2. Stability and Collapse of Molecular Clouds Flashcards
Derive the Virial Equilibrium
See notes
What is the Virial Equilibrium equation?
3VcPs = 2U + Ω
Vc = total volume of cloud
Ps = pressure at the surface
U = total thermal energy content
Ω = total gravitational energy of cloud
What are the assumptions in the Virial equilibrium equation?
Only considers thermal pressure and gravity
Spherical cloud
Ideal monoatomic gas
What forces does the Virial equilibrium equation take into account?
Gravity and internal pressure
But could include other forces e.g. magnetic fields
Virial Equation assumptions?
Constant density ρc
Constant pressure (Pc) up to Rc
Zero external (surface) pressure, Ps = 0
What is the Virial Equation?
2U + Ω = 0
What is the simplified collapse criterion?
The Virial Equation
What can be calculated under the simplified assumptions of the Virial Equation?
Ω
= 3/5 G(Mc)^2 / Rc
What is the equation for gravitational energy in the Virial equation?
Ω = 3/5 G(Mc)^2 / Rc
What is the significance of the Virial criterion / equation for cloud stability? ( 3 conditions)
2U = -Ω stable
2U > -Ω pressure wins, dispersion of cloud
2U < -Ω gravity wins, contraction of cloud
Derive Ω when in Virial Equation
See notes
How can 2U < -Ω be written?
3/5 G(Mc)^2 / Rc > 3VcPc
What equations / assumptions are used to write 2U < -Ω more usefully?
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
What is Jeans Mass?
The critical mass at which a molecular cloud will collapse to form a star
Derive Jeans Mass
See notes
How does Jeans mass depend on T and n?
Mj decreases as T decreases and n increases
Values for a typical dense core?
T = 10K
n = 10e10 m^-3
µ = 2.4
Jeans mass for a typical dense core?
5 solar masses
If a molecular cloud collapses as a whole, what happens?
Total mass Mc stays constant
Density ρ increases
Initially the cloud remains isothermal
How does a cloud initially remain isothermal after a collapse?
Gravitational potential energy released is efficiently radiated away
(this would otherwise heat the cloud)
What does the cloud being isothermal initially after collapse have an effect on?
Implications for further collapse
An increases force of gravity as well as increasing pressure force
When a cloud collapses as a whole, why does gravity increasingly dominate over pressure?
F_G ∝ M^2/R2 ∝ R^-2
and P = ρkT/µm_H ∝ R^-3
so F_P ∝ R^2P = R^2R^-3 = R-1
When a cloud collapses as a whole, how does gravitational force vary with radius?
F_G ∝ R^-2
When a cloud collapses as a whole, how does pressure force vary with radius?
F_P ∝ R-1
When a cloud collapses as a whole, when do the force vs radius relations hold?
If the collapse is isothermal
Derive free fall time
See notes
Equation for tff?
tff = (3/2πGρ_c)^1/2
What is tff independent of?
The cloud’s initial radius
What can Jeans mass be written as?
Mj = 10^5 * (T^3/2 / µ^2*n^1/2) solar masses
Is our tff the formal solution?
No, but it is very close to the formal solution
For the formation of a 5 solar mass star, what is tff?
3e5 years
How does free fall time depend on density?
tff is longer for less dense regions
and shorter for denser regions
How can cloud stability be determined?
Considering the balance of inwards vs outwards forces
What does the Virial equation do?
Sets the stability criterion of a molecular cloud supported by thermal pressure alone
When will a cloud supported only by thermal pressure collapse?
If its mass exceed the jeans mass
Mj ∝ T^3/2 ρ^-1/2
In the initial isothermal collapse, which force dominates?
Gravity
How can free fall time for collapse be estimated?
By assuming acceleration is constant
