The formation of HD 149026 b

0.44 MB·English

by  C. Broeg

Mon.Not.R.Astron.Soc.000,1–6(2002) Printed5February2008 (MNLATEXstylefilev2.2)
The formation of HD 149026 b
C. Broeg1,2,3⋆ and G. Wuchterl3
1 Astrophysical Institute and Obervatory, Friedrich-SchillerUniversity, Schillerg¨aßchen 2-3, Jena, 07745, Germany
2 Max-Planck Institute for extraterrestrial Physics, Giessenbachstrasse, Garching, 85741, Germany
3 Thu¨ringerLandessternwarte, Sternwarte 5, Tautenburg, 07778, Germany
Accepted 2007Jan9.Received2006Dec18;inoriginalform2006September 27
7 ABSTRACT
0 Today, many extrasolar planets have been detected. Some of them exhibit properties
0 quite different from the planets in our solar systemand they have eluded attempts to
2 explaintheirformation.OnesuchcaseisHD149026b.ItwasdiscoveredbySato et al.

n (2005).Atransit-determinedorbitalinclinationresultsinatotalmassof114M⊕.The
a unusuallysmallradiuscanbeexplainedbyacondensibleelementcorewithaninferred
J mass of 67M⊕ for the best fitting theoretical model.

1 In the core accretionmodel, giant planets are assumed to form around a growing
1 core of condensible materials. With increasing core mass, the amount of gravitation-
ally bound envelope mass increases. This continues up to the so-called critical core

1 mass – the largest core allowing a hydrostatic envelope. For larger cores, the lack of
v
static solutions forces a dynamic evolutionof the protoplanetin the process accreting
0
large amounts of gas or ejecting the envelope. This would prevent the formation of
4
HD149026b.
3
By studying all possible hydrostatic equilibria we could show that HD149026b
1
0 can remain hydrostatic up to the inferred heavy core. This is possible if it is formed
7 in-situ in a relatively low-pressure nebula. This formation process is confirmed by
0 fluid-dynamic calculations using the environmental conditions as determined by the
/ hydrostatic models.
h
We present a quantitative in-situ formation scenario for the massive core planet
p
HD149026b.Furthermorewepredicta widerangeofpossiblecoremassesforclose-in
-
o planets like HD149026b. This is different from migration where typical critical core
r masses should be expected.
t
s
a Key words: planets and satellites: formation – planetary systems: formation
:
v
i
X
r
1 INTRODUCTION eral migration theories predicting that the planets form at
a
largeorbitaldistances(around4-5AU)andmigrateinwards.
At the moment, there are two competing theories for gi-
Bothcontinuous(Lin et al.1996)andsuddenmigration(the
ant planet formation. In one theory, the solar nebula frag-
jumping Jupiters of Weidenschilling 1977) have been pro-
ments directly due to gravitational instability to form a
posed. On the other hand, Wuchterl (1996, 1997) showed
giant planet (Cameron 1978; Decampli & Cameron 1979).
that in-situ formation could occur if sufficent amounts of
In the second, the core accretion scenario, a core or proto-
solids and gas are available in theplanets feeding-zone.
planet forms through accretion of planetesimals and when
itsmassreachessomecriticalvaluethesurroundinggaseous
envelope is supposed to become unstable and to collapse The planet HD149026b was discovered by Sato et al.
onto the core, in the process forming the giant planet (2005) atadistancea ofonly 0.042AU.Because it wasdis-
(Perri & Cameron 1974; Mizuno et al. 1978). For an in- coveredbyboththeradial velocity andthetransit method,
depth discussion see Wuchterlet al. (2000). In this paper, its mass and density are known. It has a total mass of
wewill assumethatthenebulaisgravitationally stableand 114M⊕ andanunusuallylargedensity.Calculationsbythe
follow thesecond idea. discovery-team give themost likely core mass as 67M⊕.
Soonafterthediscoveryofthefirstexo-planet,51Pegb,
byMayor & Queloz(1995)theoristshavecomeupwithsev-
WewillshowthatthelargecoreofHD149026bcannot
be explained by migration. It has to form in-situ to allow
⋆ E-mail:[email protected] such a large subcritical core-mass.
(cid:13)c 2002RAS
2 C. Broeg and G. Wuchterl
2 MODELLING THE EQUILIBRIUM • The outer radius is given bythe Hill radius:
ENVELOPE STRUCTURES
M
r =a3 , (7)
Every protoplanet in our model consists of a solid core of Hill r3M∗
constant density (ρcore= 5500kgm−3)1 and an envelope of
hydrogen and helium with a helium mass fraction of 0.24. whereaistheplanet’ssemi-majoraxis,M itsmass,andM∗
Thecompositionoftheenvelopeisassumedtobeprotosolar. themass of thehost star.
Following therecommended procedure for the SCVH equa- • TheluminosityLisdefinedastheenergylibrationrate
tionsofstate,thatcontainnowhigh-Zelements,theireffects obtained for aconstant planetesimal accretion rateand the
are accounted for by a somewhat enhanced He-mass frac- dissipation of the planetesimal kinetic energy at the core
tion. Heavy elements and their condensates are fully taken surface:
into account in the opacities to calculate radiative transfer rHill GM
efficiency in detail. U(rcore) = −Z r2rdr, and
rc
L = −(U−U0)M˙, (8)
2.1 Model equations and assumptions
where U0 is arbitrary. U0 is the gravitational potential at
The hydrostatic equilibrium configuration of the envelopes rHill.
is given bythe well known equationsof stellar structure:
dM
= 4πr2ρ(P,T), (1)
dr 2.3 Constituent relations
dP GM
= − ρ(P,T), (2) Tofully specify thedifferentialequation system, ∇s, ρ,and
dr r2
κneedtobespecified.Weusethefollowingequationofstate
dT T dP
= ∇(P,T), (3) and opacity tables.
dr P dr
dL
= 0, (4)
dr
2.3.1 Equation of state
dU GM
= , (5)
dr r2 ∇s(P,T), and ρ(P,T) are interpolated from Saumon et al.
This, together with the boundary conditions (see sec- (1995).Firsthydrogenandheliumareinterpolatedindepen-
tion2.2),determinesthevaluesforthemassM,thepressure dently,then the mixed quantity is determined. ∇s is calcu-
P, the temperature T, the luminosity L, and the gravita- lated via spline derivatives from the mixed entropy includ-
tional potential U as a function of the radius r. G is the ingthe mixingentropy term,as suggested in Saumon et al.
gravitational constant and ∇ is calculated as: (1995). The helium mass-fraction is Y =0.24.
∇=minsmooth(∇rad,∇s), (6)
i.e. the adiabatic temperature gradient ∇s or the tempera- 2.3.2 Opacity
turegradientascausedbyradiativeenergytransportinthe
diffusion approximation, ∇ – whichever is smaller. This Rosseland-mean opacities κ(ρ,T) are interpolated from
rad
a combined table: Opacities include Rosseland-mean
corresponds to the use of zero entropy gradient convection
dust opacities from Pollack et al. (1985, lgT 6 2.3),
and the application of the Schwarzschild-criterion, but is
continuously differentiable across the transition region. ∇s Alexander& Ferguson(1994)valuesinthemolecularrange,
isdirectly givenbytheequation ofstate,∇ iscalculated andWeiss et al.(1990)LosAlamoshightemperatureopac-
rad
ities.
as:
3 κLP
∇ =
rad 64πσG T4M
(seeMihalas & Weibel-Mihalas1999,chapter6foraderiva- 3 UTILIZING ALL EQUILIBRIA TO
tion of this formula and explanation of all parameters). κ DETERMINE THE ENVIRONMENTAL
is the Rosseland-mean opacity and is determined by using PROPERTIES FOR HD149026b
tabulated values(see sect. 2.3.2).
Inordertosolvethesystemofdifferentialequations,arange
of parameters must be provided,namely the:
2.2 Boundary conditions
(i) core mass Mcore,
To solve thedifferential equation system we specify thefol- (ii) pressure at thecore Pcore,
lowing boundary conditions: (iii) mass of thehost star M∗,
(iv) semi-major-axis of theplanet a,
• Thecoreradiusisdefinedusingafixedcoredensityρc:
(v) nebula temperatureT , and the
neb
3 M (vi) planetesimal accretion rate M˙.
rcore = r3 4π ρc.
Parameters(i)and(ii)areourindependentparameters,they
are varied in a scale-free way, i.e. equidistant in the loga-
1 acoredensityof10500givessimilarresults rithm.
(cid:13)c 2002RAS,MNRAS000,1–6
The formation of HD149026b 3
(iii),(iv), and (v) are determined by the host star and
the position of the planet. The nebula temperature can be
calculated in thermal equilibrium with thestar: HD149026b: Mass and Luminosity CRFD+PiB(10)
T =280· a −1/2 L∗ 1/4 K (9)
neb “1AU” „L⊙« ]n 250
u
S
(see Hayashi 1981; Hayashi et al. 1985). L
The only remaining free parameter is the planetesimal −6
0 200
accretion rate M˙. In the case of HD149026b we choose an 1
unusually large number of M˙ = 10−2M⊕a−1. This value L / [
corresponds to particle-in-a-box planetesimal accretion for d 150
n
the density at the position of HD149026b in a minimum a
mass solar nebula (Hayashi 1981; Hayashi et al. 1985) with ] h
art 100
a gravitational enhancement factor Fg =21. ME
For HD149026b thecorrect values are therefore: s / [
• M∗ =1.3M⊙, L∗ =2.72L⊙, as 50
• a=0.042AU, M
• T =1754K,
neb
• M˙ =10−2M⊕a−1. 2 5 10+4 2 5 10+5 2 5
Usingthesevalueswehavecalculatedallhydrostaticen- Age/[a]Wuchterl and Tscharnuter 2003
velope structures by solving the equation system described
in chapter 2 for a wide range of core masses and pressures. Figure 2. Possible evolution of HD149026b. The figure shows
thetime-evolutionofthreequantities:thecoremass(dashed),the
Usingthissolution manifold itiseasytodeterminethecor-
total mass(solid),andtheluminosity(solid).Theageisdefined
rect environmentthat allows such a large critical coremass
asinWuchterl&Tscharnuter(2003),CRFD+PiB(10)standsfor
of roughly 70M⊕ (see figure 1).
fully hydrodynamic simulation with Safronov number 10. The
The results show that a hydrostatic equilibrium of
smoothluminositycurveisastronghintforquasistaticevolution
gaseous envelope and solid core is indeed possible at the which is confirmed by looking at the data. The dynamic terms
specified position, if the nebula pressure is P = 103.6Pa = arenegligible.
3980Pa. In that case the envelope mass is lgMenvkg−1 =
26.45 or roughly 47M⊕ leading to a total of 117M⊕ for
theprotoplanet.2 Thisis veryclose totheobserved mass of we use a gravitational enhancement factor Fg = 21 (or a
114M⊕. Safronov number θ = 10) which leads to a peak accretion
It should benoted,that thisisthecase only veryclose rate of M˙ =10−2M⊕a−1 as required.
tothehoststar.Withincreasingdistance,theaccretionrate In spite of the slight differences regarding the con-
M˙ decreases, and even more importantly the hill radius in- stituent relations, thedynamiccalculations confirm thehy-
creases for constant planet mass. Both effects lead to a re- drostatic model. The evolution of HD149026b is plotted
ducedcriticalcoremass.Therefore,suchalargecoreisonly in figure 2. The dynamic calculation confirms quasistatic
allowed for veryclose-in planets like HD149026b. evolution for the entire formation process of HD149026b,
no instabilities occur. This formation scenario explains the
highcoremassandshowsnodynamicaccretionphase–the
planet grows hydrostatically all the way to its final mass.
4 FLUID-DYNAMIC FORMATION OF
HD149026b
Knowingtheenvironmentalconditions,especiallythevalues
5 SOLVING THE FEEDING-ZONE PROBLEM
of the nebula pressure P and accretion rate M˙ we tried
neb
to reproduce the in-situ formation of HD149026b with a So far we have shown how HD149026b could have formed
thorough fluid-dynamiccalculation. providedthatthereisenoughmaterialavailabletoformthe
The algorithm for the dynamical calculations is de- planet at its current position. The lack of building material
scribed in Wuchterl (1990, 1991a,b). For this paper we use isusually considered thestrongest argument against in-situ
amodifiedconvectiontheoryasinWuchterl& Tscharnuter formation of close-in planets.
(2003). These calculations use a different equation of state It is true that in a classical feeding zone, i.e. 3–4 hill
(not SCVH, see Wuchterl 1989) and slightly different dust radii(equation(7))on bothsidesoftheorbit oftheplanet,
opacities, namelyforinterstellar dustinsteadofthePollack there is not enough material available in a gravitationally
et al. protosolar dust,see Wuchterl& Tscharnuter (2003). stabledisk.Thisproblemcanbecircumventedbyassuming
Because the dynamical calculation uses a particle-in-a- acontinuousflowofmaterialontothestarasisthecasefor
box accretion scheme instead of a constant accretion rate, accretion disks. According to Hartmann et al. (1998) typi-
cal T Tauri disks with an age of 1Ma have relatively low
accretion rates, they say:
2 Repeating these calculations with a core density of ρcore =
10500kgm−3andacriticalcoremassof67M⊕leadstothesame ”The median accretion rate for T Tauri stars of age ∼1Ma is
nebula pressure (P = 103.6Pa), a slightly larger envelope mass 10−8M⊙a−1;theintrinsicscatteratagivenagemaybeaslarge
(lgMenvkg−1=26.5,or53M⊕)andatotal massof120M⊕. as1orderofmagnitude.”
(cid:13)c 2002RAS,MNRAS000,1–6
4 C. Broeg and G. Wuchterl
Figure 1. The figure shows the envelope mass (logMenv) as a function of core mass and pressure at the core surface (in the figure
logMcandlogPc,respectively). ThelogarithmsaretakenfromthecorrespondingvaluesinSIunits(kg/Pa).Theresultsaregivenas
asurfaceinthree dimensions andthe surfacecolour ismapped fromtheoutside pressure(logPx). Theblow-uponthe right-hand-side
showstheregionofinterestaroundacoremassof70M⊕.
Todeterminethenebulaconditions foracriticalcoremassof70M⊕,wehavedrawnlinesofconstant nebulapressure(isobars,yellow,
in the range lgPPa−1 = 3..4, step 0.1) and lines of constant envelope mass (gray, in the range lgMenvkg−1 = 26..27, step 0.1). The
criticalcoremassforagivennebulapressureisgivenasthelargestpossiblecoremassforagivenisobar.Sotheisobarthatistangentially
touched bythelineof70M⊕ determines thenebulapressureforwhichthecriticalcoremassis70M⊕.ItislgPPa−1=3.6.
Usingthegraylineswecanimmediatelygivetheenvelopemasscorrespondingtothiscriticalcoremass.Itis47M⊕ givingatotalmass
ofroughly117M⊕.Thisisalmostexactlythetotal massofHD149026b.
For now, we will assume a very fast formation of the adiskaccretionrateof10−7M⊙a−1.Thisprovides140M⊕
planet of 105a and calculate the amount of material that of condensible material at theorbit of theplanet.
passes the orbit of the hypothetical in-situ planet. Using In the case of HD149026b we can go even fur-
Hartmann’sestimate, thematerial passingaclose-in planet ther. HD149026’s metalicity is given as [Fe/H]=0.36 (from
during its formation is therfore M = 105 × 10−8M⊙ = Sato et al. 2005). Assuming that the other heavy elements
333M⊕. For the envelope this is quite enough mass, but aresimilarly enriched,thisisanoverabundanceinheavyel-
what about the heavy element core? For solar compo- ementsof100.36 =2.3.Intotalweendupwith∼300M⊕in
sition gas we have a mass-fraction of condensible mate- heavy elements passing the planet’s orbit – This is enough
rial of ≈ 1/56 outside of the ice-line and ≈ 1/240 inside solid material for HD149026b.
(Hayashiet al.1985).Typicaldisk-lifetimesare∼1Ma.We
Thereisoneproblemremaining,concerningtheconden-
can now calculate the amount of condesible material that
sibles:thehightemperatureofthenebula.Atatemperature
passestheprotoplanet’sorbitduringthelifetimeofthedisk:
of 1754K and a pressure of 4000Pa will there be any con-
densedspecimenleft?AccordingtoDuschlet al.(1996)the
Mcond =106×10−8M⊙/240≈14M⊕. (10) mass fraction of silicon for this P-T-regime is 0.99-0.999.
So the silicates are still available. For other specimen, es-
Keeping in mind that HD149026b is an extreme case, it is pecially carbon, this is usually not the case. To answer the
quiteappropriate tousetheupperlimit givenbyHartman: questionofcondensiblemass-fractionprecisely,theexacten-
(cid:13)c 2002RAS,MNRAS000,1–6
The formation of HD149026b 5
vironmental conditions, such as pressure, temperature, and TTauristaraccretiondiskatanageof1Ma.Wecouldshow
chemical composition at the time of HD149026b’s forma- that for such a case, enough material is transported across
tion need to be taken into account. Without this informa- the orbit of a close-in planet like HD149026b to provide
tion, we can only speculate. As it is very likely that cm- sufficient material even when considering grain-evaporation
sizedandlargergrainsspiralinwardtothestar,afractionof caused bythe high temperatures this close to thestar.
theparticleswillneverreachequilibriumbeforemeetingthe The planet HD149026b could indeed have formed in-
protoplanet. Once inside the planet, the high pressure en- situ, i.e. at its present position. In that case its entire evo-
vironment prevents evaporation. Therefore we assume that lution would havebeen quasi-static – nodynamicaccretion
a fraction of the non-silicates can also be accreted onto the takesplace.FormingHD149026batlargerdistancesinthis
core of the protoplanet providing just enough material to manner – e.g. outside the ice-line – is not possible as the
form HD149026b. criticalcoremassdecreasesdramaticallywithincreasingdis-
tance.
Our model predicts a wide range of core masses for
close-inplanetswhilemigratingplanetsshouldallhavesim-
6 NOTES ON MIGRATION
ilar ’typical’ critical cores. This property could be used to
The reader might have pondered the lack of migration in distinguish between the two formation scenarios.
theabovediscussion–thisispartlyintended.Wewantedto
show that while different types of migration can take place
for differentembryomasses and nebulaproberties, theyare
not strictly necessary for theformation of close-in planets. ACKNOWLEDGEMENTS
InthecaseofHD149026bwecangoevenfurther:Our
ThisresearchwassupportedinpartbyDLRprojectnumber
calculations show that to obtain such a large core, at least
50-OW-0501.
the last phase of core accretion must have occured in close
proximitytoitspresentlocation,i.e.withoutmigration.The
’lastphase’inthiscasereferstothetimewhenthecoremass
grows beyond ∼ 30M⊕. Before that time, migration could
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