Sternentstehung - Star FormationSommersemester 2006

Henrik Beuther & Thomas Henning

24.4 Today: Introduction & Overview1.5 Public Holiday: Tag der Arbeit8.5 ---15.5 Physical processes, heating & cooling, cloud thermal structure22.5 Physical processes, heating & cooling, cloud thermal structure (H. Linz)29.5 Basic gravitational collapse models I5.6 Public Holiday: Pfingstmontag12.6 Basic gravitational collapse models II19.6 Accretion disks26.6 Molecular outflow and jets3.7 Protostellar evolution, the stellar birthline10.7 Cluster formation and the Initial Mass Function17.7 Massive star formation, summary and outlook24.7 Extragalactic star formation

More Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss06.html beuther@mpia.de

Summary last week- Isothermal sphere, Lane-Emden equation, density profile of SIS r-2 - Bonnor-Ebert mass = Jeans mass MBE = MJ = m1at

3/(01/2G3/2) = 1.0Msun (T/(10K))3/2 (nH2/(104cm-3)-1/2

Jeans length J = (at

2/G0) = 0.19pc (T/(10K))1/2 (nH2/(104cm-3)-1/2

- Virial theorem: 1/2 (2I/t2) = 2T + 2U + W + M If T, U and M very small, one can calculate free-fall time: tff = (3/32G)1/2

- Thermal energy U to small to counteract gravity, but magnetic energy M turns out to be important.- Many clouds in virial equlibrium 2T = -W virial velocity: vvir = (Gm/r)1/2

virial mass: mvir = v2r/G- Rotational energy not sufficient for cloud support- Rotation and magnetic field can increase critical masses for collapse and fragmentation but magnetic field far more important. --> if core mass below magnetically critical mass, core may be stable against collapse even with increased outer pressure.

Star Formation Paradigm

Ambipolar diffusion I

The drift velocity between ions and neutrals is vdrift = vi - vn And the drag force between ions and neutrals is: Fdrag = nn<invdrift>mnvdrift

(average number of collision per unit time nn<invdrift> times the transferred momentum mnvdrift)

The equation of motion with the Lorentz force is then:

niFdrag = j x B/c = 1/4π (rot B) x B (with Ampere’s law: rot B = 4π/c * j) vdrift = (rot B) x B / (4πninnmn <invdrift>)

Neutrals stream through the ions accelerated by gravity. There is a drag force between ions and neutrals from collisions.Furthermore, Lorentz force acts onions.

In less dense GMCs, the ionization degree is relatively large and ions andneutrals are strongly collisionly coupled. Going to denser molecular cores, the ionization degree decreases, and neutrals and ions can easier decouple.

nn: neutral density ni: number of ions in: ion-neutral cross section mn: mass of neutral

z

Ambipolar diffusion II

For a dense core with a size L, the time-scale for ambipolar diffusion is:

tad = L/|vdrift| = (4πninnmn <invdrift>)L / (|(rot B) x B|)

Approximating |(rot B) x B| = B2/L we get

tad = (4πninnmn <invdrift>)L2 / B2

Hence ambipolar diffusion time-scale is proportional to ionization degree, density and size of the cloud, and inversely proportional to magnetic field.

tad ≈ 3x106yr (nH2/104cm-3)3/2 (B/30µG)-2 (L/0.1pc)2

It is still much under discussion whether this time-scale sets the rate wherestar formation takes place or whether it is too slow and other processes like turbulence are required.

Ambipolar diffusion III

Sequence of cloud equilibria can be described as continuous change of dM/dB.

The mass enclosed by given flux M(B) increases with time at fixed B.For a centrally condensed core, the force balance can be approximated by

1/4π (rot B) x B = GMinner

And one gets for the ambipolar diffusion timetad = /|vdrift| = (4πninnmn <invdrift>) 2/ (4πGMinner)

= ni<invdrift>3/(GMinner)

In the central region for small ni 3 --> tad short, whereas further out withlarger ni and larger radius tad also increases

z

Ambipolar diffusion IV: Numerical modeling

- Start with a uniform density cylinder and fixed boundary conditions.- Height of cylinder exceeds Jeans-length --> immediate collapse- After 6x106yrs everything has settled in oblate configuration.- Then density increases due to ambipolar diffusion.- At 1.5x107yrs, surface density that high that collapse accelerates. --> Collapsing cloud core effectively separates from outer cloud. B at the center increases.

1.02x107 yr 1.51x107yr

1.60x107yr 1.61x107yr

Magnetic reconnection

- Field lines of opposite direction are dragged together --> antiparallel B field lines annihilate and magnetic energy is dissipated as heat.- This process was first invoked to explain large luminosities observed in solar flares.

Ambipolar diffusion caveat

Star Formation timescale: Observations indicate rapid star formationon the order 1-2 million years. Ambipolar diffusion usually requireslonger cloud life-times.

Maybe gravo-turbulent fragmentation necessary

Interstellar Turbulence- Supersonic --> creates network of shocks- Shock interactions create density fluctuations M2 which can be relatively quiescent.- In these higher-density regions, H and H2 may form rapidly- tform = 1.5x109yr / (n/1cm-3) (Hollenbach et al. 1971)

--> either molecular clouds form slowly in low-density gas or rapidly in ~105yr in n=104cm-3

- Decays on time-scales of order the free-fall time-scale --> needs to be replenished and continuously driven Candidates: Protostellar outflows, radiation from massive stars, Supernovae explosions

k=2 large-scale

k=4 intermediate scale

k=8 small scale

MacLow 1999

Gravo-turbulent fragmentation

2 phases:-- Turbulent fragmentation-- Collapse of individual gravitational unstable cores

Resulting clumb-mass spectraresembling the stellar initial massfunction.

Freelydecayingtubulencefield

Driventurbulencewith wave-number k(perturbationsl= L/k)

Klessen2001

Simulation exampleSPH simulation withgravity and super-sonic turbulence.

Initial conditions:

Uniform density1000Msun

1pc diameterTemperature 10K

QuickTime™ and aBMP decompressor

are needed to see this picture.

Collapse of a core IInitial conditions: spherical self-gravitating isothermal sphere, simplified without magnetic field and rotation reaches r-2

r-2

Shu 1977

Ward-Thompson et al. 1999

Motte et al. 1998

Collapse of a core II Important equations:Mass within radius r: Mr = ∫4πr2dr --> ∂Mr/∂r = 4πr2Continuity equation: ∂/∂t = -1/r2 ∂(r2u)/∂r --> ∂Mr/∂t = -4πr2uHydrostatic momentum eq.: ∂u/∂t + u∂u/∂r = -1/grad(P) - grad()

With P = at for the ideal isothermal gas and grad() = -GMr/r2

∂u/∂t + u∂u/∂r = -at2/∂/∂r - GMr/r2

This set of equations can be solved numerically for given boundary conditions.

Log10(r) [cm]

Log

10(

) [g

cm

-3]

Log10(r) [cm]

Log

10(|

u|)

[cm

s-1]

- After forming a central hydrostatic core --> protostar, the collapse starts from the inside-out.- Density and velocity profile in the inner free-fall region r-3/2 & u r-1/2

- Density profile in the outer envelope r-2

Singular isothermal sphere (Shu 1977)

Collapse of a core III

Mass accretion rate: M = ∂Mr/∂t = lim -4πr2u = m0 at3/G

r--> 0

.For a singular isothermal sphere (SIS) one gets

For a typical sound speed of 0.2km/s at T=10K:M ~ 2 x 10-6 x(T/10K) Msun/yr

--> Hence 1Msun star would form in 5 x 105 yr.

Using different initial conditions, e.g. Bonnor-Ebert spheres, one can get:

.

One finds an initialfree-fall phase until the protostar has formed. Thenthe collapse continues in an inside-out mode but with varying accretion rate.

Myers 2005

Collapse of a core IV

The inner free-falling gas causes an inner pressure decrease, and ararefaction wave moves outward.

Accretion shock and Accretion luminosity

The gravitational energy released per unit accreted mass can beapproximated by the gravitational potential GM*/R* (with M* and R* the mass and radius of the central protostar)

Hence the released accretion luminosity of the protostar can be approximated by this energy multiplied by the accretion rate:

Lacc = GMM*/R*

= 61Lsun (M/10-5Msun/yr) (M*/1Msun) (R*/5Rsun)-1

..

Rotational effects ICentrifugal force: Fcen j2/3 Gravitational force: Fgrav Gm/2

( radius on cylindrical coordinates)

Since the initial ratio of rotational to gravitational energy is

Trot/W ≈ 10-3

Rotation gets important after shrinkage of the core by a factor 1000.

However, this is only valid in non-magnetic cloud.B-field anchored to rarifiedouter cloud. Spin-up twiststhe field, increases magn.tension and creates magn.torque counteracting spin-up.

--> magnetic breaking: dense cores approximately corotate with cloud.

Rotational effects IIIn dense core centers magnetic bracking fails because neutral matter and magnetic field decouple with decreasing ionization fraction. matter within central region can conserve angular momentum.Since Fcen grows faster than Fgrav each fluid element veers away fromgeometrical center. --> Formation of disk

The larger the initial angular momentum j of a fluid element, the further away from the center it ends up --> centrifugal radius cen

cen = m03at0

2t3/16 = 0.3AU (T/10K)1/2 (0/10-14s-1)2 (t/105yr)3

cen can be identified with disk radius. Increasing with time because in inside-out collapse rarefaction wave moves out --> increase of initial j.

Observable Spectral Energy Distributions (SEDs)

Infall signatures I

1. Rising Tex along line of sight2. Velocity gradient3. Line optically thick4. An additional optically thin line peaks at center

From Evans 1999

Ovals are loci of constantline-of-sight for v(r) r-0.5

Infall signatures IIModels Spectra and fits (Myers et al. 1996)

In a relatively simple model with two uniform regions along the line of sightwith velocity dispersion and peak optical depth 0, one can estimate the infall velocity vin: vin ≈ 2/(vred-vblue) * ln((1+eTBD/TD)/(1+eTRD/TD))

In low-mass regions vin is usually of the order 0.1 km/s.

Summary today

Important missing ingredients are molecular outflows and flattenedAccretion disks --> subject of the next weeks.

- Ambipolar diffusion- Magnetic reconnection- Turbulence- Cloud collapse of singular isothermal sphere and Bonnor-Ebert sphere- Inside out collapse, rarefaction wave, density profiles, accretion rates- Accretion shock and accretion luminosity- Rotational effects, magnetic locking of “outer” dense core to cloud, further inside it causes the formation of accretion disks- Observational features: SEDs, Infall signatures

Sternentstehung - Star FormationSommersemester 2006

Henrik Beuther & Thomas Henning24.4 Today: Introduction & Overview1.5 Public Holiday: Tag der Arbeit8.5 ---15.5 Physical processes, heating & cooling, cloud thermal structure22.5 Physical processes, heating & cooling, cloud thermal structure (H. Linz)29.5 Basic gravitational collapse models I5.6 Public Holiday: Pfingstmontag12.6 Basic gravitational collapse models II19.6 Accretion disks26.6 Molecular outflow and jets3.7 Protostellar evolution, the stellar birthline10.7 Cluster formation and the Initial Mass Function17.7 Massive star formation, summary and outlook24.7 Extragalactic star formation

More Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss06.htmlbeuther@mpia.de