7. Hydrostatic Core

Question 1

Stages after fragmentation stops?

Answer 1

Formation of a protostar: first hydrostatic core

Shock fronts

Further collapse

Question 2

Why does fragmentation stop?

Answer 2

Collapsing core becomes so dense that it becomes opaque to its own radiation

And can no longer efficiency radiate energy released during collapse

Question 3

What happens when fragmentation stops?

Answer 3

Internal temp and therefore pressure increase

Eventually equilibrium is reached between internal pressure and gravity

Halting free fall collapse

Question 4

What happens after free fall collapse is halted?

Answer 4

A hydrostatic core develops

Shock front develops at interface between core and envelope

Question 5

How can infall velocity at shock front be approximated?

Answer 5

By free-fall velocity

Question 6

Size of first hydrostatic core?

Answer 6

~ 5au

Question 7

Structure of molecular cloud once hydrostatic core develops?

Answer 7

Centre: hydrostatic core, high T and P

Collapsing core and outer envelope surrounding - still being fed by accretion

Question 8

What does the shock front do?

Answer 8

Feeds onto hydrostatic core but is no longer collapsing

Question 9

How to derive free fall velocity of inflating material to hydrostatic core?

Answer 9

Equate KE to GPE

Question 10

With a shock front at Rs, what happens to the KE of infalling material?

Answer 10

Most of KE converted to radiation (with certain accretion luminosity)

Question 11

Where does accretion luminosity come from?

Answer 11

The KE from infalling material being converted into radiation at the shock front with accretion luminosity

Question 12

Equation for accretion luminosity?

Answer 12

L_acc ~ 1/2Ṁ(v_ff)^2

~ GMṀ/Rs

Ṁ = mass infall / accretion rate

Question 13

Why does accretion luminosity help to find embedded protostars in a molecular cloud?

Answer 13

Accretion luminosity is mostly absorbed by dust and re-radiated at IR wavelengths

Question 14

Derive the density structure of infalling material to the hydrostatic core

Answer 14

See notes

Question 15

Mass continuity equation?

Answer 15

Ṁ = 4π r^2 ρ v

ρ = density of material
v = free fall velocity

Question 16

How does density of infalling matter to the hydrostatic core depend on radius?

Answer 16

ρ ∝ r^-3/2

Question 17

When considering further collapse onto the hydrostatic core, what conditions are applied?

Answer 17

Adiabatic - no heat lost or gained from the system

Question 18

What are the conditions for further collapse onto the hydrostatic core?

Answer 18

γ > 4/3 core is stable

γ <= 4/3 core is unstable (further collapse)

(γ = heat capacity ratio)

Question 19

Why does γ > 4/3 mean core is stable?

Answer 19

MJ increases as T increases

Question 20

Why does γ <= 4/3 mean core might collapse further?

Answer 20

MJ decreases as T increases - unstable

Question 21

What relation relates hydrostatic core temperature to density?

Answer 21

T ∝ ρ^(γ-1)

Question 22

What determines the ratio of specific heats at constant pressure to constant volume?

Answer 22

Number of degrees of freedom f, i.e., diatomic or monoatomic

Question 23

When will γ be greater than 4/3?

Answer 23

For a diatomic molecule, monoatomic gas

Question 24

What is the equation for γ?

Answer 24

γ = f + 2 / f

Question 25

How many degrees of freedom for a diatomic molecule?

Answer 25

5; 3 translational, 2 rotational

Question 26

How many degrees of freedom for a monoatomic gas?

Answer 26

3 translational only

Question 27

Under high temp and pressure conditions of molecular hydrogen, what does gamma equal?

Answer 27

H2 dissociates into its constituent atoms

γ = 3+2/3 = 5/3

Question 28

Is H2 stable under high temp and pressure conditions?

Answer 28

Yes, γ = 5/3 > 4/3

Question 29

Does dissociation of dust and molecules affect stability?

Answer 29

Yes

Question 30

Does dissociation of dust and molecules trigger second phase of collapse?

Answer 30

Yes

Question 31

How is 2nd phase of collapse triggered?

Answer 31

H2 dissociation absorbs energy, that would otherwise provide pressure sufficient to maintain hydrostatic equilibrium

Question 32

After the second collapse, when is hydrostatic equilibrium reestablished?

Answer 32

When core radius reaches 2 solar radii

Question 33

When do we have dissociation of molecular hydrogen in the hydrostatic core?

Answer 33

T ~ 2000K and high pressures

Question 34

After second collapse of the hydrostatic core, is it a star?

Answer 34

No - it is still a protostar

Question 35

What happens after the second collapse?

Answer 35

Protostar accretes through first shock at first hydrostatic core - second shock front is established as the envelope continues to accrete infalling material

Question 36

When do we have adiabatic conditions in the core?

Answer 36

When it becomes optically thick

Question 37

When will hydrogen fusion occur?

Answer 37

After second hydrostatic core is establishing and temperature and pressure continues to increase at it accretes more material