Question
Derive an expression for the free-fall time. Calculate the free-fall time for a collapsing cloud of neutral hydrogen whose initial number density is 106 m-3. Sketch the rotation curve of the Milky Way Galaxy and discuss its significance
Answer :
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Free-Fall Time Expression:
The free-fall time (\(t_{\text{ff}}\)) for a collapsing object can be derived from basic principles of gravity. Considering a spherically symmetric cloud of mass \(M\) and radius \(R\), the free-fall time is given by the formula:
\[ t_{\text{ff}} = \sqrt{\frac{3\pi}{32G\rho}} \]
where \(G\) is the gravitational constant and \(\rho\) is the average density of the cloud.
Free-Fall Time Calculation:
For a neutral hydrogen cloud with an initial number density (\(n\)) of \(10^6 \,
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