Download Gravitational Collapse

Lecture 4 - Interstellar Cloud Collapse
o
Topics to be covered:
o The Interstellar Medium
o Gravitational collapse
o Virial Theorem
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o Jeans Instability
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Interstellar Medium (ISM)
o
Interstellar space is not empty - there
is gas and dust in the galaxy.
o
Chemical composition of gas is mostly
hydrogen (~70%), some metals (~few
%) and rest helium.
o
Clouds can be mainly
o Neutral hydrogen (HI)
o Ionized hydrogen (HII)
o Molecular hydrogen (H2)
o Because the ISM is diffuse, it does not
radiate like a blackbody, so we don't
see a continuous spectrum (like stars).
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Interstellar Medium (ISM) - HI regions
o
Far from stars, no energy available to ionize/excite HI - therefore no optical
emission lines.
o
Because electrons have charge and spin, have a magnetic dipole moment. Pauli
Exclusion Principle: electron and proton “want” to be anti-aligned - this is their
ground state. If aligned, have slightly more energy. If change spin can release
energy difference as a photon.
o Energy difference is small ~6x10-6 eV, which is ~21 cm (radio waves).
o
From radio surveys of our galaxy find lots
of HI gas: about 1 solar mass of gas for
every 10 solar masses of stars.
Radio
IR
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Interstellar Medium (ISM) - HII regions
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HII regions are found near hot stars.
o
UV photons from nearby stars ionize H atoms.
o
When electrons and protons recombine, they
generally emit Balmer lines.
o
Also observe lines from other atoms (e.g.,
oxygen, sulfur, silicon, carbon, …).
o
HII regions are associated with active star
formation.
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Interstellar Medium (ISM)
- H2 regions
o
If density of gas is high, and temperature is low
enough, H atoms can form H2. Unfortunately, no
emission from H2 in the optical or radio bands.
Therefore use,
1. Other molecules. For example, CO emits
radiation at a wavelength of 2.6mm microwaves.
2. Dust. Dust (and molecules) are often found
together - dense, dark clouds (called Bok
globules) are regions of very dense molecular
gas.
o
Where there is lots of gas - in particular molecular
and ionized gas => lots of stars formation …
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The Virial Theorem
o
Applies to any system of particles with pair interactions for which the distribution
of particles does not vary with time.
o
Theorem states that total energy of system E is related to gravitational potential
energy U by:
E = 1/2 U
But we know that total energy is sum of the kinetic and potential energy=>
K + U = 1/ 2 U
or
2K + U = 0
Can be applied to a system of gravitationally interacting bodies such as stars
forming a cloud, clusters of stars, clusters of galaxies, etc.
o
o
o
3GMc2
3GMc2
 2K  U  3NkT 
0
As K = 3/2 NkT and U  
5RC
5RC

Eqn. 1
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Application of the Virial Theorem
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VT can be used to estimate conditions for cloud collapse:
1. If 2K > U => gas pressure (energy) will exceed
gravitational potential energy and expand.
2. If 2K < U => gravitational energy will exceed gas
pressure and collapse.
2K
U
o
o
The boundary between these two cases describes the
critical condition for stability.
We know that  = Mc/Vc = Mc / (4/3Rc3) =>
3M C 1/ 3
Rc  

4



and N = Mc/mH where  is the mean molecular weight
and mH is the mass of a proton. Can we derive an
expression for the critical mass?

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Application of the Virial Theorem - The Jeans Mass
1/ 3
o
Substituting for R and N in Eqn. 1:
3M c kT 3GMc2 3M c 



m H
5 4 
 5kT 3 / 2 3 1/ 2
 M J  
 

Gm H  4 
o
The Jeans Criterion is:
o
If Mc > MJ => cloud will collapse.
o
 5k 3 / 2  3 1/ 2 T 3 1/ 2
Rearranging Eqn. 2 gives: M J  
    
Gm H  4     
Eqn. 2
The Jeans Mass
Mc > M J

 375k 3 1/ 2T 3 1/ 2
 

3 3
3  
4  m H G    
T 
 M J  z 
  
3 1/ 2
2K
U
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
Application of the Virial Theorem - The Jeans Radius
o
Is there a critical radius that corresponds to the critical mass?
o We know Mc = 4/3  Rc3 . Equating this to the Jeans Mass gives:
 375k 3  T 3 1/ 2
4 3
R   

3 3
3  
3
4 m H G    
1/ 2
3/2
3/2
1/ 2






 3 
375
k
T
 RJ3    
    
 4   m H G    4 

375x32 1/ 2 k  T 3 / 2
R   3 3  
  
 4   m H G   
3/2
3
J
3/2
3/2
15 3 / 2  k  T 
   
  
4   m H G   

 15k  T 1/ 2
 RJ  
  
4m H G   
1/ 2

o
The Jeans Radius
If RC>RJ => stable. If RC <RJ => unstable and collapse.

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Gravitational Collapse in the ISM
o
Properties of the ISM:
Diffuse HI Cloud
H2 Cloud Core
T
50 K
150 K

500 cm-3
108 cm-3
Mc
1-100 Msun
10-1000 Msun
o We know from the Jeans Criterion that if Mc>MJ collapse occurs.
 5kT   3 1/ 2
o Substituting the values from the table into M J  
 

G

m
4





H
gives:
3/2
o Diffuse HI cloud: MJ ~ 1500 MSun => stable as Mc<MJ.
o Molecular cloud core: MJ ~ 15 Msun => unstable as Mc>MJ.

o So deep inside molecular clouds the cores are collapsing to form stars.
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Cloud Collapse and
Star/Planet Formation
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Jeans cloud collapse equations
describe the conditions required for
an ISM cloud to collapse.
o
As a cloud collapses,
temperature increases.
o
This is accompanied by spinning-up
of the central star (to conserve AM).
o
Disk also flattens into an oblate
spheroid.
central
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Time-scale for collapse
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The collapse time-scale tff when M >MJ is given by the time a mass element at the
cloud surface needs to reach the centre.
o
In free-fall, an mass element is subject to acceleration g 
o
The time to cover a distance R can therefore be estimated from:
GM
R2
GM
R 1/2gt 2ff 1/2 2 t 2ff
R
o
By approximating R using R3 ~ M/ => tff ≈ (G )-1/2
o
Higher density at cloud center = > faster collapse.
o
For typical molecular cloud, tff ~ 103 years (ie very short).

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Nebular Collapse
o
Solar system was formed from a giant
molecular cloud, known as the proto-solar
or primordial nebula.
o Similar to the Orion Nebula (right).
o
Nebula may have only contained only 1020% more mass than present solar system.
o
Due to some disturbance, perhaps a nearby
supernova, the gas was perturbed, causing
ripples of increased density.
o
The denser material began to collapse
under its own gravity…
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Proto-star and Disk Formation
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Nebula must have possessed some rotation.
Due to the spin, the cloud collapsed faster
along the ‘poles’ than the equator.
o
The result is that the cloud collapsed into a
spinning disk.
o Disk material cannot easily fall all the
remaining way into the center because of its
rotational motion, unless it can somehow
lose some energy, e.g. by friction in the disk
(collisions).
o The initial collapse takes just a few 100,000
years.
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