Predicting the frequencies of diverse exo-planetary systems
Mon. Not. R. Astron. Soc. 000, 1–6 (2006) Printed 25 October 2018 (MN LATEX style file v2.2)
Predicting the frequencies of diverse exo-planetary systems
J.S. Greaves1⋆, D.A. Fischer2, M.C. Wyatt3, C.A. Beichman4,5 and G. Bryden4
1Scottish Universities Physics Alliance, Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, Fife KY16, UK
2Department of Astronomy, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA
3Institute of Astronomy, Madingley Rd, Cambridge CB3 0HA, UK
4Michelson Science Center, California Institute of Technology, M/S 100-22, Pasadena, CA 91125, USA
5Jet Propulson Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA.
Accepted 2007. Received 2007; in original form 2006
ABSTRACT
Extrasolar planetary systems range from hot Jupiters out to icy comet belts more
distant than Pluto. We explain this diversity in a model where the mass of solids in
the primordial circumstellar disk dictates the outcome. The star retains measures of
the initial heavy-element (metal) abundance that can be used to map solid masses
onto outcomes, and the frequencies of all classes are correctly predicted. The differing
dependences on metallicity for forming massive planets and low-mass cometary bodies
are also explained. By extrapolation, around two-thirds of stars have enough solids
to form Earth-like planets, and a high rate is supported by the first detections of
low-mass exo-planets.
Key words: circumstellar matter – planetary systems: protoplanetary discs – plan-
etary systems: formation
1 INTRODUCTION
Extrasolar planetary systems have largely been identified by
a change in the line-of-sight velocity in spectra of the host
star, the ‘Doppler wobble’ method (Cumming et al. 1999,
e.g.). This technique detects inner-system gas-giant plan-
ets out to a few astronomical units. Contrasting larger-scale
systems are those with ‘debris’ disks, rings of dust parti-
cles produced in comet collisions, whose presence indicates
that parent bodies exist at least up to a few kilometres in
size (Wyatt & Dent 2002). Most images of debris disks show
central cavities similar to that cleared by Jupiter and Sat-
urn in our own Solar System (Liou & Zook 1999), and also
sub-structure attributed to dust and planetesimals piled up
in positions in mean motion resonance with a distant giant
planet (Wyatt 2003). Beichman et al. (2005) have discov-
ered a few systems with both debris disks and inner giants,
linking these divergent outcomes.
These various planetary systems could reflect differ-
ent initial states, in particular the quantity of planet-
forming materials in the circumstellar disk around the
young host star. In core-growth models (Pollack et al. 1996;
Hubickyj et al. 2005, e.g.), a large supply of refractory el-
ements (carbon, iron, etc.) should promote rapid growth
of planetesimals that can then amalgamate into planetary
cores. If gas still persists in the disk, the core can attract
a thick atmosphere and form into a gas giant planet; the
⋆ E-mail: jsg5@st-andrews.ac.uk (JSG)
disk is also viscous so that the planet tends to migrate in-
wards (Nelson et al. 2000). For sparse refractories, however,
only small comets up to Neptune-like ‘ice giants’ may have
formed when the gas disperses; Greaves et al. (2004a) sug-
gested that such systems will be characterised by planetesi-
mal collisions and hence debris emission.
Here we classify these outcomes from most to least suc-
cessful, and postulate that the dominant agent is the initial
mass in refractories. We aim to test whether one underlying
property has a highly significant effect on the outcome, and
so intentionally ignore other properties that could affect the
planetary system formed around an individual star. Such
properties include the disk size, geometry, composition and
lifetime, as well as the stellar accretion rate, emission spec-
trum and environment. Stochastic effects are also neglected,
such as inwards migration caused by inter-planet encoun-
ters (Marzari & Weidenschilling 2002) and debris brighten-
ing after collisions of major planetesimals (Dominik & Decin
2003). Such factors are beyond the scope of our simple
model, but are important for detailed understanding of the
formation of particular kinds of planetary system.
2 HYPOTHESIS
The hypothesis explored here is that the mass of solid ele-
ments in a primordial circumstellar disk can be quantified,
and linked to an outcome such as a detectable debris disk or
radial velocity planet. The only piece of relevant ‘relic’ in-
c© 2006 RAS
http://arxiv.org/abs/0704.0873v1
2 J. S. Greaves, D. A. Fischer, M. C. Wyatt, C. A. Beichman & G. Bryden
Figure 1. Cumulative functions of [Fe/H] for the host stars of
hot Jupiters (at 6 0.1 AU), cool Jupiters (> 0.1 AU), debris-and-
planet systems, and debris disks only. One cool Doppler compan-
ion with [Fe/H] of –0.65 is not shown, as it is suspected to be
above planetary mass (Fischer & Valenti 2005).
formation for a particular star is the metallicity, quantified
by [Fe/H], the logarithmic abundance of iron with respect to
hydrogen and normalised to the Solar value, and this quan-
tity is taken here to track refractories in general. In com-
bination with a distribution of total (gas-plus-dust) masses
of primordial disks, masses of solids can then be inferred
statistically. Our basic hypothesis is that the metallicity is
a relic quantity originally common to the young star and
its disk, and that higher values of the trace refractory iron
should correlate with more effective planet growth.
It is well-known that gas giant detection rates rise
at higher [Fe/H] (Fischer & Valenti 2005, e.g), while
Greaves et al. (2006) have noted that debris detection is es-
sentially independent of metallicity. Robinson et al. (2006)
have shown that the growth of gas giant planets can be re-
produced in simulations taking disk mass and metal content
into account. Here we extend such ideas to include systems
with Doppler planets, with debris disks, and with both phe-
nomena.
Metal-based trends are now identified (Figure 1) for
four outcomes1 ranked from most to least successful: a) hot
Jupiters, b) more distant Doppler planets, with semi-major
axis beyond 0.1 AU, c) systems with both giant planets and
debris, and d) stars with debris only. Here we base ‘success’
(implicitly, via core-accretion models) on a large supply of
refractories and so fast evolutionary timescales, with hot
Jupiters rapidly building a core, adding an atmosphere and
migrating substantially towards the star. In systems with
progessively less success: planets form more slowly so they
migrate less over the remaining disk lifetime (cool Jupiters);
only some of the material is formed into planets (planet-and-
debris systems); and mainly planetesimals are formed, per-
haps with planets up to ice-giant size (debris). The 0.1 AU
1 The separation of hot and cool Jupiters in [Fe/H] has been pre-
viously noted as having modest significance (Santos et al. 2003;
Sozzetti 2004); a K-S test here gives P=0.35 for the null hypothe-
sis using the Fig. 1 data (errors in [Fe/H] of typically 0.025 dex).
The overlap of the two distributions is in fact an integral part of
our model, where two factors contribute to the outcome.
divide of hot/cool Jupiters is not necessarily physical, and is
simply intended to give reasonable source counts. Notably,
in systems with debris plus Jupiters, the planets orbit at >
0.5 AU, consistent with even lower success in migration.
The plotted ranges of [Fe/H] shift globally to higher
values with more success, as expected – however, the ranges
also overlap indicating that the fraction of metallic elements
is not the only relevant quantity. The hypothesis to be tested
here is that the underlying dominant factor is the total mass
of metals in the primordial disk. This mass is the product
of the total mass M of the disk (largely in H2 gas) and the
mass-fraction Z of refractory elements. The range of Z for
each outcome is broad because disks of different M were
initially present; thus, a massive low-metal disk and a low-
mass metal-rich disk could both lead to the same outcome.
3 MODEL
The distribution of the product MS ∝ M Z was constructed
from a mass distribution measured for primordial disks
(Andrews & Williams 2005) and our metal abundances mea-
sured in nearby Sun-like stars (Valenti & Fischer 2005).
With the canonical gas-to-dust mass ratio of 100 for present-
day disks, the mass of solid material is then
MS = 0.01M × 10
[Fe/H]
, (1)
with the modifying Z-factor assumed to be traced by iron.
Padgett (1996) and James et al. (2006) find that the metal-
licities of present-day young star clusters are close to Solar
(perhaps lower by ∼ 0.1 dex), so the reference-point values
of 100 for the gas-to-dust ratio and 0 for [Fe/H] are self-
consistent to good accuracy.
Our model has the following assumptions.
• There are contiguous bands of the MS distribution, cor-
responding to different planetary system outcomes. Any one
disk has an outcome solely predicted by its value of MS . The
ranges of M and Z can overlap, as only the MS bands are
required to be contiguous.
• A particular outcome will arise from a set of M Z
products, with the observable quantity being the metallicity
ranging from Z(low) to Z(high). This Z-range sets the MS
bounds, as described below.
• For the stellar ensemble, all possible products are as-
sumed to occur, and to produce some outcome that is actu-
ally observed. There are no unknown outcomes (novel kinds
of planetary system not yet observed).
• M and Z are taken to be independent, as the disk mass
is presumably a dynamical property of the young star-disk
system and the mass in solids is always a small component.
The MS bounds were determined using the minimum
number of free parameters. For a particular outcome, the
observed Z-range is assumed to trace the bounds of MS , i.e.
MS(high)/MS(low) = f × Z(high)/Z(low) (2)
where f is a constant, and this relation sets solid-mass
bounds for each outcome. Results presented here are mainly
for f = 1, but more complex relationships could exist, and
other simple assumptions without further variables could
also have been made – alternative forms of Eq. 2 are ex-
plored in section 4.1
c© 2006 RAS, MNRAS 000, 1–6
Predicting the frequencies of diverse exo-planetary systems 3
Table 1. Results of the planetary system frequency calculations. The ranges of solid mass MS(low) to MS(high) were determined based
on the input data of the observed [Fe/H] range for each outcome. The total disk masses given are consistent with these two boundary
conditions. The ranges of observed frequency include different surveys and/or statistical errors as discussed in the text. The predicted
total is less than 100 % because of a few ‘missing outcomes’ (see text). The null set represents stars searched for both planets and debris
with no detections.
outcome [Fe/H] range solid mass total disk mass predicted observed
(MJup [M⊕]) (MJup) frequency frequency
hot Jupiter –0.08 to +0.39 1.7–5 [500–1600] 70–200 1 % 2 ± 1 %
cool Jupiter –0.44 to +0.40 0.24–1.7 [75–500] 10–200 8 % 9 (5–11) %
planet & debris –0.13 to +0.27 0.10–0.24 [30–75] 5–30 5 % 3 ± 1 %
debris –0.52 to +0.18 0.02–0.10 [5–30] 1–30 16 % 15 ± 3 %
null –0.44 to +0.34 < 0.02 [< 5] < 15 62 % ∼ 75 %
Absolute values for MS were derived iteratively, work-
ing downwards from the most successful outcome. The upper
bound for the hot Jupiter band was derived from the high-
est disk mass observed and highest metallicity seen for this
outcome (under the assumption that all M Z products are
observed). Other outcomes were then derived in turn assum-
ing f = 1, and working down in order of success, with each
lower bound MS(low) setting the upper bound MS(high)
for the next most successful outcome. Both M and Z have
log-normal distributions, and 106 outcomes were calculated,
based on 1000 equally-likely values of log(M) combined with
1000 equally-probable values of log(Z), i.e. [Fe/H].
3.1 Input data
The M-distribution is taken from a deep millimetre-
wavelength survey for dust in disks in the Tau-
rus star-forming region by Andrews & Williams (2005).
Nürnberger et al. (1998) have found similar results from
(less deep) surveys of other regions, so the Taurus results
are taken here to be generic, under the simplification that
local environment is neglected. The total disk masses M were
found from the dust masses multiplied by a standard gas-
to-dust mass ratio of 100. The mean of the log-normal total
disk masses is 1 Jupiter mass (i.e. log M = 0 in MJup units)
with a standard deviation of 1.15 dex, and detections were
actually made down to −0.6σ. The Z-distribution is from
the volume-limited sample from Fischer & Valenti (2005) of
main sequence Sun-like stars (F, G, K dwarfs) within 18 pc.
These authors also list 850 similar stars out to larger dis-
tances that are being actively searched for Doppler planets.
In the 18 pc sample, the mean in logarithmic [Fe/H] is –0.06
and the standard deviation is 0.25 dex.
Both M and Z distributions have an upper cutoff at
approximately the +2σ bound: for M this is because disks
are less massive than ∼ 20 % of the star’s mass (presumably
for dynamical stability), and for Z because the Galaxy has a
metal threshold determined by nucleosynthetic enrichment
by previous generations of stars. The upper cutoff for M is
200 MJup for ‘classical T Tauri’ stars, and for Z the adopted
cutoff in [Fe/H] is +0.40 from the 18 pc sample (with a few
planet-hosts of [Fe/H] up to 0.56 found in larger volumes).
3.2 Observed frequencies
The Doppler-detection frequencies quoted in Table 1 are
mostly from the set of 850 uniformly-searched stars with
[Fe/H] values. The statistics for hot Jupiters range from
16/1330 = 1.2 % in a single-team search (Marcy et al.
2005) up to 22/850 = 2.6 % in the uniform dataset
(Fischer & Valenti 2005). For cool Jupiters (outside 0.1
AU semi-major axis), the counts are 76/850 = 8.9 %
(Fischer & Valenti 2005), with Marcy et al. (2005) finding
a range from 72/1330 = 5.4 % within 5 AU up to an extrap-
olation of 11 % within 20 AU. The upper limit is based on
long-term trends in the radial velocity data, and these as-yet
unconfirmed planetary systems could have [Fe/H] bounds
beyond those quoted here.
The debris counts are based on our surveys with
Spitzer. The debris-only statistics (Beichman et al. 2006;
Bryden et al. 2006) comprise 25 Spitzer detections out of
169 target stars in unbiased surveys, i.e. 15 ± 3 % with Pois-
sonian errors2. For planet-plus-debris systems, the 3 % fre-
quency is estimated from 6/25 detections (24± 10 %) among
stars known to have Doppler planets (Beichman et al. 2005),
multiplied by the 12 % total extrapolated planet frequency
(Marcy et al. 2005). In Table 1, the planet-only rates should
strictly sum to only up to 9 % if this estimate of 3 % of
planet-and-debris systems is subtracted.
The null set quoted has an [Fe/H] range derived from
our Spitzer targets where no debris or planet has been
detected. The MS(low) bound for the null set has been
set to zero rather than the formal limit derived from
Z(low)/Z(high), as lower values of MS(low) presumably also
give no presently observable outcome.
4 RESULTS
The predicted frequencies (Table 1) agree closely with the
observed rates in all outcome categories. This is remarkable
when very different planetary systems are observed by inde-
pendent methods, and ranging in scale from under a tenth
to tens of AU. The good agreement suggest that solid mass
may indeed be the dominant predictor of outcome.
The model is also rather robust. In particular, because
the M-distribution is an independent datum, obtaining a
good match of predicted and observed frequencies is not
inevitable. For example, artificially halving the standard
deviation of the M distribution would yield far too many
2 Figure 1 also includes a few prior-candidate systems
(www.roe.ac.uk/ukatc/research/topics/dust/identification.html)
confirmed by Spitzer programs.
c© 2006 RAS, MNRAS 000, 1–6
4 J. S. Greaves, D. A. Fischer, M. C. Wyatt, C. A. Beichman & G. Bryden
stars with detectable systems (75 %), and many more cool
Jupiters than hot Jupiters (around 20:1 instead of 5:1).
The model is also reasonably independent of outlier data
points. For example, adding in the low-metallicity system
neglected in Figure 1 would raise the prediction for cool
Jupiters from 8 % to 12 %, or extending the upper Z-cutoff
to the [Fe/H] of the distant Doppler systems would raise
this prediction to 11 %. However, the resulting effects on the
less-successful system probabilities are less than 1 %. This
suggests that although small number statistics may affect
the predictions, the results would not greatly differ if large
populations were available, that could be described by sta-
tistical bounds rather than minimum- to maximum-[Fe/H].
The model also accounts for nearly all outcomes, as re-
quired if each MS is to result in only one planetary system
architecture. For a few MS products, one of the outcomes
would be expected except that the Z value involved lies out-
side the observed ranges. These anomalous systems sum to
9 % (hence the Table 1 predictions add to < 100 %). These
‘missing’ systems are predominantly debris and debris-plus-
planet outcomes. The latter class has the smallest number
of detections (Figure 1), and more examples are likely to be
discovered with publication of more distant Doppler planets.
A further check is that the disk masses are realistic, in
terms of producing the observed bodies. Doppler systems are
predicted here to form from 5-200 Jupiter masses of gas in
the disk, enough to make gas giant planets. Also, the MS val-
ues of 30-1600 Earth-masses could readily supply the cores
of Jupiter and Saturn (quoted by Saumon & Guillot (2004)
as ≈ 0− 10 and ≈ 10− 20 M⊕ respectively) or the more ex-
treme ∼ 70 M⊕ core deduced for the transiting hot Jupiter
around HD 149026 (Sato et al. 2005). Similarly, populations
of colliding bodies generating debris have been estimated at
∼ 1–30 Earth-masses (Wyatt & Dent 2002; Greaves et al.
2004b), a quantity that could reasonably be produced from
5–75 Earth-masses in primordial solids (Table 1).
To make Jupiter analogues within realistic timescales,
core growth models (Hubickyj et al. 2005, e.g) need a few
times the MinimumMass Solar Nebula, which comprised ap-
proximately 20 MJup (Davis 2005). The Solar System itself
would thus have contained a few times 0.2 MJup in solids,
of which 0.15 MJup (50 M⊕) has been incorporated in plan-
etary cores. These primordial disk masses would place the
Solar System in the cool Jupiter category, and this is in fact
how it would appear externally (the dust belt being more
tenuous than in detected exo-systems).
4.1 Alternative models
Equation (2) was further investigated with f 6= 1. Reason-
able agreement with the observations was obtained only for
f in a narrow range, ≈ 0.8− 1.1. For higher f , systems with
debris are over-produced, while for lower f there are too few
Doppler detections. This suggests that the hypothesis that
the Z-range directly traces the MS range for the outcome
to occur is close to correct, although not well understood.
Qualitatively, there is a locus of points in M, Z parameter
space that lie inside the appropriate MS bounds for an out-
come, and at some mid-range value of M it is likely that
all the Z(low) to Z(high) values are appropriate, and so this
traces the product MS(high) to MS(low) (provided no more
extreme values of Z are suitable at the extrema of M). Z
seems to have the most constrictive effect on outcome be-
cause the distribution is much narrower than that of M, and
thus if M changes by a large amount, there is no correspond-
ing value of Z than can preserve a similar MS . Simulations
of planetesimal growth as a function of mass of solids in the
disk are needed to further explore why the ranges of Z and
MS match so closely.
One even simpler model was tested, with a constant
range of MS(high)/MS(low) for each observable outcome.
Assuming that ∼ 1 M⊕ of solids is needed for the minimum
detectable system (planetesimals creating debris), and that
the most massive disks contain 1600 M⊕ of solids (from the
maximum M and Z), then this mass range can be divided
into equal parts for the four observable outcomes, with a
range of log-M = 0.8 for each. This simple model fails, in
particular greatly over-producing debris-and-planet systems
at 12 %. Thus, it seems that the Z-range does in fact contain
information on the outcomes.
4.2 Distributions within outcomes
The dependences on metallicity of planet and debris detec-
tions arise naturally in the model. Doppler detections are
strongly affected by metallicity: Fischer & Valenti (2005)
find that the probability is proportional to the square of the
number of iron atoms. Our model predicts P(hot Jupiter) ∝
N(Fe)2.7 and P(cool Jupiter) ∝ N(Fe)0.9, above 80 %- and
30 %-Solar iron content respectively. The observed exponent
of 2 for all planets is intermediate between the two model
values, and as predicted by Robinson et al. (2006), a steeper
trend for short-period planets is perhaps seen, at least at
high metallicities (Figure 1). These dependences arise be-
cause large solid masses are needed for fast gas giant forma-
tion with time for subsequent migration, and so when [Fe/H]
is low a large total disk mass is required, with rapidly de-
creasing probability in the upper tail of the M-distribution.
In contrast, the model predicts a weak relation of
P(debris) ∝ N(Fe)0.2 (for 30-160 %-Solar iron), agreeing
with the lack of correlation seen by Bryden et al. (2006).
As fewer solids are needed to make comets, a wide variety
of parent disks will contain enough material, so little metal-
dependence is expected. For example (Table 1), MS(low) can
result from a 6 MJup disk with [Fe/H] of –0.5 or a 1.2 MJup
disk with [Fe/H] of +0.2. As these masses are both central
in the broad M-distribution, they have similar probability,
and so the two [Fe/H] values occur with similar likelihood.
5 DISCUSSION
The excellent reproduction of observed frequencies and de-
pendencies on metallicity both suggest that the method is
robust. Hence, rather surprisingly, the original search for a
single parameter that has a dominant effect on outcome has
succeeded. Under the assumption that the metallicity ranges
track the different outcomes, the primordial solid mass of the
circumstellar disk is identified as this parameter.
One implication is that for an ensemble of stars of
known metallicity, the proportions of different kinds of plan-
etary system may be predicted, without the need for detailed
models of individual disks. As main-sequence stars retain
no relic information on their primordial disk masses, this
c© 2006 RAS, MNRAS 000, 1–6
Predicting the frequencies of diverse exo-planetary systems 5
Figure 2. Distribution of solid masses. The shaded areas rep-
resent (from right to left) systems producing hot Jupiters, cool
Jupiters, gas giants plus debris disks, debris only, and disks with
1 Earth-mass or more of solids. Disks further to the left have no
presently predicted observable outcome.
is a very useful result, leading to estimates of the frequency
and variety of planetary systems, for example among nearby
Sun-analogues of interest to planet-detection missions.
The model may also be used to examine other plan-
etary system regimes that have just opened up to experi-
ment. For example, transiting hot Jupiters have not been
detected in globular clusters, although the stellar density
allows searches of many stars. In 47 Tuc, no transits were
found amongst ∼ 20, 000 stars, although 7 detections would
be expected based on the typical hot Jupiter occurrence
rate (Weldrake et al. 2005). Assuming these old stars formed
with disks following the standard M-distribution, but with
metallicities of only one-fifth Solar in a narrow range with
σ ≈0.05 dex (Caretta et al. 2004), then the model predicts
that less than 1 in 106 disks can form a hot Jupiter – the
solid masses are too small for fast planet growth and subse-
quent migration.
Finally, an upper limit can be estimated for the number
of young disks that could form an Earth-mass planet. The
maximum fraction (Figure 2) is set by disks of MS > 1 M⊕,
summing to two-thirds of stars. Thus one in three stars
would not be expected to host any Earth-analogue, and irre-
spective of metallicity since the occurrence of this minimum
mass is rather flat with P ∝ N(Fe)0.25. However, if the up-
per two-thirds of disks may be able to form terrestrial plan-
ets, this would agree with the large numbers predicted by
the simulations of Ida & Lin (2004), and also with the first
planetary detections by the microlensing method. Two bod-
ies of only around 5 and 13 Earth-masses have been detected
around low-mass stars, and Gould et al. (2006) estimate a
frequency of 0.37 (–0.21, +0.30) in this regime of icy planets
orbiting at ∼ 3 AU. Our model finds that 40 % of stars have
disks with MS greater than 5 M⊕, so the minimum materi-
als to form such planets are present at about the observed
frequency. While very preliminary, this result supports the
prediction that many stars could have low-mass planets.
6 CONCLUSIONS
We tested the hypothesis that a single underlying parameter
could have the dominant effect on the outcome of planetary
system formation from primordial circumstellar disks. An
empirical model with the mass of solids as this parameter
produces a very good match to the observed frequencies and
to the dependence on host-star metallicity, when the metal-
licity range within each outcome is used to estimate the
solid-mass range. This model may be very useful for making
statistical predictions of planetary system architectures for
ensembles of stars of known metallicity, such as nearby Solar
analogues of interest to exo-Earth detection missions.
ACKNOWLEDGMENTS
JSG thanks PPARC and SUPA for support of this work.
We thank the referee, Philippe Thebault, for comments that
greatly helped the paper.
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c© 2006 RAS, MNRAS 000, 1–6
6 J. S. Greaves, D. A. Fischer, M. C. Wyatt, C. A. Beichman & G. Bryden
This paper has been typeset from a TEX/ L
ATEX file prepared
by the author.
c© 2006 RAS, MNRAS 000, 1–6
	Introduction
	Hypothesis
	Model
	Input data
	Observed frequencies
	Results
	Alternative models
	Distributions within outcomes
	Discussion
	Conclusions
