Suppose two worlds, each having mass M and radius R, coalesce into a single world. Due to gravitational contraction, the combined world has a radius of only

3

4

R.

What is the average density of the combined world as a multiple of ρ_{0}, the average density of the original two worlds? ρ_{0}

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Density = mass/volume

Volume =V = 4/3Pi R^3

initially density of each world = ρ_{0}, = M/V

after coalesce mass = 2M and radius = 3/4R

volume of combined world (sphere ) = 4/3 x Pi x (3/4 R)^3 = 27/64 V

hence Density = 2M/(27/64V) = 128/27 ρ_{0} = 4.74 ρ_{0}

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