How common are Earth-Moon planetary systems?

Abstract

The Earth's comparatively massive moon, formed via a giant impact on the
proto-Earth, has played an important role in the development of life on our
planet, both in the history and strength of the ocean tides and in stabilizing
the chaotic spin of our planet. Here we show that massive moons orbiting
terrestrial planets are not rare. A large set of simulations by Morishima et
al., 2010, where Earth-like planets in the habitable zone form, provides the
raw simulation data for our study. We use limits on the collision parameters
that may guarantee the formation of a circumplanetary disk after a protoplanet
collision that could form a satellite and study the collision history and the
long term evolution of the satellites qualitatively. In addition, we estimate
and quantify the uncertainties in each step of our study. We find that giant
impacts with the required energy and orbital parameters for producing a binary
planetary system do occur with more than 1 in 12 terrestrial planets hosting a
massive moon, with a low-end estimate of 1 in 45 and a high-end estimate of 1
in 4.

Full text

Page 1

How common are Earth-Moon planetary systems?
S.Elsera,∗, B.Moorea, J.Stadela, R.Morishimab
aUniversity of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
bLASP, University of Colorado, Boulder, Colorado 80303-7814, USA
Abstract
The Earth’s comparatively massive moon, formed via a giant impact on the proto-Earth, has played an important role
in the development of life on our planet, both in the history and strength of the ocean tides and in stabilizing the chaotic
spin of our planet. Here we show that massive moons orbiting terrestrial planets are not rare. A large set of simulations
by Morishima et al. (2010), where Earth-like planets in the habitable zone form, provides the raw simulation data for
our study. We use limits on the collision parameters that may guarantee the formation of a circumplanetary disk after
a protoplanet collision that could form a satellite and study the collision history and the long term evolution of the
satellites qualitatively. In addition, we estimate and quantify the uncertainties in each step of our study. We find that
giant impacts with the required energy and orbital parameters for producing a binary planetary system do occur with
more than 1 in 12 terrestrial planets hosting a massive moon, with a low-end estimate of 1 in 45 and a high-end estimate
of 1 in 4.
Keywords: Moon, Terrestrial planets, Planetary formation, Satellites, formation
1. Introduction
The evolution and survival of life on a terrestrial planet
requires several conditions. A planet orbiting the central
star in its habitable zone provides the temperature suit-
able for the existence of liquid water on the surface of
the planet. In addition, a stable climate on timescales of
more than a billion years may be essential to guarantee a
suitable environment for life, particularly land-based life.
Global climate is mostly influenced by the distribution of
solar insolation (Milankovitch, 1941; Berger et al., 1984;
Berger, 1989; Atobe and Ida, 2006). The annual-averaged
insolation on the surface at a given latitude is, beside the
distance to the star, strongly related to the tilt of the ro-
tation axis of the planet relative to the normal of its orbit
around the star, the obliquity. If the obliquity is close to
0◦, the poles become very cold due to negligible insolation
and the direction of the heat flow is poleward. With in-
creasing obliquity, the poles get more and more insolation
during half of a year while the equatorial region becomes
colder twice a year. If the obliquity is larger than 57◦,
the poles get more annual insolation than the equator and
the heat flow changes. Therefore, the equatorial region
can even be covered by seasonal ice (Ward and Brown-
lee, 2000). Thus, the obliquity has a strong influence on a
planet’s climate. The long-term evolution of the Earth’s
obliquity and the obliquity of the other terrestrial planets
∗Corresponding author
Email addresses: selser@physik.uzh.ch (S.Elser),
moore@physik.uzh.ch (B.Moore), stadel@physik.uzh.ch
(J.Stadel), ryuji.morishima@lasp.colorado.edu (R.Morishima)
in the solar system, or planets in general, is controlled by
spin-orbit resonances and the tidal dissipation due to the
host star and satellites of the planet. Thus, the evolution
of the planetary obliquity is unique for each planet.
Earth’s obliquity fluctuates currently ±1.3◦around 23.3◦
with a period of ∼ 41,000 years (Laskar and Robutel, 1993;
Laskar, 1996). The existence of a massive (or close) satel-
lite results in a higher precession frequency which avoids
a spin-orbit resonance. Without the Moon, the obliquity
of the Earth would suffer very large chaotic variations.
The other terrestrial planets in the solar system have no
massive satellites. Venus has a retrograde spin direction,
whereas a possibly initial more prograde spin may have
been influenced strongly by spin-orbit resonances and tidal
effects (Goldreich and Peale, 1970; Laskar, 1996). Mars’
obliquity oscillates ±10◦degree around 25◦with a period
of several 100,000 years (Ward, 1974; Ward and Rudy,
1991). Mercury on the other hand is so close to the sun
that its rotation period is in an exact 3 : 2 resonance with
its orbital period. Mercury’s spin axis is aligned with its
orbit normal.
On larger timescales, the variation of the obliquity can
be even more dramatic. It has been shown that the tilt
of Mars’ rotation axis ranges from 0◦to 60◦in less than
50 million years and 0◦to 85◦in the case of the obliquity
of an Earth without the Moon (Laskar and Robutel, 1993;
Laskar, 1996).
The main purpose of this report is to explore the giant
impact history of the planets in order to calculate the prob-
ability of having a giant Moon-like satellite companion,
based on simulations done by Morishima et al. (2010). A
Preprint submitted to ElsevierMay 25, 2011
arXiv:1105.4616v1 [astro-ph.EP] 23 May 2011

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giant impact between a planetary embryo called Theia, the
Greek titan that gave birth to the Moon goddess Selene,
first named by Halliday (2000), and the proto-Earth is
the accepted model for the origin of our Moon (Hartmann
and Davis, 1975; Cameron and Ward, 1976; Cameron and
Benz, 1991), an event which took place within about 100Myr
after the formation of calcium aluminum-rich inclusions in
chondritic meteoroids, the oldest dated material in the so-
lar system (Touboul et al., 2007). After its formation, the
Moon was much closer and the Earth was rotating more
rapidly. The large initial tidal forces created high tidal
waves several times per day, possibly promoting the cyclic
replication of early bio-molecules (Lathe, 2004) and pro-
foundly affecting the early evolution of life. Tidal energy
dissipation has caused the Moon to slowly drift into its cur-
rent position, but its exact orbital evolution is still part of
an on-going debate (Varga et al., 2006; Lathe, 2006). Cal-
culating the probability of life in the Universe (Ward and
Brownlee, 2000) as well as the search for life around nearby
planets may take into account the likelihood of having a
massive companion satellite.
This report is structured as follows: In section 2, we
give a brief review on the evolution of simulating terres-
trial planet formation with N-body codes during the last
decades and present the method we used. In section 3, we
study the different parameters of a protoplanet collision to
identify potential satellite forming events. In section 4, we
summarize the different uncertainties from the simulations
and our analysis that may affect the final results. Finally,
we give a conclusion, we present our results and compare
them with previous works in section 5.
2. Simulating terrestrial planet formation
There are good observational data on extra-solar gas gi-
ant planets, but whilst statistics on extra-solar rocky plan-
ets will be gathered in the coming years, for constraints
on the formation of the terrestrial planets we rely on our
own solar system. The established scenario for the forma-
tion of the Earth and other rocky planets is that most of
their masses were built up through the gravitational colli-
sions and interactions of smaller bodies (Chamberlin, 1905;
Safranov, 1969; Lissauer, 1993).
(1989) observed the phase of run-away growth. This phase
is characterized by the rapid growth of the largest bodies.
While their mass increases, their gravitational cross sec-
tion increases due to gravitational focusing. When a body
reaches a certain mass, the velocities of close planetes-
imals are enhanced, the gravitational focusing decreases
and so does the accretion efficiency. This is called the oli-
garchic growth phase, first described by (Kokubo and Ida,
1998). During this phase, the smaller embryos will grow
faster than the larger ones. At the end, several bodies
of comparable size are embedded in a planetesimal disk.
These protoplanets merge via giant impacts to form the
final planets. Dones and Tremaine (1993) showed that
most of a terrestrial planet’s prograde spin is imparted
Wetherill and Stewart
by the last major impactor and can not be accumulated
via the ordered accretion of small planetesimals. Giant
impacts with a certain impact angle and velocity gener-
ate a disk of ejected material around the target which is
a preliminary step in the formation of a satellite. Usu-
ally, the simulations assume perfect accretion in a colli-
sion. Tables of the collision outcome can help to improve
the simulations or to estimate the errors in the planetary
spin, (Kokubo and Genda, 2010). Until recently simula-
tions were limited in the number of planetesimal bodies
that could be self-consistently followed for time spans of
up to billions of years, but recent algorithmic improve-
ments by Duncan et al. (1998) in his SyMBA code and by
Chambers (1998) in his Mercury code have allowed them
to follow over long time spans a relatively large number of
bodies (O(1000)) with high precision, particularly during
close encounters and mergers between the bodies, where
individual orbits must be carefully integrated (Chambers
and Wetherill, 1998; Agnor et al., 1999; Raymond et al.,
2004; Kokubo et al., 2006). Raymond et al. (2009) have
also recently conducted a series of simulations where they
varied the initial conditions for the gas giant planets and
also track the accretion of volatile-rich bodies from the
outer asteroid belt, leading either to “dust bowl” terres-
trial planets or “water worlds” and everything between
these extremes. All prior simulation methods with full in-
teraction among all particles have however been limited in
number of particles since their force calculations scale as
O(N2).
We have developed a new parallel gravity code that
can follow the collisional growth of planetesimals and the
subsequent long-term evolution and stability of the result-
ing planetary system. The simulation code is based on
an O(N) fast multipole method to calculate the mutual
gravitational interactions, while at the same time follow-
ing nearby particles with a highly accurate mixed vari-
able symplectic integrator, which is similar to the SyMBA
(Duncan et al., 1998) algorithm. Since this is completely
integrated into the parallel code PKDGRAV2 (Stadel, 2001),
a large speed-up from parallel computation can also be
achieved. We detect collisions self-consistently and also
model all possible effects of gas in a laminar disk: aero-
dynamic gas drag, disk-planet interaction including Type-
I migration, and the global disk potential which causes
inward migration of secular resonances with gas dissipa-
tion.In contrast to previously mentioned studies, this
code allows us to self-consistently integrate through the
last two phases of planet formation with the same numer-
ical method while using a large number of particles.
Using this new simulation code we have carried out 64
simulations which explore sensitivity to the initial condi-
tions, including the timescale for the dissipation of the so-
lar nebula, the initial mass and radial distribution of plan-
etesimals and the orbits of Jupiter and Saturn (Morishima
et al., 2010). All simulations start with 2000 equal-mass
particles placed between 0.5 and 4AU. The initial mass
of the planetesimal disk md is 5 or 10m⊕. The surface
2

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density Σ of this disk and of the initial gas disk depends
on the radius through Σ ∝ r−p, where p is 1 or 2. The
gas disk dissipates exponentially in time and uniformly in
space with a gas dissipation time scale τgas = 1,2,3 or
5Myr. After the disappearance of the gas disk (more pre-
cisely after time τgasfrom the beginning of the simulation),
Jupiter and Saturn are introduced on their orbits, e.g. cir-
cular orbits or the current orbits with higher eccentricities.
Figure 1 shows two merger trees: a 1.8 m⊕ planet
formed in the simulation with (τgas,p,md)=(1Myr,1,5m⊕)
and gas giants on the present orbits and a 1.1 m⊕planet
formed in the simulation with (τgas,p,md)=(1Myr,2,10m⊕)
and gas giants on circular orbits. The red branches are
satellite forming impactors both with a mass 0.3m⊕and
represent two events of the final sample in figure 7. These
merger trees with their different morphologies reveal the
variety of collision sequences in terrestrial planet forma-
tion. They show that a large set of impact histories is
generated by these simulations despite the relatively nar-
row parameter space for the initial conditions.
3. Satellite formation
During the last phase of terrestrial planet formation,
the giant impact phase, satellites form. Collisions between
planetary embryos deposit a large amount of energy into
the colliding bodies and large parts of them heat up to sev-
eral 103K, e.g. Canup (2004). Depending on the impact
angle and velocity and the involved masses, hot molten
material from the target and impactor can be ejected into
an circumplanetary orbit. This forms a disk of ejecta, the
disk material is in a partially vapor or partially molten
state, around the target planet. The proto-satellite disk
cools and solidifies. Solid debris form and subsequently
agglomerate into a satellite (Ohtsuki, 1993; Canup and
Esposito, 1996; Kokubo et al., 2000).
The giant impact which resulted in the Earth-Moon
system is a very particular event (Cameron and Benz,
1991; Canup, 2004). The collision parameter space that
describes a giant impact can by parametrized by γ =
mi/mtot, the ratio of impactor mass mito total mass in
the collision mtot, by v ≡ vimp/vesc, the impact velocity
in units of the escape velocity vesc=
where rtand riare the radii of target and impactor. Fur-
thermore, it is described by the scaled impact parameter b,
where b = 0 indicates a head-on collision and b = 1 a graz-
ing encounter, and the total angular momentum L. Recent
numerical results (Canup, 2008) obtained with smoothed
particle hydrodynamic (SPH) simulations for the Moon-
forming impact parameters require: γ ∼ 0.11, v ∼ 1.1,
b ∼ 0.7 and L ∼ 1.1LEM, where LEMis the angular mo-
mentum of the present Earth-Moon system. These sim-
ulations also include the effect of the initial spins of the
colliding bodies, but the explored parameter space is re-
stricted to being close to the Moon-forming values given
above.
?2Gmtot/(ri+ rt),
Figure 1: Two merger trees. They illustrate the accretion from the
initial planetesimals to the last major impactors that merge with
the planet. Every ’knee’ is a collision of two particles and the length
between two collisions is given by the logarithm of the time between
impacts. The thickness of the lines indicates the mass of the particle
(linear scale). The red branch is the identified satellite forming im-
pact in the planet’s accretion history. Top: a 1.1 m⊕planet formed
in the simulation with (τgas,p,md)=(1Myr,2,10m⊕) and gas giants
on circular orbits and a 0.3m⊕ impactor. In this case, the moon
forming impact is not the last collision event but it is followed by
some major impacts. Right: a 0.7 m⊕ planet formed in the simula-
tion with (τgas,p,md)=(3Myr,1,5m⊕) and gas giants on the present
orbits and a 0.2m⊕impactor. It is easy to see that the moon forming
impact is the last major impact on the planet. Although this planet
is smaller than the upper one, it is composed out of a similar number
of particles. In the case of the more massive disk (md= 10), the
initial planetesimals are more massive since their number is constant.
Therefore, fewer particles are needed to form a planet of comparable
mass than in the case of md= 5.
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If one does not focus on a strongly constrained system
like the Earth-Moon system but just on terrestrial planets
of arbitrary mass with satellites that tend to stabilize their
spin axis, the parameter space is broadened. It becomes
difficult to draw strict limits on the parameters because
collision simulations for a wider range of impacts were not
available for our study. Hence, based on published Moon-
forming SPH simulations by Canup (2004, 2008), we use a
semi-analytic expression to constrain the mass of a circum-
planetary disk that can form a satellite. In addition, we
include tidal evolution and study the ability of the satellite
to stabilize the spin axis of the planet.
3.1. Satellite mass and collision parameters
We do not know the exact outcome of a protoplanet
collision, but certainly the satellite mass is related to the
collision parameters of the giant impact. Hence, we can
draw a connection from these parameters to the mass of
the final satellite. Based on the studies of the Earth’s
Moon formation, we can start with a simple scaling re-
lation: a Mars-size impactor gives birth to a Moon-size
satellite. Their mass ratio is mMars/mMoon∼ 10. Thus,
to very first approximation, we can assume that an im-
pactor mass is usually 10 times larger than the final mass
of the satellite. Of course, this shows that only a small
amount of material ends up in a satellite, but this state-
ment is only valid for a certain combination of mass ratio,
impact parameter and impact speed and usually gives an
upper limit on the satellite mass.
In order to get a better estimation of the satellite mass,
we use the method obtained in the appendix of Canup
(2008).
There, an expression is derived that describes the mass
of the material that enters the orbit around the target after
a giant impact:
?mpass
where the prefactor Cγ ∼ 2.8(0.1/γ)1.25has been deter-
mined empirically from the SPH data. mpass/mtotis mass
of the impactor that avoids direct collision with the tar-
get. It depends mainly on the impact parameter b and on
the mass ratio γ and can be computed by studying the
geometry of the collision. The total impactor volume that
collides with the target is:
?π
with
mdisk
mtot
∼ Cγ
mtot
?2
(1)
VT=
0
A(φ)dφ,(2)
A(φ) = r2
tθt(φ) + r2
iθi(φ) − Drisinφsinθi(φ), (3)
where riand rtare impactor and target radius, D = b(ri+
rt) gives the distance between the centers of the bodies and
?D2+ r2
θi(φ) = cos−1
isin2φ − r2
2Drisinφ
t
?
, (4)
θt(φ) = cos−1
?D2+ r2
t− r2
2Drt
isin2φ
?
.(5)
Assuming a differentiated impactor with rcore∼ 0.5ri
and repeating the above integration for this radius, the
colliding volume of the impactor mantle is Vmantle= VT−
Vcore. If we assume that the core is iron and the mantle
dunite, the core density ρcorehas roughly twice the density
of the mantle ρmantle. The mass of the impactor that hits
the target is mhit= ρcoreVcore+ρmantleVmantle. Therefore,
the mass that passes the target is mpass= mi− mhit.
We use the full expression derived by Canup (2008),
equation (1), to estimate the disk mass resulting from the
giant impacts in our simulation. Equation (1) is correct to
within a factor 2, if v < 1.4 and 0.4 < b < 0.7 or if v < 1.1
and 0.4 < b < 0.8. Figure 2 illustrates the amount of
material that is transported into orbit for the parameter
range 0.4 < b < 0.8 (see figure B2 in Canup (2008) for
more details). It shows that a small impact parameter b
reduces the material ejected into orbit significantly. The
same holds for a reduction of γ, because those collisions
are more grazing. Based of simple arguments, we use the
limits above to identify the moon forming collision in the
(v,b)-plane. Details on how this assumption affects the
result are given in section 4.
Ida et al. (1997) and Kokubo et al. (2000) studied the
formation of a moon in a circumplanetary disk through N-
body simulations. They found that the final satellite mass
scales linearly with the specific angular momentum of the
disk. The fraction of the disk material that is finally in-
corporated into the satellite ranges form 10 to 55%. Thus,
we assume that not more than half of the disk material is
accumulated into a single satellite. The angular momen-
tum of the disk is unknown and we can not use the more
exact relationships.
3.2. Spin-orbit resonance
Spin-orbit resonance occurs when the spin precession
frequency of a planet is close to one of the planet’s or-
bital precession frequencies. It causes large variation in the
obliquity (Laskar, 1996), the angle between this spin axis
and the normal of the planet’s orbital plane. An obliquity
stabilizing satellite increases the spin precession frequency
to a non-resonant (spin-orbit) regime. To ensure this, one
can set a rough limit on the system parameters (Atobe et
al., 2004) through
ms
a3
s
?m∗
a3
p
, (6)
where msis the mass of the satellite, m∗the mass of the
central star and as the semi-major axis of the satellite’s
orbit and apthe semi-major axis of the planet.
If the left term of the inequality is much larger than
the right one, the spin precession frequency of the planet
should be high enough to ensure that it is over the up-
per limit of the orbital precession frequency so that spin-
orbit resonance does not occur. Although this inequality
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0.1 0.20.30.4 0.5
b
0.60.70.80.9
0.00
0.02
0.04
0.06
0.08
0.10
mdisk/mtot
Figure 2: The disk mass resulting after a giant impact in units of the
total mass of the colliding system relative to the impact parameter
b based on equation (1). The different solid lines belong to different
mass ratios γ. From bottom to top: γ = 0.05,0.1,0.15,0.2,0.3 and
0.5. These mass ratios result in different disk masses. The equation
is valid up to a factor 2 in between the two dash lines (0.4 < b < 0.8)
for small velocities. The disk mass is an upper bound on the satellite
mass.
is very simplified, we try to estimate the minimum mass
of a satellite such that it is able to stabilize the obliquity
of its planet.
The exact semi-major axis of a satellite after forma-
tion is unknown but the Roche limit is the lower bound of
its semi major axis. The Roche limit aRof the planet is
(Murray and Dermott, 1999):
?3mp
where rsis the radius of the satellite and msits mass and
the mass of the planet is given by mp. Ohtsuki (1993)
and Canup and Esposito (1996) provided detailed ana-
lytic treatments of the accretion process of satellites in an
impact-generated disk. Based on those studies, Kokubo
et al. (2000) have shown that the true value of the ra-
dius of satellite accretion will not diverge much from the
Roche radius in the case of the Earth-Moon system, a
typical satellite orbit semi-major axis in their simulations
was a ? 1.3aR. We used this approximation to estimate a
lower bound on the satellite-planet mass ratio. We rewrite
the Roche limit as
?3
where we used r3
in our model. We insert this in equation (6) instead of as
and get the condition
aR= rs
ms
?1
3
, (7)
aR=
2
?2
s= (4π
3?mp
πρp
?1
3
(8)
3ρs)−1msand the fact that ρp= ρs
ms
mp
?
9m∗
4πρpa3
p
. (9)
Inserting a density of 2g cm−2, the density of the bodies in
the Morishima simulations, and planet semi-major axis of
1AU, we get a mass ratio of ∼ 10−5, which is smaller than
the minimum ratio of the smallest and largest particles in
our simulations. This is a lower limit on the stabilizing
satellite mass but the tidal evolution of the planet-satellite
system can alter this limit dramatically.
The orbital precession frequencies of a planet depend
on the neighbouring or massive planets in its system. In
the case of the Earth, Venus, Jupiter and Saturn cause
the most important effects. To keep the spin precession
frequency high enough, a spin period below 12 h would
have the same effect as the present day Moon (Laskar and
Robutel, 1993). Hence, even without a massive satellite,
obliquity stabilization is possible as long as the planet is
spinning fast enough. However, the moon forming impact
provides often a significant amount of angular momentum.
A more sophisticated analysis including the precession fre-
quencies of the all planets involved or formed in the simu-
lations and a better treatment of the collisions to provide
better estimates of the planetary spins would clearly be an
improvement but this is out of the scope of this work.
3.3. Tidal evolution
After its formation, even a small satellite is stabiliz-
ing the planet’s obliquity. Its fate is mainly controlled
by the spin of the planet, the orientation of the spin axis
of the planet relative to its orbit around the central star
and relative to the orbital plane of the satellite and by
possible spin-orbit resonance. Which satellites will con-
tinue to stabilize the obliquity as they recede from the
planet? Orbital evolution is a complicated issue (Atobe
and Ida, 2006) and there is still an ongoing debate even
in the case of the Earth-Moon system, as mentioned. To
classify the different orbital evolutions, it is helpful to in-
troduce the synchronous radius, at which a circular or-
bital period equals the rotation period of the planet. In
the prograde case, a satellite outwards of the synchronous
radius will recede from the planet as angular momentum
is tidally transferred from the planet to the satellite and
the spin frequency of the planet decreases. In this case
the synchronous radius will grow till it equals the satel-
lite orbit. Angular momentum is transferred faster if the
mass of the satellite is large, since the tidal response in the
planet due to the satellite is greater. Large mass satellites
will quickly reach this final co-rotation radius, where their
recession stops. Even though their orbital radius becomes
larger, these moons are massive enough to satisfy inequal-
ity (6) and avoid spin-orbit resonances. Small satellites
will recede very slowly compared to their heavy broth-
ers and eventually fulfill the condition (6) within the host
star’s main sequence life time. In contrast, low mass satel-
lites that form in situ far outside the Roche limit, should
this be possible, will probably not stabilize the spin axis.
Intermediate mass satellites with ms ∼ mMoon may re-
cede fast enough so that they tend to lose their obliquity
stabilizing effect during a main sequence life time. The
Earth-Moon system shows that even in this intermediate
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mass regime, long term stability can occur, since the Moon
has stabilized the Earth’s obliquity for billion of years.
On the other hand, a satellite inside the synchronous
orbit will start to spiral towards the planet, while its an-
gular momentum is transferred to the planet. Soon, it is
disrupted by tidal forces or will crash on the planet. How
do we know if a satellite forms inside or outside the syn-
chronous radius? The Roche limit (7) depends on the mass
and size of the bodies while the synchronous radius rsync
is a function of the planet mass mpand depends inversely
proportional on its rotation frequency, which is obtained
from equating the gravitational acceleration and the cen-
tripetal acceleration:
rsync= (Gmp)
1
3ω−2
3
p . (10)
Equating this formula with (7) gives:
?3
2
?2
3?mp
πρp
?1
3
= (Gmp)
1
3ω−2
3
p . (11)
The planet mass drops out and we get an lower limit on
the planet angular velocity to guarantee a satellite outside
the Roche radius:
ωp,min=2
3(πGρp)
1
2= 0.00043s−1, (12)
which equals a rotation period of 4h. The final planets of
Morishima et al. (2010) have generally a high rotational
speed, some of them rotating above break up speed. The
rotation period after the moon forming collision is usu-
ally around 2-5 hours, see figure 3. A more exact parti-
cle growth model without the assumption of perfect stick-
ing would lower the rotation frequency by roughly 30%
(Kokubo and Genda, 2010), where we can also include
the loss of rotational angular momentum due to satellite
formation (full circles). On the other hand, the mean dis-
tance of satellite formation is around 1.3aR, and the max-
imum rotation period of the planet for a receding satellite
changes to ∼ 6h (dashed line). Hence, applying this to
our final sample, 1/4 of all moon forming collisions are
excluded.
Finally, for the remaining events, we assume that the
planet spin is large enough so that the synchronous radius
is initially smaller than the Roche radius. Satellites that
form behind the Roche limit will start to recede from the
planet. We can conclude that almost every satellite-planet
system in our simulation will fulfill (6).
A special outcome of the tidal evolution of a prograde
planet-satellite system is described by Atobe and Ida (2006).
If the initial obliquity θ of the planet after the moon form-
ing impact is large, meaning that the angle between planet
spin axis and planet orbit normal is close to 90◦, results
in a very rapid evolution when compared to the previ-
ously discussed case of a moon receding to the co-rotating
radius and becoming tidally locked. The spin vectors of
the protoplanets are isotropically distributed after the gi-
ant impact phase (Agnor et al., 1999) and the obliquity
??
?
?
?
?
?
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0.2 0.40.60.8 1.0
0
10
15
mtot?mtot,final
rotation period ?d?
Figure 3:
mtot/mtot,finalrelative to the rotational period in days of the body
after the moon-forming collision. This is the final sample but with-
out excluding moon formation inside the synchronous radius. Empty
circles: The decrease of the planet spin due to escaping material or
the formation of a satellite are not included in the calculation of the
rotation period, it is based on the perfect accretion assumption. Full
circles: The angular momentum is assumed to be 30% smaller due
to a more realistic collision model (Kokubo and Genda, 2010). The
dotted line gives the position of the threshold for synchronous rota-
tion for the Roche radius aR, the dashed line gives its position for
the 1.3aR. We want to focus on the better estimate of the anuglar
momentum (full circles) and the initial moon radius (dashed line).
Hence, roughly a quarter of all moon forming collisions are excluded
in the final sample, figure 7.
The mass of the planet in units of its final mass
distribution corresponds to p(θ) =
extreme scenario, a massive prograde satellite will crash
onto the planet in a timescale of order 10,000 years after
formation, even though it initially recedes. Hence, without
a favorable initial obliquity is an added requirement for the
survivability of a close or massive moon. We include this
by discarding massive and highly oblique impacts by an
approximated inequality, based on area B in figure 13 in
Atobe and Ida (2006):
1
2sin(θ). In the most
θ <π
2−
π
0.2
ms
mp
(13)
More exactly, this holds best for planets with ap∼ 1AU,
and it has only to be taken into account if ms/mp< 0.05.
The distribution of the apof the simulated planets ranges
for 0.1 to 4AU. In the case of a smaller distance to the host
star, angular momentum is removed faster from the planet-
satellite system and the evolution timescales are shorter in
general. To use this limit properly, a more general expres-
sion for different semi-major axes has to be derived, which
is out of the scope of this work. Moreover, it depends
linearly on the satellite mass which is overestimated in
general. A smaller satellite mass would reduce the num-
ber of excluded events in general. However, when applying
this constraint on the data, only one out of seven moon
forming collisions are affected. Since it is over-simplified,
we exclude it from our analysis.
For retrograde impacts, where the impactor hits in op-
position to the target’s spin, two cases result in differing
evolution.If the angular momentum of the collision is
6

Page 7

much larger than the initial rotational angular momen-
tum, the spin direction of the planet is reversed and any
impact-generated disk will rotate in the same direction as
the planet’s spin. On the other hand, a retrograde collision
with small angular momentum will not alter the spin di-
rection of the planet significantly and it becomes possible
to be left with a retrograde circumplanetary disk(Canup,
2008). After accretion of a satellite, tidal deceleration due
to the retrograde protoplanet spin will reduce the orbital
radius of the bodies continuously till they merge with the
planet. Hence, a long-lived satellite can hardly form in
this case.
In order to find a reasonable threshold between these
two regimes based on the limited information we have, we
assume that the sum of the initial angular momenta of the
bodies?Ltand?Liand of the collision?Lcolequals the spin
angular momentum of the planet and the satellite?Lplanet
and?Lmoonafter the collision plus the angular momentum
of the orbiting satellite?Lorbitat 1.3aRparallel to the col-
lision angular momentum.
Comparing initial and final angular momenta gives:
?Lcol+?Lt+?Li=?Lorbit+?Lmoon+?Lplanet, (14)
where?Lorbitis parallel to?Lcol:
?Lorbit= |?Lorbit|
?Lcol
|?Lcol|. (15)
We assume that |?Lmoon|/|(?Lorbit+?Lplanet)| ? 1 and
?
Lorbit= msa2
sn = msa2
s
Gmp
a3
s
= ms
?Gasmp, (16)
where n is the orbital mean motion of the satellite if its
mass is much smaller than the planet mass, n ∼?Gmp/a3
?3
If
s,
and G is the gravitational constant. With as= aRwe get
Lorbit= ms(Gmp)
1
2
2π
1
ρpmp
?1
6
. (17)
|?Lcol+?Lt|
?(L2
col+ L2
t)
< 1, (18)
the collision is retrograde. Hence,
?Lplanet=?Lcol+?Lt+ Li− |?Lorbit|
?Lcol
|?Lcol|, (19)
If?Lplanet and?Lorbit, which is parallel to?Lcol, are retro-
grade,
|?Lcol+?Lorbit|
?(L2
the satellite will be tidally decelerated. Those cases are
excluded from being moon-forming events. Spin angular
momenta are in general overestimated since material is
col+ L2
orbit)
< 1, (20)
lost during collisions in general, but as mentioned above,
the simulations of Morishima et al. (2010) assume perfect
accretion. We include this consideration by reducing the
involved spins and the orbit angular momentum of the
satellite by 30%, (Kokubo and Genda, 2010).
Both scenarios described above, a large initial obliquity
or a retrograde orbiting planet, might not become impor-
tant until subsequent impactors hit the target. Giant im-
pacts can change the spin state of the planet in such a
way that the satellite’s fate is to crash on the planet. This
scenario is discussed in the next section.
3.4. Collisional history
We exclude all collisions from being satellite-forming
impacts whose target is not one of the final planets of a
simulation. A satellite orbiting an impact is lost through
the collision with the larger target.
Multiple giant impacts occur during the formation pro-
cess of a planet and it is useful to study the impact his-
tory in more detail. Subsequent collisions and accretion
events on the planet after the satellite-forming event may
have a large effect on the final outcome of the system.
We choose a limit of 5mplanetesimalto distinguish between
large impacts and impacts of small particles, which are
responsible for the ordered accretion. To stay consistent,
the same limit is used below to exclude small impactors
from our analysis. We divide all identified moon forming
events into four groups:
a) The moon forming event is the last major impact on
the planet. Subsequent mass growth happens basi-
cally through planetesimal accretion (see merger tree
at the bottom in figure 1).
b) There are several moon forming impacts, in which
the last impact is the last major impact on the planet.
c) The moon forming event is not the last giant im-
pact on the planet. The satellite can be lost due to
a disruptive near or head-on-collision (Stewart and
Leinhardt, 2009) of an impactor and the satellite. A
late giant impact on the planet can change the spin
axis of the planet and the existing satellite can get
lost due to tidal effects (see tree at the top in figure
1).
d) There are several moon forming events in the impact
history of the planet, followed by additional major
impacts. As before, the moon forming collisions can
remove previously formed satellites. On the other
hand, and existing moon can have an influence on
the circumplanetary disk formed by a giant impact
and can suppress the formation of multiple satellites
orbiting the planet.
The final states of the planet-satellite systems in group c
are difficult to estimate, since such systems might change
significantly by additional giant impacts. To a lesser ex-
tent, this holds for d, but those systems are probably more
7

Page 8

abcd
5
10
15
20
25
30
Figure 4: This bar chart shows the distribution of the moon forming
collisions in four groups with different impact history with respect
to the last major impact. a: the moon forming impact is the last
major impact on the planet. b: there are multiple moon forming
impacts, but the last one is also the last major impact. c: there
is only one moon forming impact and it is followed by subsequent
major impacts. d: there are multiple moon forming impacts, but the
last of them is followed by subsequent major impacts.
resilient to the loss of satellites by direct collisions. The
number of events per group is shown in figure 4. Group c
and d include more then 2/3 of all collisions. Group c is
the most uncertain and we use it to quantify the error on
the final sample.
Furthermore, we exclude impactors and targets that
have masses of the order of an initial planetesimal mass
(5mplanetesimal∼ 0.0025m⊕) from producing satellite form-
ing events since their masses are discretized and related to
the resolution of the simulation. In addition, small im-
pactors will probably not have enough energy to eject a
significant amount of material into a stable orbit. There-
fore, setting a lower limit on the target and impactor mass
results in the exclusion of many collisions but not in a sig-
nificant underestimation of the true number of satellites,
see figure 5.
4. Uncertainties
Our final result depends on several assumptions, lim-
itations and approximations. In this section we want to
quantify them as much as possible and summarize them.
The data of Morishima et al. (2010) we are using has
two peculiarities worth mentioning: The focus on the Solar
System and the small number of simulations per set of
initial conditions.
The simulations were made in order to reproduce the
terrestrial planets of the Solar System. The central star
has 1 solar mass and the two gas giants that are intro-
duced after the gas dissipation time scale have the mass
of Jupiter and Saturn and the same or similar orbital el-
ements. Although the initial conditions like initial disk
mass or gas dissipation time scale are varied in a certain
range, the simulated systems do not represent general sys-
tems with terrestrial planets. However, we assume that
?
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0.00.10.20.3 0.40.5
0.0
0.5
1.0
1.5
2.0
mi?m?
mt?m?
Figure 5:
volved in the satellite-forming collisions.
include impactors and targets with masses of the order of the ini-
tial planetesimal mass (mi,mt < 5mplanetesimal) indicated by the
dashed line to avoid resolution effects. The line in the case of the
target mass is not shown since it is very close the frame. Due to this
cut, the total number of satellites decreases significantly while the
number of massive satellites in our analysis increases. The shown
sample is the final set of events without applying the threshold for
small particles. Therefore, the number of accepted events in this
plot does not equal the number of satellites in figure 7, since this cut
is not applied. If the cut is used, new events are accepted, which
where neglected before because they were not the last moon forming
impacts on the target.
The target mass mt versus the impactor mass mi in-
We exclude events that
the range of impact histories is representative of the range
that would be seen in other systems. Merger trees (figure
1) reveal that those simulations cover a huge diversity of
impact histories. But future work will need to investigate
the full range of impact histories that could be relevant to
the formation of terrestrial planets in other, possibly more
exotic, extra-solar systems.
However, the set of simulation covers a broad range
of initial conditions. But for every set of initial condi-
tions, only one simulation exists since they are very time
consuming (each simulation requires about 4 months of
a quad-core CPU). Therefore, it is difficult to separate ef-
fects of the choice of certain initial parameters from effects
of stochastic processes. We grouped our moon forming
events with respect to the simulation parameter in ques-
tion. The only parameter that reveals an effect on the final
sample is the gas dissipation time scale τgas. Larger time
scales lead to less moon forming collisions. This variation
is correlated with mass and number of final planets. If
the gas disk stays for several million years, the bodies are
affected by the gas drag for a long time, spiral towards
the sun and get destroyed. Therefore, there are less giant
impacts and smaller planets. One would suppose that the
initial mass of the gas disk should be correlated to the mass
of the final planets and therefore to the number of giant
impacts, but the two initial protoplanetary disk masses of
5m⊕and 10m⊕show no essential difference.
The approach we use to identify events and estimate
the mass of the satellite is also based on various approxi-
mations and limitations.
8

Page 9

Satellite mass. The method we use to calculate the mass
of the circumplanetary disk is valid to better than a factor
of 2 within the parameter range we use (Canup, 2008).
Only 10-55% of the disk mass are embodied in the satellite
(Kokubo et al., 2000). Therefore, the mass we use is just
half of the estimated disk mass, and in the worst case, the
satellite is ten times less massive than estimated. This
uncertainty affects the number of massive satellites but
not the number of satellites in general.
In the (v,b)-plane, we use the same restrictive limits to
constrain the collision events. Figure 6 presents all moon
forming events and this parameter range. Hence, this area
gives just a lower bound. We see that it covers some of
the most populated parts, but a significant amount of the
collisions are situated outside this area. Equation (1) can
help to constrain the outcome of the collisions close outside
of the shaded region. A collision with a small impact pa-
rameter (b < 0.4) will bring very little material into orbit,
whatever the impactor mass involved. In this regime, we
have few events with high impact velocity and even with
high velocity it might be very hard to eject a significant
amount of material into orbit. Hence, we exclude them
from being moon forming events. For intermediate impact
parameter (0.4 < b < 0.8), there are high velocity events
(v > 1.4). Above a certain velocity threshold, depending
on b and γ, most material ejected by the impact will es-
cape the system and the disk mass might be too small to
form a satellite of interest. A large parameter (b > 0.8)
describes a highly grazing collision. It is difficult to extent
equation (1) for larger b, since these collisions will proba-
bly result in a hit-and-run events for high velocities. SPH
simulations (Canup, 2004, 2008) suggest that high rela-
tive impact velocities (v > 1.4) will increase rapidly the
amount of material that escapes the system. Neverthe-
less, these sets of simulations focus not on general impacts
and the multi-dimensional collision parameter space is not
studied well enough to describe those collisions in more
detail. Detailed studies of particle collisions will hopefully
be published in the near future (e.g. Kokubo and Genda
(2010)). Based on the arguments above, the events inside
of the shaded area form our final sample. Most of the colli-
sions outside the area will not form a moon. The collisions
with 0.4 < b < 0.8 and velocity slightly above v = 1.4 and
with b > 0.8 and small velocity (v < 1.4) are events with
unknown outcome. Including those events, the final num-
ber of possible moon forming events is increased by not
more than a factor of 1.5.
Tidal evolution - Rotation period. In order to separate re-
ceding satellites from satellites which are decelerated after
their formation, we check if the initial semi-major axis
of the satellite is situated outside or inside of the syn-
chronous radius of the planet. In figure 3, the sample is
plotted twice, once including a general correction for an-
gular momentum loss due to realistic collisions and once
without correction. Moreover, two thresholds for the syn-
chronous radius are shown. We choose the threshold at
1.3aR(dashed line) and the corrected rotation period (full
circles) to be the most justified case. There are two most
extreme cases: the threshold situated at 1aR(dotted line)
and a corrected rotation period (full circles) indicates that
only one in five collisions lead to receding moons. On the
other hand, the threshold situated at 1.3aR and a rota-
tion period directly obtained form the simulations (empty
circles) indicates that only one in eight collisions lead to a
non-receding moon.
Tidal evolution - Retrograde satellites. To exclude retro-
grade orbiting satellites, we use a simple relation between
the angular momenta involved in a collision. Since we have
no exact data of the angular momentum distribution af-
ter the collision, this limit is very approximate. To get
an estimate of the quality of this angular momentum ar-
gument, we study the effect of this threshold on the final
sample. First, roughly half of all moon-forming impacts
are retrograde. But in almost every case, Lorbitis much
smaller than the initial angular momentum. Only four of
the retrograde collisions have not enough angular momen-
tum to provide a significant change of the spin axis and
those planet-satellite systems remain retrograde. There-
fore, our final estimate is not very sensitive to this angular
momentum argument. A larger set of particle collision
simulations could provide better insight into the angular
momentum distribution.
Collision history. Two issues affecting the collision history
can change the final result. These being the mass cut we
choose to avoid resolution effects, where we exclude small
impactors and targets, and the uncertainty about the fi-
nal state of the planet-satellite system because of multiple
and subsequent major impacts. The first limit seems to be
well justified. If we change the threshold mass or exclude
the cut completely, we change mainly the number of very
small, non-satellite forming, impacts since in these cases
there is usually insufficient energy to bring a significant
amount of material into orbit. The second issue has a sig-
nificant effect on the final sample, however. Group c, the
single moon forming collisions followed by large impacts,
contains the most uncertain set of events: Neither all satel-
lites survive the subsequent collisions nor is it likely that
all satellites get lost. Hence, in the extreme case, almost
half of all initially formed moons, all of Group c, could be
lost and our final result reduced significantly.
An overview of the above uncertainties in given in ta-
ble 1. Additional planet formation simulations are nec-
essary to quantify a large part of existing uncertainties.
Simulations of protoplanet collisions exploring the multi-
dimensional collision parameter space are desirable and
will hopefully be published soon and might help to con-
strain the parameters for moon formation better. A study
of the effect of subsequent accretion of giant bodies after
a moon forming impact might give better insight in the
evolution of a planet-satellite system.
9

Page 10

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0.0 0.20.4 0.60.81.0
1
2
3
4
5
b
vimp?vesc
Figure 6:
darker region gives the region for with equation (1) holds up to a
factor of 2. The shown sample is the final sample without the con-
straints in the (b, v)-plane. Similar to figure 5, the number of ac-
cepted events will increase when applying the cut, since an earlier
moon forming impact on the planet can become suitable. As one
expects, there are more collisions with larger impact parameter. Ig-
noring events with smaller b or higher v (see text), there is still a
significant group of events with b > 0.8. Much material escapes in
the high velocity cases and no moon will form, but the low velocities
are hard to exclude or calculate.
The parameter space in the (b, vimp/vesc)-plane. The
Uncertainty factor
high
high
1-1.5
0.5-2
∼ 1
0.5-1
0.5-2
0.2-1
Range of initial parameters
Number of simulations
Collision parameter
Tidal evolution - Rotation
Tidal evolution - Retrograde orbit
Collision history
Satellite mass - disk mass
Satellite mass - accretion efficiency
Table 1: A list of the different conditions that affect the final number
of satellites. The uncertainty factor gives the range in which the
final result varies around the most justified. 1 equals the final value.
The first two factors can not be estimated, it shows that additional
simulations would be helpful. The estimation of the satellite mass
is separated from the rest of the list, since the uncertainty on this
estimate does not change the number of satellites in the final sample,
in contrast to the others. The exclusion of retrograde satellites is
very approximate, but it has almost no effect on the final sample
and therefore this factor is close to 1.
5. Discussion and results
Under these restrictive conditions we identify 88 moon
forming events in 64 simulations, the masses of the re-
sulting planet-satellite systems are shown in figure 7. On
average, every simulation gives three terrestrial planets
with different masses and orbital characteristics and we
have roughly 180 planets in total. Hence, almost one in
two planets has an obliquity stabilizing satellite in its or-
bit. If we focus on Earth-Moon like systems, where we
have a massive planet with a final mass larger than half
of an Earth mass and a satellite larger than half a Lunar
mass, we identify 15 moon forming collisions. Therefore,
1 in 12 terrestrial planets is hosting a massive moon. The
main source of uncertainties results from the modelling of
the collision outcomes and evolution of the planet-satellite
system as well as the small number of simulation and the
limited range of initial conditions. We do not include the
latter in our estimate. Hence, we expect the total number
of Earth-Moon like systems in all our simulations to be in
a range from 4 to 45. This results in a low-end estimate
of 1 in 45 and a high-end estimate of 1 in 4. In addition,
taking into account the uncertainties on the estimation of
the satellite mass, roughly 60 of those systems are formed
in the best case or almost no such massive satellites are
formed if the efficiency of the satellite accretion in the cir-
cumplanetary disk is very low.
There are several papers, where the authors performed
N-body simulations and searched for moon-forming colli-
sions. Agnor et al. (1999) started with 22-50 planetary
embryos in a narrow disk centered at 1AU. They esti-
mated around 2 potentially moon-forming collisions per
simulation, where the total angular momentum of the en-
counter exceeds the angular momentum of the Earth-Moon
system. They pointed out that this number is somewhat
sensitive to the number, spacing, and masses of the ini-
tial embryos. O’Brien et al. (2006) performed simula-
10

Page 11

tions with 25 roughly Mars-mass embryos embedded in
a disk of 1000 non-interacting (with each other) planetes-
imals in an annulus from 0.3 to 4.0AU. They found that
giant impact events which could form the Moon occur
frequently in the simulations. These collisions include a
roughly Earth-size target whose last large impactor has a
mass of 0.11 − 0.14ME and a velocity, when taken at in-
finity, of 4km/s, as found by Canup (2004). O’Brien et al.
pointed out that their initial embryo mass is close to the
impactor mass. Raymond et al. (2009) set up about 90 em-
bryos with masses from 0.005 to 0.1MEin a disk of more
than 1000 planetesimals, again with non-interaction of the
latter. Assuming again Canup’s requirements (v/vesc <
1.1, 0.67 < sinθ < 0.76, 0.11 < γ < 0.15), only 4% of
their late giant impacts fulfill the angle and velocity cri-
teria. They concluded that Earth’s Moon must be a cos-
mic rarity but a much larger range of late giant collisions
would produce satellites with different properties than the
Moon. The initial embryo size seems to play a role in
those results. In contrast, the simulations of Morishima et
al. (2010) start with 2000 fully interacting planetesimals
and since embryos form self-consistently out of the plan-
etesimals in these simulations, problems with how to seed
embryos are completely avoided. To constrain the simula-
tions, Morishima et al. (2010) were interested in the tim-
ing of the Moon-forming impact. To identify potentially
events, they searched for a total mass of the impactor and
the target > 0.5m⊕, a impactor mass > 0.05m⊕and a im-
pact angular momentum > LEM. They found almost 100
suitable impacts in the 64 simulations. Since their sample
also includes high velocity or grazing impacts, although
constrained through the more general angular momentum
limit, and does not take into account collision history and
tidal evolution, the difference to our result is not surpris-
ing.
Life on planets without a massive stabilizing moon
would face sudden and drastic changes in climate, posing
a survival challenge that has not existed for life on Earth.
Our simulations show that Earth-like planets are common
in the habitable zone, but planets with massive, obliquity
stabilizing moons do occur only in 10% of these.
Acknowledgments
We thank David O’Brien and an anonymous reviewer
for many helpful comments. We thank University of Zurich
for the financial support. We thank Doug Potter for sup-
porting the computations made on zBox at University of
Zurich.
References
Atobe, K., Ida, S., Ito, T., 2004. Obliquity variations of terrestrial
planets in habitable zones. Icarus. 168, 223-236
Atobe, K., Ida, S., 2006. Obliquity evolution of extrasolar terrestrial
planets. Icarus. 188, 1-17
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0.01.0
mp?m?
1.5 2.0
0.00
0.01
0.02
0.03
0.04
msat?m?
Figure 7: The masses of the final outcomes of the planets for which
we identified satellite forming collisions. msatelliteis the mass of the
satellite, assuming an accretion efficiency of 50%, and mpis the mass
of the planet after the complete accretion. The circle indicates the
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