An interplanetary spacecraft spends most of its flight time moving under the gravitational influence of a single body – the Sun. Only for brief periods, compared with the total mission duration, is its path shaped by the gravitational field of the departure or arrival planet. The perturbations caused by the other planets while the spacecraft is pursuing its heliocentric course are negligible.
The computation of a precise orbit is a trial-and-error procedure involving numerical integration of the complete equations of motion where all perturbation effects are considered. For preliminary mission analysis and feasibility studies it is sufficient to have an approximate analytical method for determining the total V required to accomplish an interplanetary mission. The best method available for such analysis is called the patched-conic approximation.
The patched-conic method permits us to ignore the gravitational influence of the Sun until the spacecraft is a great distance from the Earth (perhaps a million kilometers). At this point its velocity relative to Earth is very nearly the hyperbolic excess velocity. If we now switch to a heliocentric frame of reference, we can determine both the velocity of the spacecraft relative to the Sun and the subsequent heliocentric orbit. The same procedure is followed in reverse upon arrival at the target planet’s sphere of influence.
The first step in designing a successful interplanetary trajectory is to select the heliocentric transfer orbit that takes the spacecraft from the sphere of influence of the departure planet to the sphere of influence of the arrival planet. If you have not already done so, before continuing it is recommended that you first study the Orbital Mechanics section of this web site. It is also recommended, if you are not already familiar with the subject, that you review our section on Vector Mathematics.
Heliocentric-Ecliptic Coordinate System
Our first requirement for describing an orbit is a suitable inertial reference frame. In the case of orbits around the Sun, such as planets, asteroids, comets and some deep-space probes describe, the heliocentric-ecliptic coordinate system is convenient. As the name implies, the heliocentric-ecliptic system has its origin at the center of the Sun. The X-Y or fundamental plane coincides with the ecliptic, which is the plane of Earth’s revolution around the Sun. The line-of-intersection of the ecliptic plane and Earth’s equatorial plane defines the direction of the X-axis. On the first day of spring a line joining the center of Earth and the center of the Sun points in the direction of the positive X-axis. This is called the vernal equinox direction. The Y-axis forms a right-handed set of coordinate axes with the X-axis. The Z-axis is perpendicular to the fundamental plane and is positive in the north direction.
It is known that Earth wobbles slightly and its axis of rotation shifts in direction slowly over the centuries. This effect is known as precession and causes the line-of-intersection of Earth’s equator and the ecliptic to shift slowly. As a result the heliocentric-ecliptic system is not really an inertial reference frame. Where extreme precision is required, it is necessary to specify that the XYZ coordinates of an object are based on the vernal equinox direction of a particular year or epoch.