Problem#1
The spectrum of the sodium atom is detected in the light from a distant galaxy. (a) If the 590.0-nm line is redshifted to 658.5 nm, at what speed is the galaxy receding from the earth? (b) Use the Hubble law to calculate the distance of the galaxy from the earth.
Answer:
(a) We use
v = {[(658.5 nm/590.0nm)2 – 1]/[(658.5 nm/590.0nm)2 + 1]}c
v = 0.1094c
v = 3.28 x 107 m/s
(b) the distance of the galaxy from the earth is
r = v/H = (3.28 x 104 km/s)/[71 (km/s)/Mpc)(1Mpc/3.26 Mly)]
r = 1510 Mly
Problem#2
Redshift Parameter. The definition of the redshift parameter z is given in Example 44.8. (a) Show that Eq. (44.13) may be written as
1 + z = [(1 + β)/(1 - β)]1/2
where β = v/c. (b) The observed redshift parameter for a certain galaxy is Find the speed of the galaxy relative to the earth, if the redshift is due to the Doppler shift. (c) Use the Hubble law to find the distance of this galaxy from the earth.
Answer:
(a) The Hubble constant has a value of H0 = 7.1 x 104 (m/s)/Mpc
z is defined as
z = (λ0 – λS)/λS
1 + z = 1 + (λ0 – λS)/λS
1 + z = λ0/λS
Then from
λ0 = λS[(c + v)/(c – v)]1/2
λ0/λS = [(c + v)/(c – v)]1/2
so that
1 + z = [(c + v)/(c – v)]1/2
1 + z = [(1 + β)/(1 - β)]1/2
Where β = v/c.
(b) the speed of the galaxy relative to the earth, if the redshift is due to the Doppler shift, we use
1 + z = [(1 + β)/(1 - β)]1/2
(1 + z)2 = (1 + β)/(1 - β)
(1 + z)2 – β(1 + z)2 = 1 + β
(1 + z)2 – 1 = β[(1 + z)2 + 1]
β = [(1 + z)2 – 1]/[(1 + z)2 + 1]
β = (1.52 – 1)/(1.52 + 1) = 0.3846
so, v = 0.3846c = 1.15 x 108 m/s
(c) We can use Eq.
r = v/R
to find the distance to the given galaxy,
r = (1.15 x 108 m/s)/(7.1 x 104 (m/s)/Mpc) = 1.6 x 103 Mpc
or
r = 1.6 x 103 Mpc x 3.26 Mly/Mpc = 5.2 x 109 Mly
Problem#3
A galaxy in the constellation Pisces is 5210 Mly from the earth. (a) Use the Hubble law to calculate the speed at which this galaxy is receding from earth. (b) What redshifted ratio λ0/λS is expected for light from this galaxy?
Answer:
(a) A galaxy in the constellation Pisces is r = 5210 Mly, then
v = H0r
with H0 = 71(km/s)/(Mpc) = 71(km/s)/(3.26 Mly) = 21.779 (km/s)/Mly, so that
v = 21.779 (km/s)/Mly x 5210 Mly = 1.13 x 105 km/s
(b) Use v from part (a),
λ0/λS = [(1 + v/c)/(1 – v/c)]1/2
with v/c = (1.13 x 108 m/s)/(2.988 x 108 m/s) = 0.367
λ0/λS = [(1 + 0.367)/(1 – 0.367)]1/2
λ0/λS = 1.5
Problem#4
(a) According to the Hubble law, what is the distance from us for galaxies that are receding from us with a speed c? (b) Explain why the distance calculated in part (a) is the size of our observable universe (ignoring any change in the expansion rate of the universe due to gravitational attraction or dark energy)
Answer:
H0 = 71(km/s)/(Mpc) = 71(km/s)/(3.26 Mly) = 21.779 (km/s)/Mly, so that
(a) we use
r = c/H0 = (2.988 x 105 km/s)/21.779 (km/s)/Mly = 1.4 x 104 Mly
(b) This distance represents looking back in time so far that the light has not been able to reach us.
Problem#5
The critical density of the universe is 9.5 x 10-27 kg/m3. (a) Assuming that the universe is all hydrogen, express the critical density in the number of H atoms per cubic meter. (b) If the density of the universe is equal to the critical density, how many atoms, on the average, would you expect to find in a room of dimensions 4 m x 7 m x 3 m. (c) Compare your answer in part (b) with the number of atoms you would find in the same room under normal conditions on the earth.
Answer:
Given: ρ = 9.5 x 10-27 kg/m3, V = 84 m3, mH = 1.67 x 10-27 kg, 1 mol = 6.02 x 1023 atoms, normal pressure is p = 1.013 x 105 Pa, and normal temperature is about T = 270C = 300K.
(a) the critical density in the number of H atoms per cubic meter is
(9.5 x 10-27 kg/m3)/(1.67 x 10-27 kg/atom) = 5.69 atoms/m3
(b) the number of atoms in the room dimensions V = 4 mx 7 mx 3 m = 84 m3 is
5.69 atoms/m3 x 84 m3 = 477.8 atoms
(c) the ideal gas law gives the number of moles to be
n = PV/RT = (1.013 x 105 Pa)(84 m3)/(300 K x 8.3145 J/mol.K) = 3.4 x 103 moles
then, the number of atoms you would find in the same room under normal conditions on the earth is
(3.4 x 103 moles)(6.02 x 1023 atoms) = 2.0 x 1027 atoms
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