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arXiv:0707.2733v1 [astro-ph] 18 Jul 2007
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 1 February 2008 (MN L
A
T
E
X style file v2.2)
Origin of the Metallicity Dependence of Exoplanet Host
Stars in the Protoplanetary Disk Mass Distribution
M. C. Wyatt1⋆, C. J. Clarke1and J. S. Greaves2
1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
2Scottish Universities Physics Alliance, University of St. Andrews, Physics & Astronomy, North Haugh, St Andrews KY16 9SS, UK
1 February 2008
ABSTRACT
The probability of a star hosting a planet that is detectable in radial velocity surveys
increases Ppl(Z)∝(10Z)2, where Zis stellar metallicity. Models of planet formation
by core accretion reproduce this trend, since the protoplanetary disk of a high metal-
licity star has a high density of solids and so forms planetary cores which accrete gas
before the primordial gas disk dissipates. This paper considers the origin of the form
of the metallicity dependence of Ppl (Z). We introduce a simple model in which de-
tectable planets form when the mass of solid material in the protoplanetary disk, Ms,
exceeds a critical value. In this model the form of Ppl(Z) is a direct reflection of the
distribution of protoplanetary disk masses, Mg, and the observed metallicity relation
is reproduced if P(Mg> M ′
g)∝(M′
g)−2. We argue that a protoplanetary disk’s dust
mass measured in sub-mm observations is a relatively pristine indicator of the mass
available for planet-building and find that the disk mass distribution derived from
such observations is consistent with the observed Ppl (Z) if a solid mass Ms>0.5MJis
required to form detectable planets. Any planet formation model which imposes a crit-
ical solid mass for detectable planets to form would reproduce the observed metallicity
relation, and core accretion models are empirically consistent with such a threshold
criterion. While the outcome of planet formation in individual systems is debatable,
we identify 7 protoplanetary disks which, by rigid application of this criterion, would
be expected to form detectable planets and may provide insight into the physical con-
ditions required to form such planets. A testable prediction of the model is that the
metallicity dependence should flatten both for Z > 0.5 dex and as more distant and
lower mass planets are discovered. Further, combining this model with one in which
the evolution of a star’s debris disk is also influenced by the solid mass in its proto-
planetary disk, results in the prediction that debris disks detected around stars with
planets should be more infrared luminous than those around stars without planets in
tentative agreement with recent observations.
Key words: circumstellar matter – stars: planetary systems: formation – stars:
planetary systems: protoplanetary discs – stars: pre-main-sequence.
1 INTRODUCTION
The study of how planetary systems form and evolve was
revolutionised when the first extrasolar planet was discov-
ered in radial velocity studies of the star 51 Peg (Mayor &
Queloz 1995). Over 200 extrasolar planets are now known
(Butler et al. 2006), and studying these planets has yielded
enormous advances in our understanding of how they formed
(Papaloizou & Terquem 2006; Udry et al. 2007). Perhaps the
most telling discovery was that of a correlation in the prob-
ability of a star hosting a planet, Ppl, which is found to
⋆Email: wyatt@ast.cam.ac.uk
increase with stellar metallicity (Gonzalez 1997). Fischer &
Valenti (2005; hereafter FV05) found that, for stars with a
metallicity Z= [Fe/H] between -0.5 and 0.5 dex, the metal-
licity dependence of the fraction of stars with planets with
orbital periods <4 years and with amplitudes in radial ve-
locity studies in excess of K > 30 m s−1(i.e., Saturn-Jupiter
mass planets, depending on orbital period) is
Ppl(Z) = 0.03 ×102Z,(1)
which corresponds to a planet fraction which increases with
the square of the number of iron atoms in the stellar atmo-
sphere. Similar trends have been found to apply to all species
including Si and Ni (e.g., Ecuvillon et al. 2004; Robinson et
2M. C. Wyatt et al.
al. 2006; Gonzalez 2006). The origin of this metallicity de-
pendence is thought to be intrinsic to the planet formation
process (FV05), and not caused by contamination from plan-
etesimals falling onto the star, as is believed to be the cause
of the high metallicities of DAZ white dwarfs (Jura 2006;
Kilic & Redfield 2007), although the recent discovery that
planet hosting giant stars do not favour metal rich systems
is currently reigniting this debate (Pasquini et al. 2007).
Since the discovery of the extrasolar planet metallicity
correlation, much work has gone into considering how stellar
metallicity could affect different aspects of the planet for-
mation process in the various models (e.g., Livio & Pringle
2003). It has been found that forming planets by gravita-
tional instability does not introduce any significant metallic-
ity dependence (Boss 2002; Cai et al. 2006), whereas models
of planet formation by core accretion seem to readily re-
produce the observed trend (Ida & Lin 2004b; Kornet et
al. 2005; Benz et al. 2006; Robinson et al. 2006). This is
because, in the core accretion models, planetesimals grow
into planet cores through collisions, subsequently accreting
gas from the surrounding gas disk once they become large
enough, and then interacting with that disk so as to migrate
inward (e.g., Lin & Papaloizou 1986; Papaloizou et al. 2007).
The core accretion models predict a metallicity dependence
because a higher metallicity implies higher solid mass and
hence faster core growth, which means that the critical core
mass for gas accretion can occur before the gas disk dissi-
pates on ∼6 Myr timescales (Haisch, Lada & Lada 2001;
Clarke, Gendrin & Sotomayer 2001). However, it remains
to be explained why the metallicity dependence has a form
∝102Zas opposed to, e.g., ∝10Z. The origin of the depen-
dence found in these models is hidden somewhere within the
large number of model components of which they are com-
prised, although it has been shown that a large solid disk
mass is required if planets are to form (Ida & Lin 2004b).
In this paper we consider the origin of the form of the
metallicity dependence using a simple heuristic model in
which detectable planets form as long as the solid mass of
material in the protoplanetary disk exceeds a critical value
(e.g., Greaves et al. 2007). That model is described in §2,
where it is shown how the metallicity relation is then directly
related to the initial disk mass distribution. This section also
compares the disk mass distribution required to reproduce
the observed planet-metallicity trend in this model with that
inferred from sub-mm observations of star forming regions.
The implications of this model are discussed in §3, along
with a discussion of why the solid mass should provide such
a strong constraint on whether a system goes on to form a
detectable planet. The conclusions are given in §4.
2 CRITICAL SOLID MASS MODEL
This model assumes that stars form surrounded by a pro-
toplanetary disk which is made up of both solids and gas.
We denote the mass of each of these components by Msand
Mg, respectively. The gaseous component dominates the to-
tal mass of the disk, and it is assumed that the outcome of
the star formation process results in some universal distri-
bution of disk masses (i.e., gas masses), which we define by
the probability of any given star having had a protoplane-
tary disk with a gas mass larger than M′
gas P(Mg> M′
g).
The solid mass of any given disk is assumed to be directly
related to the mass of the gaseous component through the
final metallicity of the star (e.g., Greaves et al. 2007):
Ms= 0.01Mg10Z.(2)
Here we have assumed that the ratio of gas to solids is 100 for
stars formed in a Z= 0 environment, consistent with that
seen in nearby star forming regions (James et al. 2006). Thus
it is assumed that stellar metallicities are indicative of the
conditions present prior to the formation of the star that
continued to be reflected in the composition of the proto-
planetary disk, and that exerted no influence over the re-
sulting distribution of protoplanetary disk masses.
The most important assumption is then that all of the
stars that have disks with Mslarger than some critical value
Ms,crit go on to form planets which can be detected in ra-
dial velocity surveys, i.e., Ppl =P(Ms> Ms,crit). The phys-
ical origin for this critical value is not part of this heuristic
model, although it does have a physical motivation based on
core accretion models (e.g., Ida & Lin 2004b), as discussed
in §1 and in more detail in §3.
2.1 Analytical solution
Since the probability of forming a planet depends only on
the solid mass, the critical mass above which the total disk
mass (i.e., gas mass) must be to form a planet is dependent
on metallicity:
Mg,crit = 100Ms,crit10−Z.(3)
The gas mass distribution is assumed to be independent
of metallicity, and so the probability of any star forming
a planet is metallicity dependent, since Ppl =P(Mg>
Mg,crit). Thus to reproduce equation (1) requires a gas mass
distribution in which:
P(Mg> M
′
g) = 0.03(100Ms,crit/M
′
g)2,(4)
where the critical solid mass required to form a planet,
Ms,crit, is some as yet undefined constant. Since the prob-
ability of any star hosting a planet given in equation (1)
is only known to apply for Ppl <0.25 (due to the lack of
surveys at higher Z), it follows that the distribution given
in equation (4) is also only valid for P(Mg> M′
g)<0.25,
and so for Mg>√1200Ms,crit. Thus, in this model the ob-
served Ppl(Z) in equation (1) is telling us about the mass
distribution of the most massive 25% of disks.
2.2 Gas disk distribution from observations
The gas mass distribution required by this model in order
to match the observed Ppl(Z) (equation 4) can now be com-
pared with the observed gas mass distribution. The gas mass
distribution of protoplanetary disks is not well-known be-
cause the majority of that mass is in molecular hydrogen
which is difficult to detect, especially in the cold outer re-
gions of the disks where most of the mass resides (Thi et al.
2001; Sheret, Ramsay-Howat & Dent 2003). Species such
as CO are easier to detect (e.g., Dent, Greaves & Coul-
son 2005; Dutrey, Guilloteau & Ho 2007), however there
is uncertainty in the CO/H2ratio because some of this gas
Origin of Metallicity Dependence of Exoplanets 3
ends up frozen onto dust grains or photo-dissociated (Dulle-
mond et al. 2007; Najita et al. 2007). On the other hand, the
dust mass distribution of protoplanetary disks is well char-
acterised, since this can be measured with relatively few un-
certainties from sub-mm and mm wavelength observations
(Andr´e & Montmerle 1994; Beckwith, Henning, & Nakagawa
2000).
Here we make the assumption that dust mass can be
used as a proxy for the total gas mass in protoplanetary
disks (for a fixed Z), and we derive the gas mass distri-
bution from the dust mass distribution in Taurus-Auriga,
which was measured using sub-mm photometry of 153 pre-
main sequence stars by Andrews & Williams (2005; hereafter
AW05). Since the stars in the AW05 sample are at a range
of evolutionary stages we chose to use only the disk masses
of the 75 class II objects (i.e., T Tauri stars) in their sam-
ple to ensure that the disk mass distribution is indicative of
that at the epoch of planet formation. Class I sources were
omitted because of a potential contribution to the sub-mm
flux from a remnant circumstellar envelope. Class III sources
were omitted because of the possibility that their currently
low disk masses are a consequence of the disks being at an
advanced evolutionary stage, and so are not necessarily in-
dicative of a low mass present at the planet forming epoch.
To obtain the gas mass distribution, the gas/dust ratio was
assumed to be 100 for all stars, based on the metallicities in
nearby star forming regions being close to solar with a small
dispersion for each region (Padgett 1996; Vuong et al. 2003;
James et al. 2006). The mass distribution of class II ob jects
is shown in Fig. 1. Ten objects from this sample have only
upper limits to their disk masses, which were set to zero in
Fig. 1. Since these upper limits are ≤1MJ, we infer that the
disk mass distribution is accurate for the most massive 69%
(52/75) of disks that are above this limit (≥1MJ). 1
The critical solid mass model (§2.1) was used to deter-
mine the metallicity relation predicted from the observed
gas mass distribution:
Ppl = [N(Mg> M
′
g)±pN(Mg> M′
g)]/Ntot,(5)
where Poisson counting statistics were used to determine the
uncertainty in the number of disks larger than a given limit
in the distribution and Ntot = 75. The probability deter-
mined from equation (5) could be assigned a corresponding
metallicity, Z′, from the relation M′
g=Mg,crit. Equation (3)
means that
Z
′
=−log 0.01M
′
g/Ms,crit.(6)
The value of Ms,crit was constrained to achieve a mean
planet probability for the metallicity range Z= 0.25 −0.5
dex in agreement with that found by FV05, i.e., Ppl =
14.8±3.5%, giving2
1We note that, even though 10 of the AW05 class II sources
were not detected in individual sub-mm photometry observations,
co-addition of this data-set leads to a net positive detection of
2.7±0.9 mJy, corresponding to a mean disk mass of 0.14MJwhich
is consistent with that expected from the log-normal distribution
plotted in Fig. 1.
2The value derived in equation 7 differs slightly from the 0.24MJ
derived by Greaves et al. (2007) because that paper included disks
from AW05 of both class II and III in their primordial gas mass
distribution.
Figure 1. Distribution of protoplanetary disk gas masses. The
gas mass distribution inferred from the dust mass distribution of
class II ob jects in Taurus-Auriga (Andrews & Williams 2005) is
shown with a dashed line. The dotted line is a log-normal fit to
this distribution centred on 2.5MJof width 0.77 dex. The distri-
bution required in the critical solid mass model to fit the extraso-
lar planet metallicity relation (equations 1 and 4) is shown with
a solid line, assuming Ms,crit = 0.5MJ.
Figure 2. Probability that a star of given metallicity has an
extrasolar planet that is detectable in the current radial velocity
surveys. The predictions of the critical solid mass model based
on the Andrews & Williams (2005) distribution of dust masses
of class II protoplanetary disks in Taurus-Auriga is shown with
a dashed line, with errors indicated by diamonds with √Nerror
bars. The asterisks shows the results of the radial velocity survey
of Fischer & Valenti (2005) with √Nerror bars. The fit to the
FV05 data (equation 1) is shown with a solid line.
Ms,crit = (0.5±0.1)MJ.(7)
The extrasolar planet-metallicity relation predicted by this
model is plotted in Fig. 2, and shows good agreement with
the observed relation (equation 1).
We have also inverted the problem by deducing the re-
quired disk mass distribution that would lead to the solid
line in Figure 2 (i.e., Ppl (Z) parameterised according to
4M. C. Wyatt et al.
equation 1). In Figure 1 we compare this required distribu-
tion with the observed gas mass distribution. Noting that
this comparison can only be made over the upper quar-
tile of disk masses (since current planet detection statis-
tics only extend to metallicities <0.5 and, in the model,
it is only this range of disk masses which can form planets
in this metallicity regime), it is evident that there is also
good agreement between the model and observed distribu-
tions when plotted in this way. To quantify this, we per-
formed a one sided Kolmogorov-Smirnov test to compare
the distribution of gas masses inferred from AW05, when
converted into metallicity (equation 6), with that inferred
from equation (1) for the range Z′=−0.5 to 0.5 dex. We
found that discrepancies as large as or greater than those ob-
served occur in 69% of samples of 75 members drawn from a
population with a cumulative distribution function in which
P(Z < Z ′) = 0.03 ×102Z
′
; i.e., we conclude that the gas
mass data are not unlikely to be drawn from such a distri-
bution, since at least 2 out of 3 times one would expect data
at least as discrepant as observed.
3 DISCUSSION
We have shown, under the assumption that a critical solid
mass in the protoplanetary disk is required to form a planet
that is detectable in radial velocity surveys, that the ob-
served frequency of planet detections as a function of metal-
licity, Ppl(Z), is compatible with the observed disk mass dis-
tribution (as derived from sub-mm dust mass measurements
of Classical T Tauri stars in local star forming regions). We
now discuss the physical basis for this simple model and
further observational tests.
3.1 Comparison with core accretion models
To consider the physical basis for the outcome of planet for-
mation being determined solely by dust mass, we appeal
to the core accretion models of Ida & Lin (2004a, 2004b;
hereafter IL04). The IL04 models are local, in the sense that
planet formation depends on local quantities such as gas and
solid surface density. Therefore we expect any threshold ef-
fect to involve surface density rather than mass. We first as-
sess whether the results of IL04 are compatible with the hy-
pothesis that planet formation requires a critical metal licity
independent solid surface density and return to a discussion
of the relationship between solid surface density normalisa-
tion and dust mass in §3.2. We can assess this hypothesis in
two ways. Firstly, we can simply take the distribution of disk
surface densities assumed by IL04 (a log-normal distribution
of width 1.0 dex that is centred on the surface density of the
minimum mass solar nebula and truncated at >1.48σ), ap-
ply a threshold solid surface density for planet formation
that is independent of metallicity and see whether we can
reproduce their numerical results. Figure 3 shows that this
is indeed the case: the nominal model from IL04b is well
reproduced by assuming a critical solid surface density of 8
times the minimum mass solar nebula, whereas their variant
models where the rate of core accretion is enhanced or re-
duced by a factor of three are well reproduced by models in
which the critical solid surface density is respectively 4 and
Figure 3. Prediction of the critical solid mass model for the prob-
ability that a star of given metallicity has an extrasolar planet
that is detectable in the current radial velocity surveys assuming
the disk mass distribution used in Ida & Lin (2004b) with critical
solid masses of 4 (dashed line), 8 (solid line) and 22 (dotted line)
times the minimum mass solar nebula. The numerical results of
the nominal model in IL04b are shown with filled circles, and the
results of their models in which the core accretion rate is 3 times
faster and slower than the nominal model shown with triangles
and crosses (see their fig. 2b).
22 times the minimum mass solar nebula. We stress that
the IL04 models contain a large number of ingredients and
do not explicitly impose a threshold criterion. Nevertheless,
we see that their results are empirically equivalent to the
imposition of a simple threshold.
In a second approach, we can now attempt to under-
stand why the IL04 models behave in this way. Examination
of these models shows that the formation of gas giant planets
hinges on rocky cores being able to grow to a critical mass (a
few M⊕) before the gas disk is dispersed. The requirement of
sufficiently rapid core growth implies that they have to form
inside a critical radius, aig, which depends on both gas and
solid surface densities. On the other hand, inward of a second
critical radius, atg (which depends on solid surface density),
a critical core mass is not achievable because the required
core mass exceeds the local isolation mass (at which point
the core has consumed all the material in its local feeding
zone). Evidently, the formation of gas giant planets is possi-
ble only for the case atg < aig and we can derive a condition
on the gas and solid surface densities corresponding to the
critical case where atg =aig. This translates into a condi-
tion on the minimum surface density of solids as a function of
metallicity. We find that the critical surface density of solids
scales as 10−0.06Z(assuming, as in IL04, that a disk’s surface
density scales Σ ∝r−pwhere ris radius and p= 1.5). This
very weak dependence on metallicity results from the fact
that the growth rate of solid cores is much more strongly
dependent on orbital radius than on the gas column density
and hence aig is only very weakly dependent on gas column
density. Therefore, the threshold criterion aig =atg is nearly
independent of gas column density and thus the dependence
of critical dust column on metallicity is extremely weak. It is
this extremely weak dependence of the critical solid surface
Origin of Metallicity Dependence of Exoplanets 5
density on metallicity which we believe to account for the
excellent correspondence between the numerical results of
IL04 and the application of our simple threshold hypothesis
(see Fig. 3).
A further test of this hypothesis would be to exam-
ine how Ppl(Z) predicted by the core accretion models de-
pends on the assumed distribution of disk surface densities,
since if the outcome is governed by a critical surface density
of solids for planet formation then using a narrower dis-
tribution of disk surface densities as input would result in
a steeper metallicity dependence (since in the critical solid
surface density model the metallicity dependence simply re-
flects the disk surface density distribution used as input). In
contrast to IL04, Robinson et al. (2006) did vary this quan-
tity and indeed found that Ppl (Z) rose more gently when a
larger range of disk surface densities was employed.
3.2 Why sub-mm dust mass determines outcome
Regardless of the comparison with core accretion models, it
is notable that the critical solid mass model fits the planet-
metallicity relation found in nature. It is, however, surprising
that sub-mm dust mass should be such a good indicator of
whether planets are going to form in a disk, since sub-mm
measurements probe the current mass in mm- to cm-sized
dust and so are not necessarily representative of the primor-
dial inventory of solid or gas mass. Indeed, class II objects in
Taurus-Auriga have a range of ages and so we would expect
the oldest stars to have already lost a significant quantity of
gas through accretion onto the star (Clarke et al. 2001). We
may also expect some loss of detectable dust mass with age
through grain growth and accretion onto the star with the
gas. However, there is no evidence that sub-mm dust mass
changes with age on the pre-main sequence (e.g., Wyatt et
al. 2003) suggesting that the mass in mm- to cm- sizes is
constant. This is to be expected, since the total dust mass
Mdust ∝r2−p
out , where rout is the disk outer edge, so that as
long as p < 2 the sub-mm dust mass is concentrated in the
outer regions of the disk. Since typically observed values for
protoplanetary disks are p≈0.85 and rout ≈200 AU (An-
drews & Williams 2007), the timescale for grains containing
most of the disk mass to grow to larger than 1 metre, and
so become invisible in the sub-mm, may be expected to be
longer than the 10Myr period over which planet formation
(in the inner regions) must take place (e.g., Dullemond &
Dominik 2005). Indeed some disks cannot harbour signif-
icant quantities of ”unseen” dust mass (i.e., with particle
sizes either much larger or smaller than 1 mm), since, even
in the absence of such unseen contributions, the gas mass
inferred from mm dust measurements is in some cases al-
ready ∼0.2 times the central star’s mass, and thus close
to the limit for gravitational instability. Given the evidence
that grain growth to mm and cm scales has occurred in the
outer regions of disks (Wilner et al. 2006), we are confident
that this grain size scale contains the majority of the disk
solid mass at these radii, and thus, by implication, the ma-
jority of the solid mass in the disk. Thus, while the dust seen
in the sub-mm is not contributing to the planet formation
process (because it is mainly at radii where it has not had
time to grow to large - greater than metre - size scales), we
are suggesting that it is nevertheless a good measure of the
primordial inventory of solids in the disk.
The fact that the sub-mm dust mass distribution fits
the observed planet-metallicity relation so well is because
there is an order of magnitude difference between the highest
and lowest masses of the top ∼25% most massive gas disks
(e.g., Figs 1 and 2). This result is not specific to the Taurus-
Auriga star forming region, since class II disks in ρOph also
exhibit an order of magnitude range for the most massive
25% of those disks (see Fig. 9 of Andr´e & Montmerle 1994).
If this distribution had been much narrower or broader then
we would have been able to rule out the critical solid mass
model.
One further requirement of nature for the critical solid
mass model to work is for a disk’s outer radius to be less im-
portant than its solid mass in setting the outcome of planet
formation. As noted in §3.1, models such as those in IL04
rely on a critical surface density (rather than mass). For the
surface density profile assumed by IL04, the surface density
normalisation (fd, where Σ ∝fd), disk outer radius (rout)
and total solid mass (Ms) are related via Ms∝fdr0.5
out. Thus
the mapping between critical surface density and critical
mass is (weakly) dependent on rout. While disk radii have
been measured using sub-mm interferometry (Kitamura et
al. 2002; Andrews & Williams 2007), these samples are bi-
ased toward the most massive disks so that it is not clear
how representative the observed distribution is of the pop-
ulation as a whole. However, there is no evidence that the
distribution of rout is as broad as that of disk masses seen
by AW05. We therefore expect the surface density of solids
in the planet formation region to be mainly controlled by
Msrather than rout, thus explaining the apparent success of
sub-mm flux as a predictor of planet forming potential.
3.3 Disks forming detectable planets
One implication of this study is that we can predict which
of the disks in the AW05 sample will go on to form plan-
ets like those detected in the current radial velocity surveys.
The class IIs in their sample with more than 0.5MJof dust
are 04113+2758, DL Tau, GG Tau and GO Tau. However,
we disqualify GG Tau as a planet-forming candidate, since
its disk is circumbinary (Guilloteau, Dutrey & Simon 1999),
and so its high sub-mm flux does not equate with a high sur-
face density of solids in the inner disk. Massive circumbinary
disks are rare (Jensen, Mathieu & Fuller 1996), so the ma-
jority of the more massive disks are not circumbinary disks
and so would not be unsuitable for forming planets. Apply-
ing the same 0.5MJdust mass limit to the ρOph study of
Andr´e & Montmerle (1994) indicates that of the class IIs in
this region, AS205, EL24, GSS39 and SR24S may go on to
form detectable planets.
While we do not claim that we can unambiguously pre-
dict the outcome of planet formation for any one of these
systems, we do suggest that studying the disks that are pre-
dicted to form planets, the characteristics of which we can
constrain at least statistically, may provide a valuable way
of probing the environments in which such planets form.
The fact that it is the most massive disks which go on
to form detectable planets means that these disks must
be close to being gravitationally unstable, since the ratio
Mdisk/M⋆>0.05 for Z= 0 and M⋆= 1M⊙. This suggests
that instability could play a role in the formation process.
However, this cannot be the only determining factor, since
6M. C. Wyatt et al.
the gravitational instability process itself is not affected by
metallicity (Cai et al. 2006), and there would be no metal-
licity dependence if Mg,crit is a constant and not dependent
on metallicity. Thus, this suggests that some degree of in-
stability may help speed up the core accretion process, e.g.,
through concentration of particles in spiral structures (Rice
et al. 2004) or instability in a thin dust layer (Youdin & Shu
2002).
3.4 Observational tests
Here we suggest three observational tests of the critical solid
mass model:
Firstly, if the model is correct, we would expect Ppl(Z)
to rise much less steeply with Zat metallicities above 0.5
dex than implied by an extrapolation of equation (1), since
at higher metallicities the model predicts that planets would
be able to form in lower mass disks, and that Ppl(Z) in this
regime would reflect the disk mass distribution of interme-
diate mass disks. A discrepancy between the observed disk
mass distribution and that resulting from an extrapolation
of equation (1) to Z > 0.5 dex is readily apparent by con-
sidering how the solid curve on Figure 1, if extrapolated to
lower disk masses, would compare with the dashed line on
that Figure. Whether a suitable high metallicity sample can
be found to test this prediction remains to be seen (e.g.,
Laughlin 2000; Valenti & Fischer 2005; Taylor 2006).
Secondly, one of the key assumptions of the model was
that the distribution of protoplanetary disk masses is uni-
versal in that it is independent of metallicity. This can be
tested by measuring the distribution of dust masses in low
(or high) metallicity star forming regions using sub-mm pho-
tometry, since these masses should be correspondingly lower
(or higher) than those of nearby regions like Taurus-Auriga
where Z≈0. While ALMA can detect the brightest known
class II disks out to 20 kpc, we are not aware of any young
(<10 Myr) cluster within the Milky Way which has a mea-
sured metallicity that is sufficiently sub- or super-solar for
the predicted difference in disk mass distribution in com-
parison with Taurus-Auriga to be confidently detected, al-
though star forming clusters such as those found by Santos
et al. (2000) and Yun et al. (2007) may be suitable candi-
dates if their large Galactocentric distances (15-16.5 kpc)
are indicative of a low metallicity as suggested by observa-
tions Cepheids which indicate a metallicity gradient in the
Milky Way of −0.06 dex/kpc (Luck et al. 2006).
Thirdly, we will be able to test in due course an ad-
junct hypothesis, i.e., that the incidence of planets of lower
masses (and at greater orbital distances) is also regulated by
a (lower) critical solid mass threshold. For example, extrapo-
lation of the exoplanet semi-major axis distribution to 20 AU
suggests that surveys able to detect planets to that distance
would double the fraction of stars known to have planets to
12% (Marcy et al. 2005). The simplest hypothesis we can ap-
ply to this population would simply be that the progenitor
disks corresponded to the top 12% of the disk mass distri-
bution, implying a critical solid mass of ∼0.3MJ. We plot
in Figure 4 the predicted dependence of planet frequency
on metallicity in this case. Although this ”prediction” will
eventually be compared with observational data, we empha-
sise that it is not entirely clear how this adjunct hypothesis
(i.e., that the critical solid mass is lower for planets located
Figure 4. Prediction of the critical solid mass model for the prob-
ability that a star of given metallicity has an extrasolar planet.
Assuming that surveys with different detection thresholds corre-
spond to different critical solid masses, the prediction for planets
that are detectable in the current radial velocity surveys is shown
with a solid line, and the prediction for a survey with an overall
planet detection frequency of 12% for the metallicity range -0.5
to 0.5 dex is shown with a dashed line.
at larger distances) can be squared with the expectations of
core accretion models.
4 CONCLUSIONS
We have presented a simple analytical model which can be
used to predict the outcome of planet formation, in which
the formation of a planet that is detectable in radial veloc-
ity studies depends only on the mass of solids in the pro-
toplanetary disk. We showed that this model predicts that
the observed planet-metallicity relation is a reflection of the
disk mass distribution. We also argued that the sub-mm
dust mass seen in the protoplanetary disk phase is a good
tracer of the initial mass budget available close to the star
for planet formation, and showed that the observed planet-
metallicity relation is consistent with the disk mass distribu-
tion estimated from sub-mm observations of protoplanetary
disks if the critical solid mass required to form detectable
planets is 0.5MJ.
We suggested that the detailed physics of the IL04 core
accretion models boil down to a critical solid mass required
to form detectable planets, although it needs to be confirmed
that the good empirical agreement with the IL04 models is
more than a coincidence. However, the value of this model
is not just in its relevance to specific core accretion mod-
els, but in its general applicability, since it shows how the
observed planet-metallicity relation would be reproduced by
any planet formation model which imposes a critical solid
mass for the formation of detectable planets. Other reasons
for imposing a threshold on a disk’s solid mass before de-
tectable planets can form include the possibility that such
conditions are required for the formation of >km-sized plan-
etesimals through gravitational instabilities (e.g., Johansen,
Klahr & Henning 2006).
The value of this model is also in its simplicity, since
Origin of Metallicity Dependence of Exoplanets 7
Figure 5. Distribution of infrared luminosities (f=Lir/L⋆) of
the debris disks of A stars in the model of Wyatt et al. (2007). The
distribution for the debris disks formed from the most massive
6% of protoplanetary disks (the planet bearers) is compared with
those formed from the least massive 94% of protoplanetary disks
(the non-planet bearers).
this means that it can be readily applied to predict other
observable properties of stars with and without detectable
planets, should those properties also depend on the solid
mass of the protoplanetary disk. For example, the statistics
for the incidence of debris disks around A stars as a func-
tion of stellar age (e.g., Rieke et al. 2005) can be explained
by a model in which all stars form planetesimal belts, the
initial mass of which is determined by the solid mass in
the protoplanetary disk (a distribution which is taken from
AW05), and which are subsequently eroded by steady state
collisional processing (Wyatt et al. 2007). We ran the A
star debris disk model in order to predict the distribution of
f=Lir/L⋆for the ensemble, computing separate distribu-
tions for the planet bearers (corresponding to the top 6% of
the input mass distribution) and the non-planet bearers (cor-
responding to the remaining 94% of the population). Fig. 5
shows how the debris disks of the planet bearers are, on av-
erage, more luminous than those of the non-planet bearers;
specifically, the mean luminosity of the most luminous 10%
in both distributions (in Lir/L⋆) differ by a factor of ∼6.
While we do not know whether it is only the top 6% of A
star protoplanetary disks that form detectable planets, be-
cause the A star exoplanet population is poorly known at
present, here we predict that if A star planets form in a sim-
ilar manner to those of sun-like stars (i.e., with a threshold
solid mass criterion), then this will be seen in the luminosity
distributions of their debris disks (Fig. 5). Similarly, while
we do not know whether the luminosity distributions of the
debris disks of sun-like stars behave as shown in Fig. 5, be-
cause the model of Wyatt et al. (2007) has yet to be applied
to that population, here we predict that if the luminosities
of sun-like star debris disks are governed by steady state
processes, then the distributions of those luminosities will
exhibit a trend similar to that in Fig. 5. Indeed observations
of the debris disks of sun-like stars both with and without
planets do show a trend in their luminosities of comparable
magnitude to that suggested by Fig. 5 (Bryden et al. 2007).
Application of this model to known systems implies that
the disks of 04113 + 2758, DL Tau, GO Tau, AS205, EL24,
GSS39 and SR24S will form (or have formed) gas giant plan-
ets. While the outcome of planet formation in individual
systems is uncertain, we suggest that studying these disks
may help constrain the physical conditions of disks in which
we know, at least statistically, what the outcome of planet
formation will be. Observational tests of the model include
a flattening of the metallicity relation for Z > 0.5 dex and
also a flattening as planet searches continue.
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