Download Lecture 21: Planet formation III. Planet

Lecture 21: Planet formation III.
Planet-disk interactions
Outline
• Definition of a planet
• Properties of exoplanets
• Formation models for exoplanets
• gravitational instability model
• core accretion scenario
• migration
Armitage 2010, “Astrophysics of Planet Formation” & arXiv notes
Udry, Fischer and Queloz 2007, PPV
PPVI: Chapters by Raymond, Baruteau, Benz, Fischer etc.
Main formation scenarios
• Core accretion scenario
1. Coalescence of solid particles. Growth from
dust to rocky planets.
2. Big rocky planets (>= 10 M⊕) accrete gas and
form gas planets
Core Accretion Scenario
The core accretion model for gas giant formation rests on one assumption: a
seed of a planet or core forms rapidly enough that it can exceed a certain
critical mass prior to the dissipation of the gas disk.
If this condition is satisfied, it can be shown (e.g. Perri & Cameron 1974) that
the core triggers a hydro-dynamical instability that results in the onset of rapid
gas accretion on to the core.
Since the critical core mass is typically of the order of 10 Mearth, the end result
is a largely gaseous but heavy element enriched planet that at least
qualitatively resembles Jupiter or Saturn.
The main phases in the formation of giant planets via core accretion are:
Core formation
Hydrostatic growth
Runaway growth
Termination of accretion
Core Accretion Scenario
Armitage 2010
Core Accretion Scenario
Core formation: A solid protoplanet (“core”) grows via a succession of twobody collisions until it becomes massive enough to retain a significant
gaseous atmosphere or envelope (similar to terrestrial planet formation).
Hydrostatic growth: Initially the envelope surrounding the solid core is in
hydrostatic equilibrium. Over time, both the core and the envelope grow until
the core exceeds a critical mass.
Runaway growth: Once the critical mass is exceeded a runaway phase of
gas accretion ensues. The rate of growth is “supply-limited” and defined by
the hydrodynamic interaction between the growing planet and the disk.
Termination of accretion: Eventually the supply of gas is exhausted, either
as a consequence of the dissipation of the entire protoplanetary disk or, more
probably, as a consequence of the planet opening up a local gap in the disk.
Accretion tails off and the planet commences a long phase of cooling and
quasi-hydrostatic contraction.
Core Accretion Scenario
The first fully consistent time-dependent models of giant-planet growth were
published by Pollack et al. (1996):
Truncated
by gap
formation
Σ
1. Embryo
formation
(runaway)
2. Embryo
isolation
3. Rapid gas
accretion
Pan
Planetary Migration
The presence of a planet in a protoplanetary disk (PPD)
modifies the distribution of gas in the planet vicinity.
Gravitational interaction between the planet and the non-uniform
arrangement of gas generates torques that alter the planet’s
orbit.
This causes a planet to migrate toward or away from the star
and also dumps or excites the orbital eccentricity and inclination.
The direction and rate of migration vary depending on the mass
of the planet and the local properties of the gas disk.
Planetary Migration
(Chambers 2009)
Type I migration
The perturbation the planet causes is
small enough that it does not alter the
background structure of the gas disk.
It affects Earth-mass planets, which
induce a linear perturbation in the
surrounding disk. The migration rate is
proportional to the mass of the planet
and the surface density of the disk.
Time scale of inward Type I
migration (1 solar mass star):
~105 yr.
Planetary Migration
(Chambers 2009)
Type II migration
As planets become more massive, the Type I
torque that they exert on the disk increases
and eventually starts to modify the disk
structure in the neighborhood of the planet.
Since the interaction adds angular
momentum to the disk exterior to the planet
and removes it from the interior gas, the
overall effect is that a strong torque repels
gas from the vicinity of the planet orbit,
creating a gap.
It affects Jupiter-mass planets. The planet’s
motion becomes tied to the viscous evolution
of the disk.
9
Viscous torques
Since all molecules revolve around the protostar in ~Keplerian
orbits, those closer to the center move faster than those
farther away.
Collisions speed up the outer molecules, driving them
outwards and slow down the inner molecules, which falls
towards the center.
The net effect is that most of the matter diffuses inwards,
angular momentum is transferred outwards and the disk as a
whole spreads.
Viscous torques
The disk evolves on a diffusion timescale (Armitage II, C, 4):
t diff
2
rdisk
=
νv
rdisk is the disk radius and νv is the molecular viscosity.
Considering thermal motions:
€
ν v = l fp c s
l fp ≡ mean free path of the molecule =
1
nσ mol
c s ≡ sound speed
€
Molecular viscosity is far too small to produce significant
viscous evolution over the lifetime of a PPD.
€
Viscous torques
If turbulence is present, turbulent fluid motions will result in the macroscopic
mixing of fluid elements at neighboring radii, which can act as an “effective” or
“turbulence” viscosity.
The magnitude of this turbulent viscosity can be estimated from dimensional
arguments:
• If the turbulence is approximately isotropic, the outer scale of the turbulent
flow is limited to be no larger than the smallest scale in the disk (the disk
scale-height, Hz).
• The velocity of the turbulent motions can be limited to be no larger than the
sound speed cs, since supersonic motions result in shocks and rapid
dissipation.
Viscous torques
We therefore write the turbulent viscosity in the form:
ν v = α v c sH z
αv is a dimentionless quantity, known as the Shakura-Sunyaev α
parameter, that measures the efficiency of angular momentum transport
due to turbulence (typically between 10-2 and 10-4; Shakura & Sunyaev
€
1973).
With the above equation, it is possible to specify the viscosity in terms of
local disk quantities and then compute the local disk structure.
Limitations: α may vary with temperature, density and composition of the
disk gas, but it is assumed constant to proceed with the calculation!
Planetary Migration
15
Disk-Planet Interactions
Disk-Planet Interactions
Model Comparison
Outstanding Questions
Lissauer & Stevenson 2007, PPV:
!How rapid do solid cores accrete in the giant planet formation region?
!How does a (growing or formed) giant planet interact with the surrounding
protoplanetary disk ? (predicted migration rates are too rapid for the survival
of all the giant planets known - are disks significantly more massive??).
(Chambers 2009)
20
Some Recent Exoplanet Results: 2011:
the Earth-like exoplanet!
http://upload.wikimedia.org/wikipedia/commons/0/07/Kepler-22b_System_Diagram.jpg
Potentially Habitable Worlds
http://phl.upr.edu/projects/habitable-exoplanets-catalog
Kepler 62
Kepler 62e
1.61 Earth Radii
1.2 times solar flux
Kepler 62f
1.41 Earth Radii
0.41 times solar flux
NASA/Ames/JPL-Caltech
Gliesse 667C
4 – 8 Earth
masses
2 – 4 Earth
masses
3 – 5 Earth
masses
Recent Exoplanet Results: 2012:
http://www.huffingtonpost.com/2012/07/18/alien-planet-ucf-101-nasa-telescope-lava_n_1683793.html
28
Alpha Centauri B has an Exoplanet
Dumusque et al. (2012)
Frequency of Planets
- One in 6 stars has an Earth or Super-Earth sized
planet with a period of less than 50 days (Petigura
et al. 2013)
- 50% of all Sun like stars have a planet, of any
size, with a period less than 85 days – roughly the
period of Mercury (Fressin et al. 2013)
- Twice as many planets in the galaxy as there are
stars (Cassan et al. 2012)
- 94% chance of an Earth like planet within 10 light
years (Kopparapu 2013)