In this problem we are going to explore the evidence for the expansion of the un

Question

In this problem we are going to explore the evidence for the expansion of the universe that was first discovered by Edwin Hubble. In the 1920s, Hubble discovered that far-off galaxies were all redshifted by an amount that increased with their distance from us. In other words, it appeared that the further away a galaxy was, the faster it was receding from us. This became known as Hubble's law.

Suppose that you attempt to measure the hydrogen spectral lines from three distant galaxies. You get the following values for what corresponds to the 656.46-nm hydrogen line. For galaxy 1 you get a line at 656.50 nm, for galaxy 2 the line is at 656.57 nm, and for galaxy 3 the line is at 656.64 nm. Using a few independent techniques (which we will not go into here) you are able to pin down the distances to these three galaxies. Galaxy 1 is determined to be approximately 10^6 light years away (or 1 Mly), galaxy 2 is 2.5 Mly away, and galaxy 3 is 4 Mly away. Recall that a light year is the distance that light travels in a year or roughly 9.46*10^15 m.

For this problem we can use the standard special relativistic Doppler shift to determine the galaxies' recessional velocities. One should note that this is not strictly correct because, unlike velocities of objects passing through space, Hubble's law is due to the expansion of space itself, which requires the more complicated theory of general relativity. For cases in which the galaxies are relatively close to earth (such as this problem) and thus have recessional speeds v << c, the Doppler shift equations from general relativity become the same as those from special relativity.

A) What is the recessional velocity v_1 of the first galaxy? (in km/s)

B) What is the recessional velocity v_2 of the second galaxy? (in km/s)

C) What is the recessional velocity v_3 of the third galaxy? (in km/s)

D) Let's first determine the relationship between the recessional velocity of a galaxy and the distance to it. It might help to use some graph paper to plot the velocity (calculated from the Doppler shift) versus distance.

Which of the following statements is correct?

a) The recessional velocity varies as the square of the distance.
b) The recessional velocity varies linearly with the distance.
c) The recessional velocity varies as the square root of the distance.
d) The recessional velocity varies as the inverse of the distance.

E) Use the given information and the answers to Parts A to D to determine the Hubble constant H_0. (in km/s/Mly)

F) What is the relation between the Hubble constant H_0 and the age of the universe t? Assume that the Hubble constant has not changed over the history of the universe.

G) Use the information from the previous part to estimate the age of the universe t_age. (in billion years)

Explanation / Answer

You can use the Hubble Constant in v = H d with H = 72.6 km/s Mpc
To get d in Mpc 1 Mpc (megaparsec) = 3.262 x 10^6 light years
so v = 72.6 x 10^6/(3.262 x10^6) = 22.3 km/s

You could also you Doppler shift problem to find v. f = fo/(1 + v/c) so lambdazero(1 + v/c) = lambda

1 + v/c = 656.50/656.46 = 1.000060933

v/c = 0.00006093 and v = 3 x10^5 km/s x 0.00006093 = 18.27 km/s

I don't know which one you are supposed to use ??

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.