to improve
the treatment of mixing and instability growth. Despite our approach
offering favourable conditions and spanning a wide parameter space,
dilute cores were not produced in any of our simulations. This result,
reinforced by observations that suggest that dilute cores are not
unique to Jupiter, offers no support for the hypothesis that a single,
extreme giant impact is the origin of dilute cores in giant planets.
ACKNOWLEDGEMENTS
TDS acknowledges support from STFC grants ST/T506047/1 and
ST/V506643/1. VRE and RJM are supported by Science and
Technology Facilities Council (STFC) grant ST/X001075/1. JAK
is supported by a UKRI/STFC Ernest Rutherford Fellowship and a
NASA Postdoctoral Program Fellowship administered by Oak Ridge
Associated Universities. The research in this paper made use of
the SWIFT open-source simulation code (Schaller et al. 2024). This
work used the DiRAC@Durham facility managed by the Institute
for Computational Cosmology on behalf of the STFC DiRAC HPC
Facility (www.dirac.ac.uk). The equipment was funded by BEIS cap-
ital funding via STFC capital grants ST/K00042X/1, ST/P002293/1,
ST/R002371/1, and ST/S002502/1, Durham University and STFC
operations grant ST/R000832/1. DiRAC is part of the National e-
Infrastructure.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable request
to the corresponding author.
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APPENDIX A: NUMERICAL ASPECTS OF
IMPACT SIMULATIONS
Here, we consider some effects of the numerical aspects of the
fiducial giant impact simulation presented in Section 4.1. We briefly
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Figure A1. Snapshots from REMIX simulations of head-on impacts on to Jupiter, carried out with different numerical resolutions between 105 and 108 SPH
particles. The five times plotted illustrate material mixing at different points during core-disruption and after the heavy elements have settled to form an undiluted
core. The plotted region is centred on the centre of mass of the heavy elements. Particles are coloured by their material-type and density.
investigate how the numerical resolution influences the outcome of
the simulations. Additionally, we present results from a simulation
that was performed with tSPH to allow for a direct comparison
between the treatment of mixing in simulations performed with
different SPH formulations.
Numerical resolution determines not only the minimum length-
scale probed in simulations, but can also significantly influence
the accuracy of simulated fluid behaviour at all length-scales. In
particular, even large-scale outcomes of SPH simulations of giant
impacts can remain unconverged at standard resolutions of 105
–106
particles (Genda et al. 2015; Hosono et al. 2017; Kegerreis et al.
2019, 2022). Not only does the computational efficiency of the SWIFT
code allow us to simulate giant impacts at far higher resolutions, but
the REMIX scheme has been demonstrated to improve numerical
accuracy such that convergence can be achieved at lower resolutions
than in equivalent tSPH simulations (Sandnes et al. 2025).
Here, we test whether we achieve numerical convergence in dilute
core production – or lack thereof – in our head-on fiducial impact.
We carry out simulations at resolutions of 10n
with n = 5–8, in steps
of 0.5, SPH particles. All these simulations were performed using
REMIX. The mixing of heavy elements into the H–He envelope in
simulations of different resolutions is shown in Fig. A1. We present
five snapshots in time: two illustrating the initial disruption of the
core by the impactor; a third capturing a moment of significant
material mixing; a fourth at a time where heavy elements have
largely re-settled, although more mixing is still present in the higher
resolution simulations; and a fifth depicting the later stage when
post-impact oscillations have dissipated and a distinct, undiluted
core has been produced for all resolutions. The mixing of particles
of different materials is observed for all resolutions. Increasing
resolution allows the simulation to capture mixing and instability
growth at smaller length-scales, with KHI growing at the shearing
interface as predicted in Section 3.1. Because of these turbulent
effects, as resolution is increased, large-scale features become less
symmetric about the impact axis and are more significantly disrupted
by chaotic fluid behaviour at smaller scales. Since the turbulent
mixing is captured at shorter length-scales with increased resolution,
we find that the heavy elements take a slightly longer time to
settle in the higher resolution simulations and we therefore do
not achieve numerical convergence in the time-scale of demixing.
Nevertheless, in all these simulations an undiluted and settled
core is produced well within the short time-scales of these impact
simulations.
In these simulations, we do not model mixing below the reso-
lution scale of individual SPH particles, as each particle retains a
fixed material type throughout the simulation. Achieving numerical
convergence of the mixing and demixing processes in these particular
highly turbulent impact scenarios might require modelling the evolu-
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MNRAS 542, 947–959 (2025)
Figure A2. Snapshots from the fiducial, head-on impact on to Jupiter carried out using a tSPH formulation, shown as in Fig. 4. Individual SPH particles are
plotted in cutaways from 3D simulations and are coloured by material-type and density. An animation of this impact is available at www.icc.dur.ac.uk/giant i
mpacts/jupiter tsph 1e7.mp4.
tion of particle composition with time, and subsequently accounting
for the resulting changes to the particles’ EoS.
We additionally perform a simulation of the fiducial scenario
presented in Section 4.1 using the more tSPH formulation that was
also used in the comparison fluid instability tests of Section 3,
with the same resolution of 107
particles. Snapshots from this
impact are shown in Fig. A2. The panels of this figure are at the
equivalent times to those in Fig. 4. The spurious surface tension-
like effects at material boundaries here act to prevent mixing of
the different materials material. Heavy elements remain largely
unmixed throughout the course of the simulation and no dilute core is
produced.
This paper has been typeset from a TEX/L
ATEX file prepared by the author.
© The Author(s) 2025.
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License
(https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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No dilute core produced in simulations of giant impacts on to Jupiter

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