Answer to Question #97036 in Astronomy | Astrophysics for Sushma

Question #97036
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 10 ki power 6 particles /m3
1
Expert's answer
2019-10-22T10:40:00-0400


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


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

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)"


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: "\u03c1= 1.67 \u00d7 10^{-21} kg\/m3"

We put the value of cloud density ρ in (5) and get: "t = 2\u00d7 10^{15} seconds"

One year contains 3× 107 seconds

So we get, t=67.000.000 years


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