Habitable Evaporated Cores: Transforming Mini-Neptunes into Super-Earths in the Habitable Zones of M Dwarfs

1.5 MB·English

by  Rodrigo Luger

Draft version January 28, 2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
HABITABLE EVAPORATED CORES:
TRANSFORMING MINI-NEPTUNES INTO SUPER-EARTHS
IN THE HABITABLE ZONES OF M DWARFS
R. Luger1,2, R. Barnes1,2, E. Lopez3, J. Fortney3, B. Jackson4, and V. Meadows1,2
1AstronomyDepartment,UniversityofWashington,Box351580,Seattle,WA98195,USA;[email protected]
2VirtualPlanetaryLaboratory,Seattle,WA98195,USA
3DepartmentofAstronomyandAstrophysics,UniversityofCalifornia,SantaCruz,California
4CarnegieInstituteofWashington,Washington,DC
Draft version January 28, 2015
5
ABSTRACT
1
We show that photoevaporation of small gaseous exoplanets (“mini-Neptunes”) in the habitable
0
2 zones of M dwarfs can remove several Earth masses of hydrogen and helium from these planets and
transformthemintopotentiallyhabitableworlds. WecoupleX-ray/extremeultraviolet(XUV)-driven
n
escape, thermal evolution, tidal evolution and orbital migration to explore the types of systems that
a
may harbor such “habitable evaporated cores” (HECs). We find that HECs are most likely to form
J
from planets with ∼ 1M⊕ solid cores with up to about 50% H/He by mass, though whether or not
6
a given mini-Neptune forms a HEC is highly dependent on the early XUV evolution of the host star.
2
As terrestrial planet formation around M dwarfs by accumulation of local material is likely to form

planets that are small and dry, evaporation of small migrating mini-Neptunes could be one of the
]
P dominant formation mechanisms for volatile-rich Earths around these stars.
E
h. 1. INTRODUCTION surrounding gaseous disk, which induces rapid inward
p Duetotheirsmallradiiandlowluminosities,Mdwarfs migrationforplanetsintheterrestrialmassrange(Ward
- currently offer the best opportunity for the detection of 1997). Around solar-type stars, disk dispersal typically
o occurs on the order of a few Myr (Walter et al. 1988;
terrestrial planets in the habitable zone (HZ), the region
r Strom et al. 1989). While there is evidence that disk
t around a star where liquid water can exist on the sur-
s lifetimesmayexceed∼5Myrforlowmassstars(Carpen-
face of a planet (Kasting et al. 1993; Kopparapu et al.
a
ter et al. 2006; Pascucci et al. 2009), disks are no longer
[ 2013). Upcoming missions such as the Transiting Exo-
planet Survey Satellite (TESS) and the repurposed Ke- presentafter∼10-20Myraroundallstellartypes(Ribas
1 pler spacecraft (K2) will be capable of detecting po- et al. 2014). Planets migrating via interactions with the
v diskwillthussettleintotheirneworbitsrelativelyearly.
tentially habitable Earths and super-Earths around M
2 Planet-planetscattering,ontheotherhand,isbynature
dwarfs(Rickeretal.2010;Howelletal.2014). Inparticu-
7 astochasticprocessandmaytakeplaceatanypointdur-
lar,thedetectionofpotentiallyhabitableplanetsaround
5 ing a system’s lifetime. However, such interactions are
M dwarfs is easier because the habitable zones of these
6 farmorefrequentduringthefinalstagesofplanetassem-
stars can be as close in as ∼ 0.02 AU (Kopparapu et al.
0 bly (up to a few tens of Myr), after which planets relax
2013). However, such proximity implies that terrestrial
1. planets forming within the HZs of low mass stars are into their long-term quasi-stable orbits (Ford & Rasio
0 likely to be small ((cid:46) 0.3M ) and form dry (Raymond 2008).
5 et al. 2007; Lissauer 2007)⊕. Moreover, M dwarfs are Given the abundance of ices beyond the snow line,
1 extremely active when young, bombarding their plan- planets that migrate into the HZ from the outer regions
: of the disk should have abundant water, therefore satis-
v ets with high energy radiation and bursts of relativis-
fyingthatimportantcriterionforhabitabilitywithinthe
i tic particles during flaring events, which can erode their
X HZ. However, the situation is complicated by the fact
atmospheres and potentially sterilize the surface (Scalo
that these planets may have accumulated thick gaseous
r etal.2007;Seguraetal.2010). Strongtidalheatingand
a envelopes, which could render them uninhabitable. In-
orbital evolution may further impact the habitability of
vestigatingwhethertheseso-called“mini-Neptunes”can
planets around these stars (Barnes et al. 2013). Many
lose their envelopes and form planets with solid surfaces
planetsformedin situ intheHZofMdwarfsmaythere-
is therefore critical to understanding the habitability of
fore be uninhabitable.
planets around low mass stars.
However,planetsneednotformandremaininplace. It
In this study we focus on mini-Neptunes with initial
is now commonly accepted that both disk-driven migra-
masses in the range 1M ≤ M ≤ 10M with up to
tion and planet-planet interactions can lead to substan- ⊕ p ⊕
50% hydrogen/helium by mass that have migrated into
tial orbital changes, potentially bringing planets from
the HZs of mid- to late M dwarf stars. We investigate
outsidethesnowline(theregionofthediskbeyondwhich
whether it is possible for atmospheric escape processes
water and other volatiles condense into ices, facilitating
to remove the thick H/He envelopes of mini-Neptunes in
the formation of massive planetary cores) to within the
theHZ,effectivelyturningthemintovolatile-richEarths
HZ (Hayashi et al. 1985; Ida & Lin 2008a,b; Ogihara
and super-Earths (terrestrial planets more massive than
& Ida 2009). Disk-driven migration relies on the ex-
Earth),whichwerefertoas“habitableevaporatedcores”
change of angular momentum between a planet and the
2 Luger et al. 2015
(HECs). We consider two atmospheric loss processes: §6. We present auxiliary derivations and calculations in
XUV-driven escape, in which stellar X-ray/extreme ul- the Appendix.
traviolet (XUV) photons heat the atmosphere and drive
a hydrodynamic wind away from the planet, and a sim- 2. STELLARANDPLANETARYEVOLUTION
ple model of Roche lobe overflow (RLO), in which the
Inthefollowingsectionswereviewtheluminosityevo-
atmosphere is so extended that part of it lies exterior to
lution of low mass stars (§2.1), the habitable zone and
the planet’s Roche lobe; that gas is therefore no longer
its evolution (§2.2), atmospheric escape processes from
gravitationally bound to the planet. We further couple
planets(§2.3), andtidalevolutionofstar-planetsystems
the effects of atmospheric mass loss to the thermal and
(§2.4).
tidalevolutionoftheseplanets. Planetscoolastheyage,
undergoingchangesofuptoanorderofmagnitudeinra-
dius, which can greatly affect the mass loss rate. Tidal 2.1. Stellar Luminosity
forcesarisingfromthedifferentialstrengthofgravitydis- Followingtheirformationfromagiantmolecularcloud,
tort both the planet and the star away from sphericity, stars contract under their own gravity until they reach
introducing torques that lead to the evolution of the or- the main sequence, at which point the core temperature
bital and spin parameters of both bodies. is high enough to ignite hydrogen fusion. While the Sun
While many studies have considered the separate ef- is thought to have spent (cid:46) 50 Myr in this pre-main
fects of atmospheric escape (Erkaev et al. 2007; Murray- sequence (PMS) phase (Baraffe et al. 1998), M dwarfs
Clayetal.2009;Tian2009;Owen&Jackson2012;Lam- take hundreds of Myr to fully contract; the lowest mass
meretal.2013),Rochelobeoverflow(Trillingetal.1998; M dwarfs reach the main sequence only after ∼ 1 Gyr
Gu et al. 2003), thermal evolution (Lopez et al. 2012), (e.g.,Reid&Hawley2005). Duringtheircontraction,M
and tidal evolution (Jackson et al. 2008; Ferraz-Mello dwarfs remain at a roughly constant effective tempera-
et al. 2008; Correia & Laskar 2011) on exoplanets, none ture (Hayashi 1961), so that their luminosity is primar-
have considered the coupling of these effects in the HZ. ily a function of their surface area, which is significantly
For some systems, in particular those that may harbor larger than when they arrive on the main sequence. M
HECs, modelingthecouplingoftheseprocessesisessen- dwarfs therefore remain super-luminous for several hun-
tial to accurately determine the evolution, since several dred Myr, with total (bolometric) luminosities higher
feedbacks can ensue. Tidal forces in the HZ typically thanthemainsequencevaluebyuptotwoordersofmag-
act to decrease a planet’s semi-major axis, leading to nitude. As we discuss below, this will significantly affect
higher stellar fluxes and faster mass loss. The mass loss, theatmosphericevolutionofanyplanetsthesestarsmay
in turn, affects the rate of tidal evolution primarily via host.
thechangingplanetradius,whichisalsogovernedbythe XUV emissions (1˚A(cid:46)λ(cid:46)1000˚A) from M dwarfs also
coolingrateoftheenvelope. Changestothestar’sradius vary strongly with time. This is because the XUV lu-
and luminosity lead to further couplings that need to be minosity of M dwarfs is rooted in the vigorous magnetic
treated with care. fields generated in their large convection zones (Scalo
Jackson et al. (2010) considered the effect of the cou- etal.2007). Stellarmagneticfieldsarelargelycontrolled
pling between mass loss and tidal evolution on the hot by rotation (Parker 1955), which slows down with time
super-Earth CoRoT-7 b, finding that the two effects are due to angular momentum loss (Skumanich 1972), lead-
strongly linked and must be considered simultaneously ing to an XUV activity that declines with stellar age.
to obtain an accurate understanding of the planet’s evo- However, due to the difficulty of accurately determining
lution. However, an analogous study has not been per- both the XUV luminosities (usually inferred from line
formedintheHZ,ingreatpartbecausebothtidaleffects proxies) and the ages (often determined kinematically)
andatmosphericmasslossaregenerallyordersofmagni- of M dwarfs, the exact functional form of the evolution
tude weaker at such distances from the star. This is not isstillveryuncertain(forareview,seeScaloetal.2007).
necessarily true around M dwarfs, for two reasons: (a) Further complications arise due to the fact that the pro-
their low luminosities result in a HZ that is much closer cess(es) that generate magnetic fields in late M dwarfs
in, exposing planets to strong tidal effects and possible may be quite different from those in earlier type stars.
RLO; and (b) M dwarfs are extremely active early on, TherotationaldynamoofParker(1955)involvestheam-
so that the XUV flux in the HZ can be orders of magni- plification of toroidal fields generated at the radiative-
tude higher than that around a solar-type star (see, for convective boundary; late M dwarfs, however, are fully
instance, Scalo et al. 2007). convective, and have no such boundary. Instead, their
In this paper we present the results of the first model magnetic fields may be formed by turbulent dynamos
tocoupletides,thermalevolution,andatmosphericmass (Durney et al. 1993), which may evolve differently in
loss in the habitable zone, showing that for certain sys- time from those around higher mass stars (Reid & Haw-
tems the coupling is key in determining the long-term ley 2005).
evolution of the planet. We demonstrate that it is pos- In a comprehensive study of the XUV emissions of
sibleto turn mini-NeptunesintoHECs within thehabit- solar-type stars of different ages, Ribas et al. (2005)
ablezone,providinganimportantpathwaytotheforma- foundthattheXUVevolutioniswellmodeledbyasimple
tionofpotentiallyhabitable,volatile-richplanetsaround power law with an initialshort-lived“saturation” phase:
M dwarfs.
phIynsi§c2s.weInpr§o3vwideedaedscertiabileedoudresmcroipdteilo,nfoolflotwheedreblyevoaunrt LXUV =(cid:16)LLXbUoVl (cid:17)sat t≤tsat (1)
(cid:16) (cid:17) (cid:16) (cid:17)−β
creosrureltsspoinnd§4in.gWceavtehaetnsdinisc§u5s,sfoolulorwmeadinbyfinadisnugmsmanadrythine Lbol  LLXbUoVl sat tstat t>tsat,
Habitable Evaporated Cores 3
whereL andL aretheXUVandbolometriclumi- 0.8
XUV bol
nosities, respectively, and β = −1.23. Prior to t = t , 0.7 HZ @ 10 Myr
sat
the XUV luminosity is said to be “saturated,” as obser- 0.6 HZ @ 1 Gyr
vationsshowthattheratioLXUV/Lbolremainsrelatively 0.5
constant at early times.
0.4
Jackson et al. (2012) find that t ≈ 100Myr and
sat )
(L /L ) ≈10−3 for K dwarfs. Similar studies for fl
XUV bol sat M 0.3
M dwarfs, however, are still being developed (e.g., En-
(
gle & Guinan 2011), but it is likely that the saturation
timescale is much longer for late-type M dwarfs. Wright M 0.2
et al. (2011) show that X-ray emission in low mass stars
issaturatedforP /τ (cid:46)0.1,whereP isthestellarro-
rot rot
tationperiodandτ istheconvectiveturnovertime. The
extent of the convective zone increases with decreasing
0.1
stellar mass; below 0.35M , M dwarfs are fully convec-
(cid:12)
tive (Chabrier & Baraffe 1997), resulting in larger val-
0.01 0.10 1.00
ues of τ (see, e.g., Pizzolato et al. 2000). Low mass a(AU)
stars also have longer spin-down times (Stauffer et al.
1994); together, these effects should lead to significantly Fig. 1.—Locationoftheinnerhabitablezone(red),centralhab-
longer saturation times compared to solar-type stars. itable zone (green) and outer habitable zone (blue) as a function
ofstellarmassat10Myr(dashed)and1Gyr(solid). After1Gyr,
This is consistent with observational studies; West et al.
theevolutionoftheHZisnegligibleforMdwarfs.
(2008) find that the magnetic activity lifetime increases
from (cid:46) 1Gyr for early (i.e., most massive) M dwarfs
to (cid:38) 7Gyr for late (least massive) M dwarfs, possibly tion of the HZ is not fixed. In Figure 1 we plot the HZ
due to the onset of full convection around spectral type at 10 Myr (dashed lines) and at 1 Gyr (solid lines), cal-
M3. Finally, Stelzer et al. (2013) find that for M dwarfs, culated from the HZ model of Kopparapu et al. (2013)
β = −1.10±0.02 in the X-ray and β = −0.79±0.05 in and the stellar evolution models of Baraffe et al. (1998).
theFUV(farultraviolet),suggestingaslightlyshallower WhileforKandGdwarfsthechangeintheHZisnegligi-
slopeintheXUVcomparedtothevaluefromRibasetal. ble,theHZofMdwarfsmovesinsubstantially,changing
(2005). by nearly an order of magnitude for the least massive
stars. Due to this evolution, planets observed in the HZ
2.2. The Habitable Zone
of M dwarfs today likely spent a significant amount of
The habitable zone (HZ) is classically defined as the time interior to the inner edge of the HZ, provided they
region around a star where an Earth-like planet can sus- either formed in situ or migrated to their current posi-
tainliquidwateronitssurface(Hart1979;Kastingetal. tions relatively early. Luger & Barnes (2015) explore in
1993). InteriortotheHZ,highsurfacetemperatureslead detailtheeffectsoftheevolutionoftheHZonterrestrial
to the runaway evaporation of a planet’s oceans, which planets.
increases the atmospheric infrared opacity and reduces Finally, we note that the location of the HZ is also a
the ability of the surface to cool in a process known as functionoftheeccentricitye. Thisisduetothefactthat
therunaway greenhouse. ExteriortotheHZ,greenhouse at a fixed semi-major axis a, the orbit-averaged flux (cid:104)F(cid:105)
gases—gases, like water vapor, that absorb strongly in is higher for more eccentric orbits (Williams & Pollard
the infrared—are unable to maintain surface tempera- 2002):
tures above the freezing point, and the oceans freeze
globally. Recently,Kopparapuetal.(2013)re-calculated
L
the location of the HZ boundaries as a function of stel- (cid:104)F(cid:105)= √bol . (2)
lar luminosity and effective temperature using an up- 4πa2 1−e2
dated one-dimensional, radiative-convective, cloud-free
2.3. Atmospheric Mass Loss
climate model. Their five boundaries are the (1) Recent
Venus, (2) Runaway Greenhouse, (3) Moist Greenhouse, Planetary atmospheres constantly evolve. Several
(4)MaximumGreenhouse,and(5)EarlyMarshabitable mechanisms exist through which planets can lose signifi-
zones. cant quantities of their atmospheres to space, leading to
Thefirstandlastlimitscanbeconsidered“optimistic” dramaticchangesincompositionandinsomecasescom-
empiricallimits,sincepriorto∼1and∼3.8Gyrago,re- plete atmospheric erosion. The early Earth itself could
spectively, Venus and Mars may have had liquid surface have been rich in hydrogen, with mixing ratios as high
water. The ability of a planet to maintain liquid wa- as 30% in the prebiotic atmosphere (Tian et al. 2005;
ter and to sustain life at these limits is still unclear and Hashimoto et al. 2007). A variety of processes subse-
probably depends on a host of properties of its climate. quently led to the loss of most of this hydrogen; Watson
Conversely, the other three limits are the “pessimistic” etal.(1981)arguethatontheorderof1023gofhydrogen
theoretical limits, corresponding to where a cloud-free, could have escaped in the first billion years. Similarly,
Earth-like planet would lose its entire water inventory Kasting & Pollack (1983) calculated that early Venus
due to the greenhouse effect (2 and 3) and to where the couldhavelostanEarthoceanequivalentofwaterinthe
addition of CO to the atmosphere would be incapable same amount of time. Currently, observational evidence
2
of sustaining surface temperatures above freezing (4). foratmosphericescapeexistsfortwo“hotJupiters,”HD
Because stellar luminosities vary with time, the loca- 209458b (Vidal-Madjar et al. 2003) and HD 189733b
4 Luger et al. 2015
(Lecavelier Des Etangs et al. 2010), and one “hot Nep- mal), (4) is no longer valid, and the escape rate must be
tune,” GJ 436b (Kulow et al. 2014), whose proximity to calculated from hydrodynamic models.
their parent stars leads to the rapid hydrodynamic loss
of hydrogen. 2.3.2. Hydrodynamic Escape
Atmospheric escape mechanisms fall into two major
One of the primary mechanisms for heating the exo-
categories: thermal escape, in which the heating of
sphere and decreasing λ is via strong XUV irradiation.
the upper atmosphere accelerates the gas to velocities J
XUV photons are absorbed close to the base of the ther-
exceeding the escape velocity, and nonthermal escape,
mosphere, where they deposit energy and heat the gas
which encompasses a variety of mechanisms and may
via the ionization of atomic hydrogen. This heating is
involve energy exchange among ions or erosion due to
balanced by the adiabatic expansion of the upper atmo-
stellar winds. While nonthermal processes certainly do
sphere, downward heat conduction, and cooling by re-
play a role in the evaporation of super-Earth and mini-
combinationradiation. ForsufficientlyhighXUVfluxes,
Neptune atmospheres, the high exospheric temperatures
the expansion of the atmosphere accelerates the gas to
resulting from strong XUV irradiation probably make
supersonic speeds, at which point a hydrodynamic wind
thermal escape the dominant mechanism for planets
is established similar to the solar Parker wind (Parker
around M dwarfs at early times. However, the escape
1965). Oncethegasreachestheexosphere,itwillescape
canbegreatlyenhancedbyflaresandcoronalmassejec-
theplanetprovideditskineticenergyexceedstheenergy
tions, which can completely erode the atmosphere of a
required to lift it out of the planet’s gravitational well.
planet lacking a strong magnetic field (Lammer et al.
Since the kinetic energy of a hydrogen atom at the
2007; Scalo et al. 2007). For a review of the nonthermal
exobase is 3kT , the condition λ < 1.5 implies that
mechanisms of escape, the reader is referred to Hunten 2 exo J
the kinetic energy of the gas is greater than the absolute
(1982).
value of its gravitational binding energy, and it should
therefore begin to escape in bulk in a process commonly
2.3.1. Jeans Escape
referred to as “blow-off.” Unlike in the Jeans escape
In the low temperature limit, atmospheric mass loss regime, where the mass loss occurs on a per-particle ba-
occurs via the ballistic escape of individual atoms from sis, blow-off leads to the rapid loss of large portions of
thecollisionlessexosphere,wherethelowdensitiesensure theupperatmosphere, irrespectiveofparticlespecies, as
that atoms with outward velocities exceeding the escape atoms and molecules heavier than hydrogen are carried
velocity will escape the planet. Originally developed by along by the hydrodynamic wind. However, contrary to
Jeans (1925), who assumed a hydrostatic, isothermal at- what O¨pik (1963) suggests, the mass loss in this stage is
mosphere, the mass loss rate of a pure hydrogen atmo-
not arbitrarily high, since once blow-off begins the up-
sphere is given by (O¨pik 1963) per atmosphere can no longer be treated as isothermal.
As the exosphere expands it also cools, so that in the
dMdtp =4πRe2xonmHvt(1+2λ√Jπ)e−λJ (3) ainbcsreenacsee,otfhaenreebnyemrgoydseorautricnegtthheevballouwe-oofff.λJThweilml taesnsdlotsos
is,inthissense,“energy-limited,”andmaybecalculated
where R is the radius of the exobase, n is the number
exo by equating the energy input to the energy required to
density of hydrogen atoms at the exobase, m is the
H drive the escape.
mass of a hydrogen atom, v is the thermal velocity of
t Originally proposed by Watson et al. (1981), the
the gas, and λ is the Jeans escape parameter, defined
J energy-limited model assumes that the XUV flux is ab-
as the ratio of the potential energy to the kinetic energy
sorbed in a thin layer at radius R where the optical
of the gas and given by XUV
depthtostellarXUVphotonsisunity. Recentlyupdated
GM m toincludetidaleffectsbyErkaevetal.(2007),thismodel
λJ ≡ kT pRH , (4) approximates the mass loss as
exo exo
where G is the gravitational constant, Mp is the mass of dMp ≈ (cid:15)XUVπFXUVRpRX2UV (5)
the planet, and Texo is the temperature in the (isother- dt GMpKtide(ξ)
mal) exosphere. Since in the Jeans regime the thermal
velocity of the gas is less than the escape velocity, the where (cid:15) is the heating efficiency parameter (see be-
XUV
bulk of the gas remains bound to the planet, and only low),F istheincidentXUVflux,R istheplanetary
XUV p
atoms in the tail of the Maxwell-Boltzmann distribution radius,andK isatidalenhancementfactor,account-
tide
are able to escape. Jeans escape is thus slow. As an ing for the fact that for sufficiently close-in planets, the
example, the present Jeans escape flux for hydrogen on stellar gravity reduces the gravitational binding energy
Venus is on the order of 10 cm−2s−1 (Lammer et al. of the gas such that it need only reach the Roche radius
2006), corresponding to the feeble rate of ∼ 10−4 g/s, to escape the planet. Erkaev et al. (2007) show that
which would remove only one part in 1011 of the atmo-
(cid:18) (cid:19)
sphere per billion years. 3 1
K (ξ)= 1− + , (6)
For higher exospheric temperatures and/or larger val- tide 2ξ 2ξ3
ues of R , however, corresponding to low values of λ ,
exo J
the atmosphere enters a regime in which the velocity of where the parameter ξ is defined as
theaverageatomintheexosphereapproachestheescape
R
velocity of the planet. In this regime, the bulk of the ξ ≡ Roche (7)
upper atmosphere ceases to be hydrostatic (and isother- RXUV
Habitable Evaporated Cores 5
with rates lower than those due to a hydrodynamic flow, but
significantly higher than those predicted by the hydro-
R ≡(cid:18) Mp (cid:19)1/3a, (8) static Jeans equation (3).
Roche 3M
(cid:63)
2.3.4. Jeans Escape or Hydrodynamic Escape?
whereM isthemassofthestarandaisthesemi-major
(cid:63)
axis. For simplicity, as in Lopez et al. (2012), we re- Since the location of the habitable zone is governed
placeR withR in(5),whichisapproximatelyvalid primarily by the total (bolometric) flux incident on a
p XUV
giventhatR istypicallyonly10-20%largerthanR planet, the higher ratio of L to L of M dwarfs
XUV p XUV bol
(Murray-Clay et al. 2009; Lopez et al. 2012). implies a much larger XUV flux in the HZ compared to
Since the input XUV energy is balanced in part by solar-type stars. The present-day solar XUV luminosity
cooling radiation (via Lyman α emission in the case is L /L ≈ 3.4×10−6 (see Table 4 in Ribas et al.
XUV bol
of hydrogen) and heat conduction, only a fraction of 2005),whileforactiveMdwarfsthisratiois∼10−3(e.g.,
it goes into the adiabatic expansion that drives escape. Scalo et al. 2007). Therefore we should expect planets
Rather than running complex hydrodynamic and radia- in the HZ of M dwarfs to experience XUV fluxes sev-
tive transfer models to determine the net heating rate, eral orders of magnitude greater than the present Earth
manyauthors(Jacksonetal.2010;Leitzingeretal.2011; level (F ≈ 4.64erg/s/cm2). Recent papers (Lam-
XUV⊕
Lopez et al. 2012; Koskinen et al. 2012; Lammer et al. mer et al. 2007, 2013; Erkaev et al. 2013) show that
2013) choose to fold the balance between XUV heating terrestrial planets experiencing XUV fluxes correspond-
aansdthceoforlainctgioinntoofatnheeffiinccioemncinygpXarUaVmeetneerr,g(cid:15)yXUthVa,tdiseficnoend- iflnogwtroeg1i0m×e,aanndd1w0e0×maFyXtUhVus⊕eaxrpeecinttthheesahmyderofodrysnuapmeric-
verted into PdV work. Because of the complex depen- Earths/mini-Neptunes in the HZ of active M dwarfs.
denceof(cid:15)XUV ontheatmosphericcompositionandstruc- In Figure 2 we plot the evolution of the XUV flux re-
ture, its value is still poorly constrained. Lopez et al. ceived by a planet located close to the inner edge of the
(2012) estimate (cid:15)XUV = 0.2±0.1 for super-Earths and HZ(definedat5Gyr),forthreedifferentMdwarfmasses
mini-Neptunesbasedonvaluesfoundthroughoutthelit- and two different XUV saturation times (see §2.1). The
erature. Earlier work by Chassefi`ere (1996) estimates dashed lines correspond to the critical fluxes in Erkaev
(cid:15)XUV (cid:46)0.30forVenus-likeplanetsbuttheauthorargues et al. (2013) above which hydrodynamic escape occurs,
that the actual value may be closer to 0.15. Recently, for 1 and 10 M and two values of (cid:15) . Earth-mass
Owen & Jackson (2012) found X-ray efficiencies (cid:46) 0.1 planets remain ⊕in the hydrodynamic XesUcVape regime for
for planets more massive than Neptune, but higher effi- at least 1 Gyr in all cases and in excess of 10 Gyr for
ciencies (∼ 0.15) for terrestrial planets. Moreover, She- active M dwarfs. The duration of this regime is shorter
matovichetal.(2014)arguethatstudiesthatassumeef- for 10 M planets, but still on the order of several 100
⊕
ficiencieshigherthanabout0.2probablyleadtooveresti- Myr.
matesintheescaperate. Ontheotherhand,somestud- Wenotealsothattidaleffectscansignificantlyincrease
ies suggest higher heating efficiencies: Koskinen et al. thecriticalvalueofλ belowwhichhydrodynamicescape
J
(2012) use hydrodynamic and photochemical models of occurs (see discussion in Erkaev et al. 2007). Hydrody-
the hot Jupiter HD209458b to calculate (cid:15)XUV =0.44. namic escape ensues when the thermal energy of the gas
As we have already implied, unlike Jeans escape, hy- exceeds its potential energy, which occurs when
drodynamic blow-off is fast. Chassefi`ere (1996) calcu-
lates the maximum hydrodynamic escape rate from the 1.5≥ GMpKtidemH
early Venusian atmosphere to be ∼ 106 g/s, ten orders kT R
exo exo
of magnitude higher than the present Jeans escape flux
or
(see § 2.3.1). Although there has been debate over the
validityoftheblow-offassumption(see,forinstance,the 1.5
λ ≤
discussion in Tian et al. 2008), recently Lammer et al. J K
(2013)showedthatforsuper-Earthsexposedtohighlev- tide
≡λ (9)
elsofXUVirradiation,theenergy-limitedapproximation crit
yields mass loss rates that are consistent with hydrody-
providedwemaintaintheoriginaldefinitionoftheJeans
namic models to within a factor of about two, which is
parameter (4). Due to the strong tidal forces acting on
within the uncertainties of the problem.
the planets we consider here, this effect should greatly
increase the critical value of the escape parameter, ef-
2.3.3. Controlled Hydrodynamic Escape
fectively reducing the value of F for which hydrody-
XUV
It is also worth noting that there may be an interme- namic escape occurs.
diate regime between Jeans escape and blow-off known
as“modifiedJeansescape”or“controlledhydrodynamic 2.3.5. Energy-Limited or
escape” (Erkaev et al. 2013). In this regime, which oc- Radiation/Recombination-Limited?
curs for intermediate XUV fluxes and/or higher plan-
Hydrodynamic escape from planetary atmospheres
etary surface gravity, blow-off conditions are not met
need not be energy-limited. In the limit of high extreme
buttheatmospherestillexpands,sothatthehydrostatic
ultraviolet (EUV) flux (low-energy XUV photons with
Jeans formalism is not valid. In order to calculate the
100˚A (cid:46) λ (cid:46) 1000˚A), Murray-Clay et al. (2009) showed
escape rate, one must replace the classical Maxwellian
that the escape is “radiation/recombination-limited,”
velocity distribution with one that includes the bulk ex-
pansion velocity of the atmosphere. This yields escape scaling roughly as M˙ ∝ (F )1/2. In this regime, the
EUV
6 Luger et al. 2015
5
0.1M , 0.1 Gyr
fl
0.2M , 0.1 Gyr
fl
0.3M , 0.1 Gyr
4 fl
0.1M , 1.0 Gyr
⊕ 0.2Mfl, 1.0 Gyr
V fl
U 0.3M , 1.0 Gyr
X3 fl
F
/
V
U
X
F 10M †=0.15
2 ⊕
g 10M †=0.40
o ⊕
L
Fl
1 1M⊕ †=0.15 ux at
1M †=0.40 E
⊕ art
h
0
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
Log Time (years)
Fig. 2.—EvolutionoftheXUVfluxreceivedbyplanetsclosetotheinneredgeoftheHZ(at1Gyr)forstarsofmass0.1,0.2,and0.3
M(cid:12). SolidlinescorrespondtoanXUVsaturationtimeof0.1Gyr;dashedlinescorrespondto1Gyr. ThefluxatEarthisindicatedbythe
blackline. ThedotcorrespondstotheearliesttimeforwhichRibasetal.(2005)hasdataforsolar-typestars;thisisalsoroughlythetime
atwhichEarthformed. AnXUVluminositysaturatedat10−3L isroughlyindicatedbythedottedline. Finally,thedashedgraylines
bol
indicatetheminimumXUVfluxesrequiredtosustainblowoffaccordingtothestudyofErkaevetal.(2013). Super-EarthsintheHZsof
Mdwarfsremainintheblowoffregimeforatleastafew100Myr; Earthsundergoblowoffformuchlonger. ForanXUVsaturationtime
of1Gyr,blowoffoccursforseveralGyrforallplanets.
upper atmosphere thermostats to T ∼ 104 K, photoion- where the quantity (1+x) is the radius of the ioniza-
ization is balanced by radiative recombination (as op- tion front in units of R , f is the Mach number of
p parker
posed to PdV work), and a large fraction of the gas en- theflow,Φ isthestellarEUVluminosityinphotonsper
(cid:63)
ergy budget is lost to Lyman α cooling. The mass loss second,Aisageometricalfactorandc istheisother-
EUV
rate is found by solving the mass continuity equation, mal sound speed of the gas. Taking x = 0, f = 1,
parker
yielding A=1/3, c =10 km/s, and an average EUV photon
EUV
energy hν =20 eV, this becomes
M˙ =4πr2ρ c (10)
s s s
(cid:18) F (cid:19)1/2(cid:18)R (cid:19)3/2
wwhinedreverlsociistytheequraaldsiutsheoflotchael ssoounnicd pspoeinetd (cws)hearnedthρes M˙RR ≈7.11×107 g/s erg/EcUmV2/s R⊕p .
is the density at r . Since the bulk of the flow is ion- (13)
s
ized,thedensityisfixedbyionization-recombinationbal-
ance, scaling roughly as (F r )1/2. For a 0.7M hot The transition from energy-limited to radiation/
EUV s J
recombination-limited escape is found by equating the
Jupiter, the radiation/recombination limited mass loss
two expressions, namely (5) and (13), and solving for
rate is (Murray-Clay et al. 2009)
the critical EUV flux. For hot Jupiters and mini-
(cid:18) F (cid:19)1/2 Neptunesalike, thetransitionoccursatroughlyFEUV ∼
M˙ ≈4×1012 g/s EUV . (11) 104erg/cm2/s. Below this flux, the escape is energy-
RR 5×105 erg/cm2/s
limited; above it, the escape is radiation/recombination-
limited and thus increases more slowly with the flux.
Owen & Jackson (2012) re-derive this expression with
Mini-Neptunes that migrate early into the HZ of M
explicit scalings on several planet properties:
dwarfsareexposedtoEUVfluxesuptoanorderofmag-
(cid:18) Φ (cid:19)1/2 nitudelargerthanthiscriticalvalue. Duringthisperiod,
M˙ ≈2.4×1011 g/s (1+x)3/2f (cid:63) which lasts on the orderof a few hundred Myr, the mass
RR parker 1040 s−1 loss rate may be radiation/recombination-limited.
(cid:16) a (cid:17)(cid:18) R (cid:19)3/2(cid:18) A (cid:19)1/2(cid:18) c (cid:19) However, whether the high-flux mass loss rate is more
× p EUV , accurately described by (5) or (13) will depend on
0.1 AU 10R 1/3 10 km/s
⊕ whether the flux is dominated by X-ray or EUV radi-
(12)
ation. Owen & Jackson (2012) show that the mass loss
Habitable Evaporated Cores 7
rateforX-raydrivenhydrodynamicwindsscaleslinearly significantly over a single orbit. To account for this, we
with the X-ray flux; this is because the sonic point for may calculate the time-averaged mass loss rate over the
X-ray flows tends to occur below the ionization front. course of one orbit, (cid:104)M˙ (cid:105) , such that the total amount of
t
Even though recombination radiation still removes en- mass lost in time ∆t is ∆M = (cid:104)M˙ (cid:105) ∆t. To this end, in
ergy from the flow, it does so once the gas is already t
the Appendix we derive the eccentric version of K :
supersonic and thus causally decoupled from the planet, tide
such that it cannot bottleneck the escape. Although the
authors caution that the dependence of the mass loss 1 1 (cid:90) 2π(cid:20) 3 1 (cid:21)−1
≡ (1−ecosE)− + dE
rate on planet mass and radius must be determined nu- K 2π 2ξ 2ξ3(1−ecosE)2
merically, this regime is analogous to the energy-limited ecc 0
(15)
regime and can be roughly approximated by (5).
For X-ray luminosities L (cid:38) 10−6L , close-in plan- where ξ is the time-independent parameter given by (7)
X (cid:12)
ets undergo X-ray driven hydrodynamic escape (see Fig- and (8) and E is the eccentric anomaly. The average
ure 11 in Owen & Jackson 2012). If X-rays contribute mass loss rate is then simply
significantly to the XUV emissions of young M dwarfs,
their X-ray luminosities may exceed this value for as (cid:104)M˙ (cid:105) = M˙0 , (16)
long as 1 Gyr, and close-in mini-Neptunes will undergo t K
ecc
energy-limited escape. Unfortunately, given the lack of
where
observational constraints on the X-ray/EUV luminosi-
tiesofyoungMdwarfs,itisunclearatthispointwhether R3 (cid:15) L
M˙ ≡ XUV XUV XUV (17)
thehydrodynamicescapewillbeEUV-driven(radiation/ 0 4GM a2
recombination-limited)orX-ray-driven(energy-limited). p
is the zero-eccentricity mass loss rate in the absence of
2.3.6. The Effect of Eccentricity
tidal e√nhancement. Note that the flux-enhancement fac-
Mostoftheformalismthathasbeendevelopedtoana- tor 1/ 1−e2 is already folded into K , since it must
ecc
lytically treat hydrodynamic blow-off only considers cir- be incorporated when integrating (A7).
cular orbits. For planets on sufficiently eccentric orbits, For ξ (cid:38) 10, the integral may be approximated by the
neitherthestellarfluxnorthetidaleffectsmaybetreated analytic expression
as constant over the course of an orbit. Due to this fact,
(cid:115)
an expression analogous to (5) for eccentric orbits seems 3 9
K ≈ 1− − −e2, (18)
to be lacking in the literature. In this section we derive ecc ξ 4ξ2
such an expression. 0 0
There are two separate effects that enhance the mass which greatly reduces computing time.
loss for planets on eccentric orbits. Most papers account As we show in the Appendix, the decreased Roche
for the first effect, which is the incre√ase of the orbit- lobe distance for eccentric orbits has a large effect on
averaged stellar flux by a factor of 1/ 1−e2 (see, for the amount of mass lost, particularly for low values of
instance, Kopparapu et al. 2013). However, for e (cid:46) 0.3, ξ and for high e (see Figure 13). Moreover, higher ec-
this effect is quite small. The second, more important centricities result in Roche lobe overflow at larger values
effect is that the Roche lobe radius is no longer constant of a compared to the circular case, since the planet may
over the course of an orbit, and (8) is not valid. Instead, overflow near pericenter, leading to mass loss rates po-
wemustreplaceawiththeinstantaneousplanet-starsep- tentiallyordersofmagnitudehigher. WediscussRLOin
aration r(t): detail in the Model Description section (§3.3).
R (t)=(cid:18) Mp (cid:19)1/3r(t). (14) 2.4. Tidal Theory
Roche 3M(cid:63) Thefinalaspectofplanetaryevolutionwediscussisthe
effect of tidal interactions with the host star. Below we
One might wonder whether this replacement is valid.
review two different approaches to analytically calculate
Specifically, if R (t) changes faster than the atmo-
Roche
the orbital evolution of the system.
sphere is able to respond to the changes in the gravita-
tional potential, we would expect that the time depen-
2.4.1. Constant Phase Lag
denceofthemasslossratewouldbeacomplicatedfunc-
tion of the tides generated in the atmosphere. On the Classicaltidaltheorypredictsthattorquesarisingfrom
other hand, if the orbital period is very large compared interactions between tidal deformations on a planet and
tothedynamicaltimescaleoftheplanet,theatmosphere itshoststarleadtothesecularevolutionoftheorbitand
willhavesufficienttimetoassumetheequilibriumshape the spin of both bodies. In this paper we employ the
dictated by the new potential. This limit is known as “equilibrium tide” model of Darwin (1880), which ap-
thequasi-static approximation (Sepinskyetal.2007). In proximates the tidal potential as a superposition of Leg-
the Appendix, we show that all the planets in our runs endre polynomials representing waves propagating along
with eccentricities e (cid:46) 0.4 are in the quasi-static regime the surfaces of the bodies; these add up to give the fa-
and that (14) is therefore valid. In some runs, we allow miliar tidal “bulge.” Because of viscous forces in the
the eccentricity to increase beyond 0.4. We discuss the bodies’ interiors, the tidal bulges do not instantaneously
implications of this in §5.9. align with the line connecting the two bodies. Instead,
Since R =R (t), ξ, K , and dM/dt (Equa- theNth waveontheith bodywilllagorleadbyanangle
Roche Roche tide
tions 5, 6 and 7) are now also functions of t, varying ε , assumed to be constant in the constant phase lag
N,i
8 Luger et al. 2015
(CPL) model. In general, different waves may have dif-
TABLE 1
ferentlagangles, anditisunclearhowtheε varyasa
N,i Free Parameters and Their Ranges
function of frequency. A common approach (see Ferraz-
Mello et al. 2008) is to assume that the magnitudes of
Parameter Range Default Notes
the lag angles are equal (see Goldreich & Soter 1966),
while their signs may change depending on the orbital M(cid:63)(M(cid:12)) 0.08−0.4 - Late-midMD
and rotational frequencies involved. This allows us to Mp(M⊕) 1−10 - -
RXUV(Rp) 1.0−1.2 1.2 See§2.3.2
introduce the tidal quality factor a IHZ-OHZ - See§3.1
e 0.0-0.95 - -
1
Q ≡ , (19) P0,(cid:63) (days) 1.0−100 30.0 Initialrot. per.
i ε0,i fH 10−6−0.5 - Hmassfraction
(cid:15)XUV 0.1−0.4 0.3 -
which in turn allows us to express the lags (in radians) ξmin 1+10−5−3 3 See§3.3
as Atmos. esc. R/R-Lim/E-Lim - See§3.3
Tidalmodel CPL/CTL CTL -
ε =± 1 . (20) Q(cid:63) 105−106 105 CPLonly
N,i Qi Qp 101−105 104 CPLonly
τ(cid:63) (s) 10−2−10−1 10−1 CTLonly
The parameter Qi is a measure of the dissipation within τp (s) 10−3−103 10−1 CTLonly
the ith body; it is inversely proportional to the amount β 0.7−1.23 1.23 SeeEq. (1)
of orbital and rotational energy lost to heat per cycle, in tsat (Gyr) 0.1−1.0 1.0 XUVsat. time
analogy with a damped-driven harmonic oscillator. The t0 (Myr) 10.0−100.0 10.0 Integrationstart
meritofthisapproachisthatthetidalresponseofabody tstop (Gyr) 0.01−5.0 5.0 Integrationend
canbecapturedinasingleparameter. Planetswithhigh
values of Q have smaller phase lags, dissipate less en-
p thatitisderivedtoeighthorderine(versussecondorder
ergy and undergo slower orbital evolution; planets with
in the CPL model).
low values of Q have larger phase lags, higher dissipa-
p The tidal quality factors Q do not enter the CTL cal-
tion rates, and therefore faster evolution. Measurements i
culationsatanypoint; instead,thedissipationischarac-
in the solar system constrain the value of Q for ter-
p terized by the time lags τ . Although there is no general
restrial bodies in the range 10-500, while gas giants are i
conversionbetweenQ andτ ,Leconteetal.(2010)show
consistentwithQ ∼104−105 (Goldreich&Soter1966). i i
p that provided annual tides dominate the evolution,
Values of Q for the Sun and other main sequence stars
(cid:63)
are poorly constrained but are likely to be (cid:38) 105−106 1
τ ≈ , (21)
(Schlaufman et al. 2010; Penev et al. 2012). Intuitively, i nQ
i
this makes sense, given that the dissipation due to inter-
nal friction in rocky bodies should be much higher than wherenisthemeanmotion(ortheorbitalfrequency)of
that in bodies dominated by gaseous atmospheres. One the secondary body (in this case, the planet).
shouldbearinmind,however,thattheexactdependence For a planet with Q = 104 in the center of the HZ
p
of Q on the properties of a body’s interior is likely to of a late M dwarf, τ ≈ 10 s; rocky planets with lower
i p
be extremely complicated. Given the dearth of data on Q mayhavevaluesontheorderofhundredsofseconds.
p
the composition and internal structure of exoplanets, it Since τ ∝ n−1, close-in planets should have much lower
is at this point impossible to infer precise values of Qp timelags. Forreference,Leconteetal.(2010)arguethat
for these planets. hot Jupiters should have 2×10−3s(cid:46)τ (cid:46)2×10−2s.
p
By calculating the forces and torques due to the tides The tidal evolution expressions are reproduced in the
raised on both the planet and the star, one can arrive at Appendix. For a more detailed review of tidal theory,
the secular expressions for the evolution of the planet’s thereaderisreferredtoFerraz-Melloetal.(2008),Heller
orbital parameters, which are given by a set of coupled etal.(2011),andtheAppendicesinBarnesetal.(2013).
nonlinear differential equations; these are reproduced in
the Appendix. 3. MODELDESCRIPTION
2.4.2. Constant Time Lag Ourmodelevolvesplanet-starsystemsforwardintime
inordertodeterminewhetherHECscanformfrommini-
UnliketheCPLmodel,whichassumesthephaselagof
Neptunes that have migrated into the HZs of M dwarfs.
the tidal bulge is constant, the constant time lag (CTL)
We perform our calculations on a grid of varying plane-
model assumes that it is the time interval between the
tary,orbital,andstellarpropertiesinordertodetermine
bulge and the passage of the perturbing body that is
the types of systems that may harbor HECs. The com-
constant. Originally proposed by Alexander (1973) and
plete list is provided in Table 1, where we indicate the
updatedbyLeconteetal.(2010), thismodelallowsfora
ranges of values we consider as well as the default values
continuumoftidalwavefrequenciesandthereforeavoids
adopted in the plots in §4 (unless otherwise indicated).
unphysical discontinuities present in the CPL model.
Integrations are performed from t = t (the time at
However, implicit in the CTL theory is the assumption 0
which the planet is assumed to have migrated into the
that the lag angles are directly proportional to the driv-
HZ)tot=t (thecurrentageofthesystem)usingthe
ing frequency (Greenberg 2009), which is likely also an stop
adaptive timestepping method described in Appendix E
oversimplification. We note, however, that in the low
of Barnes et al. (2013).
eccentricity limit, both the CPL and the CTL models
arrive at qualitatively similar results. At higher eccen-
3.1. Stellar Model
tricities, the CTL model is probably better suited, given
Habitable Evaporated Cores 9
We use the evolutionary tracks of Baraffe et al. (1998)
for solar metallicity to calculate L and T as a func- 100 1M
bol eff
tion of time. We then use (1) to calculate L , given ⊕
XUV
(L /L ) =10−3 and values of t and β given in
XUV bol sat sat
Table 1.
10
Using L and T , we calculate the location of the
bol eff
HZfromtheequationsgiveninKopparapuetal.(2013),
adding the eccentricity correction (2). Given the uncer-
taintyintheactualHZboundariesandtheirdependence 1
on a host of properties of a planet’s climate, we choose
our inner edge (IHZ) to be the average of the Recent
VenusandtheRunawayGreenhouselimitsandourouter ) 100 2M
edge (OHZ) to be the average of the Maximum Green- ⊕ ⊕
R
houseandtheEarlyMarslimits. Throughoutthispaper
(
we will also refer to the center of the HZ (CHZ), which s
we take to be the average of the IHZ and OHZ. Since u 10
i
we are concerned with the formation of ultimately hab- d
a
itable planets, we take the locations of the IHZ, CHZ, R
and OHZ to be their values at 1 Gyr, at which point the
1
stellar luminosity becomes roughly constant.
3.2. Planet Radius Model 100 5M f = 0.01
H
To determine the planetary radius Rp as a function ⊕ fH = 0.10
of the core mass Mc, the envelope mass fraction fH ≡ fH = 0.25
Me/Mp, and the planet age, we use the planet struc- fH = 0.50
ture model described in Lopez et al. (2012) and Lopez 10
& Fortney (2014), which is an extension of the model of
Fortney et al. (2007) to low-mass low-density (LMLD)
planets. These models perform full thermal evolution
calculations of the interior as a function of time. In our 1
107 108 109 1010
runs, the core is taken to be Earth-like, with a mixture
t (yr)
of 2/3 silicate rock and 1/3 iron, and the envelope is
modeled as a H/He adiabat. A grid of values of R
is then computed in the range 1M ≤ M ≤ 10M p, Fig. 3.— Evolution of the radius as a function of time due to
⊕ c ⊕ thermalcontractionoftheenvelope,intheabsenceoftidaleffects
10−6 ≤ f ≤ 0.5, and 107years ≤ t ≤ 1010years. For
H and atmospheric mass loss. From top to bottom, the plots corre-
valuesbetweengridpoints,weperformasimpletrilinear spond to planets with initial total masses (core + envelope) of 1,
interpolation. For gas-rich planets, Rp is the 20 mbar 2, and 5 M⊕. Line styles correspond to different initial hydrogen
radius; for gas-free planets, it corresponds to the surface mass fractions: 1% (red, solid), 10% (green, dashed), 25% (blue,
dot-dashed), and 50% (black, dotted). For comparison, the grey
radius. The evolution of R with age due solely to ther-
p shadedregionsinthebottomtwoplotsarethespreadinradiical-
mal contraction is plotted in Figure 3 for a few different culated by Mordasini et al. (2012b) for fH (cid:46)0.20. See text for a
core masses and values of f . discussion.
H
We note that the models of Fortney et al. (2007) and
Lopez et al. (2012) are in general agreement with those
of Mordasini et al. (2012a,b) and, by extension, Rogers value of f at a given mass; instead, we allow it to vary
H
etal.(2011). Mordasinietal.(2012a)presentedavalida- in the range 10−6 ≤ f ≤ 0.5 for all planet masses. At
H
tion of their model against that of Fortney et al. (2007), masses (cid:46) 5M , planets accumulate gas slowly and are
⊕
showing that for planets spanning 0.1 to 10 Jupiter typically unable to accrete more than ∼ 10-20% of their
masses,thetwomodelspredictthesameradiustowithin mass in H/He (Rogers et al. 2011; Bodenheimer & Lis-
afewpercent. Atthelowermassesrelevanttoourstudy, sauer 2014); values of f ≈0.5 may thus be unphysical.
H
the two models are also in agreement. To demonstrate However, as we argue in §5.1, the longer disk lifetimes
this, in Figure 3 we shade the regions corresponding to around M dwarfs (Carpenter et al. 2006; Pascucci et al.
the spread in radii at a given mass and age in Figure 2009) allow more time for gas accretion, potentially in-
9 of Mordasini et al. (2012b). Since those authors used creasing the maximum value of f . Nevertheless, and
H
a coupled formation/evolution code, at low planet mass more importantly, if a planet with f =0.5 loses its en-
H
the maximum envelope mass fraction f is small; for a tireenvelopeviaatmosphericescape,any planet with the
H
total mass of 4M , Mordasini et al. (2012b) find that same core mass and f <0.5 will also lose its envelope.
⊕ H
all planets have f < 0.2. At 2M , most planets have Below, where we present integrations with f = 0.5,
H ⊕ H
f (cid:46)0.1. We can see from Figure 3 that at these values our results are therefore conservative, as planets with
H
of f , the two models predict very similar radius evolu- f (cid:28)0.5 will in general evaporate more quickly.
H H
tion. NotethatMordasinietal.(2012b)didnotconsider While our treatment of the radius evolution is an im-
planets less massive than 2M . provement upon past tidal-atmospheric coupling papers
⊕
Themaximumenvelopefractionmeritsfurtherdiscus- (Jacksonetal.2010,forinstance,calculateR forsuper-
p
sion. Since we do not model the formation of mini- Earths by assuming a constant density as mass is lost),
Neptunes, we do not place a priori constraints on the there are still issues with our approach: (1) We do not
10 Luger et al. 2015
accountforinflationoftheradiusduetohighinsolation. implications of this choice in §5.3.
Instead, we calculate our radii from grids corresponding Giventhelargeplanetaryradiiatearlytimes,manyof
to a planet receiving the same flux as Earth. While at the planets we model here are not stable against Roche
late times this is justified, since planets in the HZ by lobe overflow in the HZ. During RLO, the stellar gravity
definition receive fluxes similar to Earth, at early times causes the upper layers of the atmosphere to suddenly
this is probably a poor approximation; recall that plan- becomeunboundfromtheplanet;thisoccurswhenR >
p
ets in the HZ around low mass M dwarfs are exposed R , where R is given by (8). For a planet that
Roche Roche
to fluxes up to two orders of magnitude higher during forms and evolves in situ, RLO never occurs, since any
the host star’s pre-main sequence phase. The primary gas that would be lifted from the planet in this fashion
effect of a higher insolation is to act as a blanket, de- would have never accreted in the first place. However,
laying the planet’s cooling and causing it to maintain an an inflated gaseous planet that forms at a large distance
inflated radius for longer. This will result in mass loss fromthestarmayinitiallybestableagainstoverflowand
rates higher than what we calculate here. (2) Since we enter RLO as it migrates inwards (since R ∝ a).
Roche
are determining the radii from pre-computed grids, we This is particularly the case for planets in the HZs of M
also do not model the effect of tidal dissipation on the dwarfs, since a and consequently R are small.
Roche
thermal evolution of the planet. Planets undergoing fast Ideally, the tidally-enhanced mass loss rate equation
tidal evolution can dissipate large amounts of energy in (5) should capture this process, but instead it predicts
their interiors, which should lead to significant heating an infinite mass loss rate as R → R (or as
XUV Roche
and inflation of their radii. (3) The radius is also likely ξ → 1) and unphysically changes sign for ξ < 1. This
todependon themassloss rate. Setting R equalto the is due to the fact that the energy-limited model implic-
p
tabulated value for a given mass, age, and composition itly assumes that the bulk of the atmosphere is located
isvalidonlyaslongasthetimescaleonwhichtheplanet at R (the single-layer assumption). Realistically, we
XUV
is able to cool is significantly shorter than the mass loss would expect the planet to quickly lose any mass above
timescale. Otherwise, the radius will not have enough the Roche lobe and then return to the stable hydrody-
time to adjust to the rapid loss of mass and the planet namicescaperegime. However,uponlossofthematerial
will remain somewhat inflated, leading to a regime of aboveR , theportionoftheenvelopebelowthenew
Roche
runawaymassloss(Lopezetal.2012). Whiletheplanets XUV absorption radius R(cid:48) will not be in hydrostatic
XUV
considered here are probably not in the runaway regime equilibrium; instead, anoutwardflowwillattempttore-
(Lopez et al. (2012) found that runaway mass loss oc- distribute mass to the evacuated region above, leading
curred only for H/He mass fractions (cid:38) 90%), we might to further overflow.
stillbesignificantlyunderestimatingtheradiiduringthe Several models exist that allow one to calculate the
early active phase of the parent star. mass loss rate due to RLO (e.g., Ritter 1988; Trilling
All points outlined above lead to an underestimate of et al. 1998; Gu et al. 2003; Sepinsky et al. 2007). These
the radius at a given time. Since the mass loss rate is often involve calculating the angular momentum ex-
proportional to R3 (5) or R3/2 (13), calculating the ra- changebetweentheoutflowinggasandtheplanet,which
p p can lead to its outward migration, given by
diusinthisfashionleadstoalowerbound ontheamount
of mass lost and on the strength of the coupling to tidal 1da 2 dM
effects. Becauseourpresentgoalistodeterminewhether =− p, (22)
a dt M dt
it is possible to form habitable evaporated cores via this p
mechanism, this conservative approach is sufficient. Fu- for a planet on a circular orbit (Gu et al. 2003; Chang
tureworkwillincorporateaself-consistentthermalstruc- et al. 2010). This leads to a corresponding increase in
ture model to better address the radius evolution. R until it reaches R and the overflow is halted.
Roche XUV
By differentiating the stability criterion R (M ) =
XUV p
3.3. Atmospheric Escape Model R (M ), one may then obtain an approximate ex-
Roche p
pressionfordM /dtintermsofthedensityprofiledM(<
WeassumethattheescapeofH/Hefromtheplanetat- p
R)/dR of the envelope.
mosphere is hydrodynamic (blow-off) at all times, which
However, for mini-Neptunes that migrate into the HZ
is valid at the XUV fluxes we consider here (see Erkaev
early on, RLO should occur during the initial migration
et al. 2013, and Figure 2). We run two separate sets of
process, which we do not model in this paper. Instead,
integrations: one in which we assume the flow is energy-
we begin our calculations by assuming that our planets
limited (5) for all values of F , and one in which we
XUV arestabletoRLOintheHZ.Ifaplanet’sradiusinitially
switch from energy-limited to radiation/recombination-
exceeds the Roche lobe radius, we set its envelope mass
limited (13) above the critical value of the flux (see
equaltothemaximum envelopemassforwhichitcanbe
§2.3.5). For planets whose orbits are eccentric enough
stableatitscurrentorbit;thedifferencebetweenthetwo
thattheyswitchbetweenthetworegimesoverthecourse
envelopemassesistheamountofH/Heitmusthavelost
of one orbit, we make use of the expressions derived in
priortoitsarrivalintheHZ.Itisimportanttonotethat
§A.3 in the Appendix. These two sets of integrations
theseplanetswillinitiallyhaveR =R ,whichas
should roughly bracket the true mass loss rate. XUV Roche
wementionedabove,leadstoaninfinitemasslossratein
For eccentric orbits, we calculate the mass loss in the
energy-limited regime from (16), with Kecc evaluated (5). An accurate determination of M˙p in this case prob-
from (15). We vary (cid:15) and R in the ranges given ably requires hydrodynamic simulations. However, the
XUV XUV
in Table 1. We choose (cid:15) = 0.30 as our default case. mass loss rate can be approximated by imposing a mini-
XUV
Whilethisisconsistentwithvaluescitedintheliterature mumvalueξmin in(5). Forξ <ξmin,wesetthemassloss
(see §2.3.2), it could be an overestimate. We discuss the rate equal to M˙ (ξ =ξ ). This is equivalent to impos-
p min