Table Of ContentDraft version January 26, 2010
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THE RUNTS OF THE LITTER: WHY PLANETS FORMED THROUGH GRAVITATIONAL INSTABILITY CAN
ONLY BE FAILED BINARY STARS
Kaitlin M. Kratter
DepartmentofAstronomyandAstrophysics,UniversityofToronto,50St. GeorgeStreet,TorontoON,M5S3H4,Canada
Ruth A. Murray-Clay
Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,MS-51,Cambridge,MA02138,USA
Andrew N. Youdin
0
CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto,60St. GeorgeStreet,Toronto,ON,M5S3H8,Canada
1
Draft version January 26, 2010
0
2
ABSTRACT
n
Recent direct imaging discoveries suggest a new class of massive, distant planets around A stars.
a
These widely separated giants have been interpreted as signs of planet formation driven by gravita-
J
tionalinstability,buttheviabilityofthismechanismisnotclearcut. Inthispaper,wefirstdiscussthe
6
local requirements for fragmentation and the initial fragment mass scales. We then consider whether
2
the fragment’s subsequent growth can be terminated within the planetary mass regime. Finally, we
placedisksinthelargercontextofstarformationanddiskevolutionmodels. Wefindthatinorderfor
]
P gravitationalinstabilitytoproduceplanets,disksmustbeatypicallycoldinordertoreducetheinitial
E fragment mass. In addition, fragmentation must occur during a narrow window of disk evolution,
after infall has mostly ceased, but while the disk is still sufficiently massive to undergo gravitational
.
h instability. Undermoretypicalconditions,disk-bornobjectswilllikelygrowwellabovethedeuterium
p burning planetary mass limit. We conclude that if planets are formed by gravitational instability,
- they must be the low mass tail of the distribution of disk-born companions. To validate this theory,
o
on-going direct imaging surveys must find a greater abundance of brown dwarf and M-star compan-
r
t ions to A-stars. Their absence would suggest planet formation by a different mechanism such as core
s
a accretion, which is consistent with the debris disks detected in these systems.
[ Subject headings: planet formation, accretion disks, binaries
2
v 1. INTRODUCTION stars have yet to turn up similar companions (Nielsen
4
& Close 2009). Standard core accretion models cannot
4 Motivated by the recent discovery of massive planets
form these planets, though further investigation is war-
6 on wide orbits, we explore the requirements for mak-
ranted.
2 ing gas giants at large separations from their host star
Infavorofthispossibility,allthreesystemsshowsome
. via gravitational instability, hereafter, GI. In particu-
9 evidence of processes related to core accretion: all have
lar,weconsidertheformationmechanismforthesystem
0 infraredexcessduetomassivedebrisdisksatlargeradii.
HR 8799 which contains three ∼10M objects orbit-
9 Jup Thisisatleastpartiallyaselectioneffectasthesesystems
ing at distances between ∼ 30 and 70 AU (Marois et al.
0 were targeted due to the disks’ presence. Nevertheless,
2008). Thestandardcoreaccretionmodelforplanetfor-
v: mation, already strained in the outer solar system, has these debris disks are composed of reprocessed grains
i difficultyexplainingthepresenceoftheseobjects. While from collisions of planetesimals. The disks’ long life-
X timesprohibitaprimordialoriginforsmallgrains—they
GI is an unlikely formation mechanism for close in plan-
r ets(Rafikov2005),formorewidelyseparatedplanets,or areremovedquicklybyradiationpressureandPoynting-
a Roberston drag (Aumann et al. 1984) and so must be
sub-stellar companions, the viability GI-driven fragmen-
regenerated from collisions between larger bodies that
tation deserves further investigation.
formed through the coagulation of solids at early times.
In the inner regions of a protoplanetary disk, gas can-
Therefore, planetesimal formation, a necessary ingre-
notcoolquicklyenoughtoallowagravitationallyunsta-
dient in core accretion models, has taken place (e.g.,
bledisktofragmentintoplanets(Rafikov2005;Matzner
Youdin & Shu 2002; Chiang & Youdin 2009).
&Levin2005). Forthisreason,coreaccretion—inwhich
In addition, other A-stars host planets at 1–2 AU
solid planetesimals collide and grow into a massive core
(Johnson et al. 2007) which, although they have a dis-
which then accretes a gaseous envelope—has emerged as
tinct semi-major axis distribution from planets orbit-
the preferred mechanism for forming planets at stellar
separations (cid:46)10 AU. Planets at wider separations have ing G and M stars, are likely formed by core accretion.
If future surveys demonstrate that this distribution ex-
only recently been discovered by direct imaging around
tendssmoothlytowideseparationplanets,thensimplic-
the A-stars HR 8799 (Marois et al. 2008), Fomalhaut
ity would argue against a distinct formation mechanism
(Kalasetal.2008),andpossiblyBetaPic(Lagrangeetal.
for the wide giants.
2009). Searches at large radii surrounding solar-type
Yet the standard core accretion model faces a serious
2
problematlargedistances. Theobservedlifetimesofgas The planets around HR 8799 probe a previously un-
disks are short, at most a few Myr (Hillenbrand et al. explored region of parameter space (Marois et al. 2008;
1992; Jayawardhana et al. 2006). In contrast, typical Lafreni`ere et al. 2009) because they are more distant
core accretion times increase with radius and exceed 10 from their host star. The three companions to HR
Myrbeyond20AU(e.g.Levison&Stewart2001;Goldre- 8799 are observed at separations from their host star
ichetal.2004). Whetherornotthistheoreticaldifficulty of 24, 38, and 68 AU. Their masses, estimated using
can be overcome will require careful modeling of the in- the observed luminosities of the planets in conjunction
teractionsbetweenplanetesimalsandtheyounggasdisk. with cooling models, have nominal values of 10, 10,
Could wide orbit planets have formed at smaller radii, and 7 M , respectively. A range in total mass of
Jup
and migrated outwards? Dodson-Robinson et al. (2009), 19–37 M is derived from uncertainties in the age of
Jup
have investigated the possibility of forming the HR 8799 the host star (Marois et al. 2008). Interpretation of
system via scattering in the absence of dynamically im- the cooling models generates substantial additional sys-
portant gas, but find that putting three massive plan- tematic uncertainty—recent measurements suggest that
ets into such closely spaced yet wide orbits is unlikely. thesemodelsmayoverpredictthemassesofbrowndwarfs
Crida et al. (2009) have suggested that under favorable by ∼25% (Dupuy et al. 2009). Fabrycky & Murray-Clay
circumstancesoutwardmigrationinresonancesmightbe (2008) have demonstrated that for planetary masses in
feasible. Alternatively, the core of a giant planet could thestatedrange,orbitalstabilityovertheageofthesys-
be scattered outward by a planet, or migrate outward tem requires that the planets occupy at least one mean-
before accreting its gas envelope, either by interactions motion resonance, and that for doubly-resonant orbital
with the gas disk (Type III migration; e.g. Masset & configurations total masses of up to at least 54M can
Jup
Papaloizou2003)orwithplanetesimalsembeddedinthe be stable.
gas (Capobianco, Duncan, & Levison, in prep). Neither HR 8799 has been called a “scaled-up solar system”
mechanism has yet been shown to move a core to such in terms of the stellar flux incident on its giant planets
largedistances,thoughthispossibilityhasnotbeenruled (Marois et al. 2008; Lafreni`ere et al. 2009). However for
out. understanding the formation of this system it is more
Giventhesedifficulties,itisnaturaltosearchforother useful to consider dynamical times, and disk mass re-
formation mechanisms, and GI (Boss 1997) stands out quirements. Because the dynamical time at fixed radius
as a promising alternative. If any planets form by GI, scalesonlyasM1/2,thedynamicaltimesarelargeratthe
∗
the recently discovered directly imaged planets are the locationsoftheHR8799planetsthatatthesolarsystem
most likely candidates (Rafikov 2009; Boley 2009; Nero giants. Since the total mass in planets greatly exceeds
&Bjorkman2009). Inthispaper,weexaminethispossi- the ∼ 1.5 M in the solar system, we can infer that
Jup
bilityinmoredetail,consideringtheexpectedmassscale (as with some other extrasolar systems) the primordial
offragmentsandtheeffectofglobaldiskevolutiononthe disk around HR 8799 was more massive than the solar
formation process. nebula and/or there was greater efficiency of planet for-
The inferred masses for the HR 8799 planets are close mation, especially in the retention of gas. Compared to
to the deuterium burning limit of 13MJup (Chabrier & solar system giants, longer dynamical times make core
Baraffe2000). Forsimplicity,wetakethisasathedivid- accretion more difficult and larger disk masses make GI
ing line between planets and brown dwarfs and we refer more plausible.
totheHR8799objectsasplanetsthroughout. However,
there is no reason for a given formation mechanism to
3. IDEALCONDITIONSFORGI-DRIVENFRAGMENT
function only above or below this threshold, and in fact, FORMATION
we will argue that if the HR 8799 planets formed by
We first determine where, and under what local disk
GI, their histories are more akin to those of higher-mass
conditions, fragmentation by GI is possible. Following
brown dwarfs than to lower-mass planets.
Gammie (2001), Matzner & Levin (2005) and Rafikov
To constrain GI as a mechanism for wide giant planet
(2005), we argue that for a disk with surface density Σ
formation, we set the stage by describing the HR 8799
andtemperatureT tofragment,itmustsatisfytwocrite-
system in §2. We review the standard requirements for
ria. First, itmusthaveenoughself-gravitytocounteract
fragmentation in §3 and discuss the initial mass scale of
thestabilizingforcesofgaspressureandrotationalshear,
fragments in §4. In §5 we show that under typical disk
as quantified by Toomre’s Q:
conditions, fragments will continue to accrete to higher
masses. We then discuss important global constraints c Ω
on planet formation provided by star formation models, Q≡ πGs Σ <Qo ∼1 (1)
disk evolution timescales, and migration mechanisms in
§6, and §7. We compare predictions of our analysis with (Safronov 1960; Toomre 1964), where c = (cid:112)kT/µ is
s
the known wide substellar companions and exoplanets theisothermalsoundspeedofthegaswithmeanparticle
in §8, suggesting that future observations will provide weight µ = 2.3m appropriate for a molecular gas, G is
H
a definitive answer to the formation mechanism for HR the gravitational constant, k is the Boltzmann constant,
8799. In the appendices we re-examine the heating and andΩistheorbitalfrequency. Equation(1)specifiesthe
coolingpropertiesofdisksthatarepassivelyandactively onset of axisymmetric instabilities in linear theory that
heated, with special attention to the implications for ir- can give rise to bound clumps (Goldreich & Lynden-Bell
radiated disks, which become increasingly relevant for 1965).1
more massive stars.
1Inarealisticdiskmodel,clumpslikelyformwithinspiralarms
2. THEHR8799SYSTEM
formedvianon-axisymmetric,non-linearinstabilities,althoughthe
3
The second criterion that must be satisfied for frag- (Levin 2007), where H = c /Ω is the disk scaleheight.
s
mentation to proceed is the so-called cooling time cri- Cossins et al. (2009) have shown that even when the
terion. The heat generated by the release of gravita- GIisnon-axisymmetric,themostunstableaxisymmetric
tional binding energy during the contraction of the frag- wavelength λ is one of the dominant growing modes,
Q
ment must be radiated away on the orbital timescale so suggesting that this is a reasonable estimate. While
thatincreasedgaspressuredoesnotstallfurthercollapse more numerical follow up will be necessary to pin down
(Gammie 2001). This implies the true distribution of fragments born through GI, at
present simulations show that this estimate may well be
t = 3γΣc2s f(τ) (cid:46)ζΩ−1. (2) a lower limit, but is the correct order of magnitude (Bo-
cool 32(γ−1) σT4 ley 2009; Stamatellos & Whitworth 2009).
Using equation(3), we rewrite the fragment mass ex-
Here ζ is a constant of order unity, γ is the adiabatic
plicitly as a function of temperature and location:
indexofthegas,andσistheStefan-Boltzmannconstant.
We take f(τ)=1/τ +τ (Rafikov 2005) for disk vertical (cid:18)H(cid:19)3 T3/2
optical depth τ = κΣ/2 and gas opacity κ. Numerical Mfrag ≈4π r M∗ =fm Ω (6)
modelsofcollapseinbarotropicdisksmeasurethecritical
valueζ throughtheinclusionofalosstermu/tcool inthe wherefm ≡(2π)2fqk/µ. Equation(6)demonstratesthat
equation for the internal energy, u. Estimates of ζ range atagivendisklocation,fragmentmassesdependonlyon
from ∼ 3−12, depending on γ (Rice et al. 2005), the temperature,withlowertemperaturesgeneratingsmaller
numericalimplementationofcooling(Clarkeetal.2007), fragments,subjecttotheminimumtemperaturerequired
andtheverticalstratificationinthedisk. Weassumeγ = for fragmentation by equation(4).
7/5, appropriate for molecular hydrogen, and we adopt Rafikov (2005) pointed out that there exists an abso-
ζ = 3 here. Although ζ was measured in disks whose lute minimum fragment mass at any disk location, when
temperatureiscontrolledbyviscousheating, weshowin the disk satisfies the equalities in equation(1) and equa-
Appendix A that the same expression (modulo slightly tion(2) and has τ = 1. The minimum temperature re-
different coefficients) should apply when irradiation sets quiredforfragmentationscalesasT ∝(τ+1/τ)2/5(equa-
the disk temperature, as will be the case in disks prone
tion4),sothecriticaltemperatureandfragmentmassare
to fragmentation (see Appendix B).
minimized at τ = 1, the optical depth for which cooling
A disk satisfying Toomre’s criterion for instability
is most efficient. The corresponding minimum mass as a
(equation 1) but not the cooling time criterion (equa-
function of location is:
tion 2) experiences GI-driven angular momentum trans-
port which regulates the surface density of the disk so Mf,min=fm(fqft)3/5Ω1/5 (7)
that Q ∼ Qo ∼ 1 and Q does not reach substantially (cid:16) r (cid:17)−3/10(cid:18) M (cid:19)1/10
smaller values (c.f. Appendix A). We can therefore use =1.5M ∗ (8)
Toomre’scriteriontodefinearelationshipbetweenΣand Jup 100 AU 1.5M(cid:12)
T at fragmentation, as a function of period:
which occurs for disk temperatures:
c Ω √
Σ= πGsQ =fq TΩ (3) T =7K(cid:16) r (cid:17)−6/5(cid:18) M∗ (cid:19)2/5 (9)
o f,min 100 AU 1.5M
(cid:12)
whereforconveniencewedefinef ≡(k/µ)1/2(πGQ )−1.
q o Equation (7) corresponds to Q = 1, Ωt = 3, and
We shall hereafter set Q =1. cool
o τ = 1. This minimum mass is only achieved for tem-
Givenequation(3),wecanrewriteequation(2)togen-
peratures consistent with T . Once an opacity law
erate a single criterion for fragmentation which depends f,min
is specified which relates T and τ, the problem becomes
on temperature and location:
overconstrained— these three criteria can only be satis-
Ωt Ω2 fied at a single disk location, and equations 7-9 are valid
ζcool =(fqft) T5/2f(τ)≤1, (4) at only one radius in the disk. We now proceed to eval-
uate the critical disk temperatures and fragment masses
where f ≡(3/32)γ(γ−1)−1k(µσζ)−1. Somewhat coun- for realistic opacity laws, demonstrating that planetary-
t
terintuitively,thecriticalcoolingconstraintrequiresthat mass fragments can only form at large separations from
a disk be sufficiently hot to fragment. The value of f(τ) their host star.
dependsonbothΩandT. Evaluatingthiscriterionrelies
on the disk opacity, which we return to in §4.2. 4.2. Opacity
At low temperatures, when T (cid:46) 155 K, ice grains are
4. MINIMUMFRAGMENTMASSESANDSEPARATIONS the dominant source of opacity (Pollack et al. 1985).
4.1. Initial masses of GI-born fragments Above ∼155 K, ices begin to sublimate. In the cold
regime applicable to the outer regions of protoplanetary
We take the initial mass of a fragment to be the mass
disks,weconsidertworealisticopacitylaws,onewhichis
enclosed within the radius of the most unstable wave-
characteristicofgrainsintheinterstellarmedium(ISM),
length, λ =2πH in a Q=1 disk, or:
Q andonewhichischaracteristicofgrainsthathavegrown
M ≈Σ(2πH)2 (5) to larger sizes due to processes within the disk. We as-
frag
sume a Rosseland mean opacity which scales as
critical value of Q at which fragmentation occurs should remain
similar. κ≈κβTβ (10)
4
where the both the exponent, β, and the constant κ
β 4
are not well constrained in protoplanetary disks. They 10
dependonthenumber-sizedistributionandcomposition
ofthedustgrains,andonthedust-to-gasratio. ForISM
3
10
like grains, the Rosseland mean opacity may be approx-
imated by an opacity law with β =2, or
κ≈κ T2 (11) 102
2 R=40 AU
(Pollack et al. 1996; Bell & Lin 1994; Semenov et al. ol
2003) as long as the ice grains dominating the opacity co101
are smaller than a few tens of microns. This opacity law Ω t
is observationally confirmed in the ISM (Beckwith et al.
2000, and references therein). For our fiducial model we 100 Fragmentation ↓ R=100 AU
useκ ≈5×10−4cm2/gforT inKelvin,afittothestan-
2
dwaerdqudoutsetκmopdeerlgbryamSemofengoavseftorala. (d20u0st3-)t.o-Tgharsouragthioouotf, 10−1 40 AU 100 AUκ=0.24 g/ cm2
10−2. κ=5 × 10−4 T2 g/ cm2
Asgrainsgrowlarger,theyeventuallyexceedthewave-
−2
lengthoftheincidentradiation,andsotheopacityisde- 10
0 1 2
10 10 10
termined by the geometric optics limit. In this case the
Temperature
Rosseland mean opacity is independent of temperature,
so the exponent β =0. In this limit: Fig. 1.— The disk cooling time as a function of tempertature
for different opacity laws at radii of 40 AU (dashed) and 100 AU
κ≈κ0 (12) (solid). ThecoolingtimeiscalculatedassumingthatQ=1. The
temperatureindependent(largegrain)opacitylawisshowninred,
For a fixed dust mass in grains of size s, κ0 ∝ s−1. For while the ISM opacity law: κ ∝ T2 is shown in blue. The line
our fiducial model we choose κ ≈ 0.24 cm2/g which is thicknessindicatestheopticaldepthregime. Whenlinesdropbe-
0
validforT (cid:38)20Kandtypicalgrainsizesoforder300µm lowthecriticalcoolingtime(grey),diskfragmentationcanoccur.
ThebendintheISMopacitycurveindicatesthatintheoptically
(see Figure 6 of Pollack et al. 1985).
thick regime, the cooling time becomes constant as a function of
Observationsofemissionfromopticallythinprotoplan- temperature.
etary disks show evidence of grain growth at millimeter
wavelengths. Specificallythemeasuredopacitiesκ ∝να 4.2.1. Small grain opacity law
ν
with α (cid:39) 0.5—1.5 in the Rayleigh-Jeans tail imply par- For an optically thin disk with β = 2, equation(14)
ticle growth toward the millimeter wavelengths of the implies that in order to fragment, the disk must have
observations (D’Alessio et al. 2001). Although most ob- temperatures in excess of:
serveddiskshavehadmoretimeforgraingrowthtopro-
(cid:16) r (cid:17)−3/10
ceed, Class 0 sources also show evidence of grain growth T>T =9K (15)
(D’Alessioetal.2001). Alternatively,theseobjectscould thin 100 AU
beopticallythickatmillimeterwavelengths,whichwould (cid:18) κ (cid:19)−1/5(cid:18) M (cid:19)1/10
2 ∗
mimic the effects of grain growth.
5×10−4cm2/g 1.5M
Using these opacity laws we see that cooling proceeds (cid:12)
with a different functional form in the optically thick Colder disks, even when optically thin, cannot cool
and optically thin limits. In the optically thick regime, quickly enough to fragment.
f(τ) ≈ τ = (κβTβΣ)/2, which when combined with For an optically thick disk, β = 2 turns out to be
equation(4) indicates that to fragment, the disk must a special case: the temperature dependence drops out
have temperature of equation(13), giving instead a critical radius beyond
which fragmentation can occur, independent of the disk
T >(cid:0)ftfq2/2(cid:1)1/(2−β)(cid:0)Ω3κβ(cid:1)1/(2−β) (13) temperature:
for β (cid:54)= 2. In the special case β = 2, the fragmentation (cid:18) M (cid:19)1/3(cid:18) κ (cid:19)−2/9
constraint is temperature-independent, as discussed in r (cid:38)70 AU ∗ 2 . (16)
1.5M 5×10−4cm2/g
§4.2.1. (cid:12)
Intheopticallythinregime,f(τ)=1/τ =2/(κ TβΣ).
β Matzner & Levin (2005) first pointed out the existence
The cooling time is independent of Σ, so we can rewrite
ofaminimumcriticalradiusforfragmentation. InFig.2
equation(4) as:
we illustrate how the two fragmentation criteria create
(cid:18) Ω (cid:19)1/(3+β) a radius rather than temperature cutoff. At radii larger
T >(2f )1/(3+β) , (14) than the critical radius defined above, any Σ−T combi-
t κβ nation which gives Q ≤ 1 will fragment (so long as the
opacitylawremainsvalid). Atsmallerradii,nocombina-
Fig. 1 shows the dependence of the cooling time on
tionofΣandT whichgivesQ≤1willfragmentbecause
disk temperature for each opacity law at two different
the disk cannot simultaneously satisfy the cooling time
radii. Since fragmentation can only occur when Ωt <
cool criterion.
ζ, the minimum temperatures at which fragmentation is
allowed are specified by the intersection of the cooling
4.2.2. Large grain opacity law
curves with the Ωt boundary.
cool
5
4.3. Initial fragment masses with astrophysical disk
a = 100 AU
11000000 temperatures
Q < 1 To consider the case favorable to GI planet formation,
) weconsiderthelowestplausibledisktemperaturesinor-
2
m der to minimize the fragment masses. We estimate the
g/c 110000 disk temperature using the passive flared disk models of
( Chiang&Goldreich(1997). Thismodellikelyunderesti-
y
t matesthetemperaturesinactivelyaccretingsystemsbe-
i
s
n cause it ignores significant “backheating” from the infall
De t < 3 Ω−1 envelope and surrounding cloud (Chick & Cassen 1997;
e 1100 c Matzner & Levin 2005). Although viscous heating will
c
a also contribute to the temperature, we ignore its modest
urf τ < 1 contribution to obtain the lowest reasonable tempera-
S
tures and fragment masses. Disk irradiation dominates
over viscous heating in this regime (see Appendix B).
11
We consider the inner region where the disk is opti-
1100 110000 cally thick to blackbody radiation. In this regime, the
Temperature (K) temperature of a flared disk in radiative and hydrostatic
equilibrium is:
Fig. 2.—Fragmentationcanonlyoccurintheregionofparam-
eotpearcistpieasceatinraddiciiaotefd10b0yAtUh.eTohveerulappppeirn,gshhaadsehdedregreiognio(nrsedf)orshIoSwMs Tm =(cid:16)α4F(cid:17)1/4(cid:18)Rr∗(cid:19)1/2T∗ ∝L2/7r−3/7 (19)
whereToomre’sparameterQ<1. Thelower,shadedregion(blue)
indicateswheret ≤3Ω−1. Atradiilessthan70AU,fragmenta-
cool where α measures the grazing angle at which starlight
tionisprohibitedbecausethetworegionsnolongeroverlap. That F
theboundariesoftheseregionsareparallellinesreflectstheκ∝T2 hits the disk; αF is dependent on the degree of disk flar-
formoftheice-grain-dominatedopacityatlowtemperatures. ingmeasuredattheheightofthephotosphere(Chiang&
Goldreich1997). Grainsettlingmayreducetheheightof
thephotosphere(sethereto4timesthescaleheight)and
As shown in Fig. 1, for the large grain opacity, τ > 1
thus α . For the standard radiative equilibrium model,
for all relevant temperatures. In this case equation(13) F
requires: the disk flaring scales approximately as H/r ∝r2/7. We
shall find when we calculate the disk temperature that
(cid:18) M (cid:19)3/4(cid:16) r (cid:17)−9/4(cid:18) κ (cid:19)1/2 a gravitationally unstable disk remains optically thick,
T >65K ∗ 0 . justifying the use of this formula.
1.5M 43 AU 0.24 cm2/g
(cid:12) To estimate the the stellar luminosity we use the stel-
(17)
lar evolution models of Krumholz & Thompson (2007),
The corresponding minimum mass for these tempera-
whichincludebothnuclearburningandaccretionenergy.
tures is:
The accretion luminosity depends on both the current
(cid:18) M (cid:19)5/8(cid:16) r (cid:17)−15/8 accretion rate and the accretion history ( in so far as
M =13M ∗ (18) it effects the stellar radius), so we obtain the lowest lu-
min Jup 1.5M 43AU
(cid:12) minosity estimates by allowing the star to accrete at a
(cid:18) κ (cid:19)3/4 constant, low accretion rate. We use thestellar luminos-
0 . ityafteraccretingto90%ofitscurrentmass(or1.35M
0.24cm2/g (cid:12)
assuming roughly 10% is still in the disk). We choose an
We have scaled equations (17) and (18) to an effective accretion rate of 10−7M(cid:12)/yr as a lower bound because
critical radius for this opacity law. Although fragmen- a star accretingat a lower accretion rate throughout its
tation can occur inside 43 AU at sufficiently high tem- historyhasaformationtimescalethatistoolong. Accre-
peratures,thefragmentsexceed13M ,makingitirrel- tion rates an order of magnitude larger give comparable
Jup
evant for planet formation. Smaller values of κ0 move luminosity (when the star has only reached 1.35M(cid:12)) to
this boundary inward, allowing for fragmentation into the present day luminosity of 5L(cid:12) (Marois et al. 2008).
lower mass objects at smaller radii, although the scal- Lowering the accretion rate below this value does not
ing with opacity is shallow. Grain growth to larger sizes significantly lower the stellar luminosity because the ac-
could plausibly reduce κ . If, for example, grains domi- cretion energy contribution is small.
0
nating the disk opacity have grown up to 1mm without The luminosity calculated for an accretion rate of
altering the dust-to-gas ratio, equation(12) implies that 10−7M(cid:12)/yr translates to temperatures of:
in the geometric optics limit κ = 0.072cm2/g. In this
0 (cid:16) r (cid:17)−3/7
case,theminimumradiusispushedinwardto26AU(see T ≈40K , (20)
also Nero & Bjorkman 2009). 70AU
Thus far we have determined the minimum masses al- which we shall use as our fiducial temperature profile.
lowed as a function of radius with the temperature as a In the outer regions of the disk where fragmentation is
free parameter. We now calculate actual disk tempera- allowed, the disk temperatures are of order 30 − 50K.
tures, which at large radii are typically higher than the These temperatures exceed the minimum threshold for
minima. In this case we must evaluate fragment masses fragmentation, and so the mass of fragments will be set
using equation(6). by equation (6).
6
102 102
Temperature 1.000 0.300 0.100 0.030
0.10 0.010
erature (K)101 d c → fragmentation, T > Tmin Mfrag 101 M/Mpuj H/R 0.100 0.030 0.003
mp b
e 0.01
T M (10 K) 1 10
frag Q
Fig. 4.— Contours of the ratio of planetary isolation mass to
stellarmassasafunctionofToomre’sQandthediskaspectratio
Low M. Model frrag mfoern κta ∝tio Tn2 → Hble/rd,isilklus.stFraotrindgiskthsawtitthhehiigsohleartiQon’smcoansssisisteanltwwayitshlacrogreeainccurnetsitoan-
100 20 40 60 8c0rit 100 120 140 100 models,theisolationmassremainssmall. Theshadedregionindi-
Radius (AU) cateswheretheisolationmassexceedsthestellarmass.
Halting the growth of planetary mass objects is a rele-
Fig. 3.—DepictionofthecurrentconfigurationofHR8799and
vant problem independent of the formation mechanism.
formation constraints for realistic disk temperatures. We show
the lowest expected irradiated disk temperatures (blue) and cor- However the GI hypothesis requires that the disk is (or
responding fragment masses (grey), as a function of radius. The was recently) sufficiently massive to have Q ∼ 1, imply-
lower bound on both regimes (burgundy) is set by the irradia- ingthatthediskisactivelyaccreting. Thecoreaccretion
tion model described in §4.3, with M˙ = 10−7M(cid:12)/yr. The upper scenario does not face this restriction.
boundaryissetbythecurrentluminosityofHR8799,∼5L(cid:12). The
green dashed-dotted line shows the mass with disk temperatures
of 10 K, a lower limit provided by the cloud temperature. The 5.1. Isolation Mass
vertical line shows the critical fragmentation radius for the ISM
We estimate an upper mass limit for fragments by as-
opacity law; fragmentation at smaller radii requires grain growth.
Fragmentmassesareshownforradiiatwhichtheirradiationtem- sumingthattheyaccreteallofthematterwithinseveral
peraturesarehighenoughtosatisfythecoolingtimeconstraintof Hill radii:
equation(17). Atsmallerradii,fragmentationispossibleathigher M ≈4πf ΣR r. (21)
disk temperatures, but the resulting fragments have correspond- iso H H
inglyhighermasses,andplanetformationisnotpossible.
Here f ∼3.5 is a numerical constant representing from
H
how many Hill radii, R = r(M /3M )1/3, the planet
Other analytic and numerical models of stellar irradi- H iso ∗
can accrete (Lissauer 1987).
ation predict temperatures in agreement with or higher
It is instructive to compare the ratio of the isolation
than our estimate. (Rafikov & De Colle 2006; Offner
mass to the stellar mass:
etal.2009). Similarly,modelsofdisksinOphiuchushave
similar temperatures for 1 Myr old stars of lower mass M (cid:18)H(cid:19)3/2
(and thus luminosity), implying that our model temper- iso =4.6f3/2Q−3/2 . (22)
M H r
atures are low, though not unrealistic (Andrews et al. ∗
2009). We find that large isolation masses are always expected
In Fig. 3 we illustrate the constraints on fragment in gravitationally unstable disks. Fig. 4 illustrates the
masses from this irradiation model, calculated using scalingofequation(22)withQandthediskaspectratio,
equation(6). We show the fiducial disk temperatures of H/r. For our fiducial temperature profile, H/r ≈ 0.09
equation(20), along with temperatures consistent with at 70 AU. For low values of Q and comparable H/r, the
luminosities up to the present-day luminosity. For our isolation mass exceeds 10% of the stellar mass. For the
fiducial opacity laws, the expected fragment masses are ideal disk values cited above (equation 16), the isolation
only marginally consistent with GI planet formation— mass is:
fragments form near the upper mass limit of 13M .
Lower opacities produced by grain growth mightJuapl- M ≈400M (cid:16) r (cid:17)6/5(cid:18) M∗ (cid:19)1/10. (23)
low fragmentation into smaller objects at closer radii. iso Jup 70 AU 1.5M
(cid:12)
Whether grain-growth has proceeded to this extent in
such young disks is unclear. This mass is nearly two orders of magnitude larger than
theminimummass. Growthbeyondtheisolationmassis
5. GROWTHOFFRAGMENTSAFTERFORMATION possible either through mergers or introduction of fresh
materialtoaccretebyplanetmigrationordiskspreading.
For realistic disk temperatures, it is conceivable that
Objects which grow to isolation mass cannot be plan-
fragments may be born at several M . We now con-
Jup ets, and so we turn to mechanisms that truncate frag-
sider their subsequent growth, which may increase their
ment growth below the isolation mass.
expected mass by an order of magnitude or more.
The final mass of a planet depends sensitively on nu-
5.2. Gap opening mass
merous disk properties (effective viscosity, column den-
sity, scaleheight) and the mass of the embedded object. Massiveobjectsopengapsintheirdiskswhengravita-
In order to constrain the mass to which a fragment will tional torques are sufficiently strong to clear out nearby
grow,wecancompareittotworelevantmassscales: the gas before viscous torques can replenish the region with
disk isolation mass and the gap opening mass. new material. (Lin & Papaloizou 1986; Bryden et al.
7
1999). The gap width is set by the balance between the order unity coefficients (cf. Crida et al. 2006).
two torques: If we make the simplifying assumption that gap accre-
∆ (cid:18)f q2 r2 (cid:19)1/3 tionterminateswhenthegapwidthreachesafixednum-
= g , (24) ber of Hill radii, f , we can calculate a gap starvation
r 3πα H2 S
mass. We expect that for a gap to truncate accretion,
where∆isthegapwidth,fg ≈0.23isageometricfactor fS (cid:38) fH = 3.5, the width in Hill radii used to calculate
derivedinLin&Papaloizou(1993),qistheplanettostar theisolationmass(§5.1). Terminatingaccretionisanun-
mass ratio, and α is the Shakura & Sunyaev (1973) ef- solved problem for Jupiter in our own solar system, and
fectiveviscosity. Thiscanbeusedtoderivethestandard so to provide a further constraint on fS, we refer to the
minimum gap opening mass by requiring that ∆>H: numerical simulations of Lissauer et al. (2009) (See also
D’Angelo et al. (2003) for a detailed explanation of the
numerical work). Their runs 2l and 2lJ exhibit asymp-
(cid:115)
(cid:18)H(cid:19)5/2 3πα totic mass growth after ∼2.5 Myr for a constant-mass,
q> (25)
r f lowviscosity(α=4×10−4)diskunderconditionsappro-
g
priate to the formation of Jupiter. Using equation(26),
(cid:16) α (cid:17)1/2(cid:18) T (cid:19)5/4(cid:16) r (cid:17)5/4 we solve for the width of the gap generating this fall-off
≈4×10−3 inaccretionrateandfindf =∆/R ∼5. Theneedfor
0.1 40 K 70 AU S H
anextremelylargeandwell-clearedgapreflectstheinte-
(cid:18) M (cid:19)−5/4 grated effects of low-level accretion through the gap and
∗ .
ontotheplanetoverthedisklifetimeofafewMyr. Even
1.5 M
(cid:12)
a slow trickle of material onto the planet can contribute
Gap opening requires ∆ > R and R > H. The lat- to significant growth.
H H
ter requirement is automatically satisfied for fragments Usingf =5, thegap-openingstarvationmassforHR
S
formed by GI. 8799 scaled to both the simulated solar-system viscosity
WhiletheeffectsofGIareoftenparameterizedthrough and to the expected GI viscosity is:
anαviscosity, Balbus&Papaloizou(1999)havepointed
out that α, a purely local quantity, may not adequately (cid:18) α (cid:19)(cid:18) ∆ (cid:19)3
M ≈8M (27)
describe GI driven transport, which is inherently non- starve Jup 4×10−4 5R
H
local. Lodato & Rice (2005) have shown that for suf- (cid:18) T (cid:19)(cid:16) r (cid:17)
ficiently thin disks, the approximation is reasonable: in
ordertoformobjectsofplanetarymass,thediskmustbe 40 K 70 AU
relatively thin and at least marginally within this limit. (cid:16) α (cid:17)(cid:18) ∆ (cid:19)3
However, even in this thin-disk limit, it is not clear that ≈2000M
Jup 0.1 5R
GIdriventorqueswillexactlymimicviscousonesatgap- H
opening scales. (cid:18) T (cid:19)(cid:16) r (cid:17)
.
Equation (25) implies that the gap opening mass is 40 K 70 AU
less than or equal to the fragment mass for effective vis-
cosities consistent with GI. We use α ≈ 0.1, as this is In order to limit growth to planetary masses, the effec-
consistent with active GI (Gammie 2001; Lodato & Rice tive viscosity must be two orders of magnitude below
2005;Krumholzetal.2007). Ifthelocaldiskviscosityis that expected in GI unstable disks, roughly α ∼ 10−3.
lower,fragmentswillalwaysformabovethegap-opening More restrictively this requires that other local trans-
mass. port mechanisms such as the MRI be weaker than cur-
rentlypredictedbysimulations—theyproduceα∼10−2
5.2.1. Gap-opening starvation mass at least in disks with a net magnetic flux (Fleming et al.
2000; Fromang et al. 2007; Johansen et al. 2009). Fig. 5
Gap-openingslowsaccretionontotheplanet, butdoes
illustrates the scaling of gap starvation mass with radius
not starve it of material completely. Accretion rates
for several values of α. It appears that active disks face
through gaps remain uncertain for standard core ac-
severeobstaclesinproducingplanetarymassobjectsun-
cretion models, and numerical models are not available
less the disk disappears promptly after their formation.
for accretion onto the distended objects formed through
Althoughweexpectfragmentationtomakethediskmore
GI fragmentation. Nevertheless simulations of accretion
stablebyloweringthelocalcolumndensity,thereislittle
through gaps in low viscosity disks (Lubow et al. 1999)
reason to expect a recently massive disk to be so quies-
demonstratethataccretionisslowerthroughlargergaps,
cent.
and this qualitative conclusion likely remains valid as
long as gaps form.
5.2.2. Planet starvation through gap overlap
Analogous to the isolation mass, we consider a “gap
starvation mass” that is related to the ratio of the gap Althoughitisunlikelythatthediskwillfragmentinto
width to planet Hill radius. Rewriting equation(24) we a sufficiently large number of planetary mass objects to
find the ratio of gap width to Hill radius is: completelydepletethediskofmass(Stamatellos&Whit-
worth 2009), the formation of multiple fragments simul-
∆ (cid:18)f q r2 (cid:19)1/3 taneously may limit accretion through competition for
= g (26) disk material by opening overlapping gaps. The cur-
R πα H2
H rent separation between the planets is such that gaps
Note that ∆ > R recovers the canonical gap opening larger than roughly three Hill radii overlap, so depend-
H
estimateappropriateforJupiter: q >40ν/(r2Ω),modulo ing on their migration history, this could limit growth
8
1994). Fragmentationatearliertimesleadstotheforma-
1000 tionofmoremassivecompanions,whilefragmentationat
later times is unlikely because disks are too low in mass
(Andrews et al. 2009).
100
6.1. Ongoing accretion and the formation of binaries
Mjup and multiples
M/ 10 Because the cooling time constraint is easily satisfied
forexpecteddisktemperatures,disksarelikelydrivento
fragmentation by lowering Q. Even when Q is above the
threshold for fragmentation, torques generated by self-
1
gravity (e.g. spiral arms) can drive accretion. When
the infall rate onto the disk is low, a self-gravitating
10 100
Radius (AU) disk regulates its surface density and hence Q so that
the torques are just large enough to transport the sup-
Fig. 5.— The gap starvation mass as a function of disk radius. plied mass down to the star, thereby avoiding fragmen-
Weshowcurvesforseveralvaluesforα,andindicatetheplanetary tation. However, GI cannot process matter arbitrarily
massregime,andtheregioninwhichdiskfragmentationislikely. quickly because the torques saturate. Thus the disk will
We use fS = 5, scaled to simulation 2lJ of Jupiter formation in bedriventowardfragmentationiftheinfallratebecomes
Lissaueretal.(2009)(labeledL09inthefigure). Theradialscaling
is derived assuming H/r ∝ r2/7. For the low viscosity case, we toohigh. Thiscriticalaccretionrateisafunctionofdisk
normalizethescaleheighttoJupiterat5.2AUina115Kdiskfor temperature:
comparisonwithL09. Forthehigherviscosities,wenormalizethe 3c3α
disk scale height to the lowest expected temperatures (equation M˙ ≈ s sat (28)
20). ForcomparisonweshowtheHR8799planetsasblackcircles. crit GQ
(see also §7). From equation(27) we see that if gaps (Gammie2001;Matzner&Levin2005). Numericalsimu-
are forced to be smaller by a factor of two due to com- lationsshowthatGIsaturatesatα ∼0.3−1(Gammie
sat
petition with another planet, the expected masses are 2001;Krumholzetal.2007;Lodato&Rice2005;Kratter
decreased by a factor of 8. This effect would imply that et al. 2010). Ifthe infall rate onto the diskexceeds M˙
crit
fragments in multiple systems should be lower in mass. the disk can no longer regulate the surface density to
Note that when simulated disks fragment into multiple keep Q just above unity, and fragmentation will occur.
objects simultaneously, they generally have orbital con- Because the conversion of accretion energy to thermal
figurations like hierarchical multiples rather than plan- energy at large radii is inefficient, disks cannot restabi-
etary systems (Stamatellos & Whitworth 2009; Kratter lize through heating to arbitrarily high accretion rates
et al. 2010). Whether the same conditions required to (Kratter & Matzner 2006).
limit fragment growth—reduced disk mass and/or vis- If fragmentation occurs due to rapid accretion, as de-
cosity after fragmentation—can allow the retention of a scribedabove, itisdifficulttolimitsubsequentfragment
planetary system of fragments remains to be simulated. growth. Asdemonstratedin§5.2.1,gapopeningdoesnot
limit accretion efficiently when the effective viscosity, in
5.3. Disk dispersal as a means to limit fragment growth this case, α , is high.
sat
Mechanisms for gas dispersal such as photoevapora- Amoregeneralbarriertomakingsmallfragmentsdur-
tion may be necessary to stunt planetary growth, even ing rapid infall is the large reservoir of material passing
for models of Jupiter in our own solar system (Lissauer by the fragment as star formation proceeds (Bonnell &
et al. 2009). Dissipation timescales for A-star gas disks Bate 1994). The specific angular momentum, j, of ac-
are thought to be short, less than 2-3 Myr (Carpenter cretingdiskmaterialtypicallyincreaseswithtime(mod-
et al. 2006), which could halt growth before the gap- ulosmallrandomfluctuationsinaturbulentcore), land-
opening starvation mass is achieved. Radiative transfer ing at a circularization radius, r = j2/GM , which
circ ∗
models such as Gorti & Hollenbach (2009) have calcu- is larger than the fragment’s orbit. Since the fragment’s
lated that photoevaporation by the central star will be- Hill radius is roughly 10% of the disk radius, newly ac-
comeimportantatradiiof∼100AUaroundoneMyrfor creted material undergoes many orbits in the fragment’s
an A star (see also Ercolano et al. 2009). This timescale sphere of influence as it tries to accrete onto the central
coincides with the expected fragmentation epoch, and star, and some fraction will accrete onto the fragment
mayaidinshuttingoffaccretionbothontothedisk,and itself. This process is less efficient if global GI modes
onto the planets. drive fragments to smaller radii, because their Hill radii
shrink. However,growthtostellar(orsub-stellar)masses
6. GIPLANETFORMATIONINTHECONTEXTOFSTAR may still occur as long as migration timescales are not
FORMATION faster than accretion timescales. The latter scenario im-
Wenowconsiderhowthediskcanreachthefragmenta- plies that the disk cannot fragment at early times and
tion conditions described in §3 in the context of a model still reproduce a single-star system like HR 8799.
for star formation. Due to the effects of infall onto the The trend toward continued growth and even mass
disk,wefindthatplanetformationviaGIcanonlyoccur equalization following disk fragmentation is observed in
whenthefragmentationepochisconcurrentwiththeend numerous simulations with ongoing accretion (Bonnell
of the main accretion phase, as the protostar transitions & Bate 1994; Bate 2000; Matsumoto & Hanawa 2003;
from a Class I to a Class II object (Andre & Montmerle Walch et al. 2009; Kratter et al. 2010; Krumholz et al.
9
2009). SimulationsofplanetformationbyGIwithongo- where we have used ν = αc2/Ω. This timescale is short
s
ing accretion also illustrate this behavior (Boley 2009). enoughtoallowsubstantialmigrationduringthelifetime
Consequently, if the HR 8799 planets formed by GI, of the gas disk.
they must have fragmented at the tail end of accretion If fragments migrated inward independently, they
fromtheprotostellarcoreontothedisk. Mostlikely,this could never become captured in resonance, as the inner-
requires that the protostellar core have properties such most planet would migrate farther and farther from its
that its infall rate reaches the critical value in equation neighbor. However, as shown in §5, planetary gaps may
(28)justasthecoreisdrainedofmaterial. Ifthiscoinci- overlapiftheyarewithinseveralHillradiiofeachother,
denceintimingdoesnotoccur,fragmentationproducesa comparable to the current separations between the HR
substellar,ratherthanaplanetary,companion. Whether 8799planets. Thisoverlapaltersthetorquesfeltby,and
fragmentationatthisepochcanproducenon-heirarchical thus the migration of, the planets. As shown by Kley
orbits like HR 8799 remains to be explored. (2000), if multiple planetary gaps interact, convergent
migration is possible. Gap interaction allows an outer
6.2. Reaching Instability in the absence of infall: FU planet to shield an inner planet from the material, and
Orionis outbursts thus torques, of the outer disk, slowing or halting its in-
ward migration and allowing the outer planet to catch
Driving the disk unstable with external accretion cor-
up. ThismechanismisinvokedbyLee&Peale(2002)to
responds to excessive growth of fragments. It is there-
generate convergent migration and resonance capture in
foretemptingtoconsidermechanismstolowerQthrough
the planets orbiting GJ 876.
diskcooling,whileholdingthecolumndensityfixed. Be-
Under the assumption that gap overlap allows conver-
cause the disks are dominated by irradiation, changes
gentmigrationandresonancecapture,wenowask: What
in the viscous dissipation due to accretion are unlikely
is the overall direction of the subsequent migration? We
to affect a significant temperature change, and so lower-
note that if the planets migrated a substantial distance
ing Q requires order of magnitude changes in the stellar
afterresonance capture, eccentricitydampingbythegas
luminosity due to the weak scaling of T ∝ L2/7. FU
diskwaslikelynecessary(c.f.,Lee&Peale2002). Wedo
Orionis type outbursts (Hartmann & Kenyon 1996) can
notconsidereccentricitydampingfurtherhere. Oncetwo
causerapidchangesinluminosity. Toresultinplanetary
planetsarecaughtinmean-motionresonance,thetorque
massfragments,theluminositydropfollowinganFUOri
onanindividualplanetfromthegasdiskcancauseboth
outburstwouldneedtoreachatleasttheminimumlumi-
planetstomigrate,withangularmomentumtransferme-
nosity used in equation (20) on timescales shorter than
diatedbytheresonance. Masset&Snellgrove(2001)(see
an outer disk dynamical time.
alsoCridaetal.2009)havearguedthatthetorqueimbal-
TheaccretionofGIformedembryosontotheprotostar
ance on a pair of gap-opening resonant planets can even
is a proposed source of the outbursts (Vorobyov & Basu
reversethedirectionofmigration,althoughthisrelieson
2006). If this occurs, perhaps a final generation of grav-
a significant difference between planet masses.
itational fragments, the so-called “last of the Mohicans”
Nevertheless,understandingtheplanets’overallmigra-
(Gonzalez 1997), would remain as detectable compan-
tion requires understanding how overlapping gaps alter
ions. Although the lack of infall in this scenario might
the torque balance on the group of planets. Guided by
ease the gap opening constraints, fragments face all of
ourinterestincleangapsthatlimitthegrowthofplanets
the other difficulties discussed above in remaining low in
(§5–6), we consider the following simplified problem.
mass.
We imagine that the three planets have cleared, and
are embedded in, a single large gap which is sufficiently
7. MIGRATIONINAMULTI-PLANETSYSTEM
clean that any disk gas passing through is dynamically
A final consideration for making wide orbit planets unimportant. Because the system is locked in a double
through GI is their subsequent migration history. If mean-motionresonance,weassumethatanimbalancein
formed via GI, the HR 8799 planets had to migrate in- the torques acting on the two edges of the gap can cause
wardtotheircurrentlocations. Thereisanindependent all three planets to migrate. We can then ask: is there a
reason to believe that migration did in fact take place sufficient flux of angular momentum from the outer disk
in this system. As discussed by Fabrycky & Murray- to cause the planets to migrate inward with the viscous
Clay (2008), the long term stability of HR 8799 requires accretion of the disk?
a resonant orbital configuration, most plausibly a 4:2:1 When in the 4:2:1 resonance, the total angular mo-
mean motion resonance, which likely resulted from con- mentum of the planets is roughly 2.4M Ω r2, where we
p p p
vergent migration in the protoplanetary gas disk. We haveassumedthattheplanetsareroughlyequalinmass,
demonstratebelowthatwhilethishistoryisplausible, it M , while r and Ω are the separation and Keplerian
p p p
requires special disk conditions. angular velocity of the outermost planet. The angular
InwardTypeIImigration(appropriateforgapopening momentum flux from the outer disk is large enough to
planets)is expectedatformation, because thefragment- move the planets on a viscous time r2/ν when:
p
ing region is within the part of the disk accreting onto
the star. Type II migration of a single planet occurs on
the disk viscous timescale: M˙ Ω r2 (cid:38)2.4M Ω r2(ν/r2), (31)
p p p p p p
r2 r2Ω
τ ≈ = (29)
ν ν αc2s where M˙ =3πΣν is the mass flux through radius rp.
(cid:16) r (cid:17)13/14(cid:18) M (cid:19)1/2(cid:16) α (cid:17)−1 Using the above inequality, we can calculate a critical
≈0.4Myr ∗ (30) disk surface density at the current location of the outer
70 AU 1.5M 0.1
(cid:12)
10
planet such that the disk can push the planets inward: of different masses because disk fragmentation becomes
more likely for higher mass stars (Kratter et al. 2008).
Σ(cid:38) Mp. (32) At present, neither the population of substellar com-
4r2 panions nor the population of exoplanets is continuous
p
out to HR 8799. Many selection biases are reflected in
For M = 10M and our fiducial disk temperatures Fig. 6, and these gaps in particular may be due to se-
p Jup
(equation 20), this constraint is always satisfied when lection effects; resolution and sensitivity make it diffi-
Q = 1. The disk is unable to cause inward migration cult to detect both wide orbit planets, and close-in low
when Σ(cid:46)4g/cm2 at 70 AU, which is equivalent to Q∼ mass brown dwarfs. We note that while there are actu-
20. ally fewer planets at distances less than 1 AU, the cutoff
If the planets do share a clean common gap, a large above5AUisunphysical. Thereisnotyetanyindication
fraction of the disk would be effectively cleared of gas of an outer cut off radius in the exoplanets: if they con-
while a massive outer disk is still present. A simi- tinued out to larger separation, the distribution would
lar mechanism has been invoked to explain transitional easily encompass the HR 8799 and Fomalhaut systems.
disks, which contain holes at radii of a few tens of AU Data from ongoing surveys like that which found HR
and smaller (Calvet et al. 2002). Transport of disk gas 8799 are necessary to verify the true companion distri-
through a less well-cleared gap could substantially alter bution as a function of mass and radius. If these planets
this picture. are formed through GI, we would expect observations
In summary, it is possible to envision a scenario in to fill in the gap between HR 8799 and higher mass ra-
which the HR 8799 planets migrate inward to their cur- tio objects to show a continuous distribution. If these
rent locations in such a way that their orbits converge, planets are formed via core accretion, than observations
allowing resonance capture. This scenario is consistent may fill in the plot on the opposite side of HR 8799, oc-
with other constraints on GI planet formation: shortly cupying a region of parameter space for which neither
after formation, the disk must have low accretion rates core accretion nor GI is currently a successful formation
and decline in mass in order to (a) limit the growth of mechanism.
fragments, (b) allow for large, overlapping gaps.
More stringent constraints will require future work on 9. SUMMARY
migration in gravitationally unstable disks, particularly We have demonstrated that while GI-driven fragmen-
in the presence of multiple planets massive enough to tation is possible at wide distances from A stars, frag-
clear large, overlapping gaps. ment masses typically exceed the deuterium burning
“planet” limit. In contrast, the formation of sub-stellar
8. CURRENTOBSERVATIONALCONSTRAINTS and stellar companions is more likely because moderate
While there is a regime of parameter space in which disktemperaturesandactiveaccretionontoandthrough
planet formation is possible, typical conditions produce the disk drive disk-born objects to higher masses.
moremassive(>13M )companions. IfGIfragmenta- If the HR 8799 planets did form by GI, the following
Jup
tion ever forms planets, then fragmentation should typ- criteria had to be met:
ically form more massive objects. Consequently, these
1. Fragments should form beyond 40−70 AU: inside
planets would constitute the low mass tail of a distribu-
ofthislocationthediskwillnotfragmentintoplan-
tion of disk-born companions. If the mass distribution
etary mass objects even if Q(cid:46)1. Grain growth is
is continuous there should be more sub-stellar compan-
required for fragmentation at the lower end of this
ions than planets at comparable distances of 50-150 AU.
range.
Observing this population is a strong constraint on the
formation mechanism, but current data are insufficient
2. Temperatures must be colder than those of typical
to draw conclusions.
disks to limit the initial fragment masses.
Zuckerman & Song (2009) have compiled the known
sub-stellarcompanionsinthisrangeofradiitodate. This 3. Thediskmustbedrivenunstableataspecialtime:
range of separations falls beyond the well-established infall onto the disk must be low, but the disk must
inner brown dwarf “desert” (McCarthy & Zuckerman remain massive (e.g. the end of the Class I phase).
2004),andhasnotbeenwellprobedduetoobservational Thediskmustonlybecomeunstabletofragmenta-
difficultiesattheselowmassratios. Notethattheoverall tion at this point because earlier episodes of insta-
dearth of brown dwarf companions to solar mass stars bility should lead to sub-stellar or stellar compan-
is not a selection effect (Metchev & Hillenbrand 2009; ion formation.
Zuckerman & Song 2009).
We illustrate the observational constraints by plotting 4. The subsequent growth of fragments must be lim-
the companions from Zuckerman & Song (2009) along itedthroughefficientgapclearingnecessitatinglow
with the known exoplanets compiled by the Exoplanet disk viscosity or early gap overlap. Disk dispersal
Encyclopedia2 as a function of mass ratio and projected via photoevaporation may also be necessary.
separationinFig.6a,andasafunctionofminimumfrag-
mentmassesandfragmentationradiiinFig.6b. Wecom- 5. The three fragments must form at the same epoch
pare these with the HR 8799 planets, Fomalhaut b, and separated by several Hill radii, implying that the
the solar system giants. We distinguish between stars entire outer disk becomes unstable simultaneously.
2 October 2009, http://exoplanet.eu, compiled by Jean Schnei- 6. Migrationmustbeconvergent. Thislikelyrequires
der the gaps of the planets to overlap so as to starve
