Thermal mass loss of protoplanetary cores with hydrogen-dominated atmospheres: The influences of ionization and orbital distance

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by  N. V. Erkaev

Mon.Not.R.Astron.Soc.000,1–??(2015) Printed24February2016 (MNLaTEXstylefilev2.2)
EUV-driven mass loss of protoplanetary cores with
hydrogen-dominated atmospheres: The influences of
ionization and orbital distance
6
1
0
2 N. V. Erkaev,1,2 H. Lammer,3 P. Odert,3 K. G. Kislyakova,3

b C. P. Johnstone,4 M. Gu¨del,4 M. L. Khodachenko3
e
F 1Institute of Computational Modelling SB RAS, 660036, Krasnoyarsk, Russian Federation
2Siberian Federal University,Krasnoyarsk, Russian Federation
3 3Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, A-8042, Graz, Austria
2 4Institute for Astronomy, University of Vienna, Tu¨rkenschanzstrasse 17, 1180 Vienna, Austria

]
P
E Released2015
.
h
p ABSTRACT
- We investigatethelossratesofthe hydrogenatmospheresofterrestrialplanetswitha
o
rangeofmassesandorbitaldistancesbyassumingastellarextremeultraviolet(EUV)
r
t luminosity that is 100 times stronger than that of the current Sun. We apply a 1D
s
a upper atmosphere radiation absorption and hydrodynamic escape model that takes
[ into account ionization, dissociation and recombination to calculate hydrogen mass

loss rates. We study the effects of the ionization, dissociation and recombination on
2
the thermal mass loss rates of hydrogen-dominated super-Earths and compare the
v
2 results to those obtained by the energy-limited escape formula which is widely used
5 for mass loss evolution studies. Our results indicate that the energy-limited formula
4 canto agreatextentover-orunderestimatethe hydrogenmasslossratesbyamounts
0 that depend on the stellar EUV flux and planetary parameters such as mass, size,
0 effective temperature, and EUV absorption radius.
.
1
Key words: planets and satellites: atmospheres – planets and satellites: physical
0
evolution – ultraviolet: planetary systems – stars: ultraviolet – hydrodynamics
6
1
:
v
i
X 1 INTRODUCTION firstdiscoveredplanetwithinthissize-massregimeCoRoT-
r 7b (L´eger et al. 2009).
a Duringtheearly stagesof planetformation, protoplanetary Owen & Jackson (2012) were the first who also stud-
cores that are still embedded in the circumstellar disk can ied the evaporation of hydrogen by stellar extreme ultra-
accumulate hydrogen-dominated primordial envelopes from violet (EUV) and X-ray radiation from planets within the
thegasdisk(e.g.,Hayashietal.1979;Nakazawaetal.1985; hot‘super-Earth’and hotNeptunedomain byapplyinghy-
Wuchterl1993; IkomaandGenda2006;Rafikov2006; St¨okl drodynamical equations at orbit locations <0.1 AU.In this
et al. 2015a; 2015b). The amount of gas captured by the work, the authors assumed X-ray luminosities similar to
planetary core strongly depends on its mass. Sufficiently thoseobservedaroundyoungsolar-likestars(e.g.,L ≈1030
X
massive cores can end up in a runaway accretion regime erg s−1) and discovered that close-in H -dominated planets
2
leading to subsequent formation of gas giants. couldexperienceaX-rayandanextremeultraviolet(EUV)
Asurpriseisthatalargenumberoflowmassexoplanets drivenevaporationregime.WhenX-raysdrivethehydrogen
discovered to date by ground-based and space-based facili- escape, theflow passes through a sonic surface and a shock
tiessuchasHATNed(Bakosetal.2004),SuperWASP(Pol- may build up before the ionization front where than EUV
laccoetal.2006),CoRoT(Auvergneetal.2009)andKepler heating occurs. When EUV drives the hydrogen escape, a
(Boruckietal.2010) wasthediscoveryofalargenumberof subsonicX-rayflowpassesthroughtheionization front and
hydrogen-dominatedsub-Neptune-typeplanetsatveryclose the EUV heated flow then is either supersonic or proceeds
orbitaldistances.Beforethediscoveryoftheseplanet’swith to pass through an EUV heated sonic point.
massesbetween1M⊕ and10M⊕ atorbitlocations<0.1AU Owen & Jackson (2012) found also that the upper at-
fromtheirhoststars,itwasexpectedthatplanetswithsuch mosphere heating by X-rays, which is related to the photo-
low masses should not have primordial hydrogen envelopes electronproductionbytheK-shellsofmetalswherethepres-
and resemble large Mercury-type rocky bodies such as the enceofOandCareimportantisrelevantatorbitlocations
2 N. V. Erkaev et al.
that are <0.1 AU. At orbits that are >0.1 AU, heating by stars most likely remain or evolve to sub-Neptunes instead
EUV photons is the dominant driver for hydrogen escape. ofEarth-likeplanetsandkeeplargefractionsoftheirhydro-
From the results of their study, one can see that the tran- gen envelopes during their whole life times. As pointed out
sition from X-ray driven to EUV-driven hydrogen escape above,inthemeantime,thesetheoreticalresultsseemtobe
occursatlowerX-rayluminositiesforplanetsclosertotheir confirmedbydetailedanalysesofobservations(Marcyetal.
hoststarsandforplanetswithlowerdensities.Thehydrogen 2014; Rogers 2015).
masslossrateforatypicalsub-Neptunewith≈1.6M⊕ and Rogers(2015)analyzedmanyplanetsdiscoveredbythe
≈5M⊕at0.1AUaroundayoungsolarlikestarwasinOwen Kepler satellite with both radius and mass measurements
& Jackson (2012) ≈ 3×1010 g s−1. Such a mass loss rate andconcludedthatmost‘super-Earths’withradiiof1.6R⊕
duringthe early stage of a solar-like star is similar to those have densities that are too low to be composed of silicates
derived by several other groups (Yelle 2004; Garcia-Mun˜oz and iron alone. The majority of these low density sub-
2007; Penzet al. 2008; Murry-Clay et al. 2009; Koskinenet Neptunes are discovered at closer orbital distances than 1
al. 2013a; 2013b; Shaikhislamov et al. 2014; Khodachenko AU. Taking this into account, in this work we study the
et al. 2015) of ≈ 4−7×1010 g s−1 for the hot Jupiter hydrogen loss rates from captured gas envelopes with core
HD 209458b. From these escape models one can conclude masses of 1M⊕, 2M⊕, 3M⊕ and 5M⊕ orbiting a moderate
that most close-in exoplanets start to evaporate within the rotatingyoungG-starwhichis100timesmoreactiveinex-
X-ray regime but switch to the EUV-driven regime when treme ultraviolet (EUV) radiation compared to the present
the X-ray flux falls below a critical value. The X-ray flux Sunat orbital distances between 0.1 - 1.0 AU.
is related to the age-activity relation of the planet’s host The main aim of this study is to investigate how ion-
star and the orbital location. From this pioneering study ization,dissociation,recombination,andLy-αcoolinginflu-
onecanalso concludethatthermalevaporation ismore im- encesthehydrogenmasslossratesofsub-Neptunesdepend-
portant forlower mass planets,especially forthose whoare ing on the orbital location between 0.1 and 1 AU around
in thehot Neptuneand sub-Neptunedomains. young solar-like stars. Furthermore, we study how the re-
Dependingonnebulaconditionsandtheformationsce- sults obtained by the upper atmosphere EUV absorption
nariosoflowmasshydrogendominatedplanets,someclose- hydrodynamic escape model differ from those provided by
in hot sub-Neptunesand Neptunesmay get rid of their ini- thewidelyusedenergy-limited formula(e.g., Lammeretal.
tiallycapturedhydrogenenvelopes,butsimilarplanetsmay 2009;EhrenreichandD´esert2011;Sanz-Forcadaetal.2011;
haveaproblemtolosethematorbitlocationsthatare>0.1 Leitzingeretal.2011;Lopezetal.2012;LopezandFortney
AU.Becausesuchremnantsofnebulagasenvelopesaround 2013;Valenciaetal.2013;KurokawaandKaltenegger2013;
super-Earths would be a problem for habitability, Lammer Luger et al. 2015). In Section 2, we describe the modeling
etal.(2014)investigatedtheoriginandlossofcapturedhy- approach while the detailed code description is given in an
drogen envelopes from protoplanetary cores with masses in appendix.InSection3,wediscusstheresultsandsummarize
the range of 0.1 to 5.0M⊕ orbiting in the habitable zone ourconclusions in Section 4.
at 1 AU of a Sun-like G star. In this study, the authors
also applied a 1D hydrodynamic upper atmosphere model
and concluded that depending on nebula properties, pro-
2 MODELING APPROACH
toplanetary cores with masses 61 M⊕ orbiting within the
habitable zone of a solar-like star most likely cannot lose To study the EUV-heated upper atmosphere structure and
their captured hydrogen envelopes. Their results have been thermal escape rates of the hydrogen atoms, we apply an
recently confirmed by Luger et al. (2015) and Owen & Mo- EUV energy absorption and 1-D upper atmosphere hydro-
hanty(2016)whostudiedthepossibilityofatransformation dynamicmodelappliedbeforeinamoresimplerwayinsev-
ofsub-Neptunesintosuper-Earthsinthehabitablezonesof eral studies (Erkaev et al. 2013; 2014; 2015; Lammer et al.
M dwarfs determined by the loss of atmospheric hydrogen 2013; 2014), and described in detail in the appendix. The
and Johnstone et al. (2015) who investigated the mass loss model solves the system of thehydrodynamicequations for
of hydrogen envelopes around similar cores as assumed by mass, momentum, and energy conservation. In addition to
Lammer et al. (2014) along various stellar rotation related the mechanisms included in our previous models, the sim-
activity evolution tracks in thehabitable zones of solar-like ulations in this study also include the effects of ionization,
stars. From the recent study of Owen & Mohanty (2016) dissociation, recombination and Ly-αcooling.
one finds in agreement with Tian et al. (2005), Erkaev et In the present parameter study, which focusses on the
al. (2013) and Chadney et al. (2015) that the lost envelope mass loss rates, we use an integrated EUV flux and do not
mass could be significant lower if one neglects the transi- consider a wavelength dependence of the incoming stellar
tion to Jeans escape. Moreover, Owen & Mohanty (2016) EUVradiation. Thisapproach isjustifiedbyarecentstudy
found that cores with masses that are > 1M⊕ with initial of Guo and Ben-Jaffel (2016), who studied the influence
H2/He envelope mass fractions > 1% will not loose their of the EUV spectral energy distribution on the upper at-
gaseousenvelopesduringtheirlifetimes.Thisfindingisalso mospherestructure,composition andatmosphericescapeof
in agreement for similar planets in G-star habitable zones HD 189733b, HD209458b, GJ 436b and Kepler-11b. These
(Lammer et al. 2014). authorsappliedanEUVspectralenergydistributionintheir
Theresults ofthesestudiesindicate thatdependingon modelandfoundthatthetotalhydrogenmasslossratesare
the initial nebular properties, such as the dust grain deple- only moderately affected by the spectral dependenceof the
tionfactor,planetesimalaccretionrates,andresultinglumi- EUVflux(seeFig.13inGuo&Ben-Jaffel2016).Fromtheir
nosities,protoplanetarycoreswithmasses>1.0M⊕ orbiting study one can see that if one considers the hydrogen mass
inside the habitable zones of M, K, G and F-type dwarf loss rates of the sub-Neptune Kepler-11b, the variation of
Hydrogen-loss rates of protoplanetary cores 3
the total mass loss rate with the variations of the spectral we assume fixed lower boundary densities and homopause
indexremainswithinafactor1.33(Guo&Ben-Jaffel2016). levels z shown in Table 1. The corresponding 1 bar lev-
0
Although no large changes in the total mass loss rates are els (i.e. R ) are also given as a reference. In the present
1bar
expected, for understanding the effect of the spectral de- study,we also compare the mass loss rates obtained by the
pendence on the volume heating rate and distribution of above described upper atmosphere model with the widely
different species, we plan also to apply EUV spectra in the used energy-limited escape formula
modelforfutureapplications.Inthepresenthydrogenmass
πηR R2 I
losscalculationsweassumetheEUVluminosityofamoder- L = 0 EUV EUV, (1)
en GM
aterotatingyoungsolar-likestar(Tuetal.2015; Johnstone pl
et al. 2015) that is enhanced by a factor of 100 compared whereI isthestellarEUVfluxoutsidetheatmosphereat
EUV
to the Sun’s present value with a flux at Earth’s orbit of theorbitallocationoftheplanet,η istheheatingefficiency,
≈4.64 erg cm−2 s−1 (Ribas et al. 2005. R is the effective radius corresponding to the absorp-
EUV
One should note that in H -dominated atmospheres, tion of the stellar EUV flux in the upper atmosphere, and
2
the main molecular infra-red (IR) emitting coolant is H+. G is Newton’s gravitational constant. If one considers pure
3
As shown by Shaikhislamov et al. (2014) and Chadney et H2 atmospheres the planetary radius would be caused by
al. (2015), this efficient IR-cooling mechanism vanishes or Rayleigh scattering and H2-H2 collisional absorption close
is negligible in hydrogen-dominated upper atmospheres at tothe1 bar level (Brown 2001). However,in real planetary
small orbital distances or high EUV flux values, because atmospheres clouds and hazes may be present which could
due to molecular dissociation preventing the balancing of alsoextinctvisiblelightatpressurelevelsbetweenthe1bar
the stellar heating by IR cooling. Since the assumed EUV andthehomopauselevel(e.g.,Lopezetal.2012;Lopezand
fluxvaluesrelatedtothemoderaterotatingyoungsolar-like Fortney2013).NotethatIEUV doesnotneedtobeaveraged
starafteritsarrivalattheZero-Age-Main-Sequence(ZAMS) over the planet’s surface because eq. (1), cast in this form,
is assumed to be 100 times higher compared to that of to- already accounts for this(cf. Erkaev et al. 2007). REUV de-
day’ssolarvaluethefluxishighenough,evenat1AU,that pends on the density distribution and can be determined
efficient H+ IR cooling plays a negligible role for the time from the following equation (Erkaev et al. 2014; Erkaev et
3
periodofthestudiedtestplanets.After≈100Myr,whenthe al. 2015)
activitydecreasesH+-coolingwillplayanimportantrolebe-
3 ∞ 0.5
yond orbital locations >0.1 AU (Chadney et al. 2015) and R =R 1+2 [1−J (r,π/2)/I ]rdr , (2)
EUV 0 EUV EUV
the losses will switch from the hydrodynamical regime to a (cid:20) Z1 (cid:21)
fastJeansandJeans-typeloss(Tianetal.2005;Erkaevetal.
where J (r,θ) is the stellar EUV flux in the atmosphere
EUV
2013;Owen&Mohanty2016).Theselossratescanbeorders
as function of the dimensional radius r =R/R and spher-
0
ofmagnitudelowercomparedtothehydrodynamical-driven
ical angle. As shown by Watson et al. (1981), R can
EUV
loss rates in theearly period of theplanets life time.
exceed the planetary radius quite substantially, especially
The lower boundary of our simulation domain,
for hydrogen-dominated low mass bodies with low gravity
R =R +z , is chosen in a similar way as in Lammer et
0 c 0 fields and high EUV fluxes. For gas giants and other mas-
al. (2014) where R is the core radius and z the altitude
c 0 sive and compact planets, R is close to R . Therefore,
EUV 0
of the gas envelope up to the homopause level that is lo- R isoftenapproximatedwithR ≈R intheliterature
EUV pl 0
catedinthelowerpartofthethermosphere.Fromformation
(e.g., Ehrenreich and D´esert 2011; Luger et al. 2015)
andstructuremodels ofhydrogen-dominatedlowmassexo-
planets it is known that the optical radius which is caused L∗ ≈ πηR03IEUV. (3)
mainly by H2 Rayleigh scattering and H2-H2 collisional ab- en GMpl
sorption could lie hundreds or thousands of kilometers or
To see the difference between both approaches, we com-
even a few Earth-radii above the core radius (e.g., Rogers
paretheresultsofbothassumptionswiththehydrodynamic
et al. 2011; Mordasini et al. 2012). The homopause level
model results. We use the model described in the appendix
z has lower pressures and lies therefore above the opti-
0 andlocatehydrogen-dominatedprotoplanetswithmassesof
cal radius. However, the homopause level lies at the base
1M⊕, 2M⊕ , 3M⊕ and 5M⊕ and at orbital locations of 0.1
of the thermosphere where above this level the bulk of the
AU, 0.3 AU, 0.5 AU, 0.7 AU and 1 AU and expose the hy-
EUV photons is absorbed and very little penetrates below
drogen envelopes to EUV flux values scaled corresponding
it. R can therefore be considered as a natural boundary
0 totheorbital locations with theEUVluminosity of amod-
between the troposphere-stratosphere-mesosphere and the
erate rotating young solar-like young star (Tu et al. 2015)
thermosphere-exosphere.
thatisenhancedbyafactorof100comparedtotoday’ssolar
Weassumeahydrogenmoleculenumberdensityat the
value.
lower boundary R of 5×1012 cm−3 (e.g. Kasting & Pol-
0
lack1983;Atreya1986,1999;Tianetal.2005;Erkaevetal.
2013; Lammer et al. 2014). This density valuecan neverbe
3 RESULTS AND DISCUSSIONS
arbitrarily increased or decreased by as much as an order
of magnitude, even if the captured envelope mass fractions Table 1 summarizes the thermal hydrogen mass loss rates
fenv of a particular planet are different. The reason for this fromthedifferentprotoplanetsatdifferentorbitallocations.
is that the value of n0 is strictly determined by the EUV Itisimportanttonotethattheassumedgasenvelopemasses
absorption optical depth of thethermosphere. are negligible compared to the core masses. Depending on
However, to enable comparison of the hydrogen loss theformationscenariosandnebularconditions,similarcores
rates at different orbital distances between the test planets cancapturedifferentamountofnebulargas(e.g., Rogerset
4 N. V. Erkaev et al.
Table 1.Hydrogenmasslossratesforprotoplanets withmassesof1M⊕ (a), 2M⊕ (b),3M⊕ (c)and5M⊕ (d)withassumedhydrogen
envelopemassfractionsfenv asmentionedinthemaintext.ThehydrogenenvelopesareexposedtoastellarEUVfluxthatis100times
stronger comparedtopresentsolarvalues at0.1–1AU.L isthehydrogen lossratecalculated withthehydrodynamicmodel neglecting
ionization, dissociation and recombination; Ln,i is the total escape rate of hydrogen ions and neutrals if ionization, dissociation and
recombination are not neglected; Ln and Li correspond to the losses of neutral H atoms or H+ ions only. Len and L∗en are the energy
limitedlossratecasesrelatedtoeq.(1)andeq.(3)thathavebeenmultipliedbyaheatingefficiencyη of15%.
Mpl/M⊕ R0/R⊕ |R1bar/R⊕ REUV/R⊕ L[gs−1] Ln,i [gs−1] Ln [gs−1] Li [gs−1] Len [gs−1] L∗en [gs−1]
d=1.0AU EUV=100
Teff=250K 464ergcm−2 s−1
1 1.15|≈1.0 2.87 2.1×108 2.1×108 2.0×108 8.3×106 3.3×108 6.8×107
2 2.26|1.43 5.2 8.5×108 8.6×108 8.2×108 5.0×107 1.3×109 2.7×108
3 2.44|1.72 5.12 5.8×108 5.9×108 5.6×108 3.0×107 9.7×108 2.2×108
5 2.71|2.12 5.69 6.5×108 6.7×108 6.0×108 6.7×107 7.2×108 1.8×108
d=0.7AU EUV=200
Teff=275K 928ergcm−2 s−1
1 1.15|≈1.0 2.64 3.6×108 3.3×108 3.2×108 1.3×107 7.3×108 1.3×108
2 2.26|1.38 5.1 1.4×109 1.2×109 1.1×109 1.0×108 2.7×109 5.2×108
3 2.44|1.67 5.12 8.9×108 7.7×108 7.0×108 6.7×107 1.9×109 4.3×108
5 2.71|2.07 4.87 1.1×109 1.0×109 9.6×108 1.0×108 1.8×109 3.7×108
d=0.5AU EUV=400
Teff=325K 1856ergcm−2 s−1
1 1.15|≈1.0 2.87 5.5×108 4.8×108 4.6×108 2.5×107 1.7×109 2.8×108
2 2.26|1.29 5.2 2.0×109 1.8×109 1.6×109 1.8×108 5.5×109 1.0×109
3 2.44|1.59 5.12 1.5×109 1.8×109 1.3×109 1.8×108 3.8×109 8.3×108
5 2.71|2.0 5.69 2.3×109 2.0×109 1.7×109 3.2×108 3.2×109 6.7×108
d=0.3AU EUV=1111
Teff=420K 5166ergcm−2 s−1
1 1.15|≈1.0 2.41 2.5×109 2.8×109 2.2×109 6.7×108 3.3×109 7.7×108
2 2.26|1.16 4.52 1.1×1010 1.2×1010 7.7×109 3.8×109 1.2×1010 2.8×109
3 2.44|1.45 5.36 3.5×109 3.1×109 2.5×109 6.7×108 1.2×1010 2.5×109
5 2.71|1.85 5.96 4.2×109 3.5×109 2.8×108 6.7×108 9.7×109 2.0×109
d=0.1AU EUV=10000
Teff=730K 46500ergcm−2 s−1
1 1.15|≈1.0 2.41 1.5×1010 1.8×1010 9.9×109 8.2×109 3.0×1010 6.7×109
2 2.26|≈1.0 3.84 5.7×1010 7.7×1010 3.9×1010 3.9×1010 7.5×1010 2.7×1010
3 2.44|1.141 4.63 2.5×1010 3.5×1010 1.5×1010 2.0×1010 8.0×1010 2.2×1010
5 2.71|1.54 5.15 1.0×1010 1.7×1010 7.3×109 1.0×1010 6.5×1010 1.8×1010
al.2011;Mordasinietal.2012;St¨okletal.2015).Ifthecap- using the described hydrodynamic model for all protoplan-
tured envelope mass was larger, then R would also move etsatthestudiedorbitallocations.Onecanalsoseethatthe
0
to larger distances. It was shown in Lammer et al. (2014) 1 bar level for a H envelope with a homopause distance at
2
thatinsuchcases,themasslossrateswouldalsobehigher. 1.15R⊕ and a core mass of 1M⊕ would be located near the
As the gas envelope evaporates, R shrinks and as a con- coreradiusifoneassumethattheatmosphereisisothermal
0
sequence the mass loss rate also decreases. Because of this below the homopause level. In such cases, with the corre-
effectifonemodelsthemasslossovertimetheshrinkingof sponding loss rates the thin hydrogen envelope would be
R has to be considered (Johnstone et al. 2015). Further- lostimmediatelyifnoadditionalsourcefromothervolatiles
0
more, it was shown by Tu et al. (2015) that depending on suchasH Oand/orCH ispresent.Foreachorbitallocation
2 4
the initial rotation rate the EUV activity levels of young and planetary mass, we run our hydrodynamic simulations
solar-like stars can evolve very differently during the first twice with different physical mechanisms included. In the
Gyr of their life time. We do not study here the mass loss first set of simulations, we neglect ionization, dissociation,
ofthetestplanetsforthewholerangeofpossibleEUVevo- and recombination, with the mass loss rates being given by
lutionary scenarios. The hydrogen mass loss rates shown in L. In the second set of simulations, we include the effects
Table1representthereforeonlyaphaseduringtheplanet’s of ionization, dissociation, and recombination, with (L )
n,i
evolution. giving the total mass loss rates, and L and L giving the
n i
mass loss rates for neutrals and ions separately.
Table 1 shows hydrogen mass loss rates calculated by
Hydrogen-loss rates of protoplanetary cores 5
Figure 1.Hydrogenmasslossrates as afunctionof orbitaldistance forrockyprotoplanetary coreswith1M⊕ (a), 2M⊕ (b), 3M⊕ (c)
and5M⊕ (d)calculatedforthestellarEUVflux100timeshighercomparedtothepresentsolarvalueinvariousorbitlocationsbetween
0.1-1AU.TheplanetaryradiirelatedtoassumedhydrogenenvelopefractionsaregiveninTable1.Dash-dottedlinescorrespondtothe
hydrogen loss rate L of the hydrodynamic model by neglecting ionization, dissociation and recombination; dash-dotted-dotted-dotted
lines correspond to loss rates Ln,i if ionization, dissociation and recombination processes are not neglected; dashed lines correspond to
the lossrates Li of ionized hydrogen atoms only; solidlines correspondto the loss rates Ln ofneutral hydrogen atoms only; the upper
dottedlinescorrespondtotheenergy-limitedlossformulamultipliedbyaheatingefficiencyη of15%,Len accordingtoeq.(1)andthe
lowerdotted linescorrespondtothelossratesL∗ accordingtoeq.(3).
en
For comparing the mass loss rates with the energy- more accurate formula of eq. (1). As it is obvious also from
limitedformula,weshowalsomasslossrates,L ,obtained Figs 1., eq. (3) tends to underestimate the mass loss rates
en
bythisapproach, butmultiplied with a heatingefficiency η because of the assumption that the EUV flux is absorbed
of15%.Dependingontheplanetaryparameters,theorbital closetotheplanetaryradius,whereasitisactuallyabsorbed
distance,correspondingEUVfluxandeffectivetemperature, at larger altitudes. For the small planets considered here,
the hydrogen mass loss rates are between ∼108 g s−1 at 1 R maybelocatedat2−3R ,muchhigherthanformore
EUV 0
AU and ∼1010 g s−1 at 0.1 AU.Onecan also see that L massive giant planets (Erkaev et al. 2007; Murray-Clay et
n,i
yieldsnegligibledifferencesthatarelessthanafactoroftwo al. 2009). Therefore, the application of the energy-limited
for all test planet loss rates between the hydrodynamic up- formulaegivenineqs.(1)and(3)areoflimiteduseforlow-
peratmospheremodelandtheenergy-limitedformulaofeq. mass planets.
(1) at orbital distances of 1 AU. For closer orbits such as
Moreover, the discrepancy between the mass loss rates
0.5 AU or 0.1 AU, depending on planetary parameters, the
calculatedwiththehydrodynamiccodeandtheenergylim-
differences between the energy-limited formula given in eq.
(1) and theresults obtained for L are factors of ≈1.5–3.5 ited escape formula arises because eq. (1) yields the max-
n,i
and ≈3.0–9.0, respectively. Therefore, ionization, dissocia- imum EUV-driven mass-loss rate that a planet can have,
even if multiplied by the heating efficiency η. The numer-
tionandrecombinationcannotbeneglectedforcloseorbital
ator represents the integrated EUV heating rate provided
distances or highly active younghost stars.
to the planet, i.e. the total stellar EUV flux absorbed at
Fig. 1 shows the hydrogen mass loss rates for the four R multiplied by the heating efficiency, i.e., the fraction
EUV
planetary masses as a function of orbital distance. Onecan of incident energy converted to heating. Since the denomi-
seethattheenergy-limitedformulaunderestimatesthemass nator represents the potential energy of the planet, eq. (1)
loss rates when one assumes R ≈ R ≈ R (eq. 3) assumes that the total absorbed EUV energy is used to lift
EUV 0 pl
and overestimates the mass loss rates when one uses the the planetary atmosphere out of the planet’s gravitational
6 N. V. Erkaev et al.
Figure 2.Hydrogenmasslossrates,normalizedtothatobtained bytheenergy-limitedformulaLen ofeq.(12). Dash-dottedlines:the
loss rates L of the hydrodynamic model by neglecting ionization and recombination processes; Dash-dotted-dotted lines: loss rates of
Ln,i;dashedlines:ionlossratesLi only;solidlines:neutralatomlossratesLn only.
well (Lammer et al. 2016). However, in transonic escape, high that the atmosphere’s thermal energy overcomes the
some fraction of the absorbed energy is also converted to gravitational potential of the planet in regions lower than
kinetic and thermal energy. As shown by Johnstone et al. whereEUVheatingistakingplace(Owen&Wu2015;Lam-
(2015), one of the reasons for this is that in a transonic mer et al. 2016; Owen & Subhanjoy 2016).
wind,alargepartoftheinputenergyisabsorbedinthesu- One can also see that the inclusion of ionization, dis-
personicpartofthewindandthereforecannotcontributeto sociation and recombination has only a small effect at the
themass loss rate. In thesecases, additional termsincrease assumed orbital distances if one assumes that all neutral
thedenominatorandreducetheatmosphericmass lossrate atoms and ions can escape from the planets. Only for the
(Sekiyaet al. 1980; Johnson et al. 2013; Erkaev et al. 2007; moremassiveplanetsandextremehighEUVfluxesatclose
2015).Forcertainplanetaryandstellarparametercombina- orbital distance (< 0.15 AU), the number density of ions
tions, these terms are not negligible leading to a true mass reachesthesamevalueastheneutrals.Ourresults indicate
loss rate, as determined with a hydrodynamicmodel, lower also that by including collisional ionization additionally to
by a factor of a few than those from eq. (1). ionizationcausedbythestellarEUVflux,themasslossrates
Koskinen et al. (2014) therefore suggested to replace η arenotaffectedsignificantly fororbitallocations >0.1AU.
with a mass loss efficiency factor to account for these dis- Thesame can besaid for Lyman-αcooling. Aninclusion of
crepancies. However, it is difficult to estimate this factor Lyman-α cooling has only a small effect on the mass loss
since it depends on planetary and stellar parameters, as il- rates for the closest test planets. Different mass loss rates
lustrated by the variation of this discrepancy for different related to neutrals and ions depend strongly on the plan-
planets and orbits. However, for hot Jupiters such a scal- etary parameters. Ionization becomes more relevant if the
inghasbeenimplementedrecentlybySalzetal.(2016).On planet is massive and, as a consequence, the upper atmo-
the other hand,for increasing T the hydrodynamicmass- sphereis more compact. Ionization also influencesthetotal
eff
loss rates increase, which is also not accounted for in eq. mass loss rates because a high degree of ionization reduces
(1) (Johnson et al. 2013; Erkaev et al. 2015). The energy- the area of the neutral atoms where the stellar EUV flux
limited mass loss rates should always be higher than those can be absorbed and heat transferred to neutrals. Figs. 1
obtained by the hydrodynamic model. The only exception shows also a clear break in the slope of the mass loss rates
isiftheplanetaryequilibriumtemperatureduetothestar’s withaseparation around0.5AU.Thereason forthisbreak
entire radiation field (i.e. its bolometric luminosity) is so is a strong nonlinear dependence related to the EUV flux
Hydrogen-loss rates of protoplanetary cores 7
I andtheriseof T ≈T .Theincrease ofI andT withcompactupperatmospheresatcloserorbitaldistances,
EUV eff 0 EUV 0
at closer orbital distances lead to a strong increase of the higher effective temperatures and higher EUV fluxes com-
hydrogen loss rates. Because we study only two orbit loca- paredtolower massplanetswith less compact upperatmo-
tions around 0.5 AU, the behavior of the curve looks like a spheres.
break of the slope, while in reality the behavior would be Afterhavingsomeideahowionization,dissociationand
smoother. Moreover, one should note that the above men- recombinationinfluencetheatmosphericmasslossofhydro-
tioned effect depends strongly on the stellar EUV flux and gen envelopesaroundvariousprotoplanetary cores, onecan
planetary parameters. investigatetheorbitallocations where‘naked’super-Earths
Themasslossratesofthefourtestplanetsconsideredat or sub-Neptunes which lost their captured nebular gas can
0.1AUarecomparabletothoseofhotJupitersat0.045AU beexpected.Ifweusethemassloss rates from Table1 and
(e.g., Yelle 2004; Koskinen et al. 2013; Shaikhislamov et al. estimateroughlyhowmuchatmospherecouldbelostduring
2014; Khodachenkoet al. 2015). Ifwe compare for instance the first 100 Myr after the protoatmosphere capture (Lam-
the loss rate of our 3M⊕ test planet, we obtain a hydrogen meretal.2014)onefindsthatdependingonnebulaparame-
lossrateof≈3.5×1010 gs−1whichisinagreementwiththe terssuchasthedustdepletionfactorf ≈0.01andassumed
lossrateofasimilarplanetgiveninFig.5ofOwen&Jackson relative accretion rates M˙acc (yr−1) of ≈ 10−6, cores with
Mpl
(2012).RecentstudiesbyHoweandBurrows(2015)studied masses of 62M⊕ can lose their captured envelopes related
mass loss rates from low density exoplanets with a coupled to the assumed f , EUV flux most likely within orbital
env
planetary structure and mass loss model which is based on distancesthatare60.3AU.Iftheaccretionrateis≈10−7,
the energy limited formula given in eq (3). These authors moremassive envelopes can becaptured,which would then
exposed sub-Neptune’s with similar EUV fluxes as in our only be lost at orbital locations 6 0.1 AU. A higher dust
study and obtained mass loss rates for hydrogen envelope depletionfactorf ≈0.1incombinationwithaccretionrates
mass fractions 6 0.01 in the order of 6 109 g s−1 at 0.1 that are < 10−6 could remove less massive envelopes from
AUand6108 gs−1 at0.3AU.Inourhydrodynamicmodel a protoplanetary core with 6 2M⊕ even at Venus orbit at
simulations we obtained for such scenarios mass loss rates 0.7AU.Moremassivecoresiftheyoriginatewithintheneb-
which are an order of magnitude higher, but the estimates ula lifetime will keep a fraction of their captured hydrogen
with eq.(3) yield comparable results. envelopeeven at orbital locations of 0.1 AU.
As mentioned above, our results represent only a finite However,adetailedstudytakingintoaccountthecom-
window of possibilities and would be different if the young plete parameter space to determine where one can expect
star was a slow or fast rotator, meaning lower or higher that‘naked’super-Earthstobefoundatorbitsthatare<1
EUV fluxes than assumed in this study. Furthermore, dif- AU,hastoapplyhydrodynamicmass loss calculations that
ferent accumulated nebula gas masses would also change do not neglect ionization, dissociation and recombination
the mass loss rates. If the planets had magnetic moments and consider all possible stellar EUV evolutionary tracks
andresultingmagnetospheres, thehigh degreeofionization (Tu et al. 2015), as it was done for the habitable zone by
at close orbital distances would also reduce the total mass Johnstone et al. (2015). This effort is beyond the scope of
lossrates(Khodachenkoetal.2015).Thediscoveryofmany the present study but is planned to be carried out in the
small close-in low density planets at orbital distances <0.2 future.
AU (Marcy et al. 2014; Rogers 2015) indicates that these
objects may haveevolved from initially more massive plan-
etstosub-Neptunesandhydrogen-dominatedsuper-Earths,
buthaveneverlosttheirenvelopescompletely.Ontheother
4 CONCLUSION
hand, their host stars could also havebeen less active stars
when they were young. Weappliedan1DupperatmosphereEUVradiationabsorp-
Fig. 2 shows the mass loss rates of neutrals only, ions tion and hydrodynamic escape model that includes ioniza-
only and the sums of neutrals and ions, and hydrody- tion, dissociation and recombination to hydrogen envelopes
namic loss rates that consider no ionization, dissociation captured from protoplanetary nebulae surrounding rocky
andrecombination,normalizedtothatcorrespondingtothe cores with masses between 1–5M⊕ at orbital locations of
energy-limited mass loss rate L (eq. 12). For very high 0.1–1AU.Thesedifferenttestplanetshavebeenexposedto
en
EUVfluxes,ionizationaltersthemasslossratesbecausethe a stellar EUV flux of a young solar-like star emitting 100
increasingnumberofelectronsenhancesrecombinationlead- times more EUV radiation compared to present Sun. De-
ingtoalargefractionoftheenergybeinglostbycoolingra- pending on the assumed planetary parameters, the orbital
diation (Murray-Clayet al. 2009; Guo2011). Inthesestud- distance,thecorrespondingEUVfluxandtheeffectivetem-
ies,forJupiter-typeplanetsthisbecomesimportantforEUV perature, our model yields hydrogen escape rates of ≈1032
fluxes>104 erg cm−2 s−1.Forconsideredlowmassplanets s−1 to 1034 s−1 and corresponding atmospheric mass loss
the mass loss rates with (L ) and without ionization (L) ratesof≈108 gs−1 to1010 gs−1between1AUand0.1AU,
n,i
areverysimilarandverysmalldeviationsoccuronlyforthe respectively. Our study also shows that the energy-limited
closestorbits.Thiseffectwouldlikelybecomemorerelevant formula can overestimate the atmospheric mass loss rates
forevencloserorbitsorhigherstellarEUVemission.Insuch of hydrogen-dominated low mass planets such as ‘super-
case, eqs. (12) or (14) are not applicable and approximate Earths’orsub-Neptunesespeciallyatcloserorbitaldistances
estimates for mass loss rates in radiation/recombination- up to a factor of ≈ 4. For cooler planets with more com-
limited regime can beused (Murray-Clayet al. 2009; Owen pact atmospheres thedifference between thehydrodynamic
and Jackson 2012; Luger et al. 2015). One can also see modelandthelossrateobtainedfromeq.(1)issmaller.By
that the rise in ionization occurs for more massive planets assumingthatR ≈R ≈R theenergy-limitedformula
pl 0 EUV
8 N. V. Erkaev et al.
yields mass-loss rates too low by up to a factor of three in Howe, A. R., Borrows, A., 2015, Astrophys.J., 808, 150
thestudied parameter space. Ikoma, M., Genda, H., 2006, ApJ, 648, 696
Jackson, A. P., Davis, T. A., Wheatley, P. J., 2012, MN-
RAS,422, 2024
Johnstone, C. P., Gu¨del, M., St¨okl, A., Lammer, H., Tu,
ACKNOWLEDGMENTS
L.,Lu¨ftinger,T.,Kislyakova,K.G.,Erkaev,N.V.,Odert,
The authors acknowledge the support by the FWF NFN P., Dorfi,E., 2015, Astrophys.J., submitted
project S11601-N16 ‘Pathways to Habitability: From Disks Johnson, R. E., Volkov, A. N., Erwin, J. T., 2013, Astro-
to Active Stars, Planets and Life’, and the related FWF phys.J. Lett. 768, L4
NFNsubprojects,S11604-N16’Radiation&WindEvolution Khodachenko, M. L., Shaikhislamov, I. F., Lammer, H.,
fromTTauriPhasetoZAMSandBeyond’,andS11607-N16 Prokopov, P.A., 2015, Astrophys.J., 813, 50, 18pp
‘Particle/RadiativeInteractionswithUpperAtmospheresof Koskinen, T. T., Yelle, R. V., Harris, M. J., Lavvas, P.,
Planetary Bodies Under Extreme Stellar Conditions’. H. 2013a, Icarus, 226, 1695
Lammer, P. Odert and N. V. Erkaev acknowledges also Koskinen, T. T., Harris, M. J., Yelle, R. V., Lavvas, P.,
support from the FWF project P25256-N27 ‘Characteriz- 2013b, Icarus, 226, 1678
ing Stellar and Exoplanetary Environments via Modeling Koskinen, T. T., Lavvas, P., Harris, M. J., Yelle, R. V.,
of Lyman-α Transit Observations of Hot Jupiters’. N. V. 2014, Phil. Trans. R.Soc. A,372, 20130089
Erkaev acknowledges support by the RFBR grant No 15- Kurokawa, H.,Kaltenegger, L., 2013, Mont.Not. Roy.As-
05-00879-a. Finally, we thank an anonymous referee for his tron.Soc., 433, 3239
suggestionsandrecommendationsthathelptoimprovethis Lammer, H.,et al., 2009, Astron.Astrophys., 506, 399
work. Lammer, H., Erkaev, N. V., Odert, P., Kislyakova, K. G.,
Leitzinger,M.,2013,Mont.NotesRoy.Astron.Soc.,430,
1247
Lammer, H., St¨okl, A., Erkaev, N. V., Dorfi, E. V.,
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As in Murray-Clay et al. (2009), Erkaev et al. (2013),
Lammeretal.(2013)andLammeretal.(2014),weassumea
singlewavelengthforallphotons(hν =20ev)anduseanav-
APPENDIX A: MODEL DESCRIPTION erage EUV photoabsorption cross sections σEUV for hydro-
genatomsandmoleculesabout2×10−18cm2and1.2×10−18
Forstudyingthehydrogenlossofthetestplanets,weapplya cm2,respectively.Theappliedvaluesareinagreement with
time-dependent1-Dhydrodynamicupperatmospheremodel experimentalandtheoreticaldataofBates(1963),Cookand
that solves thesystem of the fluidequations for mass, Metzger (1964), and Beynon and Cairns (1965).
∂ρ ∂(ρvR2) The continuity equations for the number densities of
∂t + R2∂R =0, (A1) theatomicneutralhydrogennH,atomichydrogenionsnH+,
momentum, hydrogenmoleculesnH2,andhydrogenmolecularionsnH+,
2
can thenbe written as
∂ρV ∂ R2(ρV2+P) ∂U P
∂t + (cid:2) R2∂R (cid:3) =−ρ∂R +2R, (A2) ∂(nH) + 1 ∂ nHvR2 =−ν n −ν n n +
∂t R2 (cid:0) ∂R (cid:1) H H Hcol e H
and energy conservation,
∂ 1ρV2+E+ρU ∂VR2 1ρV2+E+P +ρU αHnenH+ +2αH2nenH+2 +2νdisnH2n−2γHnn2H, (A9)
2 + 2 =
(cid:2) ∂t (cid:3) (cid:2) R2∂R (cid:3)
QEUV−QLα + R2∂∂R R2χ∂∂RT .(A3) ∂(n∂Ht+) + R12∂(cid:0)nH∂+RvR2(cid:1) =νHnH+νHcolnenH−
(cid:16) (cid:17)
Here Q is the stellar EUV volume heating rate, which αHnenH+, (A10)
EUV
depends on the stellar EUV flux at the orbital distance of
thetestplanetsandontheatmosphericdensity,andisgiven ∂(nH2) + 1 ∂ nH2vR2 =−ν n −
by ∂t R2 (cid:0) ∂R (cid:1) H2 H2
ν n n+γ nn2, (A11)
Q =ησ (n +n )φ , (A4) dis H2 H H
EUV EUV H H2 EUV
QQLLαα =is 7th.5e·L1a0y−m19anne-nalHpehxapc(o−o1li1n8g3,4g8i/vTen),by (A5) ∂(cid:16)n∂Ht+2(cid:17) + R12∂(cid:16)nH∂+2RvR2(cid:17) =νH2nH2−αH2nenH+2.(A12)
10 N. V. Erkaev et al.
Theelectron densityisdeterminedforquasi-neutralitycon- Q˜ =7.5·10−19N R /(m V3 ), (A26)
Lα 0 0 H2 T0
ditions ν˜H =νHR0/VT0, ν˜H2 =νH2R0/VT0, (A27)
ne =nH+ +nH+ (A13) α˜H =αHN0R0/VT0, α˜H2 =αH2N0R0/VT0, (A28)
2
ν˜ =ν N R /V , ν˜ =ν N R /V , (A29)
and the total hydrogen numberdensity Hcol Hcol 0 0 T0 diss diss 0 0 T0
γ˜ =γ N2R /V , χ˜=χT /(ρ V3 R ). (A30)
n=nH+nH+ +nH2 +nH+. (A14) H H 0 0 T0 0 0 T0 0
2 Subscript “0” denotes lower boundary values. The normal-
αH is the recombination rate related to the reaction ized equationscan be written as follows
H++e→H of 4 × 10−12(300/T)0.64 cm3 s−1, α
is the dissociation rate of H++e→H + H: α =2.3H×2 ∂ρ˜ ∂ r2ρ˜V˜
2 H2 + =0, (A31)
10−8(300/T)0.4 cm3 s−1, ν is the thermal dissociation ∂t (cid:0)r2∂r (cid:1)
diss
rate of H2 → H + H: 1.5 · 10−9 exp(−49000/T), γH is the ∂ρ˜X ∂ r2ρ˜XV˜
rate of reaction H + H → H2: γH = 8.0 · 10−33 (300/T)0.6 ∂t + (cid:0)r2∂r (cid:1) =
(Yelle, 2004).
−ν˜ Xρ˜−ν˜ T˜1/2ρ˜2X(2X++Y+)+
ν isthehydrogenionizationrate,andν istheioniza- H Hcol
tion rHateof molecular hydrogen(StoreyandHH2ummer1995; α˜Hρ˜2X+(2X++Y+)+
Murray-Clay et al. 2009), α˜H2ρ˜2Y+(2X++Y+)+νdissρ˜2Y(1+X+X+)
νH =5.9·10−8φEUVs−1, νH2 =3.3·10−8φEUVs−1, (A15) −γ˜HHρ˜3(1+X+X+)X2. (A32)
andν isthecollisionalionizationrate(Black,1981),ν ∂ρ˜X+ ∂ r2ρ˜X+V˜
Hcol Hcol + =
= 5.9·10−11T1/2exp(−157809/T) . ∂t (cid:0) r2∂r (cid:1)
φEUV is the function describing the EUV flux absorp- ν˜HXρ˜+ν˜HcolT˜1/2ρ˜2X(2X++Y+)
tion in the atmosphere −α˜ ρ˜2X+(2X++Y+), (A33)
H
1 π/2+arccos(1/r) ∂ρ˜Y+ ∂ r2ρ˜Y+V˜
φEUV = 4π Z JEUV(r,θ)2πsin(θ)dθ. (A16) ∂t + (cid:0) r2∂r (cid:1) =
0
Here,JEUV(r,θ)isthefunctionofsphericalcoordinatesthat ν˜H2ρ˜Y −α˜H2ρ˜2Y+(2X++Y+),
describes the spatial variation of the EUV flux due to the ∂ρ˜V˜ ∂ r2(ρ˜V˜2+P˜)
atmospheric absorption (Erkaev et al. 2015), r corresponds ∂t + (cid:2) r2∂r (cid:3) =
totheradialdistancefromtheplanetarycenternoramalized ∂U˜ P˜
to R . −ρ˜ +2 , (A34)
0 ∂r r
In the hydrodynamic equations, the mass density, ρ,
∂ 1ρV˜2+E˜+ρ˜U˜
and the pressure, P,can then be written as 2
(cid:2) ∂t (cid:3)
ρ=mH(nH+nH+)+mH2(cid:16)nH2+nH+2(cid:17), (A17) +∂V˜r2 21ρV˜2+E˜+P˜+ρ˜U˜ =
(cid:2) r2∂r (cid:3)
P =(cid:16)nH+nH+ +nH2 +nH2+ +ne(cid:17)kT, (A18) Q˜ −Q˜ + ∂ r2χ˜∂T˜ , (A35)
EUV Lα r2∂r (cid:18) ∂r(cid:19)
where T is the upper atmosphere temperature and k is the
Boltzmannconstant,andmHandmH2 arethemassesofthe P˜ =ρ˜T˜(1+X+3X++Y+)/2.0, (A36)
hydrogen atoms and molecules, respectively. E˜ =ρ˜V˜2/2+ρ˜T˜(5+X+7X++3Y+)/4.0. (A37)
For atmospheres that are in long-term radiative equi-
librium, the temperature T near the lower boundary of The obtained equations (A31-A37) make a self-
0
the simulation domain is quite close to the planetary effec- consistent closed system withrespect to6unknownquanti-
tiveandequilibriumtemperaturesT ≈T .Thehydrody- tiesρ,X,X+,Y+,T,V.TheseventhquantityY (ratioofthe
eff eq
namic model is only applicable as long as enough collisions molecular hydrogen mass to the total mass) is determined
occur,whichisthecaseiftheKnudsennumberis<0.1.We bysimple equation
set the upperboundary conditions in the supersonic region Y =1−X−X+−Y+. (A38)
assuming the radial derivatives of the density, temperature
and velocity are zero. WeapplythefinitedifferencenumericalschemeofMacCor-
Forcomputationalconvenienceweintroducedimension- mack to integrate the system of equations in time, which
less quantities can bewritten in a vectorform
ρ˜=ρ/ρ0, ρ0=N0mH2, (A19) ∂U + ∂Γ(U) =Ψ(U) (A39)
r=R/R , U˜ =m U/(kT ), (A20) ∂t ∂r
0 H 0
Finite difference approximation of this equation is the fol-
V˜ =V/V , V = kT /m , (A21)
T0 T0 0 H lowing
T˜=T/T ,P˜ =pP/(ρ V2 ), (A22)
0 0 T0 U¯n+1 =Un− ∆t[Γ(Un )−
X =mHnH/ρ, X+ =mH+nH+/ρ, (A23) i i ∆r i+1
Y =mH2nH2/ρ, Y+ =mH+nH+/ρ, (A24) Γ(Uin)]+∆tΨ(Uin), (A40)
2 2 1 ∆t
Q˜EUV =ησEUVφEUVR0/(mH2VT30), (A25) Uin+1 = 2(U¯in+1+Uin)− 2∆r[Γ(U¯in+1)−