2. Stability and Collapse of Molecular Clouds

Question 1

Derive the Virial Equilibrium

Answer 1

See notes

Question 2

What is the Virial Equilibrium equation?

Answer 2

3VcPs = 2U + Ω

Vc = total volume of cloud
Ps = pressure at the surface
U = total thermal energy content
Ω = total gravitational energy of cloud

Question 3

What are the assumptions in the Virial equilibrium equation?

Answer 3

Only considers thermal pressure and gravity

Spherical cloud

Ideal monoatomic gas

Question 4

What forces does the Virial equilibrium equation take into account?

Answer 4

Gravity and internal pressure

But could include other forces e.g. magnetic fields

Question 5

Virial Equation assumptions?

Answer 5

Constant density ρc

Constant pressure (Pc) up to Rc

Zero external (surface) pressure, Ps = 0

Question 6

What is the Virial Equation?

Answer 6

2U + Ω = 0

Question 7

What is the simplified collapse criterion?

Answer 7

The Virial Equation

Question 8

What can be calculated under the simplified assumptions of the Virial Equation?

Answer 8

Ω

= 3/5 G(Mc)^2 / Rc

Question 9

What is the equation for gravitational energy in the Virial equation?

Answer 9

Ω = 3/5 G(Mc)^2 / Rc

Question 10

What is the significance of the Virial criterion / equation for cloud stability? ( 3 conditions)

Answer 10

2U = -Ω stable

2U > -Ω pressure wins, dispersion of cloud

2U < -Ω gravity wins, contraction of cloud

Question 11

Derive Ω when in Virial Equation

Answer 11

See notes

Question 12

How can 2U < -Ω be written?

Answer 12

3/5 G(Mc)^2 / Rc > 3VcPc

Question 13

What equations / assumptions are used to write 2U < -Ω more usefully?

Answer 13

Pc = NkT / Vc (assuming ideal gas)

Rc = (Mc / 4/3π ρc)^1/3

N = Mc / µ m_H

where

µ m_H = mean molecular mass of gas
N = total number of particles in the cloud

Question 14

What is Jeans Mass?

Answer 14

The critical mass at which a molecular cloud will collapse to form a star

Question 15

Derive Jeans Mass

Answer 15

See notes

Question 16

How does Jeans mass depend on T and n?

Answer 16

Mj decreases as T decreases and n increases

Question 17

Values for a typical dense core?

Answer 17

T = 10K
n = 10e10 m^-3
µ = 2.4

Question 18

Jeans mass for a typical dense core?

Answer 18

5 solar masses

Question 19

If a molecular cloud collapses as a whole, what happens?

Answer 19

Total mass Mc stays constant

Density ρ increases

Initially the cloud remains isothermal

Question 20

How does a cloud initially remain isothermal after a collapse?

Answer 20

Gravitational potential energy released is efficiently radiated away

(this would otherwise heat the cloud)

Question 21

What does the cloud being isothermal initially after collapse have an effect on?

Answer 21

Implications for further collapse

An increases force of gravity as well as increasing pressure force

Question 22

When a cloud collapses as a whole, why does gravity increasingly dominate over pressure?

Answer 22

F_G ∝ M^2/R2 ∝ R^-2

and P = ρkT/µm_H ∝ R^-3

so F_P ∝ R^2P = R^2R^-3 = R-1

Question 23

When a cloud collapses as a whole, how does gravitational force vary with radius?

Answer 23

F_G ∝ R^-2

Question 24

When a cloud collapses as a whole, how does pressure force vary with radius?

Answer 24

F_P ∝ R-1

Question 25

When a cloud collapses as a whole, when do the force vs radius relations hold?

Answer 25

If the collapse is isothermal

Question 26

Derive free fall time

Answer 26

See notes

Question 27

Equation for tff?

Answer 27

tff = (3/2πGρ_c)^1/2

Question 28

What is tff independent of?

Answer 28

The cloud’s initial radius

Question 29

What can Jeans mass be written as?

Answer 29

Mj = 10^5 * (T^3/2 / µ^2*n^1/2) solar masses

Question 30

Is our tff the formal solution?

Answer 30

No, but it is very close to the formal solution

Question 31

For the formation of a 5 solar mass star, what is tff?

Answer 31

3e5 years

Question 32

How does free fall time depend on density?

Answer 32

tff is longer for less dense regions

and shorter for denser regions

Question 33

How can cloud stability be determined?

Answer 33

Considering the balance of inwards vs outwards forces

Question 34

What does the Virial equation do?

Answer 34

Sets the stability criterion of a molecular cloud supported by thermal pressure alone

Question 35

When will a cloud supported only by thermal pressure collapse?

Answer 35

If its mass exceed the jeans mass

Mj ∝ T^3/2 ρ^-1/2

Question 36

In the initial isothermal collapse, which force dominates?

Answer 36

Gravity

Question 37

How can free fall time for collapse be estimated?

Answer 37

By assuming acceleration is constant