One motion, two parts
Stars are not fixed. Each one moves. Its real velocity is called the space velocity (v). From Earth we cannot see it whole. We can only see two parts of it:
- Radial velocity (vr): along our line of sight, towards us or away from us.
- Tangential velocity (vt): across the sky, at right angles to our line of sight.
The two parts are at right angles, so we add them like the sides of a right triangle:
v² = vr² + vt²
Radial velocity from the Doppler shift
A star that moves away stretches its light waves: the colours shift to the red end. A star that comes closer squeezes them: the colours shift to the blue end. Astronomers look at dark lines in the star's spectrum. If a line has moved by Δλ from its normal wavelength λ, then
vr = c × Δλ / λ (c = 3 × 10⁵ km/s)
Red shift means vr is positive (moving away). Blue shift means vr is negative (coming closer).
Tangential velocity from proper motion
Over many years a near star slides slowly across the sky against far stars. This slide per year is called the proper motion (μ), in arcseconds per year. It is small: even the fastest star changes by only about 10″ in a year.
A slide of the same angle means more km if the star is far. So we need the distance (see Distance to Stars):
vt = 4.74 × μ × d (μ in ″/year, d in pc, vt in km/s)
Now join the two: v = √(vr² + vt²).
Try it: split a motion
- In the 3D, set the angle θ = 0°. All the motion is radial. The green part is zero.
- Set θ = 90°. Now all of the motion is tangential and the star keeps its yellow colour.
- Predict first: at θ = 45° are vr and vt equal? Then check the numbers.
At home: walk past a friend who is holding a toy car on a string. Notice what changes if the car moves straight at you or across you.
Key formulas and definitions
- v² = vr² + vt²
- vt = 4.74 × μ × d (μ in ″/yr, d in pc, vt in km/s)
- vr = c × Δλ / λ, c = 3 × 10⁵ km/s
- red shift: vr > 0 (away); blue shift: vr < 0 (towards)
Worked examples
1. A star has vr = 30 km/s and vt = 40 km/s. Find its space velocity.
v = √(30² + 40²) = √(900 + 1600) = √2500 = 50 km/s.
2. A star at 10 pc has proper motion 0.2″ per year. Find vt.
vt = 4.74 × 0.2 × 10 = 9.48 km/s.
3. A spectral line has shifted by Δλ / λ = 0.0001 towards red. Find vr and say what the star is doing.
vr = 3 × 10⁵ × 0.0001 = 30 km/s. Red shift means it is moving away.
4. A star 2 pc away moves 10″ every year across the sky. Find vt.
vt = 4.74 × 10 × 2 = 94.8 km/s.
5. A star has vr = −60 km/s and vt = 80 km/s. What is v and is it coming or going?
v = √(60² + 80²) = 100 km/s. The minus sign in vr means it is coming closer (blue shift).
Common mistakes
- Adding vr and vt directly. They are at right angles, so use v² = vr² + vt².
- Thinking a star moving straight across the sky has a Doppler shift. Its radial velocity is zero, so there is no shift from that motion.
- Using proper motion in degrees or in arcseconds per century. The formula needs ″ per year.
- Mixing up the sign: red shift is moving away (positive), blue shift is coming closer (negative).