Final Project

John J.

Legislator:  Leo Berman, Representative

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Final Project - Orbital Transfers

In space travel it is often necessary for a spacecraft to change its orbit around a larger body.  This change of orbit can be used to position the spacecraft properly or to bring the spacecraft into orbit around another celestial body.  These orbital transfers are very useful in all types of space travel.  Dr. Walter Hohmann devised a means of efficiently moving a spacecraft between two circular orbits through the use of an intermediate orbit that intersects the other two.(1)  Similar types of orbital transfers can be used for elliptical orbits.  An efficient method of transferring a spacecraft between two orbits can be achieved through the use of intermediate orbits that intersect the two orbits.

To understand orbital transfers, one must first understand orbits.  When one celestial body orbits another, the orbit is in the shape of an ellipse, as shown by Johannes Kepler and Sir Isaac Newton.(2)  An ellipse contains two foci, and the sum of the distances from any point on the ellipse to the two foci is constant. (See Figure 1)  In mathematical terms, if P is a point on the ellipse and F₁ and F₂ are the foci, PF₁ + PF₂ is constant for any point P on the ellipse.  The major axis on an ellipse is the segment that joins two points on the ellipse and passes through both foci; it is the longest segment that can be drawn through the ellipse.  The minor axis of an ellipse is the perpendicular bisector of the major axis, and it is the shortest segment that can be drawn through the ellipse.  The focal length is the distance between the two foci, and the eccentricity is the focal length divided by the major axis.  Thus, the eccentricity of an elliptical orbit can be between zero and one, with an eccentricity of zero meaning that the orbit is actually a circle.

Apart from the normal mathematical properties of an ellipse, there are other properties that must be considered when orbital ellipses are observed.  When a satellite orbits a larger body (known as the primary) in an elliptical orbit, the primary is one focus of the ellipse.  The point where the satellite is closest to the primary, which occurs at one end of the major axis, is called the periapsis.  The point at which the satellite is farthest from the primary, at the other end of the major axis, is the apoapsis.  When more than one orbit is being considered, it is also useful to know the angle formed by the periapsis and a given longitude on the primary.  In addition, the angle formed by the orbital plane of the satellite and the equatorial plane of the primary is the inclination of the orbit.  This report, however, only discusses transfers between coplanar orbits.

The simplest type of orbital transfer, a Hohmann transfer, occurs when a spacecraft moves from one circular orbit to another circular orbit of a different altitude.  In this case, the transfer orbit is an ellipse, the periapsis of the transfer orbit intersects with the inner circular orbit, and the apoapsis of the transfer orbit intersects with the outer circular orbit.  If the spacecraft is moving to the outer orbit, it will accelerate into the transfer orbit and accelerate out of it.  If the spacecraft is moving to the inner orbit, it will decelerate into the transfer orbit and decelerate out of it.  Similarly, an orbital transfer between two orbits that share a major axis can be achieved with two velocity changes.  If the periapses of the orbits are on the same side of the body, then the transfer orbit will intersect the apoapsis of the inner orbit and the periapsis of the outer orbit.  If, however, the periapses are on opposite sides of the body, the transfer orbit will intersect the periapses of the two orbits.  Once again, if the spacecraft is moving to the outer orbit, it will accelerate into the transfer orbit and accelerate out of it, and if the spacecraft is moving to the inner orbit, it will decelerate into and out of the transfer orbit.

If the spacecraft needs to travel between two orbits that do not share a major axis, a more complex orbital transfer is needed. (See Figure 2)  A simple way of doing this consists of two transfer orbits, one of which is circular.  The spacecraft will first transfer to a circular orbit. (Fig. 2-A)  Then, when the spacecraft is along the major axis of the desired orbit, it will enter a second transfer orbit whose apoapsis or periapsis intersects the desired orbit. (Fig. 2-B)  Then, the spacecraft will transfer into the final orbit. (Fig. 2-C)  This method is simple because the transfer problem has been reduced to that of two orbits with a common major axis once the spacecraft enters the circular orbit, where any diameter of the circle can be considered the major axis.

All of the above orbital transfers rely upon velocity changes when a spacecraft changes orbits.  If the spacecraft is moving to a larger orbit, the spacecraft must accelerate to enter the larger orbit.  Conversely, if the spacecraft is moving to a smaller orbit, the spacecraft must decelerate.  To find the desired velocity change, one can use the vis viva equation developed by Gottfried Wilhelm Leibniz.(3)  This equation is v = (GM(2/r-1/a))^1/2, where "G" is the universal gravitational constant (6.673x10^-11Nm²kg^-2), "M" is the mass of the primary, "r" is the distance from the central body, and "a" is the semi-major axis of the orbit.  This equation can be easily used to determine the necessary velocity change by finding the new velocity using the semi-major axis of the new orbit, then subtracting the old velocity using the semi-major axis of the old orbit.  The mass of the central body and the distance of the central body remain the same for this calculation.

Once the required velocity changes are known, it is necessary to know the appropriate time for each velocity change.  To find these times, the period of each transfer orbit is needed.  The period on an orbit can be found with the equation p = (4(pi)²a³/GM)^1/2, where "G" is the universal gravitational constant, "M" is the mass of the primary, and "a" is the semi-major axis.  In the orbital transfers described above, all velocity changes take place when the satellite is in a circular orbit, at the apoapsis of an elliptical orbit, or at the periapsis of an elliptical orbit.  Thus, if the transfer orbit is an ellipse, the time spent in the transfer orbit will be half of the period of the orbit as the satellite moves from periapsis to apoapsis or vice versa.  If the transfer orbit is a circle, the time spent in the orbit will be a fraction of the period proportional to the fraction of the total circle that is traveled.

Using the above information, a simple algorithm can be used to find the required information for coplanar orbital changes.  First, determine the properties of the initial, final, and transfer orbits that will be used.  Then, use the vis viva equation to calculate the required velocity changes to move between orbits.  Finally, determine the period of each transfer orbit, then use that information to find the length of time between velocity changes.  Thus, when the spacecraft is in orbit, it can execute these velocity changes at the predetermined intervals, placing the satellite in the designated orbit around the primary.

As man's exploration and colonization of space continues, the frequency of space flights will increase, and orbital transfers will become more and more important to space travel.  This report outlines one method of efficient orbital transfers, but there are many other means of altering a satellite's orbit.  Some are more energy efficient, some are more time efficient, and some require less distance to be traveled.  As there are more and more types of missions that will be launched, each mission that requires orbital transfers will require a certain type of transfer that is the best for that mission, based on the speed of the spacecraft, the cost of the mission, and the amount of time and fuel that can be used to move the satellite to another orbit.

Sources:

1.          Orbital Mechanics 101

http://aerospacescholars.org/scholars/earthstationmoon/unit2/Orb1.html

2.          Orbital Mechanics

http://users.commkey.net/Braeunig/space/orbmech.htm

http://users.commkey.net/Braeunig/space/orbmech1.htm

http://users.commkey.net/Braeunig/space/orbmech2.htm

3.          Mission: Mission Possible

http://aerospacescholars.org/scholars/earthmars/unit5/Mission_Mission_Possible.htm

Last Updated:  09/07/01