Download L2 Star formation Part I

Lecture 2
Star formation
Lecture Universität Heidelberg WS 11/12
Dr. Christoph Mordasini
Based partially on script of Prof. W. Benz
Mentor Prof. T. Henning
Lecture 2 overview
1. Stars
2. Observed features of star formation
3. Overview of the sequence of star formation
4. The ISM
5. Mean molecular weight
6. The problem of angular momentum
7. The virial theorem
8. Linearized hydrodynamic equations
1. Stars
Stars
- definition: burn hydrogen in the core
characteristics: - mass range: ~0.08 (Coulomb barrier) ≤ M/Msun ≤ 100 (radiation barrier)
- luminosity: 10-4 ≤ L/Lsun ≤ 106 (much larger domain: 10 o.o.M)
- mass-luminosity: L ~ M3.5 (strong dependence)
- lifetime: τ = M/L ~ M-2.5
- star formation
production rate: 3-5 Msun / year in our galaxy (about 1011 stars). Our galaxy is calm.
- efficiency gas → star: ~ 2% (small fraction only)
- initial mass function (IMF): nb of stars born with mass between m and m+dm:
f (m)dm ∝ m−x dm
x=2.35 for the Salpeter IMF (M> 1Msun)
most stars which form are low mass.
- the total mass in stars of mass between m1 and m2 is given by:
� m2
� m2
� −0.35
�
−0.35
−2.35
f (m)dm =
m
mdm = 2.86A m1
− m2
m1
m1
which shows that most of the mass is in form of small stars and most of
the luminosity in the more massive stars.
-what we observe is the PMF (present MF): high mass stars disappear very
quickly.
2. Observed features of star
formation
Giant molecular clouds (GMC)
Star formation takes place in cold, dense gas clouds: The molecular clouds.
Stars form in groups or clusters. The largest GMC in Orion is about 1000
light years away. Hot young stars (25-50 million year old) ionize their
surroundings and are therefore easily visible.
● main characteristics:
- molecular hydrogen
- ~1% of dust (Si and C)
- organic and non-organic molecules ammoniac
NH3, formaldehyde H2CO, acetylene HC3N
- ~ 105-106 solar masses
- ~ 1000 atoms per cm3
- clumpy: 103-104 solar masses clumps
- T ≈ 10 K
● structure
- emission of the H2 molecule but difficult because small dipole moment.
Molecules with larger dipole moment (CO) are better but do not represent
the mass...
- dust emission (radio band). Interpretation difficult since many parameter
unknown (departure from LTE, opacity, dust temperature, etc.)
- IR cameras on large telescopes measure the absorption on thousands of
background stars → map the dust distribution in the cloud
Bok globules
Example characteristics:
- 410 light years away
- diameter ~ 12'500 AU
- T ~ 16 K
- ~ 2.1 solar masses
- in gravitational equilibrium
with Pbound=12.5 × 10-12 Pa
Bok globule: Barnard 68
Star formation: observations
HH-30
- Herbig-Haro objects
- first discovery by Herbig and Haro in 1951.
- patchy nebulous structures at an extended jet (0.1-0.3 ly=6000-18000 AU)
- formed through collision of jet launched by young stars colliding with ISM
- frequent in star forming regions
- super-sonic flows: 200 to 400 km/s
- 50% bipolar, knots in jets, variable on year timescales
- mass loss: 10-10 to 10-6 solar mass / year
- hot bow shocks with molecular emission lines (CO, OH, H2O)
knots
collimation
by B fields
HR 4796
DG Tau
- T-Tauri stars
- pre-main sequence precursors of F, G, K, M stars
- energy generation mostly by contraction
- IR excess (protoplanetary disk)
- variable (irregular, accretion phenomena), active
- emission lines (Ca, H, Balmer lines, Li)
- disk + bipolar outflows
- age: 1 to 10 million years
- 2 classes: - classical T-Tauri CTTS
(with disks, strong IR excess)
- weak-line T-Tauri WTTS
(without disks, weak/no IR excess)
- Disks
- mass: 2 × 10-3 to 10-1 solar masses
- no correlation mass vs. age
- no disks around stars older than 10 Myrs
- temperature: ≈ 50-300 K at 1 AU
T ∝ r q with q = -1/2 to -3/4
- outer radius by about 100 AU
- sometimes gap in the center
- UV excess at the inner boundary (accretion shock)
1206
HERNÁND
Hernandez et al. 2008
3. Overview of the sequence of
star formation
Young stellar objects (YSO)
Classification of YSO according to evolutionary
stage based upon spectral and physical characteristics.
0) GMC
*Dust is a strong absorber in VIS and UV->heats
up->radiates thermally at ~300 K ->IR excess
radiation (over what is expected from the star) at
10-100 microns.
1)
Protostar
2)
Main accretion phase
The cloud collapse
is very rapid!
Late accretion phase
3)
*
PMS star
4)
5)
Classical T Tauri
Optically thick disk
Giant planets ?
(direct instability)
Planet formation
Giant planets
(core accr.)
Weak line T Tauri
Optically thin disk
Rocky planets
Star formation sequence
•
•
•
•
•
•
•
•
•
•
jeans instability, cloud collapse begins (~105 M☉)
isothermal collapse (free-fall time 108 yr)
fragmentation
homologous contraction
center becomes optically thick: adiabatic compression (~10-13 g cm-3)
first hydrostatic core forms, ~170 K (1st equilibrium phase)
H2 dissociation (T~2000 K) drops γ below 4/3
γ = (f+2)/f
and core collapses again
H2: 1.4 close to 4/3
ionized, atomar H in second core: dynamically stable again
(critical value for
(0.001 M☉, 20000 K; 2nd equilibrium phase)
dynamical stability)
pre-main sequence contraction
zero age main sequence ZAMS (Luminosity produced by H fusion; if
degeneracy does not stop temperature rise before fusion, threshold~0.08
M☉)
Overview star formation sequence
Protostar
adiabatic compression
efficient cooling by molecular
hydrogen and grains
First core
Size: several AU
Second core
2000 K
The temperature and density of a spherically symmetric cloud change during collapse.
Density must increase by a factor of 1024 and temperature by 106 to form a star.
Overview star formation sequence:
Nomenclature
•Protostar
•optically thick stellar core
•forms during the end of the
adiabatic contraction phase and
grows during the accretion phase.
•large accretion rates ~10-5 Msun/yr
•Pre Main Sequence Star
•visible in the optical
•small accretion rates ~10-7 Msun/yr
•energy generation mainly via
contraction
•ZAMS: Zero age main sequence
•PMS contraction => center heats up
•~3 Mio K: H burning ignites
•contraction stops, energy production
mainly via fusion
(via protoplanetary disk)
4. The ISM
Composition of the interstellar medium
Nuclear abundances
The atomic composition of the interstellar medium is the result of Big Bang nucleosynthesis (H, He, some Li and Be) followed by
enrichment due to processes connected with stellar evolution. Galactic chemical evolution can be summarized by the following
drawing:
gas
infall
stars
enriched gas
wind
galaxy
remnants:
neutron stars
white dwarfs
Since there is no compelling evidence of strong fractionation at the
time of the formation of the solar system, we can suppose that the
chemical composition of the solar system reflects the composition
of the interstellar matter at 8.5 kpc of the galactic center 4.6 billion
years ago.
closed box models (if no exchange)
Note:
•abundances decrease on average with
increasing nucleon number
•nucleons with even numbers are more
stable and hence more abundant
•nucleons with magic numbers of protons
(2,8,20) are also more abundant
•iron (A=56) is the nuclei with the largest
binding energy
Arnett 1996
Big bang
nucleosynt.
stellar nucleosynt.
supernova nucleosynt. (neutron capt.)
Note:
Complex organic
molecules are found in the
interstellar medium. This is
not evidence for life
e.g.
-HC3N: Cyanoacetylene, a
“life molecule”
-H2CO: Formaldehyde
-NH3: Ammoniac
(important for amino acid
formation)
Chemical reactions in ISM
Typical chemical reactions in the interstellar medium:
type
• molecule formation
radiative association
grain surface formation
• molecule destruction
photodissociation
dissociative recombination
collisional dissociation
• chemical reactions
ion-molecule exchange
charge transfer
neutral-neutral
formula
reaction rate
X+Y ⇔ XY + photon
X+Y:grain ⇔ XY + grain
∼ 10-16-10-9 cm3s-1
∼ 10-18 cm3s-1
XY + photon ⇔ X + Y
XY+ + e ⇔ X + Y
XY + M ⇔ X + Y + M-
~10-10-10-8 s-1
~10-6 cm3s-1
X+ + YZ ⇔ XY+ + Z
X+ + YZ ⇔ X + YZ+
X + YZ ⇔ XY + Z
~10-9 cm3s-1
~10-9 cm3s-1
~10-12 cm3s-1
This is a cooling process if photons can escape.
example: ion-molecule exchange:
H3+ + HD ⇔ H2D+ + H2 + Q
with Q=0.02 eV
Dust formation in the ISM
Density must be sufficiently large for dust formation.
1) condensation from the ISM: molecules serve as seed for accretion of other molecules.
Typical growth rates are of order 100 million years (slow).
2) formation in cold dense stellar atmospheres (giant stars with very large, relatively cold
atmospheres): silicates, carbon clusters, PAHs (Polycyclic aromatic hydrocarbons) etc.
3) condensation in cold collapsing cloud envelopes: density is high enough to form grains.
5. Mean molecular weight
The mean molecular weight
k: Boltzmann constant
n: number density
T: temperature
p: pressure
ρ: density
: mean part. mass
The equation of state of an ideal gas is:
Defining the mean molecular weight
we can write the EOS as
.
a) neutral gas
setting Nj: number of particles of type j
mj: mass of particles of type j
Mass fractions
or in general
with
atomic
mass
X= mass of H / total mass (Ax=1 (atomar) Ax=2 (mol.))
X+Y+Z=1
Y= mass of He / total mass (A=4)
Z= mass of metals / total mass (mean solar composition A = 15.5)
then we have
protosolar fractions: X=0.7110, Y=0.2741, Z=0.0149 (Lodders 2003)
μ=2.353 (molecular hydrogen)
μ=1.281 (atomic hydrogen)
The mean molecular weight II
b) fully ionized gas
No additional mass, but more particles (free electrons), so we have a lower mean mass.
where Zj is the number of free electrons when the element is fully ionized
using
we obtain (with Z=1-X-Y)
Nprotons =
ca. Nneutrons
μ=0.612 (molecular hydrogen)
6. The problem of angular
momentum
The problem of angular momentum
Structure formation in the universe (galaxies, stars, planets, ...) one must get rid of angular
momentum, as structure formation is associated with size decrease.
grav. only
grav. “centripetal force”
(ang. momentum)
Disk formation
How to shrink the disk?
One possibility is to get the angular momentum redistributed. Give all L to a small fraction of
mass, accrete the rest.
In the Keplerian potential, the orbital frequency is given as
so for circular orbits
which means that also a small mass, but at a large distance, can contain significant L.
We thus need to segregate mass and angular momentum.
We recall that the Sun contains 99.96% of the mass, but only 0.6% of the angular momentum.
The problem of angular momentum II
Angular momentum is an enemy of star formation. To form a star, a gas cloud must get rid of a
large amount of angular momentum.
•average density:
-of the ISM: ~10-24 g/cm3
-of a star: ~ 1 g/cm3
•to form a star, the volume must therefore shrink by 24 orders of magnitude and therefore in
radius by about 8 orders. If angular momentum is conserved, the cloud must rotate 10-8 times
slower than young stars. The latter typically rotate with 50-200 km/s which implies a cloud
rotation speed of 0.05 to 0.1 cm/s i.e. mm/s!
a) Minimal rotation from galactic differential rotation
R
v1
r
vCM
r=radius cloud (e.g. 20 pc~ 4 Mio AU)
R=galactocentric distance of the
cloud (Sun: ca 8 kpc) R>>r
r
∆v=Ωcr
to GC
Differential rotation of the galaxy leads to
angular momentum relative to the CM at
decoupling of the cloud.
angular frequency Ω = 2π/T
vrot(R)= Ω R
v1: rotational velocity around the galaxy center of a gas parcel at R+r
vCM: rotational velocity around the galaxy center of a gas parcel at R
(at the mass center of the gas cloud)
The problem of angular momentum III
If the cloud collapse, the velocity difference must be transformed into cloud rotation
Examples
•Solid body rotation
prograde
•Keplerian rotation
retrograde
The problem of angular momentum IV
b) Measuring galactic rotation
(subtract solar
velocities)
Ro
GC
Sun
l
R
d
α
α
l
Ω0R0
From simple geometry
Star
ΩR
l=galactic longitude
We further assume d<<R (and d<<R0) because we measure stars in the solar neighbourhood,
so we obtain the following expression (again from geometry)
(w. Taylor)
Inserting this in the expressions for the velocities yields
with the Oort constant
with the Oort constant
All quantities measurable from radial velocity (vrad), astrometry (proper motion μ, parallax d).
Numerical values:
A=14.5 ± 1.5 km/s kpc-1
B=-12 ± 3 km/s kpc-1
The problem of angular momentum V
c) Resulting cloud rotation
Recalling that the cloud rotation is given by
We obtain by inserting the expressions for the Oort constants:
For a typical GMC, R~20 pc, and the rotation velocity is therefore vrot ~ 50 m/s.
This is a factor 104 to 105 larger than the velocity deduced from stellar rotation assuming
conservation of angular momentum. There must be mechanism to loose angular
momentum.
Loss mechanism:
• magnetic fields (but needs a certain ionisation, problem at very low T)
• accretion disk (friction)
7. The virial theorem
Euler equations
a) Continuity equation
Temporal change of fluid mass in a volume V
must be due to in/outflow
Gauss
So for an
infinitesimal volume:
i.e.
b) Momentum equation
Control volume V of fluid. Momentum content (i=1,2,3):
Temporal change:
In the absence of gravity and pressure (and viscosity), the only way of changing the
momentum is to transport it in our out:
Gauss
therefore
(using Einsteinsche
Summenkonvention
everywhere)
Euler equations II
Additionally, pressure and gravity are acting on the particles in the control volume:
Gauss
Newtons law apply to particles, not control volumes. The momentum change in the control
volume thus contains the flow and the Newtonian contributions
Using the continuity equation we can also write this as
where the second part on the LHS reads as
We also note
as it must be.
The virial theorem
The equation of motion of a self-gravitating inviscid fluid is written as:
Dotting this expression with
and integrating over all mass yields:
First term
r
where we have defined the moment of inertia as
so
.
Second term
With the Satz von Gauss, an ideal gas equation of state in the form
where u = spec. int. energy [erg/g], a spherical cloud with a constant surface pressure pS and
from calculus the general relation
The virial theorem II
Gauss
Third term
or
so
where we switched the labels which must yield the same. Adding and dividing by 2 gives
with W the potential energy (The factor of 1/2 is present because each pair of interacting
fluid elements occurs twice in the integration, but should only be counted once.)
The virial theorem III
Putting all three expressions together yields the Virial Theorem
I: moment of inertia
Ekin: (bulk macroscopic) kinetic energy
Utherm: internal energy
W: potential energy
ps: surface pressure
V: volume
In case of a static configuration, no surface pressure:
In absence of internal energy (mechanics)
In the absence of bulk kinetic energy (for example no bulk rotation), we can write
(total energy)
Solving for the total energy yields
An example is a non-rotating
star in hydrostat. equilibrium
The virial theorem IV
Notes:
•γ>4/3
=> Etot < 0 : The system is bound. For example, for a monoatomic gas,
γ=5/3 so
This means that for a slow (quasi-static) contraction, half the potential energy
goes into internal energy (heat) while the other half must be lost in form of
radiation.
Note the somewhat counter-intuitive result that in such a contraction, the total
energy becomes more negative, while the temperature is increasing. This is a
general characteristic of self-gravitating systems. By analogy to normal gases,
where u=cvT, self-gravitating systems are said to have a negative heat capacity
•γ=4/3
=> Etot = 0 : Radiation dominated gas and the total energy is independent
from the radius of the configuration.
•γ<4/3
=> Etot > 0 : The system is unbound.
8. Linearized hydrodynamic
equations
Linearized hydrodynamic equations
The basic equations for an inviscid, self-gravitating fluid are:
1)
Continuity equation (mass conservation)
2)
3)
Euler equation (momentum conservation)
Poisson equation
(Laplace operator)
+ an equation of state. Often taken as isothermal gas, so p=p(ρ) cf
Otherwise, one must also consider an energy conservation equation.
.
We linearize this system of equations, assuming that we know a time dependent solution
to this set of equations which we call
We now consider small, time dependent perturbations to these quantities and write the
new variables:
where ε << 1.
Linearized hydrodynamic equations II
Plugging back these variables in the equations listed above, taking into account that the time
independent base solution solves the equations, and neglecting terms higher than first order in
the perturbation ε, we get
1)
lin. continuity equation
2)
lin. momentum equation
3)
lin. Poisson equation
Note that all quantities with the subscript 0 are know constants, therefore these linearized
equations do not contain any non-linear terms (as e.g.
in the original equations) any
more.
We recall that these linearized equations are valid in the limit of very small perturbations.
They allow to discuss the stability of a configuration and the early growth of perturbations
but loose their validity as soon as the perturbations leave the linear regime.
Application: sound waves
a) non-self gravitating fluid
Let us consider a fluid of constant density and temperature (and thus also pressure), initially at
rest:
The linearized equations then simplify to:
1)
2)
Let us further assume an equation of state of the type p=p(ρ): =>
Inserting this into eq. 2 yields:
Take the time derivative of this expression (keeping in mind that all 0 quantities are constants):
Application: sound waves II
Inserting eq. 1 in this expression, using div grad = laplace, and
with
We recognize the standard wave equation indicating that these small perturbations propagate
as a wave called a sound wave through the medium at a speed c called the sound speed.
Stability analysis
Formally, we can try a solution of the type
Inserting this into the wave equation, we get the so called dispersion relation for a
sound wave in a non self-gravitating medium:
We see, as expected, that the solution is always oscillatory in nature without any sign of
instability (ω is always real).
Application: sound waves III
b) self-gravitating fluid
Here again we consider a uniform fluid of constant density and temperature initially at rest.
Taking the time derivative of the lin. continuity equation, and the divergence of the lin.
momentum equation, and combining this with the lin. Poisson equation yields now for the
density perturbation:
Stability analysis
Assuming a similar plane wave solution as above gives the new dispersion relation
We see that the new term due to gravity can make instabilities possible, since if ω2<0,
the perturbations will grow exponentially in time. Note that for a given density, the
instabilities occur at small wave number k, i.e. at large wavelengths.
Further reading
R. Kippenhahn & A. Weigert
Stellar structure and evolution,
Springer Verlag, Berlin, 3rd Ed. 1994
S. Stahler, F. Palla
The formation of stars,
Wiley-VCH, Weinheim, 2004
Questions?