A star is born

A large, spherical, and very diffuse cloud of molecular hydrogen sits in empty space. The radius of the cloud is \(r_0\) and the molecules are initially at rest. Let \(t_0\) be the time it takes for this cloud to completely collapse, i.e. the time it takes for a molecule at \(r_0\) to free-fall into the origin. Similarly, let \(t_1\) be the time it takes for a cloud of radius \(r_1=4r_0\) to collapse. What is \(\frac{t_1}{t_0}?\)

Details and Assumptions:

  • The entire cloud is collapsing in this process.
  • Both clouds have the same mass.

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