Hydrostatic Core

Question 1

Why does fragmentation stop?

Answer 1

• the collapsing core becomes so dense that it becomes opaque to its own radiation
• it can no longer efficiently radiate the energy released during collapse

Question 2

What happens when fragmentation stops?

Answer 2

• the internal temperature, and therefore pressure, increases
• eventually an equilibrium is reached between internal pressure and gravity
• this halts free-fall collapse

Question 3

Where does the shock front arise form?

Answer 3

• a hydrostatic core develops
• a shock front develops at the interface between the core and the envelope
• in-fall velocity at the shock front can be approximated by the free-fall velocity

Question 4

Free-Fall Velocity

Answer 4

-equate kinetic energy with gravitational potential energy:
1/2 m vff² = GMm/r
-rearrange for vff:
vff = √ [2GM/r]

Question 5

Accretion Luminosity

Answer 5

-if the shock front is at Rc, then most of the kinetic energy of the in-falling material is converted into radiation where the luminosity is given by:
Lacc = 1/2 M’ vff² = GMM’/Rs
-where M’ is the mass in-fall rate
-and Lacc is the accretion luminosity which will mostly be absorbed by dust and remitted at infrared wavelengths

Question 6

Mass Continuity Equation

Answer 6

M’ = 4πr²ρv

Question 7

Density of In-Falling Matter

Answer 7

-from the mass continuity equation:
M’ = 4πr²ρv
-for a constant mass accretion rate and free-falling material:
v ∝ r^(-1/2)
-because M’ is a constant and independent of radius, we find that the density of in-falling matter:
ρ ∝ r^(-3/2)

Question 8

Potential for Further Collapse

Answer 8

-as the hydrostatic core heats up there is potential for further collapse:
Mj ∝ T^(3/2) ρ^(-1/2)
-so at the point of marginal stability where M is constant;
T^(3/2) ∝ ρ^(1/2)
=>
T ∝ ρ^(1/3)
-finally:
T ∝ ρ^(γ-1), with γ=4/3=heat capacity ratio= Cp/Cv

Question 9

γ and Core Stability

Answer 9

• if γ>4/3, Mj increases as T increases and the core is stable
• if γ≤4/3, Mj decreases as T increases and the core is unstable leading to potential further collapse of the hydrostatic core

Question 10

What happens within the protostar as time goes on that might disrupt its stability?

Answer 10

• there is a hydrostatic core that is accreting molecular material from a free-falling envelope
• the temperature and density are both increasing
• a point will be reached at which molecules and dust will dissociate

Question 11

Core Stability

Dissociation of Hydrogen

Answer 11

• as long as the hydrostatic core material is mostly molecular hydrogen, γ=1.4>4/3 so the core remains stable
• as temperature increases, around 2000K molecular hydrogen dissociates to monatomic hydrogen, γ=5/3>4/3 so the core is still stable
• what triggers collapse?

Question 12

How is further collapse triggered?

Answer 12

• the process of H2 dissociation absorbs 4.5eV of energy per dissociation event, energy that would normally provide a pressure gradient significant to maintain hydrostatic equilibrium
• the core becomes dynamically unstable and a second collapse occurs
• hydrostatic equilibrium is re-established when the core radius reaches ~2R☉
• the mass enclosed remains much less than its final value
• after core collapse, a second shock front is established as the envelope continues to accrete in-falling material

Question 13

Stages of Collapse

Answer 13

• FIRST DYNAMIC COLLAPSE PHASE
• cooling phase
• isothermal phase
• optically thick at infrared wavelengths
• FIRST QUASI-EQUILIBRIUM PHASE
• heating phase
• dissociation of molecular hydrogen
• SECOND DYNAMIC COLLAPSE PHASE
• all atomic hydrogen
• SECON QUASI-EQUILIBRIUM PHASE