planet formation - | jila

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


Disks are inevitableconsequence of starformation - in principleshould be able to predicttheir masses / sizes

In practice we don’tunderstand star formationwell enough - these quantities are estimatedobservationally at laterepochs…

Q: how much “planetformation” (or earlyevolution of solids) occurs during verydisk phases?


Protoplanetary Disks

Disks of gas / dust in orbit around young (~Myr old) low massstars

Seen / inferred from infra-red observations

Inevitable consequenceof angular momentumconservation duringcollapse

Standard classification scheme for Young Stellar Objects


Basic properties: masses ~ 10-3 - 10-1 M*radii ~ 102 AU (selection effects!)accretion rates 10-10 - 10-7 Msun / yr

For planet formation interested in:• structure (density, temperature, chemical composition)• evolution - lifetime of gas and disk components• strength and nature of any turbulence


Vertical structure

Small z: vertical gravity

!

gz ="2z

!

" =GM

*

r3

where

Hydrostatic equilibrium:

!

dP

dz= "#gz

+ isothermal gas:

!

" = "0e#z 2 2h 2

where h, the vertical scale height:

!

h

r=cs

vK

(inverse Mach number)

Protoplanetary disks can cool efficiently and hence aregeometrically thin i.e. h / r << 1 (typically ~0.05)


Radial force balanceFor a thin disk the orbital velocity of the gas is very closeto the Keplerian velocity of a particle at the same radius:

!

v2" ,gas

r=GM

*

r2

+1

#

dP

dr

(steady state, axisymmetric,no B fields)

If we write:

!

P = P0

r

r0

"

# $

%

& '

(n

!

v" ,gas = vK 1# ncs2

vK2

$

% &

'

( )

1 2

order (h/r)2

Typically the gas rotates a few x 10-3 slower than the gas.Very small difference - of the order of 102 m s-1 at 1 AU…

BUT - extremely important for planetesimal formation!


Disk temperature

Two sources of heating:• reprocessed starlight: absorbed by dust and then

reradiated, usually in the IR• dissipation of gravitational potential energy as

gas spirals in to be accreted by the star

If disk absorbs fraction f of starlight (f = 1/4 for flat disk),and potential energy at surface of star is GM* / R*

Accretion heating dominates for:

!

GM*

˙ M

R*

> fL*

Plausible numbers:

!

˙ M " 2 #10$8

Msun

yr-1

Young disks are “active” (dominated by accretional heating), old disks are “passive” (dominated by reprocessing)


Detailed models: Kenyon & Hartmann (1987); Chiang & Goldreich (1997)

Kenny Wood model

Passive disk

hot dust atdisk surface

!

Tdust

= 550r

1 AU

"

# $

%

& '

(2 5

K

cooler, optically thick interior

!

Ti=150

r

1 AU

"

# $

%

& '

(3 7

K

With allowance for accretionalheating, models of this classmatch the observed spectralenergy distributions of YoungStellar Objects quite well


AA TauKenny Wood

Stellar photosphericcontribution to theobserved emission Disk contribution - dominant

in the mid to far IR


Condensation sequence

Given the pressure P(r) and temperature T(r) at the diskmidplane, can predict the make up of the solid component

Very schematically:

• start with box of gas of Solar composition at hightemperature - all elements in the gas phase

• cool the gas slowly, so that at each T find the statethat is thermodynamically preferred

!

G = H "TS

minimize:

…subject to constraint of the overall abundancesComposition as a function of T is called the condensationsequence (e.g. Lodders 2003)


Highest temperature condensates are aluminium oxide(Al2O3), perovskite (CaTiO3)… present for T ~ 1500K and below

Water ice condenses at disk pressures below about150 - 170 K - usually this is at a few AU in the disk

Examples of low T condensates: methane and argonhydrate (Ar.6H2O)

Puzzle: Argon is enriched in Jupiter’satmosphere despite the fact that thedisk temperature at 5 AU is much greater than ~40-50K condensationtemperature…


Snowline

Total surface density in condensible materials jumps from~0.5% to ~1.5% once it is cool enough to have water icepresent

log Σ

log r

~2.7 AU (Solar System)

Snowline is important for habitability, extra mass in theouter disk makes it easier to form cores of giant planets


Disk evolutionMeasure the disk fraction observationally by looking for theabundance of near-IR excess in clusters of different ages

after Haisch, Lada & Lada (2001)

Disks clearly evolve:

• fraction of stars with disks declines• masses drop• accretion rates fall

Typical disk lifetime ~3 Myr - must form the gas giants withinthis time before the gas is lost!


Physics of disk evolution

In a Keplerian disk, specific angular momentum:

!

l = GM*r

…is an increasing function of radius

For gas to accrete, must lose angular momentum. Twological possibilities:

• angular momentum redistribution within the disk, angular momentum flows to large radius while

mass flows to be accreted by the star• angular momentum loss from the system via a

wind from the disk (magnetic)

Don’t know for sure… but mechanism #1 is generally favored


Angular momentum transport is driven via a kinematicviscosity within the fluid:

!

"m~ #c

s where mean free path

!

" =1

n#m

Taking disk conditions at 10 AU, molecular cross-sectionis ~2 x 10-15 cm2, n ~ 1012 cm-3, cs ~ 0.5 km s-1

!

"m

~ 2.5 #107 cm

2 s

-1

Is this large or small?

Diffusive process, so time scale

!

t" =r

2

#m

= 3$1013 yr

Life is not so simple… molecular viscositydoes not explain the evolution of disks…


Turbulent transport

Phenomenological approach - assume that disk is turbulentand that turbulence results in an effective (“turbulent”) viscosity:

• mean free path -> eddy size (< h)• thermal velocity -> turbulent velocity (< cs or shocks form)

!

" =#csh

Shakura & Sunyeav (1973) α prescription - α dimensionless parameter

Observationally, we can obtain evolution in ~Myr for a diskof ~30 AU size if α ∼ 10-2 (e.g. Hartmann et al. 1998)

Note: one can construct a whole theory of disk evolution ifyou assume that α is a constant…


Example from Armitage et al. (2003) - beware stellar ages!Nevertheless, α ~ 0.01 definitely works… may not be unique…


Low molecular viscosity implies a very high Reynolds number:

!

Re =UL

"m

~ 1010 - terrestrial flows with high Re

are generally turbulent

BUT… in disks the linear stability is governed by theRayleigh criterion, for instability need:

!

d

drr2"( ) < 0 - specific ang momentum decrease

outwards… in disks it increases

Many other fluid possibilities:• non-linear (finite amplitude) instability• vortices• transient growth…

None demonstrated to be generally effective in disks


Failure to observe hydrodynamic instability in simulationsis important evidence against their existence, but not a proof… cannot attain the physical Reynolds numbers.

Note: astrophysicists have a much more “relaxed” attitudetoward drawing conclusions from simulations at the “wrong”Reynolds numbers than engineers…


Magnetorotational instability

With arbitrarily weak magnetic field, disk is subject to a local, linear instability with very fast growth rate if:

!

d

dr"( ) < 0 …this condition is met within disks!

Physical origin is very simple…

r

z

Start with pure Bz threading the disk




!

d



r

z

Perturb field in radial direction




!

d



r

z

Field is sheared by differential rotation




!

d



Magnetic tension removesangular momentum from innerfluid element, adds angular momentum to outer element

Angular momentum transport!


Vertical field MRIBest reference for more comprehensive case Balbus & Hawley (1998)

Setup: axisymmetric incompressible disk flow with uniform Bz

Equations of motion of a parcel of gas:

!

˙ ̇ r " r ˙ # 2 = "d$

dr+ fr

r ˙ ̇ # + 2˙ r ̇ # = f#

…where fr etc are magnetic forces (as yet unspecified)

Transform to a locally corotating cartesian patch:

!

˙ ̇ x " 2#˙ y = "xd#

2

d ln r+ fx

˙ ̇ y + 2#˙ x = fy


Perturb in the x-y plane with a displacement vector:

!

s"ei(#t$kz)

Magnetic induction equation (flux freezing) yields:

!

"B = #ikBzs

f = #(kvA)2s …magnetic tension force,

with vA the Alfven speed

Substitute into the equations of motion - derive a dispersion relation(i.e. relation between the temporal frequency of the perturbation ω and spatial wavenumber k):

!

" 4 #" 2 d$2

d ln r+ 4$2 + 2 kv

A( )2

%

& '

(

) * + kv

A( )2

kvA( )

2

+d$2

d ln r

%

& '

(

) * = 0

Instability requires that ω2 < 0


Condition:

!

kvA( )

2

+d"

2

d ln r< 0

Take limit as field goes to zero: weak-field limit is just that theangular velocity needs to decrease outward…

Note: does not reduce to the Rayleigh criterion as B -> 0!

Instability was known to Chandrasekhar (1961) and to Velikhov (1959), and their analysis was commented on by Safronov (1969). BUT… importance for disks wasonly correctly recognized by Balbus & Hawley (1991)


Numerical simulations show the magnetorational instability in ideal (no dissipation) MHD leads to turbulence and outwardangular momentum transport

Workman & Armitage (2008)


In disks around black holes etc, little doubt that the MRI is the mainphysical mechanism behind angular momentum transport:


Research topic #1: nature of disk turbulence

Is the MRI the main (or only) source of disk turbulence?

Problem in protoplanetary disks: non-ideal effects

!

"B

"t=# $ v $B( ) %&#2

B+ Hall effect (very important in detail)

Ideal term, MRI leadsto B field growth on a time scale Ω−1

In disk with diffusivity (1 / conductivity)η Ohmic losses damp fields on scaleλ on time scale λ2 / λ

Equate: find that we require the magnetic Reynolds number

!

ReM

=UL

">1

in order to generate and sustain turbulence. Hard if the conductivity islow and the diffusivity high…


T < 103 K - ionizationwithin disks is allnon-thermal (X-rays,some radioactivity,maybe cosmic rays)

Balance against recombinations (gas phase, dust surfaces)to see if ionization is “high” enough (x ~ 10-12) to sustainMHD turbulence

Result is not definitively known - but seems that ionizationdegree at the midplane at few AU may well be too smallto allow MHD turbulence…


layered disk concept (Gammie 1996)What does this mean for planet formation?

• seems disk must be turbulent to yield observed evolution• infer α ~ 0.01 (from which get δv)• don’t clearly know the origin of the turbulence (MHD?)• might be weakly turbulent regions just where we need

to form planets (coincidence?)


Research topic #2: how is the disk destroyed?

Why is the disk lifetime a few million years?

Main answer: viscous time scale

!

t" =r

2

"~ 106 yr

Gas is accreted by the star on this time scale… no first principles explanation

Observations suggest that the transition between “disk”and “disk-less” state is rather sharp… ~105 yr

Not predicted by simple models - need another mechanism


Observational clue

See what appear to be flowsfrom disks in the Orion nebula

Interpreted as gas that hasbeen photoevaporated fromthe disk by the UV radiationfield of massive stars withinthe cluster

UV (maybe X-rays) from thelow mass star itself couldalso drive such a flow


Physics: photoionize gas at disk surface to T ~ 104 K, soundspeed ~10 km s-1

Where cs > vesc gas is unbound - escapes as a thermal wind

!

rescape =GM

*

cs2

5-10 AU with this naïve estimate, ~few AU with more sophisticated analysis


Hydrodynamic simulation from Alexander, Clarke & Pringle (2006)


Predicted evolution ofthe disk when viscousevolution + effect of photoevaporation is included…

If this is what is goingon star formation environment matters!


Research topic #3: vortices

Do we have “Great Red Spots” in protoplanetary disks?

!

" =# $ v

Are vortices important for eitherangular momentum transport orplanet formation?

Define:

Fluid equations (now no B):

!

D

Dt

"

#

$

% &

'

( ) =

"

#

$

% &

'

( ) * +v,

1

#+1

#

$

% & '

( ) -+P

For weakly compressible (dv << cs) barytropic flow P=P(ρ) the vortensity is a conserved quantity. So vortices, if they exist, are hard to destroy…


Moreover, in 2D simulations show that vortices in disks are:

• long lived (inverse size cascade)• transport angular momentum outward• act to concentrate particles at their cores

A disk is effectively 2D in the upper regions due to the strong density stratification (like Jupiter). But near z = 0 disk is threedimensional. There, simulations suggest rapid vortex destructiondue to vertical shear (Shen et al. 2006)…

Mechanisms that might form vortices remain hotly debated…


What are the initial conditions for planet formation?

Composition - rocky particles in the inner few AU, mixof rock / ice further out. Initially small - µm in size.

Confidence: HIGH

Temperature / density - known for both gas and solidsfrom combination of theory + observations

Confidence: HIGH (T), MODERATE (ρ)

Turbulence - is the disk turbulent, and if so how strong isthe turbulence and what is its nature?

Confidence: LOW

planet formation - | jila

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