JuanCastillo29/space-mission-intelligence-data huggingface dataset:
though they did not take the structural evolution of the Sun into account. Observationally, there is evidence of Earth-like planets around white dwarfs (e.g. Zhang et al. 2024), which favours the survival of planets like Earth during the RGB and AGB phases. However, the exact fate of the Earth and the inner Solar System is still uncertain and requires further investigation.
When stars deplete hydrogen in their core, they evolve off the main sequence (MS) and expand during the red giant branch (RGB) and asymptotic giant branch (AGB) phases (e.g. Kippenhahn & Weigert 1990). During this evolution, the star grows in size and loses mass through a stellar wind. For the Sun, this expansion will result in a star that is larger than the current orbit of Earth (1.2–1.5 M⊙depending on the AGB massloss rate). This will have a significant impact on the orbital evolution of the Solar System planets. The fate of the inner Solar System planets, and in particular Earth, has been a topic of debate in the literature, with different studies reaching different conclusions. Some studies suggested that Earth will be engulfed by the expanding Sun during the RGB (e.g. Rybicki & Denis 2001; Schröder & Smith 2008; Lanza et al. 2023), while others suggested that Earth will survive the RGB phase (e.g. Sackmann et al. 1993; Rasio et al. 1996; Nordhaus et al. 2010; Guo et al. 2016). These studies often made different assumptions about the mass-loss rates and tidal interactions, which led to different conclusions. Additionally, some studies considered the impact of planet-planet interactions on the orbital evolution of the inner Solar System (e.g. Mogavero & Laskar 2021), al-
The effect of mass loss on the orbital evolution of the Solar System planets was first studied by Sackmann et al. (1993), who assumed conservation of angular momentum when the star loses mass (effectively widening the orbit). Rybicki & Denis (2001) analysed effects in addition to the mass loss, such as tidal interactions, mass accretion by the wind, evaporation by stellar irradiation, and wind drag. The authors found that mass loss and tidal interactions dominate the orbital evolution. Schröder & Smith (2008) and Nordhaus et al. (2010) combined the effects of mass loss and tidal interactions and arrived at different conclusions (engulfment for Schröder & Smith 2008 and survival for Nordhaus et al. 2010). The difference in the conclusions of these studies can be attributed to the different assumptions made about the mass-loss rates during the RGB, where Nordhaus et al. (2010) assumed a higher RGB mass-loss rate than Schröder & Smith (2008). This dependence was further analysed by Guo et al. (2016), who performed a detailed analysis of the effect of different mass-loss rates during the
⋆Corresponding author: mats.esseldeurs@kuleuven.be
Article number, page 1
der & Smith 2008 and survival for Nordhaus et al. 2010). The difference in the conclusions of these studies can be attributed to the different assumptions made about the mass-loss rates during the RGB, where Nordhaus et al. (2010) assumed a higher RGB mass-loss rate than Schröder & Smith (2008). This dependence was further analysed by Guo et al. (2016), who performed a detailed analysis of the effect of different mass-loss rates during the
⋆Corresponding author: mats.esseldeurs@kuleuven.be
Article number, page 1

RGB on the orbital evolution. For low mass-loss rates during the RGB, the star not only spends more time in the RGB phase, but also more crucially, the star grows more significantly. This extends the time and enables stronger tidal interactions to shrink the orbit of the Earth, leading to engulfment, and vice versa. Based on the observationally calibrated RGB mass-loss rates of McDonald & Zijlstra (2015), Guo et al. (2016) concluded that Earth will survive the RGB phase. Lanza et al. (2023) added the effect of residual eccentricity, which is a stochastic process in which density fluctuations around the planet lead to eccentricity fluctuations of about 10−5 during the MS that are enhanced to the order of 10−2 during the RGB, making the survival of the Earth a stochastic process. These previous studies either neglected the effect of tidal interactions altogether (e.g. Sackmann et al. 1993) or used simplified prescriptions for the tidal interactions based on the equilibrium tide theory of Zahn (1966, 1977) (e.g. Rybicki & Denis 2001; Schröder & Smith 2008; Guo et al. 2016; Lanza et al. 2023). These prescriptions are based on the assumption that the tidal dissipation is dominated by the equilibrium tide and use parametrisations for stellar properties. However, recent studies have improved these prescriptions by using ab initio modelling of the tidal dissipation based on the latest understanding of the internal structure and dynamics of evolved stars and including dynamical tides associated with progressive internal gravity waves (see Esseldeurs et al. 2024, and references therein). Along the same lines, mass loss during the AGB phase is also still uncertain (Decin 2021) and can affect the orbital evolution of planetary or binary systems significantly. We aim to improve the modelling of the tidal interactions by using the prescriptions of Esseldeurs et al. (2024) for the tidal dissipation. We also use the tools developed in Esseldeurs et al. (2026) to model the orbital evolution, which allows us to take the tidal interactions and the changes through stellar winds into account. Using these tools, we model the orbital evolution of the inner Solar System planets during the Sun’s RGB and AGB phases and determine their fate during these phases. In Sect. 2 we describe the tools we used to model the orbital evolution of the inner Solar System. In Sect. 3 we outline the stellar evolutionary models we used for the Sun. In Sect. 4 we present the results of the orbital evolution of the inner Solar System and discuss the effect of different tidal dissipation prescriptions and AGB mass-loss rates on the fate of the Earth.
To model the orbital evolution of the inner Solar System, we used the tools developed in Esseldeurs et al. (2026). To follow changes in planetary orbital parameters, we considered tidal interactions in the Sun as well as stellar winds, 1 a da
�����wind
�����tides
+ 1
dt = 1
. (1)
a da dt
a da dt For all simulations, the eccentricity remained negligible, but was taken into account for consistency. The equations for their evolution are given in Appendix A. These differential equations were integrated for each planet independently, and we did not take the effect of planet-planet interactions into account. Earth only crosses the 5:3 mean motion resonance with Venus during the RGB phase, but this is expected to have only a minor effect on the orbital evolution of Earth (see Appendix D). The changes through stellar winds are given by �1 a da
�������wind = −2 ˙m1
� 1 −β
� , (2)
q −η(1 −β)1 + q
dt
m1
q −1 −β
2 1 1 + q
where ⟨⟩denotes an orbit-averaged quantity, q is the mass fraction, η is the specific angular momentum of material lost in units of the orbital angular momentum of the system per reduced mass, and β =
1+e2
(1−e2)3/2 βc is the enhanced mass-accretion effi-
ciency effect accounting for the change in the orbital separation for eccentric orbits. η and βc were calculated using the analytical prescriptions proposed by Saladino et al. (2019). The changes through tides are given by
∞
∞
�5
m2 m1
�R⋆ a
�����tides
1 a da
= −5
nΩo|A2,m,n(e)|2 Im(k2,m n ) , (3)
�
�
4π
dt
n=−∞
m=0
where A2,m,n(e) are the tidal coefficients (see Ogilvie 2014 table 1 and Esseldeurs et al. 2026 for details), and Im(k2,m n ) are the imaginary parts of the tidal Love numbers, which were calculated using the prescriptions of Esseldeurs et al. (2024). These equations assume a negligible stellar rotation, which is a good approximation for evolved stars that have not been spun up through tidal interactions. This is a valid approximation for our Solar System calculations since tidal interactions are too weak to significantly spin the Sun up (Madappatt et al. 2016). The simulation either ended at the end of the stellar evolution model (a cool WD with L = 10−1 L⊙) or when the planet was engulfed by the star (when the orbital separation was smaller than the Roche radius).
We used the stellar evolutionary code MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023) to compute the evolution of the Sun from the pre-main sequence to the white dwarf phase. We used the same input physics as in Esseldeurs et al. (2024), which were based on Cinquegrana & Joyce (2022) to be consistent with the Monash code (Karakas & Lattanzio 2007) during the evolved phases. The initial mass of the Sun was set to 1 M⊙, and we used a metallicity of Z = 0.0134 (Asplund et al. 2009). For the mass-loss rate during the RGB phase, we used the Reimers mass-loss rate prescription (Reimers 1975) with ηReimer = 0.477, as calibrated from observations by McDonald & Zijlstra (2015), and we used the Blöcker mass-loss rate prescription (Blöcker 1995) with ηBlöcker between 0.01 and 0.5 (0.1 in Esseldeurs et al. 2024) for the mass-loss rate during the AGB phase, where 0.05 was taken as a reference value. A Kippenhahn diagram is shown in Appendix B.
for their constructive comments which helped to improve the quality of the
paper. M. Esseldeurs, S. Mathis and L. Decin acknowledge support from the
FWO grant G0B3823N. M. Esseldeurs and L. Decin acknowledge support from
the FWO grant G099720N, the KU Leuven C1 excellence grant MAESTRO
C16/17/007, the KU Leuven IDN grant ESCHER IDN/19/028 and the KU Leu-
ven methusalem SOUL grant METH/24/012. S. Mathis acknowledges support
from the PLATO CNES grant at CEA/DAp, from the Programme National
de Planétologie (PNP-CNRS/INSU) and from the European Research Council
through HORIZON ERC SyG Grant 4D-STAR 101071505. L. Decin acknowl-
edges support from the FWO sabbatical grant K803625N. While partially funded
by the European Union, views and opinions expressed are however those of the
ropean Research Council. Neither the European Union nor the granting authority
can be held responsible for them.

0.75
0.50
ηBl¨ocker = 0.01 ηBl¨ocker = 0.02 ηBl¨ocker = 0.05 ηBl¨ocker = 0.1 ηBl¨ocker = 0.2 ηBl¨ocker = 0.5
0.25
0.00
1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 stellar age - stellar age AGB [yr] ×107

[Figure: Fig. 3. Orbital evolution of the Earth during the evolved phases of the Sun for different AGB mass-loss rates. The different lines correspond to different values for ηBlöcker. The dashed lines correspond to the Solar radius, and the dotted lines correspond to the Solar Roche lobe radius. After the radius of the Sun exceeds the Roche lobe radius, the solid line for the orbital distance is changed to a dashed line to highlight the uncertainty in the model after this point (see text).]
Using the stellar and orbital evolution models described in the previous section, we predicted the orbital evolution of the inner Solar System planets from the PMS until the end of the WD phase. The results are shown in Fig. 1, where we show the evolution of the orbital separation of the planets as a function of time until the WD phase. During the RGB phase, the orbital separation of the planets increases due to the mass loss of the star, where the increase is larger for planets that are farther away from the star. Mercury and Venus are not able to move outward fast enough to avoid engulfment and are engulfed during the RGB phase. Earth and Mars are able to move outward fast enough to avoid engulfment during the RGB phase. During the HB phase, the orbital separation of the planets stabilises as the star contracts, while during the AGB phase, the orbital separation of the planets increases again due to the mass loss of the star. During this phase, Earth and Mars are able to move outward fast enough to avoid engulfment. During the WD phase, the orbital separation of the planets stabilises again as the star contracts.
Article number, page 2 ous studies, the stochastic eccentricity fluctuations induced by Lanza et al.
(2023) are less important for Earth.
Observationally, the tidal dissipation was tested in evolved
stars using the circularisation of binary systems. In the RGB
phase, Beck et al. (2018, 2022, 2024), and Dewberry & Wu
(2025) have shown that the equilibrium tide dominates the tidal
dissipation, but that the observed circularisation is stronger than
predicted by the equilibrium tide using the Duguid et al. (2020)
viscosity. In the AGB phase, Esseldeurs et al. (2026) have shown
expected given current tidal predictions. This suggests that tidal
dissipation in evolved stars might be stronger than predicted by
current models, making Earth’s engulfment more likely. How-
ever, these observations are based on binary circularisation,
which involves higher-frequency tides than circular migration of
orbits such as in the Earth-Sun system, and it might therefore not
be directly applicable.

PMS MS RGB HB AGB WD
R⋆ Mercury Venus Earth Mars
2.5
2.0
R⋆and a [AU]
1.5
1.0
0.5
0.0
−1010 −109 −108 −107
stellar age - stellar age WD [yr]

[Figure: Fig. 1. Orbital evolution (solid line, and the corresponding Solar Roche lobe radius, dotted line) of the inner Solar System during the evolved phases of the Sun. The different lines correspond to different planets (Mercury in orange, Venus in green, Earth in red, and Mars in purple). The current age of the Sun is indicated by the vertical dashed line.]
In the past, the importance of mass loss during the RGB
and AGB phases was highlighted by several studies (e.g.
Sackmann et al. 1993; Guo et al. 2016). Guo et al. (2016) inves-
mass-loss rate during the RGB phase) on the orbital evolution
during the RGB phase, and vice versa. As our tidal dissipation
is weaker, this value will decrease when recomputed using our
the fate of the Earth during the RGB phase is relatively well con-
strained, and the Earth is expected to survive the RGB phase, but
the engulfment is still within the uncertainty.
The AGB mass-loss rates are more uncertain. Different pre-
scriptions predict mass-loss rates that differ by more than one
order of magnitude (Decin 2021). Additionally, within the same
used in the literature. To constrain this parameter observation-
ally for our Sun, we considered the observations of the AGB
rounded by a dusty disk, inside which a potential planet of mass
mate based on the dust emission, which gives a mass-loss rate of
mate based on the CO emission, which gives a much lower mass-
to Appendix E for a discussion, but the large difference between
the two estimates already highlights the uncertainty in the AGB
mass-loss rates. With the Blöcker mass-loss rate prescription, the
between 0.01 and 1 are commonly used in the literature. We took
explored the effect of different values between 0.01 and 0.5 for
that other AGB mass-loss prescriptions exist, such as the period-
based prescription of Vassiliadis & Wood (1993), which cannot
be directly compared to the luminosity-based Blöcker prescrip-
mass-loss evolution within the explored parameter space.

PMS MS RGB HB AGB WD
Esseldeurs Zahn
1.75
1.50
1.25
R⋆and a [AU]
1.00
0.75
0.50
0.25
0.00
−1010 −109 −108 −107
stellar age - stellar age WD [yr]

[Figure: Fig. 2. Orbital evolution of the Earth during the evolved phases of the Sun for different tidal dissipation prescriptions. In orange, we plot Esseldeurs et al. (2024) and in green, we plot Mustill & Villaver (2012) (based on Zahn 1966; see Appendix C).]
To understand the effect of the different tidal dissipation pre-
scriptions on the orbital evolution, we compared the results us-
ing the tidal prescriptions of Esseldeurs et al. (2024) to a Zahn
(1966)-based prescription as used by Mustill & Villaver (2012)
(see Appendix C for a comparisson). The main difference be-
tween the prescriptions is the treatment of the frequency depen-
dence of the turbulent viscosity. At Earth’s orbit, the difference
between the viscosity prescriptions is largest, making the choice
of prescription crucial for Earth’s orbital evolution.
The results are shown in Fig. 2. The older prescriptions
of Mustill & Villaver (2012) predict stronger tidal dissipation,
which leads to a reduction in the orbital widening, and thus, to
Earth leaving the RGB phase on a much closer orbit. Earth there-
fore starts at a shorter orbital distance at the start of the AGB
phase, resulting in engulfment during the AGB phase. In con-
trast, our models predict Earth to be surviving the Sun’s RGB
to better constrain AGB mass-loss rates and hence the fate of the
inner Solar System. Additionally, the number of detected plan-
ets around red giants is expected to increase substantially in the
coming years, particularly with the PLATO mission (Rauer et al.
2025). This will enable us to conduct population studies of the
planetary orbital evolution around evolved stars and help us to
constrain the future evolution of the Earth-Sun system.

1.75
1.50
1.25
R⋆and a [AU]
1.00
Other high-energy, high-angular momentum models, such as the Synestia model (Lock & Stewart 2017; Lock et al. 2018) and the collision of two comparably massive bodies (Canup 2012) also produce excessive angular momentum, which would need to be reduced by some feasible mechanism, still a matter of debate. The high-energy, high-angular momentum model ( ´Cuk & Stewart 2012) is designed to produce a circum-terrestrial debris disk primarily composed of
Corresponding author: Wenshuai Liu 674602871@qq.com 2 to represent the materials.

After deriving the density profile for the non-rotating case (Ruiz-Bonilla et al. 2021), we use WoMa (Ruiz-Bonilla et al. 2021) to generate the target and impactor profile with a rotation period of 0 hour. We use SWIFT (Kegerreis et al. 2019) with 105 equal-mass SPH particles to simulate the giant impact. The artificial viscosity is taken to be 1.5, the physical viscosities of Theia and proto-Earth are set to be 1014Pa ∗s and 10Pa ∗s in the modified SWIFT, respectively. For comparison, the physical viscosities of Theia and proto-Earth are both set to be 10Pa ∗s in order to show the effect of the viscosity contrast on the giant impact. The simulation with such high physical viscosity is so time consuming that the simulation of giant impact is terminated after 1.68 hours in order to preliminarily illustrate the effect of viscosity contrast on the composition of the debris ejected by the giant impact. The results are shown in Figure 2. It shows from Figure 2 that giant impact between a high-viscosity Theia and a low-viscosity proto-Earth could significantly affect the composition of debris where about 70% materials originate from proto-Earth. Although the high viscosity of Theia is taken to be 1014Pa ∗s here, the composition of the debris disk will be totally made of materials of proto-Earth if Theia’s viscosity is taken to be extremely large value of 1021Pa ∗s. It’s like throwing a stone into water, the stuff that splashes out is all just water.

[Figure: Figure 1. The big circle represents the target planet with mass Mt and the small circle is the impactor planet with mass Mi. vc is the speed at first contact of the impactor with the target’s surface. The target planet rotates clockwise.]
30% of the mass, respectively. In reality, the outer portion of the proto-Earth may have existed as a magma ocean due to planetesimal accretion and multiple giant impacts. Here, we adopt the ANEOS equations of state for Fe85Si15 and forsterite (Stewart et al. 2020)

[Figure: Figure 2. Up and bottom panel represent the giant impact with same viscosity and with viscosity contrast, respectively. From left to right, the results are shown with simulation times of 0 hr, 0.56 hr, 1.12 hr and 1.68 hr.]

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