Detecting exoplanets — the radial-velocity method

What just happened?

You increased the planet's mass, and the star began to wobble more strongly around their common centre of mass. Its radial velocity oscillates with a larger amplitude, so the spectral line jumps further toward the red and toward the blue.

Common centre of mass:

The planet and star orbit the same point. The more massive the planet ($m_p$), the larger the star's orbit ($a_\star$) and the faster the star moves.

Radial-velocity amplitude:

For a fixed star and orbital period, the amplitude $K$ is directly proportional to the planet mass. That is the signal we look for.

The Doppler effect — how do we measure it?

When the star recedes, its spectral lines shift toward the red; when it approaches — toward the blue. The alternating, periodic shift betrays the presence of an unseen planet. That is how Mayor and Queloz discovered 51 Pegasi b in 1995 (Nobel Prize in Physics 2019). Fun fact: Jupiter rocks the Sun by roughly one of its own radii.

Why this matters for your exam:

It is a model example of motion around the centre of mass + the Doppler effect + conservation laws. The trick: remember that the signal $K$ grows with the planet mass, and the line shift follows directly from the radial velocity.

Remember: a more massive planet → a larger star wobble → a larger radial-velocity amplitude → a larger Doppler shift. K ∝ m_p.