## General principles

The motion of a satellite around a planet obeys the three laws of Kepler:
1. The orbit is an ellipse with the planet at one of its foci.
2. The radius vector of the satellite with respect to the planet as origin sweeps over equal areas in equal time.
3. The ratio of the squares of the periods of two satellites is equal to the ratio of the cubes of the semimajor axes of their orbits.
The configuration of two gravitationally interacting bodies constitutes a mechanical system with six degrees of freedom. The satellite follows an orbit in a plane fixed with respect to astronomical coordinates. The orbit intersects the planet's equatorial plane at two points: the ascending node when it moves from the southern to the northern hemisphere, and the descending node when it moves from the northern to the southern hemisphere. The line connecting the two nodes is called the line of nodes. The pericentre (perigee, periareion and perijove for Earth, Mars and Jupiter respectively) is the point on the major axis that is closest to the planet. The apocentre (apogee, apoareion and apojove for Earth, Mars and Jupiter respectively) is the opposite point on the major axis. The line joining the pericentre and apocentre is called the line of apsides.

Six parameters fix the orbit and the position of the satellite on the orbit:

1. Semi-major axis.
2. Eccentricity: the ratio of the focal distance to the major axis. The orbit is called elliptic, parabolic or hyperbolic when the eccentricity is smaller than, equal to, or greater than 1, respectively.
3. Time of pericentre passage T: the time elapsed since pericentre passage. If time is measured from the instant that the satellite is at pericentre, then T=0. It is equivalent to the true anomaly, the angle from the pericentre direction to the satellite direction.
4. Orbit inclination: the angle between the orbital plane and the equatorial plane, measured at the ascending node in the direction of orbital motion. The orbit is called direct when the inclination is smaller than 90 degrees and retrograde when the inclination is larger than 90 degrees.
5. Argument of pericentre: the angle measured in the orbital plane from the ascending node to the pericentre.
6. Right ascension of the ascending node: the angle in the equatorial plane between the line of nodes and the direction to the vernal equinox, measured from the vernal equinox (the direction of the intersection of the ecliptic and equatorial planes) towards the ascending node.
The parameter pair semi-major axis and eccentricity is equivalent to the pair apocentre radius and pericentre radius.

## Implementation in SPENVIS

The SPENVIS orbit generator computes trajectory osculatory orbital elements using a numerical Runge-Kutta integration method.