Katamari Damacy is a popular Japanese video game where a character rolls a ball around a world that is filled with random things. As the ball rolls it picks up objects in its path and engulfs them, resulting in a growth process.

To a first approximation, any object of volume \(V\) that is picked up is instantly and uniformly distributed to the volume of the growing ball of crap. Also, the ball rolls with constant angular velocity.

Suppose that in a simple world, the character rolls the ball in a straight line that has random crap distributed with a line density of \(\lambda_0\) kg/m. If the original radius doubles in time \(T_2\), and it increases eight-fold in time \(T_8\), what is the value of \(T_8/T_2\)?

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