Kepler’s laws

• The Kepler laws

In the Solar System, Kepler discovered laws that bear his name, based on observation of the orbits of the planets, although without understanding why. Later on, Newton was to demonstrate that these laws are the result of gravitation applied to two bodies, the Sun and a planet.

• Kepler’s first law: The orbit of a planet is an ellipse with the Sun at one of the two foci.
• Kepler’s second law: known as the areas law. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. When the planet is closer to the Sun, it travels faster.

  

• Kepler’s third law: The period $P$ f revolution around the Sun increases with the semi-major axis ^?  $a$ and $\frac {P^{2}} {a^{3}}$ is constant for all the planets of the Solar System.

If $P$ is expressed in terrestrial years and $D$ in astronomical units ^? , the constant is equal to 1.
For example, for a planet which is at 5 au ^? , $P^2=5^3=125$, and thus $P=\sqrt {125}=11.2 years$

In exoplanet systems, we often use Kepler’s laws assuming that the planets have masses well below that of the star (as if the case in the Solar System). We then approximate the mass of the system as being the mass of the star M_* and the star-planet distance as being the distance from the planet to the barycentre: $\frac {P^{2}} {a^{3}} =\frac {4\pi^{2}} {GM*}$
The motion of a planet in space is defined by a number of parameters, those which define the planet’s orbit, those which define the position of the orbit in space and those which determine the rotation of the planet on its own axis.

• The parameters of planetary orbits

The main parameters are:

• Semi-major axis ^? , $a$: half the aphelion ^? -perihelion ^? distance.
• Sidereal period of revolution, $P$: time a planet takes to complete one revolution around the Sun.
• Rotation period ^? , $T$: time a planet takes to rotate once on its own axis.
• Eccentricity, $e$: value between 0 and 1, characterising the flattening of an ellipse. Value e=0 designates a circle. The greater the value of e and the closer it comes to 1, the more the ellipse is elongated. A planet is then said to have a highly elliptic orbit.
• Inclination, $i$: angle of inclination of the orbit in relation to the ecliptic.
• Axial tilt: angle of inclination of the equator in relation to the ecliptic.