The radial velocity method
A planet and its star both orbit their common center of mass. The star moves in a small circle, alternately toward and away from us, and its spectral lines shift slightly blue and then red by the Doppler effect. Measuring that shift found the first planet around a Sun-like star, 51 Pegasi b, in 1995.
The size of the wobble is the semi-amplitude K:
K = 28.43 m/s × (m sin i ÷ M_Jup) × (P ÷ 1 yr)^(−1/3) × (M★ + m)^(−2/3) ÷ √(1 − e²)
- m sin i: planet mass times the sine of the orbit’s inclination. Because inclination is usually unknown, radial velocity gives a minimum mass.
- P: orbital period.
- M★: star mass in solar masses.
- e: orbital eccentricity.
Examples
| Planet | Period | K on a Sun-like star |
|---|---|---|
| Jupiter | 11.86 years | 12.5 m/s |
| Saturn | 29.5 years | 2.8 m/s |
| Hot Jupiter | 4 days | about 130 m/s |
| Earth | 1 year | 0.09 m/s |
Jupiter moves the Sun at about 12.5 m/s, the speed of a bicycle sprint, and pulls it around a circle about 743,000 km in radius, just outside the Sun’s surface. Earth’s effect is only 9 cm/s, slower than a crawling baby.
Detection limits
- Classic spectrographs reach about 3 to 10 m/s.
- HARPS reaches about 1 m/s and found many super-Earths.
- ESPRESSO on the VLT aims at 10 cm/s, the level needed for an Earth twin, but starspots and convection on the star itself create noise of 1 m/s or more.
Reading the result
- Short periods and low-mass stars give larger signals, which is why many radial velocity planets orbit red dwarfs closely.
- A large K with an unknown inclination might be a much heavier companion seen almost face-on.
- Combining radial velocity with a transit (see the exoplanet transit calculator) gives both mass and radius, and therefore the planet’s density.