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.

Question

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?

Answer 1

http://tap.iop.org/astronomy/cosmology/705/file_47582.doc

Answer 2

so what's the final answer?

Answer 3

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

Hubble's law states that the velocity of a galaxy, v, is proportional to its distance from us, d, as given by the equation v = H * d. Here, H is the Hubble constant.

To understand whether a galaxy will escape to infinity or not, we need to consider the gravitational force that acts on it. This force depends on the mass of the galaxy, as well as the density of matter around it.

Let's assume that the critical density is denoted by ρ_critical, and the density of matter at a distance d is denoted by ρ.

The density of matter can be described as the mass (m) of matter divided by its volume (V), which can be written as ρ = m / V.

Let's consider a spherical region around the galaxy, with a radius d and volume V = (4/3) * π * d³.

Now, we need to find the mass of matter contained within this spherical region. We can assume that the mass is uniformly distributed, so we can calculate it as the product of the density ρ and the volume V, giving us m = ρ * V.

Substituting the expression for V, we have m = ρ * (4/3) * π * d³.

Now, we need to calculate the gravitational force experienced by the galaxy due to this matter. According to Newton's law of universal gravitation, the force (F_grav) between two masses (m1 and m2) separated by a distance (r) is given by F_grav = G * (m1 * m2) / r², where G is the gravitational constant.

In this case, we consider the mass of the galaxy (m1) and the mass within the spherical region (m2), which is given by m.

The gravitational force experienced by the galaxy can be written as F_grav = G * (m * m1) / d².

For the galaxy to escape to infinity, the gravitational force should be equal to or smaller than the centripetal force acting on it. The centripetal force is given by F_cen = m1 * v² / d.

Setting the gravitational force equal to the centripetal force, we have:

G * (m * m1) / d² = m1 * v² / d.

We can now substitute the expressions for m, v (H * d), and simplify the equation:

G * (ρ * V * m1) / d² = m1 * (H * d)² / d.

Canceling out the m1 and d terms, and substituting the expression for V, we get:

G * (ρ * (4/3) * π * d³ * m1) / d² = m1 * (H * d)² / d.

Simplifying further, we have:

G * ρ * (4/3) * π * d = H² * d.

Now, we can cancel out the d terms and solve for the critical density ρ_critical:

G * (4/3) * π = H².

Since we need the critical density in terms of the number of atoms of hydrogen per cubic meter, we can further rewrite this equation in terms of the mass of hydrogen atom (m_H) and Avogadro's constant (N_A):

(4/3) * π * G * m_H * N_A = H².

Solving this equation will give us the critical density in terms of the number of atoms of hydrogen per cubic meter. You can use the known values for the gravitational constant (G), the mass of a hydrogen atom (m_H), Avogadro's constant (N_A), and the Hubble constant (H) to obtain the numerical value.