Stability and Collapse of Molecular Clouds

Question 1

Isothermic Spheres in Hydrostatic Equilibrium

Description

Answer 1

• consider a cloud that maintains equilibrium through the forces of self-gravity and thermal pressure only
• gravity inwards and thermal pressure outwards

Question 2

Isothermic Spheres in Hydrostatic Equilibrium

Gravitational Force

Answer 2

Fg = - GM(r)dM(r) / r²

Question 3

Isothermic Spheres in Hydrostatic Equilibrium

Pressure Force

Answer 3

-the definition of pressure:
P = F/A
-pressure force:
Fp = 4πr² dP(r)

Question 4

Isothermic Spheres in Hydrostatic Equilibrium

Ω

Answer 4

-symbol for total gravitational energy in the cloud

Ω = - ∫ GM(r)dM(r)/r

Question 5

Isothermic Spheres in Hydrostatic Equilibrium

Virial Equilibrium

Answer 5

3VcPs = 2U + Ω

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

Question 6

Isothermic Spheres in Hydrostatic Equilibrium

Virial Equation

Answer 6

-to simplify the collapse criterion, consider a cloud with:
-constant density ρc
-constant pressure Pc up to Rc
-zero external surface pressure Ps=0
-then we have:
2U + Ω = 0

Question 7

Significance of the Virial Equation for Cloud Stability

Answer 7

• if 2U=-Ω then the cloud is stable
• if 2U>-Ω then the pressire wins and the cloud disperses
• if 2U

Question 8

Mass Condition For Collapse

Answer 8

-the conditions for collapse:
Mc > Mj, the Jeans mass
-where:
Mj = (5kT/Gμmh)^(3/2) (3/4πρc)^*(1/2)

Question 9

Jeans Mass in Terms of Solar Mass

Answer 9

Mj = 10^5 * T^(3/2)/µ²√n * M☉

• Mj decreases with decreasing T and increasing n
• the typical Mj=5M☉

Question 10

Length Condition for Collapse

Answer 10

Rc = Rj, the Jeans length
-where:
Rj = 1/μmh √[15kT/4πGn]
-separating the constants from the variables
Rj = 1/mh √(15k/4πG) √(T/µ²n)
-the hotter the material and the more diffuse the material, the greater the Jeans length

Question 11

Jeans Length in Terms of Constants

Answer 11

-to calculate the Jeans length in parsecs:

Rj = 10^4 √(T/µ²n) parsecs

Question 12

Conditions During Cloud Collapse

Answer 12

• if the cloud collapses as a whole:
• -the total mass Mc stays constant
• -the density ρ increases as the same mass is condensed into a smaller volume

Question 13

Is cloud collapse isothermal?

Answer 13

• initially the cloud remains isothermal because:

- the gravitational potential energy that is released and would otherwise heat the cloud is efficiently radiated away

Question 14

Proportionality Relations For Collapse of a Spherical Cloud

Answer 14

Fg ∝ M²/R² ∝ 1/R²
P = ρkT/µmh
Fp ∝ R²P ∝ R²R^(-3) = 1/R
-these relations hold if the collapse is isothermal

Question 15

Equation of Motion for a Thin Shell with Initial Radius Rc

Answer 15

d²r/dt² = -GM(r)/r² = 4πGRc³ρc/3r²

-for a good approximate solution assume that acceleration is constant

Question 16

Free Fall Time

Approximate Solution

Answer 16

tff = √[3/2πGρc]

• freefall time is shorter for denser clouds
• AND independent of the initial radius of the cloud
• freefall time is longer for less dense regions as material has further to fall and shorter for denser regions

Question 17

Free Fall Time

Formal Solution

Answer 17

tff = √[3π/32Gµmhn]

= 5*10^5 / √(µn) years

Question 18

Typical Free Fall Time

Answer 18

-for a typical cloud:
tff = 3*10^5 years
-which is relatively short in astronomic terms

Question 19

Jeans Mass Proportionality

Answer 19

Mj ∝ T^(3/2) * ρ^(-1/2)

Question 20

Free Fall Time Proportionality

Answer 20

tff ∝ ρ^(-1/2)