Detection of Exoplanets using the Radial Velocity Method

In 1995, a team of Swiss astronomers detected the first known exoplanet orbiting a main sequence star, 51 Pegasi (previously a planet was discovered orbiting a pulsar, but that's a very different system). The method with which they detected this planet is called the Doppler Technique or the Radial Velocity Method. This technique combines gravity, Newton's 3rd Law, and the Doppler effect to provide information about exoplanets.

First of all we need to think about the gravitational forces acting on a star and it's planet. We all know that the sun exerts a gravitational pull on the Earth, which keeps it in orbit, but we often forget that the Earth also exerts a force on the sun due to Newton's third law. This force is modeled by the equation:

,

where G is the gravitational constant, m1 and m2 are the masses of the star and planet (order does not matter), and r is the distance from the star to the planet.

Using Newton's second law, we can remember that forces cause accelerations (F=ma). This means that stars are pulled around by the planets that orbit them! Stars orbit around the center of mass of their solar system, which usually is actually within the radius of the star itself. You can see what this would look like for a one planet solar system below.

(Wikimedia Commons)

There are 2 detection methods that result from this motion of the star one is the Astrometric Method, which directly measures the change in position of a star over time, the other is the Radial Velocity Method.

The Radial Velocity Method uses the Doppler effect to measure the velocity at which a star is moving over the course of its orbit. As per the Doppler effect, when the Star is moving towards us, the light waves appear compressed and thus more blue. When the star is moving away, the light it emits appears to shift to longer (more red) wavelengths.

(Wikimedia Commons)

The video below shows the absorption lines from a star shifting to blue as the star moves towards the observer, then to red as the star moves away.

http://www.eso.org/public/videos/eso1035g/

Now that we have the change in wavelength, we can use the following equation to determine the actual velocity of the star:

,

where vrad is the radial velocity of the star, c is the speed of light (3.00 x 10^8 m/s), λshift is the wavelength at a given time, and λemit is the wavelength emitted at no radial velocity.

Plotting the velocity over time produces a sine wave where the amplitude is the maximum radial velocity.

What if there are multiple planets?

This is where it gets really intricate, graphically. The graph becomes sine waves on sine waves...

This would be the velocity of a star over time in a hypothetical 3 planet system...Wanna see a 10 planet system?

As you can see, it gets very messy very quickly...

The radial velocity method can get us an approximate mass of the planet because the larger the planet, the larger the amplitude. Additionally, we can use the period of the sine wave to find the period of the planet. Then, using Newton's variation of Kepler's third law, we can determine the distance at which the planet orbits. If you combine this information with the radius obtained through the transit method, you can obtain the density of the planet, which provides some insight to the planet's composition.

The downside of the radial velocity method is that you can only get mass, period, and average orbital distance, which tell you very little about the planet by themselves. To get any substantial information, you need to combine this with the transit method.  Additionally, the assumption must be made that we are observing the doppler shift in the plane of the planets orbit. The chances are, we are actually looking at an inclination to the orbital plane, and therefore can only calculate a minimum planet mass.

While this method was once only good for finding "Hot Jupiters" (massive planets orbiting very close to their stars), it has been refined and is starting to find planets closer to Earth's mass. In fact, our satellites are now picking up changes in a star's velocity of as little as 50 cm/s! This is less than the maximum speed of a mosquito! As we increase the precision of our instruments, we will be able to find smaller planets at greater distances from stars.