Download ORBITAL MOTION

PH507
Astrophysics
Professor Michael Smith
1
Final exam: Major Revision Topics
Exoplanet finding techniques
Distances – cosmic ladder
Distance modulus
Kepler’s laws
Planet formation processes
Hertzsprung-Russell tracks
Planets; Density of, escape speed from.
Solar atmosphere
Molecular cloud evolution/collapse
Protostars: classes, accretion rate, envelopes, cores.
Lectures Week 8: Star Formation & Theory of Exoplanets
1. Intro: Star formation is on-going.
 What is the origin of our solar system? Descartes, Kant,
Laplace: vortices, nebular hypothesis: importance of angular
momentum.
PH507
Astrophysics
Professor Michael Smith
2
Major facts for nebula hypothesis:

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Coplanar orbits of the planets
All planets have prograde revolution (orbits)
The revolution of rings and natural moons are all prograde (some moons of the
outer planets are not prograde, but these are believed to be captured satellites)
All planets except Venus and Uranus have prograde rotation
The sun contains all the mass
The planets (especially Jupiter and Saturn) contain most of the angular
momentum in the solar system
Small, dense, iron and silicate rich planets in the inner 2 AU. Slow rotors, few
or no moons, no rings, differentiated (molten interiors)
Large, low density, gaseous planets rich in H, He and volatile elements at >= 5
AU

Rapid rotors, many moons, all have ring systems

Abundance gradient. Inner solar system is poor in light volatile gases such as H,
He, but rich in Fe & Ni. Outer solar system is rich in volatiles H, He, etc.
Abundances similar to that of the sun.
PH507
Astrophysics
Professor Michael Smith
3

In general: Gravity is fast-acting. Galaxy is old. But young stars
are still being born.

Stars don't live forever, they must continue to be "born".
Where?

Born in obscurity….needed infrared/millimeter/radio
wavelengths.
2. Molecular clouds: ingredients

Young stars are located in or near molecular clouds (the stellar
factories/nurseries).

Stars mainly form in clusters in giant molecular clouds.

Over 90% of atoms tied up in molecules. 99.99% is molecular
hydrogen: H2

Over 120 other molecules discovered, including water, carbon
monoxide CO, formaldehyde H2CO, ammonia NH3, hydrogen
cyanide HCN, formic acid HCOOH and methanol CH3CO

Admixture of dust: 1% by mass– tiny grains (less than 1 micron in
size) of silicates/graphite with ice coatingss, or soot (polycyclic
aromatic hydrocarbons or PAHs).

Cosmic rays, magnetic field.

The large amount of gas and dust in the cloud shields the
molecules from UV radiation from stars in our galaxy. The
molecules can then cool the gas down to 10-30K. Dense cold cores
can form (eggs?) in which gravity rules).

The H2 molecules cannot form by H-H collisions (excess energy
needs an outlet). H2 forms on dust, atoms stick, migrate, bind,
ejected. Other molecules form through collisions (ion-chemistry).
3. Molecular clouds: anatomy
Opaque at UV and visible wavelengths.
Bright and luminous at millimetre wavelengths: dust
continuum.
Bright rotational and vibrational molecular emission lines at
radio and infrared wavelengths.
Molecular clouds are cold: 8<= Tkin<=20 K Typical value ~10 K
Low ionization: fe =ne/n ~10-6 - 10-7 => very neutral!
PH507
Astrophysics
Professor Michael Smith
4
High density: n(H2) >= 100 cm-3
Giant molecular clouds are very massive: M~ 104 to 106 solar
masses
Giant molecular clouds are large: Diameters ~ 325 ly
They are clumpy
Supersonic gas motions are found in almost all clouds
Line widths ~ 0.5 to 2 km s-1; sound speed ~ 0.2 km s-1
indicative of nonthermal motions such as rotation, turbulence,
shocks, contraction or expansion, stellar bipolar outflows, etc.
Measures:
Atmospheric cloud:
PH507
Astrophysics
Professor Michael Smith
5
A comparison of scales between typical molecular and
atmospheric clouds.
Molecular Cloud
Size
1014 km
Mass
1036 gm
Particle density
103 cm-3
Temperature
20
K
Mol/atomic weight
2.3
Speed of sound
0.3 km/s
Turbulent speed
3
km/s
Dynamical time
Million years
Atmospheric Cloud
1
km
1011 gm
1019 cm-3
260
K
29
0.3
km/s
0.003 km/s
Five minutes
Scales & Types:
Estimated properties of individual molecular aggregates in the
Galaxy:
Phase
GMCs
Mass
(Msun) 6x104 - 2x106
Size
(parsecs) 20 - 100
Density (cm-3)
100 - 300
Temperature (K)
15 - 40
Magn. Field(G)
1 - 10
Line width (km/s)
6 - 15
Dynamic life (years) 3 x 106
Clumps/Globules
102
0.2 - 4
103 - 104
7 - 15
3 - 30
0.5 - 4
106
Cores
1 - 10
0.1 - 0.4
104 - 105
10
10 - 50
0.2 - 0.4
6 x 105
Note: dynamical life defined as Size/(Line width), true lifetimes
would be considerably longer if clouds were static.
Example: Orion millimeter dust emission – clumps and cores
PH507
Astrophysics
Professor Michael Smith
6
The Horsehead (optical – dark cloud)
Summary: clouds are turbulent, possibly fractal
3. Molecular clouds: their origin
Agglomeration: collisions and merging/coalescence of smaller
clouds – not sufficient small clouds. Spiral arm density-wave
focusing.
Gravitational instability followed by fragmentation
Condensation: out of atomic clouds.
PH507
Astrophysics
Professor Michael Smith
7
Accumulation: gas swept up into supershells, focused in
turbulent interstellar medium.
Answer: combination of these.
4. Molecular cloud evolution
PH507
Astrophysics
Professor Michael Smith
8

Observed: Giants, clumps, cores, eggs

Gravitational Collapse: When a fragment of a molecular cloud
reaches a critical mass – the Jeans mass (after Sir James Jeanss
(1877-1946) - it collapses to form a star. Gas and dust are then
pulled together by gravity until a star is formed.

Balance forces: gravity and pressure: GMJ2/R ~ MJcs2
PH507

Astrophysics
Professor Michael Smith
9
Eliminate R in favour of the density, yields the Jeans Mass, which
more precisely calculated is
MJ 
  
 
6 G
3/ 2
c s3  1 / 2

Fragmentation: The molecular cloud does not collapse into a single
star. It fragments [through the Jeans instability - into many clumps.

As the density rises, the Jeans mass falls. This means the cloud
continues to fragment into smaller clumps.

What makes it reach/exceed the critical mass?

Mechanisms: sequential, spontaneous, turbulence, triggers
What are the conditions that favour the initiation of star
formation?
Decrease internal pressure: By decreasing the temperature or
the density or both
Increasing the mean mass per particle by transforming from
an atomic gas to a molecular gas.
PH507
Astrophysics
Professor Michael Smith
10
Decrease the ionization fraction, fe = ne/n to < 10-7 => gas
decouples from any magnetic field present so that magnetic
pressure cannot support the cloud.
Increase the external pressure: By partially focused shocks.
By ionization of the gas around a molecular clump: radiativelydriven implosion.
Collapses
Methods
Collapse: Method 1

Accretion- coalescence:
Build up of small clouds of gas and dust into
clumps.

Clumps "stick" together and grow.

Very slow - due to low interstellar densities
Collapse: Method 2
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Gravity and Radiation Pressure
Collapse: Method 3: sequential, triggered
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Compression by supernova blast waves
Evidence that the Solar System/Sun was triggered by a supernova –
PH507
Astrophysics
Professor Michael Smith
11
(radioactive isotopes so short-lived that they no longer exist were
trapped in chondrules within meteorites).
PH507
The Difficult
Path
to Collapse
Astrophysics
Professor Michael Smith
12

Gravity makes parts of a the cloud collapse.

Hindrances to collapse which favour expansion:
1. Internal heating

Causes pressure build-up
2. Angular momentum

Causes high rotation speeds

(exemplified by a figure skater)
3. Magnetic support
Internal
Heating

Cloud fragments collapse

Potential energy => Kinetic Energy
o
Gas particles speed up and collide.

The temperature increases.

This causes a pressure build-up which slows (or
stops) the collapse.

Solution: Energy is radiated away.
PH507
Angular
Momentum
Astrophysics
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Angular momentum
o
A = mass x vel. of rotation x radius
o
A=mvr
Conservation of angular momentum.
o
Magnetic
support
Professor Michael Smith
A = constant for a closed system.

As the cloud fragment shrinks due to gravity, it
spins faster.

Collapse occurs preferentially along path of least
rotation.

The cloud fragment collapses into a central core
surrounded by a disk of material.

Further collapse: magnetic braking – winding
and twisting of magnetic field lines connected to
external gas.

There is a critical mass, for which gravity is held up by
magnetic pressure.

A cloud can be super-critical – free to collapse

The field diffuses out: ambipolar diffusion –
since the magnetic field is only tied to the ions,
and the ions slip through the molecules.
The Final Collapse: approaching birth
Final adjustments. The thermodynamics now take on supreme importance.
Much of what occurs is still theory:
Stage 1. The density shields the core from external radiation, allowing the
temperature to drop slightly. Dust grains provide efficient cooling. The hydrogen
is molecular.
Stage 2. An isothermal collapse all the way from densities of 10 4 cm-3 then
proceeds. The gravitational energy released goes via compression into heating
the molecules. The energy is rapidly passed on to the dust grains via collisions.
The dust grains re-radiate the energy in the millimeter range, which escapes
PH507
Astrophysics
Professor Michael Smith
14
the core. So long as the radiation can escape, the collapse remains
unhindered.
Stage 3. At densities of 1011 cm-3 and within a radius of 1014 cm the gas
becomes opaque to the dust radiation even at 300microns. The energy
released is trapped and the temperature rises. As the temperature ascends, the
opacity also ascends. The core suddenly switches from isothermal to adiabatic.
Stage 4. The high thermal pressure resists gravity and this ends the first
collapse, forming what is traditionally called the first core at a density of 1013
cm-3 - 1014 cm-3 and temperature of 100-200 K.
Stage 5. A shock wave forms at the outer edge of the first core. The first core
accretes from the envelope through this shock. The temperature continues to
rise until the density reaches 1017 cm-3.
Stage 6. The temperature reaches 2000 K. Hydrogen molecules dissociate at
such a high temperature if held sufficiently long. The resulting atoms hold less
energy than the molecules did (the dissociation is endothermic), tempering the
pressure rise. The consequence is the second collapse.
Stage 7. The molecules become exhausted and the cooling stops at the centre
of the first core. Protostellar densities of order 1023 cm-3 are reachedand with
temperatures of 10,000 K, thermal pressure brakes the collapse. This brings a
second and final protostellar core into existence. The mass of this core may
only be one per cent of the final stellar mass.
Stage 8. The first shock from Stage 5 disappears while a second inner shock
now mediates the accretion onto the protostellar core. A star is born.
Further Collapse with Angular Momentum into a Disk

All astronomical objects spin, even if very slowly.

The original collapsing cloud will have some small amount of spin.

During a collapse, angular momentum is conserved.

Angular momentum is J = a x W R2
o
a = a constant whose value we aren't interested in
o
W = Angular velocity = 2 pi/P
o
P = Spin Period
o
R = Radius of the star cloud

If angular momentum is conserved then
Wfinal = W0 x (R0/Rfinal)2

Since R0/Rfinal is much larger than 1
Final angular velocity can be very high, even if the initial angular
velocity is very low.
PH507
Astrophysics
Professor Michael Smith

Centrifugal force and gravity approach equilibrium

A very rapidly rotating cloud will get flattened into a disk.
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This disk can then fragment into protoplanets.
15
Disk
Formation
Planet
Formation

The disk around the central core will
fragment further, producing rings of
material.

The particles in these rings can accrete
together to form planetesimals and
planets!
PH507
Making the
Stars
Visible
Astrophysics
Professor Michael Smith
16
Making the Stars Visible

After a star is born it heats the gas
and dust around it.
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Jets of gas are ejected: bipolar
outflows are observed.
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Eventually the gas and dust are
accreted or dispersed.

The star is then "visible."
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Prior to this it could be seen only in
the radio and the infrared.
Spectral energy distributions define the classes of protostars
and Young Stellar Objects……
PH507
Astrophysics
Professor Michael Smith
17
PH507
Astrophysics
Professor Michael Smith
18
Young Stars: accretion, contraction, jets and outflow
Accretion through disc: bolometric luminosity of protostar is
.
L
GM M
R
Where M*dot* is the mass accretion rate and GM/R is the energy released per unit mass
onto the protostar of (accumulating) mass M and radius R.
Star accumulates gas from envelope through the disc, releases some through jets back
into cloud. The jets are thought to be the channels
for the extraction of angular momentum.
Jets: extend parsecs from source. They are seen through their impact with cloud: shock
wave heating: Herbig-Haro Objects. They create large reservoirs of outpouring and
swept-up gas: bipolar outflows or molecules
outflows
HH46/47:
Optical: HST
PH507
Astrophysics
Professor Michael Smith
19
Infrared: Spitzer
Massive Stars & Clusters:
Massive stars should not form: hydrogen burning begins while accreting: radiation
pressure should resist the infall.
Produce hot molecular cores, masers, compact/extended H II regions.
Accretion must be high and through a disk: suffocate the feedback.
Most stars are in multiple bound systems.
PH507
Astrophysics
Professor Michael Smith
20
Frequency of occurrence:
single:binary:triple:quadruple is 58:33:7:1
Theory: capture, fission, core/collapse/disk fragmentation
Capture: unlikely
Fission: leads only to close binaries
Fragmentation plausible.
90% of stars are born in clusters.
Cluster: over 35 stars, at least 1 Msun/pc3
Embedded clusters: 1000 Msun with a density 10,000 Msun/pc3
Segregation: Massive stars tend to form in centre (form in situ, don’t migrate)
Relaxation: Cloud evolves and cluster disperses in a few million years.
Clusters dissolve: most stars are NOT in clusters, they become field stars.
All suggests: Hierarchical fragmentation within a turbulent medium
.
Star formation efficiency, the amount of cloud gas transformed into stars, is 3%20%.
The mass function (the IMF: initial mass function): most star are of low mass.
Questions:
Power law? Scale-free hierarchy.
Is there a brown dwarf desert? Planets form in disks, stars in collapse.
PH507
Astrophysics
Turbulence v. Gravity
Professor Michael Smith
21
PH507
Astrophysics
Proplyds – protoplanetary disks:
Professor Michael Smith
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PH507
Astrophysics
Professor Michael Smith
23
Debris Disks
Debris disks are remnant accretion disks with little or no gas left (just
dust & rocks), outflow has stopped, the star is visible.
Theory: Gas disperses, “planetesimals” form (100 km diameter rocks),
collide & stick together due to gravity forming protoplanets).
Protoplanets interact with dust disks: tidal torques cause planets to
migrate inward toward their host stars. Estimated migration time ~ 2 x
105 yrs for Earth-size planet at 5 AU.
Perturbations caused by gas giants may spawn smaller planets:
Start with a stable disk
around central star.
Jupiter-sized planet forms
& clears gap in gas disk.
Planet accretes along spiral Disk fragments into more
arms, arms become unstable. planetary mass objects.
Spiral density waves continuously produced by the gravity of embedded or
external perturber.
PH507
Astrophysics
Professor Michael Smith
24
Debris Disks – Outer Disk
AB Aurigae outer
debris disk nearly
face on – see
structure &
condensations
(possible protoplanet formation
sites? Very far
from star) .
(Grady et al. 1999)
Debris: not from original nebula but from recent collisions.
The Birth of the Solar System
The properties of the Solar System hold important clues to its origin

Orbits of the planets and asteroids.

Rotation of the planets and the Sun.

Composition of the planets, especially the strong distinction between
Terrestrial, Jovian, and Icy planets.
Clues from planetary motions:

Planets orbit in nearly the same plane.

Planet orbits are nearly circular.

Planets & Asteroids orbit in the same direction.

Rotation axes of the planets tends to align with the sense of their orbits,
with exceptions.

Sun rotates in the same direction in the same sense.

Jovian moon systems mimic the Solar System.
Clues from planet composition:
Inner Planets & Asteroids:

Small rocky bodies

Few ices or volatiles
PH507
Astrophysics
Professor Michael Smith
25
Jovian Planets:

Deep Hydrogen & Helium atmospheres rich in volatiles.

Large ice & rock cores
Outer solar system moons & icy bodies:

Small ice & rock mixtures with frozen volatiles.
Formation of the Sun: back to the Primordial Solar Nebula
Stars form out of interstellar gas clouds:

Large cold cloud of H2 molecules and dust gravitationally collapses and
fragments.
Rotating fragments collapse further:

Rapid collapse along the poles, but centrifugal forces slow the collapse
along the equator.

Result is collapse into a spinning disk

Central core collapses into a rotating proto-Sun surrounded by a rotating
"Solar Nebula"
Primordial Solar Nebula
The rotating solar nebula is composed of

~75% Hydrogen & 25% Helium

Traces of metals and dust grains
Starts out at ~2000 K, then cools:

As it cools, various elements condense out of the gas into solid form as
grains or ices.

Which materials condense out when depends on their "condensation
temperature".
Condensation Temperatures
Temp (K)
Elements
>2000 K
1600 K
Condensate
All elements are gaseous
Al, Ti, Ca
Mineral Oxides
PH507
Astrophysics
Professor Michael Smith
26
1400 K
Iron & Nickel
Metallic Grains – Refractory, Rocky
1300 K
Silicon
Silicate Grains - Rocky
300 K
Carbon,
Oxygen
Carbonaceous grains -Volatiles
300-100 K
Hydrogen,
Nitrogen
Ices (H2O, CO2, NH3, CH4)
The "Frost Line"
Rock & Metals can form anywhere it is cooler than about 1300 K.
Carbon grains & ices can only form where the gas is cooler than 300 K.
Inner Solar System:

Too hot for ices & carbon grains.
Outer Solar System:

Carbon grains & ices form beyond the "frost line".
The location of the "frost line" is also a matter of some debate, but current
thinking holds that it is probably about 4 AU . A great deal depends on how
much solar radiation can penetrate deep into the outer parts of the primordial
Solar Nebula.
From Grains to Planetesimals to Planets
Grains that have low-velocity collisions can stick together, forming bigger
grains.

Beyond the "frost line", get additional growth by condensing ices onto
the grains.

Grow to where their mutual gravitation assists in the aggregation
process, accelerating the growth rate. Can form km-sized planetesimals
after a few 1000 years of initial growth.

Aggregation of planetesimals into planets
Terrestrial vs. Jovian planet formation.
Terrestrial Planets
Only rocky planetesimals inside the frost line:

Collisions between planetesimals form small rocky bodies.

It is hotter closer to the Sun, so the proto-planets cannot capture H and
He gas.
PH507

Astrophysics
Professor Michael Smith
27
Solar wind is also dispersing the solar nebula from the inside out,
removing H & He.
Result:

Form rocky terrestrial planets with few ices.
Jovian Planets
The addition of ices to the mix greatly augments the masses of the
planetesimals
These collide to form large rock and ice cores:.

Jupiter & Saturn: 10-15 MEarth rock/ice cores.

Uranus & Neptune: 1-2 MEarth rock/ice cores.
As a consequence of their larger masses & colder temperatures:

Can accrete H & He gas from the solar nebula.

Planets with the biggest cores grow rapidly in size, increasing the
amount of gas accretion.
Result:

Form large Jovian planets with massive rock & ice cores and heavy H
and He atmospheres
Moons & Asteroids
Some of the gas attracted to the proto-Jovians forms a rotating disk of material:

Get mini solar nebula around the Jovians

Rocky/icy moons form in these disks.

Later moons added by asteroid/comet capture.
Asteroids:

Gravity of the proto-Jupiter keeps the planetesimals in the main belt
stirred up.

Never get to aggregate into a larger bodies.
Icy Bodies & Comets
Outer reaches are the coldest, but also the thinnest parts of the Solar Nebula:

Ices condense very quickly onto rocky cores.

Stay small because of a lack of material.
Gravity of the proto-Neptune also plays a role:
PH507
Astrophysics
Professor Michael Smith
28

Assisted the formation of Pluto-sized bodies in 3:2 resonance orbits
(Pluto and Plutinos)

Disperses the rest into the Kuiper Belt to become Kuiper Belt Objects.
Comets and other Trans-Neptunian objects are the leftover icy planetesimals from the
formation of the Solar System.
Mopping up...
The entire planetary assembly process probably took about 100 Million years.
Followed by a 1 Billion year period during which the planets were subjected to heavy
bombardment by the remaining rocky & icy pieces leftover from planet formation.
Light from the Sun dispersed the remaining gas in the Solar Nebula gas into the
interstellar medium.
Planetary motions reflect the history of their formation.
Planets share the same sense of rotation, but have been perturbed from perfect
alignment by strong collisions during formation.
The Sun "remembers" this original rotation. Rotates in the same direction with its
axis aligned with the plane of the Solar System.
Planetary compositions reflect the formation conditions.
Terrestrial planets are rock & metal:

They formed in the hot inner regions of the Solar Nebula.

Too hot to capture Hydrogen/Helium gas from the Solar Nebula.
Jovian planets contain ice, H & He:

They formed in the cool outer regions of the Solar Nebula.

Grew large enough to accrete lots of H & He.
.
Two obvious differences between the exoplanets and the giant planets in
the Solar System:
• Existence of planets at small orbital radii, where our previous theory
suggested formation was very difficult.
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Astrophysics
Professor Michael Smith
29
• Substantial eccentricity of many of the orbits. No clear answers to
either of these surprises, but lots of ideas...
It is very difficult to form planets close to the stars in a standard theory of planet
formation using minimum mass solar nebula, because

it's too hot there for grain condensation, or

there's too little solid material in the vicinity to built protoplanet's core of 10 ME
(applies to r~1 AU as well).

problematic to build it quickly enough (< 3 Myr)

there's too little gas to build a massive envelope
Most conservative (accepted) possibility:
• Planet formation in these extrasolar systems was via the core accretion model –
i.e. same as dominant theory for the Solar System
• Subsequent orbital evolution modified the planet orbits to make them closer to
the star and / or more eccentric
We will focus on this option. However, more radical options in which exoplanets form
directly from gravitational instability are also possible.
Settling and growth of dust grains: well-coupled to gas, rapid only if turbulent?
Pebbles to planetesimals (km size): inward drift due to gas drag. Gas orbits
slightly slower than Keplerian, so must grow quickly to avoid spiraling in. Twobody collisions, rather than accretion?
PH507
Astrophysics
Professor Michael Smith
30
Planetesimal to rocky planet/gas-giant core: independent of gas, slow –
gravitational dynamics (gravity increase the collision cross-section).
Gas accretion onto core, grav. Torques - migration
Orbital evolution – migration
Giant planets can form at large orbital radii.
Need a migration mechanism that can move giant planets from formation at ~5 AU to a
range of radii from 0.04 AU upwards.
Three theories have been proposed:
• Gas disc migration: planet forms within a protoplanetary disc and is swept
inwards with the gas as the disc evolves and material accretes onto the star. The
most popular theory, as by definition gas must have been present when gas giants
form.
• Planetesimal disc migration: as above, but planet interacts with a disc of rocks
rather than gas. Planet ejects the rocks, loses energy, and moves inwards.
• Planet scattering: several massive planets form – subsequent chaotic orbital
interactions lead to some (most) being ejected with the survivors moving inwards
as above.
Gas disc migration
Planet interacts with gas in the disc via gravitational force.
Strong interactions at resonances, e.g. where disc = nplanet, with n an integer.
For example the 2:1 resonance, where n = 2, which lies at 2-2/3 rp = 0.63 rp
Resonances at r < rp: Disc gas has greater angular velocity than planet. Loses
angular momentum to planet -> moves inwards
Resonances at r > rp: Disc gas has smaller angular velocity than planet. Gains
angular momentum from planet -> moves outwards.
PH507
Astrophysics
Professor Michael Smith
31
Migration type I - no gap
If the object has too small a mass to open a gap, it will drift inwards. The
analysis of Type I migration relies on the (near) exact cancelling of the various
torques
It is thought that the intrinsic imbalance of torques from the inner and outer disk
determines this.
It is very rapid, and may shift the protoplanetary core to arbitrarily small
distance from the star in the allotted ~3 Myr time frame.
Migration type II - inside an open gap
Interaction tends to clear gas away from location of planet.
Result: planet orbits in a gap largely cleared of gas and dust.
Tidal locking of the planet in the gap. The planet, unless more massive than the
surrounding disk, follows the disk's viscous flow.
This process occurs for massive planets (~ Jupiter mass) only.
Earth mass planets remain embedded in the gas though gravitational torques can be
very important source of orbital evolution for them too.
How does this lead to migration?
PH507
Astrophysics
Professor Michael Smith
32
1. Angular momentum transport in the gas (viscosity) tries to close the gap (diffusive
evolution of an accretion disc).
2. Gravitational torques from planet try to open gap wider.
3. Gap edge set by a balance:
-> Internal viscous torque = planetary torque
4. Planet acts as a angular momentum ‘bridge’:
• Inside gap, outward angular momentum flux transported by viscosity within disc
• At gap edge, flux transferred to planet via gravitational torques, then outward
again to outer disc
• Outside gap, viscosity again operative
Typically, gap extends to around the 2:1 resonances interior and exterior to the planet’s
orbit.
As disc evolves, planet moves within gap like a fluid element in the disc – i.e. usually
inwards.
Inward migration time ~ few x 105 yr from 5 AU.
Mechanism can bring planets in to the hot Jupiter regime.
This mechanism is quantitatively consistent with the distribution of exoplanets at
different orbital radii – though the error bars are still very large!
Eccentricity generation mechanisms
Substantial eccentricities of many exoplanets orbits do not have completely satisfactory
explanation. The theories can be divided into groups corresponding to different
formation mechanisms:
(A) Direct molecular cloud fragmentation
(B) Protostellar disk fragmentation theories
(C) Companion star-planet interaction (in double star like 16 Cyg)
PH507
Astrophysics
Professor Michael Smith
33
(D) Classical giant planet formation w/planet-planet interaction
(E) Resonant disk-planet interaction
(D) Scattering among several massive planets
Assumption: planet formation often produces a multiple system
which is unstable over long timescales:
• Chaotic evolution of a, e (especially e)
• Orbit crossing
• Eventual close encounters -> ejections
• Eccentricity for survivors
Advantages:
• Given enough planets, close together, definitely works
• Can produce very eccentric planets cf e=0.92 example discovered
• Some (stable) multiple systems are already known
Disadvantages:
• Requires planets to form very close together.
Is it plausible that unstable systems formed in a large fraction of
extrasolar planetary systems?
• Collisions may produce too many low e systems
(E) Disc interactions
Assumption: gravitational interaction with disc generates eccentricity
Advantages:
• Same mechanism as invoked for migration
• Works for just one planet
• Theoretically, interaction is expected to increase eccentricity
if dominated by 3:1 resonance
Disadvantages:
• Gap is only expected to reach the 3:1 resonance for brown dwarf type
masses, not massive planets. Smaller gaps definitely tend to circularize the
orbit instead.
• Seems unlikely to give very large eccentricities
(3) Protoplanetary disc itself is eccentric
Assumption: why should discs have circular orbits anyway?
Eccentric disc -> eccentric planet?
Not yet explored in much depth. A possibility, though again seems unlikely to lead to
extreme eccentricities.
Scattering theory is currently most popular, possibly augmented
by interactions with other planets in resonant orbits.
THE END