6. Fragmentation

Question 1

How do we know another process must occur to form solar mass (and lower) stars?

Answer 1

Jeans mass for a typical dense core is 5 solar masses

So there must be another process that means collapse occurs at lower solar masses

Question 2

Under what condition will the Jeans mass decrease as density increases?

Answer 2

If core temperature remains constant during the initial stages of the collapse

Question 3

Equation for cooling time of a molecular cloud?

Answer 3

t_cool = u / Λ

where

u = 3/2nkT thermal energy
Λ = rate of loss of thermal energy

Question 4

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

Answer 4

There must be a heating process - Cosmic-ray heating

Question 5

Why is the ionisation fraction in molecular clouds > 0?

Answer 5

Cosmic-ray proton scatters inelastically with H2 (ionising it)

Question 6

What do cosmic rays consist of?

Answer 6

Mostly relativistic (v ~ c) protons, with a mix of heavy elements and electrons

[all particles are charged so subject to magnetic deflection]

Question 7

How are cosmic rays produced?

Answer 7

Particle acceleration within the magnetised shocks created by supernova remnants

Question 8

How do cosmic rays provide heat?

Answer 8

Cosmic-ray proton scatters inelastically, ionising H2

Secondary electron collides with another H2 = dissociation

[e- + H2 -> H + H + e-]

Resulting 2 H molecules have higher KE, colliding with other molecules and creating heating mechanism

Question 9

What type of collisions are involved with hydrogen and cosmic-ray protons?

Answer 9

Inelastic

Question 10

How does dissociation relate to the heating mechanics within a molecular cloud?

Answer 10

An electron dissociates a H2 molecule, so 4.5eV goes into KE of the 2 H atoms

Question 11

Why do cosmic-ray protons behave different to other protons (H+) in the gas?

Answer 11

They are relativistic so much higher energy

Question 12

Equation for heat deposition in a molecular cloud per unit volume?

Answer 12

Γcr(H2) = ζ(H2)n(H2)ΔE(H2)

where

ζ = ionisation rate
n = number density
ΔE = energy deposition

Question 13

Typical ionisation rate?

Answer 13

(1-10)e17 s^-1

Question 14

What is meant by ionisation rate?

Answer 14

Probability per unit time of ionisation

Question 15

Why do we have an equilibrium temperature of around 10K in a typical molecular cloud?

Answer 15

Cosmic-ray heating is balanced by cooling

Question 16

Net energy provided by a single 10MeV proton?

Answer 16

ΔE(H2) ~ 7eV (very small)

Question 17

What is the main cooling mechanism in molecular clouds?

Answer 17

CO rotational emission

Question 18

How is energy removed and transferred to radiation in a molecular cloud?

Answer 18

CO is easily excited in low temperature gas; it will spontaneously decay, emitting photons as radiation and removing KE

Question 19

What does rate of loss of thermal energy ΛCO depend on?

Answer 19

Number density of CO molecules in the cloud, the energy of the transition, and the optical depth of the emitted lines

Question 20

ΛCO value in a typical cloud?

Answer 20

10e-23 J/s/m^3

Question 21

Is cooling rate sensitive to temperature?

Answer 21

Yes

Question 22

How does cooling time compare to free fall time at the same density?

Answer 22

Free fall time is longer than cooling time by 2 orders of magnitude

Question 23

Why do we compare cooling time and free fall time?

Answer 23

Cooling time is shorter than free fall time, therefore cooling time is fast enough that the cloud can maintain an isothermal temperature

Question 24

How does density affect ratio of cooling time to free fall time?

Answer 24

Higher density: cooling time wrt free fall time is faster

Question 25

Why can we say that the molecular gas stays isothermal during collapse?

Answer 25

Cooling time is fast compared to free fall time

(despite some cosmic-ray heating)

Question 26

What implication does cooling time being fast compared to free fall time have on Jeans mass?

Answer 26

Means initial collapse of cloud is isothermal (removing T dependence)

Therefore MJ decreases (since MJ prop to n^-1/2)

But initial cloud MJ doesn’t decrease, instead subunits within cloud reach MJ for that given density

Question 27

When we say MJ decreases in isothermal collapse, what does this mean specifically?

Answer 27

The initial MJ of the cloud doesn’t decrease, instead subunits within cloud reach MJ for that given density (fragmentation)

Question 28

How does fragmentation occur?

Answer 28

A single entity reaches MJ and begins to collapse

Cloud cools sufficiently so collapse is isothermal

As density increases in subunits within the cloud, these reach MJ for that given density and fragments collapse

Question 29

What are the stages of collapse of molecular clouds into stars?

Answer 29

Large cloud begins to collapse as it becomes sufficiently dense

Fragmentation: Localised regions collapse

When fragment collapses independently without further disruption, forms protostar

Question 30

What halts fragmentation?

Answer 30

When cloud becomes optically thick and cannot cool efficiently

[Once opaque a fragment radiates almost as a black body]

Question 31

Why must a cloud remain optically thin for it to remain isothermal?

Answer 31

If cloud becomes optically thick, it can no longer cool efficiently i.e., cannot radiate GPE released during collapse

Question 32

What effect does a cloud being optically thick have on Jeans Mass?

Answer 32

Since it is no longer isothermal, T increases does MJ

Question 33

Why does optical depth τ increase during fragmentation?

Answer 33

τ ∝ n^1/2 for constant T

So τ increases as density increases

Question 34

Derive the dependency of mass (at which the fragment becomes opaque) on temperature

Answer 34

See notes

Question 35

Once a fragment is opaque, what assumption can be used?

Answer 35

It will radiate almost as a black body

Therefore rate of energy lost due to radiation = rate in gain in GPE

Question 36

How does minimum fragment mass depend on temperature?

Answer 36

M prop to T^1/4

Question 37

For a typical molecular cloud (T~10K, µ = 2.4), what is the mass of the smallest fragment?

Answer 37

M ~ 0.002 solar masses

Question 38

How is minimum fragment mass estimated?

Answer 38

Equating rate of energy lost via radiation with that gained in GPE