11institutetext: Observatoire de Genève, Université de Genève, Chemin Pegasi
51, CH$-$1290 Sauverny, Suisse
11email<EMAIL_ADDRESS>
# The key impact of the host star’s rotational history on the evolution of
TOI-849b

C. Pezzotti 11 O. Attia 11 P. Eggenberger 11 G. Buldgen 11 V. Bourrier 11

(Received …; accepted …)

###### Abstract

Context. TOI-849b is one of the few planets populating the hot-Neptune desert
and it is the densest Neptune-sized one discovered so far. Its extraordinary
proximity to the host star, together with the absence of a massive H/He
envelope on top of the $40.8\leavevmode\nobreak\ M_{\oplus}$ rocky core, calls
into question the role played by the host star in the evolution of the system.

Aims. We aim to study the impact of the host star’s rotational history on the
evolution of TOI-849b, particularly focussing on the planetary migration due
to dynamical tides dissipated in the stellar convective envelope, and on the
high-energy stellar emission.

Methods. Rotating stellar models of TOI-849 are coupled to our orbital
evolution code to study the evolution of the planetary orbit. The evolution of
the planetary atmosphere is studied by means of the JADE code, which uses
realistic X-ray and extreme-ultraviolet (XUV) fluxes provided by our rotating
stellar models.

Results. Assuming that the planet was at its present-day position ($a_{\rm
in}=0.01598$ AU) at the protoplanetary disc dispersal, with mass
$40.8\leavevmode\nobreak\ M_{\oplus}$, and considering a broad range of host
star initial surface rotation rates ($\rm\Omega_{\rm
in}\in[3.2,18]\leavevmode\nobreak\ \Omega_{\odot}$), we find that only for
$\Omega_{\rm in}\leq 5\leavevmode\nobreak\ \Omega_{\odot}$ do we reproduce the
current position of the planet, given that for $\Omega_{\rm
in}>5\leavevmode\nobreak\ \Omega_{\odot}$ its orbit is efficiently deflected
by dynamical tides within the first $\rm\sim 40\leavevmode\nobreak\ $Myr of
evolution. We also simulated the evolution of the orbit for values of $\rm
a_{\rm in}\neq 0.01598$ AU for each of the considered rotational histories,
confirming that the only combination suited to reproduce the current position
of the planet is given by $a_{\rm in}=0.01598\leavevmode\nobreak\ $AU and
$\rm\Omega_{\rm in}\leq 5\leavevmode\nobreak\ \Omega_{\odot}$. We tested the
impact of increasing the initial mass of the planet on the efficiency of
tides, finding that a higher initial mass ($M_{\rm in}=1\leavevmode\nobreak\
M_{\rm Jup}$) does not change the results reported above. Based on these
results we computed the evolution of the planetary atmospheres with the JADE
code for a large range of initial masses above a core mass of
$40.8\leavevmode\nobreak\ M_{\oplus}$, finding that the strong XUV-flux
received by the planet is able to remove the entirety of the envelope within
the first 50 Myr, even if it formed as a Jupiter-mass planet.

###### Key Words.:

Planet-star interaction - Planetary systems - Planets and satellites:
atmospheres - Stars: evolution - Stars: rotation - Stars: solar-type

##  1 Introduction

Armstrong et al. (2020) recently announced the discovery of TOI-849b, a planet
having a size comparable to the one of Neptune ($R_{\rm
pl}=3.45^{+0.16}_{-0.12}\leavevmode\nobreak\ R_{\rm\oplus}$), but an
anomalously larger mass ($M_{\rm pl}=40.8^{+2.4}_{-2.5}\leavevmode\nobreak\
M_{\oplus}$) and a density similar to the one of the Earth ($\rho=5.5\pm
0.8\leavevmode\nobreak\ \rm g/cm^{3}$). This is the densest Neptune-sized
planet discovered to date. Interior structure models suggest that for this
kind of planet, any H/He envelope would consist of no more than $\rm 3.9\%$ of
the total mass (Armstrong et al. 2020). TOI-849b orbits around a late G-type
star, with an orbital period of $P=18.4$ hours. Its equilibrium temperature is
$T_{\rm eq}=1800$ K (for a Bond Albedo 0.3). With such properties, TOI-849b
represents one of the few planets populating the hot Neptune desert, a region
on the radius-orbital distance plane characterised by a surprising deficit of
Neptune-sized planets on very short orbits (e.g. Lecavelier Des Etangs 2007;
Beaugé & Nesvorný 2013; Mazeh et al. 2016). Growing evidence suggests that the
evaporation of hot Neptunes due to stellar irradiation represents a major
process in shaping the desert (Owen 2019). Some of the planets within the
desert or at its lower-radius border could thus be the remnant cores of larger
gas-rich progenitors that lost most of their atmosphere (e.g. Lecavelier des
Etangs et al. 2004). Alternatively, orbital migration may have also played a
part in sculpting the desert, with different classes of planets forming
differently (e.g. Batygin et al. 2016), or following different dynamical
tracks (e.g. Matsakos & Königl 2016). Even so, the origin of this key feature
in the demographics of close-in planets remains debated (Zahnle & Catling
2017; Owen & Lai 2018), and investigating the past history of planets such as
TOI-849b contributes to the global effort towards disentangling the important
mechanisms at the root of the desert. Since the mass of the planet is larger
than the threshold value for runaway gas accretion (roughly
$10-20\leavevmode\nobreak\ M_{\oplus}$, Mizuno et al. 1978; Rafikov 2006; Piso
et al. 2015), TOI-849b might have been a gas giant before undergoing extreme
mass loss via thermal self-disruption, collisions with other giant planets, or
it was prevented from accreting gas because of gap openings in the
protoplanetary disc, or because of late formation (Armstrong et al. 2020). In
their work, Armstrong et al. (2020) found that their estimated
photoevaporation rates cannot provide the mass-loss rates needed to remove a
roughly $280\leavevmode\nobreak\ M_{\oplus}$ envelope from a Jupiter-like gas
giant. Nevertheless, in their estimations the planetary atmosphere is not
self-consistently monitored, and neither is the luminosity emitted by the host
star relative to its rotational history. This is critical, considering how
recent works emphasise the interplay between all these elements in driving the
evolution of close-in worlds (e.g. Owen & Alvarez 2016; Kubyshkina et al.
2018; King & Wheatley 2021; Pezzotti et al. 2021).

In this work we aim to study the orbital evolution of the system,
investigating the impact of the host star’s rotational history on the exchange
of angular momentum between the star and the orbit through the dissipation of
tides, and on the X-ray and extreme-ultraviolet (XUV) luminosities predicted
for the host star. In this context, we also aim to follow the evolution of the
planetary atmosphere and to provide estimations of the photo-evaporation rates
using the XUV-luminosity evolutionary tracks derived from our rotating stellar
models.

##  2 Stellar model

We computed rotating stellar models of TOI-849 by means of the Geneva stellar
evolution code (GENEC) (Eggenberger et al. 2008). We used the values for the
effective temperature ($T_{\rm eff}=5329\pm 48\leavevmode\nobreak\ \rm K$),
the iron abundance ($\rm[Fe/H]=0.2\pm 0.03\leavevmode\nobreak\ dex$), and the
surface gravity ($\rm log\leavevmode\nobreak\ g=4.43\pm 0.3$) provided in
Armstrong et al. (2020) as inputs and constraints to the models. Using the
parallax values given by Gaia (Gaia Collaboration et al. 2018), we computed
the bolometric luminosity ($L/L_{\odot}=0.568\pm 0.027$) as in Buldgen et al.
(2019), applying the bolometric correction derived by the code of Casagrande &
VandenBerg (2014, 2018b, 2018a) and the extinction inferred by the Green et
al. (2018) dust map. We therefore derived a stellar mass of
$0.93\leavevmode\nobreak\ M_{\odot}$, and a stellar radius of
$0.91\leavevmode\nobreak\ R_{\odot}$ at an age of 6.45 Gyr, finding that these
values are in good agreement with the ones in Armstrong et al. (2020).

While a thorough description of the physics included in GENEC is provided in
Eggenberger et al. (2008), in the following we briefly recall the fundamental
properties that are of interest for the present study. Rotational effects are
accounted for in the hypothesis of shellular rotation (Zahn 1992). The
evolution of the star is followed simultaneously with the transport of angular
momentum (AM), accounting for meridional currents, and transport by shear
instabilities and magnetic fields. The transport by magnetic fields is taken
into account in the context of the Tayler-Spruit dynamo (Spruit 2002). In
addition to the internal AM transport, we also account for the braking of the
stellar surface rotation due to magnetised winds (Matt et al. 2015, 2019).

With the rotational history of TOI-849 being unknown, we computed models
representative of slow, medium, and fast rotators by using different values
for the initial surface rotation rate ($\rm\Omega_{in}=3.2,5$ and $\rm
18\leavevmode\nobreak\ \Omega_{\odot}$, respectively) and for the disc
lifetimes during the pre-main sequence (PMS) phase ($\rm\tau_{\rm dl}=6$ Myr
for the slow and medium rotators, $\rm\tau_{\rm dl}=2$ Myr for the fast one)
111The choice of these values for $\Omega_{\rm in}$ and $\tau_{\rm dl}$ was
determined from the distribution of surface rotation rates observed for stars
in star-forming regions and young open clusters at various ages (Eggenberger
et al. 2019).. In the top panel of Fig. 1, we show the evolution of the
surface rotation rate for each of the considered rotational histories. In the
bottom panel of Fig. 1, the corresponding XUV-luminosity evolutionary tracks
are represented. We computed the high-energy luminosity by recalibrating the
prescription of Wright et al. (2011) for the X-ray and by using the one of
Sanz-Forcada et al. (2011) for the EUV luminosity (Pezzotti et al. 2021).

Figure 1: _Top panel:_ Evolution of the surface rotation rate of TOI-849 for a
fast, medium, and slow rotator $\rm\Omega_{in}=18,5,3.2\leavevmode\nobreak\
\Omega_{\odot}$ (green, magenta, and black line, respectively). The red-dashed
line indicates the evolution of the critical rotation velocity ($\Omega_{\rm
crit}$), defined as the velocity at which the centrifugal acceleration at the
equator becomes equal to the gravity. _Bottom panel:_ Evolution of the XUV
luminosities computed for each of the considered rotators. In both panels the
starting point of the tracks corresponds to the dissipation of the
protoplanetary disc (6 Myr for the slow and moderate rotators, 2 Myr for the
fast one).

##  3 Impact of the host star’s rotational history on the evolution of the
orbit

In order to study the orbital evolution of TOI-849b, we coupled the rotating
stellar models to our orbital evolution code (Privitera et al. 2016b, a; Rao
et al. 2018), and computed the total change of the orbital distance accounting
for the dissipation of equilibrium and dynamical tides222Equilibrium tides
represent the large-scale, non-wave-like flow induced in the host star by the
presence of a planetary companion (Zahn 1966); dynamical tides instead
designate the wave-like solution of the gravitational response. In convective
envelopes of rotating stars, these are constituted by inertial waves excited
by the tidal potential, whose restoring force is the Coriolis force (Zahn
1975; Ogilvie & Lin 2007). in the host star convective envelope along the
evolution of the system. Thanks to this coupling, it is possible to study the
AM exchange between the star and the planetary orbit and to test whether the
dissipation of tides might have played a significant role in shaping the
architecture of the system. Given the proximity of TOI-849b to the host star
($a=0.01598^{+0.00013}_{-0.00013}$ AU) and its relatively large mass
($40.8^{+2.4}_{-2.4}\leavevmode\nobreak\ M_{\oplus}$), we may expect
especially dynamical tides to have an impact in the early evolution of the
system, with respect to a certain host star’s rotational history

In our orbital evolution code, we consider the planet on a circular, coplanar
orbit. The equation for the change of the orbital distance due to equilibrium
tides is the same as in Privitera et al. (2016b). The contribution due to
dynamical tides is included in the form of a frequency-averaged dissipation of
inertial waves in the convective envelope of a rotating star and for their
expression we refer to the works of Ogilvie (2013) and Mathis (2015). The
impact of tides is to widen (respectively shrink) the orbit of the planet when
it is beyond (respectively inside) the corotation radius, defined as the
distance at which the orbital period of the planet is equal to the stellar
spin period, namely $a_{\rm corot}=[G(M_{\star}+M_{\rm
pl})/\Omega_{\star}^{2}]^{1/3}$, where $G$ is the gravitational constant,
$M_{\star}$ is the mass of the star, $M_{\rm pl}$ is the mass of the planet,
and $\rm\Omega_{\star}$ is the host star surface rotation rate. In the context
of the formalism used for dynamical tides, at each evolutionary time step,
these are considered to be active only when the orbital distance is larger
than a minimum critical value, defined as $a_{\rm
min}=4^{-1/3}\leavevmode\nobreak\ a_{\rm corot}$ (Ogilvie & Lin 2007). For a
more detailed description of the tidal model used in the present work, we
refer the interested reader to Appendix A. In the following we focus on
describing the impact of dynamical tides on the planetary orbit since they
represent the dominating mechanism governing the early evolution of the system
when the star rotates much faster compared with more advanced evolutionary
stages.

As a first step, we computed the orbital evolution of the planet at a constant
planetary mass ($40.8\leavevmode\nobreak\ M_{\oplus}$), assuming that it
started the evolution with an initial orbital distance equal to its present-
day value ($a_{\rm in}=0.01598$ AU). Since the rotational history of the star
is unknown, we simulated the evolution of the orbit considering different
values for the host star’s initial surface rotation rate ($\Omega_{\rm
in}=3.2,5,18\leavevmode\nobreak\ \Omega_{\odot}$). In this way, we tested for
the considered rotational histories whether the planet would remain at its
initial (and present-day) orbital distance until an age of approximately 6.45
Gyr. We found that for $\Omega_{\rm in}=3.2,5\leavevmode\nobreak\
\Omega_{\odot}$, there is no significant impact on the evolution of the orbit;
instead for $\Omega_{\rm in}=18\leavevmode\nobreak\ \Omega_{\odot}$, the orbit
of the planet rapidly shrinks on timescales of the order of $\rm\sim 40$ Myr.
At this stage, given the proximity of the planet to the star, it might get
totally disrupted by tides or get engulfed by the host star at $\sim 400$ Myr.
Given the significant change in the dynamical tides efficiency occurring
between the moderate and the fast rotator case, we considered a series of
rotators with $\Omega_{\rm in}\in[5,18]\leavevmode\nobreak\ \Omega_{\odot}$,
in steps of $1\leavevmode\nobreak\ \Omega_{\odot}$, to find the value of
$\Omega_{\rm in}$ (and the corresponding rotator) below which dynamical tides
leave the planetary orbit unperturbed. We found that for $\rm\Omega_{in}\leq
5\leavevmode\nobreak\ \Omega_{\odot}$, there is no significant impact on the
evolution of the orbit; instead, for $\rm\Omega_{in}>5\leavevmode\nobreak\
\Omega_{\odot}$, the orbit of the planet shrinks on timescales of the order of
$\rm 40$ Myr, leading us to obtain final values that do not reproduce the
current position of the planet. The results related to these simulations are
shown in Fig. 2. We recall that in the context of the formalism used in the
present work, at each time step dynamical tides are considered to be active
only for $a\geq a_{\rm min}$. Given the dependency of $a_{\rm min}$ on
$\rm\Omega_{\star}$, we notice that for values of $\rm\Omega_{in}\leq
5\leavevmode\nobreak\ \Omega_{\odot}$, if we start the simulation at $a_{\rm
in}=0.01598$ AU, the orbit of the planet remains below $a_{\rm min}$ for the
whole evolution of the system, and consequently dynamical tides never become
active. Instead, for values of $\rm\Omega_{in}>5\leavevmode\nobreak\
\Omega_{\odot}$, either the value of $a_{\rm min}$ is smaller than $\rm
0.01598$ AU from the beginning of the evolution (for $\rm\Omega_{in}\geq
16\leavevmode\nobreak\ \Omega_{\odot}$), or it becomes smaller at $\rm\sim
20-30$ Myr, allowing dynamical tides to be active. Therefore, regarding the
hypothesis that TOI-849b started its evolution at $\rm 0.01598$ AU and evolved
at constant mass, our results show that only for $\rm\Omega_{in}\leq
5\leavevmode\nobreak\ \Omega_{\odot}$ are we able to reproduce its current
position.

We may ask whether for $\rm\Omega_{in}>5\leavevmode\nobreak\ \Omega_{\odot}$
can the present position of the planet be reproduced, this time changing its
initial orbital distance (namely for $a_{\rm in}\neq 0.01598$ AU).
Nevertheless, we found that there is no value of $a_{\rm in}$ for which this
is possible. This is shown in Fig. 3 for the specific case of the fast rotator
($\rm\Omega_{in}=18\leavevmode\nobreak\ \Omega_{\odot}$), for which the impact
of tides is the most evident.

In the previous computations, we assumed that the planet evolved at a constant
mass ($40.8\leavevmode\nobreak\ M_{\oplus}$). Nevertheless, TOI-849b might
have formed with a significantly higher mass (roughly $1\leavevmode\nobreak\
M_{\rm Jup}$, Armstrong et al. 2020). Therefore, for each of the considered
rotational histories, we recomputed the evolution with $M_{\rm
in}=1\leavevmode\nobreak\ M_{\rm Jup}$, and different values of $a_{\rm in}$,
finding that assuming a higher initial mass does not change the results we
reported in the previous paragraphs. This shows that for the assumed
configuration of the system, the rotational history of the host star mainly
determines the efficiency of tides.

The reproducibility of the planetary position at $\rm\sim
6.45\leavevmode\nobreak\ Gyr$ for values of $\rm\Omega_{in}\leq
5\leavevmode\nobreak\ \Omega_{\odot}$ helps to reduce the degeneracy on the
host star’s rotational history, indicating that an evolution as a slow or
moderate rotator might be favoured over the fast one. This result also
provides constraints on the high-energy emission history of the host star,
with its activity and rotational history being directly linked. In this
context, given that the planet likely remained its entire life at its present-
day position if it migrated early on, and that its initial mass does not
represent a dominating factor for the efficiency of tides, it is possible to
simulate the atmospheric evolution of the planet separately, considering
different initial masses at a fixed orbital distance.

Figure 2: Evolution of the orbital distance for a planet having the same mass
as TOI-849b and $a_{\rm in}=0.01598\leavevmode\nobreak\ \rm AU$. From top to
bottom, the solid lines show the evolution of the orbital distance for
$\rm\Omega_{in}=5,6,7,8,9,10\leavevmode\nobreak\ \Omega_{\odot}$ (green,
black, red, cyan, blue, and magenta). The dotted and dashed lines indicate the
evolution of $a_{\rm min}$ and $a_{\rm corot}$ (or Corad), respectively, for
different values of $\rm\Omega_{in}$ (same colour code). The cyan and magenta
shaded areas represent the extension of the stellar radiative and convective
regions, respectively. Figure 3: Evolution of the planetary orbital distance
(solid lines) in the case of a fast rotator
($\rm\Omega_{in}=18\leavevmode\nobreak\ \Omega_{\odot}$), spanning a range of
initial values $\rm 0.01598-0.025\leavevmode\nobreak\ AU$. The red-dashed line
represents the current orbital distance of TOI-849b. The red-dotted and black-
dashed lines show the evolution of $a_{\rm min}$ and $a_{\rm corot}$,
respectively. The cyan- and magenta-shaded areas represent the extension of
the stellar radiative and convective regions, respectively.

##  4 Impact of evaporation

We study the impact of atmospheric processes on the evolution of the mass of
the planet. To this effect, we employed the JADE code (Attia et al. 2021). We
refer to that paper for details, but briefly summarise the main ingredients
here. The JADE code allows for one to simulate the evolution of a planet over
secular timescales under the coupled influence of complex dynamical and
atmospheric mechanisms. The planet is composed of a rocky core topped by a
H/He gaseous envelope. The planet’s radius is self-consistently derived by
integrating its thermodynamical structure as it varies following the evolution
of the orbit, the amount of stellar irradiation, and inner planetary heating.
The atmosphere is prone to erode due to XUV-induced photo-evaporation, with a
mass-loss rate calculated as (e.g. Watson et al. 1981; Lammer et al. 2003; Jin
et al. 2014)

$\dot{M}_{\mathrm{env}}=\epsilon\left\langle
F_{\mathrm{XUV}}\right\rangle\frac{S_{\mathrm{XUV}}}{\Phi_{0}K_{\mathrm{tide}}},$
(1)

where $M_{\rm env}\equiv{}M_{\rm pl}-M_{\rm core}$ is the mass of the gaseous
envelope, $\epsilon$ the efficiency of photo-evaporation, $F_{\rm XUV}$ the
stellar XUV flux, $S_{\rm XUV}$ the cross-section surface that collects the
high-energy irradiation, $\Phi_{0}$ the planetary gravitational potential, and
$K_{\rm tide}$ a correction factor accounting for atmospheric loss enhancement
due to the action of tidal forces (Erkaev et al. 2007). One of the novelties
of the JADE code is that the XUV flux is consistently secularised with the
dynamical equations of motion, so as to better account for the orbital
evolution (Attia et al. 2021). The factors $\epsilon$ and $R_{\rm XUV}$ are
computed using analytical formulae derived from self-consistent simulations of
upper atmospheric structures (Salz et al. 2016), which allows a better
agreement with observed mass-loss rates than with the simple energy-limited
approach.

To investigate the sole impact of photo-evaporation in accordance with the
results found in the previous section, we conducted purely atmospheric
simulations at a fixed circular orbit ($a_{\rm in}=0.01598\leavevmode\nobreak\
{\rm AU}$), for two sets of simulations corresponding to the star being a slow
or an intermediate rotator ($\Omega_{\rm in}=3.2,5\times\Omega_{\odot}$). We
used an atmospheric helium fraction $Y=0.15$ as a typical value for Neptune-
sized planets (e.g. Hubbard et al. 1995). Since Armstrong et al. (2020)
constrained a maximum H/He envelope mass fraction of 3.9%, we considered a
rocky core mass of $M_{\rm core}=40.8\leavevmode\nobreak\ M_{\oplus}$, which
corresponds to the currently measured value of the total mass. We simulated
the atmospheric evolution of the planet for a large range of initial masses
above this value.

Figure 4 illustrates the results. Simulated planets with an initial mass
higher than $\sim 46\leavevmode\nobreak\ M_{\oplus}$ are affected by tidal
disruption. Indeed, for these large atmospheric masses and at such short
ranges from the star, the planetary radius is highly inflated ($R_{\rm
pl}\gtrsim 1\leavevmode\nobreak\ R_{\rm Jup}$). The atmosphere then extends
behind the Roche limit of the star (e.g. Gu et al. 2003; Jackson et al. 2017)
and we assume it is completely depleted by disruption due to tidal
interactions. For initial masses below this threshold, the planet always loses
the entirety of its atmosphere within $\sim 30\leavevmode\nobreak\ {\rm Myr}$
from the dissipation of the protoplanetary disc due to photo-evaporation
leaving a rocky bare core. This result is valid for both rotators, with a
slightly faster evaporation for the intermediate rotator as in this case the
high-energy flux received by the planet is higher (see Fig. 1).

Tidal atmospheric disruption is likely more complex than the simple threshold
condition used in our simulations. To investigate whether a more massive
planet could keep its atmosphere if it was more resilient to disruption, we
considered the extreme case where tidal disruption is inactive and the planet
only evolves under photo-evaporation. We simulated the case where TOI-849b
formed as a Jupiter-mass planet, unaffected by disruption. The result,
illustrated as dotted lines in Fig. 4, shows that the gaseous envelope still
completely erodes after $\sim 40$ Myr. Consequently, the efficient removal of
the atmosphere, even for a high initial planetary mass, is a robust result
regardless of whether the Roche limit is considered or not.

Given the very close orbit of TOI-849b and high stellar emission of its host
star after formation, the XUV flux received by the planet is one order of
magnitude higher than the range of validity of our mass loss formalism from
Salz et al. (2016). However, given the speed of TOI-849b’s full atmospheric
erosion, a possible overestimation of the mass loss rate does not change our
conclusion that the present-day planet would have lost its atmosphere long ago
if it had evaporated following its formation. Several scenarios can be put
forward to explain its possible residual gaseous envelope. A simple solution
could be a different planetary composition (e.g. water-dominated, Lopez 2017;
Armstrong et al. 2020), which would allow for a wider range of initial masses
with compact atmospheres more resilient to tidal disruption and photo-
evaporation. Alternatively, a different atmospheric history can be envisioned
in the framework of late orbital migration. If the planet formed far enough
from its star and did not migrate early on, it would have been largely spared
from photo-evaporation during the first $\sim 100\leavevmode\nobreak\ {\rm
Myr}$, when stellar emission is the highest (see Fig. 1). Migrating billions
of years later to its current orbit, the planet would have then been subjected
to a stellar irradiation decreased by several orders of magnitude. We will
investigate this scenario, which could explain how TOI-849b still holds a
volatile atmosphere, in a follow-up paper.

Figure 4: Evolution of the planetary mass under the influence of photo-
evaporation for different initial mass values. Blue (respectively orange)
curves correspond to the star being a slow (respectively intermediate)
rotator. The red-dashed line indicates the mass threshold above which the
planet would be immediately tidally disrupted. The black-dashed line indicates
the mass of the rocky core. The dotted lines correspond to simulations of a
$1\leavevmode\nobreak\ M_{\rm Jup}$ initial mass planet without accounting for
the tidal disruption limit. We point out the broken $y$-axis. The starting
time of the evolution ($t=0$) corresponds to the protoplanetary disc
dispersal.

##  5 Conclusion

In this work we aimed to study the impact of the stellar rotational history on
the past evolution of the TOI-849b, considering both the impact of tides on
the planetary orbit and the irradiation of the planetary atmosphere. Coupling
the rotating stellar models computed with the Geneva code to our orbital
evolution code, assuming that the planet was at $0.01598$ AU and had its
present-day mass ($40.8\leavevmode\nobreak\ M_{\oplus}$) at the dispersal of
the protoplanetary disc, we found that the impact of dynamical tides on the
evolution of the orbit becomes significant only for values of the surface
rotation rate of the star ($\Omega_{\rm in}$) above $\rm 5\leavevmode\nobreak\
\Omega_{\odot}$. For these values of $\Omega_{\rm in}$, the planet migrates
towards the host star during the first $1-100$ Myr and it is eventually
engulfed later on. Relaxing the assumption of an initial orbital distance at
$0.01598$ AU, we explored a range of values for $a_{\rm in}$ and $\Omega_{\rm
in}$ in order to find a combination for which it is possible to reproduce the
current position of the planet thanks to the impact of tides. We found that
such a configuration is only retrieved for $a_{\rm in}=0.01598$ AU and
$\Omega_{\rm in}\leq 5\leavevmode\nobreak\ \Omega_{\odot}$. Simulations
performed for a higher initial planetary mass, up to $1\leavevmode\nobreak\
M_{\rm Jup}$, and various values of $a_{\rm in}$ and $\Omega_{\rm in}$ do not
change this conclusion.

Thanks to the constraints derived on the host star’s rotational history, we
selected a set of XUV-luminosity evolutionary tracks computed consistently
with the evolution of the surface rotation of the star to study the impact of
photo-evaporation on the planetary atmosphere. We performed atmospheric
simulations with the JADE code assuming a fixed circular orbit at $0.01598$
AU, a H/He envelope, and an in situ formation or early-on migration, and we
computed the evolution of the planetary atmosphere for a large range of
initial masses above a core mass of $40.8\leavevmode\nobreak\ M_{\oplus}$. The
strong irradiation associated with the saturated regime of emission of the
host star leads TOI-849b to completely evaporate its atmosphere in less than
50 Myr, even if it had formed as a Jupiter-mass planet. In addition, initial
planetary masses above $46\leavevmode\nobreak\ M_{\oplus}$ are associated with
highly inflated radii, so much so that the atmosphere would be affected by
tidal disruption as it extends beyond the stellar Roche limit. We conclude
from these atmospheric simulations that TOI-849b could not have retained a
fraction of its gaseous envelope if it reached its current orbit just after
the dissipation of the protoplanetary disc. If TOI-849b still has a residual
atmosphere today, it likely has a peculiar atmospheric composition, or it
underwent a delayed migration. In the latter scenario, the planet would have
avoided erosion by remaining far from the star during the first $\sim 100$
Myr, when the stellar emission is the strongest.

This study demonstrates how thoroughly considering the rotational history of a
host star together with the evolution of the planet may lead to alternative
results that would be unexplored otherwise. We indeed found that in the
configuration assumed for this system, it is possible for a planet such as
TOI-849b to lose the entirety of its envelope even if it formed as a Jupiter-
mass planet. A further detailed investigation of this system, which will be
carried out in a follow-up paper, will eventually help to shed light on the
relevant mechanisms shaping the planets populating the hot Neptune desert and
determine their possible filiation.

## Acknowledgements

We thank the referee for the useful comments that helped to improve the
quality of the manuscript. G.B. acknowledges funding from the SNF AMBIZIONE
grant No 185805 (Seismic inversions and modelling of transport processes in
stars). P.E. has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement No 833925, project STAREX). This project has received funding from
the ERC under the European Union’s Horizon 2020 research and innovation
program (project SPICE DUNE, grant agreement No 947634). V.B. & O.A.
acknowledge support by the Swiss National Science Foundation (SNSF) in the
frame of the National Centre for Competence in Research “PlanetS”.

## References

  * Armstrong et al. (2020) Armstrong, D. J., Lopez, T. A., Adibekyan, V., et al. 2020, Nature, 583, 39
  * Attia et al. (2021) Attia, O., Bourrier, V., Eggenberger, P., et al. 2021, A&A, 647, A40
  * Barker (2020) Barker, A. J. 2020, MNRAS, 498, 2270
  * Barker & Ogilvie (2010) Barker, A. J. & Ogilvie, G. I. 2010, MNRAS, 404, 1849
  * Batygin et al. (2016) Batygin, K., Bodenheimer, P. H., & Laughlin, G. P. 2016, ApJ, 829, 114
  * Beaugé & Nesvorný (2013) Beaugé, C. & Nesvorný, D. 2013, ApJ, 763, 12
  * Bolmont & Mathis (2016) Bolmont, E. & Mathis, S. 2016, Celestial Mechanics and Dynamical Astronomy, 126, 275
  * Buldgen et al. (2019) Buldgen, G., Rendle, B., Sonoi, T., et al. 2019, MNRAS, 482, 2305
  * Casagrande & VandenBerg (2014) Casagrande, L. & VandenBerg, D. A. 2014, MNRAS, 444, 392
  * Casagrande & VandenBerg (2018a) Casagrande, L. & VandenBerg, D. A. 2018a, MNRAS, 479, L102
  * Casagrande & VandenBerg (2018b) Casagrande, L. & VandenBerg, D. A. 2018b, MNRAS, 475, 5023
  * Eggenberger et al. (2019) Eggenberger, P., Buldgen, G., & Salmon, S. J. A. J. 2019, A&A, 626, L1
  * Eggenberger et al. (2008) Eggenberger, P., Meynet, G., Maeder, A., et al. 2008, Ap&SS, 316, 43
  * Erkaev et al. (2007) Erkaev, N. V., Kulikov, Y. N., Lammer, H., et al. 2007, A&A, 472, 329
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Goldreich & Nicholson (1977) Goldreich, P. & Nicholson, P. D. 1977, Icarus, 30, 301
  * Goodman & Dickson (1998) Goodman, J. & Dickson, E. S. 1998, ApJ, 507, 938
  * Green et al. (2018) Green, G. M., Schlafly, E. F., Finkbeiner, D., et al. 2018, MNRAS, 478, 651
  * Gu et al. (2003) Gu, P.-G., Lin, D. N. C., & Bodenheimer, P. H. 2003, ApJ, 588, 509
  * Hubbard et al. (1995) Hubbard, W. B., Podolak, M., & Stevenson, D. J. 1995, in Neptune and Triton, 109–138
  * Jackson et al. (2017) Jackson, B., Arras, P., Penev, K., Peacock, S., & Marchant, P. 2017, ApJ, 835, 145
  * Jin et al. (2014) Jin, S., Mordasini, C., Parmentier, V., et al. 2014, ApJ, 795, 65
  * King & Wheatley (2021) King, G. W. & Wheatley, P. J. 2021, MNRAS, 501, L28
  * Kubyshkina et al. (2018) Kubyshkina, D., Fossati, L., Erkaev, N. V., et al. 2018, ApJ, 866, L18
  * Lammer et al. (2003) Lammer, H., Selsis, F., Ribas, I., et al. 2003, ApJ, 598, L121
  * Lecavelier Des Etangs (2007) Lecavelier Des Etangs, A. 2007, A&A, 461, 1185
  * Lecavelier des Etangs et al. (2004) Lecavelier des Etangs, A., Vidal-Madjar, A., McConnell, J. C., & Hébrard, G. 2004, A&A, 418, L1
  * Lopez (2017) Lopez, E. D. 2017, MNRAS, 472, 245
  * Mathis (2015) Mathis, S. 2015, A&A, 580, L3
  * Matsakos & Königl (2016) Matsakos, T. & Königl, A. 2016, ApJ, 820, L8
  * Matt et al. (2015) Matt, S. P., Brun, A. S., Baraffe, I., Bouvier, J., & Chabrier, G. 2015, ApJ, 799, L23
  * Matt et al. (2019) Matt, S. P., Brun, A. S., Baraffe, I., Bouvier, J., & Chabrier, G. 2019, ApJ, 870, L27
  * Mazeh et al. (2016) Mazeh, T., Holczer, T., & Faigler, S. 2016, A&A, 589, A75
  * Mizuno et al. (1978) Mizuno, H., Nakazawa, K., & Hayashi, C. 1978, Progress of Theoretical Physics, 60, 699
  * Ogilvie (2013) Ogilvie, G. I. 2013, MNRAS, 429, 613
  * Ogilvie & Lin (2007) Ogilvie, G. I. & Lin, D. N. C. 2007, ApJ, 661, 1180
  * Owen (2019) Owen, J. E. 2019, Annual Review of Earth and Planetary Sciences, 47, 67
  * Owen & Alvarez (2016) Owen, J. E. & Alvarez, M. A. 2016, ApJ, 816, 34
  * Owen & Lai (2018) Owen, J. E. & Lai, D. 2018, MNRAS, 479, 5012
  * Pezzotti et al. (2021) Pezzotti, C., Eggenberger, P., Buldgen, G., et al. 2021, A&A, 650, A108
  * Piso et al. (2015) Piso, A.-M. A., Youdin, A. N., & Murray-Clay, R. A. 2015, ApJ, 800, 82
  * Privitera et al. (2016a) Privitera, G., Meynet, G., Eggenberger, P., et al. 2016a, A&A, 593, L15
  * Privitera et al. (2016b) Privitera, G., Meynet, G., Eggenberger, P., et al. 2016b, A&A, 593, A128
  * Rafikov (2006) Rafikov, R. R. 2006, ApJ, 648, 666
  * Rao et al. (2018) Rao, S., Meynet, G., Eggenberger, P., et al. 2018, A&A, 618, A18
  * Salz et al. (2016) Salz, M., Schneider, P. C., Czesla, S., & Schmitt, J. H. M. M. 2016, A&A, 585, L2
  * Sanz-Forcada et al. (2011) Sanz-Forcada, J., Micela, G., Ribas, I., et al. 2011, A&A, 532, A6
  * Spruit (2002) Spruit, H. C. 2002, A&A, 381, 923
  * Villaver & Livio (2009) Villaver, E. & Livio, M. 2009, ApJ, 705, L81
  * Watson et al. (1981) Watson, A. J., Donahue, T. M., & Walker, J. C. G. 1981, Icarus, 48, 150
  * Wright et al. (2011) Wright, N. J., Drake, J. J., Mamajek, E. E., & Henry, G. W. 2011, ApJ, 743, 48
  * Zahn (1966) Zahn, J. P. 1966, Annales d’Astrophysique, 29, 489
  * Zahn (1975) Zahn, J. P. 1975, A&A, 41, 329
  * Zahn (1977) Zahn, J. P. 1977, A&A, 500, 121
  * Zahn (1992) Zahn, J. P. 1992, A&A, 265, 115
  * Zahnle & Catling (2017) Zahnle, K. J. & Catling, D. C. 2017, ApJ, 843, 122

##  Appendix A Equilibrium and dynamical tides model

In our orbital evolution code (Privitera et al. 2016b, a; Rao et al. 2018), we
computed the total change of the planetary orbital distance by accounting for
the dissipation of equilibrium and dynamical tides in the host star. While
equilibrium tides constitute the non-wave-like large-scale flow induced in the
star by the presence of the planet, dynamical tides instead represent the
wave-like solution of the gravitational response. We coupled rotating stellar
models that were computed with the Geneva stellar evolution code to the
orbital evolution code in order to take into account the evolution of the
fundamental stellar parameters and their impact on the tidal dissipation
efficiency.

Equilibrium tides are mainly dissipated in the low-mass star convective
envelopes through turbulent friction induced by convection (Zahn 1966), while
their damping is less efficient in other regions (Zahn 1977). In our approach,
we therefore account for the dissipation of equilibrium tides only when a
convective envelope is present. The equation determining the change of the
orbital distance due to equilibrium tides reads as (Privitera et al. 2016b)

$\left(\dot{a}/a\right)_{\rm eq}=\frac{f}{\tau}\frac{M_{\rm
env}}{M_{\star}}q(1+q)\left(\frac{R_{\star}}{a}\right)^{8}\left(\frac{\Omega_{\star}}{\omega_{\rm
pl}}-1\right),$ (2)

where the term $a$ is the orbital distance and $\dot{a}$ is its derivative
with respect to time. The term $f$ is a numerical factor obtained from
integrating the viscous dissipation of the tidal energy across the convective
zone (Villaver & Livio 2009), which is equal to 1 when the convective turnover
timescale is larger than half of the orbital period ($\tau>P_{\rm orb}/2$),
otherwise $f=(P_{\rm orb}/2\tau)^{2}$ (Goldreich & Nicholson 1977). The factor
$M_{\rm env}$ is the mass of the convective envelope, $q$ is the ratio between
the mass of the planet and the one of the star ($q=M_{\rm pl}/M_{\star}$),
$\Omega_{\star}$ is the stellar surface rotation rate, and $\omega_{\rm
pl}=2\pi/P_{\rm orb}$ is the orbital frequency of the planet.

Dynamical tides might be efficiently dissipated both in convective and
radiative regions of a star. In convective regions, they are constituted by
inertial waves driven by the Coriolis force. These waves are excited when the
values of the tidal frequency ($\omega_{t}$) are comprised in the range
$[-2\Omega_{\star},2\Omega_{\star}]$ (Ogilvie & Lin 2007). In our code the
impact of dynamical tides propagating in convective zones is accounted for in
the form of a frequency-averaged tidal dissipation of inertial waves (Ogilvie
2013; Mathis 2015; Bolmont & Mathis 2016). This expression is considered
assuming a two-layer (core and envelope) model for the host star, each of them
characterised by a uniform density (Mathis 2015). We note that assuming this
schematic stratification for the stellar structure, together with the use of a
frequency-average dissipation rate for the dynamical tide, adds a certain
degree of uncertainty on our results. Nevertheless, while this approach
suffers from some schematic simplifications, on the other hand it has the
advantage of providing us with relevant orders of magnitude of tidal
dissipation rates accounting for the evolution of the structural and
rotational parameters of the host star (Mathis 2015; Bolmont & Mathis 2016;
Rao et al. 2018; Barker 2020). If we assume that the planet is on a circular-
coplanar orbit, dynamical tides are considered to be active whenever the
condition $\rm\omega_{pl}<2\leavevmode\nobreak\ \Omega_{\star}$ is satisfied,
since in this configuration the planet is able to excite inertial waves in the
stellar convective envelope only when the orbital frequency is lower than
twice the stellar rotation rate (Ogilvie & Lin 2007). The equation defining
the change of the orbital distance due to the dissipation of dynamical tides
(Ogilvie 2013; Mathis 2015) reads as

$\left(\dot{a}/a\right)_{\rm dyn}=\left(\dfrac{9}{2Q^{\prime}_{\rm
d}}\right)q\omega_{\rm
pl}\left(\frac{R_{\star}}{a}\right)^{5}\dfrac{(\Omega_{\star}-\omega_{\rm
pl})}{\mid\Omega_{\star}-\omega_{\rm pl}\mid},$ (3)

where $Q^{\prime}_{\rm d}=3/(2D_{\omega})$ and $\rm
D_{\omega}=D_{0\omega}D_{1\omega}D_{2\omega}^{-2}$. The ‘D’ terms are defined
as

$\begin{cases}D_{0\omega}=\dfrac{100\pi}{63}\epsilon^{2}\dfrac{\alpha^{5}}{1-\alpha^{5}}(1-\gamma)^{2},\\\
D_{1\omega}=(1-\alpha)^{4}\left(1+2\alpha+3\alpha^{2}+\frac{3}{2}\alpha^{3}\right)^{2},\\\
D_{2\omega}=1+\frac{3}{2}\gamma+\frac{5}{2\gamma}\left(1+\frac{\gamma}{2}-\frac{3\gamma^{2}}{2}\right)\alpha^{3}-\frac{9}{4}\left(1-\gamma\right)\alpha^{5},\end{cases}$
(4)

where $\alpha=R_{\rm c}/R_{\star}$, $\beta=M_{\rm c}/M_{\star}$,
$\gamma=\dfrac{\alpha^{3}(1-\beta)}{\beta(1-\alpha^{3})}$,
and$\epsilon=\dfrac{\Omega_{\star}}{\sqrt{\dfrac{GM_{\star}}{R_{\star}^{3}}}}$.
The terms $M_{c}$ and $R_{c}$ indicate the mass and radius of the radiative
core. The term $\rm D_{\omega}$ was computed in Ogilvie (2013) as the
frequency-averaged tidal dissipation.

It is possible that under specific conditions, waves are excited at the
convective-radiative interface and that they propagate in the radiative
regions as internal-gravity waves, eventually contributing to the dissipation
of energy and the exchange of angular momentum between the planet and the host
star (e.g. Goodman & Dickson 1998; Ogilvie & Lin 2007; Barker & Ogilvie 2010;
Barker 2020). In the current version of our orbital evolution code, we do not
account for the dissipation of dynamical tides in stellar radiative regions.
In order to understand the implications of not considering this dissipation
mechanism in the orbital history of TOI-849b, we analysed the results recently
obtained in the work of Barker (2020). In this work, the dissipation of
dynamical tides in radiation zones is computed accounting for the evolution of
the relevant stellar quantities. Barker (2020) assumes that the internal
gravity waves are launched from the convective-radiative interface, and that
these are subsequently fully damped. Such an efficient damping is likely
justified when wave breaking occurs; also, in order for this mechanism to be
triggered, it is necessary that the planetary mass is above a certain critical
value ($M_{\rm critic}$). Barker (2020) studied the evolution of $M_{\rm
critic}$ with respect to the evolution of solar-like and low-mass stars,
finding that for stars with masses lower than $1\leavevmode\nobreak\
M_{\odot}$, the planetary critical mass goes below $1\leavevmode\nobreak\
M_{\rm Jup}$ only at ages larger than about 10 Gyr. This result indicates that
only planets more massive than $1\leavevmode\nobreak\ M_{\rm Jup}$ are able to
trigger wave breaking during the PMS and most of the MS phases for these
stars. Nevertheless, as mentioned before, the value of $M_{\rm critic}$
rapidly drops below $1\leavevmode\nobreak\ M_{\rm Jup}$ when these stars are
close to the end of the MS, and subsequently super-Earths and mini-Neptunes
may be massive enough to trigger wave breaking once their host stars are in
the sub-giant and red giant branch phases.

We recall that in the case of the TOI-849 system, the host star is a main
sequence star with mass $0.93\leavevmode\nobreak\ M_{\odot}$ and
$\rm[Fe/H]=0.2\pm 0.03\leavevmode\nobreak\ dex$. In order to get an estimation
of the order of magnitude required for $M_{\rm critic}$ to trigger wave
breaking, and recalling that the metallicity of TOI-849 is super solar, we
examined the tracks representing the evolution of $M_{\rm critic}$ as a
function of the stellar mass and age reported in Fig. 9 of Barker (2020).
Indicatively, we focussed on the region comprised between the tracks
representing the evolution of $M_{\rm critic}$ with respect to the $1.0$ and
$0.8\leavevmode\nobreak\ M_{\odot}$ stellar models. We noticed that $M_{\rm
critic}$ reaches values larger that about $1000\leavevmode\nobreak\ M_{\rm
Jup}$ at the beginning of the evolution and it remains largely above
$1\leavevmode\nobreak\ M_{\rm Jup}$ for the whole PMS and most of the MS
phase, while it drops below $1\leavevmode\nobreak\ M_{\rm Jup}$ at
approximately $7$ and $20$ Gyr for the $1$ and $0.8\leavevmode\nobreak\
M_{\odot}$ tracks, respectively. If we assume that wave breaking is the only
mechanism allowing for a full dissipation of progressive internal gravity
waves, in the case of TOI-849 it seems unlikely that such a process occurred
during the early stages of evolution because it would have required TOI-849b
to have been formed with a mass of the order of $1000\leavevmode\nobreak\
M_{\rm Jup}$. It is also unlikely that wave breaking occurred during the first
part of the MS experienced so far by the star, because even if TOI-849b formed
with a mass of about $1-2\leavevmode\nobreak\ M_{\rm Jup}$, this would have
not been sufficient to trigger this process. On the contrary, it is possible
that this process will be triggered at more advanced evolutionary stages.

According to these considerations, it seems unlikely that the dissipation of
dynamical tides in the radiative core of TOI-849 might have influenced the
past history of the system. The definition of a model able to determine the
secular evolution of the planetary system, accounting for the dissipations of
tides in stellar radiative and convective zones in a consistent way with
respect to the evolution of the host star, will allow us to explore in more
detail the dependencies of planetary migrations with respect to these
dissipations processes in the future.