## Detecting Planets via Doppler Spectroscopy

We’ve established that an orbiting planet will gravitationally pull on its star in a barycentric motion around the system’s centre of mass (see here) and that we can detect this spectroscopically by watching the movement of absorption lines give away the radial velocity of the star (see here) Let us now, therefore, put the two together into some coherence.

Throughout the orbit of the planet, the observed stellar radial velocity will range between a maximum and a minimum value, half the difference between them we call the “velocity semiamplitude,” $K$. This can be calculated with

$\displaystyle K = \frac{2\pi a_* \sin{i}}{P \sqrt{1-e^2}}$

It might seem weird that there is no coefficient for the planetary mass in this. But realise that we aren’t observing the planet’s orbit, but rather we’re observing the star’s orbit around the star+planet barycentre. By taking the semi-major axis of the star and its orbital period into account, we’re accounting for the planet’s mass.

The velocity semiamplitude is related to the masses of the two components in a planetary system through the so-called “mass function”

$\displaystyle \frac{(m_* \sin{i})^3}{(m_*+m_p)^2} = \frac{P}{2 \pi G} K^3 (1 - e^2)^{3/2}$

Which, in the typical case of the star being much more massive than the planet, can be simplified and approximated

$\displaystyle m_p \sin{i} \approx \left(\frac{P}{2 \pi G} \right)^{1/3} Km_{*}^{2/3} (1 - e^2)^{1/2}$

Putting this approximation into Kepler’s third law gives us an expression for the semi-major axis of the planet.

$\displaystyle a \approx a_p \approx \left( \frac{G}{4 \pi^2} \right)^{1/3} m^{1/3}_* P^{2/3}$

Monitoring the radial velocity of a star over time gives us the semi-amplitude. The function of the radial velocity of the star with time may be expressed as

$V_r(t) = K[\cos{(v(t) + \omega)} + e \cos{(\omega)}] + \gamma$

Where $\gamma$ is the radial velocity drift of the star. A star not under the influence of a planet will still have a radial velocity value purely because of its galactocentric motion. Other sources of radial velocity drift include additional, more distant companions. For the case of an additional planet, the changing radial velocity in response to this outer planet will result in $\gamma$ not being constant, and will therefore be expressed as a rate, $d\gamma / dt$ or $\dot{\gamma}$, frequently in m s-2. Sometimes the RV drift is not linear, but curved, if significant progress of the outer planet’s orbit has been made over the time coverage of the observations, in which case it might be expressed as $d^2\gamma / dt^2$ or $\ddot{\gamma}$.

When plotting the radial velocity measurements as a function of time on a graph, one achieves a graph that resembles something similar to the graph below for the planet 51 Peg b.

We see, therefore, that fitting radial-velocity data with a Keplerian model gives four out of six of the orbital elements of the planet, the longitude of the ascending node and orbital inclination remain unknown. The mass of the star is estimated through other means but since the inclination is unknown, the Keplerian fit gives only a lower limit to the mass and semi-major axis of the planet.

The main take-away message is that with careful monitoring of a star with Doppler spectroscopy, we can measure the mass of the planet as multiplied by the sine of the (typically unknown) inclination, $m_p \sin{i}$, which in practice turns out to be the minimum mass of the planet (as $\sin{i} \leq 1$).

Update: Fixed exponent in $m \sin{i}$ equation. I inadvertently wrote it as negative.