Problems:

1) Assume two astronauts are traveling at v = 0.95c on a journey to the system of Alpha Centauri. We on Earth would say that it takes 4.2 / 0.95c = 4.4 years to reach the system 4.2 light years distant. But the astronauts dispute this.

(a) How much time passes on the astronauts' clocks?

*Solution*

Let t(A) be the astronauts' clock and t(E) be the time recorded on an Earth clock.

Then, we have t(E) = 4.4 yrs.

And:

t(A) = t(E)[1 - v^2/c^2]^½

t(A) = (4.4 yrs.) [1 - (0.95c)^2/c^2]^½ = 1.37 yrs.

(b) What is the distance to Alpha Centauri as measured by the astronauts? (Hint: this is an exact analog of the muon path length problem (#3) from the previous problem set)

*Solution*

Since we know: t(A) = 1.37 yrs.

then the distance D(A) = (0.95c) (1.37 yrs) = 1.31 Ly

2) According to Hubble's law, the distant galaxies are receding from us at speeds proportional to their distances, d, e.g. v = Hd. Where H = 2.26 x 10^-18 s^-1, currently).

a) How far away would a galaxy be in light years whose velocity relative to the Earth is c?

*Solutions*

In this case, v = c = 3 x 10^8 m/s

d = v/H = (3 x 10^8 m/s)/ (2.26 x 10^-18 s^-1)

d = 1.32 x 10^26 m

Converting to light years:

d = (1.32 x 10^26 m)/ (9.5 x 10^15 m /Ly) = 1.4 x 10^10 Ly

b) Would it be observable from Earth?

Given that modern telescopes can penetrate to about 1.8 x 10^10 Ly, the galaxy should easily be observable to the Hubble but might be more problematical for land-based scopes.

3) A galaxy in Hydra emits light with a red shift corresponding to a recessional velocity of 6 x 10^4 km/s.

a) What is its distance according to Hubble's law?

b) What is the value of z?

c) Assume this galaxy passed Earth T years ago and has moved with constant velocity ever since, what is the value of T?

*Solutions*

We know the recessional velocity v = 6 x 10^4 km/s

By Hubble's law: v = Hd so the distance d = v/H

Then, attending to the proper units for v, H:

d = (6 x 10^7 m/s)/(2.26 x 10^-18 s^-1)= 2.6 x 10^25 m

and d = (2.6 x 10^25 m)/(9.5 x 10^15 m /Ly) = 2.8 x 10^9 Ly

(b) z = v/c = (6 x 10^7 m/s)/(3 x 10^8 m/s) = 0.2

(c) T = d/v = (2.6 x 10^25 m)/(6 x 10^7 m/s) = 4.3 x 10^17 s

But 1 yr. = 3.15 x 10^7 s

so T = (4.3 x 10^17 s)/(3.15 x 10^7 s/ yr)

T = 1.36 x 10^10 years, or 13.6 billion years

4) Some observations reported on the quasar 3C-9 suggest that when it emitted the light that just reached Earth it was receding at a velocity of 0.8c. One of the lines identified in its spectrum has a wavelength of 1200 Å (angstroms) when emitted from a stationary source.

a) At what wavelength must this spectral line have appeared in the observed spectrum of the quasar?

b) What is its red shift, z?

*Solutions*

Let L(o) be the normal wavelength = 1200 Å

and L be the red-shifted value.

We know v = 0.8c so we must use the modified Doppler version, viz.

L/L(o) = (1 - v/c)^ ½ /(1 + v/c)^ ½]

L/L(o) = (1 + 0.8)^ ½/ (1 = 0.8)^ ½ = (1.8/0.2)^ ½

L/L(o) = (9)^ ½ = 3

then:

L = 3 L(o) = 3 (1200 Å) = 3600 Å

(b) The red shift of the quasar is found from:

1 + z = (1 + v/c)^ ½]/(1 - v/c)^ ½

1 + z = (1.8/0.2)^ ½ = (9)^ ½ = 3

Then: z = 3 - 1 = 2

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