1. Lecture 4 - Interstellar Cloud Collapse
Topics to be covered:
The Interstellar Medium
Gravitational collapse
Virial Theorem
Jeans Instability
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2. Interstellar Medium (ISM)
Interstellar space is not empty - there is gas and dust in the galaxy.
Chemical composition of gas is mostly hydrogen (~70%), some metals (~few %) and rest helium.
Clouds can be mainly
Neutral hydrogen (HI)
Ionized hydrogen (HII)
Molecular hydrogen (H2)
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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3. Interstellar Medium (ISM) - HI regions
Far from stars, no energy available to ionize/excite HI - therefore no optical emission lines.
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.
Energy difference is small ~6x10-6 eV, which is ~21 cm (radio waves).
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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4. Interstellar Medium (ISM) - HII regions
HII regions are found near hot stars.
UV photons from nearby stars ionize H atoms.
When electrons and protons recombine, they generally emit Balmer lines.
Also observe lines from other atoms (e.g., oxygen, sulfur, silicon, carbon, …).
HII regions are associated with active star formation.
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5. Interstellar Medium (ISM) - H2 regions
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,
Other molecules. For example, CO emits radiation at a wavelength of 2.6mm - microwaves.
Dust. Dust (and molecules) are often found together - dense, dark clouds (called Bok globules) are regions of very dense molecular gas.
Where there is lots of gas - in particular molecular and ionized gas => lots of stars formation …
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6. The Virial Theorem
Applies to any system of particles with pair interactions for which the distribution of particles does not vary with time.
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.
As K = 3/2 NkT and
Eqn. 1
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7. Application of the Virial Theorem
VT can be used to estimate conditions for cloud collapse:
If 2K > U => gas pressure (energy) will exceed gravitational potential energy and expand.
If 2K < U => gravitational energy will exceed gas pressure and collapse.
The boundary between these two cases describes the critical condition for stability.
We know that  = Mc/Vc = Mc / (4/3Rc3) =>
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? 2K U
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8. Application of the Virial Theorem - The Jeans Mass
Substituting for R and N in Eqn. 1:
The Jeans Criterion is: Mc > MJ
If Mc > MJ => cloud will collapse.
Rearranging Eqn. 2 gives:
Eqn. 2
The Jeans Mass 2K U
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9. Application of the Virial Theorem - The Jeans Radius
Is there a critical radius that corresponds to the critical mass?
We know Mc = 4/3  Rc3 . Equating this to the Jeans Mass gives:
If RC>RJ => stable. If RC <RJ => unstable and collapse.
The Jeans Radius
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10. Gravitational Collapse in the ISM
Properties of the ISM:
We know from the Jeans Criterion that if Mc>MJ collapse occurs.
Substituting the values from the table into
gives:
Diffuse HI cloud: MJ ~ 1500 MSun => stable as Mc<MJ.
Molecular cloud core: MJ ~ 15 Msun => unstable as Mc>MJ.
So deep inside molecular clouds the cores are collapsing to form stars.
Diffuse HI Cloud
H2 Cloud Core T
50 K
150 K 
500 cm-3
108 cm-3 Mc
1-100 Msun
Msun
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11. Cloud Collapse and Star/Planet Formation
Jeans cloud collapse equations describe the conditions required for an ISM cloud to collapse.
As a cloud collapses, central temperature increases.
This is accompanied by spinning-up of the central star (to conserve AM).
Disk also flattens into an oblate spheroid.
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12. Time-scale for collapse
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.
In free-fall, an mass element is subject to acceleration
The time to cover a distance R can therefore be estimated from:
By approximating R using R3 ~ M/ => tff ≈ (G )-1/2
Higher density at cloud center = > faster collapse.
For typical molecular cloud, tff ~ 103 years (ie very short).
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13. Nebular Collapse
Solar system was formed from a giant molecular cloud, known as the proto-solar or primordial nebula.
Similar to the Orion Nebula (right).
Nebula may have only contained only 10-20% more mass than present solar system.
Due to some disturbance, perhaps a nearby supernova, the gas was perturbed, causing ripples of increased density.
The denser material began to collapse under its own gravity…
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14. Proto-star and Disk Formation
Nebula must have possessed some rotation. Due to the spin, the cloud collapsed faster along the ‘poles’ than the equator.
The result is that the cloud collapsed into a spinning disk.
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).
The initial collapse takes just a few 100,000 years.
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