Download Anomalous diffusion in generalised Ornstein

Planet Formation – an unresolved
puzzle
Michael Wilkinson (Open University)
Bernhard Mehlig (Gothenburg University)
Overview
Lecture 1:
Particles suspended in a turbulent fluid flow can cluster together. This
surprising observation has been most comprehensively explained
using models based upon diffusion processes.
Lecture 2:
Planet formation is thought to involve the aggregation of dust particles
in turbulent gas around a young star. Can the clustering of particles
facilitate this process?
The final lecture will discuss the problems of planet formation, and
consider whether aggregation of particles is relevant. Clustering
does not help to explain planet formation, but planet formation does
introduce new applications of diffusion processes.
However, no version of the dust aggregation model appears to be
satisfactory. An alternative hypothesis is introduced.
The solar system
Eight
planets, numerous smaller bodies, orbiting a typical star, mass
:
M,V,E,M are rocky planets
J,S are gas giants
U,N are ice giants
The structure has some striking features:
The orbits are nearly circular, and nearly coplanar with equator of sun.
The planets rotate about axes close to the rotation axis of the sun.
Several planets have moons, again with circular, coplanar orbits.
These features suggest origin in a disc.
Formation of stars by gravitational collapse
The interstellar medium includes large molecular clouds: mainly molecular
hydrogen,
some helium. There are also heavier materials which were made
:
inside stars, and which where blown away from the surface, or dispersed by a
supernova explosion. This includes water, small organic molecules, dust.
The density and temperature are very low:
The cloud collapses due to gravitational self-attraction. The collapse is limited by
build up of pressure (which can be relieved as the collapsing gas cools) and by
the conservation of angular momentum.The cores of the collapsed regions
become dense and hot enough to start nuclear reactions (hydrogen to helium..).
They become stars.
Artist’s impression of a nascent
star (a ‘protostar’)
http://www.daviddarling.info/images/protostar.jpg
Circumstellar discs
Some material has too much angular momentum to fall in to the stars, remaining
:
behind
as circumstellar nebula. It forms a structure with as little internal motion and
potential energy as possible for the amount of angular momentum: a circumstellar
disc.
Beta Pictoris at 0.5 micron
Artist’s impression
http://www.daviddarling.info/images/dust_disk_080205.jpg
http://astro.berkeley.edu/~kalas/disksite/pages/bpic.html
Accretion in circumstellar discs
Material in the circumstellar disc rotates with a speed close to the Kepler velocity:
:
Because the Kepler rotation rate decreases with radius, there is a velocity gradient.
If there are viscous forces acting, they cause a torque which speeds up material at
higher radius, and slows material at smaller radius.
Angular momentum is transferred from the centre to the outside of the disc. As the
material at small radius slows, gravity draws it inwards. Viscous torque causes
material to accrete onto the star by transferring angular momentum outwards.
Viscosity causes the circumstellar disc to disappear. But molecular viscosity is much
too slow. Surveys of stars indicate that the disc lifetime is order one million years. It
is surmised that turbulence causes an effective viscosity which is much larger than
the molecular viscosity.
The standard model for planet formation
Model
:
Dust particles in the accretion disc collide and stick together, making larger and
larger clusters.
Eventually the clusters become large enough that their mutual gravitational
interaction is significant. These planetessimals continue to collide with each other
until they form planets.
Supporting evidence
Because the planets form within a disc, the model explains why the planets in our
solar system have nearly circular and nearly coplanar orbits.
Theoretical Questions
We need to understand the rate of collision of particles in a turbulent environment,
which is determined by their relative velocities.
We need to estimate the intensity of turbulence in the circumstellar disc.
Turbulence
Fluid motion is determined by the Navier-Stokes equation.
When the effects of viscosity are weak, the fluid flow is very complex:
Image from M.A.Green et al, J. Fluid. Mech.
572, 111 (2007).
Statistical ideas must be used to
describe turbulence.
Turbulence is a multi-scale flow:
large-scale eddies interact to make
smaller scale eddies. Energy is
dissipated by viscosity on the
smallest scale of the cascade.
Kolmogorov theory of turbulence
All modern theories of turbulence are based upon work by Kolmogorov (1941). He
argued that the on shorter length scales the turbulence ‘forgets’ how it was
generated. The only variables that enter are the energy dissipation per unit mass
and the kinematic viscosity
(which is a diffusion constant for the transfer of
momentum). The kinematic viscosity can only influence properties of the smallest
eddies, where energy is dissipated.
Dissipation of energy occurs at the Kolmogorov lengthscale
These are deduced by dimensional analysis:
and timescale
Many properties of motion one scales larger than the dissipation scale can be
deduced from dimensional analysis alone:
.
Motion of dust particles
Model the dust grains as spheres of radius
moving in a gas with a turbulent
velocity field
. The equation of motion is
Damping rate for a spherical particle of radius
given by Stokes
in a conventional fluid was
In gases with very low density (mean free path large compared to the particle), the
same equation of motion hold. The damping rate was obtained by Epstein (1923):
Collision rates
For spherical particles the collision rate of a particle in a dilute gas is:
It is important to estimate the relative velocity of the dust particles.
When the dust grain clusters become sufficiently large, their inertia becomes
important, and their velocity becomes different from that of the surrounding gas.
We model the evolution of the relative positions and velocities of particles by a
diffusion process.
Diffusive model for relative velocities
Equation of motion for separation of dust grains:
Langevin model:
Corresponding Fokker-Planck equation:
Predictions for relative velocities
An approximate solution of the Fokker-Planck equation yields the following
approximate distribution for relative velocities:
From this we obtain an estimate for the variance:
Here
is dimensionless. This can also be surmised from the Kolmogorov
theory of turbulence. We expect that the motion of weakly-damped particles
depends only upon the rate of dissipation, and not on the kinematic
viscosity. The result then follows from dimensional analysis alone.
Modelling circumstellar gas discs
We
: will find that the properties of accretion discs are (with some simplifying
assumptions) determined by the accretion rate
. Surveys indicate disc
lifetimes of roughly a million years, and it is estimated that the disc initially
contains about 10 percent of the mass of the star. We therefore use
Other parameters for model:
Accretion discs – conservation principles
The structure is determined by the conservation equations for mass, angular
:
momentum and energy, combined with equations for hydrostatic equllibrium, and
an estimate of turbulent viscosity.
Viscous drag
leads to a torque
Conservation of mass:
Transport of angular momentum:
Steady state solution, valid at large radius:
Viscous drag causes dissipation, at a rate per unit area
Find:
Accretion disc structure - equations
Mass, angular momentum and energy balance give:
:
We make an ansatz for the effective kinematic viscosity due to turbulent motion:
Equations are closed by relating disc height to temperature and speed of sound:
These yield an estimate:
These equations describe a thin, optically thick accretion disc which is heated by
viscous dissipation. The solution is specified by a single parameter, the accretion
rate
.
Some properties of circumstellar discs
Our
: steady-state model predicts that quantities have a power-law scaling:
Values at radius of
based upon
Collision velocity as a function of size
The Stokes trap
The relative velocity of clusters with Stokes number larger than unity is
sufficiently large that the clusters will usually fragment upon impact: for solartype stars we estimate (at
radius):
When dust clusters become large enough that their Stokes number is order
unity, they start to fragment upon collision. This is the Stokes trap. It occurs
for particles of approximate size:
It is not clear how to escape this ‘Stokes trap’ and proceed to larger clusters.
Extra-solar planets : detection
:
About 300 extra-solar planets have been detected, orbiting other stars. Several
methods have been used:
1. The ‘wobble’ in motion of a star due to an orbiting planet can be detected by
the Doppler effect.
Velocity of star
Limit of sensitivity:
Apparent amplitude
Signal due to Jupiter:
2. The transit of a planet across a star can be detected as a slight reduction in
luminosity of the star.
3. Gravitational lensing effects from planets have been observed.
With these techniques it is easier to detect large planets with small rotational
periods.
Extra-solar planets : observations
: The extra-solar planets turned out to have some surprising features. Some
observations:
1. There are many planets with large eccentricity.
2. There are numerous examples of ‘hot Jupiters’, that is gas-giant planets
which orbit very close to their star.
3. There are many examples with more than one planet. Resonances between
periods are common in these cases:
Data on exo-solar planets : eccentricity
:
Data on exo-solar planets : masses
:
Concurrent collapse
We propose that planets are formed by gravitational condensation of a cloud
of interstellar gas, at the same time as their star is being formed by the same
process.
This mechanism does not rely upon colliding particles adhering to each
other.
The objects which are produced by gravitational collapse will be termed
juvenile planets. They must initially have a composition which is similar to
that of the interstellar medium. They would have to undergo a radical
transformation to become rocky or icy planets. We propose that such
transformations can occur due to interaction with the residual gas in the
protostellar nebula, or by collisions.
Gravitational collapse
Simulations of gravitational collapse by Matthew Bate,
http://www.astro.ex.ac.uk/people/mbate/Research/pr.html)
Interaction of juvenile planet with gas disc
Juvenile planet may pass through the gas disc at a high velocity, of order
Material can be removed from its outer layers, reducing the fraction of lighter
elements (because heavier elements may sink to its core).
.
Also, momentum is transferred, and the motion of the juvenile planet might
become entrained to the disc (reaching a nearly circular, nearly in-plane orbit). Can
this happen?
The planet becomes entrained to the disc if the mass of gas displaced is
comparable to the mass of the juvenile planet. Estimate the number of orbits for
this to be achieved:
Planet mass, density, size
Gas disc density, orbit radius
Mass of gas displaced per orbit
planet entrained in
The numbers in this estimate are uncertain by orders of magnitude. This implies that
in some cases juvenile planets would be entrained to circular orbits, in other cases
the orbits remain elliptical.
Collisions between juvenile planets
Let us estimate the probability for collision of two juvenile planets, in eccentric orbits
of approximate radius
:
Approximate area swept out by orbit of one planet:
Approximate cross sectional area of a planet:
Probability of collision per orbit:
We conclude that collisions between juvenile planets in elliptical orbits are possible.
Depending upon parameters, they may be unlikely or almost inevitable.
What are the consequences? Large amounts of debris would be produced. If two
Jupiter size planets collide, large amounts of debris will be scattered into orbits which
will reach the star. The first fragments will reach the star in a fraction of the orbital
period (perhaps 1-10 years), and material will continue to reach the star at a
significant rate for several multiples of the orbital period (perhaps 10-100 years). The
total amount of material is
solar masses, which gives a rate of accretion three
orders of magnitude higher than the typical rate. Collisions can explain the
magnitude and timescale of outbursts.
FU Orionis outbursts
FU Orionis is a star which started to increase in
luminosity in 1937. Over a few years the
luminosity increased by three orders of
magnitude, before starting to decrease.
The star appears to be surrounded by a dusty
cloud, consistent with it being a young star.
http://www.daviddarling.info/images/FU_Orionis.jpg
Since them many other outburst events have been observed in other young stars.
The events are very diverse. Common features are: quick rise time (1-10 years),
relatively slow decay (10-100 years), increase in magnitude is very large (2-6
orders). There is no apparent change preceding the outburst. Surveys suggest that
most young stars experience a few outbursts.
Most published explanations assume an instability of the accretion disc. These
instabilities require the gas to become opaque due to ionisation of hydrogen. This
requires very high temperatures.
Summary on concurrent collapse
The concurrent collapse hypothesis avoids the problems of having to build planets
by aggregation of dust in a turbulent environment. It suggests simple models or
explanations for some observations:
1. The juvenile planets are formed in eccentric orbits. The existence of circular
orbits can be explained by entrainment to the gas disc.
2. Collisions between juvenile planets could explain ‘outbursts’.
3. The model is consistent with planetary systems being highly variable in
structure.
The juvenile planets are created with a composition reflecting that of the interstellar
medium. Rocky or icy planets can be formed if heavier elements sink to the core of
the juvenile planet, and if motion through the gas in the circumstellar disc strips
away the outer layers.