Answer to Question #88783 in Astronomy | Astrophysics for Navdha

Question #88783
Write down the condition under which a large molecular cloud collapses to give rise
to new stars. Calculate the time of free fall of an HI cloud of density 106 particles/m3
1
Expert's answer
2019-04-30T09:46:39-0400

Write down the condition under which a large molecular cloud collapses to give rise to new stars.

Answer:

For collapse of a cloud with radius R to occur, need either

"M_{cloud}>M_{Jeans}=\\frac{3kTR}{2G \\mu m_H}"


Calculate the time of free fall of an HI cloud of density 106 particles/m3

Answer:

The acceleration g felt by a test particle for a spherically symmetric distribution of mass M and radius r.

From Newton's second law, the equation of motion for a test particle at the edge of the cloud is then


"m \\times g=G \\times m \\frac{M}{r^2} (1)"

The mass M is equal to:


"M= \\frac{4 \\pi}{3}\\times r^3 \\times \u03c1 (2)"

where ρ is the cloud density

We put (2) in (1):


"g=\\frac{4 \\pi}{3}G \\times r \\times \\rho (3)"

If it starts initially at rest, then (if acceleration is constant) it will reach the center when


"\\frac{ g \\times t^2}{2}=r (4)"

We put (3) in (4) and solve for t:


"t=\\sqrt \\frac {3}{2\\times \\pi \\times G \\times \\rho} (5)"

The cloud density ρ is equal to:


"\\rho=n \\times m_H (6)"

where n=106 particles/m3 ; mH = 1.67 × 10-27 kg


Using (6) we calculate the value of cloud density ρ: ρ= 1.67 × 10-21 kg/m3

We put the value of cloud density ρ in (5) and get:

t=2× 1015 seconds

One year contains 3× 107 seconds

In this case, t=67.000.000 years


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