The Planetary Accretion Shock: I. Framework for Radiation-hydrodynamical Simulations and First Results

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by  Gabriel-Dominique Marleau

DRAFTVERSIONJANUARY12,2017
PreprinttypesetusingLATEXstyleAASTeX6v.1.0
THEPLANETARYACCRETIONSHOCK:
I.FRAMEWORKFORRADIATION-HYDRODYNAMICALSIMULATIONSANDFIRSTRESULTS
GABRIEL-DOMINIQUEMARLEAU1,2,HUBERTKLAHR2,ROLFKUIPER3,CHRISTOPHMORDASINI1
1PhysikalischesInstitut,Universita¨tBern,Sidlerstr.5,3012Bern,Switzerland
2Max-Planck-Institutfu¨rAstronomie,Ko¨nigstuhl17,69117Heidelberg,Germany
3InstituteforAstronomyandAstrophysics,EberhardKarlsUniversita¨tTu¨bingen,AufderMorgenstelle10,72076Tu¨bingen,Germany
7
1
ABSTRACT
0
2 The key aspect determining the post-formation luminosity of gas giants has long been considered to be the

n energeticsoftheaccretionshockattheplanetarysurface. Weuseone-dimensionalradiation-hydrodynamical
a simulationstostudytheradiativelossefficiencyandtoobtainpost-shocktemperaturesandpressuresandthus
J
entropies. Theefficiencyisdefinedasthefractionofthetotalincomingenergyfluxwhichescapesthesystem

0
(roughlytheHillsphere),takingintoaccounttheenergyrecyclingwhichoccursaheadoftheshockinaradia-
1
tiveprecursor. Wefocushereonaconstantequationofstatetoisolatetheshockphysicsbutuseconstantand

] tabulatedopacities.Whilerobustquantitativeresultswillrequireaself-consistenttreatmentincludinghydrogen
P
dissocationandionization, theresultspresentedhereshowthecorrectqualitativebehaviorandcanbeunder-
E
stoodsemi-analytically.Theshockisfoundtobeisothermalandsupercriticalforarangeofconditionsrelevant
.
h to core accretion (CA), with Mach numbers M (cid:38)3. Across the shock, the entropy decreases significantly,
p
by a few entropy units (k /baryon). While nearly 100 percent of the incoming kinetic energy is converted
- B
o toradiationlocally, theefficienciesarefoundtobeaslowasroughly40percent,implyingthatameaningful
r fractionofthetotalaccretionenergyisbroughtintotheplanet. ForrealisticparametercombinationsintheCA
t
s scenario, a non-zero fraction of the luminosity always escapes the system. This luminosity could explain, at
a
[ leastinpart,recentobservationsintheyoungLkCa15andHD100546systems.

1 Keywords:planetsandsatellites: formation—planetsandsatellites: gaseousplanets—planetsandsatellites:
v physicalevolution
7
4
7 1. INTRODUCTION constraintsonthemassdistributionofplanetaryorvery-low-
2 masscompanions(e.g.Billeretal.2013;Brandtetal.2014;
Starting with the discovery of planetary and low-mass
0
Clanton&Gaudi2015).
. companions to 2M 1207, 1RXS 1609, HR 8799, and β Pic
1 Whiletheuncertaintyontheageoftheparentstaroftenre-
in the last decade (Chauvin et al. 2004; Lafrenie`re et al.
0
mainsconsiderable, itis, tofirstorder, presumablyrandom.
7 2008;Maroisetal.2008;Lagrangeetal.2009),photometric
However, theconversion ofa luminosityto a massentails a
1 andspectroscopicdirectobservationsofseveraldozenyoung
v: ((cid:46)20–100-Myr-old) objects have challenged and enriched theoretical, probably systematic uncertainty: that of the lu-
minosityofaplanetorlow-massobjectattheendofitsfor-
i our knowledge about exoplanets, providing access to their
X mation, as it enters into the evolutionary ‘cooling’ phase1.
theirmodynamicstate,chemicallycomplexatmospheres,and
ar otherwiseunobtainableinformationontheouter((cid:38)20au)ar- Thisisamajorsourceofuncertainty(Bowler2016). Indeed,
atthoseyoungages,coolinghasnotyeterasedtracesofthe
chitectureofplanetarysystems. Onemajorlimitation,how-
formation process, as reflected in a planet’s luminosity and
ever, has been the difficulty of determining the masses of
radius(andthusalsospectrum);thishappensontheKelvin–
these objects, which is of particular importance in the light
Helmholtztimescalet ≡GM 2/RL∼107–109 yr. Forma-
of recent or upcoming surveys expected to detect several KH p
tion models up to now (e.g., Marley et al. 2007; Mordasini
young objects (e.g., LEECH, SPHERE, GPI, Project 1640,
etal.2012)haveonlymadepredictionsinthelimitingcases
CHARIS;seeSkemeretal.2014;Zurloetal.2014;Macin-
of ‘hot’ and ‘cold starts’, as discussed below, without how-
tosh et al. 2014; Oppenheimer et al. 2012; Peters-Limbach
everattemptingtomodeltheshockindetail.
et al. 2013 and references therein) as they seek to provide
1Deuteriumburningmightslowdownthecoolingbuttheargumentre-
[email protected]
mainsthesame.
2
What is thought to be the key process setting the entropy 2000)
ofthegasistheaccretionshockintherunawaygasaccretion
1 1 1
phase (Marley et al. 2007; Spiegel & Burrows 2012). This = + , (1)
R k R R
accretionshockistraditionallyassociatedwithcoreaccretion acc Lissauer Hill Bondi
butitmightalsooccurinsomecircumstancesinthecontext where
ofgravitationalinstability(seethediscussioninSection8.1
ofMordasinietal.2012). Whentheplanetbecomesmassive (cid:18) M (cid:19)1/3 GM
p p
R =a , R = (2)
enough, it detaches from the local disk and gas falls freely Hill 3M Bondi c 2
∗ ∞
ontoit. Thequestionisusuallyputintermsofwhathappens
to the kinetic energy of the gas, namely whether it is radi- are the Hill and Bondi radii, respectively, with a the semi-
ated away at the shock or whether it gets added as thermal major axis of the planet of mass M around a star of mass
p
energy to the planet. The extreme outcome of full radiative M∗ and c∞ the sound speed in the disk at the planet’s loca-
lossleadsto‘coldstarts’,whilethelimitingcaseofnoradia- tion. The factor k =1/3 accounts for the findings of
Lissauer
tivelossleadsto‘hotstarts’,astheresultingplanetsarethen Lissaueretal.(2009)thatonlytheinnerpartofthematerial
respectivelycolderorhotter(Marleyetal.2007). in the planet’s Roche lobe is bound to it, most of the gas in
Mordasini(2013)foundthatthemassofthesolidcore(in thevolumeflowingwiththematerialinthedisk(Mordasini
thecore-accretionframework)correlateswithpost-formation etal.2012).ThesphereofradiusR isthustheapproximate
acc
luminosity but, as explained there, this is due to a self- regionwheregasshouldbecomeboundtotheplanet,bothin
amplifyingprocessbasedontheshock.Thustheshock(orat terms of gravitational force compared to that from the star
least its computational treatment in formation calculations) (R )andthermalenergycomparedtotheplanet’spotential
Hill
iscrucialissettingthepost-formationradiusandluminosity. energy(R ). Intherunawayphase,theR termusually
Bondi Hill
It has been shown (Marleau & Cumming 2014) how to (thoughmarginally)dominates.
place joint constraints on the mass and initial entropy of an While global and local disk simulations have shown that
objectfromaluminosityandagemeasurement. Now, how- theaccretionontotheprotoplanetishighlythree-dimensional
ever, we take a first step towards predicting this initial en- (Ayliffe & Bate 2009; Tanigawa et al. 2012; D’Angelo &
tropybypresentingsimulationsoftheshockefficiency,con- Bodenheimer 2013; Ormel et al. 2015; Fung et al. 2015;
sideringsnapshotsoftheformationprocess. Szula´gyietal.2016)andpossiblyaffectedbymagneticfields
Inthisfirstpaper,wefocusonthephysicsattheaccretion in the gap and protoplanetary disk (e.g., Uribe et al. 2013;
shock and the upstream region. Since ionization and disso- Keith & Wardle 2015), we take a first step here by using a
ciation act as energy sinks (Zel’dovich & Raizer 1967), we spherically-symmetricset-upandneglectingmagneticfields.
focus on an ideal-gas equation of state (EOS) with constant This allows us to model in detail the last stages of the ac-
heatcapacityandmeanmolecularweighttoisolatetheshock cretion process on small scales around the proto-planetary
physics from the microphysics. However, we use both con- object(r(cid:46)30R ).Thisstagesshouldremainsimilarinmore
J
stantandmorerealisticopacities. Alsotosimplifytheanal- complexgeometries.
ysis, we assume here that the gas and the radiation couple Note that, in the detached runaway phase, the continued
perfectlyandthereforeuse‘one-temperature’(1-T)radiation accretion of solids (dust and planetesimals) by the planet is
transport (discussed below). A forthcoming paper will be important for setting the final mass of the core (Mordasini
concernedwiththeeffectsofdissociationandionizationand 2013). However, this accretion rate of solids is several or-
will also address the importance of 2-T radiation transport. ders of magnitude smaller than that of gas and is therefore
Finally, a subsequent work will also discuss how the shock neglectedhere.
resultscanbeusedinformationcalculationsandperformthis
coupling.Onlywiththiswillitbepossibletopredictdirectly
2.2. Methods
post-formation entropies and thus the luminosities and radii
For our one-temperature radiation hydrodynamics sim-
ofyounggasgiants.
ulations, we use the static-grid version of the modular
(magneto-)hydrodynamicscodePLUTO(version3;Mignone
2. MODEL
et al. 2007, 2012) combined with the grey, 1-T flux-limited
2.1. Physicalpicture diffusion (FLD) radiation transport module Makemake de-
scribed and tested in Kuiper et al. (2010) and Kuiper &
Each simulation is meant as a snapshot of the accretion
Klessen (2013), without ray tracing. We use the HLL hy-
processwhentheplanetisataradiusR =r ,theshock
p shock
drodynamical solver and the flux limiter λ from Levermore
radius. Tofollowgasaccretionontoagrowingplanetwhich
&Pomraning(1981)givenby
isdetachedfromthenebula,weletoursimulationboxextend
from the top-most layers of the planet to a large fraction of
2+R
itsaccretionradiusR ,definedthrough(Bodenheimeretal. λ(R)= , (3)
acc 6+3R+R2
3
wheretheradiationparameterRisdefinedthrough where m is the mass interior to r (dominated by M ), P,
r p
T, and S are respectively the pressure, the temperature, and
F =−D ∇E , (4a)
rad F rad the entropy per mass, L=4πr2F is the luminosity, G the
rad
λ(R)c universalgravitationalconstant,andε theenergygeneration
D ≡ , (4b)
F
κRρ rate. Theactual,adiabatic,andradiativegradientsaregiven
(cid:107)∇lnE (cid:107) respectivelyby
R≡ rad , (4c)
κ ρ
R ∇ =min(∇ ,∇ ) (8a)
act ad rad
whereF andE aretheradiationfluxandenergydensity,
rad rad γ−1
respectively, κR the Rosseland mean opacity, ρ the density, ∇ad= γ , (8b)
andcthespeedoflight. Thereissomefreedominthechoice
3LPκ
of the flux limiter’s functional form but it is required to be- ∇rad= 64πσGm T4. (8c)
r
haveasymptoticallyas(Levermore1984)
Equation(8a)istheSchwarzschildcriterion. (Notethatcon-

1, R(cid:28)1 (diffusionlimit) vectionthereforeplaysaroleonlyintheinitialprofile;inthe
λ(R)→ 3 (5)
radiation-hydrodynamicalsimulationsproperthereisnocon-
1, R(cid:29)1 (free-streaminglimit)
R vection because of the assumption of spherical symmetry.)
to recover the limiting regimes of pure diffusion, where We use an adaptive step size for the integration to resolve
F = 1c∇E /κ ρ,andfree-streaming,whereF =cE accurately the pressure and temperature gradients. This at-
rad 3 rad R rad rad
inthedirectionoppositetotheE gradient. mosphereisthensmoothlyjoinedontoacalculatedaccretion
rad
ThelocalradiationquantityR(ρ,T,E )definedinEqua- flow for ρ and v. (See Equations (11ff) below.) The goal
rad
tion (4c) compares the photon mean free path λ = of these efforts is (i) to provide a numerically sufficiently
phot
1/κρ to the ‘radiation energy density scale height’ H = smooth initial profile while (ii) beginning with a certain at-
Erad
E /(∂E /∂r);insphericalcoordinatesitisgivenby mosphericmasstospeedupthecomputation.
rad rad
Thegridisuniformfromr tor +∆r andhasahigh
(cid:12) (cid:12) (cid:12) (cid:12) min min
R= 1 (cid:12)(cid:12)∂Erad(cid:12)(cid:12)=(cid:12)(cid:12)∂lnErad(cid:12)(cid:12) (6a) resolution to resolve sufficiently well the pressure gradient
κρE (cid:12) ∂r (cid:12) (cid:12) ∂τ (cid:12)
rad intheinnermostpart,usingbydefault∆r=0.5R andN=
J
λphot 500 cells there. The other grid patch is a stretched segment
= . (6b)
HErad out to rmax, with usually also N =500, i.e., a much smaller
resolution. Thishasproventobestableandaccurate.
LargeRvaluesmeanthattheradiationenergydensity—and
As gas is added to the simulation domain, quasi-
thus, in the 1-T approximation, the temperature—changes
hydrostatic equilibrium establishes below the shock. The
over a shorter distance than photons get absorbed and re-
shock position r defining the top of the planet’s atmo-
emitted. shock
sphereissimplygivenbythelocationwherethegaspressure
is equal to the ram pressure. The shocks moves in time as
2.3. Set-up
gas is added (inward or outward depending on the simula-
Weuseasemi-openboxfixedatsomeheightintheatmo-
tion),usuallyatanegligiblespeed,i.e.,dr /dt (cid:28)v ,
shock shock
sphere of the planet, with a closed left, inner edge (towards
where v is the pre-shock velocity. Nevertheless, we al-
shock
the centre of the planet) at r = r , and start with an at-
min ways take this term into account when calculating mass or
mosphereofsomearbitrarysmallheight(e.g., 0.5R ), onto
J energyfluxes;thispossiblyleadstoslightlynon-nominalef-
which gas falls from the outer edge of the grid at r . For
max fective accretion rates but allows for a more accurate verifi-
the initial set-up, we calculate an atmosphere in hydrostatic
cation of energy conservation. We consider only data from
equilibrium with a constant luminosity L =10−3 L using
p (cid:12) afteranearlyadjustmentphase,oncethelab-frameaccretion
theusualequationsofstellarstructure(butEquation(7d)as
rate at the shock is equal to the one set through the outer
appropriateforanatmosphere):
boundaryconditions,describedbelow.
dm
r =4πr2ρ, (7a)
dr 2.4. Boundaryconditions
dT T dP
For the hydrodynamics, reflective (zero-gradient) bound-
=∇ , (7b)
act
dr P dr ary conditions are used at r in the density, pressure, and
min
dP Gm
=−ρ r, (7c) velocity,i.e.,
dr r2 dP dρ dv
dL dm (cid:18) dS(cid:19) = = =0 (9)
= r ε−T , dr dr dr
dr dr dr
Sincetheconditiondv/dr=0ensuresthatnomassflows
=0, (7d)
overtheboundary, itisnotnecessarytoenforcehydrostatic
4
equilibriumatr .Intheradiationtransportalso,weprevent et al. (2014) gas opacities combined with the dust opacities
min
theflowofenergyoverr byusing fromSemenovetal.(2003)andcomparetheseinFigure1.
min
Note that the Bell & Lin (1994) lacks water opacity
dE
rad =0. (10) lines just above the dust destruction temperatures (M. Ma-
dr
lygin, priv. comm.; see also Figure 1 of Arras & Bild-
Theouteredgeofthegridr issetwelloutsideoftheat-
max sten 2006); as a consequence, their opacities reach down to
mosphereandawayfromtheshock. Forthehydrodynamics, κ ∼10−6 cm2g−1 for ρ =10−11 gcm−3, where κ is the
R R
wechooseanaccretionrateandapproximatethevelocityas
Rosselandmean,somefourordersofmagnitudesmallerthan
thefree-fallvelocity:
in more recent calculations (Freedman et al. 2008; Malygin
(cid:115) (cid:18) (cid:19) etal.2014).
1 1
v(r)=vff(rmax)= 2GMp r −R , (11) Figure A1 shows that for low masses and accretion rates,
max acc theshocktemperatureshouldbeT (cid:46)1500K,inwhichcase
s
with R defined in Equation (1). Mass conservation then the dust is not destroyed and the opacity is relatively high.
acc
determinesthe‘free-falldensity’: When higher temperatures are reached, the opacity is lower
byordersofmagnitude,sothatthetotal(gasanddust)Rosse-
M˙
ρff(rmax)= 4πr2|v(r )|. (12) landopacitiesrangefromκR∼10−2 to10cm2g−1 overall.
max Thisprovidesapproximatevalueswhenconsideringconstant
Thepressuregradientheretooisrequiredtovanish: opacities. Since 1-T requires only the Rosseland mean, the
dP(cid:12)(cid:12) subscriptonκRwillbedroppedhereafter.
(cid:12) =0. (13)
dr(cid:12) 2.6. Quantitiestobemeasured
rmax
2.6.1. Efficiencies
We considered for some simulations a Dirichlet boundary
condition with P=P(ρ (r ), T ) for a nebula temper- The main goal of this study is to determine the radiative
ff max neb
atureT , takenasT =150K(e.g.,Mizuno1980). This lossefficiencyoftheaccretionshock.Thereareseveralways
neb neb
didnotchangetheresultssignificantly. ofdefiningthis. Theclassicaldefinition,ηkin,indicateswhat
Finally, the radiation outer boundary condition is usually fraction of the inward-directed kinetic energy flux is con-
settotheflux-divergence-freecondition vertedintoajumpinoutgoingradiativeflux(e.g.,Hartmann
et al. 1997; Baraffe et al. 2012; Zhu 2015). This kinetic-
∂r2E
rad =0, (14) energyluminosityisatmost
∂r
1 1
which corresponds to a constant luminosity if the reduced L =4πR 2 ρv3= M˙v2 (15)
acc,max p
2 2
flux f , defined in Section 3.2, is sufficiently close to 1.
red GM M˙
However, even when the flux at the outer edge is rather in ≈ p , (16)
R
the diffusion regime, we obtain similar results for a simple p
Dirichletboundaryconditionontheradiationtemperature. whereM˙ =4πr2ρvisthemassaccretionrate(neglectingthe
signofv)andthelastexpressionisvalidforfreefallfroma
2.5. Microphysics
largeradius. Therefore,theenergyactuallyradiatedawayat
To isolate the shock physics, we consider in this work a theshockiswrittenas
constant equation of state (EOS). The EOS enters into the GM M˙
L =ηkin p (17)
radiation-hydrodynamical simulations through the effective acc
R
p
heatcapacityratioγ ≡c /c =P/E +1,whereE isthe
p v int int
internal energy per volume, and through the mean molecu- and it is usually assumed that ηkin =100 percent (i.e., full
lar weight µ. Estimates presented in Appendix A suggest loss). This is called ‘cold accretion’. Note that this ηkin
thatthehydrogenwillbeinmolecularoratomicformatthe corresponds to the quantity (1−η) of Spiegel & Burrows
shock. Accordingly, γ =1.44 and γ =1.1 bracket the ex- (2012), αh of Mordasini et al. (2012), (1−α) of Hartmann
pected range, while µ varies from 2.353 to 1.23 (see Fig- etal.(1997),andX ofCommerc¸onetal.(2011).
ureA1). We present here and use a second definition based on the
We consider both constant and tabulated opacities. The total energy available. This efficiency ηphys measures what
contribution of the dust to the opacity dominates below ap- fraction of the total energy flowing towards the planet actu-
proximately1400–1600K,atwhichtemperaturetherefrac- allyremainsbelowtheshock,i.e.,isabsorbedbytheembryo:
tory components (olivine, silicates, pyroxene, etc.) evapo-
E˙(r )−E˙(r −)
rate(Pollacketal.1994;Semenovetal.2003). Thestandard ηphys≡ max shock , (18)
E˙(r )
opacity tables we use are the smoothed Bell & Lin (1994, max
hereafterBL94)tables.WecanalsomakeuseoftheMalygin where r − is immediately downstream of the shock and
shock
5
103 Malyginetal.(2014)
Semenovetal.(2003),‘nrm.h.s’
1) 102 Bell&Lin(1994)
−
g
2 101
m
(c 100
y
cit 10−1 ̺cgs 10−13,−11,−9
a =
op 10−2
n
a 10 3
e −
m
10 4
d −
n
a 10 5
l −
e
oss 10−6
R
10 7
−
10 8
−
102 103 104
Temperature(K)
Figure1. Gas, dust, and total Rosseland mean opacities from Malygin et al. (2014), BL94, and Semenov et al. (2003). Three densities
are shown: ρ =10−13,−11,−9 gcm−3. For the Semenov et al. (2003) opacities, we use their ‘nrm.h.s’ model, with dust grains made of
‘normalsilicates’([Fe/(Fe+Mg)]=0.4)ashomegeneousspheres.TheMalyginetal.(2014)opacitiesarekeptconstantabovethetablelimitof
T =2×104K.
the outer edge of the computation domain r is used as a whichthepotentialenergyisentirelyconvertedinkineticen-
max
proxyfortheaccretionradiusR correspondingtotheloca- ergybytheexternalpotential. Forfinitetemperatures,how-
acc
tion of the nebula. The material-energy flow rate is defined ever,a(small)pressuregradientbuildsupaheadoftheshock;
as inthiscase,onlypartofthechangeinpotentialenergyserves
toincreasethekineticenergy,theremaindergoingintointer-
E˙(r)≡−|M˙|[e (r)+h(r)+∆Φ(r,r )], (19)
kin shock nal energy and thus, outside of phase transitions, into pres-
where e = 1v2, e , and h=e +P/ρ are respectively sure.
kin 2 int int Alsobyenergyconservation,∆E˙(r ±)measuredinthe
thekineticenergy, internalenergydensity, andtheenthalpy shock
shock frame should be equal (up to a sign) to the change
perunitmassandΦtheexternalpotential. The∆Φtermin
in the luminosity ∆L across the shock. However, in the
Equation(19)accountsfortheworkdonebythepotentialon
casethattheradiativeprecursor(Zel’dovich&Raizer1967)
thegasdowntotheshock,withthepotentialdifferencefrom
is contained within the accretion region—roughly the Hill
r torgivenby
0
sphere—,itisnottrueanymorethatE˙(r +)=E˙(r ).In
(cid:18) (cid:19) shock max
1 1 fact,theluminosityupstreamoftheprecursorcanbesmaller
∆Φ(r,r )=−GM − . (20)
0 p
r r0 than downstream (i.e., the planet is invisible, at least in the
grey approximation), which would lead to a negative effi-
Thusηphysmeasureshowmuchoftheincomingenergyflow
ciency if using ∆L. Thus, ∆E˙ is a more useful numerator
E˙(r )inthe gas is stillflowing inwardonce ithas passed
max becauseitisintuitiveandapplicablebothwhentheprecursor
throughtheshock;ifbothareequal(E˙(r −)=E˙(r )),
shock max reachestor andnot.
ηphys=0andtheaccretionwouldbethoughtofas‘hot’. If max
NotefinallythatthedefinitionofEquation(18) takesinto
in the other extreme case none of the energy traverses the
accountthefactthateveniftheentirekineticenergyiscon-
shock, ηphys =100 percent, implying that the energy must
verted to luminosity, the net efficiency can still be zero if
have been entirely converted to outward-traveling radiation.
thisradiationisabsorbedbytheincomingmaterial. Thiswas
This therefore automatically reflects the fact that the (non-
seenby Vaytetet al.(2013a) inthecase ofLarson’s second
)heating of the planet is determined by the imbalance be-
coreandestimatedbyBaraffeetal.(2012)tobethecaseat
tweentheamountofkineticenergyconvertedtointernalen-
highaccretionratesinthecontextofmagnetosphericaccre-
ergyandthere-emittedradiation.
tionontostars.
Byenergyconservation,thenumeratorofηphys shouldbe
Thus,wewillfocusinthisstudyontheefficienciesasde-
equal to the difference E˙(r +)−E˙(r −) between the
shock shock finedabove: ontheclassical,‘kinetic’efficiencyηkin,which
material energy flow rate directly across the shock. This is
makesadirectstatementabouttheenergyconversionatthe
true for a zero-temperature gas (infinite Mach number), for
6
shock,withηkin<100percentforeitheranisothermalshock tothetransportofenergybyradiation, thisisamuchlarger
at Mach number M (cid:46)2.5 (Commerc¸on et al. 2011) or an compressionthantheinfinite-Machnumberlimitforahydro-
non-isothermalshock;andonthe‘physical’efficiencyηphys, dynamical shock, where ρ /ρ =(γ+1)/(γ−1)≈4 to 20
2 1
whichindicateshowmuchtheupstreamgasisabletorecycle forγ = 5 to1.1(e.g.,Mihalas&Mihalas1984;Commerc¸on
3
theenergyliberatedattheshock(Drake2006). et al. 2011). As it falls deeper in the potential well of the
planet, the gas slows down to slightly sub-free-fall speeds
2.6.2. Post-shockentropy
duetothepressuregradientcausedbytheincreasingtemper-
The post-shock temperature and thus entropy depend on atureanddensity.
thethermalprofileofthelayersbelowtheshock, whichare The post-shock pressure is given very accurately by the
expectedtoadjusttocarrytheluminosityfromdeeperdown rampressureoftheincominggas,
(Paxtonetal.2013).Sincehoweverwedonotattempttopre-
P =P =ρv2. (21)
dictthisluminosityaccuratelywithourset-upofatruncated post ram
atmosphere, thereportedtemperaturevalueswillserveonly This differs slightly from the strong-shock (high-Mach-
asanindication. Moreover,thereisanon-trivialrelationship number), non-radiating case where P = 2/(γ +1)ρv2
post
betweenthepost-shockentropyvaluesandtheirinfluenceon (Drake2006,hisEquation4.18),asweverifiedwithasimu-
the entropy of the planet’s deep adiabat; in particular, the lationusingahigherγ =5/3toincreasethedifference.
post-shock material does not simply set, weigthed by mass,
theinteriorentropy. Thisquestionisthesubjectofseparate 3.2. Opticaldepth,reducedflux,andradiationregime
studies(Berardoetal.2016,Marleauetal.,inprep.),which
Webeginbydiscussingthereducedflux. Thereducedflux
howeverrequiretheobtainedpost-shockentropiesasbound-
aryconditions.
f ≡F /(cE ) (22)
red rad rad
3. RESULTS:RADIALPROFILESANDEFFICIENCIES isalocalmeasureoftheextenttowhichradiationisstream-
Wehaveperformedalargenumberofsimulations,varying ing freely (fred →1) or diffusing (fred →0). This is thus
physical parameters (mass, radius, accretion rate) but also the more correct, physical measure of what is often loosely
computational or numerical settings (technique for accret- termed the optical depth, as discussed below. The reduced
ing gas into the domain, outer temperature boundary condi- flux, radiation quantity R, and flux limiter λ are related in
tion, resolution, Courant number, etc.). For the latter, we general by fred =λ(R)R. Note that the effective speed of
select the most stable set-up (as described in Section 2.2 propagationofthephotonsisceff= fredc.
above), and present results for a typical combination rele- Next,weconsidertheopticaldepth. Forafree-fallprofile
vant to core accretion formation calculations (Bodenheimer with Racc (cid:29)rshock (so that ρ ∝r−3/2) and a radially suffi-
etal.2000;Mordasinietal.2012). Welookattheproperties ciently constant opacity (κ ∝rα with |α|(cid:28) 1), the optical
2
oftheaccretionshockforM =1.3M ,M˙ =10−2 M yr−1, depthtotheshockis
p J ⊕
and rshock ≈1.8 RJ. The Bondi, Hill, and resulting accre- (cid:90) ∞
∆τ = κ(r)ρ(r)dr (23a)
tion radius according to Equation (1) are R ≈4200 R ,
Bondi J rshock
R ≈800 R , and R ≈250 R (for T =150 K and a
Hill J acc J neb =2κρr (const.κ), (23b)
shock
solar-massstar,whichhoweverdoesnotaffectR strongly).
acc
Figure 2 shows the detailed structure of the accretion flow where κρ is evaluated at the shock2. For the data of Fig-
neartheshockforκ =10−2 and1cm2g−1, aswellaswith ure2,ρ=1.5×10−10gcm−3upstreamoftheshock,sothat
the Bell & Lin (1994) opacities. For the EOS, we con- ∆τ =3.9(cid:0)κ/1cm2g−1(cid:1). Thisagreesverywellwiththeac-
siderahydrogen–heliummixturewithaheliummassfraction tualopticaldepths(measuredfromr =0.7R ≈250R )
max acc J
Y =0.25andcaseswherehydrogeniseverywheremolecular in the constant-opacity cases. For the simulation with the
(µ=2.353,γ=1.44)oratomic(µ=1.23,γ=1.1). Thera- Bell & Lin (1994) opacities, the estimate is moderately ac-
dialstructuresareasexpectedandshowanumberoftypical curate, if one takes for κ not the actual pre-shock value
features,whichwediscussinthefollowing. (κ ∼10−5 cm2g−1,setbythegas)butratheratypicalvalue
(κ ∼1cm2g−1)intheouterregions(r(cid:38)40R ), wherethe
J
3.1. Density,velocity,andpressure dustisnotdestroyed. Alsofornon-constantopacities, then,
The density and velocity reveal gas almost exactly free-
fallingontoanearlyhydrostaticatmosphereabruptlycutoff
at the shock. The mass in the total domain, dominated by 2Thisjustifies(withinafactorofafew)theestimateκρrshockofStahler
etal.(1980)fortheopticaldepthupstream(andnotdownstream,asthefor-
the post-shock region, is typically ∆M∼10−4 M⊕, making mulamightatfirstsuggest)oftheaccretionshockinthecontextofLarson’s
perfectly justified the neglect of the self-gravity of the gas. secondcore. Theestimateiscertainlyroughforanon-constantopacitybut
atleastitdoesestimatetheopticaldepthinthecorrect(upstream)direction.
The density jumps at the shock by a factor ρ2/ρ1 ∼ 200, AsimilarexpressionisusedbyMordasinietal.(2012)intheirboundary
whereρ andρ arethepost-andpre-shockdensity. Thanks conditions.
2 1
7
10−4 5000 7
3m)− 1100−−65 e(K) 44050000 30L−⊙56¢
Density(gc 111000−−−987 Temperatur 233505000000 Luminosity1 1234¡
10−10 2000 0
1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4
Radius(RJ) Radius(RJ) Radius(RJ)
10 101
0 100 104
1Velocity(kms)− −−−−5432100000 21Opacity(cmg)− 11110000−−−−4321 Temperature(K) 103
−
−60 1.6 1.8 2 2.2 2.4 10−5 1.6 1.8 2 2.2 2.4 10120−12 10−11 10−10 10−9 10−8 10−7 10−6
Radius(RJ) Radius(RJ) Density(gcm−3)
102 102
1
101 E max
sure(bar) 111000−−210 dflux/Fc|| 000...864 epthfromr 110001
s e d
Pre 1100−−43 Reduc 0.2 ptical 10−1
O
10−5 0 10−2
1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4
Radius(RJ) Radius(RJ) Radius(RJ)
24
1ryon)− 2202 Mp=1.3MJ,M˙ =10−2 M⊕yr−1,rshock≈1.8RJ
baB 18 γ=1.44,µ=2.253,κ=1cm2g−1
y(k 16 γ=1.1,µ=1.23,κ=1cm2g−1
Entrop 1124 γγ==11..11,,µµ==11..2233,,κBe=ll1&0−L2icnm(21g9−914)
10
1.6 1.8 2 2.2 2.4
Radius(RJ)
Figure2. Detailed shock profiles for simulations with Mp =1.3 MJ, rshock ≈1.8 RJ, and M˙ =10−2 M⊕yr−1 using a constant equation
of state and constant opacities or the Bell & Lin (1994) opacities (see legend). The simulation grids extend to 0.7 of the accretion radius
Racc≈250RJ butonlytheinnerregionisshown. Theaxislabelsdescribethequantitiesshown,andonlyafewcommentsareneeded: The
temperaturepanelalsoshowsthelowerboundestimateofEquation(28b;filleddots);intheluminositypanel,themaximalaccretionluminosity
Lacc,max(r)= 21M˙v(r)2≈GMpM˙/risshownateveryradius(graydashedcurve);thevelocitypanelalsodisplaysthefree-fallvelocityfrom
Racc(thesameforallsimulations;greydottedline);intheopacitypanel,simulationswithconstantopacityoverlap;inthetemperature–density
phasediagram,thesoliddotsmarktheup-anddownstreamconditionsofshock,thesolidlinesshowcontoursof10,50,and90percentatomic
hydrogen(relativetothehydrogenspecies),andthegreyregionhightlightswherethedustisbeingdestroyed,withκ∼1cm2g−1belowand
κ∼10−6–10−3cm2g−1above;thepressurepanelalsodisplaystherampressurePram=ρv2,thesameforallsimulations(dashedcurve);and
theentropyiscomputedself-consistentlyfromEquation(29).
8
theopticaldepthtotheshockwillberoughlygivenby Mihalas (1984) term ‘static diffusion’. Therefore, the radi-
(cid:18) κ (cid:19)(cid:18) M˙ (cid:19) ation is able to diffuse into the incoming gas, heating it up
∆τ ∼3 out to the edge of the computation grid, near the accretion
1cm2g−1 10−2M⊕yr−1
radius. Inotherwords,theshockprecursorislargerthanthe
(cid:115)
(cid:18) (cid:19)(cid:18) (cid:19)
× 1MJ 2RJ , (24) Hill radius, which implies that the radiation should be able
M r to escape from the system to at least the local disk. In this
p shock
sense, the shock for these parameter values is an optically
using that M˙ = 4πr2ρv. Since the nebula should always
thick–thin shock (down- and upstream, respectively) in the
be at temperatures lower than the dust destruction temper-
classification of Drake (2006). That despite the somewhat
atureT ≈1500K,thehighopacityofthedustwillalways
dest highopticaldepththeshockisnotequivalenttoahydrody-
contribute to the optical depth. Therefore, independently of
namicalshockisalreadyhintedatbythelargecompression
whetherdustisdestroyedintheinnerpartsoftheflow,close
ratiopointedoutinSection3.1.
totheshock,theopacitytoinsertinEquation(24)shouldbe
oforderκ ∼1cm2g−1. 3.3. Temperature
Secondly, writing E = F /(cf ) ∝ L/(r2f ), it
rad rad red red 3.3.1. Shocktemperature
is clear that when L/f is locally spatially constant
red
(L/f ∝ rβ with |β| (cid:28) 2), the radiation quantity R = Forallchoicesofκ andtheEOS(γ, µ),thetemperatures
red
immediatelyup-anddownstreamoftheshockareessentially
1/(κρ)|∂lnE /∂r|isgivenbyR≈2/(κρr)whereκρ and
rad
rareevaluatedlocally3. Thisresultappliesingeneral,inde- equal,i.e.,thereisnojumpinthetemperature. Thisisthusa
supercriticalshock(Zel’dovich&Raizer1967),inwhichthe
pendently of the radiation regime (diffusion or free stream-
downstreamgasisabletopre-heattheincominggasuptothe
ing).
post-shocktemperature.Notethatthe1-T approachtothera-
Combining these observations leads to the result that, in
diationtransportusedherecannotrevealtheZel’dovichspike
thecaseofconstantopacityandL/f ,thereducedfluxup-
red
expected in the gas temperature. This feature of radiation-
streamoftheshockis
(cid:18) (cid:19) hydrodynamical shocks consists of a sharp increase of the
4 4
f (r +)=λ × (25a) gastemperatureimmediatelybehindtheshock, followedby
red shock
∆τ ∆τ
aquickdecreaseina‘radiativerelaxionregion’,whilethera-
(cid:18)3 (cid:19)−1 diationtemperatureremainsessentiallyconstant(Zel’dovich
≈ ∆τ+1 , (25b)
4 &Raizer1967;Mihalas&Mihalas1984;Stahleretal.1980;
where the second line would be an equality for the simple see Drake 2007 and Vaytet et al. 2013b for a more detailed
flux limiter λ =1/(3+R) (LeBlanc & Wilson 1970; Lev- description). However,thisisnotofconcernsincethisspike
ermore 1984; Ensman 1994). We will return to this result isverythinbothspatially(physically,afewmolecularmean
in Section 3.3. For a non-constant opacity, Equation (25) freepaths,broadenedinsimulationstoafewgridcells;e.g.,
providesinfactanapproximatelowerboundof f (r +) Ensman 1994; Vaytet et al. 2013b; Marleau et al. in prep.)
red shock
given∆τ orviceversa: indeed,alowopacityinfrontofthe andinopticaldepth,andbelowtheZel’dovichspike,themat-
shock will drive down f (compared to the prediction of terandradiationequilibrateagain. Therefore,theZel’dovich
red
Equation(25))butwithoutdecreasingmuchthetotaloptical spikeshouldaffectneitherthepost-shocktemperatureoren-
depth. Thishighlightstheconceptualindependencebetween tropynortheshockefficiency. Apossibledisequilibriumin
the (non-local) optical depth and the (local) radiation trans- temperaturesjustupstreamoftheshockwillbeexploredina
portregime(free-streamingordiffusion). forthcomingpublication.
The optical depths from the shock out to rmax ≈Racc are We find shock temperatures of Tshock ≈ 2500 K for the
∆τ ≈2–5 for all except the κ =10−2 cm2g−1 simulation, cases with a low pre-shock opacity (κ =10−2 cm2g−1 or
whichhas∆τ ≈3×10−2. (Acomparisonrunwiththedust with Bell & Lin (1994)), but Tshock ≈3500 K for the other
opacities of Semenov et al. (2003) and the gas opacities of two cases, both with κ = 1 cm2g−1. These temperature
Malyginetal.(2014)yieldedverysimilarprofilesandopti- values (and their relatively large difference of 1000 K) can
caldepths.) Intheκ (cid:54)=10−2 cm2g−1 simulations,theshock be understood from an analytical estimate, presented next.
wouldthereforebecalled‘opticallythick’. However,theef- Firstly,onecanalwayswrite
fectivespeedoflightceff= fredc(cid:38)0.3cthroughouttheflow F(r +)=F(r −)+ηkin1ρv 3, (26)
(see below), which is still orders of magnitude larger than shock shock 2 shock
thegasflowspeedv∼10−4c. ThisistheregimeMihalas&
wherehereρ isthedensityjustaheadoftheshock,v is
shock
the velocity at the same location, and ηkin is the ‘kinetic-
energy loss efficiency’, discussed in Section 3.6. In gen-
3 ItmayseemsurprisingthatthelocalquantityRdependsonanabso-
eral, the flux on either side of the flux is F(r ±) =
lutecoordinaterbutthisisinfactasimpleconsequenceofthe(spherical) shock
geometry. f ±caT4(r ±),where f ±≡ f (r ±)andaisthe
red shock red red shock
9
radiationconstant,relatedtotheStefan–Boltzmannconstant accretion.Theirassumptionsaboutthereprocessingofshock
σ by ac=4σ. Note that here f =F /cE should be photons4 implythat,whenF (r −)(cid:28)F (r +)and
red rad rad rad shock rad shock
negative(oneusuallyimplicitlytakesthenorm)ifthedown- neglectingtheirT term,∆f ≈ f +≈1/3automatically.
d red red
streamradiationisflowinginward(F <0). Foranisother- Commerc¸onetal.(2011,theirEquation22or53)giveafor-
rad
malshockatT ,Equation(26)thenimpliesthat mula similar to Equation (27) in the limiting case ηkin =1
shock
butdonotincludethefactor1/(4∆f ).Thisisbecausethey
ηkin ρv 3 red
σT 4= shock , (27) equatethetemperatureattheshockwiththeeffectivetemper-
shock 4∆f 2
red ature needed to radiate away the kinetic energy, increasing
where∆f ≡ f +−f −. CombiningwithEquations(24) thetemperatureestimateby≈40%(afactor41/4≈1.4),or
red red red
and(25)yieldstheestimatesforanisothermalshock ≈1200KforTs≈3000K.
(cid:18)r (cid:19)−3/4 3.3.2. Temperatureprofile
T (∆τ (cid:28)1)≈2315K shock
shock 2R
J Equation (22) implies that, if the luminosity and the re-
(cid:18) M˙ (cid:19)1/4(cid:18) M (cid:19)1/4 duced flux are radially roughly constant, T ∝ r−1/2 since
10−2M⊕yr−1 1MJ (28a) L=4πr2Frad,independentlyoftheopticaldepthtotheshock.
Thisisthecasefortheconstant-κ simulationsbutnotsofor
(cid:18)r (cid:19)7/8(cid:18) κ (cid:19)1/4
T (∆τ (cid:29)1)≈2710K shock thetabulatedopacities(atlargerradialdistancesthanshown).
shock 2R 1cm2g−1
J Note that if the temperature increased solely due to adi-
(cid:18) M˙ (cid:19)1/2(cid:18) M (cid:19)1/8 abatic compression, i.e., at constant entropy in the absence
10−2M yr−1 1M , (28b) ofradiationtransport,wewouldhaveT ∝ργ−1∝r−1.5(γ−1),
⊕ J
i.e., T ∝r−0.15 or T ∝r−0.66 for γ =1.1 or 1.44, respec-
wherea(cid:0)ηkin(cid:1)1/4 factorwasleftoutontheright-handsides tively. Thus, when T ∝r−1/2, entropy decreases inward if
since we find it is ≈1 (see Section 3.6). The first expres- γ >4/3≈1.33.
sionusedthat,byEquation(25),∆τ (cid:28)1implies f +≈1,
red
and further took f − (cid:28) f +. The second case is some- 3.4. Entropy
red red
what crude for non-constant opacities. This assumes a con-
Tocomputetheentropy,weusetheSackur–Tetrodeequa-
stantluminosityintheshock’snearupstreamvicinity. Since
tion (e.g., Marleau & Cumming 2014; Berardo et al. 2016,
the post-shock region is very dense, f − is small; this is
red andreferencestherein)foranidealgascomposedofH and
2
equivalent to neglecting the downstream luminosity, which
HeorHandHe:
isrelatedinanon-trivialwaytotheinteriorluminosityofthe
(cid:18) (cid:19)
T
planet(Berardoetal.2016;Marleauetal.,inprep.). SH2–He=8.80+3.38log10 1000K
The filled circles in Figure 2 show the lower bound of
(cid:18) (cid:19)
Equation(28b). Thesimulationswithalowpre-shockopac- P
−1.01log , (29a)
ity (κ = 10−2 cm2g−1 or with Bell & Lin (1994)) have 10 1bar
f ≈1 upstream of the shock and indeed have a tempera- (cid:18) T (cid:19)
red
ture given by Equation (28b), whereas in the other cases a SH–He=13.47+4.68log10 1000K
higher temperature is needed to carry a similar luminosity. (cid:18) (cid:19)
P
Thedifferenceisquitelargeandnearly1000K.Onewayof −1.87log10 1bar , (29b)
thinkingaboutthisisthattheeffectivespeedoflightislower
thanc,sothatE mustincreaseinordertoreachthesame respectively, using Y =0.243, and where the entropies are
rad
Frad=ceffErad. in units of Boltzmann’s constant per baryon, kB/baryon.
Interestingly, the molecular- and atomic-hydrogen cases In Figure 2, we see that the entropy decreases across the
leadtoaverysimilartemperatureTshock=3500K.Thephase shock by |∆S| ≈ 2.5 and 4.0 kB/baryon for the molecu-
diagram indicates that the atomic-hydrogen simulation with lar and atomic case, respectively. (In general but for con-
κ=1cm2g−1isself-consistent,butthatthecasewithatomic stantγ and µ,thejumpinentropyatanisothermalshockis
hydrogen and detailed opacities leads to temperatures and ∆S=−2.303/µ×log10(γM2)inunitsofkB/baryon.) That
densitieswherethedissociationprocess(andthusavarying theentropydecreasesthroughthisshockisactuallyinagree-
µ andγ)wouldbeimportant. Onecanalreadyanticipatethe
resultthat,foranisothermalshock,thehydrogenshouldre-
combine in part through the shock (Marleau et al., in prep.) 4Theyassumethathalfofthephotonsgeneratedattheshockmovein-
ward,andtheotherhalfoutward;inturn,onehalfofthisoutward-moving
sinceatfixedtemperaturetheabundanceofH2increaseswith radiationisassumedtobereradiatedinwardbyanabsorbinglayeraheadof
density. theshock. Ifoneignoresthecontributionfromtheinteriorluminosity,this
impliesthatηkin=25percent. However,itseemstousthatoneneedsra-
Note that Stahler et al. (1980, their Equation 24) present
diativetransfercalculationssuchastheonespresentedhere(orusingmore
anestimatesimilartoEquation(27)inthecontextofstellar detailedradiationtransportasinDrake2007)tojustifythisaccounting.
10
ment with the statement that entropy increases across a hy- A general feature of these shock simulations is that L de-
drodynamical shock. Indeed, once it arrives at the radiative creasesradiallyoutward. Thisisnotduetoabsorptionofthe
shock found here, the gas has already seen its entropy in- lightwithoptical depthaccordingtoL∝exp(−∆τ), asone
crease from the value far outside of the precursor. (In the mightnaivelyexpect,butratherreflectsenergyconservation.
casethattheprecursorislargerthanthesimulationdomain, To derive this, we start with the total energy equation (e.g.,
asappliesforthesesimulations, this‘far-field’valuecannot Kuiperetal.2010),
beobtaineddirectly. However,alreadyatr istheentropy
max dE
muchlowerthandownstreamoftheshock.) Thustheradia- dttot +∇·([Ekin+H]v+Frad)=ρv·g, (30)
tiveshockwhichisthesubjectofthisworkcanbethoughtas
wherethetotalenergyvolumedensityisE =E +H,with
beingembeddedinausualhydrodynamicalshock, a‘shock tot kin
E = 1ρv2, and the enthalpy is H =E +P for an inter-
within a shock’ (Mihalas & Mihalas 1984), or a hydrody- kin 2 int
nalenergydensityE . ForaconstantEOS,E =ρc T =
namicalshockasbeingaradiativeshockwithaninfinitelyor int int v
ρ/(γ−1)×k T/(µm )=1/(γ−1)×P, where c is the
B H v
unresolvedthinprecursor. Separatetestsimulationswithex-
heatcapacity. Itiseasytoverifythatthethermaltimescales
tremelyhighopacityvalues(κ=102cm2g−1),suchthatthe
are much shorter than the dynamical timescales, so that the
precursoriscontainedinthesimulationdomain,confirmthat
flow is in steady state and the time derivative dE /dt can
tot
the post-shock entropy is higher than the entropy far away
be neglected. Also, M˙ is constant radially. Remembering
fromtheshock.
that∇·F=1/r2d/dr(r2F)foravectorF,Equation(30)be-
The post-shock entropies are respectively S ≈ 12 and
comes
20k /baryonforthemolecularandatomiccases. Compared
B (cid:18) (cid:19)
dL dh d 1 GM
to the range of entropies seen for cold starts to hot starts =M˙ +M˙ v2− p , (31)
(S≈8–14 k /baryon; Marley et al. 2007; Spiegel & Bur- dr dr dr 2 r
B
rows 2012; Mordasini 2013), this is an extremely large dif- whereh=H/ρ isthespecificenthalpypermassandish=
ference,whichisduemostlytothedifferentmeanmolecular γ/(γ−1)k T/(µm )foraconstantEOS.Theaccretionrate
B H
weights. Moreover, it highlights the importance of using a M˙ wastakentobepositivehere,i.e.,M˙ =|4πr2ρv|,andone
self-consistentEOSwhichfollowsinparticularthedissocia- can trivially replace GM /r by GM (1/r−1/R ). If the
p p acc
tionofhydrogen.However,theentropyvaluesdonotdepend secondtermontherighthandsideofEquation(31)issmall,
sensitivelyonthepreciseopacity(seeFigure2). Equation (31) shows that the radial decrease in L is mostly
Finally,itisimportanttorememberthattheseentropyval- due to the inward increase in enthalpy. Therefore, it is not
uesaremeanttoberatherindicativeatthisstage. Firstofall, an explicit function of the optical depth, although T(r) and
theyarenotentirelyself-consistentwiththeprobablestateof thus h(T) are indirectly set by the opacity. Note that this
the hydrogen in all parts of the domain (see the phase dia- derivationisvalidforageneralEOS(withvariableeffective
gram in Figure 2). Second of all, what they actually imply γ)andalsodoesnotdependontheopacitybeingconstant.
forthepost-formationentropyneedstobeworkedoutsepa- Equation(31)canbeintegratedtoyield,whenthesecond
rately, with a study of the post-shock settling region and its termontherighthandsideofEquation(31)isnegligible,
couplingtotheplanetinterior(Berardoetal.2016,Marleau (cid:20) M˙∆h(r) (cid:21)
etal.,inprep.). L(r)−L =∆L(r ) 1− , (32)
downstr shock ∆L(r )
shock
3.5. Luminosity where ∆L(r )=ηkinL is the jump in luminosity
shock acc,max
The luminosity increases from the imposed L=0 value at the shock, and ∆h(r)≡h(r )−h(r) is the change in
shock
at r to the shock where it jumps by a finite amount ∆L, enthalpy relative to the shock, with ∆h > 0 for outwards
min
then decreasing with radius. The value of L downstream of decreasing enthalpy. This result seems plausible: the in-
the shock reflects in part the cooling of the layers below it, ward enthalpy flux is comparable to the outward radiation
and is set in reality also by (inefficient) convective energy flux only when the infalling gas absorbs a significant frac-
transport, which we do not attempt to include in these sim- tion of the radiation and thus decreases L. The maximal
ulations. Thus the post-shock gas will probably have a dif- dropinluminosityoccursforh(r )(cid:28)h(r ),i.e.,when
max shock
ferentthermalhistorythanifthelayerswereallowedtosink theeffectivenebulatemperatureT (cid:28)T . Thisleadsto
neb shock
further down into the planet instead of stopping at most at L(r )−L(r +)=−M˙h(T ).
max shock shock
r . Nevertheless, the obtained immediate post-shock lu-
min
minositiesareroughlyL ≈3×10−4 L andthushave 3.6. Efficiencies
downstr (cid:12)
valuescomparabletothe(rough)internalluminositiesofac- NextweshowinFigure3themainresultfortheexamples
cretingplanets(Mordasinietal., submitted). Therefore, the ofFigure2,thelossefficiencyηphys oftheaccretionshock.
inclusion of convection or similar changes to the tempera- Werecallthatηphys=0wouldcorrespondtoallthekinetic
turestructureshouldnotleadtoverydifferentvaluesforthe energy of the gas being absorbed by the planet and the gas
post-shockregion. beingaccreted,whileηphys=100percentwouldcorrespond