Is a Molecular Cloud Stable?● Gravity will try to collapse any distribution of interstellar

medium.– Countering gravity is thermal support – gas pressure.

● First question then … How quickly can it happen? – 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.– Countering gravity is thermal support – gas pressure.

● First question then … How quickly can it happen? – 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.

● 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.

P = ρ k TμmH

12m v2≃k T 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, .

● Formally

c s=√ k Tm particle

The Jeans Length/Mass● See section 17.1 for details

t ff = [ 3 π32Gρcloud ]

1 /2

t sound = r originalcs

= r original √m particle

k T

r Jeans = √ 3π k T32Gρom 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 = 43

π r Jeans3 ρ0

The Jeans Length/Mass● Using the typical values to create scaling relations

r Jeans = √ 3π k T32Gρom particle

r Jeans = 2000 AU ( T10K )1 /2

( molecular hydrogens per cubic meter1012 )

−1/2

M Jeans = 0.2Msun ( T10K )3/2

( molecular hydrogens per cubic meter1012 )

−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.

● Collapse is halted, to first order, when rotational speed at the edge of the cloud reaches orbital speed

movo ro = m f v f r f

√GMr f

= v f = ( r or f ) vo

v f= ( r or f ) vo

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

● 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 = rdisk = r o voGM

= 200 AU ( vo

0.1 km / s )2

( roriginal4000AU )2

( MM sun )−1

√GMr f

= v f = ( r or f ) vo

● 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(), where 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 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)

Enceladus (Saturn)

Ariel (Uranus)

Origin of the Gasballs