Suppose that a Universe were full of baseballs. massbb=0.145kg. radiusbb=0.0369m

Question

Suppose that a Universe were full of baseballs. massbb=0.145kg. radiusbb=0.0369m. If the baseballs were distributed uniformly throughout the universe, what number density of baseballs would be required to make the density equal to the critical density? (Assume nonrelativistiv baseballs) Given this density of baseballs, how far would you be able to see, on average, before your line of sight intersected a baseball? In fact, we can see galaxies at a distance c/H0 4000 Mpc; does the transparency of the universe on this length scale place useful limits on the number density of intergalactic baseballs? [Note to readers outside North America or Japan: feel free to substitute regulation cricket balls, with mcr = 0.160 kg and rcr = 0.0360 m.]

Explanation / Answer

Here we have
Pcrit = 3H^2 / 8*pi*G
= 1.88 * 10^-20 h^2g/cm
= 1.88 * 10^-20 * 0.7^2 g/cm^3

and we set Pcrit = n.m , where   n is the requested number density.

Thus
n = 1.88 * (0.7)^2 g/cm^3 / 145g

= 6.35 * 10^-23 / cm^3

b)
1/ n*pi*r^2 = 1 / [6.35 * 10^-29 * pi * 0.0369]  
= 1.36 * 10^29 m

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