ASTR 100 (McGaugh)
Homework #8 solutions

Chapter 17, Problem # 3

(a) The cosmological principle is the assumption that the universe is homogeneous and isotropic everywhere, in other words, that every part of the universe contains the same kinds of matter and structures, such as galaxies and stars, and that these structures are distributed such that the universe looks uniform (if we look at things on a large enough scale).

We cannot be certain that the cosmological principle is true since we cannot test it directly by traveling to distant parts of the universe to see if they are the same as our local area. However, there is good evidence that the cosmological principle is true. First, the fact that the universe looks the same (ie. has the same structures arranged and moving in similar ways) in every direction which we look from our observation point on the earth. Also, the cosmic background radiation is very uniform, suggesting that the universe was isotropic and homogeneous in the past as well.

(b) The sky is dark at night primarily because the universe is of finite age. As we look farther away from the earth, we are observing light that comes from farther and farther back in the past. When we look far enough away that the light would come from a time longer ago than the age of the universe, we will see no more galaxies and stars since they had not yet formed at that time. Thus, light reaches the earth only from galaxies and stars within a finite distance of the earth (about 14 billion light years). Another effect which helps explain why the night sky is dark is the redshift of photons from galaxies very far away from us. This redshift causes the photons to be greatly reduced in energy.

(c) For the Doppler redshift, waves are moving through a medium. The source an observer of the waves are also moving through that medium at some finite speed with respect to one another. It is the relative motion of the source and observer of the waves which causes the redshift. For the cosmological redshift, the medium (space) through which the waves are traveling is stretching. Because space itself is stretching, the wavelength of the light traveling through that space becomes stretched as well.

(d) The distance to very distant galaxies is measured relative to that of nearby galaxies by comparing the relative brightness or linear size of the galaxies.

(e) The expansion of the universe slows down in big bang models because all of the matter in the universe is attracted to all other matter in the universe by the force of gravity. Thus, the force of gravity trying to pull all the matter in the universe closer together slows down the rate at which the universe expands.

Chapter 17, Problem # 6

If the density parameter is shown to be 1, then the big bang model predicts a flat universe with an age of (2/3) × (1/Ho). If the value of the Hubble constant is then determined to be 65 km/s/Mpc, the big bang model thus predicts that the age of the universe is (2/3) × (1/65 km/s/Mpc) = (2/3)(1/65)(Mpc/km)(3.09 × 19 km / 1 Mpc) (sec)(1 yr / 31557600 sec) = 10.0 billion years.

If we also know that the oldest globular clusters are 14 billion years old, then the above values of the density and Hubble parameters would be bad news for the big bang model, because using those values the big bang model predicts than the universe is significantly younger than the age of the oldest known objects within it. Thus, scientists would have to conclude that there was something wrong with the big bang model.

Chapter 13, Problem # 8

The discovery of cosmic background radiation was strong evidence in favor of the big bang theory, because scientists had previously predicted that such radiation should exist as a consequence of the big bang theory. The temperature and uniform distribution of the cosmic background radiation agreed well with what scientists predicted the radiation should look like if it had been emitted about 300,000 years after the big bang and then redshifted.

The cosmic background radiation is very uniform, suggesting that the universe was quite homogeneous and isotropic at the time when it was emitted. There are, however, some small deviations from constant temperature observed. Part of this deviation is due to the motion of our planet within the universe. This motion causes a slight blueshift of the radiation in the direction we are moving toward and a slight redshift in the direction we are moving away from. The other type of small deviation observed in the temperature of the cosmic background radiation takes the form of tiny ripples or fluctuations of less than 0.001 percent. Scientists interpret these fluctuations as indications that the matter in the universe was just a little bit non-uniform at the time the radiation was produced. In other words, there were areas of very slightly higher and lower density. The higher density regions would ultimately lead to the formation of galaxies.

Chapter 18, Problem # 4

To convert from a 4.5 billion year timescale to a one day (24 hour) timescale, you simply multiply by (24 hrs/4.5 billion years)

(a) The first primitive forms of life appeared about 3.8 billion years ago
or 4.5 billion years - 3.8 billion years = 0.7 billion years after the formation of the earth.
Using the above conversion factor, we get:
0.7 billion years × (24 hrs/4.5 billion years) = about 4 hrs after the beginning of the day,

(b) The increase in atmospheric oxygen occurred from about 2 to 3 billion years after the formation of the earth.
Again using the conversion factor, we get:
2 billion × (24 hrs/4.5 billion years) = about 10 hrs after the start of the day
to 3 billion × (24 hrs/4.5 billion years = about 16 hrs after the beginning of the day,
which is approximately 10 am to 4 pm

(c) Life emerged from the oceans about 400 million years ago
or 4.5 billion years - 0.4 billion years = 4.1 billion years after the formation of the earth.
Again using the conversion factor, we get:
4.1 billion years × (24 hrs/4.5 billion years) = about 22 hrs after the beginning of the day,
which is around 10 pm

(d) The dinosaurs died out 65 million years ago
or 4.5 billion years - 0.065 years = 4.435 billion years after the formation of the earth.
Again using the conversion factor, we get:
4.435 billion years × (24 hrs/4.5 billion years) = about 23.65 hrs after the beginning of the day = about 23 hrs and 40 minutes after the beginning of the day,
which is approximately 11:40 pm

Using the same calculation method used in parts a through d, we find:

(e) Homo sapiens appeared at 11:59 pm

(f) The earliest human cities and civilizations appeared at about 0.1 second before midnight