Fragmentation Flashcards
mechanisms controlling fragmentation of molecular clouds, heating and cooling, isothermal collapse, what stops fragmentation, mass of the smallest fragment
What is Jeans mass proportional to?
Mj ∝ T^(3/2) ρ^(-1/2)
What is free-fall time proportional to?
-the free-fall time is of the order of:
Gρ)^(-1/2
Typical Dense Core
T ~ 10K
n ~ 10^10 m^(-3)
µ ~ 2.4
Mj ~ 5M☉
If Mj ~ 5M☉ for typical clouds, how do solar-mass and lower mass stars form?
-if the core temperature remains constant during the initial stages of collapse, then as density increases the Jeans mass will decrease since:
Mj ∝ ρ^(-1/2)
Cooling Time
tcool = thermal energy / rate of loss of thermal energy
= u/Λ = (3/2nkT) / Λ
Why are molecular clouds (~10K) warmer than the Cosmic Microwave Background (2.7K)?
-starlight can’t penetrate the clouds to warm them, but cosmic rays can
Why is the ionisation fraction in molecular clouds >0 ?
-cosmic rays can ionise molecular hydrogen
What are cosmic rays?
- cosmic rays mostly consist of relativistic (v->c) protons with a mixture of heavy elements and electrons (including also positrons and anti-protons)
- all of these particles are charged and hence subject to magnetic deflection
- cosmic rays with energies up to 10^9GeV are produce by particle acceleration within the magnetised shocks created by supernova remnants
- more energetic particles are most likely of extra-galactic origin
Interaction of Cosmic Rays With Molecular Hydrogen in Molecular Clouds
Ep≥1GeV
- in a molecular cloud, a gyrating cosmic ray proton interacts with ambient nuclei and electrons through both coulomb and strong nuclear forces
- the nuclear excitations (Ep≥1GeV) principally decay through emission of gamma rays
Interaction of Cosmic Rays With Molecular Hydrogen in Molecular Clouds
Ionisation
-proton scatters inelastically with H2, mainly ionising it:
p’+ + H2 -> H2+ + e- + p’’+
-this is an inelastic interaction so the energy of the proton changes
-it is the secondary electron that provides heat through its subsequent interactions with H2
-the ionisation rate is:
1 to 10 * 10^(-17) /s
Cosmic Ray Heating of Molecular Clouds
Description
-the electron produced from cosmic ray ionisation of molecular hydrogen goes on to interact with other molecular hydrogen splitting it into atoms:
e’- + H2 -> H + H + e’’-
-this is an inelastic process so the energy of the electron changes
-the energy of the incoming electron beyond that which is required to dissociate H2 (4.52eV) goes into kinetic energy (translational motion) of the two hydrogen atoms
-subsequent collision quickly disperse this energy throughout the gas
Cosmic Ray Heating of Molecular Clouds
Energy Provided by a Single Proton
-the net energy provided by a single 10MeV proton is:
ΔE(H2) ≈ 7eV
-so this is a very inefficient process
-BUT it happens frequently enough to maintain a cloud temperature of ~10K
Cosmic Ray Heating of Molecular Clouds
Heat Deposition Per Unit Volume
-the heat deposition in the cloud per unit volume is given by:
Γcr(H2) = ζ(H2) * n(H2) * ΔE(H2)
-where:
ΔE(H2) = 7eV per event
n(H2) = number density of molecular hydrogen
ζ(H2) = ionisation rate of H2
-for a typical cloud this is around 10^(-25)J/(sm^3)
How is the equilibrium temperature of 10K reached?
-the heating by cosmic rays is balanced by cooling due to CO rotational emission giving an equilibrium temperature of around 10 K in a typical molecular cloud
Cooling of Molecular Clouds
-the main cooling mechanism is due to CO rotational emission
-rate of loos of thermal energy due to CO, Λco, depends on the number density of CO molecules in the cloud, on the energy of the transition and on the optical depth of the emitted lines
-in typical clouds:
Γcr ≈ Λco
Cooling Rate and Temperature
-cooling rate is very sensitive to temperature, e.g. doubling temperature increases the rate by more than an order of magnitude
-an indicative value for cooling time is:
tcool = (3/2nkT) / Λco
Cooling Rate vs. Free-Fall Time
-comparing cooling time and free-fall time at the same density:
tc ≈ 0.02tff
-cooling is fast
What are the stages of collapse of molecular clouds?
-the large cloud begins to collapse when it becomes sufficiently dense
-the cooling time is much shorter than the free-fall time so the initial collapse WILL be isothermal
-the Jeans mass will decrease
-smaller subunits then become unstable to collapse, localised regions in the cloud collapse independently of the molecular cloud
=» fragmentation
-when a fragment can collapse independently without further disruption, it will go on to form a protostar
What halts fragmentation?
- during collapse, the cloud remains isothermal as long as it remains optically thin so that it can efficiently radiate away gravitational potential energy
- but opacity is proportional to number density which is increasing as the cloud collapses so opacity increases as well
- when the cloud becomes sufficiently opaque, radiation cannot be radiated away and the cloud behaves as a blackbody
- temperature increases which increases mass away from the Jeans mass and towards stability
Energy Transfer in Opaque Fragments
-once a fragment is opaque, it will radiate almost as a black body:
rate of energy loss due to radiation = rate of gain in gravitational potential energy
=>
4πR²σT^4 ~ GM²/R * 1/tff
At what mass does a fragment become opaque?
-from transfer of energy in an opaque cloud, we have:
4πR²σT^4 ~ GM²/R * 1/tff
-the mass at which a fragment becomes opaque can be found by setting T=10K after which the temperature will increase
Mass of the Smallest Fragment
Dependency of Mass on Temperature
M ∝ T^(1/4)
-from other proportionality relation
- more elaborate derivation gives:
M ~ 0.007 * T^(1/4) / μ^(9/4) * M☉
Mass of the Smallest Fragment
Typical Molecular Cloud
M ~ 0.002M☉
-this is approximately of the order of two Jupiter masses
