SME
Grade 12 Science · Problem type
A galaxy’s redshift and distance
Use this for “A galaxy’s light is shifted to longer wavelengths, z = 0.03. How fast is it moving away, and how far is it?”
Diagram
Hydrogen absorbs at 656.3, 486.1, 434.0, 410.2 nm: its fingerprint.
λ = λ₀(1 + z) = 656.3 × (1 + 0.03)
λ = 676 nm
v ≈ cz = 300,000 km/s × 0.03 = 9,010 km/s away from us (redshift)
Distance: 129 Mpc
This is an example. Type your own values to start a new problem.
λ
Observed wavelength of Hα
example · nm
z
Redshift
calculated
v
Speed away
calculated · km/s
H₀
Hubble constant
example · km/s per Mpc
d
Distance
calculated · Mpc
Formulas
z
=
λ
− 656.3
656.3
0.030017
=
676 − 656.3
656.3
v
=
300,000
×
z
9,005 = 300,000 × 0.030017
v
=
H₀
×
d
9,005 = 70 × 129
Assumptions
•
v = cz only for z under about 0.1.
•
A redshift means longer wavelengths and a galaxy moving away.
•
Hydrogen’s Hα line is 656.3 nm in the lab.
Step-by-step
WE KNOW
λ = 676 nm, H₀ = 70 km/s per Mpc
FIND
redshift (z), speed away (v), distance (d)
Step 1 · Find redshift (z)
Use z = (λ − 656.3) ÷ 656.3
The shift as a fraction of the lab wavelength.
z
=
676 − 656.3
656.3
z
=
19.7
656.3
z
=
0.030017
Step 2 · Find speed away (v)
Use v = 300,000 × z
Multiply the shift by light’s speed, 300,000 km/s.
v
=
300,000
×
0.030017
v
=
9,005
km/s
Step 3 · Find distance (d)
Use v = H₀ × d
How many megaparsecs give that speed.
d
=
v
H₀
d
=
9,005
70
d
=
129
Mpc
Check
Put the numbers back in. Both sides should match.
0.030017
≈
676 − 656.3
656.3
✓
9,005 ≈ 300,000 × 0.030017 ✓
9,005 ≈ 70 × 128.64 ✓
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