A&A 394, 241-251 (2002)
DOI: 10.1051/0004-6361:20021108
D. E. Trilling1 - J. I. Lunine2 - W. Benz3
1 - University of Pennsylvania,
Department of Physics and Astronomy, David Rittenhouse Laboratory,
209 S. 33rd St., Philadelphia, PA 19104, USA
2 -
Lunar and Planetary Laboratory, University of Arizona,
Tucson, AZ 85721, USA
3 -
Physikalisches Institut, Universitaet Bern,
Sidlerstrasse 5, 3012 Bern, Switzerland
Received 30 January 2001 / Accepted 31 July 2002
Abstract
We present a statistical
study of the post-formation migration
of giant planets
in a range of initial
disk conditions. For given initial conditions we model
the evolution of giant planet orbits under the influence
of disk, stellar, and mass loss torques. We determine the mass
and semi-major axis distribution of surviving planets after
disk dissipation, for various disk masses, lifetimes, viscosities,
and initial planet masses.
The majority of planets migrate too fast
and are destroyed via mass transfer onto the central
star. Most surviving planets
have relatively large orbital semi-major axes
of
several AU or larger.
We conclude that the extrasolar planets observed to date, particularly
those with small semi-major axes,
represent only a small fraction (25%
to
33%) of a larger
cohort of giant planets around solar-type stars,
and many undetected giant planets must exist at large (>1-2 AU)
distances from their parent stars.
As sensitivity and completion of the observed
sample increase with time, this distant majority population
of giant planets should be revealed.
We find that the current distribution
of extrasolar giant planet masses implies that high
mass (more than 1-2 Jupiter masses) giant planet
formation must be relatively rare.
Finally, our simulations imply that
the efficiency of giant planet formation must be high:
at least 10%
and perhaps as many as 80%
of
solar-type stars possess giant planets during their pre-main sequence phase.
These predictions, including those for pre-main sequence stars, are
testable with the next generation of ground-
and space-based planet detection techniques.
Key words: solar system: formation - stars: circumstellar matter - stars: planetary systems - stars: statistics
Extrasolar giant planets (EGPs) have been detected
by the radial velocity method at orbital distances from
several AUs (e.g.,
47 UMa c, Gl614b, Eri b,
55 Cnc d) to several hundredths
of an AU (51 Peg b et al.)
from their central stars
(see
Mayor & Queloz 1995;
Butler & Marcy 1996;
Butler et al. 1997;
Noyes et al. 1997;
Cochran et al. 1997;
Santos et al. 2000;
Vogt et al. 2000;
Hatzes et al. 2000;
Butler et al. 2001;
Santos et al. 2001;
Fischer et al. 2002;
Marcy et al. 2002;
and many
others; see also the review by Marcy
et al. 2000a).
Unless giant planets form in place within 1 AU of low mass stars -
unlikely in the context of published formation models (see,
for example, Guillot et al. 1996) -
the observed range of orbital semi-major axes implies that dramatic
orbital changes occur
after formation.
Previous work showed that in gaseous disks and even
subsequent particulate disks,
giant planets can move from
formation distances of around 5 AU
to a wide range of final
distances, a process known as
orbital migration
(Lin & Papaloizou 1986;
Lin et al. 1996;
Takeuchi et al. 1996;
Ward 1997a,1997b;
Trilling et al. 1998;
Murray et al. 1998;
Bryden et al. 1999;
Kley 1999,2000;
Del Popolo et al. 2001;
Tanaka et al. 2002).
The first direct determination
of the radius of an extrasolar giant planet via transit measurement
(Charbonneau et al. 2000;
Henry et al. 2000)
supports rapid inward migration of giant planets, since
the large planetary radius requires close proximity to the parent star
when the planet's internal entropy was much larger than
at present (Guillot et al. 1996;
Burrows et al. 2000), and formation in place is
generally considered implausible (see, e.g., Guillot et al.
1996).
Early inward migration is commensurate with
the idea of migration caused by disk-planet interactions,
as disk lifetimes are not longer than 107 years (e.g.,
Zuckerman et al. 1995).
In this work, we address the following questions: (1) How efficient is orbital migration? (2) What population of planets survives the migration process? and (3) How does this produced population compare to the observed EGP population? We answer these questions by allowing a large population of giant planets to evolve and migrate in circumstellar disks with various initial conditions (one planet per disk) and determining the final semi-major axis and mass distributions. We do not employ any stopping mechanisms, but instead allow only those planets to survive whose migration timescales are longer than their disks' lifetimes. We compare the final semi-major axis and mass distributions of the surviving model planets to those of the observed EGPs. We reproduce the fraction of planets in small orbits, and predict the frequency of and orbital distributions for giant planets that are as yet unobservable. Of those stars which do form giant planets, a few percent should have giant planets ultimately residing in small orbits, and around one quarter of stars which form planets retain planets in orbits of semi-major axis several AU and beyond. Given the current discovery statistics and their uncertainties, our results imply that giant planet formation is very efficient around low-mass stars: we find that 10% to 80% of young stars form planets, with the uncertainty dominated by the initial planetary mass distribution (and with some poorly-known uncertainties associated with discovery statistics). The presently detectable portion of the giant planet population represents only about 25% to 33% of the total extant population of giant planets.
The model we present here is a simple one with a minimum set of physical assumptions. By neglecting specific stopping mechanisms, we provide results that represent a baseline for planet formation statistics, bereft of specific additional assumptions that any given stopping mechanism would require. It is hoped that the calculations and results presented here are transparent enough that others can use them to explore the particular effects that specific assumed stopping mechanisms would have. In short, we use the simplest possible model in order to explore the consequences of and for planet formation.
In order to approach these problems in a simple and clear way, we must make several assumptions about the problem of planetary migration. These assumptions are listed here and explained below: (1) we use the simple impulse approximation for the type II migration; (2) we employ no ad hoc mechanisms to stop planetary migration; (3) we assume that gas accretion onto planets migrating in a gap is small, and neglect this accretion; and (4) we neglect type I migration.
In the migration model presented here, we use the impulse approximation described in Lin & Papaloizou (1986) to calculate the torques between the planet and disk and determine the planet's orbital motion. We have previously shown (Trilling et al. 1998) that the impulse approximation yields quantitatively similar results to much more computationally intensive methods of calculating orbital migration (for example, the WKB approximation of Takeuchi et al. 1996) in the dissipation regime relevant to this work. Given our current knowledge of the migration process, we prefer to explore the consequences of the simplest possible model rather than to add complexity which does not guarantee more accurate results.
Halting a giant planet's inward migration at small semi-major axes is even less well understood than initiating and maintaining migration. Various stopping mechanisms have been proposed, including tides (Lin et al. 1996; Trilling et al. 1998); magnetic cavities (Lin et al. 1996); scattering by multiple planetesimals (Weidenschilling & Marzari 1996); resonant interactions by two (or more) protoplanets (Kley 2000); and mass transfer from the planet onto the star (Trilling et al. 1998). All stopping mechanisms have problems: Tides are insufficient because the magnitude of the tidal torque is hardly ever enough to counterbalance the inward torque; magnetic cavities have not been demonstrated to work in a quantitative way; the presence of more than one planetesimal or planet requires a great deal of serendipity; and mass transfer only works in a small percentage of cases. Additionally, stopping mechanisms must not only produce 51 Peg b-like planets, at small orbital distances, but must also produce planets like HD 108147b at 0.1 AU, HD 28185b at 1.0 AU, and 47 UMa c at 3.8 AU (Pepe et al. 2002; Santos et al. 2001; Fischer et al. 2002). None of the above mechanisms adequately halts planets at the wide range of observed distances from the parent stars.
In this work, we presume that none of the above mechanisms are effective in stopping more than a small fraction of migrating planets, and instead consider planets stopping for what is essentially a statistical reason: the planet stops migrating when the disk dissipates. We show that by considering a physically reasonable distribution of initial conditions for planets and disks, our model produces a range of final orbital semi-major axes for extrasolar planets - including small, 51 Peg b-like separations - without resorting to any exotic stopping mechanisms. It is not necessary to produce many planets very close to their stars: the current detection rate suggests that approximately 1% of low-mass main sequence stars (F, G, K dwarfs) have extrasolar giant planets in small orbits (Vogt et al. 2000), and extrasolar planets in all orbits exist around perhaps a few percent of all stars in the radial velocity surveys (see, e.g., Marcy et al. 2000a). We note in passing that one M-dwarf exhibits two companion objects of at least 0.5 and 2 Jupiter masses (Marcy et al. 1998; Delfosse et al. 1998; Marcy et al. 2001a); the statistics for M dwarfs are thus very poor and indeed the environment for planet formation may be very different from that of the more massive dwarfs. In what follows we confine ourselves to F through K stars and refer to these conveniently as low-mass stars.
Giant planets form gaps in their gaseous
circumstellar disks;
when this occurs,
,
the planetary mass accretion rate, drops
substantially from its pre-gap value
as the local surface density is substantially reduced.
It is an outstanding question how much, if any,
material may flow across the gap from the disk
and be accreted onto the planet
(see Bryden et al. 1999;
Kley 1999;
Bryden et al. 2000;
Kley 2000;
Lin et al. 2000;
Kley et al. 2001).
The results shown in those
works suggest
that
accretion
timescales (
)
are between
and
years
for the viscosities, mass ratios, and other parameters
we use in our models.
We have adopted a characteristic accretion timescale
of 106 years
and
performed several tests to determine the
effect
of including mass accretion in the migration models
(Fig. 1).
Note that the accretion timescale is comparable
to the migration timescale; we allow both linear
and exponential growth with this characteristic
timescale.
The results from these test cases show
that when all disk parameters are held
constant,
the range of variation
between the migration timescales
without mass accretion and
timescales with mass accretion (all cases,
linear and exponential)
is around 25% from the no-growth migration
timescale of
106 years
(Fig. 1).
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Figure 1:
Planetary semi-major axis (in AU) and planet mass (in
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Including mass accretion for a population of migrating planets has a small effect on the overall results, at best. For more massive planets, which migrate quite slowly anyway (see Sects. 3.2 and 4), the effect of considering mass accretion is very much smaller, or completely negligible. Additionally, the primary parameters which drive migration (diskmass, viscosity, disk lifetime) are not known to better than 25%, and even the mass accretion rate itself is a product of some underlying assumptions about the flow of material near, around, and onto a migrating planet. Attention to the detail of mass accretion during migration is not warranted in the face of these larger unknowns; instead, we choose to use a simpler model (ignoring post-gap formation accretion) which may neglect some details for the sake of clarity and understanding. Lastly, because migration timescales are only affected by mass accretion in a minor way, the overall semi-major axis and final mass distributions of the statistical study (see below) do not change significantly in the presence or absence of mass accretion. We therefore neglect mass accretion for the remainder of the present work; all subsequent discussions concerning the statistical study implicitly refer to model runs without mass accretion.
Giant planets, because of their large ratio of hydrogen and
helium relative to heavier elements, must form in gaseous protoplanetary
disks.
Regardless of whether a giant planet forms by
core accretion (e.g., Wuchterl et al. 2000)
or by direct collapse
(e.g., Boss 1997,1998,2000,2001),
when a planet grows to a certain mass (10-30
,
although there is some debate about what this critical
mass actually is),
a gap
is formed in the disk and spiral density waves travel
away from the planet
(see, for example, Goldreich & Tremaine 1980;
Lin & Papaloizou 1986;
Ward & Hourigan 1989;
Lin & Papaloizou 1993).
Interactions of these
waves at the planet's Lindblad resonances
transfer angular momentum between the planet
and the disk; this angular momentum exchange in turn
produces a change in orbital distance of the planet.
Usually (but not always),
these interactions lead to a net loss of angular
momentum and a decrease in the planet's orbital
semi-major axis.
Such planet-disk interactions have been
described in detail
(see, for example,
Goldreich & Tremaine 1980;
Ward & Hourigan 1989;
Lin & Papaloizou 1986,1993;
Lin et al. 2000;
Ward & Hahn 2000).
The type of migration executed with a planet in
a gap is referred to as type II (see Ward & Hahn 2000).
So-called type I migration is executed by bodies which
are not massive enough to form gaps in their disks
(i.e., less than 10-30
)
(Ward 1997a,1997b).
Ward has shown that type I migration may be very fast (less
than 106 or even 105 years) which is obviously
detrimental to planet growth and survival: if migration
timescales are so short for Earth mass objects,
it is unclear how any
planet could survive long enough to accrete enough gas mass
to open a gap
and
initiate the slower type II migration (which itself is
perilous for planet survival).
Clearly, there is a problem with planet survival in the
face of type I migration. We do not intend to solve this problem
in this paper. Instead,
we neglect type I migration for the following reasons.
If type I migration were really as dangerously
efficient as has
been proposed, then formation and survival of planets
(particular at distances greater than 0.1 AU)
would be nearly impossible, since the type I migration
timescales are shorter than any possible disk lifetime.
But, since we know that planets survived both at small
and intermediate
semi-major axes (many EGPs) and also relatively large
semi-major axes (Jupiter), type I migration cannot be
as universally destructive as proposed.
For lack of a better quantification
of the type I effect, in the present work
we neglect
the effect of type I migration on the
initial conditions used in our models.
Furthermore, all planets we consider in
this paper are too massive to undergo type I
migration.
Thus,
we can
safely neglect type I effects
in our calculations.
We reiterate that this does statement does not
refute type I migration nor the importance of
understanding and characterizing its timescales;
however, it is clear that for planets which have reached
1 Jupiter mass
via any formation mechanism, type I migration
is no longer relevant.
We have applied a model of disk-induced migration (Trilling et al. 1998) to a set of planetary systems whose initial characteristics are distributed over a representative range of planet formation environments. We use the same method and model as used in Trilling et al. (1998), which includes torques on the planet due to disk-planet interactions, due to star-planet tidal torques, and due to planetary mass loss. Mass loss significantly prolongs planetary lifetimes for only a very small population of model planets, but is included here for completeness. Star-planet tidal torques affect a larger population, but we shall see that, by far, the largest effect is the magnitude of the disk-planet torque. A planet's survival, which is enabled essentially entirely by dissipation of the gaseous disk before infall onto the star, depends on its mass, and on the disk mass, disk lifetime, and disk viscosity.
We have performed a statistical study of planet
fates in systems in which each of disk mass,
disk lifetime, and disk viscosity
is varied in turn. Initial planet masses are taken to be
integral values between 1 and 5 Jupiter masses
(1 Jupiter mass = 1
=
g), inclusive, evenly
distributed, with one planet per system.
Planetary accretion after gap formation
is neglected (see Sect. 2.3).
For the three varied parameters, we adopt
nominal values and vary each parameter
over a range spanning a factor of 30 larger
and smaller than the
the nominal value, with 200 model runs
evenly spaced
per decade in log space,
for a total of 600 model runs per
planet per varied parameter.
For example, in varying disk mass,
we adopt a nominal disk mass of
0.02
(after Beckwith et al. 2000),
and calculate
model cases between 0.02/30 and
,
or 0.0007 and 0.6
.
This range spans disk masses
from less than the
minimum mass disk for a system containing a Jupiter
to a disk so massive that direct binary star production
is more likely than giant planet formation
(see, for example, reviews by Beckwith & Sargent 1993
and Beckwith et al. 2000).
Over this entire range we test 600 model runs
in evenly spaced intervals of 0.005 log units.
Similarly, we use a nominal disk lifetime and
alpha viscosity of
years and
,
respectively (after
Zuckerman et al. 1995; Shakura & Sunyaev 1973;
and Trilling et al. 1998)
and test parameter values
from disk lifetimes of 105 to
years
and disk viscosities of 10-4 to
.
In all, we carried out 1800 model runs per planet mass
and 9000 model runs total;
each model run has one parameter varied
while the other two are given their nominal
values.
This very fine sampling of disk parameters
allows us to simulate a continuum of
physical properties of planet-forming
systems.
All planets' initial orbital semi-major axes
are 5.2 AU. Giant planets forming farther out would
spend more time migrating inward, all else being equal, and hence
would have a larger chance of survival (see below).
The result of our computations
is a three-dimensional grid of
models,
based on the nominal values and
spanning the range of
parameter space, for each initial planet mass.
To produce a
Gaussian statistical population in which
the nominal value is most likely
and the extreme values are least
likely,
the significance of each model run
was weighted by the Gaussian
probability of the value of the varied
parameter according to the standard
Gaussian formalism:
![]() |
(1) |
We use this Gaussian weighting
in all results discussed in this paper.
Weighting is achieved by assigning
a significance to each surviving
model planet as
where
is equal to 1 for
planets which survive and 0 for planets
which do not.
Thus, in production
of histograms and discussing the
ensemble results for the 9000 model
runs, we explicitly incorporate
the probability of the planet's
formation conditions and
circumstellar disk. Thus, surviving planets
from extreme disks
whose initial conditions are extremely
unlikely are not overrepresented in
the statistical results.
Note that because probabilities are assigned
after all model runs are completed
and are simple functions of the nominal
values and a description of a Gaussian
distribution,
other probability weightings
can be installed
to represent other
experiments.
Future work includes deriving both
x0 and
more directly
from observations (see, e.g.,
Gullbring et al. 1998;
Hartmann et al. 1998)
and testing non-Gaussian probability
distributions.
Our models calculate orbital evolution for 1010 model years, and planets are defined to have survived when they have non-zero semi-major axes after this time. Very few planets (much less than 1%) undergo significant orbital evolution (due to star-planet tides) after the disk dissipates.
The results from our statistical study are shown in Figs. 2 and 3. Of the 9000 initial planets, 30% survive. This survival rate assumes a flat mass function (see below), and is normalized to the total probability of all surviving planets; in this and subsequent discussions, all results are post-weighting. We find that around 0.8% of the initial planets have final semi-major axes less than 0.1 AU. Approximately 6.5% have final semi-major axes between 0.1 and 1 AU. The rest of the surviving planets (23% of the original planets) have final distances from their stars greater than 1 AU. We find that 2.5% of all surviving planets are found at less than 0.1 AU; 21% are found between 0.1 and 1 AU; and 76% of surviving planets have semi-major axes greater than 1 AU. The majority (70%) of the population of initial planets migrates too fast; these planets lose their mass onto the central star (by Roche lobe overflow), and do not survive. As noted in Sect. 2.2, other stopping mechanisms at the inner edge of the disk may operate and would increase somewhat the fraction of planets retained in orbits with very small semi-major axes. These mechanisms will have a small or negligible effect on the overall statistics, except for the possible existence of a magnetic cavity, which effect on the migration has yet to be quantified.
![]() |
Figure 2:
Planet mass versus heliocentric distance for
observed (triangles) and model
(filled circles) EGPs.
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Figure 3: Same as in Fig. 2 but for the total overall probability of planet survival in given semi-major axis and mass bins; both semi-major axis and mass bins are 0.1 log units wide, centered on 0.05 log units. The radii of the filled circles again show the probability of model planets populating that bin, but a different (arbitrary) probability normalization is used than for Fig. 2 to show circle size more clearly; the same probability normalization is used for both panels here. As before, the upper panel is shows results for model planets with a flat initial mass distribution; the lower panel is for results with the preferred initial mass distribution. |
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The most likely fate of a planet which forms near 5 AU is that it migrates inward too fast and ultimately is lost onto the central star. The vast majority of planets that do survive, though, tend to reside at greater than 1 AU from their central stars. Our statistical study is robust because we have sampled planet migration over a broad distribution of disk masses, viscosities, lifetimes and initial planet masses: the variables that are key to migration and its termination. We have also carried out an initial study of the effects of initial semi-major axis on planet migration and survival. While it is not easy or obvious to parameterize these results, it is clear that for identical initial disk conditions, planets which form farther out migrate less rapidly. This is because the surface density is lower at larger semi-major axes, so that the torque between the planet and disk is less. The effect on planet survival is that more planets, of lower masses, survive (see below). Therefore, if 5 AU is the minimum semi-major axis for giant planet formation, then the results presented here represent a minimum for planet survival; conversely, if 5 AU corresponds to the largest semi-major axis for planet formation, then the planet survival found here is the upper limit. We choose 5.2 AU as a nominal middle ground. The fraction of planets which survive, the ratio of planets at small semi-major axes compared to at large semi-major axes, and the overall shape of the surviving planet mass distribution are quite similar even when somewhat different initial semi-major axes are considered (see below).
We have analyzed our statistical results in terms
of the final distribution of orbital semi-major axes
of the planets, in comparison to the observed
EGP distribution. The results
are shown in Fig. 4.
The observed EGP distribution
is relatively flat, that is, equal numbers of
planets throughout the range of 0.05 to
around 0.7 AU; there is the suggestion of
a rise from 0.7 AU through just over 1 AU;
and a decrease at greater semi-major axes
than just over 1 AU (this decrease is
due at least in part to observational bias).
(The
exception to this apparently flat
distribution is the
slight enhancement in planets at orbital distances
around 0.05 AU. In our previous work (Trilling et al. 1998),
we described how tidal and mass loss torques can
cause a small "piling-up'' of planets near the tidal
limit, which is typically around 0.05 AU.)
In contrast, the surviving planets in our statistical
study are predominantly at large (>1 AU)
semi-major axes, including a rise at
just under 1 AU to a peak at 2-4 AU.
The hint of an upturn in the observed distribution
at less than 1 AU echoes the pattern in the
model distribution and may suggest the larger
population that we predict at greater than 1 AU.
The physical explanation for the
peak in model planets at separations greater than 1 AU
is that the time during
which a planet is migrating (105 to 106 years)
can be relatively short compared
to both the lifetime of the disk (107 years)
and the timescale required
to initiate migration (variable, but at least
106 years, or more)
(Trilling et al. 1998).
Therefore, the number of planets
which survive at small distances (<1 AU) is small.
The majority of planets that survive
do so by
creating very large gaps in their disks
and by
requiring long timescales to excavate the
farthest regions of their
large gaps, as follows.
More massive planets create larger gaps than smaller planets, everything else being equal (Takeuchi et al. 1996; Trilling et al. 1998). For the most massive planets, the length of time it takes to fully form the gap can be quite long, approaching or even surpassing the lifetime of the disk, as such gaps are many AUs wide. The gap is completely formed when it reaches its equilibrium size (see Takeuchi et al. 1996 and Trilling et al. 1998). During the stage in which a gap is growing but has not yet reached its full equilibrium size, there is no type I migration since the planet has already opened a gap; there is also little type II migration because most Lindblad resonances already fall within the growing gap. Thus, the planet is hardly migrating during this time in which the gap is still growing to its equilibrium size. If the timescale for gap growth is longer than the disk lifetime then the planet will not migrate far. Therefore we find that, overall, close-in surviving model planets have smaller masses and distant surviving models planets have larger masses; this rough trend is seen in the observed data as well (see below). Note that relatively large mass planets (compared to the disk mass) are required, in general, for these arguments about large gap formation to be applicable.
Since the vast majority of planets either migrate
too fast and are lost onto the star or remain out
near their formation location,
disk and planet parameters must
be "just right'' to halt a planet
at small distances (<1 AU).
The conclusion of this analysis is that
it is very difficult to make planets survive at
small distances from the parent star,
and that to do so requires a large population
of giant planets
which still reside at large
distances from their parent star as well as
a population which has migrated too fast
and been lost onto the central star.
From our model results, we find that for
every planet found at small semi-major axes,
around 3 planets must exist at much larger
distances, depending on the initial mass
distribution (see below).
The radial velocity technique is most sensitive to
planets at small semi-major axes, because the
magnitude of the stellar wobble is greatest for close-in
planets. Additionally, since the highest precision
radial velocity surveys have relatively short
baselines of data, planets with longer periods
are only beginning to be identified
(e.g., 55 Cnc d, with 15 year orbit,
Marcy et al. 2002).
Therefore, the majority of the extant population of
giant planets we predict
would, at present,
be beyond the
detection capabilities of the radial velocity searches.
In other
words, a large population of Jupiter-like planets is as yet undetected.
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Figure 4: Histogram showing orbital semi-major axes for observed (solid line) and model (dashed and dotted) EGPs. Each model histogram is normalized such that the number of model planets with semi-major axes less than 1 AU is equal to the number of observed EGPs with a< 1 AU. The vertical line corresponds to 1 AU, the completion limit for radial velocity surveys. Most model EGPs are to the right of the vertical line, i.e., at larger semi-major axes than would have been detected thus far by the radial velocity searches. The dashed line is for a flat mass function; the dotted line is for the preferred mass function (see text). Bins are 0.1 units in the log of semi-major axis. |
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We have also analyzed our surviving statistical
population for its distribution of final
planet mass, in comparison to the observed
EGP mass distribution. This comparison is
complicated by the fact that for most EGPs,
the known quantity is not planetary mass but
.
We assume
,
for simplicity. We shall see in any case that it is not
the absolute mass that matters but the slope and
shape of the mass distribution curve.
The slope and shape
will be roughly unchanged between
and
assuming a random distribution of
system inclinations relative to the Earth.
For a random distribution of inclinations, the
average of
is
/4.
With increasing numbers of extrasolar
planets known, the average
for the
population gets closer to the value
for a random distribution. However, because
of the detection bias of the radial velocity
technique (biased towards detection of systems
with i near 90 degrees), the average of
for the detected systems is actually between
/4
and unity (Marcy et al. 2000a).
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Figure 5:
Histogram showing final planet mass distribution
for
observed EGPs (solid line).
Also shown are the
flat model
initial mass distribution (dot-dash line),
the surviving population of model
planets produced from this flat
mass function (dotted line),
and the
preferred initial mass distribution (dashed line) required to
produce the observed mass function (see text).
For the observed EGPs,
![]() ![]() |
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The comparison among mass functions used
in this work is shown
in Fig. 5.
By assumption, in our model we
have started with equal numbers of
planets with masses 1, 2, 3, 4, and 5
(i.e., after accretion
has taken place), or a flat initial mass distribution.
As shown in Figs. 2, 3, and 5,
most of the surviving planets in the statistical
study are the more massive planets;
planets with the largest initial mass (5
)
survive
preferentially.
The observed EGP mass distribution, however, is
biased toward smaller mass planets (Fig. 5).
Since the radial velocity
technique is most sensitive to larger masses, the
relative dearth of large mass planets discovered is
significant.
The present observed EGP population
can instead be obtained with
an initial mass distribution
biased strongly toward smaller mass
giant planets. This
"preferred'' initial distribution is shown in Fig. 5
as the dashed line.
The preferred mass function is derived from the
observed mass distribution and the ratio of initial model
planets to surviving model planets:
Model planet masses and semi-major axes
produced using the preferred the
initial mass distribution are
shown in Figs. 2 and 3.
The final distributions of
semi-major axis
and planet mass
produced using the preferred initial
mass distribution are shown
in
Figs. 4
and 5.
As with a flat initial mass distribution,
in order to produce the observed extrasolar planet
characteristics from
the preferred initial mass distribution,
many planets with large
semi-major axes must be produced.
The preferred initial mass distribution
results in 0.2% of all initial
planets surviving with separations less than
0.1 AU;
1.6% of all initial planets survive
and have 0.1 AU <a< 1 AU;
and
3.8% of all initial planets survive
and have a> 1 AU.
These percentages correspond to
4.4%, 28%, and 68% of
all surviving planets to be found
in those three semi-major axis
ranges, respectively.
Thus, the ratio of as-yet undetected planets
to detected planets should be around 2:1,
compared to a ratio of 3:1 obtained with
the flat mass distribution.
Thus, the total percentage of stars with planets
must be 3-4 times greater
than the percentage given by the
current observational results.
With the preferred initial mass
distribution, 5.6% of
all initial planets survive at
any semi-major axis; with the
flat initial mass distribution,
this total survival rate is 31%.
With the preferred initial mass distribution,
fewer planets
survive than in the case of the flat initial mass
distribution
because the smaller mass planets that dominate the
preferred initial distribution preferentially
are
destroyed.
Our results imply that there
must still be a substantial population of giant planets
at large semi-major axis that has not yet been
detected. The ratio of extant planets at large semi-major
axes to extant planets at small semi-major axes is less
when the preferred initial mass distribution is used (around 2 to 1)
compared to when the nominal (flat) initial mass distribution is
used (closer to 3 to 1).
Planet forming efficiency, that is, the number
of stars that form giant planets, must be relatively
high. This is because many planets which form are
subsequently destroyed, and many planets which exist
have not yet been detected, as follows:
Our model suggests a high efficiency of planet formation. Since planets more easily survive migration in lower mass disks, planet formation in such disks leads to greater planet survival. Additionally, the shape of the preferred initial mass distribution suggests a natural cutoff towards larger mass planets: larger mass planets may be more difficult to form (in agreement with Lineweaver & Grether 2002). In other words, giant planet formation must be a relatively efficient use of the disk gas: it cannot be a process requiring primarily disks at the upper end of the mass range we have considered, but must also occur in disks with masses closer to the minimum we have selected. Our minimum mass disk is slightly less than the mass of Jupiter itself, and represents an extremum. Exclusion of extreme diskmasses from the cohort changes the overall required efficiency very little because these extreme cases have very low probabilities and therefore do not contribute much to the total survival probability. Overall we require that the small- or moderate-disk masses participate in the formation of giant planets, and these giant planets are then overrepresented in the final distribution because of slower migration rates.
Finally, it is important to emphasize that high planet forming efficiency does not mean that most low mass stars currently have Jupiter-like planets. Planetary migration is very unforgiving: our statistical study shows that around 70% to almost 95% of formed planets migrate too fast and meet their demise at their central stars. A large percentage (10% to 80%) of stars and disks must produce planets so that a few percent may survive close to their central stars and be detected by radial velocity searches. But there should then be a population of giant planets in intermediate (2-4 AU) and larger orbits, around perhaps 2%-3% or more of F, G, and K dwarf stars. This fraction agrees with the estimates of Zucker & Mazeh (2001b). Giant planet formation may be occurring around many young stars today, and searches for indications of planet formation around these stars should have a high success rate.
Our preferred initial mass distribution
and mechanism for producing close-in
giant planets requires a high frequency of planet formation
in disks, and a bias toward making giant planets with smaller
masses (in agreement
with Lineweaver & Grether 2002).
The latter conclusion is robust so long as the distribution
of observed system inclinations as seen from Earth is
roughly random; that
is, as long as the measured
yields actual masses on
average less than a factor of two larger.
To date, five systems with known inclinations (HD 209458,
55 Cnc, HD 210277,
CrB, and
Eri)
all have
degrees
and thus have mass factors less than 2
(Charbonneau et al. 2000;
Henry et al. 2000;
Trilling & Brown 1998;
Trilling et al. 2000;
Greaves et al. 1998).
Additionally, by tidal arguments,
And, HD 75289, HD 187123, 51 Peg, and
HD 217107 are all constrained to have
degrees
as well, although there are no direct observations as yet
to confirm this (Trilling 2000);
the Hipparcos data implies an inclination
25 degrees
for
And (Mazeh et al. 1999).
Additionally, from dynamical
and astrometric arguments,
the multiple-planet systems Gl876,
HD 168443,
and
And likely
have inclinations that are far from
face-on (Laughlin & Chambers 2001;
Marcy et al. 2001b;
Chiang et al. 2001).
Therefore, most systems for which inclinations
are known or suspected
appear to have mass factors less than
around 2.
Astrometric studies of EGP systems
show a range of results but
few certain inclinations, to date
(Zucker & Mazeh 2000;
Halbwachs et al. 2000;
Gatewood et al. 2001;
Han et al. 2001;
Pourbaix 2001;
Pourbaix & Arenou 2001;
McGrath et al. 2002).
Giant planet formation is as yet not well enough understood to permit prediction of a preferred initial mass distribution as a function of disk mass. Models in which core accretion precedes accumulation of gas require a substantial surface density of solids to permit accretion to occur within the disk lifetime (Lissauer 1987), with the resulting gas accretion occurring fairly rapidly once a threshold mass of solids is accumulated (Wuchterl et al. 2000). There is thus no simple and direct link between the disk gas mass and the final masses of the giant planets formed. The presence of Uranus and Neptune in our own system is suggestive of a process that is sensitive to depletion of the gas either due to small available amounts of gas beyond a certain distance from the primary, or to long core accretion times that stretch beyond the disk lifetime. However, alternative models for the formation of Uranus and Neptune exist that do not tie them directly to conditions in the 20-30 AU region of the gas disk (Thommes et al. 1999).
In sum, we do not as yet have a direct link between
the mass distribution
of giant planets and the mass of the disk. Thus our preferred
initial mass distribution
is a general prediction: whatever
mechanism forms giant planets tends to prefer the
smaller mass objects. This is at
least crudely consistent with the inference that
the combined giant planet/brown dwarf mass function
appears to show a minimum around approximately
10
(Marcy & Butler 2000;
Zucker & Mazeh 2001a),
suggestive of a planet-forming process biased to lower masses and a
brown dwarf-forming process (direct collapse or disk instability) that is most
efficient at much higher masses.
In our study we did not consider minimum masses below 1
.
The lack of data for objects significantly below
of
0.5
hampers our ability to extend the calculations,
and we must await further
and more sensitive data (e.g., SIM, see below)
to establish the frequency of lower mass
planets.
However, the radial velocity surveys have
begun to
detect planets with masses as small
as 0.12
(HD 49674b; Butler et al. 2002).
This implies that orbital evolution takes place
for Saturn-mass and smaller planets.
Further observations will show the effects
of planet mass on orbital evolution.
Likewise, we have arbitrarily imposed planet formation at 5.2 AU. Formation of giant planets out to double or even quadruple that distance is possible, based on our own solar system. We have performed a small set of model runs for planets with larger initial semi-major axes, out to 10 AU. The overall behavior and survival of these migrating planets is not qualitatively different from that of the entire suite of 9000 models. The general effect of larger initial semi-major axes is an increase in the survival rate (because migration times are longer - see Papaloizou & Larwood 2000). This would produce a lower required planet-forming efficiency and an initial mass distribution which is biased toward slightly higher masses relative to the preferred initial mass distribution, since more small mass planets survive. Eventually, however, a distance must be reached in all disks at which accretion times become too long and giant planet formation yields first to ice giants (Uranus and Neptune) and then debris. The current cohort of close-in giant planets does not provide a constraint on the outer orbital radius at which giant planet formation could occur. A more complete sample of observed EGPs at larger semi-major axes is required. It is possible that production of giant planets is actually preferred at the position in the disk where water ice first condenses out, because the surface density of ice cold-trapped there (and hence of total solids) is very high (Stevenson & Lunine 1988). However, testing this hypothesis must await a more complete sample of planetary systems.
Our model says nothing about the distribution of orbital eccentricities of the known radial velocity companions. Many of the known planets with semi-major axes beyond the tidal circularization radius have significant eccentricities. This suggests a process of orbital evolution commonly associated with the formation and earliest evolution of giant planets, and yet one which failed to act on our own four giant planets. The discussion of whether radial migration in gaseous or particulate disks produces eccentric orbits is an ongoing one (e.g., Ward 1997a,1997b; Bryden et al. 1999; Kley 2000; Papaloizou et al. 2001; Murray et al. 2002). The presence of high eccentricities suggests that multiple giant planet formation may be the norm, and that interactions among giant planets after disk migration ends pump up the orbital eccentricities. The stochastic nature of such interactions precludes their inclusion in an analysis such as the one we present. Further, until the orbital inclinations can be determined, an important clue to the genesis of the eccentricities is missing. Regardless of the mechanism(s) by which eccentricity is created, except for extreme cases, our results regarding the semi-major axis distribution are hardly affected. Our statistical results regarding planet formation frequency hinge on the frequency of the close-in giant planets versus those farther out. Additional modest semi-major axis evolution associated with eccentricity pumping does not alter our results significantly. Dramatic three-planet interactions that would leave giant planets in vastly altered orbits and greatly affect our statistics are arguably rare. In summary, while we do not calculate eccentricities induced by gas drag in our one-dimensional model, the results based on the statistics of semi-major axis distribution remain robust.
The migration behavior
and orbital evolution of
planets in a multiple-giant planet system
could differ from the results presented here.
In these cases, both the gas-planet interactions
and the planet-planet interactions could cause
orbital evolution. Since we know that systems
with multiple giant planets exist (e.g.,
And,
Butler et al. 1999;
47 UMa, Fischer et al. 2002;
HD168443, Marcy et al. 2001b;
55 Cnc, Marcy et al. 2002,
and, to a lesser extent,
our own),
quantitative studies of these systems are needed.
For some cases of multi-planet
systems, planet survival statistics
could be different from our results because of the additional
planet-planet interactions.
Additionally, if type II migration
should operate significantly more or
less efficiently than we have calculated
here (either through the presence of another
planet or some other disk-planet interaction),
the overall survival and statistics we
have derived here could change substantially.
Migrating giant planets may be detrimental to terrestrial planet survival, if terrestrial planets form coevally with giant planets. Planets interior to a migrating giant planet would be disrupted and lost from the system. This of course assumes that smaller planets do not migrate, although they too likely migrate, potentially on even shorter timescales than giant planets (Ward 1997a,1997b). If terrestrial planets form after the dissipation of gas in the protoplanetary disk, then disruption by a migrating giant planet may be less of a risk (excepting giant planet migration caused by planet-planetesimal interaction, Murray et al. 1998). Kortenkamp & Wetherill (2000) have considered the case of terrestrial planet formation when Jupiter has both its current and a larger heliocentric distance of 6.2 AU. They have found that accumulation of rocky bodies may be easier with Jupiter at a larger heliocentric distance (that is, pre-migration). It is possible that formation of the terrestrial planets in our Solar System may reveal clues about giant planet migration in our planetary system; certainly, these studies are also relevant to the formation of small, rocky planets in other planetary systems. In the context of large scale migrations, if terrestrial planet formation requires that giant planets have not migrated through the terrestrial zone around 1 AU, then only around 20% (flat initial mass distribution) or 3% (preferred) of all planet-forming systems qualify (i.e., a few percent of all late-type stars; see Sects. 3 and 5.2). If, however, the constraint is merely that there be no giant planet in the immediate vicinity of the terrestrial planet zone at the onset of terrestrial planet formation (perhaps 107 years), then the vast majority (93% for flat initial mass distribution, 99% for preferred) of planet-forming systems qualify. This number corresponds to nearly all planet-forming systems which is around 10% to 80% of late-type stars.
Although radial velocity detections of more
distant giant planets will become possible
as the time baseline of observations increases,
astrometric techniques are more sensitive to
giant planets in large orbits.
SIM, the Space
Interferometry Mission
(Danner & Unwin 1999),
will do a thorough job of detecting
giant planets, Uranus-mass objects, and even smaller bodies
from small to large semi-major axes
(10 AU),
with maximum sensitivity achieved
for planets approximately 0.3
around
3-5 AU.
SIM's
ability to test
our predictions of the
preferred initial giant planet mass
distribution
will be limited
largely by mission lifetime. However, SIM can also analyze
disks around young stars with high precision
(target resolution of 1 microarcsec), perhaps mapping
out the signature of gaps created by migrating (or
non-migrating) giant planets and
giving us a rough snapshot of the time-dependent mass distribution
of planets during the migration phase itself.
(A 1 AU gap at 100 parsecs is 10 milliarcsec.)
For giant
planets in the largest orbits, i.e., 20-30 AU from their parent
star, direct imaging techniques may be the only practical method
for detection since astrometric techniques would require baselines
of decades or more.
Our results suggest that techniques to study planet formation around young stars - radial velocity; high resolution imaging of young stellar systems; searches for gaps such as with SIM and potentially also the Space InfraRed Telescope Facility (SIRTF); searches for planetary outflows (Quillen & Trilling 1998); or searches for other indirect evidence, like cometesimals scattered onto stars (Quillen & Holman 1999) - should ultimately have a very high success rate. We anticipate that observational data will show that planet formation is taking place around 10% to 80% of low mass pre-main sequence stars, and that planet searches around main sequence stars will have a much lower success rate. Planet formation is an "easy come, easy go'' business, with many planets created and many planets destroyed, and with an important minority - including our own Jupiter - surviving.
Acknowledgements
The authors thank Doug Lin, Peter Bodenheimer, Geoff Bryden, Hubert Klahr, Geoff Marcy, Fred Adams, and Norm Murray for many useful discussions. We thank an anonymous referee for useful suggestions. D.E.T. acknowledges a NASA GSRP grant, and NASA grants to Doug Lin, Peter Bodenheimer, and Robert H. Brown. J.I.L. acknowledges support from the NASA Origins Program. W.B. acknowledges support from the Swiss National Science Foundation. D.E.T. thanks E. Stein for hosting him for useful work sessions, and NASA's IRTF which accommodated him for some of the writing of this paper.