ASTRONOMY 120: GALAXIES AND THE UNIVERSE
HOMEWORK # 8 SOLUTIONS
FALL 2016
1. Alien Astronomer (20 Points)
Figure 25-9 shows the radio galaxy Centaurus A. Is it possible that somewhere in the universe there is an alien astronomer who observes apparent
superluminal motion in the AGN jets of Centaurus A? Explain your answer
with a labeled drawing showing the relative positions of the Earth, the alien astronomers, and Centaurus A, and the jet axis. [HINT: Draw a diagram showing
the "unified AGN model", NOT a diagram of superluminal motion!]
Double radio sources, quasars and blazars may be the same type of object — a supermassive black hole with its accretion disk and relativistic jets — viewed from different
angles. If the jets are nearly perpendicular to the Earth’s line of sight, the Earth’s observer will see a double radio source. If the alien planet is located where one of the jets is
pointing to, then the alien observer will see a blazar: in this case, since the jet is pointing
to the alien and the speed of the jet is very close to the speed of light, the alien will observe
apparent superluminal motion.
Date: Spring 2014.
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ASTRONOMY 120: GALAXIES AND THE UNIVERSE HOMEWORK # 8 SOLUTIONS FALL 2016
2. Apparent Superluminal Motion (30 Points)
This problem attempts to demonstrate how jets from active galactic nuclei
can have apparent superluminal motion. Suppose the distance from point A
to point B in Figure 25-14a is 13 light years, and the blob moves at 13/15 of
the speed of light. As the blob moves from A to B, it moves 12 light-years
toward the earth and 5 light years in a transverse direction.
[For the purposes of this problem, assume (pretend!) that the
astronaut is always moving at constant velocity. Assume (pretend!)
that no acceleration is involved. This indicates relativity is
not involved.]
2.a. How long does it take the blob to travel from A to B (in years)? An object
traveling at the speed of light c travels a distance of 1 light-year in 1 year of time.
c = velocity =
distance
time
1ly
1yr
13ly
13
=⇒ c =
15
15yr
Thefore an object traveling at 13/15 c travels 13 light-years in 15 years of time. As we
know the blob has moved precisely 13 light years, so the time it takes to go from A to B
is 15 years.
=
2.b. If the light from the blob at A reaches earth in 2014, in what year does
the light from B reach Earth? The light emitted at point B occurs 15 years after it
was emitted from point A. However, the distance the light has to travel to reach the Earth
from point B is 12 fewer. So it takes a net 3 more years for the light to reach Earth via
”A =⇒ B =⇒ Earth”, i.e. in 2017.
2.c. As seen from earth, at what speed does the blob appear to move across
the sky? From the Earth we can only see the blob’s tranvserse motion across the sky. It
appears the blob has traveled 5 light years in just 3 years, so its apparent speed is 5/3 c.
Figure 1. Note that the blob moves from A to B, photons are transmitted
from A first then from B 15 years later. Earth must be in your diagram in
order to describe where the photons will be observed.
ASTRONOMY 120: GALAXIES AND THE UNIVERSE
HOMEWORK # 8 SOLUTIONS
FALL 2016 3
3. Old Photons (30 points)
Notice that once you have adopted a numerical value for H0 , you can use
Hubble’s Law to obtain the distance of an extragalactic object for which you
know only its redshift. Suppose the spiral galaxy Arp122 has a redshift z=
0.0444 = v/c.
3.a. At what wavelength is the radio HI line (rest wavelength 21.12cm) observed? The HI line has a rest wavelenght of 21.12 cm. Therefore, using λ = λ0 (1 + z)
we obtain:
λ = 21.12 cm · (1 + 0.0444) = 22.06 cm
3.b. What is the distance (in both Mpc and lightyears) given the measured
value of the Hubble “constant” of Ho = 73.0 km s-1 Mpc-1? We can determine
the distance to Arp122 using the Hubble’s Law:
zc
d=
H0
For Arp122, z = 0.0444 and, applying the Hubble’s Law with H0 = 73km sec−1 Mpc−1 ,
we have that
0.0444 × 3 × 105 km sec−1
= 182Mpc
d=
73km sec−1 Mpc−1
Since 1 pc = 3.26 ly, distance in light years is d = 5.95 × 108 ly.
3.c. How many years ago were the photons that we now detect emitted from
this galaxy? Since the distance to the object is 5.95 × 108 ly, the light we observe was
emitted 5.95 × 108 years ago.
4. Growing with the Universe (20 points)
Suppose you were not held together by electromagnetic forces. How long
would it take you to grow 2 centimeters because of the expansion of the universe?
[HINT: Apply Hubble’s Law to your head as seen by your feet. Calculate
the velocity in cm/sec between your feet and head, using v=Hd,
where H is the Hubble "constant", and d is your height. With this
"expansion" or "growth" velocity, figure out how long it will take
you to grow an additional 2cm.] [ANOTHER HINT: Take care with
units!]
Let’s say that you’re 1.73 meters (about 5 feet, 8 inches) tall. Using Hubble’s Law we
can calculate the receding velocity of your head with respect to your feet.
v = H0 d
where d = 1.73 meters or 173 cm, and H0 = 73 km/s Mpc. Now we can change the Mpc
into cm with:
1 Mpc = 3.086 × 1024 cm
and then
173 cm
v = 73 km/s Mpc
3.086 × 1024 cm/Mpc
v = 4.1 × 10−21 km/s = 4.1 × 10−16 cm/s
Then
t=
d
v
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ASTRONOMY 120: GALAXIES AND THE UNIVERSE HOMEWORK # 8 SOLUTIONS FALL 2016
is the amount of time required to grow d = 2 cm when you are growing with speed
v = 4.1 × 10−16 cm/s. Multiplying we get: t = 4.9 × 1015 s = 1.5 × 108 yr.