SME
Grade 12 Science · Problem type
A star’s Doppler shift
Use this for “A star’s Hα line is seen at 656.5 nm. How fast is it moving, and which way?”
Diagram
Hydrogen absorbs at 656.3, 486.1, 434.0, 410.2 nm: its fingerprint.
λ = λ₀(1 + z)
λ = 656.3 × (1 + 0.000305)
λ = 656.5 nm
v ≈ cz = 300,000 km/s × 0.000305 = 91.4 km/s away from us (redshift)
This is an example. Type your own values to start a new problem.
v (km/s)
=
300,000
km/s
×
λ (nm)
−
λ₀ (nm)
λ₀ (nm)
Solving the picture’s λ = λ₀(1 + z) for z gives z = (λ − λ₀) ÷ λ₀, and v = cz with c = 300,000 km/s.
z
Shift
calculated
Formulas
z
=
λ
−
λ₀
λ₀
0.00030474
=
656.5 − 656.3
656.3
v
=
300,000
×
z
91.42 = 300,000 × 0.00030474
Assumptions
•
Moving away stretches the lines red (+v); moving toward shifts them blue (−v).
•
Only motion along our line of sight shows.
•
Pick the hydrogen line you measured: its lab wavelength λ₀ is Hα 656.3, Hβ 486.1, Hγ 434 or Hδ 410.2 nm.
Step-by-step
WE KNOW
λ₀ = 656.3 nm, λ = 656.5 nm
FIND
shift (z), line-of-sight speed (v)
Step 1 · Find shift (z)
Use z = (λ − λ₀) ÷ λ₀
The shift as a fraction of the lab wavelength.
z
=
656.5 − 656.3
656.3
z
=
0.2
656.3
z
=
0.00030474
Step 2 · Find line-of-sight speed (v)
Use v = 300,000 × z
Multiply the shift by light’s speed, 300,000 km/s.
v
=
300,000
×
0.00030474
v
=
91.42
km/s
Check
Put the numbers back in. Both sides should match.
0.00030474
≈
656.5 − 656.3
656.3
✓
91.42 ≈ 300,000 × 0.00030474 ✓
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