Transcript Document 7622023

Sternentstehung - Star Formation
Sommersemester 2006
Henrik Beuther & Thomas Henning
24.4
1.5
8.5
15.5
22.5
29.5
5.6
Today: Introduction & Overview
Public Holiday: Tag der Arbeit
--Physical processes, heating & cooling, cloud thermal structure
Physical processes, heating & cooling, cloud thermal structure (H. Linz)
Basic gravitational collapse models I
Public Holiday: Pfingstmontag
12.6
19.6
26.6
3.7
10.7
17.7
24.7
Basic gravitational collapse models II
Accretion disks
Molecular outflow and jets
Protostellar evolution, the stellar birthline
Cluster formation and the Initial Mass Function
Massive star formation, summary and outlook
Extragalactic star formation
More Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss06.html
[email protected]
Summary last week
- Isothermal sphere, Lane-Emden equation, density profile of SIS r-2
- Bonnor-Ebert mass = Jeans mass
MBE = MJ = m1at3/(01/2G3/2) = 1.0Msun (T/(10K))3/2 (nH2/(104cm-3)-1/2
Jeans length
lJ = (pat2/G0) = 0.19pc (T/(10K))1/2 (nH2/(104cm-3)-1/2
- Virial theorem: 1/2 (d2I/dt2) = 2T + 2U + W + M
If T, U and M very small, one can calculate
free-fall time: tff = (3p/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
In less dense GMCs, the ionization degree is relatively large and ions and
neutrals are strongly collisionly coupled. Going to denser molecular cores,
the ionization degree decreases, and neutrals and ions can easier decouple.
z
Neutrals stream through the ions
accelerated by gravity. There is a
drag force between ions and
neutrals from collisions.
Furthermore, Lorentz force acts on
ions.
The drift velocity between ions and neutrals is vdrift = vi - vn
And the drag force between ions and neutrals is: Fdrag = nn<sinvdrift>mnvdrift
(average number of collision per unit time nn<sinvdrift> 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 <sinvdrift>)
nn: neutral density
ni: number of ions
sin: ion-neutral cross section
mn: mass of neutral
Ambipolar diffusion II
For a dense core with a size L, the time-scale for ambipolar diffusion is:
tad = L/|vdrift| = (4πninnmn <sinvdrift>)L / (|(rot B) x B|)
Approximating |(rot B) x B| = B2/L we get
tad = (4πninnmn <sinvdrift>)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 where
star formation takes place or whether it is too slow and other processes
like turbulence are required.
Ambipolar diffusion III
z
Sequence of cloud equilibria can be described as continuous change of dM/dFB.
The mass enclosed by given flux M(FB) increases with time at fixed FB.
For a centrally condensed core, the force balance can be approximated by
1/4π (rot B) x B = GMinner/w2
tad
And one gets for the ambipolar diffusion time
= w/|vdrift| = (4πninnmn <sinvdrift>)w w2/ (4πGMinner)
= ni<sinvdrift>w3/(GMinner)
In the central region for small ni w3 --> tad short, whereas further out with
larger ni and larger radius w-->tad also increases
Ambipolar diffusion IV: Numerical modeling
1.02x107 yr
1.60x107yr
1.51x107yr
1.61x107yr
- 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.
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 formation
on the order 1-2 million years. Ambipolar diffusion usually requires
longer cloud life-times.
 Maybe gravo-turbulent fragmentation necessary
Interstellar Turbulence
- Supersonic --> creates network
of shocks
- Shock interactions create density
fluctuations dM2 which can
be relatively quiescent.
- In these higher-density regions,
H and H2 may form rapidly
- tform = 1.5x109yr / (n/1cm-3)
k=2 large-scale
(Hollenbach et al. 1971)
k=4 intermediate scale
k=8 small scale
MacLow 1999
--> 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
Gravo-turbulent fragmentation
Klessen
2001
2 phases:
-- Turbulent fragmentation
-- Collapse of individual
gravitational unstable cores
Resulting clumb-mass spectra
resembling the stellar initial mass
function.
Freely
decaying
tubulence
field
Driven
turbulence
with wavenumber k
(perturbations
l= L/k)
Simulation example
SPH simulation with
gravity and supersonic turbulence.
Initial conditions:
QuickTime™ and a
BMP decompressor
are needed to see this picture.
Uniform density
1000Msun
1pc diameter
Temperature 10K
Collapse of a core I
Initial conditions: spherical self-gravitating isothermal sphere, simplified
without magnetic field and rotation reaches  r-2
Shu 1977
r-2
Ward-Thompson et al. 1999
Motte et al. 1998
Collapse of a core II
Important equations:
Mass within radius r: Mr = ∫4πr2dr --> ∂Mr/∂r = 4πr2
Continuity equation: ∂/∂t = -1/r2 ∂(r2u)/∂r
--> ∂Mr/∂t = -4πr2u
Hydrostatic momentum eq.: ∂u/∂t + u∂u/∂r = -1/grad(P) - grad(F)
With P = at for the ideal isothermal gas and grad(F) = -GMr/r2
Log10() [g cm-3]
∂u/∂t + u∂u/∂r = -at2/∂/∂r - GMr/r2
This set of equations can be solved numerically for given boundary conditions.
- After forming a
central hydrostatic
Singular isothermal sphere (Shu 1977)
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
Log10(r) [cm]
Log10(r) [cm]
Collapse of a core III
For a singular isothermal sphere (SIS) one gets
.
Mass accretion rate: M = ∂Mr/∂t = lim -4πr2u = m0 at3/G
r--> 0
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:
Myers 2005
One finds an initial
free-fall phase until the
protostar has formed. Then
the collapse continues in
an inside-out mode but
with varying accretion rate.
Collapse of a core IV
The inner free-falling gas causes an inner pressure decrease, and a
rarefaction wave moves outward.
Accretion shock and Accretion luminosity
The gravitational energy released per unit accreted mass can be
approximated 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 I
Centrifugal force: Fcen =j2/w3
(w radius on cylindrical coordinates)
Gravitational force: Fgrav =Gm/w2
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 rarified
outer cloud. Spin-up twists
the field, increases magn.
tension and creates magn.
torque counteracting spin-up.
--> magnetic breaking: dense cores approximately corotate with cloud.
Rotational effects II
In 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 from
geometrical 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 wcen
wcen = m03atW02t3/16 = 0.3AU (T/10K)1/2 (W0/10-14s-1)2 (t/105yr)3
wcen 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
Ovals are loci of constant
line-of-sight for
v(r) r-0.5
From Evans 1999
1.
2.
3.
4.
Rising Tex along line of sight
Velocity gradient
Line optically thick
An additional optically thin line peaks at center
Infall signatures II
Models
Spectra and fits
(Myers et al. 1996)
In a relatively simple model with two uniform regions along the line of sight
with velocity dispersion s and peak optical depth t0, one can estimate the
infall velocity vin:
vin ≈ s2/(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
-
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
Important missing ingredients are molecular outflows and flattened
Accretion disks --> subject of the next weeks.
Sternentstehung - Star Formation
Sommersemester 2006
Henrik Beuther & Thomas Henning
24.4
1.5
8.5
15.5
22.5
29.5
5.6
12.6
Today: Introduction & Overview
Public Holiday: Tag der Arbeit
--Physical processes, heating & cooling, cloud thermal structure
Physical processes, heating & cooling, cloud thermal structure (H. Linz)
Basic gravitational collapse models I
Public Holiday: Pfingstmontag
Basic gravitational collapse models II
19.6
26.6
3.7
10.7
17.7
24.7
Accretion disks
Molecular outflow and jets
Protostellar evolution, the stellar birthline
Cluster formation and the Initial Mass Function
Massive star formation, summary and outlook
Extragalactic star formation
More Information and the current lecture files: http://www.mpia.de/homes/beuther/lecture_ss06.html
[email protected]