Download 13 Formation

Chapter
13
Formation
The question of how the solar system formed is the most difficult question in planetary
science. This is because of the lack of data and the vast physical and chemical complexities
of the problem. However, there are certain key parameters that we do know. From the
study of meteorites we know the age of the solar system and a good deal about its initial
composition (Chapter 2). We also know much about the nature of the present-day solar
system. And we have observations of both old and young stars that inform us about the
life cycle of the sun. The goal is to use all the information in hand in combination with the
laws of physics and chemistry to fill in the blanks between the initial and the present-day
state of the solar system.
13.1 Before the Solar System: Molecular Clouds
13.1.1 Gravitational Collapse and the Jeans Criterion
Average (low-mass) stars like the sun are believed to have formed by the collapse of a
dense (by comparison to the instellar medium) molecular cloud. One basis for this belief
is that present-day young stars in the Milky Way are commonly found near the galactic
plane, within or in the vicinity of clouds of cold molecular gas. Many of the clouds that
13–1
have been observed appear to be gravitationally bound. Formation of a star from a cloud of
gas requires that the force of gravity exceed dispersive forces due to thermal gas pressure,
magnetic fields and turbulent motions. The absorption of external radiation from other
stars by dust may contribute significantly to keeping star-forming clouds cold enough to be
thermally-disrupted. It is also possible that dust absorbs ionizing radiation, enabling gas
to diffuse along magnetic field lines and thus reducing magnetic forces that also counteract
gravity. An example of a nearby star-forming region is a region in Taurus, with a mass
about 106 that of the sun and a size contains clouds of order 10× more massive than that
of the sun. Another is Orion, with a mass of 105 times that of the sun, ≈ 100 pc across
that contains very dense star-forming regions.
If the central regions of molecular clouds are gravitationally bound, as must be the
case for star formation to occur, and if thermal pressure is the principal force that opposes
gravity (i.e. no major magnetic fields or turbulent motions), then the relationship between
the mass Mc and radius Rc of the central region of the cloud can be described by
kT
GMc
≈ c2sound =
Rc
µmH
(13.1)
where G is the Universal constant of gravitation (=6.6732 × 10−11 N m2 kg −2 ), csound is the
speed of sound, k (=1.38 × 10−23 JK −1 ) is Boltzmann’s constant, T is temperature, mH is
the mass of the hydrogen atom (=1.6717 × 10−27 kg), and µ is the mean molecular weight.
Equation (13.1) is a statement of the Jeans criterion, which indicates that collapse will
occur if the gravitational potential energy of a gas cloud exceeds the internal energy. For
µ = 2.3 (a reasonable value for molecular hydrogen plus helium) and T = 10 K, a 1-solarmass molecular cloud should have a radius Rc ≈ 0.1 pc. Clouds with these approximate
characteristics are in fact observed, providing some hope that our understanding of the
physics is on the right track.
The properties of dense molecular clouds have been studied principally using cm- and
mm-wavelength spectral lines. The most abundant species, H2 , is difficult to detect unless
heated by shock or UV radiation. So molecules such as CO and N H3 are used instead.
Observations show typical hydrogen number densities of 2 × 103 to 2 × 105 cm−3 , masses
typically 10× that of the sun, an average size of order 0.1 pc, and velocity dispersions
of of 0.2 − 0.9 km s−1 . How well density can be constrained depends on the molecular
tracer, as different ones are sensitive to different temperatures and densities. The magnetic
properties of molecular clouds are difficult to measure.
13.1.2 Fragmentation and Jeans Mass
Clearly something must happen to cause a molecular cloud to form a dense core or
cores that collapse by self-gravitation and undergo star formation. A reasonable possibility
is that gravitational instabilities play a role. The Jeans mass is a concept that is often
used to provide a quantitative framework for gravitational condensation, even though its
premise of a constant background density generally does not provide a solution consistent
with hydrostatic equilibrium.
We assume that velocity (u) and density (ρ) can be represented by a mean state plus
a small perturbation, such that u = uo + δu and ρ = ρo + δρ. We take the equations of
13–2
mass and momentum conservation and linearize for the case of no magnetic pressure
∂
1
δu = − ∇c2sound δρ − ∇Φ
∂t
ρo
(13.2)
∂
δρ + δu · ∇ρo = −ρo ∇ · δu
∂t
(13.3)
where Φ is the gravitational potential and it is assumed that uo = 0 and that the equilibrium
state is
c2sound
∇ρo = −∇Φo .
(13.4)
ρo
Now the problem becomes apparent. It is appropriate to invoke Poisson’s equation (discussed in Chapter 9) to relate gravitational potential to density
∇2 Φ = 4πG ρ ,
(13.5)
but we know that ∇2 Φ = ∇ · ∇Φ and g = −∇Φ, so
∇ · g = −4π G ρ .
(13.6)
In a homoegenous, isotropic Newtonian universe, the LHS of (13.6) must vanish, but that
introduces an inconsistency if we want the RHS to give a non-zero mass density for the
medium from which molecular clouds condense. Jeans got around this problem by ignoring
zeroth order equations and dealing instead with first order solutions, a course of action
known as the Jeans swindle.
Relativity solves this problem by allowing the universe to expand. But for now we
continue with the simpler approach. We can combine (13.2) through (13.5) to yield
∂ 2 δρ
= −ρo
∂t2
µ
¶
c2sound 2
∇ δρ − 4π G ρ
ρo
(13.7)
where t is time. From (13.7) it follows that
δρ = δρo exp[i(ωt − kx)],
(13.8)
ω 2 = c2sound (k 2 − kJ2 ), kJ2 = 4π G ρo /c2sound .
(13.9)
where
When k < kJ , ω is imaginary. Thus (13.7) has both exponentially-increasing and -decaying
solutions. The exponentially-increasing solutions correspond to gravitational instability,
in which density growth is unbounded. It is common practice to define the Jeans mass
µ
MJ =
λ3J ρo
=
2π
kJ
¶3
µ
ρo =
13–3
πc2sound
G
¶3/2
ρo−1/2 .
(13.10)
For a given temperature and density, masses greater than the Jeans mass will be unstable
with respect to gravitational collapse. Here λJ is the Jeans length,
µ
λJ =
πc2sound
Gρo
¶1/2
,
(13.11)
which represents the length scale for fragmentation. But recall that this calcuation assumes a background state that does not correspond to an equilibrium state. A more
realistic solution, such as for an isothermal thin sheet, would yield a smaller length scale
for fragmentation.
A gas molecule in a cloud with a size defined by the Jeans length will not be energetic
enough to escape. As collapse progresses the density increases, as do the number of molecular collisions. The effectiveness with which the cloud radiates contributes to the extent
and velocity of collapse. In the case of a cloud that is transparent to infrared radiation the
thermal state of molecules is controlled by heat exchange with surrounding space, which
is at an approximately constant temperature. Collapse will be nearly isothermal until the
opacity increases considerably. Hydrogen gas must be very dense before it becomes opaque
so for early phases of hydrogen cloud collapse the cloud radiates away most of the collapse
energy.
13.1.3 Free-fall Collapse
A molecular cloud would be expected to become gravitationally unstable because
internal pressures do not balance gravity. The limiting case for a cloud with no internal
pressure corresponds to free-fall. The equation of motion of a shell of material that starts
at radius ro is
GMr
4πGρo ro3
d2 r
=
−
=
−
,
(13.12)
dt2
r2
3r2
where ρo is the initial density and Mr is the mass interior to radius r. Here we invoke the
result for a spherical mass configuration that only the mass interior to the point of interest
has any gravitational effect. Multiplying through by dr/dt, integrating, and making the
substitution r/ro = cos2 β we find a solution
1
β + sin2β =
2
µ
8πGρo
3
¶1/2
tc
(13.13)
where tc is the time from the beginning of collapse (r = ro ). At a given tc , β is fixed no
matter what the original starting radius ro was. Therefore shells of material within the
sphere do not cross, but all reach the center at the same time, the free-fall time
µ
tf f =
3π
32Gρo
¶1/2
≈
3.4 × 107
1/2
NH2
yr,
(13.14)
where NH2 is the number density of molecular hydrogen. If the 1-solar-mass cloud discussed
previously had internal gas pressure ’turned off’, the free-fall time would be ≈ 5×105 years.
13–4
But even with pressure forces counteracting gravitational free fall, it is apparent that the
collapse of a molecular cloud to a protostar will be rapid.
13.2 Rotation and Angular Momentum
13.2.1 Rotation
All the planets revolve in the same direction around the sun, and in practically the
same plane. For the most part they also rotate in the same direction about their own
axes, although there are notable exceptions, such as Venus. As we have discussed, the
gravitational collapse of molecular clouds is widely believed to lead to star formation,
and it is likely that our solar system condensed from a collapsed, rotating cloud of gas
and dust. Rotating disks of material are ubiquitous in space, occurring all the way from
planetary to galactic scales. A rotating disk is the signature of a self-gravitating system
that has contracted in radius and amplified its angular velocity in order to preserve its
total angular momentum. In a rotating protostar the gravitational attraction everywhere
will be towards the center of mass. But the centrifugal force will be directed normal to the
axis of rotation. The resolved force vector will move gas and dust nearer to the median
plane as the cloud contracts. This process leads to the disk shape, which dissipates energy
and minimizes collisions. Planetary ring systems like Saturn’s provide a natural testing
ground for theories of rotating disks.
13.2.1 Angular Momentum
One of the more interesting boundary conditions is the present distribution of angular
momentum. Consider a planet of mass m that orbits a central body of mass M , whose
position with respect to the central body can be described by a vector r.
The orbital angular momentum (L) of the planet can be written
L = mr2 ω = mr2
dθ
,
dt
(13.15)
where r is distance, m is the mass, ω is the angular velocity (=dθ/dt), and θ is the angle
with respect to a fixed direction in the orbit plane. It can be shown that
r2
dθ
L
dS
=
=2 ,
dt
m
dt
where S is the area swept out by r. Then
L
dS
=
,
dt
2m
(13.16)
which is a statement of Kepler’s Second Law of Motion: the line between a planet and
the sun sweeps out equal areas in equal periods of time. Equation (13.16) is a statement
13–5
of conservation of angular momentum. The total planetary energy (E), which is the
sum of the kinetic and potential contributions,
E=
1 2 GM
v +
= constant
2
r
where
µ
2
v =
dr
dt
¶2
µ
dθ
+ r
dt
(13.17)
¶2
is the planetary velocity, is also conserved. It is possible to rewrite (13.15) in terms of the
mass of the sun and doing so yields
L = mr2 ω = (GMs )1/2 mr1/2 .
(13.18)
By integrating (13.18) over all the planets we find that while the sun contains 99.8%
of the mass of the solar system, it has only about 1% of the angular momentum. About
60% of the angular momentum of the solar system is associated with the orbit of Jupiter
alone. Most models suggest that the protosun was rotating more rapidly than at present.
Helioseismological results show that deeper parts of the sun rotate faster than the surface.
The deep solar interior, which has not yet been probed, may hold the record of that
body’s relic rotation. Solar system evolution models must show how the protosun’s angular
momentum gets transported outward. Most models invoke magnetic and gravitational
torques that spin down the sun and spin up the planets. Magnetizations of meteorites are
consistent with this idea. The transfer of angular momentum could have contributed to
the chemical fractionation of the solar system, since an outwardly migrating magnetic field
would affect the ionized plasma but not condensed particles, which couple to the field only
by viscous drag. Thus higher temperature condensates would remain in the inner part of
the solar system and more volatile constituents would be transferred outward. In fact this
is observed.
13.3 Stellar Evolution
13.1.1 The Main Sequence
One method for sorting out the different kinds of stars is to make a HertzsprungRussell (H-R) diagram, which is a plot of absolute magnitude or luminosity versus
effective (blackbody) temperature. It is traditional to plot effective temperature from high
to low on the abscissa, and luminosity from dim to bright along the ordinate. For two
stars with the same effective temperature, more light will come from the larger star than
the smaller; hence the largest stars are at the top of an H-R diagram.
As each star proceeds through its life cycle it moves around on the H-R diagram. While
we can’t observe the entire life cycle of a single star, we can search through the current
“snapshot” of our galaxy and find stars at all stages of evolution. Making an ensemble
H-R plot reveals that many stars fall along a single line called the main sequence. Stars
on the main sequence are in a relatively steady state of hydrogen burning in their cores,
13–6
as is the present sun. An average G − type (yellow) star like our sun is thought to have a
lifetime (i.e. a residence time on the main sequence) of about 10 billion years.
13.3.2 T Tauri and FU Orionis Stars
The conspicuous absence of gas between the planets in the solar system must be
explained in any model of solar system formation. Before a new star reaches the main
sequence it goes through a pre-main sequence evolution of gravitational collapse from a
protostellar nebula. Our best information about this stage comes from studying a class of
young stars called T Tauri stars. T Tauri stars are thought to be still contracting and
evolving, and are believed to be less than one million years old. They are typically 0.2 to
2 solar masses in size, and they show evidence of strong magnetic activity. Some T Tauri
stars have spectra that include forbidden lines, which occur in low-density gas and are the
signature of a gaseous nebula. Rapid fluctuations in ultraviolet and x-ray emissions are
common. They also tend to show strong infrared emission and have spectra with silicon
lines indicating that they are surrounded by dust clouds.
T Tauri stars are associated with strong solar winds and high luminosities. It is
thought that our sun probably passed through a T Tauri stage in its early evolution, and
that the volatile elements in the inner solar system were blown away during this stage.
Some low-luminosity pre-main sequence stars have been observed to brighten significantly and expell shells of material in a time period less than a year, a phenomenon called
an FU Orionis outburst. In at least one case a faint T Tauri star exhibited such an outburst. It has been suggested that most young stars may go through FU Orionis outbursts.
The leading explanation is that the rate of mass accretion from a young star’s circumstellar
disk temporarily increases and causes the flare up. If our Sun had one or more FU Orionis
outbursts in its early history the volatile material in the inner solar system would have
been strongly affected.
13.4 Planetary Formation
13.4.1 Condensation and Cooling
The most widely accepted cosmogonical (formation) theory is that of V. Safronov,
who was the first to hypothesize that the solar system initially accreted from a nebular
cloud that evolved from a sphere to a disk. While details of solar system formation models
differ, a common premise is that the planets formed from particle growth in an initially
tenuous dust-gas nebula. The mechanism to trigger the initial collapse of the nebula has
been argued and hypotheses range from uniform gravitational collapse, to galactic spiral
density waves, to catastrophic suggestions such as a supernova in the solar neighborhood. A
supernova, though a low probability event, is supported by the discovery of micro-diamonds
in cosmic dust. These imply that the solar system environs achieved high pressures due to
passage of severe shock waves that would accompany only an event of this intensity. The
problem with the supernova hypothesis is that it would imply that solar system formation
is not a common phenomenon, which runs contrary to current thought.
13–7
There are a number of scenarios for planetary growth. It is possible that the planets
accumulated from small moon-sized bodies, called planetesimals, by infrequent encounters. Or instead accumulation may have occurred from groups of bodies that collectively
became gravitationally unstable. It is not clear whether planetary accumulation occurred
in a gaseous or gas-free environment. In a gaseous nebula temperatures tend to be homogeneous, but as gas clears due to the solar wind and condensation into dust grains the
opacity of the nebula decreases significantly. During this time the system establishes a
large temperature gradient.
It is generally accepted that the planets accreted from a nebula with a composition
similar to that of the sun, i.e., made mostly of hydrogen. The slowly-rotating nebula
had a pressure and temperature distribution that decreased radially outward. The density
of the nebula was probably not very great. Model estimates of typical pressures in the
vicinity of Earth’s orbit generally fall in the range of 10-100 Pa but these are not very
well constrained. The disk must have cooled primarily by radiation, condensing out dust
particles that were initially of composed of refractory elements. These high temperature
condensates first appear at temperatures of 1600◦ − 1750◦ K and consist of silicates oxides
and titanates of calcium and aluminum, such as Al2 O3 , CaTiO3 , and Ca2 Al2 Si2 O7 and
refractory metals such as those in the platinum group. These minerals are found in white
inclusions in the most primitive class of meteorites, the Type III carbonaceous chondrites,
discussed in Chapter 2. Metallic iron condenses out next, followed by the common silicate
materials forsterite (an olivine) and enstatite (a pyroxene). Iron sufide (troilite; FeS)
and hydrous minerals condense at temperatures of 700◦ -800◦ K. Volatile materials, most
notably H2 O and CO2 condense out at 300◦ -400◦ K. Planets that contain these substances
were accumulated from material that condensed in this temperature range, which provides
some clue about the early thermal structure of the solar nebula.
Time scales for the condensation of gas to dust, of accumulation of dust to planetesimals, and of accretion of planetesimals to planets and moons are also not well constrained.
If cooling occurred slowly in comparison to other processes then planets would have formed
during the cooling process and could have accreted inhomogeneously. If instead cooling occurred rapidly, then the planets would have formed from cold, generally homogeneous material. Homogeneous accretion models are favored, with planetary differentiation thought
to be mostly accomplished in the early stages after accretion.
13.4.2 Accretion
The process or processes that were responsible for the accumulation of dust and small
particles into planetesimals is a matter of debate. Sticking mechanisms such as electrostatic attraction and vacuum welding have been suggested. But as material accumulates,
more planetesimal surface area is available for adding more material so the process accelerates. When planetesimals reach sizes of order 102 km gravitational attraction begins to
dominate and accretion becomes dominated by that force. In the planetesimal accretion
stage collisional velocities are a key consideration. If relative velocities between planetesimals are too low, then planetesimals will fall into nearly concentric orbits. Collisions will
be low probability events and planets will not grow. Whereas if relative velocities between
planetesimals are too high, fragmentation rather than accumulation will occur, and again
13–8
planets won’t grow. Safronov used scaling arguments concerning energy dissipation during
collisions and an assumed size distribution of planetesimals to suggest that the mutual
gravitation causes relative velocities to be somewhat less than the escape velocities of the
largest bodies. By his estimation the system should regulate itself in a way to favor the
growth of large planetesimals. If this idea holds in a general sense, then solar systems
should form with a relatively small number of large planetary bodies rather than with
many small bodies. Monte carlo simulations bear this idea out.
As planetesimals accrete into moons and planets a significant amount of energy must
be released, much of which will be converted to heat. Various theories place the time for
the accretion of Earth from 105 -108 years, which was very rapid indeed in comparison to
the age of the solar system. If accretion occurred rapidly, then not much cooling could
have occurred between collisions.
To determine the amount of heating associated with accretion, it is necessary to take
an inventory of the various sources of energy in the system. These include the kinetic
energy of impacting projectiles, the potential energy of infalling material to the planetary
surface and the thermal energy. For simplicity we will begin by assuming that accretion
occurs sufficiently rapidly such that the process is adiabatic, i.e. with no heat lost from the
accreting planet’s surface. The total energy per unit mass of accreted material is simply a
sum of the change in kinetic and potential contributions:
Cp ∆T =
¢ GM
1¡ 2
v∞ − vp2 +
2
R
(13.19)
where ∆T is the temperature change, Cp is specific heat, v∞ is the absolute velocity of
2
the approaching projectile, vp is the planetary velocity, (∆v)2 = v∞
− vp2 is the relative
impact velocity, G is the universal constant of gravitation, M is the mass of the planet, R
is the planetary radius and
GM
= gR,
(13.20)
R
where g is the gravitational acceleration at the planetary surface. It is reasonable to assume
that the impact process is not perfectly efficient and that only a fraction h of the total
energy will be converted to heat. Taking this into account and substituting (13.20) we
may write
·
¸
¢
1¡ 2
2
Cp ∆T = h
(13.21)
v − vp + gR .
2 ∞
This expression provides an upper limit of the increase in temperature that could occur during accretion. In practice the potential energy term dominates (13.21). But this
expression isn’t very realistic because it doesn’t allow for cooling.
So we next consider the additional complication that heat is lost from the system by
cooling at the surface. It is possible to write a balance between the gravitational potential
energy of accretion, the heat lost by radiation, and the thermal energy associated with
heating of the body. This causes the problem to become time dependent:
ρ
£
¤
GM (r)
dr = ²σ T 4 (r) − Tb4 dt + ρCp [T (r) − Tb ] dt
r
13–9
(13.22)
where M (r) is the mass of the accumulating planet, ρ is the density of accreting material,
² is emissivity, σ is the Stefan-Boltzmann constant, Tb is the radiation equilibrium (blackbody) temperature, and t is time. In reality there will also be energy associated with latent
heats of melting and vaporization that are ignored here. Temperature increases associated
with the accretion of the the terrestrial planets from numerical solutions to (13.22) require
rapid accretion times, 103 to 104 years for Earth, to exceed the melting temperature. These
time scales are less than suggested by accretion models and would suggest that accretional
heating is not very important for Earth or the other terrestrial planets. But it is necessary
to consider in the more realistic sense the possible importance of radiation in ridding the
planet of heat. Radiative temperature loss goes as T 4 and so is highly efficient in the sense
that the planetary surface cools quickly. But if an impact site becomes buried by ejecta
from fallback or from nearby impacts, the surface would be covered. In this situation the
outer part of the planet is hotter than the interior and thermal convection is prohibited.
The only way to rid the planet of heat is to conduct it to the surface where it can be radiated away. As will be discussed later in the semester, conduction is a much less efficient
heat transport process and so accretional heat would be retained longer if that mechanism
dominated. If accretional energy is buried deeply enough to prohibit thermal radiation
from the surface, then temperature increases of order 2000◦ can be attained for planets
that accrete in times suggested by models (106 -107 years). But even if accretion did cause
the near surface of the Earth to melt the process does not explain the earliest heating of
the Earth’s deep interior, which occurred through the process of differentiation.
13.5 Planetary Differentiation
From moment of inertia information (C/M R2 in Table 9.1), we know that the terrestrial and giant planets and many of the large moons have a radially stratified internal
density structure. The implied increases in density with depth are greater than would be
associated with simple self compression due to an increase of pressure with depth. This
leaves compositional changes, and to a lesser extent phase changes, to explain the observations. If the Earth accreted cold, then there must have been a process of internal
differentiation to produce its radially stratified density structure. Differentiation from a
homogeneous initial state to a structure with a distinct core and mantle involves a change
in gravitational potential energy. The release of this energy was likely to have been an
important source of heat in some planetary bodies. It is believed that differentiation would
have occurred early in planetary evolution after a period of radioactive heating or in the
last stages of impact accretion in which the temperature required to melt iron is achieved
at shallow depth. Molten iron separates out from its silicate matrix and is denser than
its surroundings and sinks by gravitational settling. It is reasonable to assume that the
separation and sinking time is short compared to the time of heating. Also, the process is
taking place in the interior so to first order surface heat loss may be neglected.
Under these assumptions it is possible to estimate the increase in temperature associated with core formation. We may calculate the change in gravitational potential energy
associated with the instantaneous differentiation of a planet from a homogeneous state to
a final state with a core and mantle. We shall assume that the total mass in the system
remains constant. In addition, we will neglect contributions from other effects such as
13–10
phase changes, the latent heat of melting, rotational kinetic energy (due to the change
in moment of inertia), and strain energy. The gravitational potential energy (Ω) for a
spherical planet in hydrostatic equilibrium in which density is simply a function of radius
may be written
Z M
Gm
dm
(13.23)
Ω=
r
0
where m = 4/3πr3 ρ is the mass of accreting spherical body, and dm = 4πr2 ρdr. Substituting (13.20) we find
Z M
Ω=
g(r)rdm.
0
We then re-arrange once again to integrate over the radius so that
Z
R
g(r)ρ(r)r3 dr.
Ω = 4π
(13.24)
0
In practice ρ = ρ(r) is determined from an empirically-derived equation of state that
relates density to pressure (i.e. depth). Equation (13.24) must be evaluated numerically.
Now assume that the change in gravitational potential energy will be fully converted to
heat. Then
∆Ω = Cp ∆T
or
∆T =
∆Ω
.
Cp
(13.25)
Table 13.1
Temperature Increase Due to Core Formation
Planet
Core radius (km)
∆Ω (J)
Energy released
∆T (◦ K)
Mean temp increase
Earth
Venus
Mars
Mercury
Moon
3485
?
1400 − 2100
1840
< 400
1.5 × 1031
2300
≈ 2 × 1029
2 × 1029
≈< 1 × 1027
300-330
700
10
Table 13.1 shows the mean temperture increase associated with instantaneous core
formation for the terrestrial planets based on (13.24) and (13.25). Note that for the
Earth the increase in temperature is expected to have been great enough to have produced
extensive melting. So shortly after accretion the Earth would have been largely molten
and vigorously convecting in the interior as a consequence of differentiation. For Venus
13–11
the size of the core isn’t known but if it is similar to Earth (given that planet’s similar
radius and mass), then Venus also would have experienced significant early melting when
it formed its core. Melting also probably occurred on Mercury. But for Mars and the
Moon the temperature increase is not great enough for melt generation, even taking into
account the considerable uncertainties in core radii. Core formation could not have been
a significant heat source early in the evolution of these bodies.
13.6 Formation of the Moon
The origin of the Moon has been a long-debated topic. While moons around planets
are common in the solar system, Earth’s moon is unusual given its large size compared to
the primary. It has long been wondered whether ”special circumstances” were associated
with lunar origin.
Traditional models for lunar formation included co-accretion (the Moon formed near
the Earth), capture (the Moon strayed too near to Earth and became trapped in orbit),
and fission (the Moon formed by spinning off the Earth during an early rapid rotational
period). All of these models had serious problems in explaining important features like
the Moon’s bulk composition, the angular momentum of the Earth-Moon system, etc.
The theory that is currently is favored is the giant impact hypothesis, which has
gained support from numerical simulations and is consistent with the features above. In
this scenario, shortly after accretion during the terminal bombardment the Earth received a glancing impact from a Mars-sized asteroidal body. Smoothed particle hydrodynamical simulations from independent groups at Harvard and the University of Arizona
showed the same general features: the mantles of both the early Earth and the impactor
melted and vaporized and the core of the impacting body wrapped around Earth’s core.
Mantle material from Earth and the projectile that was ejected re-condensed in orbit to
form the Moon. This hypothesis is able to explain the puzzling lack of iron in the Moon.
If this event did indeed occur then the Earth would have been largely melted by the event.
Such a catastrophic occurrence must factor into scenarios for the post-accretional evolution
of the Earth.
13.7 Other Planetary Systems
The question of whether our solar system is unique or whether planets revolving
around other stars is a common occurrence has puzzled mankind for millenia. Technology
resulting in the ability to detect other planetary systems is now coming of age, and such
systems are now being detected at a rapid rate. The study of extrasolar planets is an
exciting new branch of planetary science.
13.7.1 Detecting Planets Around Other Stars
Detecting planets around other stars is not an easy proposition. These bodies shine
by reflected light and even planets larger than Jupiter will be about a billion times fainter
than a typical G−class star like the sun. Separations between the star and planet are tiny,
a handful of arcseconds even for systems in the solar neighboorhood. Such issues have until
now made direct detections problematic, and indirect methods have instead been used to
13–12
detect extrasolar planets. One method, transit photometry, measures a faint dimming
of the star as a giant planet periodically transits across the disk. Several groups are now
involved in trying to detect planets in this way.
A more successful method has been radial velocity spectroscopy, which measures
stellar wobbles due to gravitational perturbations of massive planets. The size of the
wobble yields the planetary mass and the period of the wobble yields the orbital period.
This method uses Doppler frequency changes in the wavelength of starlight representing
velocity changes in a (presumed) Keplerian orbit of only tens of meters per second! The
ability of such small perturbations has come of age due to very stable spectrometers and the
use of a control spectrum of iodine or hydrogen fluoride. Using this method the detection
of a planet of Jupiter’s mass and distance from the sun is still not possible, though a
detection of a Saturn-mass planet very close to its sun has been recently accomplished. A
Jupiter-induced wobble of our sun causes a Doppler perturbation of only 3 m sec−1 .
The amplitude of these Doppler velocity perturbations yields a lower bound on the
planetary mass rather than the actual mass. The actual mass is determined only when
the perturbation is in the direct line of sight between the star and observer. But if the
orbiting body of mass M is at an orbital inclination i then the Doppler shift is decreased
from the maximum. So there is an ambiguity of M sini in the inferred mass.
13.7.2 Characteristics of Other Planetary Systems
The number of extrasolar planets is increasing rapidly but there has been some commonality to the nature of the the systems detected. Interestingly, the detections made so
far have not been along the lines of pre-conceived notions of what a ”typical solar system”
(read: our own) should look like. The planets are ”Jupiter-like” in terms of mass, with
M sini from 0.4 to 12 Jupiters. The radial velocity method could easily detect larger planets, but hasn’t, indicating that for some reason nature likes to accrete planets with the
approximate mass of Jupiter.
Even more puzzling is the fact that about half of the dozen or so detections so far
indicate that Jupiter-mass planets revolve extremely close to the host star, within ≈ 0.25
AU. This is much closer than Mercury’s distance of 0.39 AU. It must be noted that these
”close-in” planets have large velocity perturbations and so are relatively easy to detect.
But theories and simulations of solar system formation do not predict such large planets
forming so close to the sun. In addition, many of these planets are in orbits that are much
more elliptical than those of planets in our solar system. Theoretical work subsequent to the
detections suggests that spiral density waves generated by large protoplanets in rotating
stellar nebulae could in fact pull material from circular orbits, but not enough work has
been done to understand whether such a mechanism could produce orbital eccentricites as
large as the largest observed (e ≈ 0.6). Another possibility is that a system with multiple
large planets could undergo a situation where these planets perturb each other’s orbits until
they overlap. The planets would then be forced into chaotic orbits in which some would
be scattered out of the system entirely, some might fall into the gravitational well of the
central star and the remaining one or ones would stabilize in eccentric orbits. Observations
exist of flattened dust/gas disks (e.g. Beta Pictoris) that are warped in a way that would
be consistent with a fairly inclined orbit.
13–13
Pre-extrasolar-planet-detection theories for solar system formation suggest that the
giant planets form at least several AU from the central star. This is because the region
close to the star is ”cleared out” of volatiles and one needs to be out at a distance where
water and methane (etc.) ice grains can collect and assemble. If this is the case then giant
planets must migrate in toward the star. Recent theoretical considerations by Douglas Lin
and colleagues suggest that it is reasonable to expect giant planets to move inward. For one
thing viscous dissipation in the solar nebula would extract energy from the orbit and cause
the planets to spiral inward. Another thing is that protoplanets must have resonant orbital
periods with the spiral density waves that they set up in the disk. This would cause the
protoplanets to lose angular momentum and be dragged inward. So given that hindsight
is 20 − 20 all of this makes good sense and it is not surprising for giant protoplanets to
form far from their stellar primary and move in. But then what causes them to stop and
why aren’t other planets such as Jupiter close in? One possibility is that many of these
planets didn’t stop and were consumed by their suns and planets like Jupiter formed later
in the accretional process when the solar nebula ”cleared out”. Another possibility is that
there is an exclusion zone close to the protostar, i.e. a zone 10X or so larger than the star
that is cleared by the young sun’s rapidly rotating intense magnetic field, which entrains
hot ionized gas and either flings it from near the sun or drags it into the sun along field
lines.
13.7.3 Brown Dwarfs
A question that comes up often is whether the confirmed detections are really giant
planets like Jupiter or whether they are brown dwarfs. Brown dwarfs are celestial objects
that are not massive enough to undergo fusion and shine due to internal nuclear processes.
This threshold occurs at ≈ 0.08Msun ≈ 80MJup . So the question is whether so-called
planet detection techniques are really showing us planets or just failed stars.
The difference between a brown dwarf and a giant planet is subtle and concerns how
each formed. A brown dwarf forms like a star does, by collapse of an interstellar gas cloud
before the cloud forms a protoplanetary disk. A giant planet forms from dust and gas in an
already accreting circumstellar disk. Current thought is that the brown dwarfs form the
higher end of the mass distribution between, in the range ≈ 10 − 70MJup , and the giant
planets are lower. Since mostly low mass detections have been made when higher mass
detections are possible suggests to many that it is indeed planets that are being detected,
but a much more systematic inventory must be undertaken.
13.6.3 Is Our Solar System Typical and Why Does It Matter?
In the current state of affairs there is not much of a statistical sample to work with.
But if the current trend holds up it could be the case that: (1) solar systems are extremely
common in the solar neighboorhood but (2) our solar system is not a typical example,
even though out G − type sun is a very average galactic inhabitant. The stakes in this
situation are high in that they have implications regarding the probability of life developing
in other solar systems. If eccentric giant planets are the norm and our solar system
had one that persisted through planet formation, then Earth and Mars would have been
long ago flung out of the habitable zone. The habitable zone is a concept defined by
13–14
James Kasting and colleagues and refers to the region near a star with conditions that are
favorable to the development and sustenance of present-day life on Earth (though not life
in extreme environments). The discovery and characterization of other solar systems thus
has significant implications for the origins in the broadest sense.
Problems
13-2 (a) Explain using basic physical principles why a rotating cloud of gas and dust
will flatten into a disk. (b) Do the terrestrial or giant planets contribute more to the total
angular momentum of the solar system? Show a simple calculation based on the observed
properties of the planets to support your conclusion.
13-3 Consider an impactor that strikes the accreted Moon with a relative velocity
2
(∆v)2 = v∞
− vp2 of 2 km s−1 . Assume that 50% of the impact energy is available for
heating and that the impact structure is covered with ejecta fallback, effectively preventing
heat radiation to space. Calculate the increase in temperature of the impact site. Relevant
parameters include RM oon =1738 km, gM oon =1.62 m s−2 , and Cp = 1 J kg−1 ◦ C.
References
Black, D. and M.S. Matthews, 1985, Protostars and Planets II, University of Arizona Press,
Tucson.
Hartmann, L., 1998, Accretion Processes in Star Formation, Cambridge Astrophys. Series,
32, Cambridge Univ. Press, New York, 237 pp.
Shu, F.H., 1992, The Physics of Astrophysics, Volume II, Gas Dynamics, ed. D.E. Osterbrock and J.S. Miller, University Science Books, Sausalito, 476 pp.
Weaver, H. and L. Danly, 1989, The Formation and Evolution of Planetary Systems, Cambridge University Press.
13–15