Transfer Calculations
  Hohman Transfer
  Free Return Trajectory
  Spiral
  Summary of Numerical Data

Launch Windows
  Calculations
  Possible Launch Dates
  Graphical Comparison

Sources

Abstract

We decided to split the mission into three separate transfers. The first transfer will contain all communications equipment necessary for the mission. This package will follow a simple Hohman transfer to Mars. The second package will contain the scientific equipment, as well as any other machinery that is not directly essential for human survival. It will follow a low power spiral transfer to Mars, although we may need to use a different transfer because of the Van Allen radiation belts. The third transfer will send the crew and lifesupport equipment to Mars using a free return trajectory. The trajectory for the return trip will be a Hohman transfer.

There are many important factors to consider when evaluating which trajectory would be best for this mission. The first consideration is the energy needed to accomplish the transfer. The sum of the changes in velocity needed to leave Earth, enter Mars, leave Mars, and enter Earth, or total delta V, can be used to compute how much fuel is needed to complete the transfer. Thrust acceleration, acceleration that is due to rocket thrust and acts in the direction tangent to the path of the rocket, is also proportional to the amount of fuel needed. Since fuel adds to the cost and the weight of the mission, decreasing the amount of fuel was a priority. Other factors that were looked at were the number of flights, the time required for each kind of transfer, and the safety of the human crew.

First, after looking at all of the factors involved in an interplanetary flight, it is evident that any spacecraft with a human crew must spend as little time in space as possible. There are health hazards as a result of exposure to zero-G and solar radiation. Unfortunately, as the time of transfer decreases, power needed to complete the transfer increases. Since fuel expenditures are also directly proportional to the mass of the payload, it is unfeasible to attempt to send all of the scientific equipment and all of the human support systems on the same flight as the humans. It was concluded that the mission must be broken into more than one flight. A two-flight plan was considered, but was discarded for two main reasons. First, the weight of communications equipment and scientific equipment was unwieldy. Second, the class felt that it would be to the mission¹s advantage to have communications satellites in place and functioning before the crew arrives. A three-flight plan has advantages in many areas, not all of them related to trajectory. In a three-flight plan, the experimental equipment, communications equipment, and machinery would be launched first, in two slower, lower power, lower cost flights. The human crew would be launched on a separate flight to Mars. This method reduces the mass in any single flight and allows a communications network to be formed prior to the departure of the manned mission. It also allows maximum efficiency without subjecting men to any greater time in space than necessary. This plan was chosen after careful consideration of trajectory options
 

Trajectory Options

There are several different kinds of trajectories from Earth to Mars. Transfers based on conic sections are the most common type. A bi-elliptic transfer is one in which two ellipses, both with the sun as one focus, are drawn, one tangent to Earth's orbit, and one tangent to Mars' orbit. At the point where the two ellipses intersect, the vehicle's rockets are fired, changing the vehicle's course from one elliptical orbit to the other. A variation on this transfer is the bi-parabolic transfer, in which the radii of the ellipses go to infinity. This requires little to no delta V, but is only a theoretical trajectory, as the time of transfer also approaches infinity.

A transfer modeled on a single ellipse that intersects both earth and Mars orbits can also be used. A single ellipse transfer that is tangent to both earth and Mars' orbits is known as a Hohman transfer. A larger ellipse, one that is tangent to earth¹s orbit, but merely crosses Mars¹ orbit, can provide faster transfer times in exchange for an increase in delta V. A free-return trajectory is a large-elliptical transfer whose period of orbit around the sun is a whole number, N, multiple of the earth¹s period of orbit. Thus, after N years, both the spacecraft and the earth are in the same position. This allows the spacecraft to be recovered after N years should it, for any reason, not enter Mars orbit.

Planetary flyby transfers were also considered. In this kind of transfer, the vehicle is launched from Earth to either Venus or the Moon. The vehicle then flies around Venus or the Moon, "borrowing" energy from the other mass's gravitational field. A final type of transfer is the spiral transfer. A low thrust ion propulsion engine applies continuous thrust to the ship, causing it to spiral slowly from low Earth orbit until it reaches escape velocity. Once escape velocity is attained, the vehicle travels from Earth to Mars and slowly spirals into Mars' gravity well.

Trajectory Requirements

There are many factors that must be taken into consideration for each flight. For the first transfer, the short life span of the communications satellites requires that a short and fast trajectory be used. It is important to note that although this is referred to as the ³first² transfer, it is quite possible that this transfer will take place second, arriving at Mars and deploying just before the human mission is launched, due to this short life span. Fortunately, the smaller mass of the satellite package means that a larger expenditure in delta V is acceptable. The package with the majority of the equipment is quite massive, so the largest consideration was decreasing fuel expenditures. As the equipment on the spacecraft is quite rugged, it can survive the longer transit times associated with decreasing power. For the human transfer, it remains important to decrease delta V, but one must also recognize the need for a quicker time of transfer. In addition, a free-return trajectory is important.

With these criteria in mind, flyby trajectories may be eliminated. A flyby past the moon gives no appreciable decrease in delta V. A flyby past Venus increases the time of transfer greatly and also brings the vehicle too close to the sun for most safety standards. A Venus flyby would subject delicate scientific equipment and human crewmembers to unreasonable extremes of heat and other radiation, and would also increase the likelihood that a solar flare would disturb the trajectory of the vehicle.

The spiral trajectory bears greater consideration. A spiral trajectory is desirable because research indicates that the low, sustained thrust associated with ion propulsion is an efficient use of fuel, but provides long transfer times. The savings in fuel expenditure are worth the time trade-off for the massive equipment flight. The increased efficiency of the engine compensates for the increase in fuel needed to move the large payload. The length of time required for this trajectory was not acceptable, however, for the communications flight.

Trajectories based on conic sections were then considered. The bi-parabolic trajectory can be immediately eliminated, because although the delta V was close to zero, the time of transit approached infinity, and this is unacceptable for both manned and unmanned flights. The bi-elliptical transfer can also be eliminated because its delta V is lower than a single-ellipse trajectory only when the radius of orbit for the outer planet was more than 15.58 times the radius of orbit for the inner planet. (from Orbital Mechanics; Conway and Prussing) The ratio of Mars' radius of orbit to that of the Earth is approximately 1.5. Also, the bi-elliptical transfer results in no time benefit.

Of the single ellipse trajectories, a Hohmann trajectory is the lowest energy transfer. Increasing the size of the ellipse decreases the time required to travel the arc of the ellipse connecting Earth and Mars. Certain large ellipses can provide free return trajectories. Both of these qualities were desirable for the manned spacecraft. After calculating the delta V¹s and time of transfer for the Hohmann trajectory and for the shortest period free return trajectory, it was determined that some variation on the elliptical transfer would be used for both the communications flight and the manned flight. For the communications flight, the large increase in delta V caused by large fast-track ellipses could not be justified. The Hohmann trajectory provides more reasonable delta V¹s than large ellipse trajectories and also provides a more rapid time of transfer than spiral trajectories. The increase in delta V of the Hohmann trajectory over that of spiral trajectories is justified by the need for rapid flight time to compensate for the short lifespan of the satellites. For the manned mission, it was decided that the large delta Vs associated with free return trajectories were justifiable because of the added safety provided by having a free return trajectory. However, there is no value in having a free return trajectory on the way back to Earth. Consider that in the extreme case the spacecraft cannot land on Earth: the astronauts would merely return to Mars. A non-trajectory contingency plan should be developed for the return trip. For the return trip a Hohmann trajectory was selected because of its balance of time and fuel efficiency.

References:

Prussing, John E. and Bruce Conway. Orbital Mechanics. New York: Oxford University Press, 1993.