planet formation - university of colorado boulder

XIII Ciclo de Cursos Especiais

Planet FormationPlanet Formation


Outline

1. Observations of planetary systems2. Protoplanetary disks3. Formation of planetesimals (km-scale bodies)4. Formation of terrestrial and giant planets5. Evolution and stability of planetary systems


Terrestrial planet formationInitial conditions: planetesimals (~10km solid bodies)

Physics: gravitational interactions

Well defined BUT - 4 x 109 planetesimals needed to buildthe terrestrial planets. Time scale ~100 Myr - too hard tosolve directly…

Conventional approach:

• treat early phases statistically (methods from the kinetic theory of gases / coagulation theory)

• final phases (100-1000 bodies) using direct N-bodysimulations


Collision physics

Collisions can have three main outcomes:• accretion - target gains mass from collision• dispersal - target is broken into unbound pieces• fragmentation + reaccumulation / shattering -

target composition is modified leading to a “rubble pile”

Fragments of comet Shoemaker-Levy9 prior to its collision with Jupiter…


For a small impactor colliding with a target define the specific energy of the collision as:

!

Q "mv

2

2M

…where m is the impactor mass, M is the target mass, and v the impact velocity

Threshold for disruption of bodies of a particular size is:

!

Q >QD

*

Two regimes:• small bodies: strength dominated - larger objects

are weaker due to more defects• large bodies: gravity dominated - resist disruption

due to their own self-gravity

At leading order, determining outcome of collisions amountsto comparing Q with calculated disruption energy


Benz & Asphaug (1999)

Weakest planetesimals are those in the 100m - 1km range:larger bodies ought to accrete unless large eccentricities /inclinations are excited


Physics of collisions: gravitational focusing

Consider 2 bodies of mass m and relative velocity (at infinity) of σ. Gravity deflects the trajectories, leading tolarger collision cross-section.

Conservation of energy and angular momentum yields:

!

" = #Rs

21+

vesc

2

$ 2

%

& '

(

) *

physical collisioncross-section

enhancement due to gravitationalfocusing: vesc = escape speed


!

" = #Rs

21+

vesc

2

$ 2

%

& '

(

) *

Simple result is largely responsible for runaway growth:

• small body will have vesc < σ: relatively low rate ofcollisions and slow growth

• if one body grows slightly larger so that vesc > σ,that body grows faster… outpaces itsneighbors. Called runaway growth…


Shear vs dispersion dominated encounters

Can we model collisions using 2-body dynamics (the twocolliding objects) or do we need 3-body dynamics (+star)?

Depends upon how close the orbits are to circular (i.e.on the velocity dispersion):

Example: Jupiter’s Trojan asteroids - don’t collide withJupiter despite sharing the same orbit (3-body effects)


Delineate the sphere of influence of a planet by comparingthe time scale for orbit of a particle around the planet withthe time scale for orbit of the planet around the star:

!

rHill =Mp

3M*

"

# $

%

& '

1 3

a …known as the Hill sphere

Defining the Hill velocity similarly:

!

vHill =GMp

rHill

Describe collisions as:• dispersion dominated if σ > vHill - 2 body dynamics• shear dominated if σ < vHill - need 3 body dynamics

Very important distinction in detailed calculations of thecollision rate and evolution of the velocity dispersion…


Particle in a box model for growth

Body of mass M, radius Rs and escape speed vescEmbedded with swarm of planetesimals with surface density Σp, velocity dispersion σ, and scale height hp

Define volume density of swarm:

!

"sw =#p

2hp

In dispersion dominated regime, growth is then:

!

dM

dt= "

sw#$R

s

21+

vesc

2

# 2

%

& '

(

) *

Note: the scale height is just hp = σ / Ω, so can simplify:

!

dM

dt=

3

2"p#$Rs

21+

vesc2

% 2

&

' (

)

* +

Prefactor depends upon the velocity distribution of the swarm


!

dM

dt=

3

2"p#$Rs

21+

vesc2

% 2

&

' (

)

* +

Gravitational focusing unimportantIn this limit:

!

dM

dt"R

s

2"M

2 3

# Rs" t

Substituting for typical disk conditions at 5 AU:

!

dRs

dt~ 0.2 cm yr

-1

This growth rate is very slow! To build planets in any reasonable time we require large gravitational focusingfactors…


!

dM

dt=

3

2"p#$Rs

21+

vesc2

% 2

&

' (

)

* +

Gravitational focusing dominant

Suppose (not very realistic) that σ is constant

Noting that:

!

vesc

2"M

Rs

!

dM

dt"MR

s

!

M =1

M0

"1 3" kt( )

3

…much more rapid growth!


Evidently the velocity dispersion of the small bodies iscritical to determining the growth rate. What sets σ?

• viscous stirring (“relaxation” in stellar dynamics) -weak gravitational scattering among the planetesimals

• dynamical friction - energy transfer from large bodies to smaller bodies

• gas drag• energy loss in inelastic collisions

EX

CIT

ED

AM

P

All of these can be calculated in detail…

Normally, velocity dispersion is set by balancing dynamicalfriction against gas drag…


Coagulation equation

Formal machinery for handling coagulation of a populationof bodies:

!

dnk

dt=1

2Aijnin j

i+ j= k

" # nk Akinii=1

$

"

At time t suppose that there are nk bodies with mass mk=km1 where m1 is some small mass (discrete approach):

Here Aij is the rate of mergers between bodies with massesmi and mj (the “kernel”)

Three analytic solutions are known, and care is neededto solve this equation numerically…


Example: Aij = constant

This example shows orderly growth - the populations smoothlyevolves to larger mean masses

Planet formation: due to gravitational focusingcollisions become morefrequent with mass - i.e.Aij is positive power ofmasses

Analytic example: Aij = Cmimj - this shows runaway growthRunaway growth eventually invalidates the use of thecoagulation equation…


Time line of terrestrial planet formation

1) Runaway growth - viscous stirring / gas drag set σ. Onebody increases in mass much more rapidly than all its

neighbors2) Oligarchic growth - eventually the large body becomes

massive enough to heat the planetesimal populationvia dynamical friction. This reduces the gravitationalfocusing, and slows growth. A number of bodies - oligarchs - come to dominate each annular region.

3) Final assembly - the oligarchs are eventually perturbedinto crossing orbits, collide and merge into the finalterrestrial planets

4) Clean-up - any surviving small bodies are accreted orejected. Water arrives from elsewhere in the system.


Process takes ~100 Myr, can reproduce the number andmasses of the terrestrial planets. Some problems (finaleccentricity too high?) but not insurmountable…


Predicted number of terrestrial planets (on average - stochastic process) depends upon the:

• disk surface density (solids)• stellar mass• presence of giant planets

Predictions for Kepler and beyond!


Giant planet formation

Disk instability model

Postulates that the gas disk is massive enough to becomegravitationally unstable:

!

Q =cs"

#G$<1

AND that the instability leads to fragmentation / collapse

If satisfied, then collapse of a disk with Σ ~ 103 g cm-2 at Q ~ 1 yields a mass:

!

Mp ~ "#$2~ 2MJ

In this model giant planet formation occurs early andquickly from disk gas. Enrichment / core formation wouldoccur later, during planetary evolution.


Does gravitational instability in the gas lead to collapse?

Progress: we now know that this depends on the coolingtime within the disk - fast cooling is needed to obtainfragmentation (long known, but demonstrated by Gammie2001; Rice et al. 2003)

Slow cooling

Fast cooling!

tcool ,crit

=3

"

+ dependenceon equation ofstate and onopacity dependenceon temperature


Does the gas in the disk cool quickly enough to allowcollapse to giant planets?

Known: radiative cooling is not efficient enough

Unknown: realistic cooling (including convection) in a detailed disk model

• simulations by Boss show rapid cooling andcollapse

• simulations by Durisen and collaborators failto obtain collapse… only spiral arms

Personal opinion: seems hard to form Uranus / Neptunefrom disk instability, and to reproduce the planet frequencymetallicity correlation. Instability may form very massive planets at large radii in some extrasolar planetary systems.


Core accretion model

Core accretion model builds on thescheme for terrestrial planet formation:

1. Build a core by same mechanismsas for terrestrial planet formation

2. Core becomes massive enough tomaintain a substantial atmosphere(vesc > cs in disk). Atmosphere isin hydrostatic balance, supported byheat generated as planetesimals collide with core / quasi-steady contraction of the gas


Papaloizou & Terquem (1999)

Plot the total planet mass (core + gasmass in envelope) against core massfor these hydrostatic solutions…

Find that there is a maximum orcritical core mass, beyond which no hydrostatic solution exists!

Once the core exceeds the maximum mass, rate of gas inflow increases rapidly

Eventually, planet stops accreting, either because thegas disk is dissipated or because the planet opens a gap in the disk which frustrates accretion


Classic calculation by Pollack et al. (1996)

Predicts:

• giant planets have cores (at least initially…)• time scale depends on the rate of building the core

(rate limiting step), faster with larger surfacedensity of solids (e.g. in a disk with higher [Fe/H])


Do giant planets have cores?Tristan Guillot

Saturn: yes

Jupiter: maybe… but if there is a core it is almost certainlyless massive than the 20 Earth masses predicted by Pollack et al. (1996)


However, Pollack et al. (1996) is an illustrative example ofa very broad class of models

Core mass depends on both the planetesimal accretion rate and on the opacity in the envelope

!

Mcrit

M"

#12˙ M

core

10$6M"yr-1

%

& '

(

) *

1 4

+

1 cm2 g-1

%

& '

(

) *

1 4

Ikoma et al. (2000)

A small core could result from:• a low envelope opacity (unknown, hard to calculate,

but why not?)• starvation of planetesimals

For Jupiter to be truly core-less would be a problem, buta small core within the current limits is not unreasonable


Time scale problem

Pollack et al. (1996) formed both Jupiter and Saturn in a modestly mass enhanced disk (above the MMSN) in about 10 Myr

Easy to find models that form these planets in 3-5 Myr, meeting astrophysical constraints

Forming Uranus and Neptune in their current locationsin a reasonable time is much harder - requires very largegravitational focusing factors and not clear that happens…

This seems a serious problem… unless one drops theassumption that Uranus and Neptune formed in situ - maybe they formed closer to the Sun where the time scale is much quicker!

planet formation - university of colorado boulder

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