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Optimizing Trajectory Scenarios in Sending Human Beings and their Cargo on a Mission to Mars
To send mankind to our neighboring planet Mars has, and always will, remain a goal of space explorers. It is not a question of whether we will send human beings to the red planet, but when and how? This discussion aims to realistically address that question from a trajectory standpoint. With the understanding that both crew and cargo need to take the journey, various mission scenarios will be assessed to determine what options exist to make this possible. What types of trajectories will be most suitable based on ΔV and the relative positions of both Earth and Mars in their orbits?
Background - The Planet Mars
Mars has always been considered "Earth's neighbor" and there are many reasons why this is so. In a literal sense, Mars is the fourth planet from the Sun, right behind Earth, and a terrestrial planet. The planet's physical characteristics, in many ways, are similar to that of Earth's, including its rotational period and seasonal changes. Above the surface resides at atmosphere, although not as thick as Earth's, does form clouds, has wind and allows weather to form. Being farther from the Sun than the Earth, Mars' temperature changes are much more substantial. The surface temperatures can drop as low as -195 degrees F near to poles to as high as 70 degrees F near the equator. Unfortunately, the planet's atmosphere contains much less oxygen in comparison to Earth: 0.13% on Mars, 21% on Earth. The majority of the air is carbon dioxide and other gases, all similar to ones on Earth, but in larger percentages.5
Figure 1 - Artist's Conception of NASA's Phoenix Lander (Image credit: NASA/JPL)
The theory that Mars might have once been home to life has been speculated for decades. Mars surely has some of the necessary products for life to foster such as organic compounds, a source of energy and water. To collect further evidence, Mars has been under observation since the late 1960s. Currently, there are satellite orbiting about Mars further analyzing the planet and probes on its surface, such as Spirit and Opportunity. It has been confirmed that water, in the form of ice and beneath the Martian surface, does it in fact exist. The more evidence that is collected, the more intriguing it becomes to send human beings to continue the exploration of this interesting planet.
Human beings, by nature, have a tendency to explore. A desire to venture out beyond the environment they know and discover. Curiosity can cause one to dream as was the case when the first person proposed the idea of sending human beings to Mars. This has been something under slow development in the aerospace and scientific communities yet still a desirable goal. Many are silently awaiting the time when this idea will come to fruition. When will we be ready for the challenge? When will time, money, politics, and technology come together to allow the human race to pursue such a feat? In the meantime, there is much work that can be done. Planning such a mission is no easy task and one of such complexity will surely take many years. One chief concern is when will we launch from Earth and upon what trajectory will we travel on our mission to Mars?
The critical problem of sending a spacecraft from one planet to another resides in the development of a trajectory that departs the first planet at one epoch and arrives at the destination planet at another. As the planets orbit the Sun, there exist trajectories that are more optimal than others in terms of fuel expended (ΔV) and time of flight. To determine these types of transfer orbits is the problem of finding the desired Lambert Arc. Lambert's problem is stated as follows: "given an initial and final position, together with a time of flight between these positions, determine the connecting orbit."3 A graphical representation of Lambert's problem may be seen in Figure 2, where r1 is the radial distance from the Sun to the departure planet, r2 is the radial distance to the arrival planet, c is the chord length, θ is the transfer angle, and v1 and v2 are the departure and arrival velocities, respectively.
Figure 2 - Lambert's problem for a transfer between two planetary orbits (source: Reference 3).
The following references were used in the motivation of this work and in the development and verification of the problem of interest. Additional references may be found towards the end.
- Shepard, Kyle, Jack Duffey, Dom D'Annible, Jeff Holdridge, Walter Thompson, and Robert C. Armstrong. "A Split Sprint Mission to Mars." American Institute of Physics (1992): 58-63. Print.
- George, L. E., and L. D. Kos. "Interplanetary Mission Design Handbook: Earth-to-Mars Mission Opportunities and Mars-to-Earth Return Opportunities 2009-2024." National Technical Information Service. National Aeronautics and Space Administration, July 1998. Web.
- Kemble, Stephen. Interplanetary Mission Analysis and Design. Berlin, Germany: Springer in Association with Praxis, 2006. Print.
Problem of Interest
Importance of aspect to the mission
The primary concerns of a human mission to Mars revolve around time and distance. Depending on the trajectory or transfer orbit chosen, which itself is dependent on the departure and arrival times chosen, the ΔV values and time of flights will be directly impacted. The second most important concern is one of distance. The Mars crew will be out on its own and only have each other and the equipment they brought with them to depend on. A radio signal traveling from Earth to Mars takes approximately 10 to 20 minutes, depending on the positions of the planets in their orbits, and that same amount of time for Earth to receive a response. Therefore, in case of an emergency, communication back to Earth will be delayed and any hope of rescue is a few months time away at best.
While several trajectories will be explored, there is a logical pattern to the trajectories that will make it into the tradespace. Some will be chosen based on time of flight, others on required ΔV. The idea is to comprise a mission scenario where "the cargo is split from the crew, [in a] 'Split Sprint' architecture. An efficient low thrust 'slow boat' is used for the cargo and a high trust 'spring' vehicle is used for the crew."1 (see Figure 3). The idea is to send the cargo ahead of the astronauts so they may rendezvous with it in Mars' orbit or have it ready and waiting on the Martian surface, prior to their arrival. These trajectory options will be explored further as the problem of interest.
Figure 3 - Split-Spring Mission Overview (source: Reference 1)
Importance of problem to the field of Astrodynamics
The desire to explore our solar system continues as does our desire for understanding the unknown. There is much we can learn by continuing to study celestial bodies that are foreign to us. Just this year a NASA probe discovered a vast amount of extractable water on our own Moon, which leads to the question, how much can we learn by exploring Mars? It is a worthy endeavor to explore our neighboring planet and according to Stephen Hawking, it is critical to the survival of the human race to colonize space. Sending probes is just not enough, and "although astronaut missions are much more expensive and risky than robotic craft," Stephen Hawking continues, "they are absolutely critical to the success of our exploration program." With this being said, it brings up many questions of how we are going to accomplish this? This paper strives to shed some light on how we can quickly and economically achieve this goal from a trajectory standpoint. Crew and cargo will both need to be ferried to Mars and in a timely manner. Certain trajectory options will be more appealing than others and tradeoffs will need to be made. The relative positions of the planets in their orbits will dictate what trajectories are feasible and will ultimately limit these options.
It is important to mathematically analyze what trajectories will produces optimum results in the framework of time of flight and fueled expended. This analysis will yield a tradespace of trajectories from which a desired one may be chosen. This aspect of the mission is most heavily influenced by the field of astrodynamics and its analysis can significantly impact the mission as a whole. Due to the orientation of the planets, trajectories at different velocities and different departure and arrival times will need to be considered that will ultimately impact how long the astronauts will be required to stay on the surface of Mars.
Development of Solution Method
The solution that will first be investigated is Lambert's Problem of minimum energy. This is a logical approach and a common method to choose the optimal and lowest cost trajectory. The method beings by determining the positions of both planets with respect to the Sun and calculating the transfer angle θ between them, using a simple cosine angle formula for vectors. Once the positions of the planets and the transfer angle have been calculated, the time of flight (TOF) may be chosen. This is an important step and one that will lead to drastic variations in the departure and arrival (velocities with respect to the corresponding planet) values. Having these values now ready, the method begins by calculating the chord c (shown in Figure 2).
where r1 and r2 are the vectors representing the positions of the planets and being the transfer angle. Then the value S may be computed:
where S is referred is referred to as the semi-perimeter of the connecting triangle between the two endpoints of vectors r1 and r2. Then two intermediate variables are found, α and β:
and incorporated into the following expression (note: values of μ are in reference to the Sun's gravitational parameter unless otherwise stated):
The problem is continued by performing an iterative technique to solve for a by continually checking if the left-hand and right-hand sides of the equations are equal. At this juncture is referred to as TOFmin (time of flight for the minimum energy solution). Once a value for a (the semi-major axis) is found, it is referred to as amin and is the semi-major axis for the minimum energy solutions based on the initial variables chosen. A check must now be done to determine whether the orbit is parabolic or hyperbolic. A simple method is to compute the time of flight for the parabolic transfer:
This value must then be compared to the time of flight initially chosen in the beginning of the problem. If TOFpar < TOF, then a Type I (Short Way) transfer is needed, otherwise, if TOFpar > TOF then a Type II (Long Way) transfer is needed (see Figure 4).
Figure 4 - Transfer methods for the Lambert Problem (source: Reference 6, pg465)
The next step is to determine whether this is a Type A or Type B transfer. If TOF < TOFmin, then it is a Type A, otherwise, if TOF > TOFmin, then it is a Type B. This is important in choosing the semi-latus rectum, p. The semi-latus rectum is calculated using the following formula:
then it is necessary to choose the larger p value if it is a Type A transfer or the smaller p value if it is a Type B transfer. The eccentricity of the orbit may be simply calculated with the following equation:
Then the true anomalies and at the departure and arrival points:
as well as the flight path angles at departure and arrival, and :
Now the circular velocities of each planet will be needed which is simply the square root of the Sun's gravitational parameter divided by the distance from the Sun to that planet.
And then the (velocity with respect to the planet of interest) values may be computed incorporating the results from all the calculations above.
One additional step may be taken to compute the required for departure and arrival which are dependent on the altitude of the parking orbit above Earth and the desired capture altitude into an orbit about Mars.
where rpo and rco refer to the radius of Earth's parking orbit and radius of Mars' capture orbit, respectively.
Analysis of Method
To model the solution method above, cargo trajectory characteristics were chosen from "A Split Sprint Mission to Mars" (Reference 3), and all the computed values (see Table 1). This trajectory profile is for a mission launched in 2018.
|Cargo Trajectory Characteristics||Departure from Earth|
|Parking Orbit Altitude||400||km||17.7807||deg|
|Capture Orbit Altitude||400||km||29.7847||km/s|
|177478954.6560||km||Arrival at Mars|
Table 1 - Computation results from Lambert's Minimum Energy Problem
The long flight time and slightly above average values can be explained by the cargo spacecraft's SP-100 derived nuclear electric propulsion system. The limitations of the Lambert method for this type of propulsion system is that the computed values are instantaneous at the departure and arrival points of the transfer arc while this particular spacecraft's propulsion system will continue thrusting for the majority of the duration of the flight. It will take the spacecraft approximately 380 days of constant thrust to fully escape Earth's gravitational pull and then it will continue thrusting for another 180 days. Then it changes to a heliocentric coast phase for a time of 130 days and then begins thrusting again for the remainder of the journey to more easily rendezvous with Mars.1
This is an acceptable trajectory option for sending cargo to Mars as it is more economical due to the long duration of flight and chosen propulsion system. For a manned mission, a shorter transfer time is desired as "this reduces crew exposure to the deleterious effects of natural space radiation (in particular galactic cosmic radiation) and zero gravity."1
While a minimum energy solution is an economical one, it might be based on an unsuitable date for various reasons. The idea, by mission planners, is to design a mission scenario where crew and cargo arrive at the planet in a timely manner. Cargo must arrive before crew in order for the crew to have the necessary supplies to assemble their habitat and home for the next several months. This type of scenario design also includes sending two cargo spacecraft due to the amount of supplies required and for redundancy purposes as well.
Based on the preliminary calculations of a minimum solution, the proposed extension is to design a Lambert solver that will calculate a variety of trajectories based on varying departure and arrival epochs. Depending on the resulting C3 values, launch energy in km2/s2 values, trajectories may be categorized. Following the design of a Lambert Solver (script LambertSolver.m), NASA's "Interplanetary Mission Design Handbook: Earth-to-Mars Mission Opportunities and Mars-to-Earth Return Opportunities 2009-2024" (Reference 2) will be used in order to verify its functionality. Following verification of the Lambert Solver for computing these trajectories, the problem will be further extended beyond NASA's tabulated values by choosing more than one option for each test departure and arrival date in order to gain further insight into why NASA chose these particular trajectories. Through this analysis, it was discovered that not all the chosen trajectories were chosen based on the minimum launch energy, so using the Lambert Solver, the minimum launch energy was found for each transfer arc through the use of pork chop plots.
Analysis of Extension
To begin the formulation of a Lambert Solver, a departure and arrival epoch are chosen and converted to a Julian date. Julian date, JD, is the interval of time measured in days from the epoch January 1, 4713 B.C., 12:00 and is a common unit of time used by astronomers.6 Then the positions and velocities of the Earth at departure and Mars at arrival are calculated based on planetary ephemerides in reference to the J2000 coordinate frame (see Equations 1 and Equations 2).
Equations 1 - Earth Planetary Ephemerides in Reference to J2000 in Degrees (source: Reference 6, pg995)
Equations 2 - Mars Planetary Ephemerides in Reference to J2000 in Degrees (source: Reference 6, pg995-996)
These position and velocity vectors may then be fed into a Lambert Solver which will determine the necessary departure and arrival velocities for a transfer orbit connecting the two planets. The values are then calculated in the following manner:
where and are the departure and arrival velocities, respectively, computed by the Lambert Solver, and and are the velocities of Earth and Mars, respectively, computed from the planetary ephemerides in reference to the J2000 coordinate frame. C3 is a term known as the launch energy and is common used as a comparison metric when choosing an interplanetary trajectory for the mission.
The Lambert Solver also takes in one additional important parameter: tm. If tm = +1, a transfer angle is used that is less than 180° and if tm = -1, a transfer angle greater than 180° is used (refer to Figure 4 ). These are known as the Short Way and the Long Way, respectively. The above steps may be used to compute this trajectory for a single chosen departure and arrival JD, however, a more efficient method is to loop the above steps through an array of departure and arrival Julian dates. Obtaining an array of selected values will enable a graphical representation (pork chop plot) assisting a mission designer in choosing a trajectory that minimizes both C3 and time of flight.
Obtained from the "Interplanetary Mission Design Handbook: Earth-to-Mars Mission Opportunities and Mars-to-Earth Return Opportunities 2009-2024" (Reference 2) is a summary of all the cargo and piloted opportunities from 2009-2014 (Table 1).
Figure 5 - Summary of all cargo and piloted opportunities, 2009-2024 (Source: Reference 2, pg14)
Using the solution method mentioned in this section, several of these trajectories were verified by choosing the launch and arrival dates and verifying that the TOF matched the C3 value. In every case that was tested, the values from the Lambert Solver were verified and in addition, the values at departure and arrival were computed. A summary of these results may be seen below in Table 2.
|TOF (days)||Arrival Date
| Type I
| Type I
| Type II
| Type II
Table 2 - Summary of results from Lambert Solver (Czerep). Launch dates, arrival dates, and TOFs obtained from Reference 2 and used as input values to the Lambert Solver.
To show the results from the verification process, Table 3 may be consulted. For each launch and arrival epoch, the C3 value from the Lambert Solver was compared to the value from NASA's handbook. For the Type I trajectories, the error percentage in comparison to NASA's values had a mean value of approximately 0.4 % and for the Type II trajectories, the error was 0 %.
Table 3 - Comparison of C3 values computed from the Lambert Solver (Czerep) to NASA's values from the "Interplanetary Mission Design Handbook: Earth-to-Mars Mission Opportunities and Mars-to-Earth Return Opportunities 2009-2024" (Reference 2).
No further information was given in the handbook as to why these particular trajectories were chosen. In some cases, it was obvious as the C3 value was at a minimum and the TOF was a reasonable value. In other cases it was not so clear as a sacrifice of a few days of travel time allowed the C3 value to be decreased by a significant amount (nearly 20% in one case). A summary of minimum C3 values and additional trajectory options is presented in Table 4.
Option 2 (Czerep)
|3.21.16||305||1.20.17||II||-||7.9955||7.9935||305||8.8675 (Type I)||181|
|10.5.24||345||9.15.25||II||-||11.1929||11.1929||345||17.7161 (Type I)||220|
Table 4 - Additional options for C3 including minimum value for the dates chosen (unless otherwise mentioned, minimum and option 2 C3 options are the same type of trajectory as mentioned in the column following Arrival Date. No suitable second option was found for 5.18.18.
For the first epoch at 1.20.14, a C3 value of 15.9269 km2/sec2 was calculated via the Lambert Solver in comparison to a value of 15.92 km2/sec2 presented in NASA's handbook. The minimum C3 value was found to be 9.0380 km2/sec2 but at a price of increasing the TOF to 208 days (see intersection of orange lines on Figure 7). To assist with finding the minimum C3 value, additional plots may be used relating C3 versus Earth departure days as well as C3 versus Mars arrival days). Figure 9 shows how the minimum C3 value may be determined from this type of graphical representation. Since this was a piloted/manned trajectory, it was most likely an unacceptable sacrifice. However, increasing the TOF by 16 days reduced the C3 value to 12.9313 km2/sec2 which was a difference of 2.9956 km2/sec2 and 18.8 %. This is a tradeoff worth considering.
In order to reduce clutter, the following legend will be used for all pork chop plots (Figure 6):
Figure 6 - Legend used for all pork chop plots.
Figure 7 - Pork Chop Plot for Departure Date 1.20.14 and Arrival Date 6.30.14 (Orange - minimum C3, Cyan - C3 Type I Option 1)
Figure 8 - C3 vs. Earth Departure Days - Constant departure date, varying arrival date for Departure Date 1.20.14 and Arrival Date 6.30.14
Figure 9 - C3 vs. Mars Arrival Days - Constant arrival date, varying departure date for Departure Date 1.20.14 and Arrival Date 6.30.14
For a departure date of 3.7.16 and an arrival date of 7.14.16, NASA's proposed trajectory takes 129 days at a launch energy of 15.92 km2/sec2. However, by spending an additional 10 days in flight, the launch energy may be reduced to 13.4114 km2/sec2, roughly a 16 % reduction (Figure 10). What's important to note in this situation is that this is a manned orbit traveling to Mars, and as with the Split-Spring mission scenario, a cargo spacecraft will be sent on its way as well. This accompanying cargo spacecraft has a launch date of 3.21.16 with an arrival date of 1.20.17 spending 305 days in flight. This Lambert arc is a Type II trajectory and Mars may also be reached, during these same epochs, with a Type I trajectory taking only 181 days with only a 10% increase in launch energy (Figure 11). The 305 TOF trajectory might have been chosen due to a strict constraint on arrival time, however, should this not be the case, this secondary option exists.
Figure 10 - Pork Chop Plot for Departure Date 3.7.16 and Arrival Date 7.14.16 (Orange - minimum C3, Cyan - C3 Type I Option 1)
Figure 11 - Pork Chop Plot for Departure Date 3.21.16 and Arrival Date 1.20.17 (Cyan - C3 Type II Option 1, Yellow -C3 Type I Option 2)
Pork Chop Plots for Remaining Epochs
Figure 12 - Pork Chop Plot for Departure Date 3.14.16 and Arrival Date 7.29.16
Figure 13 - Pork Chop Plot for Departure Date 5.17.18 and Arrival Date 1.8.19
Figure 14 - Pork Chop Plot for Departure Date 5.18.18 and Arrival Date 9.10.18
Figure 15 - Pork Chop Plot for Departure Date 9.14.22 and Arrival Date 10.2.23
Figure 16 - Pork Chop Plot for Departure Date 9.10.22 and Arrival Date 3.9.23
Figure 17 - Pork Chop Plot for Departure Date 10.5.24 and Arrival Date 9.15.25
Figure 18 - Pork Chop Plot for Departure Date 10.17.24 and Arrival Date 4.15.25
Performing an analysis of trajectories using a Lambert Solver and Pork Chop Plots yielded many interesting conclusions (Table 4). Not all the options presented in NASA's "Interplanetary Mission Design Handbook: Earth-to-Mars Mission Opportunities and Mars-to-Earth Return Opportunities 2009-2024" (Reference 2) were the best trajectories given the chosen departure and arrival epochs. By increasing the time of flights by approximately ten days resulted in measureable reductions in the launch energies.
Based on the analysis, a few trajectories were found with more optimal solutions (Table 5). There was usually a tradeoff between C3 and time of flight when recommending new trajectories. For a launch date of 1.20.14, by increasing the time of flight by 16 days, the launch energy was reduced by nearly 20%. For a launch date of 3.21.16, by increasing the launch energy by slightly more than 3% reduced the time of flight by 11 days. For a launch date of 3.7.16, the time of flight was reduced by 124 days as a result of changing the trajectory type at an expense of increasing the launch energy by roughly 10%. For a launch date of 5.17.18, the time of flight was reduced by 46 days for an increase in launch energy by slightly less than 3%.
Table 5 - Recommended changes to analyzed trajectories.
Summary and Conclusion
The passion to pursue science stems from an ever-present and unwavering sense of curiosity. In the realm of exploring celestial bodies, the Moon and Mars are the stepping stones to uncovering profound knowledge of our solar system. When will human beings step foot on Mars? Will it be this generation or the next? The time is soon and it is only a matter of when technology, the economy and political agendas align, that we will begin to construct the necessary craft to send human beings and their cargo to Mars. In the meantime, there is much planning and analysis that can be done to be better prepared for when the moment to act arrives. One aspect of this planning is trajectory analysis and optimization. The Earth and Mars are only in ideal positions within the solar system during certain years and planning years ahead is imperative to achieve optimal launch and arrival dates while simultaneously conserving the required fuel.
A minimum energy solution was initially considered, using Lambert's Minimum Energy Problem as a base for this analysis. Additionally a Lambert Solver was written to analyze many trajectories based on an array of departure and arrival epochs. These solutions were motivated by creating a split-sprint trajectory to Mars including a piloted/manned trajectory coupled with two cargo trajectories. The values from the Lambert Solver were compared to NASA's handbook on interplanetary trajectories from Earth to Mars and an error analysis showed the computed values, on average, were less than half a percent different. Having deemed the Lambert Solver to be a credible tool, additional trajectories were explored in hopes to find more optimal solutions. From the ten analyzed trajectories, four optimal replacement trajectories were discovered. Some of these recommended options increased time of flight by a small number of days and saved a significant amount of launch energy while others significantly reduced the time of flight at a small expense of launch energy. Choosing an optimal trajectory ultimately lends itself to determining whether the mission is willing to sacrifice time or energy and to what degree.
- Bate, Roger R., Donald D. Mueller, and Jerry E. White. Fundamentals of Astrodynamics. New York: Dover Publications, 1971. Print.
- Squyres, Steven W. "Mars." World Book Online Reference Center. 2004. World Book, Inc. (http://www.nasa.gov/worldbook/mars_worldbook.html.)
- Vallado, David A., and Wayne D. McClain. Fundamentals of Astrodynamics and Applications. Third ed. Hawthorne, CA: Microcosm, 2007. Print.