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 of 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?
Note: enter your answer to the nearest hundredth.
Details and assumptions
The Hubble constant is 68 km/s/Megaparsec.
Assume that all the matter is uniformly distributed throughout space.
Newton's gravitational constant is 6.67×10−11 Nm2/kg2.
To determine the critical density of the universe in terms of the number of atoms of hydrogen per cubic meter, we need to understand a few key concepts and apply Newtonian physics.
1. Hubble's Law: According to Hubble's Law, the velocity (v) of a galaxy moving away from us is proportional to the distance (d) between the galaxy and us. The proportionality constant is known as Hubble's constant (H), which is given as 68 km/s/Megaparsec. It is important to note that this law holds on large scales in the universe.
2. Escape Velocity: For a galaxy to escape to infinity, the velocity (v) required to overcome the gravitational attraction must be greater than the escape velocity. Escape velocity can be determined using Newton's law of gravitation.
3. Newton's Gravitational Law: Newton's law of gravitation states that the force (F) between two objects with masses (M1 and M2) is directly proportional to the product of their masses and inversely proportional to the square of the distance (r) between them. The formula is given as F = G(M1M2)/r^2, where G is the gravitational constant.
Now, let's calculate the critical density in terms of the number of atoms of hydrogen per cubic meter.
1. Convert the Hubble constant to SI units:
Hubble constant (H) = 68 km/s/Megaparsec
The conversion factor between km/s and m/s is 1000. The conversion factor between parsec and meters is approximately 3.09 x 10^16 m, as there are about 3.09 x 10^16 meters in a parsec.
H = (68 km/s/Megaparsec) x (1000 m/km) / (3.09 x 10^16 m/parsec)
H ≈ 2.20 x 10^(-18) s^(-1)
2. Calculate the escape velocity using Hubble's Law:
Escape velocity (v_esc) is given by v_esc = H x d, where d is the distance between the galaxy and us.
3. Set the escape velocity equal to the square root of 2 times the gravitational potential energy per unit mass (GM/r):
v_esc = √(2GM/r), where G is the gravitational constant, M is the mass, and r is the distance.
4. Rearrange the equation to solve for the mass:
M = v_esc^2 x r / 2G
5. Calculate the critical density (ρ_c) using the mass and assuming the matter is uniformly distributed throughout space:
ρ_c = M / V, where V is the volume.
6. Assume the mass of hydrogen atom (M_H) is approximately 1.67 x 10^(-27) kg.
7. Calculate the number of atoms of hydrogen per cubic meter:
Number of atoms of hydrogen per cubic meter = (ρ_c / M_H) = (M / V) / M_H
Now, plug in the values and perform the calculations to find the critical density in terms of the number of atoms of hydrogen per cubic meter.