ORBITAL MECHANICS | Page 1 |

*Orbital mechanics*, also called flight mechanics, is the study of the
motions of artificial satellites and space vehicles moving under the
influence of forces such as gravity, atmospheric drag, thrust, etc. Orbital
mechanics is a modern offshoot of celestial mechanics which is the study of
the motions of natural celestial bodies such as the moon and planets. The
root of orbital mechanics can be traced back to the 17th century when
mathematician Isaac Newton (1642-1727) put forward his laws of motion and
formulated his law of universal gravitation. The engineering applications of
orbital mechanics include ascent trajectories, reentry and landing,
rendezvous computations, and lunar and interplanetary trajectories.

Orbital Elements

To mathematically describe an orbit one must define six quantities, called
*orbital elements*. They are

- Semi-Major Axis
- Eccentricity
- Inclination
- Argument of Periapsis
- Time of Periapsis Passage
- Longitude of Ascending Node

An
orbiting satellite follows an oval shaped path known as an ellipse with the
body being orbited, called the primary, located at one of two points called
foci. An ellipse is defined to be a curve with the following property: for
each point on an ellipse, the sum of its distances from two fixed points,
called foci, is constant (see figure to right). The longest and shortest
lines that can be drawn through the center of an ellipse are called the major
axis and minor axis, respectively. The *semi-major axis* is one-half of
the major axis and represents a satellite's mean distance from its primary.
*Eccentricity* is the distance between the foci divided by the length of
the major axis and is a number between zero and one. An eccentricity of zero
indicates a circle.

*Inclination* is the angular distance between a satellite's orbital
plane and the equator of its primary (or the ecliptic plane in the case of
heliocentric, or sun centered, orbits). An inclination of zero degrees
indicates an orbit about the primary's equator in the same direction as the
primary's rotation, a direction called *prograde* (or direct). An
inclination of 90 degrees indicates a polar orbit. An inclination of 180
degrees indicates a retrograde equatorial orbit. A *retrograde* orbit is
one in which a satellite moves in a direction opposite to the rotation of its
primary.

*Periapsis* is the point in an orbit closest to the primary. The
opposite of periapsis, the farthest point in an orbit, is called
*apoapsis*. Periapsis and apoapsis are usually modified to apply to the
body being orbited, such as perihelion and aphelion for the Sun, perigee and
apogee for Earth, perijove and apojove for Jupiter, perilune and apolune for
the Moon, etc. The *argument of periapsis* is the angular distance
between the ascending node and the point of periapsis (see figure below). The
*time of periapsis passage* is the time in which a satellite moves
through its point of periapsis.

Nodes are the points where an orbit crosses a plane, such as a satellite
crossing the Earth's equatorial plane. If the satellite crosses the plane
going from south to north, the node is the *ascending node*; if moving
from north to south, it is the *descending node*. The *longitude of
the ascending node* is the node's celestial longitude. Celestial longitude
is analogous to longitude on Earth and is measured in degrees
counter-clockwise from zero with zero longitude being in the direction of the
vernal equinox.

In general, three observations of an object in orbit are required to
calculate the six orbital elements. Two other quantities often used to
describe orbits are period and true anomaly. *Period* is the length of
time required for a satellite to complete one orbit. *True anomaly* is
the angular distance of a point in an orbit past the point of periapsis,
measured in degrees.

Types Of Orbits

For a spacecraft to achieve earth orbit, it must be launched to an elevation above the Earth's atmosphere and accelerated to orbital velocity. The most energy efficient orbit, that is one that requires the least amount of propellant, is a direct low inclination orbit. To achieve such an orbit, a spacecraft is launched in an eastward direction from a site near the Earth's equator. The advantage being that the rotational speed of the Earth contributes to the spacecraft's final orbital speed. At the United States' launch site in Cape Canaveral (28.5 degrees north latitude) a due east launch results in a "free ride" of 915 mph (1,470 kph). Launching a spacecraft in a direction other than east, or from a site far from the equator, results in an orbit of higher inclination. High inclination orbits are less able to take advantage of the initial speed provided by the Earth's rotation, thus the launch vehicle must provide a greater part, or all, of the energy required to attain orbital velocity. Although high inclination orbits are less energy efficient, they do have advantages over equatorial orbits for certain applications. Below we describe several types of orbits and the advantages of each:

* Geosynchronous orbits*, also called

* Polar orbits* (PO) are orbits with an inclination of 90
degrees. Polar orbits are useful for satellites that carry out mapping
and/or surveillance operations because as the planet rotates the spacecraft
has access to virtually every point on the planet's surface.

* Walking orbits*: An orbiting satellite is subjected to a
great many gravitational influences. First, planets are not perfectly
spherical and they have slightly uneven mass distribution. These fluctuations
have an effect on a spacecraft's trajectory. Also, the sun, moon, and planets
contribute a gravitational influence on an orbiting satellite. With proper
planning it is possible to design an orbit which takes advantage of these
influences to induce a precession in the satellite's orbital plane. The
resulting orbit is called a

* Sun synchronous orbits* (SSO) are walking orbits whose
orbital plane precesses with the same period as the planet's solar orbit
period. In such an orbit, a satellite crosses periapsis at about the same
local time every orbit. This is useful if a satellite is carrying instruments
which depend on a certain angle of solar illumination on the planet's
surface. In order to maintain an exact synchronous timing, it may be
necessary to conduct occasional propulsive maneuvers to adjust the orbit.

* Hohmann transfer orbits* are interplanetary trajectories
whose advantage is that they consume the least possible amount of propellant.
A Hohmann transfer orbit to an outer planet, such as Mars, is achieved by
launching a spacecraft and accelerating it in the direction of Earth's
revolution around the sun until it breaks free of the Earth's gravity and
reaches a velocity which places it in a sun orbit with an aphelion equal to
the orbit of the outer planet. Upon reaching its destination, the spacecraft
must decelerate so that the planet's gravity can capture it into a planetary
orbit.

To send a spacecraft to an inner planet, such as Venus, the spacecraft is launched and accelerated in the direction opposite of Earth's revolution around the sun (i.e. decelerated) until in achieves a sun orbit with a perihelion equal to the orbit of the inner planet. It should be noted that the spacecraft continues to move in the same direction as Earth, only more slowly.

To reach a planet requires that the spacecraft be inserted into an
interplanetary trajectory at the correct time so that the spacecraft arrives
at the planet's orbit when the planet will be at the point where the
spacecraft will intercept it. This task is comparable to a quarterback
"leading" his receiver so that the football and receiver arrive at the same
point at the same time. The interval of time in which a spacecraft must be
launched in order to complete its mission is called a *launch
window*.

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Compiled and edited by Robert A. Braeunig, 1997.

Acknowledgements