Is a Molecular Cloud Stable?
●
Gravity will try to collapse any distribution of interstellar
medium.
–
●
First question then … How quickly can it happen?
–
●
Countering gravity is thermal support – gas pressure.
At best, in a freefall time.
How big a region has to collapse
–
A dense molecular cloud has 1012 molecular hydrogen atoms per
cubic meter.
–
Let's work it out from there beginning with determining the
number of hydrogen atoms in the Sun.
Is a Molecular Cloud Stable?
●
Gravity will try to collapse any distribution of interstellar
medium.
–
●
First question then … How quickly can it happen?
–
●
●
Countering gravity is thermal support – gas pressure.
At best, in a freefall time.
How big a region has to collapse
–
A dense molecular cloud has 1012 molecular hydrogen atoms per
cubic meter.
–
Let's work it out from there.
Next question – How long will a purely gravitational collapse
take?
–
Use Kepler's Laws! The outermost particle is on an e=1 orbit
around a 1 solar mass object (since this particle is always at the
edge of the mass distribution it always appears to the particle as
if there is a point mass down at the center).
Gas Pressure Resists Collapse
●
Although it is difficult to conceive of the pressure in interstellar
space, there is a well defined pressure described by the ideal
gas law.
ρk T
P=
μ mH
●
A good way of representing whether the feeble local pressure
can resist the weak gravitational pull is to consider the sound
speed – how fast fair warning of gravity's action propagates.
–
If collapse can occur before the sound pressure wave gives
warning then gravity can do it's evil deed before the cloud
notices.
1
2
m v ≃k T
2
cs ≃
√
2k T
m
Sound Speed
●
●
To first order the sound speed is equal to the velocity of the
individual particles in a gas.
Equipartition suggests that the particle energy is ½ kT per
degree of freedom.
–
For particles behaving like ball bearings there are just 3
translational degrees of freedom.
–
More complex particles have additional degrees of freedom (e.g.
internal vibration) putting energy into these modes as well as
translation. In the formal expression of sound speed this is
accounted for by the adiabatic index, γ.
●
Formally
√
γkT
c s=
m particle
The Jeans Length/Mass
●
See section 17.1 for details
t ff
[
3π
=
32 G ρcloud
]
1/ 2
t sound
r Jeans =
√
√
r original
m particle
=
= r original
cs
γkT
3πγk T
32 G ρo m particle
– A cloud has to be bigger than rJeans to collapse, which comes from requiring
that tsound > tff
– There is an associated “Jeans Mass” – simply the mass in a Jeans volume
given the density.
M Jeans
4 3
= π r Jeans ρ0
3
The Jeans Length/Mass
●
Using the typical values to create scaling relations
r Jeans =
r Jeans
√
3πγk T
32 G ρo m particle
T
= 2000 AU
10K
1/2
molecular hydrogens per cubic meter
12
10
)
T
10K
3/ 2
molecular hydrogens per cubic meter
12
10
−1 / 2
M Jeans = 0.2M sun
( ) (
( ) (
−1/ 2
)
Formation of a Star via Cloud Collapse
●
Conservation of angular momentum guides an initially
spherical collapse to form a disk around the forming star.
mo v o r o = m f v f r f
●
ro
vf=
vo
rf
( )
Collapse is halted, to first order, when rotational speed at the
edge of the cloud reaches orbital speed
√
ro
GM
= vf =
vo
rf
rf
( )
Formation of a Star via Cloud Collapse
●
Collapse is halted, to first order, when rotational speed at the
edge of the cloud reaches orbital speed
√
●
ro
GM
= vf =
vo
rf
rf
( )
Given typical 0.1 km/s motions in cloud cores, the centrifugal
radius of the collapsed cloud (now disk) is a couple of hundred
AU (also see section 17.1).
r f = r disk
r o vo
vo
=
= 200 AU
GM
0.1 km/ s
(
2
)(
r original
4000AU
2
−1
)( )
M
M sun
●
This process of disk formation and accretion is common to any
forming star anywhere.
–
Many, if not most, stars are likely to have planets around them.
Disks are Seen Directly
Disks are Seen Directly
Infrared Observations Reveal Disks
●
●
Warm dust emits thermal radiation at infrared wavelengths.
Infrared observations of forming stars reveal excess infrared light
coming from these stars.
–
The distribution of infrared light is consistent with the material being
spread out in a disk.
What is the expected disk temperature profile with
radius?
●
●
If the disk were optically thin, that is, if every particle could see
the Sun unhindered, then we already know temperature
should fall off with radius inversely as the square root of the
distance.
However, now we are talking about an optically thick disk that
is behaving more like a sheet of cardboard.
–
Incident flux needs to be calculated accounting for the Sun
shining obliquely on each square centimeter of surface.
–
Whereas before flux dropped off as R2, it now drops off a
R2*sin(θ), where θ is the apparent angular radius of the sun (and
thus it's altitude in the “sky”) for that square centimeter.
●
●
For a “cardboard” disk T α R3/4.
Ultimately “spectral energy distributions” can reveal the mass
distribution and structure of a circumstellar disk.
Flared Disks
Disks Should Fade As Planets Grow
●
●
As planets grow the disk
should begin to clear.
Simulations/calculations
say that within 10 million
years most of the small
dust should be gone.
–
This age corresponds well
with observations of
young stars showing the
infrared excesses go
away before an age of 10
million years.
Reading the Tea Leaves
Faint Disks Should Linger
●
Leftover comets and asteroids should continue to collide and
generate faint dust signatures.
–
We see it in our solar system and elsewhere.
Beta Pictoris
See this article
Beta Pic is a star with 1.8 times the mass
of the Sun (spectral type A6) located 20
parsecs away.
Fomalhaut (Alpha Piscis Austrini)
●
Structure in debris disks suggests the gravitational influence of
planets.
Hubble
ALMA
Go looking for Fomalhaut due South in the evening (declination -30).
Structured Debris Disks
●
●
Using wavelength of peak
excess as a clue, it is becoming
evident that some systems
have inner and outer debris
zones just like the solar system.
Vega shows excesses at
temperatures of 170K (asteroid
belt analog) and 50K (Kuiper
belt analog)
ALMA
●
●
Nearby forming stars are typically 100 parsecs away. At a distance
of 100 parsecs a typical disk with size 100AU = one arcsecond.
No wonder it is so difficult to study the solar system formation
process directly..... until now.
ALMA
Getting Material Onto the Star
●
●
Turbulence and magnetic fields lead to viscosity that permits
material to lose angular momentum and reach the star.
At small radius magnetic pressure begins to dominate other
forces. Accretion flows likely follow magnetic field lines onto the
star. Click on the right figure for an excellent review article.
Timescales
●
●
For “revealed” T-Tauri stars we can use evolutionary models to
estimate the age of the star.
–
“Disked” stars tend to have ages up a few hundred thousand to a few
million years.
–
Few T-Tauri stars (a.k.a. YSO's for “young stellar objects) older than 3
million years have disks.
–
Protostars are about 10 times less common and thus must evolve to
T-Tauri phase in about 1/10th the time (10-100 thousand years),
consistent with collapse predictions.
But what about cloud support in general. If protostars are forming
on a freefall time, why are there molecular clouds at all?
–
Turbulence, magnetic fields...
Timescales
●
●
For “revealed” T-Tauri stars we can use evolutionary models to
estimate the age of the star.
–
“Disked” stars tend to have ages up a few hundred thousand to a few
million years.
–
Few T-Tauri stars (a.k.a. YSO's for “young stellar objects) older than 3
million years have disks.
–
Protostars are about 10 times less common and thus must evolve to
T-Tauri phase in about 1/10th the time (10-100 thousand years),
consistent with collapse predictions.
But what about cloud support in general. If protostars are forming
on a freefall time, why are there molecular clouds at all?
–
Turbulence, magnetic fields...
Disk Masses
●
●
●
The disk is the conduit for material on its way into the star.
At any given time the disk mass is limited to a few hundredths
of a solar mass by gravitational instabilities.
Indeed, sub-millimeter measurements are used to infer disk
masses of order 0.01-0.05 solar masses around young stars.
Star Formation Efficiency
●
Possibly illuminated by noting the similarity of the “molecular
core” mass function to the stellar initial mass function.
Alves et al.
2007
Making Planets via Accretion
●
●
●
●
The disk concentrates small grains (sub-micron sized) so that
they will collide (gently) frequently.
Initially, electrostatic “sticking” builds millimeter sized clumps.
Gravity then comes into play and permits the assembly of true
“planetesimals” - asteroid size.
Gravity and collisions build larger bodies until collisions
become rare.
Golden Rule: Only solid
particles take part in the
accretion process.
Making Planets via Accretion
●
●
●
●
The disk concentrates small grains (sub-micron sized) so that
they will collide (gently) frequently.
Initially, electrostatic “sticking” builds millimeter sized clumps.
Gravity then comes into play and permits the assembly of true
“planetesimals” - asteroid size.
Gravity and collisions build larger bodies until collisions
become rare.
Golden Rule: Only solid
particles take part in the
accretion process.
Making Planets via Accretion
●
●
●
●
The disk concentrates small grains (sub-micron sized) so that
they will collide (gently) frequently.
Initially, electrostatic “sticking” builds millimeter sized clumps.
Gravity then comes into play and permits the assembly of true
“planetesimals” - asteroid size.
Gravity and collisions build larger bodies until collisions
become rare.
Making Planets via Accretion
●
●
●
●
The disk concentrates small grains (sub-micron sized) so that
they will collide (gently) frequently.
Initially, electrostatic “sticking” builds millimeter sized clumps.
Gravity then comes into play and permits the assembly of true
“planetesimals” - asteroid size.
Gravity and collisions build larger bodies until collisions
become rare.
Making Planets via Accretion
●
●
●
●
The disk concentrates small grains (sub-micron sized) so that
they will collide (gently) frequently.
Initially, electrostatic “sticking” builds millimeter sized clumps.
Gravity then comes into play and permits the assembly of true
“planetesimals” - asteroid size.
Gravity and collisions build larger bodies until collisions
become rare.
Interesting Subtleties
●
●
●
●
The gas that goes along for the ride (but that does not directly
participate in accretion) likely plays an important role.
The gas orbits at less than Keplerian velocity because it has
pressure and thus requires less centripetal acceleration to
maintain an orbit.
Particles themselves must follow Kepler's laws – all particles
feel a headwind.
–
Smaller particles are more significantly influenced than larger
ones.
–
Differential drift velocity aids in collisions needed for accretion
until those relative velocities become too large.
–
Around a meter in size those relative velocities begin to lead to
disruptive collisions.
Gravity is still weak for these particles so crossing the meter
barrier requires some collective gravitational instability –
collapsing clumps of planetesimals.
The Jovian/Terrestrial Distinction
●
●
●
Planets accrete from the solid particles in the disk.
The local temperature of the disk determines what types of
particles can survive in solid form.
The obvious temperature gradient is likely connected to
compositional differences with radius in the Solar System.
The Jovian/Terrestrial Distinction
●
●
Planets accrete from the solid particles in the disk.
The local temperature of the disk determines what types of
particles can survive/condense in solid form.
Condensation/Survival vs. Temperature
The Jovian/Terrestrial Distinction
●
Close to the Sun, rock and metal are the primary building
materials. Ice dominates at greater distance (and in a big way,
note the relative abundance of silicates vs. ices).
Icy Moons (likely with significant volatile content)
Ariel (Uranus)
Enceladus (Saturn)
Origin of the Gasballs