SME
University Physics III: Waves, Optics & Modern
Particle physics and cosmology
Use this for “A galaxy is 150 Mpc away. With H₀ = 68 km/s per Mpc, how fast is it receding, what is its redshift, and what is the Hubble time?”
Diagram
The farther a galaxy, the faster it moves away: 10,200 km/s at 150 Mpc.
This is an example. Type your own values to start a new problem.
v (km/s)
=
H₀ (km/s per Mpc)
×
d (Mpc)
z
Redshift
worked out (not typed)
tH
Hubble time
worked out (not typed) · billion years
Formulas
v
=
H₀
×
d
10,200 = 68 × 150
z
=
v
299,800
0.03402
=
10,200
299,800
tH
=
977.8
H₀
14.3794
=
977.8
68
Assumptions
•
Hubble’s law v = H₀d: space stretches evenly, so a galaxy twice as far recedes twice as fast; 1 Mpc = 3.086 × 10¹⁹ km.
•
The redshift z ≈ v ÷ c only while v is far below c (z under about 0.1), with c = 2.998 × 10⁵ km/s.
•
The Hubble time 1 ÷ H₀ estimates the age of the universe if the expansion rate had never changed.
Step-by-step
WE KNOW
H₀ = 68 km/s per Mpc, d = 150 Mpc
FIND
speed away (v), redshift (z), Hubble time (tH)
Step 1 · Find speed away (v)
Use v = H₀ × d
Each
megaparsec
of
distance
adds
H₀
km/s
to
the
speed
away.
v
=
68
×
150
v
=
10,200
km/s
Step 2 · Find redshift (z)
Use z = v ÷ 299,800
For speeds far below light’s, the redshift is the speed over c = 299,800 km/s.
z
=
10,200
299,800
z
=
0.03402
Step 3 · Find Hubble time (tH)
Use tH = 977.8 ÷ H₀
1
H₀
is
a
time:
977.8
turns
1
km/s per Mpc
into
billions
of
years.
tH
=
977.8
68
tH = 14.3794 billion years
Check
Put the numbers back in. Both sides should match.
10,200 = 68 × 150 ✓
0.03402
=
10,200
299,800
✓
14.3794
=
977.8
68
✓
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