Great Migrations

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AST3020 _at_ UofT. Lecture L5 - Scenarios of Planet
formation 0. Student presentation by Libby 1.
Top-down (GGP hypothesis) and its
difficulties 2. Standard (core-accretion)
scenario ( a ) from dust to planetesimals - two
ways ( b ) from planetesimals to planetary
cores how many planetesimals were
there?( c ) Safronov number, runaway growth,
oligarchic growth of protoplanets
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Or some other way??
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Note the standard scenario on the left also
looks like the r.h.s. pictures. With one major
difference time of formation of giant
protoplanets 3-10 Myr (left panel) 0.1 Myr
(right panel)
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There are two main possible modes of formation of
giant gaseous planets and exoplanets
bottom-up, or accumulation scenario for rocky
cores (a.k.a. standard theory) predicts
formation time (3-10) Myr
(V.Safronov, G.Kuiper,
A.Cameron) _at_ top-down, by accretion disk breakup
as a result of gravitational instability of the
disk. A.k.a. GGP Giant Gaseous Protoplanets
formation time (I.Kant, G.Kuiper, A.Cameron) To understand
the perceived need for _at_, we have to
consider disk evolution and observed time scales
cores may not be ready! To understand the
physics of _at_, we need to study the stability of
disks against self-gravity waves.
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Gravitational Instability and the Giant
Gaseous Protoplanet hypothesis
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Stability vs. fragmentation of disks
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Self-gravity as a destabilizing force for the
epicyclic oscillations (radial excursions) of
gas parcels on slightly elliptic orbits
To study waves in disks, we substitute into the
equations of hydrodynamics the wave in a WKBJ
(a.k.a. WKB) approximation, also used in quantum
mechanics it assumes that waves are sinusoidal,
tightly wrapped, or that kr 1. All quantities
describing the flow of gas in a disk, such as
the density and velocity components, are
Fourier-analyzed as
,
Some history WKB applied to
Schrödinger equation (1925)
Gregor Wentzel (1898-1978) German/American
physicist Hendrick A. Kramers (18941952)
Dutch physicist
1926 Léon N. Brillouin (1889-1969) French
physicist Harold Jeffreys (1891 1989) English
mathematician, geophysicist, and astronomer,
established a general method of approximation of
ODEs in 1923
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An example of a crest of a spiral wave X1
exp for k const 0 , m 2,
const.
This spiral pattern has a constant shape and
rotates with an angular pattern speed equal to
The argument of the exponential function, is
constant on a spiral wavecrest
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Dispersion Relation for non-axisymmetric waves in
disks tight-winding (WKB) local approximation
Doppler-shifted frequency
epicyclic frequency
self- gas gravity pressure
In Keplerian disks, i.e.disks around point-mass
objects
- for derivation see Binney and Tremaines book
(1990) Galactic Dynamics
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As elsewhere in physics, the dispersion relation
is the dependence between the time- and spatial
frequencies, Though it looks much more
frightening than the one describing the simple
harmonic (sinusoidal) plane wave of sound in the
air you can easily convince yourself that
in the limit of vanishing constant G (no
self-gravity in low-mass disks!) and vanishing
epicyclic frequency (no rotation!), the full
dispersion relation assumes the above form.
Therefore, the waves in a non-rotating medium w/o
gravity are simply pure pressure (sound) waves.
The complications due to the differential
rotation lead to a spiral shape of the sound- or
the fully self-gravitating density wave.
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Dispersion Relation in disks with axisymmetric
(m0) waves
(1960,1964)
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Gravitational stability requirements
Local stability of disk, spiral waves may grow
Local linear instability of waves, clumps
form, but their further evolution depends on
equation of state of the gas.
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Question Do we have to worry about self-gravity
and instability?
Ans No
z/r0.1 qd0.1
Ans Yes
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Recently, Alan Boss revived the half-abandoned
idea of disk fragmentation
Clumps forming in a gravitationally unstable
disk (Q GGPs?
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It turns out that even at Q1.5 there are
unstable global modes.
Disk in this SPH simulation initially had Q 1.5
1 The m-armed global spiral modes of the
form grow and compete with each other. But
the waves in a stable Q2 disk stop growing and
do not form small objects (GGPs).
From Laughlin Bodenheimer (2001)
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Two examples of formally unstable disks not
willing to form objects immediately Durisen et
al. (2003)
Break-up of the disk depends on the equation of
state of the gas, and the treatment of boundary
conditions.
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Armitage and Rice (2003)
Simulations of self-gravitating objects forming
in the disk (with grid-based hydrodynamics)
shows that rapid thermal cooling is crucial
Disk not allowed to cool rapidly (cooling
timescale 1 P)
Disk allowed to cool rapidly (on dynamical
timescale, 18
Mayer, Quinn, Wadsley, Stadel (2003)

SPH Smoothed Particle Hydrodynamics with 1
million particles
Isothermal (infinitely rapid cooling)
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GGP (Giant Gaseous Protoplanet) hypothesis disk
fragmentation scenario (A. Cameron in the
1970s) Main Advantages forms giant planets
quickly, avoids possible timescale paradox
planets tend to form at large distances amenable
to imaging. MAIN DIFFICULTIES 1.
Non-axisymmetric and/or non-local spiral modes
start developing not only at Qwhen Q decreases to Q1.52 They redistribute
mass and heat the disk increase Q (stabilize
disk). 2. Empirically, this self-regulation of
the effects of gravity on disk is seen in disk
galaxies, all of which have Q2 and yet do not
split into many baby gallaxies. 3. The only way
to force the disk fragmentation is to lower
Qc/Sigma by a factor of 2 in just one orbital
period. This seems impossible. 4. Any clumps in
disk (a la Boss clumps) may in fact shear and
disappear rather than form bound objects. Durisen
et al. Have found that the equation of state
and the correct treatment of boundary conditions
are crucial, but could not confirm the
fragmentation except in the isothermal E.O.S.
case. 5. GGP is difficult to apply to Uranus and
Neptune final masses Brown Dwarfs not GGPs 6.
Does not easily explain core masses of planets
and exoplanets, nor the chemical correlations
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OBSERVATIONS OF CORES IN EXOPLANETS
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Comparison of gas and rock masses (in ME)
in giant planets and exoplanets (1980s)
Planet Core mass Atmosph. Total mass
Radius _________(rocks, ME )___(gas,_ME
)____(ME )_______(RJ) _ Jupiter
0-10 313 318
1.00 Saturn 15-20 77
95 0.84 Uranus
11-13 2 - 4 14.6
0.36 Neptune 13-15
2 - 4 17.2 0.34

core
envelope (atmosphere)
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Comparison of gas and rock masses (in ME)
in giant planets and exoplanets (Oct. 2005)
Planet Core mass Atmosph. Total mass
Radius _________(rocks, ME )___(gas,_ME
)____(ME )_______(RJ) _ Jupiter
0-10 313 318
1.00 Saturn 15-20 77
95 0.84 Uranus
11-13 2 - 4 14.6
0.36 Neptune 13-15
2 - 4 17.2 0.34
HD 209458b 0 220 204-235
1.32 0.05 (disc. 1999) HD 149026b
70 45 105-124 0.73
0.03 (disc. 7/2005) HD 189733b 10-20(?)
350 351-380 1.26 0.03 (disc.
10/2005)
core
envelope (atmosphere)
?
?
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Video of density waves in a massive
protoplanetary disk The shocks at the surface are
suggested as a way to heat solids and form
chondrules, small round grains inside meteorites.
Durisen and Boss (2005)
but that is another issue.
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Standard Accumulation Scenario
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Two-stage accumulation of planets in disks
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Planetesimal solid body 1 km
Mcore10 ME(?) contraction of the atmosphere
and inflow of gas from the disk
(issues not fully addressed in the standard
theory, so far)
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Two scenarios proposed for planetesimal formation
Particles settle in a very thin sub-disk, in
which Q dust layer forms planetesimals
Particles in a turbulent gas not able to achieve
Qgas
gas
QQ1
Solid particles (dust, meteoroids)
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How many planetesimals formed in the solar nebula?
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How many planetesimals formed in the solar
nebula? Disk mass 0.02 0.1 Msun
2e33g0.024e31 g 1e32 g Dust mass (0.5 of
that) 1e30 g 100 Earth masses (Z0.02 but
some heavy elements dont condense) Planetesimal
mass, s 1 km 1e5 cm, (4pi/3) s3(1 g/cm3)
1e16 g Assuming 100
efficiency of planet formation N 1e30g /1e16g
1e14, s1 km N 1e30g /1e19g 1e12, s10
km (at least that many initially) N 1e30g
/1e22g 1e8, s100 km N 1e30g /1e25g 1e5,
s1000 km N 1e30g /1e28g 100, s10000 km
(rock/ice cores planets) N 1e30g /1e29g
10, (10 ME rock/ice cores of giant
planets)
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Gravitational focusing factor
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Oligarchs rule their vicinity but do not
interfere with each other
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This following 10 slides are a digression on
celestial mechanics
Non-perturbative methods (energy constraints,
integrals of motion, Roche Lobe, stability of
orbits)
Joseph-Louis Lagrange (1736-1823)
Karl Gustav Jacob Jacobi (1804-1851)
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Solar sail problem again
A standard trick to obtain energy integral
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Energy criterion guarantees that a particle
cannot cross the Zero Velocity Curve (or
surface), and therefore is stable in the Jacobi
sense (energetically). However, remember that
this is particular definition of stability which
allows the particle to physically collide with
the massive body or bodies -- only the escape
from the allowed region is forbidden! In our
case, substituting v0 into Jacobi constant, we
obtain
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Allowed regions of motion in solar wind (hatched)
lie within the
Zero
Velocity Curve
f0
f0.051 particle cannot escape from the planet located
at (0,0)
f0.063 (1/16)
f0.125
particle can (but doesnt always do!) escape from
the planet (cf. numerical cases B and C,
where f0.134, and 0.2, much above the limit of
f1/16).


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Circular Restricted 3-Body Problem (R3B)
L4
L1
L3
L2
Joseph-Louis Lagrange (1736-1813) born Giuseppe
Lodovico Lagrangia
L5
Restricted because the gravity of particle
moving around the two massive bodies is neglected
(so its a 2-Body problem plus 1 massless
particle, not shown in the figure.)
Furthermore, a circular motion of two massive
bodies is assumed. General 3-body problem has no
known closed-form (analytical) solution.
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NOTES The derivation of energy (Jacobi) integral
in R3B does not differ significantly from the
analogous derivation of energy conservation law
in the inertial frame, e.g., we also form the dot
product of the equations of motion with velocity
and convert the l.h.s. to full time derivative
of specific kinetic energy. On the r.h.s.,
however, we now have two additional accelerations
(Coriolis and centrifugal terms) due to frame
rotation (non-inertial, accelerated frame).
However, the dot product of velocity and
the Coriolis term, itself a vector perpendicular
to velocity, vanishes. The centrifugal term can
be written as a gradient of a centrifugal
potential -(1/2)n2 r2, which added to the
usual sum of -1/r gravitational potentials of
two bodies, forms an effective potential
Phi_eff. Notice that, for historical reasons, the
effective R3B potential is defined as positive,
that is, Phi_eff is the sum of two 1/r terms
and (n2/2)r2
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R3B
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Effective potential in R3B
mass ratio 0.2
The effective potential of R3B is defined as
negative of the usual Jacobi energy integral.
The gravitational potential wells around the two
bodies thus appear as chimneys.
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Lagrange points L1L5 are equilibrium points in
the circular R3B problem, which is formulated in
the frame corotating with the binary system.
Acceleration and velocity both equal 0 there.
They are found at zero-gradient points of the
effective potential of R3B. Two of them are
triangular points (extrema of potential). Three
co-linear Lagrange points are saddle points of
potential.
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Jacobi integral and the topology of Zero Velocity
Curves in R3B
rL Roche lobe radius Lagrange points
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Sequence of allowed regions of motion (hatched)
for particles starting with different C values
(essentially, Jacobi constant energy in
corotating frame)
High C (e.g., particle starts close to one of
the massive bodies)
Highest C
Low C (for instance, due to high init.
velocity) Notice a curious fact regions near
L4 L5 are forbidden. These are potential
maxima (taking a physical, negative gravity
potential sign)
Medium C
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0.1
C R3B Jacobi constant with v0
Édouard Roche (18201883),
Roche lobes
terminology
Roche lobe Hill
sphere sphere of
influence (not
really a sphere)
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Is the motion around Lagrange points stable?
Stability of motion near L-points can be studied
in the 1st order perturbation theory
(with unperturbed motion being state
of rest at equilibrium point).
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Stability of Lagrange points Although the L1,
L2, and L3 points are nominally unstable, it
turns out that it is possible to find stable and
nearly-stable periodic orbits around these points
in the R3B problem. They are used in the
Sun-Earth and Earth-Moon systems for space
missions parked in the vicinity of these
L-points. By contrast, despite being the maxima
of effective potential, L4 and L5 are stable
equilibria, provided M1/M2 is 24.96 (as in
Sun-Earth, Sun-Jupiter, and Earth-Moon
cases). When a body at these points is perturbed,
it moves away from the point, but the Coriolis
force then bends the trajectory into a stable
orbit around the point.
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Observational proof of the stability of
triangular equilibrium points
Greeks, L4
Trojans, L5
From Solar System Dynamics, C.D. Murray and
S.F.Dermott, CUP
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Roche lobe radius depends weakly on R3B mass
parameter
0.1
0.01
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Computation of Roche lobe radius from R3B
equations of motion ( , a
semi-major axis of the binary)




L
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Roche lobe radius depends weakly on R3B mass
parameter
m2/M 0.01 (Earth Moon) r_L 0.15 a m2/M
0.003 (Sun- 3xJupiter) r_L 0.10 a m2/M
0.001 (Sun-Jupiter) r_L 0.07 a m2/M
0.000003 (Sun-Earth) r_L 0.01 a
0.1
0.01
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Hill problem
George W. Hill (1838-1914) - studied the small
mass ratio limit of in the R3B, now called the
Hill problem. He straightened the azimuthal
coordinate by replacing it with a local Cartesian
coordinate y, and replaced r with x. L1 and L2
points became equidistant from the planet. Other
L points actually disappeared, but thats
natural, since they are not local (Hills
equations are simpler than R3B ones, but are good
approximations to R3B only locally!) Roche lobe
Hill sphere sphere of influence (not really
a sphere, though)
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Hill problem
Hill applied his equations to the Sun-Earth-Moon
problem, showing that the Moons Jacobi constant
C3.0012 is larger than CL3.0009 (value
of effective potential at the L-point), which
means that its Zero Velocity Surface lies inside
its Hill sphere and no escape from the Earth is
possible the Moon is Hill-stable. However, this
is not a strict proof of Moons eternal stability
because (1) circular orbit of the Earth was
assumed (crucial for constancy of Jacobis C) (2)
Moon was approximated as a massless body, like in
R3B. (3) Energy constraints can never exclude the
possibility of Moon-Earth collision
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How wide a region is destabilized by a planet?
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Hill stability of circumstellar motion near the
planet
C
CL
The gravitational influence of a small body (a
planet around a star, for instance) dominates
the motion inside its Roche lobe, so particle
orbits there are circling around the planet, not
the star. The circumstellar orbits in
the vicinity of the planets orbit are affected,
too. Bodies on disk orbits (meaning the disk of
bodies circling around the star) have Jacobi
constants C depending on the orbital separation
parameter x (r-a)/a (rinitial circular
orbit radius far from the planet, a planets
orbital radius). If x is large enough, the
disk orbits are forbidden from approaching L1 and
L2 and entering the Roche lobe by the energy
constraint. Their effective energy is not enough
to pass through the saddle point of the
effective potential. Therefore, disk regions
farther away than some minimum separation x
(assuming circular initial orbits) are guaranteed
to be Hill-stable, which means they are
isolated from the planet.
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Hill stability of circumstellar motion near
the planet
C
CL
On a circular orbit with x (r-a)/a, At the
L1 and L2 points Therefore, the Hill stability
criterion C(x)CL reads or Example What is
the extent of Hill-unstable region around
Jupiter? Since Jupiter is at a5.2 AU, the
outermost Hill-stable circular orbit is at
r a - xa a - 0.24a 3.95 AU. Asteroid belt
objects are indeed found at r group at 4 AU is the outermost large group of
asteroids except for Trojan and Greek
asteroids)
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Back to the formation scenario Isolation,
Giant Impacts
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Stopping the runaway growth of planetary
cores Roche lobe radius
grows non-linearly with the mass of the
planet, slowing down the growth as the mass
(ratio) increases. The Roche lobe radius rL is
connected with the size of the Hill-stable disk
region via a factor 2sqrt(3), which, like the
size of rL, we derived already in this
lecture. This will allow us to perform a thought
experiment and compute the maximum mass to which
a planet grows spontaneously by destabilizing
further regions.
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Isolation mass in different parts of the Minimum
Solar Nebula
Based on Minimum Solar Nebula (Hayashi nebula)
a disk of just enough gas to contain the same
amount of condensable dust as the current
planets total mass 0.02 Msun, mass within 5AU
0.002 Msun Conclusions (1) the inner outer
Solar System are different critical
core10ME could only be achieved in the outer
sol.sys. (2) there was an epoch of giant impacts
onto protoplanets when all those semi-isolated
oligarchs where colliding.