VIRIAL THEOREM
In the case of an isolated, self-gravitating system:
2 KE - PE = 0
mV2 = GmM/r
M = V2r/G
V
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Gravitational Collapse
•
Overdense regions will tend to become more overdense because of self-gravity.
Matter will move out of underdense regions leaving them even more evacuated.
•
Jeans criterion for collapse:
inward pull of gravity (Potential Energy or PE)
dominates
resistance to collapse from thermal pressure (Kinetic Energy or KE)
M/R>
PE = GmM
R
2
V
KE=mV2
2
•
Rewrite M/R>V2, replacing mass with density (ρ),
where mass = density x volume (M ∝ ρ R3)
so ρR2 > Vs2
•
and replace sound speed with temperature, mV2~kT => Vs ∝ √T,
so ρR2 > T for collapse
•
The “Jeans criteria” are usually presented as either the radius or the mass at the
critical threshold between collapse and support
•
Rj ~ √(T/ρ)
Jeans Radius at threshold for collapse
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or replacing radius using the relation between mass, density, and volume (M ∝ ρ R3)
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Mj ~ √(T3/ρ) Jeans Mass at threshold for collapse
Mj ~ T3/2 / ρ1/2
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Lower Mj by decreasing temperature (T) or increasing density (ρ)
o smaller masses can collapse
•
Masses greater than Mj: the pull of gravity is greater than the
support pressure  the mass collapses
Masses less then Mj: the pull of gravity is not enough to overcome
the support pressure.
•
•
The Jeans mass depends on temperature (T) and density (ρ)
o Higher temperature
o  more support pressure
o  need more mass to overcome pressure
o Higher density
o more pull of gravity
o do not need as much total mass to overcome
pressure
dense cold clouds collapsing to form stars
Star
StarFormation
Formation
Star Formation
Energy Conservation
• In the absence of dissipation, energy equilibrium
o Potential Energy  PE ∝ KE  kinetic Energy
o Equilibrium prevents gas clouds from collapsing very far
o PE of gravity balanced by KE (pressure from particle motions)
• Energy is conserved in a closed system
* examples: a cluster of stars or our current Solar System
In these cases, the bodies gravitationally attract each other but almost never
collide. The cluster or our Solar System cannot get rid of energy, so relaxes
into an equilibrium between PE and KE
PE~GmM/R
KE~mV2~kT
Energy Dissipation
• But KE can be lost from a collapsing cloud of gas:
o Gas heats up with collapse because of collisions  shocks  gas
ionizes  as electrons recombine with atoms, radiation is released,
which escapes from gas cloud.
• Hence, as energy is lost
(KE decreases) then cloud
can continue to shrink
=> Runaway cycle: cloud shrinks
 increased PE causes increased KE
 collisions heat and ionize gas
 electrons recombine with atoms
radiating energy away, cooling gas
 KE lowered by cooling; less support
 cloud shrinks, density increases
Energy Dissipation
optical image
infrared image penetrates the dust to reveal newborn stars
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As the gas cloud shrinks, the density, ρ, goes a way up
•
The Jeans Mass (Mj) threshold goes down since Mj ∝ √(T3/ρ), where T is temperature and ρ is density
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Fragmentation: locally denser regions in cloud collapse faster than the cloud as an ensemble
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Million solar mass cloud of gas breaks into thousands of proto stars
The Role of Chemical Abundances
• Cooling to low temperatures is much more efficient
if there are elements heavier than H (hydrogen) and
He (helium).
✔ reason: there are many more transitions available
✔ more ways for gas to radiate energy, hence cool
• If there are heavy elements and molecules then gas
can cool to very low temperatures
✔ Jeans mass can get small
✔ can fragment into small stars
Scenario for Populating Galaxy With Stars
•
First generation: only Hydrogen (H) and Helium (He) present
✔Collapsing clouds only fragment to masses of a few times Sun
✔ Lifetimes of first stars short compare with age of universe
✔ Most of the mass in these stars is returned to interstellar medium via novae, supernovae, or more gentle mass loss
✔ Enrich interstellar medium in heavier elements
•
Later generations: increasing amounts of heavier elements
✔ Clouds cool more; Jeans mass can be smaller (down to around 0.1 M sun)
✔ Wide spectrum of star types form; many with lifetimes much greater than the age of the universe
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This scenario explains why we never see stars with only H and He.
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The earliest population has disappeared: massive stars  short lifetimes
Key concepts concerning gravitational collapse
 PE (force of attraction) is resisted by KE (pressure)
 A balance is reached between PE and KE in a closed system
with GmM/R ~ mV2 ~ kT
 Dissipation: a process that allows energy to escape from a
closed system (mainly from radiation from hot, ionized gas)
 Energy loss reduces pressure and allows the system to shrink