The Astrophysical Journal, 928:105 (13pp), 2022 April 1

https://doi.org/10.3847/1538-4357/ac558c© 2022. The Author(s). Published by the American Astronomical Society.

Orbital Evolution of Close-in Super-Earths Driven by Atmospheric Escape1

Naho Fujita1, Yasunori Hori2,3, and Takanori Sasaki1

Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan; fujita@kusastro.kyoto-u.ac.jp
2
Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
3
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Received 2021 October 19; revised 2022 January 28; accepted 2022 February 14; published 2022 March 30

Abstract
The increasing number of super-Earths close to their host stars have revealed a scarcity of close-in small planetswith 1.5–2.0 R⊕ in the radius distribution of Kepler planets. The atmospheric escape of super-Earths byphotoevaporation can explain the origin of the observed “radius gap.” Many theoretical studies have considered thein situ mass loss of a close-in planet. Planets that undergo atmospheric escape, however, move outward due to thechange in the orbital angular momentum of their star–planet systems. In this study, we calculate the orbital
evolution of an evaporating super-Earth with a H2/He atmosphere around FGKM-type stars under stellar X-rayand extreme-UV irradiation (XUV). The rate of increase in the orbital radius of an evaporating planet isapproximately proportional to that of the atmospheric mass loss during a high stellar XUV phase. We show thatsuper-Earths with a rocky core of 10 M⊕ and a H2/He atmosphere at 0.03–0.1 au (0.01–0.03 au) aroundG-type stars (M-type stars) are prone to outward migration driven by photoevaporation. Although the changes in
the orbits of the planets would be small, they would rearrange the orbital conﬁgurations of compact, multiplanet
systems, such as the TRAPPIST-1 system. We also ﬁnd that the radius gap and the so-called “Neptune desert” inthe observed population of close-in planets around FGK-type stars still appear in our simulations. On the other
hand, the observed planet population around M-type stars can be reproduced only by a high stellar XUVluminosity model.
Uniﬁed Astronomy Thesaurus concepts: Exoplanets (498); Exoplanet evolution (491); Exoplanet formation (492);
Exoplanet atmospheres (487); Super Earths (1655)
mechanisms that cause planets to lose their atmospheres.4 Thiscan occur through (1) ion pick-up by high-energy charged
particles from a stellar wind or coronal mass ejection (e.g.,
Khodachenko et al. 2015), and (2) thermal processes in theupper layer driven by X-ray or extreme-UV (XUV) heating,e.g., the Jeans escape and hydrodynamic escape (Watson et al.
1981). For a nonmagnetized or weakly magnetized planet, likeMars, the nonthermal process strongly affects atmosphericmass loss. Although a super-Earth may have a weak and shortlived magnetic ﬁeld (Gaidos et al. 2010; Tachinami et al.
2011), the lifetime of a dipole magnetic ﬁeld can be as long as∼1 Gyr for a rapidly rotating super-Earth (Zuluaga &Cuartas 2012). The magnetic protection against the invasionof high-energy charged particles into a close-in super-Earth islikely maintained for ∼1 Gyr (Zuluaga et al. 2013). Thus, the
XUV-driven atmospheric escape is considered the dominantmechanism for sculpting the atmospheres of close-in superEarths. In fact, the atmospheric escape of super-Earths byphotoevaporation can explain the radius gap observed in the
population of Kepler planets (e.g., Owen & Wu 2017).
Atmospheric escape from a planet causes a change in its
orbital radius. A planet that loses its mass should moveoutward5 owing to the change in the orbital angularmomentum. Pioneer studies investigated the orbital evolutionof a planet that undergoes its atmospheric loss and gas drag due
to a stellar wind from an evolved star, using simpliﬁed

1. Introduction
Since the ﬁrst discovery of a hot Jupiter orbiting a Sun-likestar was reported in 1995 (Mayor & Queloz 1995), over 4500
exoplanets have been discovered. Exoplanet statistics show thatsuper-Earths account for the majority of exoplanetary systems(e.g., Mayor et al. 2011; Howard et al. 2012), especially aroundlow-mass stars (Bonﬁls et al. 2013; Dressing & Charbonneau2015). Interestingly, there is a lack of close-in small planetswith 1.5–2.0 R⊕ and orbital periods of <100 days in the radius
distribution of Kepler planets around FGK-type stars (Fulton
et al. 2017; Van Eylen et al. 2018) and M-type stars (Van Eylen
et al. 2021). This “radius gap/valley” is understood as theradius boundary between bare rocky planets (Weiss &Marcy 2014; Rogers 2015) and planets with a substantial
amount of atmosphere (at most ∼1%–10% of their total
masses; e.g., Wolfgang & Lopez 2015). The transmission
spectra of close-in super-Earths above the radius gap, such as
GJ 1214b (e.g., Kreidberg et al. 2014) and GJ 3470b (e.g.,
Benneke et al. 2019), revealed that they have either a (hazy/cloudy) hydrogen-rich atmosphere or a metal-rich atmosphere.
The hydrogen-rich atmosphere of a super-Earth can beinterpreted as the remnant of an accreted disk gas, namely,
the primordial atmosphere. A metal-rich atmosphere of a superEarth is likely to originate from post-formation processes, suchas volcanic activities and outgassing from the impact material.
Super-Earths close to a host star are in danger of atmosphericloss under an intense stellar radiation ﬁeld. There are two main

4
The silicate mantle and iron core inside a super-Earth can play a role in thehydrogen budget. The equilibrium chemistry between an interior and ahydrogen-rich atmosphere of a super-Earth can reduce the hydrogen massfraction in the atmosphere (e.g., see Schlichting & Young 2021).
5
Note that a close-in planet also experiences orbital decay by tidal torques(e.g., Jackson et al. 2016; Ginzburg & Sari 2017).

Original content from this work may be used under the termsof the Creative Commons Attribution 4.0 licence. Any further
distribution of this work must maintain attribution to the author(s) and the titleof the work, journal citation and DOI.

1

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

expressions (e.g., Livio & Soker 1984). Subsequently, Villaver& Livio (2007, 2009) studied how a gas giant survives aroundan evolved star, including the orbital change of a planet driven
by photoevaporative mass loss. These studies considered theisotropic mass loss from a planet driven by stellar XUVirradiation. Photoevaporative mass loss from an irradiatedplanet, however, may be associated with an anisotropic winddriven by the day–night heating contrast (e.g., Stone &Proga 2009; Guo 2013) and nonspherically symmetric outﬂow
controlled by magnetic ﬁelds (Adams 2011). Anisotropic
ejection by evaporation of a hot Jupiter (Boué et al. 2012)
and a super-Earth/hot Neptune (Teyssandier et al. 2015) can
induce orbital migration, depending on the geometry of themass ejection, such as the ejection direction of the material andthe orbital velocity. The mass transfer from a planet to its hoststar via Roche lobe overﬂow also causes its orbital expansion.
Trilling et al. (1998) investigated the effect of mass loss via the
Roche lobe overﬂow on the orbital evolution of a migrating gas
giant. Valsecchi et al. (2014) demonstrated the orbital
expansion of a close-in gas giant accompanied by Roche lobe
overﬂow in order to explain the observed population of small
ultrashort-period planets (Sanchis-Ojeda et al. 2014; Steffen &Coughlin 2016). Recently, Jackson et al. (2016) formulated theorbital evolution of a gas giant due to the mass loss via Roche
lobe overﬂow, incorporating the effect that part of the orbital
angular momentum of the atmosphere ﬂowing out of a planet
remains in the star–planet system. Thus, the orbital evolution ofa gas giant that undergoes mass loss has been discussed in thecontext of either stellar evolution or Roche lobe overﬂow.
Many theoretical studies on the photoevaporation of superEarths, however, assume that they stay in situ while losing theiratmospheres for ∼Gyr. Note that most of the evaporation
driven by stellar XUV irradiation occurs in the ﬁrst few
100 Myr. The strength of a stellar XUV ﬂux that a planet
receives determines its mass-loss rate. As a planet migrates
outward, a reduced stellar XUV irradiation should mitigate
mass loss from the planet; that is, the atmospheric escape froma super-Earth and its orbital evolution occur simultaneously.
Therefore, the coevolution of the orbit and the atmosphere ofan evaporating, close-in planet is crucial to better understanding the formation history and orbital evolution of superEarths under atmospheric conditions.
In this study, we calculate the orbital evolution of close-insuper-Earths that undergo atmospheric loss driven by stellar
XUV irradiation. Using a one-dimensional model for the massloss from a super-Earth, we investigate how the angularmomentum change due to atmospheric escape inﬂuences its
orbital migration. The remainder of this paper is structured asfollows. In Section 2, we describe our model of theatmospheric escape and orbital evolution of a super-Earth. In
Section 3, we demonstrate that the atmospheric escape from aclose-in super-Earth never strands its orbit and strongly affects
the ﬁnal orbital conﬁguration of the planetary systems. We also
compare the period–radius distribution of the simulated planetswith that of the observed planets. In Section 4, we discuss theimpact of the geometry of an evaporative wind, tidal torques,
and the eccentricity on the outward migration of close-in planet
systems. Finally, in Section 5, we summarize our results.

momentum of a planet due to atmospheric mass loss causes
outward migration. We consider the orbital evolution of closein super-Earths with a H2/He atmosphere onto a rocky core of
0.5–20 M⊕ around FGKM-type stars (0.2–1.4 Me) for 1 Gyrafter disk dispersal. The initial mass fraction of a H2/He
atmosphere relative to the core mass ranged from 1% to 10%.
2.1. Stellar Evolution
We considered a close-in planet with a H2/He atmospherearound FGKM-type stars. The evolution of a planet-hostingstar was simulated using the general-purpose stellar evolution
code MESA (Paxton et al. 2011) to derive the stellar intrinsic
luminosity, temperature, and radius. The time evolution of thestellar XUV luminosity was estimated from the X-ray-tobolometric luminosity relations of stars in open clusters aged
5–740 Myr,
(LXUV L bol )sat
LXUV
=⎧
-a
⎨
L bol
⎩(LXUV L bol )sat (t tsat )for t  tsat
for t > tsat ,(1 )where t is the age of the star (t = 0 is deﬁned as the time when
the star enters the pre-main-sequence phase), τsat is thesaturation turn-off time, Lbol is the stellar bolometricluminosity, and α is the power-law index of 0.790
(B - V )0 < 0.935, 0.565  (B - V )0 < 0.675, and 0.290 (B − V )0 < 0.450 for K-, G-, and F-type stars, respectively(Jackson et al. 2012). The (B − V ) color index of a star varies
with time. In this study, we calculate the orbital evolution ofclose-in super-Earths via photoevaporation after disk dispersal(t = 0.04 Gyr) until t = 1 Gyr. Since we do not consider the
pre-main-sequence phase of the host star, we adopt the (B − V )value of a main-sequence star with a given stellar mass for thestellar XUV luminosity. The upper limit values forlog (LXUV L bol ) given by Jackson et al. (2012) were used inour simulations. The XUV luminosity of M-type stars follows
that of Penz and Micela (2008), where the XUV luminosity fort < 0.6 Gyr is assumed to be at the saturation level.
2.2. Atmospheric Escape
We consider two regimes of atmospheric escape driven
by stellar irradiation (Murray-Clay et al. 2009; Owen &Jackson 2012), an X-ray-driven radiation recombinationlimited regime, and an EUV-driven energy-limited mass loss.
A high stellar XUV ﬂux fully photoionizes the hydrogen atomin the upper layer. The temperature in the X-ray heated regionis almost constant (∼10,000 K) because the radiative cooling
due to H+
3 ion and Lyα emission from the recombination ofhydrogen ions and electrons can be balanced with thephotoionization of hydrogen. The mass ﬂux of the X-ray
heated ﬂow (i.e., isothermal wind) passing through the sonicpoint is given byM p = - 4prs cs rs2,

(2 )where the subscript “s” denotes the sonic point, cs is theisothermal sound speed, and ρs is the gas density at the sonicpoint. The location of the sonic point rs is deﬁned as

2. Methods
Close-in super-Earths experience atmospheric escape driven
by stellar XUV irradiation. The change in the orbital angular2cs2 =

2

GMp
rs

-

3GM rs2
,
a3

(3 )

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki(Erkaev et al. 2007). Since the heating efﬁciency forhydrogen-rich upper atmospheres was lower than 20%(Shematovich et al. 2014; Ionov & Shematovich 2015), we
adopted η = 0.1. Ktide is the potential energy reduction factor
owing to the effect of stellar tidal forces:

where Mp is the planetary mass, Må is the mass of the host star,a is the orbital radius, and G is the gravitational constant. If thepenetration depth of stellar XUV photons in the atmosphere islocated above the sonic point, we assume that rs = RXUV,
where RXUV is the location at which the H2/He atmospherebecomes optically thick to stellar XUV photons. This situation
occurs on highly inﬂated planets. The detailed modeling of theﬂow pattern of escaping hydrogen in such a scenario is beyond
the scope of this study. At a high UV ﬂux, the effect ofradiation recombination overwhelms the advection of hydrogenions. In ionization equilibrium, if we neglect the effect of H+3
ion cooling, a balance between radiation recombination andphotoionization at the photoionization base (i.e., the UV opticaldepth is unity) approximately yields
FUV
sn 0 n 0 ~ n+2 a rc ,hn 0

Ktide (x ) = 1 -The radius of a close-in planet with a H2/He atmosphere is
calculated by its interior structure in hydrostatic equilibrium
under time-dependent stellar radiation (see Hori & Ogihara2020). The planetary temperature is given by the equilibrium
temperature: Teq = [(1 - A) L  (16psa2 )]1 4 . Teq is the equilibrium temperature of a planet at its orbital radius, Lå is thetime-dependent stellar luminosity, σ is the Stefan–Boltzmann
constant, and A is the Bond albedo of the planet. For simplicity,
A was assumed to be zero in this study. The rocky core inside aclose-in planet consists of a silicate mantle and an iron core.
The silicate mantle was described by the third-order Birch–
Murnagham equation of state (EoS) for MgSiO3 perovskite(Karki et al. 2000; Seager et al. 2007). The Vinet EoS for ò-Fe
(Anderson et al. 2001) was used for the iron material in thecore. The Thomas–Fermi–Dirac EoS (Salpeter &Zapolsky 1967) was applied to high-pressure EoS for
MgSiO3 at P 4.90 TPa and Fe at P 2.09 × 104 GPa (Seager
et al. 2007; Zeng & Sasselov 2013). The iron material in the
rocky core was 30% of the core mass. The pressure–temperature proﬁle in the H2/He envelope was calculatedusing the SCvH EoS (Saumon et al. 1995). This study assumed
that Rp(t) = RXUV for close-in super-Earths, where Rp(t) is theplanetary radius at a given time t, and RXUV is the location of
the photoionization base. We deﬁne the planetary radius as thephotosphere, which is given by Rp(t) = Rbc + Ratm, where Rbc is
the radiative–convective boundary derived from the interior
modeling, and Ratm is the photospheric correction given in
Lopez & Fortney (2014). The thermal evolution of a superEarth with a H2/He atmosphere is computed by calculating its
interior structure at each time step, followed by the mass loss
and subsequent orbital evolution. The atmospheric mass-loss
rate of a super-Earth is determined by Equation (6), using Rp(t).
If RRLO is smaller than Rp, a planet can no longer
gravitationally bind its upper atmosphere, leading to the onsetof Roche lobe overﬂow. We considered the initial H2/He
atmosphere of a planet that satisﬁes Rp < RRLO. In this study,
the transition point from the radiation recombination-limited tothe energy-limited regime was determined by calculating the
mass-loss rate of each regime and adopting the regime yielding
a smaller rate (see Appendix A).

where n0 and n+ are the number density of neutral hydrogenatoms and hydrogen ions at the base, respectively, therecombination coefﬁcient for hydrogen ions (Storey &Hummer 1995) is arc = 2.7 ´ 10-13(T 10 4 K)-0.9 cm3 s-1,
the photoionization cross section of hydrogen is sn0 = 6´
10-18 (hn0 13.6 eV)-3 cm2 (e.g., Spitzer 1978), FUV is the UV
ﬂux, and T is the temperature. The number density of neutralhydrogen atoms at the base is n 0 ~ 1 (sn0 H ) using the scaleheight H ∼ kT/(gmH/2), where k is the Boltzmann constant, g
is the gravitational acceleration, and mH is the mass of thehydrogen atom. The density at the sonic point can be related tothat at the photoionization base in the isothermal, hydrostatic
structure:(5 )where ρbase is the density at the photoionization base (RXUV).
This study assumed that an incoming UV photon (hν0) has acharacteristic energy of 20 eV, which results in T ∼ 10,000 K in
the upper layer (Murray-Clay et al. 2009).
If stellar UV ﬂux is low, a fraction of the hydrogen atoms in
the upper atmosphere are ionized by UV photons. The gas
temperature in the EUV-heated region is mainly determined by
the balance between UV heating and cooling due to adiabatic
expansion. In the EUV-driven regime, energy-limited hydrodynamic escape becomes the dominant mechanism of atmospheric loss from a planet. Three-dimensional simulations of anoutﬂow from a hot Jupiter (Tripathi et al. 2015) suggest that the
average mass-loss rates are similar to those predicted in onedimensional models. The one-dimensional hydrodynamic
mass-loss rate of a H2/He atmosphere of a planet M p is givenbyM p = - h

R p3LXUV (t )4GMp a2Ktide (RRLO R p)

,

(7 )

2.3. The Interior Structure of a Planet(4 )GMp -1
rs = r base exp ⎡ - 2 (RXUV- rs-1)⎢ cs⎣
3GM 22(rs - RXUV
) ⎤,+⎥2a 3cs2⎦

3
1
R+ 3 < 1, x = RLO .
2x
2x
Rp

2.4. Orbital Evolution of an Evaporating Planet
We modeled the orbital evolution of a planet by photoevaporation based on Jackson et al. (2016)ʼs approach for theorbital evolution of a close-in gas giant driven by Roche lobe
overﬂow. Although Jackson et al. (2016) assumed that theplanet is in a circular orbit around its host star, we consider thata star and a planet have a Keplerian circular orbit around theircommon center of mass (i.e., the barycenter of the star–planet
system). Note that the nonzero eccentricity of a planet has anegligible effect on the orbital evolution in our simulations (see(6 )where η is the heating efﬁciency due to stellar XUV irradiation,
LXUV is the stellar XUV luminosity, Rp is the planetary radius,
and the Roche lobe radius is RRLO » (Mp 3M )1 3a
3

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

momentum of the star. Thus, we simply used M  = -cMp 7
for Equation (9) to eliminate M  to obtain Equation (10). This
simpliﬁed treatment would hardly affect our results.

Section 4). We considered the isotropic mass loss from a planetin this study. The effects of the anisotropic mass loss are also
discussed in Section 4.
The orbital angular momentum of the star–planet system L isgiven by
L = M a 2W + Mp a p2W = M Mp

Ga,
M

3. Results(8 )3.1. Orbital Evolution of an Evaporating Super-Earth

Although χ may vary with time due to the geometry of theevaporative wind, χ is assumed to be constant in this study.
The evaporative wind from a planet would produce a cometary
tail or torus of a fast H I atom due to charge exchange between
the H I atom escaping from a planet and a proton in a stellar
wind (e.g., Shaikhislamov et al. 2016; Debrecht et al. 2020);
however, the detailed behavior and geometry of the escaping
atmosphere is also not considered in this study. Note that theescaping atmospheric mass of -cMp , which is conserved in thestar–planet system, does not necessarily have to accrete onto
the star; what is important for the orbital evolution is not themass accretion onto the star but whether the escaping mass isbound by the system just after it ﬂows out of the planet. If astar–planet system conserves part of the orbital angularmomentum of an escaping atmosphere, the system needs tocompensate for the angular momentum by the orbital expansionof a planet. On the other hand, the change in the stellar mass
has almost no effect on the orbital evolution because Må ? Mp,
and aå is too small to signiﬁcantly increase the orbital angular

Figure 1 demonstrates the mass loss of a close-in super-Eartharound FGKM-type stars and subsequently its orbital migration. A super-Earth initially has a rocky core of 3 M⊕
surrounded by a H2/He atmosphere of 10% relative to thecore mass. A planet continues to move outward from 0.05 au
while losing its atmosphere. Once the atmosphere of a planet iscompletely lost or the stellar XUV ﬂux decreases signiﬁcantly(typically, t  τsat), the outward migration ceases. The increasein the orbital radius of an evaporating super-Earth isdetermined by atmospheric mass loss caused by photoevaporation. The evaporative loss of a 10 wt% H2/He atmosphere froma planet results in an increase in its orbital radius byapproximately 10% (see also Equation (10)).
The mass-loss rate from a planet is controlled by thestrength of the stellar XUV ﬂux. A high stellar luminosity alsoenhanced the inﬂation of the planetary atmosphere. The more
massive the star a planet orbits around is, the more violent andrapid atmospheric escape occurs due to intense stellar
irradiation. As a planet moves outward rapidly, the incident
stellar XUV ﬂux quickly decreases as FXUV ∝ a−2. The
integrated XUV ﬂux that a planet receives for ∼1 Gyrdetermines the atmospheric mass loss from itself. As shownin the upper panels of Figure 1, planets around FG-type stars
rapidly lose their entire atmospheres, while for those aroundK-type stars, the mass-loss rate decreases signiﬁcantly after
the stellar XUV ﬂux starts to decrease at ∼0.1 Gyr. A planetaround an M-type star maintains the initial atmosphere over
∼1 Gyr because of a lower XUV ﬂux. We also show a casewhere the stellar XUV luminosity is 10 times as high as ourstandard model (Figures 1(c) and (d)) because of its large
uncertainty. In a high-XUV model, planets around K-type andFG-type stars rapidly lose their entire atmospheres. Even aplanet around an M-type star loses one third of its initialatmosphere in 1 Gyr. Hereafter, results are observed instandard XUV models.
Figure 2 shows the change rate in the atmospheric mass andthe orbital radius of super-Earths with different core masses
around a G-type (1.0 Me) and an M-type star (0.3 Me). Less
massive cores are prone to lose a H2/He atmosphere byphotoevaporation because the inﬂation effect makes their upper
atmospheres loosely bound. The existence of a massive corehelps the retention of a H2/He atmosphere. Small planets witha rocky core of 3 M⊕ around a G-type star completely lose
their atmosphere in 0.1 Gyr, regardless of the initial atmospheric mass. Massive cores with 10 M⊕ can retain theiratmospheres over 1 Gyr, especially if they have a substantial
amount of the initial atmosphere.
As a planet is initially farther away from its host star, theH2/He atmosphere is likely to survive photoevaporation for
1 Gyr. Figure 3 shows the orbital evolution of super-Earths
with different initial locations. A close-in super-Earth with
0.03 au around an M-type star avoids the complete loss of
H2/He atmosphere. The outward migration of a super-Earth6

7and the time derivative of L is given by
M p
L
M
1 a
1 M
,
=  +
+
L
M
Mp
2a
2M

(9 )

where Ω is the Keplerian angular velocity, Må is the stellar
mass, and M is the total mass of the star–planet system (i.e.,
M = Må + Mp), a (=aå + ap) is the orbital separation between
the planet and star, and aå and ap are the orbital distances of thestar and the planet from the barycenter of the system,
respectively. If L = 0 and M  = -M p , we obtain a a=-2M p (M - Mp ) (M Mp ) > 0 , which corresponds to theconservative mass transfer from a planet to its host star. On
the other hand, when the material escaping from a planet carries
away its orbital angular momentum, that is, L = Mp a p2 W,
Equation (9) yields a a = -Mp M > 0. Thus, the mass loss
from a planet via atmospheric escape causes orbital expansion(see also Alexander et al. 1976; Livio & Soker 1984).
The change in the orbital angular momentum of the systemdetermines the orbital migration of a planet. The change in theorbital angular momentum of the system depends on the
dynamics of an evaporative wind, such as the velocity of theatmospheric particles and the direction of the ﬂow. The detailed
behavior of a planetary wind launched by stellar XUVradiation, however, is beyond the scope of this study. Instead,the change rate in the angular momentum of a star–planet
system is deﬁned as L = (1 - c ) Mp a p 2W, where χ (0 χ  1) is the fraction of the angular momentum conserved in
the system.6 Substituting this into Equation (9), we obtainM - MpM pa
= - 2cM p- (1 - c )
> 0.
M Mp
M
a

(10)

Note that χ nearly corresponds to 1 − γδ in Jackson et al. (2016).

4

Jackson et al. (2016) assumed no mass accretion onto a host star (Må = 0).

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 1. Time evolution of the mass and orbital radius of a close-in super-Earth that experiences the atmospheric escape for 1 Gyr after disk dispersal. A super-Earthwith a rocky core of 3 M⊕ was initially located at 0.05 au around FGKM-type stars: 0.3 Me (solid lines), 0.7 Me (dashed lines), 1.0 Me (dotted lines), and 1.2 Me(dotted–dashed lines). The initial mass fraction of the H2/He atmosphere was 10% of the core mass. χ is assumed to be 0.5. Crosses represent the point where the H2/
He atmosphere is completely lost. The bottom panels show the results of the case where the stellar XUV luminosity is 10 times higher than that of our standard model.

around an M-type star driven by its atmospheric escape isnegligible unless it has a small core of 3 M⊕ with a massive
H2/He atmosphere at 0.03 au. The critical orbital radius atwhich a super-Earth around an M-type star loses its entire
atmosphere is ∼0.01–0.02 au. This suggests that an ultrashortperiod (USP) planet around an M-type star should have no
atmosphere unless they have either a massive core or asubstantial amount of the initial atmosphere, which isconsistent with the mass–radius relation of observed USP
planets (USPs; see Figure 20 in Hirano et al. 2021). A G-typestar blows off most of the H2/He atmosphere that a super-Earthinitially possesses. Because a close-in super-Earth with a 1 wt%(10 wt%) H2/He atmosphere at 0.03 au around a G-type starcompletely loses the initial H2/He atmosphere by photoevaporation, its orbital radius increases by 1% (10%) of its initial
location. This implies that the observed close-in super-Earths
around a G-type star might have migrated outward for ∼1 Gyrif they had a primordial atmosphere. A super-Earth with amassive core of 10 M⊕ and a massive H2/He atmosphere at>0.1 au around a G-type star can maintain almost all of theinitial atmosphere at its current location.
Figure 4 shows the orbital evolution of a 3 M⊕ and 10 M⊕
super-Earth with ﬁve different χ values (χ = 0, 0.3, 0.5, 0.7,
and 1). We found that the larger χ is, the further the planet
migrates. A star–planet system conserves part of the orbital
angular momentum of an escaping atmosphere. Because
atmospheric escape reduces the total mass of a planet, theorbital expansion of a planet compensates for the mass loss tosatisfy the conservation of angular momentum. The fractionof the leftover orbital angular momentum, which is deﬁned by
χ, regulates the orbital migration of a planet (see also
Section 2.4). Thus, a large χ results in an efﬁcient outward
migration of the planet. Two super-Earths with a given χ
value follow the same path on the atmospheric mass fractionorbital radius plane, as predicted from Equation (10).
However, the migration rate of a planet strongly depends on
its initial conditions; e.g., the core mass and orbital radius. Asa planet migrates outward, it should undergo a less violent
atmospheric escape. As a result, a different χ value affects theﬁnal atmospheric mass of a planet. The net angularmomentum loss by the atmospheric escape plays a key rolein accurately determining the orbital evolution and mass-loss
process of a close-in super-Earth.
3.2. Period–Radius Relationship of Close-in Super-Earths
We demonstrate the period–radius distribution of planets
around a Sun-like star (an FGK-type star) and an M-type star.
We simulate the orbital evolution of an evaporating super-Earthin ∼105 single-planet systems for each type of star. The massof an FGK-type star follows a Gaussian distribution with amean of 1 Me and a standard deviation of 0.15 Me. The orbital
period distribution of a planet between 1 and 100 days is givenby the Kepler planet sample, following the population synthesis
5

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 2. Time evolution of the atmospheric mass fraction and the orbital radius of super-Earths with four different core masses for 1 Gyr around an M-type star with
0.3 Me (top panels) and a G-type star with 1.0 Me (bottom ones): 1 M⊕ (black lines), 3 M⊕ (blue ones), 10 M⊕ (green ones), and 20 M⊕ (red ones). The initial massfraction of the H2/He atmosphere was 1% (solid lines) and 10% (dashed lines) of the core mass. Both the initial location of a planet and χ are the same as those in
Figure 1. Matm is the mass of the H2/He atmosphere, and ainitial is the initial location of the planet. Crosses represent the point that a H2/He atmosphere iscompletely lost.

model in Owen & Wu (2017):
constantdNµ ⎧ 1.9⎨
d log P⎩P

for P > 7.6 days,for P  7.6 days.

around a massive star are more vulnerable to photoevaporative
mass loss. Close-in super-Earths with P  3–5 days around
FGK-type stars lose almost all the H2/He atmospheres and thenmove outward, whereas mass-loss processes are less efﬁcientfor those around M-type stars. Small planets with 2 R⊕ arebare rocky cores. Detected planets with 4 R⊕ and >4 R⊕
around the FGKM-type stars are also represented by black andgray ﬁlled circles in Figure 6.8 The period–radius distributionsof simulated planets agree with observations, except for thoseof planets with P  10 days with 2–4 R⊕ around FG-type stars
in a high-XUV ﬂux model. We can see the Neptune desert in
the planet population around the FGK-type stars. Note that
there are no USPs at 1 day and large planets with 4 R⊕ in
our simulations because of the initial distributions of planets.

(11)

A close-in super-Earth has a rocky core whose mass
distribution takes the following Rayleigh distribution:
M2 ⎞dNµ Mcore exp ⎛⎜ - core,
2 ⎟dMcore⎝ 2s M ⎠(12)

where Mcore is the core mass, and σM is set to 3 M⊕. The initialatmospheric mass fraction of a planet is given by a log-uniform
distribution in the range of 1%–10% with respect to its coremass. The χ value was set to 0.5.
Figure 5 shows the orbital period distribution of planets
around G-type stars (0.8–1.0 Me) in the initial state and after
atmospheric escape for 1 Gyr. Many super-Earths having no
atmospheres appear at 30 days. Close-in super-Earths withP  10 days lose a large fraction of H2/He atmosphere andmove outward. As a result, there is a lack of close-in superEarths with radii larger than 2 R⊕ after 1 Gyr.
The ﬁnal period–radius distributions of planets around
FGKM-type stars are shown in Figure 6. We simulated theorbital evolution of planets in two XUV ﬂux models because of
uncertainties in the stellar XUV ﬂux: (left) the standard XUVmodel, and (right) a 10× standard XUV model. As stellar XUV
ﬂux increases with increasing stellar mass, close-in planets

3.3. Occurrence Rates of Close-in Super-Earths
Figure 7 shows period–radius distributions of planets around
FGK-type stars and M-type stars after atmospheric escape with
χ = 0.5 for 1 Gyr. We see the lack of planets with radii of
1.5–2.0 R⊕ for P ∼ 10 days around FGK-type stars on the
period–radius plane. As a high XUV ﬂux enhances the massloss from a planet at 10 days, small planets with 2 R⊕ aremore common. On the other hand, there is no clear radius gap
around M-type stars because of less efﬁcient mass loss from
8
The data come from NASA Exoplanet Archive (https://exoplanetarchive.
ipac.caltech.edu).

6

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 3. Same as the right panels in Figure 2, but for different initial locations: 0.03 au (blue lines), 0.05 au (green lines), and 0.1 au (red lines). The upper panels
show the orbital evolution of a 3 M⊕ (left panels) and 10 M⊕ super-Earth (right panels) around an M-type star (0.3 Me) that initially has a H2/He atmosphere of 1 wt%(solid lines) and 10 wt% (dashed lines) of the core mass, respectively. The lower panels are similar to the upper panels, but for super-Earths around a G-type star
(1.0 Me). χ is assumed to be 0.5. Crosses represent the point that a H2/He atmosphere is completely lost.

close-in planets, although a high-XUV model produces barerocky planets at 10 days. These results suggest that thepresence of a radius gap is sensitive to the intensity of thestellar XUV ﬂux.
Figure 8 shows the weighted sum of planet size distributionsin the two stellar XUV models (the standard XUV model and10× standard XUV model) with P < 100 days after 1 Gyr forχ = 0 (blue), 0.5 (green), and 1.0 (red). Gray histograms show
the completeness-corrected distribution of planets with P < 100days around FGK-type stars (Fulton & Petigura 2018) andM-type stars (Cloutier & Menou 2020). We ﬁnd that there is aradius gap near 1.6–1.8 R⊕ in the bimodal size distribution ofplanets around FGK-type stars in our simulations. The
occurrence rate ratio of planets within and beyond a radius
gap increases with χ because the outward migration of a planet
can mitigate atmospheric escape. The peak magnitudes and thelocation of a gap in the planet size distribution are sensitive tothe composition of a rocky core (see Figure 8 in Owen &Wu 2017). In contrast, we can see no clear deﬁcit of planetswith 1.5–2.0 R⊕ around M-type stars. Many planets aroundM-type stars retain their H2/He atmospheres after 1 Gyr in our
simulation, while in observation there exists a lot of small
planets which may have no atmospheres. Our results of M-typestars are not consistent with observation, which shows a radius
gap near 1.5–2.0 R⊕, because of their less intensive stellar
XUV ﬂux. This implies that a young M-type star may havehigher XUV luminosity. Indeed, we conﬁrmed that a stronger
XUV ﬂux (dashed line for the M-type star in Figure 8) canblow off the atmosphere of planets near 2 R⊕ to increase theproportion of planets with smaller radii in the size distributionof planets around M-type stars. Note that the size distributionof close-in super-Earths with a paucity in their occurrence rates
changes with their initial atmospheric mass fractions, thedistribution of their core masses, and the range of their orbital
period. Observations show that there exist a lot of small planetswith P  20 days (i.e., they had no primordial atmosphere), so
in practice, we may need to set up initial distributions of planets
which include bare rocky planets. For example, the initial
distribution following the updated core-mass function andinitial atmospheric mass function in Rogers & Owen (2021)
would alter our results, especially for the radius gap; the radius
gap can be reproduced even if we adopt a standard XUV model
because there exist small planets without even atmosphericescape.
4. Discussions
In this study, we considered the atmospheric escape from aplanet as an isotropic ejection; that is, no torque caused by massejection acts on a planet. However, a torque driven by ananisotropic evaporative wind may have a nonnegligible effecton the orbital evolution of a planet. If a planet loses its
atmosphere anisotropically, the planet can move outward more
7The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 4. Atmospheric and orbital evolution of super-Earths with rocky cores of 3 M⊕ (blue lines) and 10 M⊕ (green lines) around a G-type star (1.0 Me). Each superEarth initially has a H2/He atmosphere of 10% of its core mass at 0.075 au. Each line style corresponds to a different χ value:0 (dashed lines), 0.3 (dotted lines), 0.5(solid lines), 0.7 (dotted–dashed lines), and 1.0 (dotted–dotted–dashed lines), respectively. The open circles denote the ﬁnal locations of each planet.

Figure 5. Orbital period–planetary radius distribution of planets around G-type stars (a) in the initial state and (b) after 1 Gyr. The color contour shows theatmospheric mass fraction of the planet.

signiﬁcantly depending on the ejection direction (Boué et al.
2012; Teyssandier et al. 2015). Detailed three-dimensionalhydrodynamic simulations of atmospheric particles ﬂowing out
of a planet are essential for scrutinizing the geometry of theevaporative wind such as the ejection direction and the angularmomentum loss by the evaporative wind.
Although χ was treated as a parameter in this study, ourtheoretical prediction of planetary mass orbital distributions has
the potential to better understand the dynamics of evaporative
wind from future observations of close-in super-Earths. Moresuper-Earths around M-type stars are expected to be discoveredby ongoing high-precision Doppler surveys, such as the
Canada-France-Hawaii Telescope (CFHT)/SPectropolarimètre
InfraROUge (SPIRou), Calar Alto high-Resolution search for
M dwarfs with Exoearths with Near-infrared and optical

Echelle Spectrographs (CARMENES), Subaru/InfraRed Doppler(IRD), Very Large Telescope (VLT)/Echelle SPectrographfor Rocky Exoplanets and Stable Spectroscopic Observations(ESPRESSO), Gemini/M dwarf Advanced Radial velocity
Observer Of Neighboring eXoplanets (MAROON-X), andHobby–Eberly Telescope (HET)/Habitable-zone Planet Finder(HPF). Observed planet populations around FGKM-type stars
help to retrieve information about the atmospheric escape andorbital evolution of close-in planets.
The mass loss from a planet increases the orbital radius by
1%–10% of its initial location. The orbital expansion of aplanet in a compact, multiplanet system, such as the
TRAPPIST-1 and Kepler-186 systems, likely rearranges theorbital conﬁgurations of planets. Furthermore, the outward
migration of a planet in a nonresonant system may trigger the8The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 6. Period–radius distribution of the planets around F-type stars (1.1–1.4 Me), G-type stars (0.8–1.0 Me), K-type stars (0.6–0.7 Me), and M-type stars(0.2–0.5 Me) after atmospheric escape with χ = 0.5 for 1 Gyr. The detected planets with 4 R⊕ and > 4 R⊕ are plotted as black and gray ﬁlled circles, respectively.
The left panels show the results of our standard XUV model, and the right panels show results in the case where a stellar XUV ﬂux is 10 times as strong as the standard
model.

et al. (2015) suggests that outward migration driven byanisotropic evaporative wind would have pushed the systems
deeper inside resonances. Thus, the orbital evolution by themass loss of close-in planets due to the photoevaporation
would alter the fraction of resonant systems expected by thepresent simulation. The detailed orbital evolution of multiple

trapping of outer planets into mean-motion resonance. In
addition, the mass-loss processes of close-in planets in a
multiplanet system may cause dynamical instability. Planetarysystems are apt to undergo orbital instability with increasing
multiplicity and mass loss of planets and their host star(Matsumoto & Ogihara 2020). On the other hand, Teyssandier
9The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 7. Period–radius distribution of planets around FGK-type stars (Top: 0.6–1.4 Me) and M-type stars (Bottom: 0.2–0.5 Me) after atmospheric escape for 1 Gyr.
The left panels show the results of our standard XUV model, and the right panels show those in the 10 times XUV ﬂux model. The color contour represents the relative
occurrence rate of the planets. The mass of an FGK-type star follows a Gaussian distribution with a mean of 1 Me and a standard deviation of 0.15 Me.

Figure 8. Final size distribution of planets with P < 100 days around FGK-type stars and M-type stars. Gray histograms show the completeness-corrected distributionof planets with P < 100 days around FGK-type stars (Fulton & Petigura 2018) and M-type stars (Cloutier & Menou 2020). Each color histogram represents theweighted sum of the planet size distributions after 1 Gyr in the two stellar XUV models for χ = 0 (blue), 0.5 (green), and 1.0 (red). For the M-type star, the higher
XUV case (weighted sum of 10× and 100× standard XUV) is also shown by the dashed line.

10The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

planets due to the mass loss will be an important future work.
The tidal interaction between a close-in planet and its hoststar causes an orbital decay. As the orbit of a planet expands
because of its atmospheric mass loss, the strength of tidal
interactions decreases quickly as ttide (=a p ap ) µ a p13 2 (seeAppendix B for more details). For a close-in super-Eartharound a main-sequence star, the mass loss should be
experienced during a high stellar XUV ﬂux (t  0.1 Gyr).
The tidal damping of a planet’s orbit occurs at >1 Gyr. As aresult, the inward movement of an evaporating super-Earth due
to tidal interactions has a negligible effect on its orbital
expansion due to mass loss. In this study, we neglect the tidal
effects on the orbital evolution of a close-in super-Earth.
Following Villaver & Livio (2009), we simulated the orbital
evolution of a close-in super-Earth that undergoes mass loss,
including tidal effects, and conﬁrmed the validity of ourassumption (see also Appendix B).
This study considered a close-in planet in a circular orbit.
The nonzero eccentricity of a planet causes time variations instellar XUV irradiation, which are synchronized with orbital
motion. The tidal circularization timescale for a close-in superEarth is >1 Gyr (see Equation (B4) in the Appendix). As
shown in Figures 1–3, a planet loses its H2/He atmosphere
during a high-XUV phase (t  0.1 Gyr). An initially eccentric
super-Earth may have nonzero eccentricity while losing its
atmosphere. An eccentric planet receives the stellar ﬂux
averaged over the entire orbit of áFXUVñ = FXUV 1 - e 2 .
For e  0.1, as predicted by the formation of close-in superEarths via giant impacts (e.g., Matsumoto & Kokubo 2017), theaverage stellar ﬂux is greater by only <0.5%, compared to aplanet in a circular orbit. The eccentric orbit of a planet decays
rapidly owing to the tidal interactions. The eccentricity of aplanet damps faster than its orbital radius if 0 < e = 1 (see also
Equation (B4)). Consequently, the orbit of an eccentric planet
decays in a similar fashion to a planet in a circular orbit.
Therefore, the nonzero eccentricity of a planet has a negligible
effect on mass-loss-driven orbital evolution in our simulations.

most of their atmospheres in 1 Gyr, they should have no
atmospheres. On the other hand, USPs around an M-type star
have the possibility of retaining a little atmosphere because of aless intensive stellar XUV ﬂux in case that a planet has either amassive core or a substantial amount of the initial atmosphere.
The observed close-in super-Earths around FGKM-type stars
should have experienced orbital migration for ∼1 Gyr if they
had a primordial atmosphere. The atmospheric escape of superEarths by photoevaporation can affect the orbital conﬁgurationof compact, multiplanet systems, such as the TRAPPIST-1 andKepler-186 systems. The orbital separations between eachplanet in these systems are ∼10% to their planetary orbital
radius, while a planet with an initial atmosphere of 10 wt%
increases its orbital radius by approximately 10% via atmospheric escape. Planets can be kicked out or pushed into meanmotion resonances.
We performed population synthesis studies on the orbital
evolution of an evaporating super-Earth around FGKM-typestars. Close-in planets with P  3–5 days around FGK-typestars lose almost all of the H2/He atmospheres and then moveoutward, which is consistent with the “Neptune desert” in theobserved planet population. We found that there is a radius gap
near 1.6–1.8 R⊕, as seen in observations, in the size distributionof planets around FGK-type stars after the orbital migration via
atmospheric escape occurs. This supports the hypothesis thatthe radius gap reﬂects the bimodal distribution of the amount of
atmosphere, which is caused by atmospheric escape (e.g.,
Owen & Wu 2017). The period–radius distribution of close-insuper-Earths is highly sensitive to a stellar XUV luminosity,
the composition of a rocky core, the initial atmospheric massfraction, and the range of the orbital period. The observed
planet population around M-type stars can be reproduced only
by a high stellar XUV luminosity model. The detection of more
super-Earths around M-type stars by high-precision nearinfrared Doppler surveys, and the Transiting Exoplanet Survey
Satellite mission will enable us to verify our photoevaporation
model.
We gratefully acknowledge the referee’s helpful comments.
We thank Teruyuki Hirano for his valuable advice on theselection of observational data. We also thank Ryan Cloutier
for providing us with useful data on planet occurrence rates
around low-mass stars. This work was supported by JSPS
KAKENHI grant No. 19K03950. Y.H. was supported by aGrant-in-Aid for Scientiﬁc Research on Innovative Areas (JSPS
KAKENHI grant No. 18H05439).

5. Summary
We examined the effect of atmospheric escape driven bystellar X-ray and UV irradiation on the orbital evolution ofclose-in super-Earths around FGKM-type stars. This study was
the ﬁrst comprehensive attempt to evaluate the orbital
migration of close-in super-Earths via hydrodynamic escape.
The orbital expansion of a close-in planet compensates for themass loss to satisfy the conservation of the orbital angularmomentum in the system. The less angular momentum theescaping atmospheric material carries away from the system,
the further the planet migrates. The rate of increase in theorbital radius of an evaporating planet is approximately
proportional to that of the atmospheric mass loss during ahigh stellar XUV phase. The more massive the star a planet
orbits around is, the more violent and rapid atmospheric escapeoccurs due to intense stellar irradiation. Less massive cores are
more vulnerable to photoevaporative mass loss because theinﬂation effect loosely binds their upper atmospheres. Small
planets around massive stars are likely to experience mass-lossdriven orbital migration.
Super-Earths with a rocky core of 10 M⊕ and a H2/He
atmosphere at 0.03–0.1 au (0.01–0.03 au) around G-typestars (M-type stars) are prone to the outward migration driven
by photoevaporation. Since USPs around an FGK-type star lose

Appendix A
Transition of Atmospheric Escape Regime
Two mechanisms of atmospheric escape from a planet
depend on the core mass and H2/He atmospheric mass. The
pressure, density, and velocity of the X-ray heated ﬂow change
steeply in the subsonic region from the photoionization base(i.e., the planetary radius) to the sonic surface. The propertiesof the sonic point for the evaporative wind in the radiation
recombination-limited regime are sensitive to the planetary
mass and radius. The transition from the radiation recombination-limited to the energy-limited regime occurs when thephotoionization base, which corresponds to RXUV, is equal to
the sonic point. If a planetary core mass is sufﬁciently small(20 M⊕), rs can become smaller than the core radius Rcore. We
adopt the energy-limited regime if rs < Rcore because the11The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 9. Time evolution of (a) mass and (b) orbital radius of a 20 M⊕ super-Earth. The ﬁlled circles represent the transition point from radiation recombinationlimited to the energy-limited regime. The planets initially have a H2/He atmosphere of 10% of their core mass at different initial locations: 0.014 (black lines), 0.016(blue lines), 0.018 (green lines), and 0.020 au (red lines) around a G-type star. Crosses represent the point that a H2/He atmosphere is completely lost. χ is assumed tobe 0.5. Planets closer to their host stars undergo the transition from the radiation recombination-limited to the energy-limited regime in the later stage because of ahigher XUV ﬂux.

where ta,  º (a a )tide,  . This indicates that a close-in superEarth around a Sun-like star can be locked into a synchronous
rotation by tidal forces in >1 Gyr. The orbit of a super-Eartharound an M-type star decays more slowly because a low-massstar has a fully convective interior and a small radius.
Tidal dissipation in a planet also damps orbit andeccentricity. The orbital decay rate is given byradiation recombination-limited regime is not expected tooccur.
Figure 9 demonstrates that a 20 M⊕ planet undergoes thetransition from the radiation recombination-limited to theenergy-limited regime for mass loss. For planets closer to their
host stars, the transition from the radiation recombinationlimited to the energy-limited regime occurs in the later stage
because of a higher ﬂux of stellar XUV photons. They lose alarge fraction of the H2/He atmosphere in the radiation
recombination-limited regime. The radiation recombination
processes should be effective in the upper atmosphere of closein planets, especially, close-in massive planets.

2e2 ⎛ e ⎞2a⎛ a ⎞==E ,2
GM Mp1 - e ⎝ e ⎠tide,p⎝ a ⎠tide,pwhere e is the eccentricity, and E is the energy loss due to tidal
dissipation given by E ~ -(63e 2Wp 4Qp m˜ p )(Rp a)5GM2 awhen e = 1 (e.g., Goldreich & Soter 1966), where Rp is theplanetary radius, Qp is the tidal dissipation factor of the planet,and m˜ p is the bulk modulus of the planet. Note that the tidal
circularization of a planet occurs prior to orbital decay when0 < e = 1. From above the equations, we yield

Appendix B
Tidal Evolution of a Super-Earth
In a star–planet system, the orbital change of a planet ismainly caused by the tidal deformation of its host star and thetidal dissipation itself. The orbital decay rate of a planet by tidal
interactions (Villaver & Livio 2009) can be written asf MenvR⎛ a ⎞=
q (1 + q ) ⎛  ⎞ ,td M
⎝ a ⎠tide, 
⎝ a ⎠

ta,p

8

(B1)

.

(B2)

The dissipation factor is f = (P 2td )2 when τd >P/2;
otherwise, we take f = 1, where P is the orbital period of theplanet. Equation (B1) gives us
f ⎞-1⎛ td ⎞ ⎛ Menv M ⎞-1ta,  ~ 3 ´ 1013⎛⎜⎟⎝ 0.01 ⎠ ⎝ 10 days ⎠ ⎝ 0.025 ⎠-8

q -1 Ra ⎞8´ ⎛ -5 ⎞ ⎛⎜  ⎞⎟ ⎛yr,
⎝ 10 ⎠ ⎝ R ⎠ ⎝ 0.03 au ⎠Qp ⎞ ⎛ k 2,p ⎞

-5

13

-1

2
⎛ q ⎞ ⎛⎜ M ⎞⎟5⎝ 100 ⎠ ⎝ 0.3 ⎠ ⎝ 10 ⎠ ⎝ M ⎠(B5)

where ta,p º (a a )tide,p , and the Love number is k 2,p=
3 1
. The tidal decay of a close-in super-Earth around2 1 + m˜ p
FGKM-type stars due to tidal dissipation has a negligible effecton the orbital evolution for 1 Gyr.
Figure 10 demonstrates the orbital evolution of a 10 M⊕
super-Earth around a G-type star due to mass loss and tidal
interactions. The size and heat ﬂux in the stellar convective
envelope vary with time. The turnover timescale in theconvective zone was calculated using the stellar evolution
code MESA (Paxton et al. 2011). A planet initially has a 1 wt%
and a 10 wt% H2/He atmosphere at 0.03 and 0.05 au,
respectively, in a circular orbit. The orbital evolution of aplanet in the absence of tidal interactions is also shown in
Figure 10. We can see that tidal forces hardly change the orbital
evolution of a planet for 1 Gyr. The orbital decay of a planet
due to tidal forces had little effect on our simulations.

1 3

Menv (R  - R env )2 ⎤
⎥
3L 
⎦

~ 2 ´ 1011⎛

Rp
a ⎞ 2 ⎛ e ⎞-2yr,´ ⎛⎜ ⎞⎟ ⎛⎝ RÅ ⎠ ⎝ 0.03 au ⎠ ⎝ 0.01 ⎠where q = Mp/Må, Mp is the mass of a planet, a is thesemimajor axis of the planet, Må and Rå are the mass and radiusof a star, Lå is the stellar luminosity, Menv is the mass enclosed
in the convective envelope of a star, τd is the eddy turnovertimescale (Rasio et al. 1996) given by
td = ⎡
⎢
⎣

(B4)

(B3)

12

The Astrophysical Journal, 928:105 (13pp), 2022 April 1

Fujita, Hori, & Sasaki

Figure 10. Time evolution of the atmospheric mass fraction and the orbital radius of super-Earths without and with tidal effects (thin solid and thick dashed lines). A
10 M⊕ super-Earth around a G-type star initially has a 1 wt% and 10 wt% H2/He atmosphere at 0.03 au (orange and red) and 0.05 au (cyan and blue). Crosses
represent the point that a H2/He atmosphere is completely lost. χ is assumed to be 0.5.

ORCID iDs

Mayor, M., & Queloz, D. 1995, Natur, 378, 355
Murray-Clay, R. A., Chiang, E. I., & Murray, N. 2009, ApJ, 693, 23
Owen, J. E., & Jackson, A. P. 2012, MNRAS, 425, 2931
Owen, J. E., & Wu, Y. 2017, ApJ, 847, 29
Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3Penz, T., & Micela, G. 2008, A&A, 479, 579
Rasio, F. A., Tout, C. A., Lubow, S. H., & Livio, M. 1996, ApJ, 470,
1187
Rogers, J. G., & Owen, J. E. 2021, MNRAS, 503, 1526
Rogers, L. A. 2015, ApJ, 801, 41
Salpeter, E. E., & Zapolsky, H. S. 1967, PhRv, 158, 876
Sanchis-Ojeda, R., Rappaport, S., Winn, J. N., et al. 2014, ApJ, 787, 47
Saumon, D., Chabrier, G., & van Horn, H. M. 1995, ApJS, 99, 713
Schlichting, H. E., & Young, E. D. 2021, arXiv:2107.10405
Seager, S., Kuchner, M., Hier-Majumder, C. A., & Militzer, B. 2007, ApJ,
669, 1279
Shaikhislamov, I. F., Khodachenko, M. L., Lammer, H., et al. 2016, ApJ,
832, 173
Shematovich, V. I., Ionov, D. E., & Lammer, H. 2014, A&A, 571, A94
Spitzer, L. 1978, Physical Processes in the Interstellar Medium (New York:
Wiley)
Steffen, J. H., & Coughlin, J. L. 2016, PNAS, 113, 12023
Stone, J. M., & Proga, D. 2009, ApJ, 694, 205
Storey, P. J., & Hummer, D. G. 1995, MNRAS, 272, 41
Tachinami, C., Senshu, H., & Ida, S. 2011, ApJ, 726, 70
Teyssandier, J., Owen, J. E., Adams, F. C., & Quillen, A. C. 2015, MNRAS,
452, 1743
Trilling, D. E., Benz, W., Guillot, T., et al. 1998, ApJ, 500, 428
Tripathi, A., Kratter, K. M., Murray-Clay, R. A., & Krumholz, M. R. 2015,
ApJ, 808, 173
Valsecchi, F., Rasio, F. A., & Steffen, J. H. 2014, ApJL, 793, L3
Van Eylen, V., Agentoft, C., Lundkvist, M. S., et al. 2018, MNRAS, 479,
4786
Van Eylen, V., Astudillo-Defru, N., Bonﬁls, X., et al. 2021, MNRAS,
507, 2154
Villaver, E., & Livio, M. 2007, ApJ, 661, 1192
Villaver, E., & Livio, M. 2009, ApJL, 705, L81
Watson, A. J., Donahue, T. M., & Walker, J. C. G. 1981, Icar, 48, 150
Weiss, L. M., & Marcy, G. W. 2014, ApJL, 783, L6
Wolfgang, A., & Lopez, E. 2015, ApJ, 806, 183
Zeng, L., & Sasselov, D. 2013, PASP, 125, 227
Zuluaga, J. I., Bustamante, S., Cuartas, P. A., & Hoyos, J. H. 2013, ApJ,
770, 23
Zuluaga, J. I., & Cuartas, P. A. 2012, Icar, 217, 88

Naho Fujita https://orcid.org/0000-0002-5791-970X
Yasunori Hori https://orcid.org/0000-0003-4676-0251
Takanori Sasaki https://orcid.org/0000-0003-1242-7290
References
Adams, F. C. 2011, ApJ, 730, 27
Alexander, M. E., Chau, W. Y., & Henriksen, R. N. 1976, ApJ, 204, 879
Anderson, O. L., Dubrovinsky, L., Saxena, S. K., & LeBihan, T. 2001, GeoRL,
28, 399
Benneke, B., Knutson, H. A., Lothringer, J., et al. 2019, NatAs, 3, 813
Bonﬁls, X., Delfosse, X., Udry, S., et al. 2013, A&A, 549, A109
Boué, G., Figueira, P., Correia, A. C. M., & Santos, N. C. 2012, A&A, 537, L3
Cloutier, R., & Menou, K. 2020, AJ, 159, 211
Debrecht, A., Carroll-Nellenback, J., Frank, A., et al. 2020, MNRAS,
493, 1292
Dressing, C. D., & Charbonneau, D. 2015, ApJ, 807, 45
Erkaev, N. V., Kulikov, Y. N., Lammer, H., et al. 2007, A&A, 472, 329
Fulton, B. J., & Petigura, E. A. 2018, AJ, 156, 264
Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109
Gaidos, E., Conrad, C. P., Manga, M., & Hernlund, J. 2010, ApJ, 718, 596
Ginzburg, S., & Sari, R. 2017, MNRAS, 469, 278
Goldreich, P., & Soter, S. 1966, Icar, 5, 375
Guo, J. H. 2013, ApJ, 766, 102
Hirano, T., Livingston, J. H., Fukui, A., et al. 2021, AJ, 162, 161
Hori, Y., & Ogihara, M. 2020, ApJ, 889, 77
Howard, A. W., Marcy, G. W., Bryson, S. T., et al. 2012, ApJS, 201, 15
Ionov, D. E., & Shematovich, V. I. 2015, SoSyR, 49, 339
Jackson, A. P., Davis, T. A., & Wheatley, P. J. 2012, MNRAS, 422, 2024
Jackson, B., Jensen, E., Peacock, S., Arras, P., & Penev, K. 2016, CeMDA,
126, 227
Karki, B. B., Wentzcovitch, R. M., de Gironcoli, S., & Baroni, S. 2000,
PhRvB, 62, 14750
Khodachenko, M. L., Shaikhislamov, I. F., Lammer, H., & Prokopov, P. A.
2015, ApJ, 813, 50
Kreidberg, L., Bean, J. L., Désert, J.-M., et al. 2014, Natur, 505, 69
Livio, M., & Soker, N. 1984, MNRAS, 208, 763
Lopez, E. D., & Fortney, J. J. 2014, ApJ, 792, 1
Matsumoto, Y., & Kokubo, E. 2017, AJ, 154, 27
Matsumoto, Y., & Ogihara, M. 2020, ApJ, 893, 43
Mayor, M., Marmier, M., Lovis, C., et al. 2011, arXiv:1109.2497

13