Stability and Collapse of Molecular Clouds Flashcards Preview

PHYS3281 Star and Planet Formation > Stability and Collapse of Molecular Clouds > Flashcards

Flashcards in Stability and Collapse of Molecular Clouds Deck (20)
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1
Q

Isothermic Spheres in Hydrostatic Equilibrium

Description

A
  • consider a cloud that maintains equilibrium through the forces of self-gravity and thermal pressure only
  • gravity inwards and thermal pressure outwards
2
Q

Isothermic Spheres in Hydrostatic Equilibrium

Gravitational Force

A

Fg = - GM(r)dM(r) / r²

3
Q

Isothermic Spheres in Hydrostatic Equilibrium

Pressure Force

A

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

4
Q

Isothermic Spheres in Hydrostatic Equilibrium

Ω

A

-symbol for total gravitational energy in the cloud

Ω = - ∫ GM(r)dM(r)/r

5
Q

Isothermic Spheres in Hydrostatic Equilibrium

Virial Equilibrium

A

3VcPs = 2U + Ω

Vc = cloud volume
Ps = pressure at the surface of the cloud
U = total kinetic energy of the cloud
Ω = gravitational energy
6
Q

Isothermic Spheres in Hydrostatic Equilibrium

Virial Equation

A

-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

7
Q

Significance of the Virial Equation for Cloud Stability

A
  • if 2U=-Ω then the cloud is stable
  • if 2U>-Ω then the pressire wins and the cloud disperses
  • if 2U
8
Q

Mass Condition For Collapse

A

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

9
Q

Jeans Mass in Terms of Solar Mass

A

Mj = 10^5 * T^(3/2)/µ²√n * M☉

  • Mj decreases with decreasing T and increasing n
  • the typical Mj=5M☉
10
Q

Length Condition for Collapse

A

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

11
Q

Jeans Length in Terms of Constants

A

-to calculate the Jeans length in parsecs:

Rj = 10^4 √(T/µ²n) parsecs

12
Q

Conditions During Cloud Collapse

A
  • 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
13
Q

Is cloud collapse isothermal?

A
  • initially the cloud remains isothermal because:

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

14
Q

Proportionality Relations For Collapse of a Spherical Cloud

A

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

15
Q

Equation of Motion for a Thin Shell with Initial Radius Rc

A

d²r/dt² = -GM(r)/r² = 4πGRc³ρc/3r²

-for a good approximate solution assume that acceleration is constant

16
Q

Free Fall Time

Approximate Solution

A

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
17
Q

Free Fall Time

Formal Solution

A

tff = √[3π/32Gµmhn]

= 5*10^5 / √(µn) years

18
Q

Typical Free Fall Time

A

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

19
Q

Jeans Mass Proportionality

A

Mj ∝ T^(3/2) * ρ^(-1/2)

20
Q

Free Fall Time Proportionality

A

tff ∝ ρ^(-1/2)