Download 4. Star Formation

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4. Star Formation
Fig. 27. cartoon of the formation of an isolated single solar mass star (M.D. Smith)
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Class 0 source (André et al. 2000):
– central source, identified either by free-free emission at cm wavelength or outflows
– extended envelope in sub-mm and millimetre maps
– Lsmm / Lbol > 0.005, for forming solar mass stars this might be equivalent to M∗ <
Menv or Tbol < 80K; bolometric temperature is the temperature of a black body
with the same mean frequency as the observed spectral energy distribution.
Class 1 source (Lada 1987, Myers & Ladd 1993):
– 0 < α < 3, with α = [dlogλ Fλ ]/[dlogλ ]; 2µ m < λ < 25µ m
– 80K < Tbol < 650K
Both, Class 0 and Class 1 objects are called protostars. They generate their luminosity
mainly by accretion of material onto the central object. Lifetimes of these phases are
a few 104 and 105 yrs for Class 0 and Class 1, respectively.
Flat spectrum sources, Class 1/2 (Lada 1987, Myers & Ladd 1993):
– −0.3 < α < 0.3
– 650K < Tbol < 1000K
Class 2 objects (Lada 1987, Myers & Ladd 1993):
– Identical to classical T-Tauri stars
– −2 < α < 0
– Tbol > 1000K
Class 3 objects (Lada 1987, Myers & Ladd 1993):
– Identical to weak-line T-Tauri stars
– −3 < α < −2
Both, Class 2 and Class 3 sources are T-Tauri stars. They generate energy mainly from
contraction towards the main sequence (R ≈ 3R⊙ → R ≈ R⊙ ). They are usually
called pre-main sequence stars. Lifetimes of these phases are a few 106 and 107 yrs
for Class 2 and Class 3, respectively.
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It is now well established that stars form from gas and dust in the ISM. Young stars
and massive stars are preferably found close to/within molecular clouds. This obscuration hence requires observations at infrared, millimetre, or radio wavelength.
The problems of star formation: Create a star (radius 7·108 m) out of a cloud with
a density of 103 cm−3 . This requires to change the size of the object by a factor of
107 , and hence the density by 21!!!! orders of magnitude. Clouds usually rotate, due
to differential rotation in the disc of the Galaxy (Ω ∼ 10−15 s−1 ). If collapsing to
a star while conserving the angular momentum, this would mean rotation periods
of well under a second!. The potential energy of the cloud (E pot = − GM2 / R) has
to be released. For the Sun this amounts to 3.8·1041 J, equivalent of 31.6·106 yrs of
solar luminosity. But star formation timescales are much shorter than this. Hence the
energy has to be radiated/transported away, despite the high opacities. Magnetic
fields will tend to inhibit the collapse, even for small ionisation fractions, and only
large masses can overcome this, but how did then all the low mass stars form?
4.1. Classical Theory
Stars form from gravitational contraction of gas and dust in molecular clouds. Under
which conditions can such a cloud collapse? In the simplest case one can assume
viral equilibrium, hence balance of potential and kinetic energy, for stability of a
cloud (E pot + 2Ekin = 0). The system collapses when E pot + 2Ekin < 0.
If one sets the potential energy equal to the gravitational potential of a spherical
cloud and the kinetic energy as the internal thermal energy, then a criterion for collapse can be determined. This is the so-called Jeans criterion, where clouds with
masses Mc > M J collapse, with:
MJ =
5kT
Gµ m H
3/2 3
4πρ0
1/2
(12)
This equation can also be expressed in terms of sound speed cs since c2s ∝ T for an
isothermal gas. As can be seen, an increasing density and decreasing temperature
lead to lower jeans masses → fragmentation into smaller lower mass clumps.
The observed turbulent motions in cloud cores certainly contribute as a counterforce
to the gravitational pull. If the turbulence is isotropic, then the jeans mass can simple
be computed using an effective sound speed:
D E
c2s,e f f = c2s + 1/3 ν 2
(13)
One can ask how long it will take for a cloud to collapse. This is the so called free fall
time. It can be determined as:
tf f =
3π 1
32 Gρ0
1/2
(14)
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Observational evidence shows that pre-stellar cores do not posses a constant density. They rather increase their density towards the centre and reach a plateau.
Analytically such functions can be described by:
ρ (r ) = ρ 0
"
R0
2
( R0 + r 2 )1/ 2
#η
(15)
with η = 5 the classical Plummer sphere and η = 4 used by Whitworth and WardThompson (2001). This allows to determine the velocity and density profile during
the collapse, as well as the mass accretion rate onto the central protostar.
Density Profile
Velocity profile
Accretion rate
Before core formation
(t < 0)
ρ ∝ (r2 + r20 )−1
(r0 → 0 as t → 0− )
flattened isothermal sphere
ν ∝ r/t as t → 0−
ν ≈ −3.3cs , r → ∞
After core formation
(t > 0)
ρ ∝ r− 3/ 2 , r → 0
ρ ∝ r− 2 , r → ∞
ν ∝ r− 1/ 2 , r → 0
ν ≈ −3.3cs , r → ∞
Ṁ = 47c3s /G
Table 2. Properties of the Larson-Penston solution of isothermal collapse
Problems: so far no taking into account of angular momentum conservation. furthermore, observations show that the ISM is thread by magnetic fields (typical strength
are B ∼ 3µ G). Gravitational attraction can overwhelm magnetic repulsion only if
n −2
5 3/2 B3
6
M > Mcr =
=
4
·
10
M
⊙
48π 2 G 3/2ρ2
1cm−3
B
3µ G
3
(16)
or when the mass to flux ratio (which is constant during collapse) is larger than a
critical value of
M
Φ
cr
ζ
=
3π
1/2
5
G
(17)
A cloud is called sub − critical if it is magneto-statically stable, and supercritical if
it is not. The very large critical magnetic jeans mass forms a crucial objective to the
classical theory since it cannot explain the fragmentation into clouds forming final
stellar masses in the range of 0.01 – 100M⊙ .
4.2. Standard Theory
As a solution to the magnetic jeans mass problem the process of ambipolar diffusion
was proposed. There the neutral gas can gravitational condense and fragment across
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Fig. 28. Accretion rate, in units of 10−5 M⊙ / yr, plotted against time, in units of 104 yr. The
borderline between pre-stellar cores and class 0 protostars occurs when the accretion rate
first becomes non-negligible. The tick marks indicate the fraction of the total mass that has
reached the central protostar. The class 0/I borderline occurs when this fraction is ∼50%.
the field lines, while the ions are frozen to the magnetic field. The local density of
material can thus increase while the mass to flux ratio is lowered. This requires that
the neutrals can move independent of the ions in the gas, hence a balance between
Lorenz forces and ion-neutral drag. For a low ionisation fraction χ one can determine
the ambipolar diffusion timescale (τ AD ), as the time required for neutrals to move a
distance R against the ions.
τ AD ≈ 25Myrs
B
3µ G
−2 n H 2
102 cm−3
R
1pc
2 χ 10−6
(18)
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For ambipolar diffusion to solve the magnetic flux problem on an astrophysical relevant timescale, the ionisation fraction has to be very low. At low densities (below
104 cm−3 ) the ionisation increases due to the external UV radiation field and the gas
is tightly coupled to the magnetic field. Above this densities th ionisation is mostly
caused by cosmic rays and can be determined by:
χ ≈ 5 · 10
−8
− 1/2
n
105 cm−3
(19)
at densities above 107 cm−3 the ionisation fraction is constant. For typical molecular
cloud parameters the ambipolar diffusion timescale is about 10 – 20 times longer
than the free fall time.
These consideration lead to the investigation of star formation models based on ambipolar diffusion as a dominant process rather than relying solely on gas-dynamical
collapse. → Shu 1977 proposed the self-similar collapse of initially quasi-static singular isothermal spheres as the most likely description of the star formation process. He assumed that ambipolar diffusion in an originally sub-critical isothermal
cloud core would lead to the buildup of a quasi-static 1/r2 density structure that
contracts on times scales of the order of the ambipolar diffusion timescale. This is
called quasi-static since τ AD >> τ f f . When a singularity is formed in the centre,
the system becomes unstable and undergoes an inside-out collapse. During this period the magnetic fields are no longer dynamically important. A rarefaction wave
moves outward with sound speed behind which the material is free falling towards
the centre. Similar to the Larson-Penston solution we can determine the density and
velocity profile, as well as the accretion rate.
Density Profile
Velocity profile
Accretion rate
Before core formation
(t < 0)
ρ ∝ r− 2
singular isothermal sphere
ν=0
After core formation
(t > 0)
ρ ∝ r− 3/ 2 , r ≤ cs t
ρ ∝ r− 2 , r > cs t
ν ∝ r− 1/ 2 , r ≤ cs t
ν = 0, r > cs t
Ṁ = 0.975c3s /G
Table 3. Properties of the Shu solution of isothermal collapse
ALL basic assumptions of the ’standard’ theory can be questioned!!!
Singular isothermal spheres (SIS)
– SIS are the astrophysically most unlikely members of a whole family of self-similar
solutions to the 1D collapse problem (e.g. Whitworth & Summers 1985)
– To reach SIS from initially stable hydrostatic equilibrium conditions (BE
spheres), the system goes through a sequence of unstable equilibriums → collapse is likely to start before reaching the SIS state
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– Any external perturbation will tend to break spherical symmetry in the inner
part and thus the system will evolve away from SIS conditions rather than towards them
– Ambipolar diffusion in initially magnetically supported gas clouds results in
Larson-Penston-type collapse of the central region (while the outer part is still
hold up by B-fields)
– Star formation from SIS is biased against binary and multiple stellar systems
Observations of clouds and cores
– Most if not all protostellar cores are magnetically supercritical. There is NO convincingly sub-critical core found today (Crutcher 1999, Bourke et al. 2001)
– Density contrasts of observed protostellar cores are too high for them being subcritical (Nakano 1998)
– Molecular clouds as a whole are supercritical (McKee 2000)
– Infall motions too extended to be in agreement with inside-out collapse (e.g. André
et al. 2000)
– Density profiles of pre-stellar cores exhibit flat inner core more resembling BonnorEbert-type objects and thus following Larson-Penston-type collapse behavior (e.g.
André et al. 2000)
– Chemical ages of molecular cloud substructure (MC cores) are much smaller than
ambipolar diffusion times. → MC structure is transient and short lived (105 - 106 yr
as opposed to >107 yr) (Bergin & Langer 1997)
Observations of protostars and young stars
– Accretion rates appear strongly time varying as inferred from proto- stellar outflow strengths (Shu 1977 predicts a constant value of dM/dt = 0.975 c3 /G)
– The fraction of protostellar cores with embedded objects is high → roughly equal
time spent in pre-stellar and in class 0/1 phase (e.g. André et al. 2000)
– Age spread in stellar clusters is small and compares to the dynamical timescale.
If stars form on ambipolar diffusion timescales, then the expected timescale in
clusters were expected to be 10 times larger.
– Most stars form in clusters (Adams & Myers 2001). Magnetic mediation cannot
possibly account for the formation of star clusters, so clusters were always thought
to from from supercritical molecular cloud regions (see Shu, Adams, & Lizano
1987).
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Fig. 29. B vs. N(H2 ) from Zeeman measurements. (from Bourke et al. 2001) → cloud cores
are magnetically supercritical!!!
4.3. New Paradigm
Star formation is controlled by the interplay between self-gravity and supersonic
turbulence.
– Molecular cloud dynamics and star formation: turbulent fragmentation
– Basic concept of turbulently mediated star formation
– Some quantitative predictions (SF rates and efficiencies, stellar mass spectra,
chemical mixing properties, characterizing spatial structure of molecular clouds,
SEDs from protostars, PMS tracks, etc.)
Molecular cloud dynamics:
– MC masses vastly exceed thermal Jeans limit. (Mcloud = 103 ...106 M⊙ — M Jeans =
1...100M⊙ )
– MC’s should collapse globally and form stars on a free-fall timescale τ f f ≈ 106 yr.
– MC’s are ’thought’ to life longer (few 106 to 107 yr)
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What stabilizes MC’s against global collapse?
– Broad spectral lines indicate the presence of supersonic random motions carrying
enough energy to counterbalance gravity on global scales.
MC’s are stabilized by supersonic turbulence!
Timescale problem: Turbulence decays on timescales comparable to the free-fall time
ė ∝ t−1 (Mac Low et al. 1998, Stone et al. 1998, Padoan & Nordlund 1999); Magnetic
fields (static or wavelike) cannot prevent loss of energy.
→ Interstellar turbulence needs to be constantly replenished or SF proceeds on a few
free-fall times (see earlier lecture).
How does star formation occur in globally stable molecular clouds? What are the
properties and the relevant physical parameters?
Supersonic turbulence: Collapse occurs localized in shock compressed regions.
– Shock interactions create density fluctuations: δρ ∝ M2 (in isothermal shocks)
– Local Jeans mass: M J ∝ ρ−1/2 T 3/2
– Some fluctuations may exceed the local Jeans limit, even if the cloud is globally
stabilized by turbulence.
– If not turbulently supported on smaller scales, compressed regions will collapse
locally.
– Local collapse → star formation
What can those models predict?
Efficiency of star formation: Star formation efficiency is high for large-scale turbulence and low if most energy resides on small scales. Efficiency decreases with increasing turbulent kinetic energy. Local collapse can only be prevented completely if
turbulence is driven on scales below the Jeans length. Interstellar turbulence is able
to maintain low global star formation efficiencies in molecular clouds over many free
fall times.
Influence of magnetic field: Magnetic fields reduce the fraction of mass in collapsed
objects, but do not prevent local collapse.
Modes of tar formation: Spatial distribution and timescale of SF depend on strength
and power distribution of turbulence.
– Small wavelength driving: independent collapse of isolated clumps with low efficiency at random locations in the cloud → wide distribution of formation times
(age spread >> τ f f ) ... isolated mode of star formation!
– Long wavelength driving, or complete loss of turbulent support: highly efficient
formation of coherent structure → coeval stellar population (age spread ≈ τ f f ) ...
clustered mode of star formation!
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– Influence of magnetic fields is moderate
The Initial Mass Function and binary frequency of stars can as well be reproduced
by the models. Mass accretion rates are time varying and strongly influenced by the
cluster environment.
Star formation is a highly stochastic process !!!
4.4. Collapse
This is partially still theory, but here is what we think happens in the central core
during the collapse:
– High density shields centre from external heating (radiation, and/or cosmic rays);
cooling through dust grains, molecular lines; hydrogen is molecular
– collapse starts; gravitational energy goes via compression into heating; energy
transfered on to the dust grains via collisions, re-radiation of the energy at millimetre wavelength; as long as radiation can escape core unhindered the isothermal collapse continues.
– at densities of 1011 cm−3 and at a radius of 1014 cm the gas becomes optically thick
for dust radiation at even 300µ m. Hence, energy is trapped and the temperature
rises; increasing temperature increases opacity further; collapse turns adiabatic
– The high thermal pressure resists gravity and end this first collapse, forming the
first core with densities of 1013 ...1014 cm−3 and temperatures 100–200 K
– A shock wave forms at the outer edge of the first core; direct accretion of material
onto the core through this shock; temperature rises until density reaches 1017 cm−3 ;
– As temperature reaches 2000 K the hydrogen molecules dissociate; the atoms hold
less energy, tempering the pressure rise → second phase of collapse
– cooling stops in the centre of the first core when all molecules are dissociated;
protostellar densities of 1023 cm−3 and temperatures of 10000–20000 K are reached
and thermal pressure brakes the collapse; formation of second and final core →
protostar;
– the first shock disappears and a second inner shock mediates accretion onto the
core;
– angular momentum leads to creation of a disc around the central object (disc is the
configuration of lowest energy for a given angular momentum); further collapse
requires accretion through this disc.
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4.5. Angular Momentum Transport
The accreting material has to lose angular momentum in order to be able to accrete
onto the central object. If material of mass m starts with angular momentum L and
is being pulled inwards by the mass M of the protostar, then it only can get as close
as the radius:
r=
L2
GMm
(20)
If this radius is larger than the radius of the protostar accretion has to stop. For
a solar mass protostar forming from a cloud that rotates with typical velocities of
Ω = 10−15 s−1 , only material inside 4700 AU can in principle reach the central object.
But even if, the addition of the angular momentum would speed up the central star
beyond break up.
There are two main possibilities of how material is losing angular momentum:
– viscosity in the disc
– jets and outflows
Viscosity: collisions in the disc transfer energy and angular momentum between particles in the disc. the particle with the lower angular momentum after a collision is
able to move closer to the central object, the other particle is moving away from
the centre. Hence effectively material is transported inward, while angular momentum is transported outward. This helps to understand while 99.8 % of the mass in
the Solar System are in the Sun, while 98 % of the angular momentum in the Solar
System are in the planets (which form out of disc material in the final stages of mass
accretion onto the central star).
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4.6. Jets and Outflows from young stars
Fig. 30. examples of jets and outflows from young stars
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One manifestation of the star formation process was not predicted by theory before it was actually observed. Since the late 19th century some HH-objects were
known as emission nebula. In the 1940’s Herbig and Haro independently discovered a larger number of these objects, cataloged them and realised that they possess
a rather strange electromagnetic spectrum. Ambartsumian named them HH-objects
and found that many of them were placed in the vicinity of young stars. This lead to
the conclusion that they might represent an early stage of T-Tauri star evolution.
In the 1980’s it became evident that they are usually jet-like, emerging from young
stars. It was Snell et al. (1980) who draw the right conclusions of what HH-objects
and the associated jets are (see Figure.)
Fig. 31. Model of outflows from young stars from Snell et al. 1980.
It is still unclear how exactly those jets are launched, accelerated and collimated.
Current theories involves MHD, winds from the disc, star, or X-winds. It is observationally shown that jets are launched accelerated and collimated within a very small
region from the star (few AU) and that they rotate. Hence, they are able to transfer
angular momentum away from the system.
The jets from the young stars typically have speeds of up to several hundred kilometers per second, hence they are highly supersonic,
given the sound speed in the
√
interstellar medium of about 200 m s−1 (cs = γ RT). Interaction of material with
such speeds with the interstellar medium results in shocks. They represent discontinuities where certain boundary conditions have to be satisfied. These are e.g. the
conservation of mass, energy and momentum before (index 1) and after (index 2) the
shock front. They can be written down as:
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Fig. 32. Cartoon of the X-Wind model of jet launching
ρ1 ν 1 = ρ2 ν 2
(21)
p1 + ρ1 ν12 = p2 + ρ2 ν22
(22)
ν12
ν2
γ p1
γ p2
= 2+
+
2
γ − 1 ρ1
2
γ − 1 ρ2
(23)
where ρ is the density, ν the velocity, p the pressure and γ the adiabatic index (γ =
f +2
f ; f degrees of freedom; γ = 5/3 for monatomic gas; γ = 7/5 for diatomic gas).
One can further express the speed of the shock front in units of the Mach number
M = ν1 /cs . This allows us to determine the changes of the physical properties of
the gas before and after the shock front. One obtains (after a bit of algebra):
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(γ + 1)M2
ρ2
ν1
=
=
ρ1
ν2
(γ − 1)M2 + 2
(24)
2γ M2 γ − 1
p2
−
=
p1
γ+1
γ+1
(25)
T2
[2γ M2 − (γ − 1)][(γ − 1)M2 + 2]
=
T1
(γ + 1)2 M2
(26)
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As we can see there is an upper limit for the change in density. For very strong
shocks we obtain a ratio of the densities of 4 and 6, for monatomic and diatomic gas,
respectively. For the pressure and temperature there is no limit, and hence very high
values can be reached. these temperatures can easily reach several thousand Kelvin
and are high enough to excite and/or ionise atoms or even destroy molecules. This
energy is radiated away and can be used to determine the physical conditions in the
pre-shock medium and the jet itself.
In the ISM (esp. away from the denser clouds) the densities are generally low enough
that the mean free path’s of particles are several thousand AU. Hence there should
be no collisions to carry the shock. Since they are observed, e.g. in supernova remnants (see later in course), there must be another force to carry the shocks. These
are magnetic fields. Hence: adiabatic shocks turn into MHD shocks; we have to add
magnetic field terms to the conservation equations and also conserve the magnetic
flux at the shock front. These are however, beyond the scope of this lecture.
The shocks not just excite material, they also transfer momentum into the pre-shock
medium and hence entrain material. This is what generally is observed in protostellar outflows (molecular material moving with moderate velocities; 10–50 km s−1 ).
How much energy is put into the ISM? The luminosity generated by accretion onto a
protostar is L = GM Ṁ / R; Observational evidence and theoretical calculations show
that about 1 – 10 % of this energy is going into the bipolar outflows. The energy transfered into the ISM is then Lturb = 1/2 Ṁ jet ν 2jet ; this is an upper limit since some of the
energy is radiated away. The forward momentum supply is given by Fturb = Ṁ jet ν jet ;
as above, integrated over the entire mass loss phase of the forming star. This transfer of energy and momentum seems to be sufficient enough to support the level of
turbulent energy on local scales within molecular cloud cores.
Some shocks apparently move with too high velocities in order that molecular hydrogen molecules can survive (∼ 20 km s−1 ). But still emission from this molecule
is observed. This can have two reasons; 1) The shock is an internal working surface. The pre-shock material was passed by a/several previous shock-fronts and has
been accelerated. The shock speed is then only the difference in the velocities between pre-shock medium and jet, while the apparent proper motion is the jet speed.
2) There are magnetic fields connected to the shock front. If there is a fraction of
ions in the pre-shock medium, then the magnetic fields will accelerate the pre-shock
ions before the shock front arrives. due to collisions between ions and neutrals, most
of the pre-shock material will be accelerated prior to the arrival of the shock front,
hence weakening the shock. In such cases there is a continuous change of the physical properties of the gas through the shock front (C-shock) compared to the ’jump’
in conditions for very strong shocks or shocks without magnetic fields (J-shocks).
4.7. Modes of Star Formation
We have so far described the formation of an isolated, single solar mass star. This
is NOT the dominant mode of star formation in our Galaxy. Most stars (90 %; Lada
& Lada 2003) are formed in clusters (cluster: more than 35 stars; at least one so-
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lar mass per cubic parsec; embedded clusters – up to 1000 solar masses and 10000
solar masses per cubic parsec). This is e.g. the dominant mode in Orion, Serpens,
Ophiuchus, Perseus. More isolated star formation is e.g. found in Taurus. Many stars
form in multiple systems. The more massive a star, the more likely it is in a multiple system. Esp. the most massive stars (OB) are always part of multiple systems.
The frequency of single/binary/triple/quadruple systems is about 58/33/7/1. In
principle capture, fission and core/collapse/disc fragmentation could be responsible. Capture is very unlikely; Fission leads only to very close binaries; Fragmentation
is the most plausible mechanism.
In clusters the most massive stars tend to form in the centre (in situ, no migration);
The radiation of the massive stars and the outflows disperse the surrounding cloud.
Due to the velocity distribution of the formed stars, the cluster will disperse in a
few crossing times, i.e. millions of years. Thus: most stars are not in clusters but
become field stars. All this evidence suggests that hierarchical fragmentation within
a turbulent medium is what determines star formation.
Sequential Star Formation: e.g.: A molecular cloud forms dense clumps due to fragmentation. They form stars and disperse the gas around them locally. The radiation
pressure of the young stars and/or outflows compresses the gas in the remaining
cloud. The strong compression leads to the formation of a cluster of massive OB
stars. Some of the OB stars will explode as supernovae, some of the older less massive stars turn into red giants. The formation of stars will proceed through the entire
cloud. This process is observable as a gradient in ages of young stars ranging from
older objects (Class 3) at a distance from the cloud → middle aged Class 2 sources
close to the cloud edge → protostars just emerging from or still deeply embedded
within the cloud.
Triggered Star Formation: In principle like above. But e.g. also Supernova from binaries SNIa, that happen close to clouds, can trigger new phases of star formation.
Spontaneous Star Formation: As described in the past lectures.
How much of the cloud material is converted into stars: The Star formation
Efficiency; Observationally one finds in between 3 and 20 %; The models can reproduce any number we want!
4.8. Open Questions
– Angular momentum problem. (How do collapsing objects loose their angular momentum?)
– What terminates star formation locally? (Feedback processes? Like ionizing radiation from O stars? Yes, in dense clusters. But in small associations without O and
B stars?)
– What drives turbulence in the ISM? Is driving always on large scales? (Molecular
cloud structure dominated by largest-scale modes.) (Is SN driving really the dominant mechanism? Maybe MRI in LSB galaxies?)
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– How do molecular clouds form? (Fast process, or slow?) (Turbulent compression
vs. thermal instability vs. gravitational instability, or others?) (What is the role of
dust?)
– Star formation in the early universe? (no/few metals → different cooling → different EOS) (formation of Pop III stars, formation of globular clusters)
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