A star is born

A large, spherical, and very diffuse cloud of molecular hydrogen sits in empty space. The radius of the cloud is r0r_0 and the molecules are initially at rest. Let t0t_0 be the time it takes for this cloud to completely collapse, i.e. the time it takes for a molecule at r0r_0 to free-fall into the origin. Similarly, let t1t_1 be the time it takes for a cloud of radius r1=4r0r_1=4r_0 to collapse.

What is t1t0?\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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