Fragmentation

Question 1

What is Jeans mass proportional to?

Answer 1

Mj ∝ T^(3/2) ρ^(-1/2)

Question 2

What is free-fall time proportional to?

Answer 2

-the free-fall time is of the order of:

Gρ)^(-1/2

Question 3

Typical Dense Core

Answer 3

T ~ 10K
n ~ 10^10 m^(-3)
µ ~ 2.4
Mj ~ 5M☉

Question 4

If Mj ~ 5M☉ for typical clouds, how do solar-mass and lower mass stars form?

Answer 4

-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)

Question 5

Cooling Time

Answer 5

tcool = thermal energy / rate of loss of thermal energy

= u/Λ = (3/2nkT) / Λ

Question 6

Why are molecular clouds (~10K) warmer than the Cosmic Microwave Background (2.7K)?

Answer 6

-starlight can’t penetrate the clouds to warm them, but cosmic rays can

Question 7

Why is the ionisation fraction in molecular clouds >0 ?

Answer 7

-cosmic rays can ionise molecular hydrogen

Question 8

What are cosmic rays?

Answer 8

• 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

Question 9

Interaction of Cosmic Rays With Molecular Hydrogen in Molecular Clouds
Ep≥1GeV

Answer 9

• 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

Question 10

Interaction of Cosmic Rays With Molecular Hydrogen in Molecular Clouds
Ionisation

Answer 10

-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

Question 11

Cosmic Ray Heating of Molecular Clouds

Description

Answer 11

-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

Question 12

Cosmic Ray Heating of Molecular Clouds

Energy Provided by a Single Proton

Answer 12

-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

Question 13

Cosmic Ray Heating of Molecular Clouds

Heat Deposition Per Unit Volume

Answer 13

-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)

Question 14

How is the equilibrium temperature of 10K reached?

Answer 14

-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

Question 15

Cooling of Molecular Clouds

Answer 15

-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

Question 16

Cooling Rate and Temperature

Answer 16

-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

Question 17

Cooling Rate vs. Free-Fall Time

Answer 17

-comparing cooling time and free-fall time at the same density:
tc ≈ 0.02tff
-cooling is fast

Question 18

What are the stages of collapse of molecular clouds?

Answer 18

-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

Question 19

What halts fragmentation?

Answer 19

• 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

Question 20

Energy Transfer in Opaque Fragments

Answer 20

-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

Question 21

At what mass does a fragment become opaque?

Answer 21

-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

Question 22

Mass of the Smallest Fragment

Dependency of Mass on Temperature

Answer 22

M ∝ T^(1/4)
-from other proportionality relation
- more elaborate derivation gives:
M ~ 0.007 * T^(1/4) / μ^(9/4) * M☉

Question 23

Mass of the Smallest Fragment

Typical Molecular Cloud

Answer 23

M ~ 0.002M☉

-this is approximately of the order of two Jupiter masses