The galaxies in the universe are all flying away from each other. The speeds of nearby galaxies are proportional to the distance the galaxy is away from us. This relation, v=Hd is known as Hubble's law and the constant H is known as Hubble's constant. The evolution of our universe is determined by general relativity and the amount of matter, dark matter, and dark energy in our universe. If we ignore dark energy pretend there is none) one can determine the "critical density" of the universe. If the universe is more dense than the critical density the universe the universe will eventually crash back together, whereas if the density is less than the critical density the universe will fly apart forever. This was a big question up until the discovery of dark energy.

Interestingly enough, one can determine the critical density through Newtonian physics. Consider a galaxy a distance d away from us, moving radially away with a velocity given by Hubble's law. If the galaxy is not to escape to infinity, what is the critical density in # atoms of hydrogen per cubic meter?

To determine the critical density in terms of the number of atoms of hydrogen per cubic meter, we can use Newtonian physics and Hubble's law.

First, let's understand the concept of escape velocity. The escape velocity is the minimum velocity an object needs to escape the gravitational pull of another object. In this case, we're considering a galaxy that is moving radially away from us. If the galaxy is not to escape to infinity, the velocity given by Hubble's law (v = Hd) should be less than the escape velocity.

The escape velocity can be calculated using Newton's law of universal gravitation and conservation of energy. However, since we are considering a galaxy moving away from us, we can ignore the gravitational potential energy term. This leaves us with only the kinetic energy term:

(1/2)mv^2 = GmM/r

where m is the mass of the galaxy, v is the velocity, G is the gravitational constant, M is the mass of our universe (which we can approximate with the critical density times the volume), and r is the distance between the galaxy and us.

Rearranging the equation, we get:

v^2 = 2GM/r

Now, let's substitute Hubble's law into this equation. Since v = Hd:

(Hd)^2 = 2GM/r

Simplifying further:

H^2d^2 = 2GM/r

Now, we need to find the mass of our universe, M. We know that the mass of the universe is equal to the critical density (ρ_critical) times the volume (V) of the universe.

M = ρ_critical * V

To find the volume V of the universe, we can consider a sphere with radius d (distance to the galaxy). So, V = (4/3)πd^3.

Substituting this back into the equation:

H^2d^2 = 2G(ρ_critical * (4/3)πd^3)/r

Now, we can cancel out the common terms:

H^2d^2 = (8/3)G(ρ_critical * π)/r * d^3

Simplifying further:

H^2 = (8/3)G(ρ_critical * π)/r * d

Now, we can solve for the critical density (ρ_critical):

ρ_critical = H^2 * r / (8Gπ)

Finally, we can substitute the known values for H (Hubble's constant), r (distance to the galaxy), G (gravitational constant), and π (pi) to calculate the critical density in terms of the number of atoms of hydrogen per cubic meter.

Please note that the actual value of the critical density may vary depending on the currently accepted values for these constants.