AST3020 @ 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
Or some other way??
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)
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)
@ top-down, by accretion disk breakup as a result of gravitational
instability of the disk. A.k.a. GGP = Giant Gaseous Protoplanets
formation time < 0.1 Myr
(I.Kant, G.Kuiper, A.Cameron)
To understand the perceived need for @, we have to consider
disk evolution and observed time scales: cores may not be ready!
To understand the physics of @, we need to study the
stability of disks against self-gravity waves.
Gravitational Instability
and the Giant Gaseous
Protoplanet hypothesis
Stability vs. fragmentation of disks
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
X ( r , , t ) ~ X 0  X1 ( r ) exp[i ( m   k dr   t )]
   ( k )  wave' s frequency in the inertial frame
k  ( radial ) wavenumber or vector;
Some history
,
m  number of arms
WKB applied to Schrödinger equation (1925)
Gregor Wentzel (1898-1978) German/American physicist
Hendrick A. Kramers (1894–1952) Dutch physicist
Léon N. Brillouin (1889-1969) French physicist
1926
Harold Jeffreys (1891 –1989) English mathematician, geophysicist, and
astronomer, established a general method of approximation of ODEs in 1923
This spiral
pattern has a constant
shape and rotates with
an angular pattern speed
equal to  p   / m
An example of a crest of a spiral
wave ~X1 exp[ … ]
for k = const >0 , m = 2,
 = const.

m/r
k
k
The argument of the
exponential function,
m  k r   t
is constant on a spiral wavecrest
2

pitch angle   1 :
tan   m / kr
Dispersion Relation for non-axisymmetric waves in disks*
tight-winding (WKB) local approximation
Doppler-shifted
frequency
epicyclic
frequency
selfgravity
gas
pressure
( m    ) 2   2  2 G  | k |  c 2 k 2
m  number of arms (azimuthal number )
  angular orbital speed
  frequency of the wave in the inertial frame
  epicyclic frequency ( natural radial freq . in disk )
  surface density of gas
c  gas soundspeed
k  wave vector ( length)
In Keplerian disks, i.e.disks
around point-mass objects
    angular Keplerian speed
* - for derivation see Binney and Tremaine’s book (1990) “Galactic Dynamics”
As elsewhere in physics, the dispersion relation is the
dependence between the time- and spatial frequencies,
   (k )
Though it looks much more frightening than the one describing
the simple harmonic (sinusoidal) plane wave of sound in the air:
   (k )  ck  2 c / 
  wavelength of the wave
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.
Dispersion Relation in disks with axisymmetric (m=0) waves
( m  0)
 2   2  2G | k |  c 2 k 2
 2  min   k  2  0  | kcr | G / c 2
and if we plug the above most unstable (or critical ) k ,
and take   , then the smallest  is
2
 2   2  (G) 2 / c 2
Finally ,  2  0 correspond s to the loss of stability
c
(1960,1964)
Q 
Safronov  Toomre number
G
decides about the stability : Q  1 means gravit . instability
Gravitational stability requirements
c
Q 
G
Q 1
Q 1
Safronov  Toomre number
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.
Question: Do we have to worry about self-gravity and instability?
Ans: No
z/r~0.1
qd>0.1
Ans: Yes
Recently, Alan Boss revived the half-abandoned idea of
disk fragmentation
Clumps forming in
a gravitationally
unstable disk
(Q < 1)
GGPs?
It turns out that even at Q~1.5 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
exp[i (m   k dr  t )]
grow and compete with
each other.
But the waves in a stable
Q~2 disk stop growing
and do not form small
objects (GGPs).
From: Laughlin & Bodenheimer (2001)
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.
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, <0.5 P)
Mayer, Quinn, Wadsley, Stadel (2003)
SPH =
Smoothed
Particle
Hydrodynamics
with 1 million
particles
Isothermal
(infinitely
rapid cooling)
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 Q<1 but already when Q decreases to Q~1.5…2
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 Q~2 and yet do not split into many baby gallaxies.
3. The only way to force the disk fragmentation is to lower Q~c/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
OBSERVATIONS OF CORES IN EXOPLANETS
Comparison of gas and rock masses (in ME)
in giant planets and exoplanets (1980s)
envelope
core
(atmosphere)
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
Comparison of gas and rock masses (in ME)
in giant planets and exoplanets (Oct. 2005)
envelope
core
(atmosphere)
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
~0
~220
204-235
1.32 ± 0.05
~45
105-124
0.73 ± 0.03
~350
351-380
1.26 ± 0.03
HD 209458b
?
(disc. 1999)
HD 149026b
~70
(disc. 7/2005)
HD 189733b ~10-20(?)
(disc. 10/2005)
?
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.
Accumulation
Scenario
Two-stage accumulation of planets in disks
Planetesimal = solid
body >1 km
Mcore=10 ME(?) =>
contraction of the
atmosphere and inflow
of gas from the disk
(issues not fully addressed
in the standard theory,
so far)
Two scenarios proposed for planetesimal formation
Particles settle in a very thin
sub-disk, in which Q<1, then
gravitational instability in
dust layer forms planetesimals
gas
Q<1
Solid particles
(dust, meteoroids)
Particles in a turbulent gas
not able to achieve Q<1,
stick together via non-gravity
forces.
gas
Q>1
How many planetesimals formed in the solar nebula?
How many planetesimals formed in the solar nebula?
Disk mass ~ 0.02 … 0.1 Msun > 2e33g*0.02=4e31 g ~ 1e32 g
Dust mass ~ (0.5% of that) ~ 1e30 g ~ 100 Earth masses
(Z=0.02 but some heavy elements don’t condense)
Planetesimal mass, s =1 km = 1e5 cm, ~(4*pi/3) s^3*(1 g/cm^3)
=> 1e16 g
Assuming 100% efficiency of planet formation
N = 1e30g /1e16g = 1e14, s=1 km
N = 1e30g /1e19g = 1e12, s=10 km (at least that many initially)
N = 1e30g /1e22g = 1e8, s=100 km
N = 1e30g /1e25g = 1e5, s=1000 km
N = 1e30g /1e28g = 100, s=10000 km (rock/ice cores & planets)
N = 1e30g /1e29g = 10,
(~10 ME rock/ice cores of giant planets)
Gravitational focusing
factor
Oligarchs rule their vicinity but do not interfere with each other
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)
Solar sail problem again
A standard trick
to obtain energy
integral
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 v=0 into Jacobi constant,
we obtain:
Allowed regions of motion in solar wind (hatched) lie within the
f=0
f=0.051 < (1/16) Zero Velocity Curve
particle cannot
escape from the
planet located at
(0,0)
f=0.063 > (1/16)
f=0.125
particle can (but
doesn’t always do!)
escape from the
planet
(cf. numerical
cases B and C, where
f=0.134, and 0.2, much
above the limit of f=1/16).
Circular Restricted 3-Body Problem (R3B)
L4
L3
Joseph-Louis Lagrange (1736-1813)
[born: Giuseppe Lodovico Lagrangia]
L1
L2
L5
“Restricted” because the gravity of particle moving around the
two massive bodies is neglected (so it’s 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.
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)n^2 r^2, 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 +(n^2/2)r^2
R3B =
n
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.
Lagrange points L1…L5 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.
Jacobi integral and the topology of Zero Velocity Curves in R3B
  m1 /( m1  m2 )
rL = Roche lobe radius
+ Lagrange points
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
Medium 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)
  m1 /( m1  m2 )
= 0.1
C = R3B Jacobi constant with v=0
Édouard Roche (1820–1883),
Roche lobes
terminology:
Roche lobe ~
Hill sphere ~
sphere of influence
(not really a sphere)
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).
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.
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
Roche lobe radius depends weakly on R3B mass parameter
  m1 /( m1  m2 )
= 0.1
  m1 /( m1  m2 ) = 0.01
Computation of Roche lobe radius from R3B equations
of motion (   x / a , a = semi-major axis of the binary)
L
Roche lobe radius depends weakly on R3B mass parameter
 1/ 3
rL  ( 3 ) a
  m1 /( m1  m2 )
= 0.1
  m1 /( m1  m2 ) = 0.01
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
Hill problem
rL  rL  ( 3 )1/ 3 a
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 that’s natural, since they are not
local (Hill’s 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)
Hill problem
rL  rL  ( 3 )1/ 3 a
Hill applied his equations to the Sun-Earth-Moon problem, showing
that the Moon’s Jacobi constant C=3.0012 is larger than CL=3.0009 (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 Moon’s eternal stability because:
(1) circular orbit of the Earth was assumed (crucial for constancy of Jacobi’s C)
(2) Moon was approximated as a massless body, like in R3B.
(3) Energy constraints can never exclude the possibility of Moon-Earth collision
How wide a region is destabilized by a planet?
Hill stability of
circumstellar motion
near the planet
C
rL  rL  ( 3 )1/ 3 a
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 planet’s 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 (r=initial circular orbit
radius far from the planet, a = planet’s 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.
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
C  3  34 x
2
 1/ 3
rL  ( 3 ) a
CL  3  9(rL / a) 2
2
2
x

12
(
r
/
a
)
Therefore, the Hill stability criterion C(x)=CL reads
L
or
x  2 3(rL / a)  3.5 rL / a
Example:
What is the extent of Hill-unstable region around Jupiter?
x  3.5 rL / a  3.5  (0.001/ 3)1/ 3  0.24
Since Jupiter is at a=5.2 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 < ~4 AU
(Thule group at ~4 AU is the outermost large group of
asteroids except for Trojan and Greek asteroids)
Back to the formation scenario:
Isolation, Giant Impacts
Stopping the runaway growth of planetary cores
 1/ 3
Roche lobe radius
rL  ( 3 ) a
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 2*sqrt(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.
2 3
2 3 rL 2 3( rL  drL )
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 core=10ME 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.