Investigation of Asteroid Rendezvous Trajectories

Emma Young

ASEN 5050 Project - Fall 2014


The first uncrewed Exploration Test Flight (EFT-1) for the Orion vehicle occurred on December 5th, 2014. This marked the beginning of the return of crewed missions beyond Low Earth Orbit (LEO) for the first time since the conclusion of the Apollo program in 1972. The first crewed flight of Orion is scheduled for 2021, however, the destination of that mission is still uncertain. One proposed destination for the maiden Orion flight is a near-Earth asteroid, and the Asteroid Redirect Mission (ARM) aims to provide a novel opportunity for astronauts to explore an asteroid and study samples (Klotz).

The Asteroid Redirect Mission (ARM) is an upcoming NASA mission split into two sub-missions: the Asteroid Redirect Robotic Mission (ARRM) and the Asteroid Redirect Crewed Mission (ARCM). The ARRM will characterize, capture, and redirect a near-Earth asteroid to a stable, Distant Retrograde Orbit (DRO) around the moon, likely making use of a lunar gravity assist. This placement of an asteroid in a lunar orbit is a feasible, cost-effective solution to allow astronauts during the subsequent ARCM to visit, explore, and study an asteroid while demonstrating technologies that will enable future manned missions to Mars.

    Two conceptual mission plans for the robotic segement, ARRM, currently exist:
  • Plan A: Capture a small, 7 - 10 meter diameter asteroid to redirect to lunar orbit.
  • Plan B: Retrieve a 1 - 10 meter diameter sample from a larger asteroid to redirect to lunar orbit.

In mid-December 2014, NASA will select one of these mission concepts with which to move forward. The final selection of the target asteroid from a list of candidates will occur in the future, at a minimum of one year before the expected launch date (Gates).

This project explores the characteristics and orbits of the proposed candidate asteroids for each of the two plans in order to examine possible rendezvous trajectories and mission timeline options.

A short video below produced by NASA showcases artistic concepts of mission highlights for the robotic capture and human exploration of a redirected asteroid.

Video courtesy of NASA.


The Asteroid Redirect Mission (ARM) concept has been in development since 2011, when it was first introduced by the Keck Institute for Space Studies (KISS) at the California Institute of Technology (Brophy). ARM will help fulfill the goal for the United States to send humans to a near-Earth asteroid in order to demonstrate new technologies and develop the capabilities necessary for travel beyond Low Earth Orbit (LEO) for a future manned mission to Mars. The mission is split into two sub-missions: the Asteroid Redirect Robotic Mission (ARRM) and the Asteroid Redirect Crewed Mission (ARCM).

Instead of sending a crew into deep space to an asteroid, which is very expensive, dangerous, and time-intensive, it is more feasible to capture an asteroid and redirect it to a lunar orbit where it could be more easily explored and studied by astronauts. There are two main mission concepts being considered for the robotic ARRM mission: Plan A, capture a small asteroid of 7-10 meters in diameter, or Plan B, take a 1-10 meter diameter sample from a larger asteroid (Moore). In both cases, the captured asteroid material will be transferred to a stable lunar orbit via a lunar gravity assist to await rendezvous with a crewed Orion capsule during the crewed ARCM.

Two capture options: Plan A (Capturing an entire small asteroid) or Plan B (Collecting a sample from a larger asteroid)
Figure 1. Two robotic capture options: Plan A (Capturing an entire small asteroid) or Plan B (Collecting a sample from a larger asteroid) (NASA).

The Asteroid Redirect Robotic Mission (ARRM) is the second segment of a larger sequence of missions:


The first segment is an observation campaign to identify asteroid candidates and increase the knowledge of near-Earth asteroids.

Figure 2. Asteroid identification observation campaign (NASA)

Asteroid Redirect Robotic Mission (ARRM)

The second segment is the robotic capture of the asteroid or asteroid sample, the transit to the Moon, and the placement of the asteroid into a stable lunar Distant Retrograde Orbit (DRO).

Figure 3. Asteroid Redirect Robotic Mission (NASA)

Asteroid Redirect Crewed Mission (ARCM)

The third segment is a crewed mission for astronauts to rendezvous with the captured asteroid in order to explore and study it.

Figure 4. Asteroid Redirect Crewed Mission (NASA)

The Asteroid Redirect Mission will provide, as a stepping stone for a human mission to Mars, an opportunity for the maiden crewed Orion flight in the early to mid-2020’s to journey to a destination beyond LEO while gaining experience in human spaceflight, in-situ resource utilization (ISRU), and advanced Extra-Vehicular Activity (EVA) and spacesuit design for manipulation of regolith in a microgravity environment. The robotic ARRM mission in itself acts as a technology demonstration for a 40 kW solar electric propulsion system. Future crewed follow-on missions to deep space, the Martian moon Phobos, and the Martian surface have been proposed, all taking advantage of the technologies and capabilities developed through both the robotic and crewed segments of ARM. The Asteroid Redirect Mission is aligned with NASA’s overall vision for exploration and plans to send humans to explore the solar system beyond LEO, specifically Mars (Lightfoot).

Description of Investigation

This investigation will focus on the trajectories for the rendezvous of the robotic Asteroid Redirect Vehicle (ARV) with the target asteroid in the second segment of the mission sequence described above.

This investigation will:

  • propagate heliocentric orbits of each candidate asteroid by converting osculating classical orbital elements obtained from the JPL Small Bodies Database to radius vectors.
  • Determine transfer trajectories from the Earth to the asteroids, and the delta-V requirements for each will be calculated. These will be calculated for eccentric, inclined orbits.
  • Calculate the synodic periods of each asteroid with respect to the Earth and examine the dates of closest approach to determine a timeline for a notional mission to occur.
  • Examine and compare results for each asteroid candidate in order to determine which of the candidates is the optimal choice based on the notional mission timeline and delta-V requirements to reach the asteroid.
This investigation assumes the following:

  • Tangential impulsive maneuvers. (While the ARRM concept utilizes electric propulsion, for simplification this study will assume traditional chemical propulsion.)
  • Orbits are propagated using osculating orbital elements and perturbations are ignored, including 3rd body effects, solar radiation pressure, thermal effects, etc.
  • Asteroids are modeled as point masses and have negligible gravitational effects on the spacecraft.


Asteroid Candidate Orbits

A total of eight candidate asteroids are currently being considered as viable options, with four candidates for Plan A and four candidates for Plan B.

Plan A:

Candidate asteroids for Plan A were evaluated based on the following criteria: A natural approach near Earth in the early 2020's, a total mass between 45,000 - 800,000 kg, a size that can fit within the capture system, and a spin-rate that can be matched by the vehicle during capture. With these criteria in mind, a pool of thousands of NEAs were narrowed down to four candidate targets, as outlined below:

  • 2009 BD
  • 2011 MD
  • 2013 EC20
  • 2008 HU4
The following osculating orbital parameters have been found for each of the candidate target asteroids using the JPL Small Bodies Database, as of 11/25/14. Note that these values have been rounded for presentation purposes.

Orbital Element 2009 BD 2011 MD 2014 EC20 2008 HU4
Semi-major axis (AU) 1.008614 1.060217 1.111168 1.092993
Eccentricity 0.040818 0.041638 0.120007 0.073242
Inclination (deg) 0.3851643 2.5836981 1.3042058 1.2575784
Right Ascension of the Ascending Node (deg) 58.487991 274.092053 165.836142 221.330237
Argument of Perigee (deg) 110.503917 4.679965 33.086030 341.464789
Mean Anomaly (deg) 294.295423 9.105932 154.569864 297.230091
Orbital Period (days) 369.986527 398.740895 427.826854 417.372983

Plan B:

Candidate asteroids for Plan B were evaluated based on the following criteria: A natural approach near Earth in the early 2020's and the existance of boulders of diameter 1-10 m on its surface. With these criteria in mind, a pool of thousands of NEAs were narrowed down to four candidate targets, as outlined below:

  • Itokawa (Hayabusa's target asteroid)
  • Bennu (OSIRIS-REx's target asteroid)
  • 1992 JU3 (Hayabusa 2's target asteroid)
  • 2008 EV5

Again, the osculating orbital parameters below have been found for each of the candidate target asteroids using the JPL Small Bodies Database, as of 11/25/14. Note that these values have been rounded for presentation purposes.

Orbital Element Itokawa Bennu 1992 JU3 2008 EV5
Semi-major axis (AU) 1.324120 1.126391 2.338304 0.958308
Eccentricity 0.280049 0.203745 0.135451 0.083577
Inclination (deg) 1.621460 6.034937 4.825968 7.436835
Right Ascension of the Ascending Node (deg) 69.081436 2.060868 191.744934 93.398923
Argument of Perigee (deg) 162.810332 66.223074 157.048141 234.802826
Mean Anomaly (deg) 333.941822 101.703945 352.969938 242.931396
Orbital Period (days) 556.531011 436.648728 1306.019802 342.654213

The osculating classical orbital elements above were converted to Cartesian elements (radius and velocity vectors in an inertial X, Y, and Z direction) in order to propagate the orbit in time. Code based on Algorithm 9 from Vallado's Fundamentals of Astrodynamics and Applications was used for these calculations.

Hohmann Transfers and Plane Changes

While timing is important for the Asteroid Redirect Mission, the following discussion deals solely with maneuvers that are designed to minimize the required delta-V.

Hohmann transfers consist of a pair of orbital maneuvers that change the size of an orbit from one heliocentric radius to another; in this investigation, the initial orbital radius is that of the Earth, and the second is that of the candidate asteroid. Plane change maneuvers are required to change a spacecraft's inclination with respect to the ecliptic plane when the initial and final orbits have different inclinations.

In order to minimize the required delta-V and most efficiently perform both a Hohmann transfer and a plane change for eccentric orbits, the first maneuver to initiate the Hohmann transfer from the Earth to the candidate asteroid should be performed at Earth's perihelion. The combined plane change maneuver should replace the second Hohmann transfer maneuver, as it combines the Hohmann transfer maneuver with the one to change inclination. This second maneuver should be performed at the asteroid's aphelion.

The equations for the Hohmann transfer, combined plane change, and time of flight are outlined below:

The semi-major transfer axis in kilometers is given by:

The velocity of the Earth at perihelion in kilometers per second is given by:

It is assumed that the spacecraft also has this velocity at the beginning of the Hohmann transfer.

The velocity of the asteroid at aphelion in kilometers per second is given by:

It is assumed that this is the velocity which the spacecraft will match at the conclusion of the Hohmann transfer.

The velocity of spacecraft needed to escape the Earth's sphere of influence and travel on the transfer trajectory in kilometers per second is given by:

The first maneuver's delta-V in kilometers per second is given by:

The second maneuver's delta-V, which is the combined plane change, in kilometers per second is given by:

The sum of the first delta-V and second delta-V results in the total delta-V in kilometers per second:

The time of flight for the transfer trajectory in days is given by:

Synodic Periods

The synodic period describes time between repetitions of a specific orbital geometry of two bodies. The equation below describes the synodic period between the Earth and an asteroid:

where n is the mean motion in radians per second, as described below:


Candidate Orbits

Given the osculating orbital elements described above, the orbits for each asteroid were propagated for an entire orbital period ignoring purturbations. In the diagrams below, the sun is represented by a red star, and the Earth's orbit is shown in green.

Figure 5. Propagated orbits for Plan A candidates compared to the Earth's orbit.

For the Plan A asteroid candidates, the asteroid 2009 BD (shown in blue) has an orbit very similar to the Earth's with little inclination and a similar semi-major axis and eccentricity. The other asteroid candidates all have inclined orbits of varying sizes and eccentricities.

Figure 6. Propagated orbits for Plan B candidates compared to the Earth's orbit.

For the Plan B asteroid candidates, there is much greater variation in the asteroids' orbits. The orbit of Itokawa (shown in blue) is the most similar to that of the Earth, with a relatively small semi-major axis and small inclination. The remaining asteroids' orbits are much larger and are very inclined.

Orbital Trajectories

The tables below describes the semi-major axis of the transfer orbit (kilometers), the magnitude of each delta-V maneuver (kilometers per second), the total delta-V (kilometers per second), and the time of flight (days) for the transfer orbit between the Earth and the candidate asteroid.

Plan A:
2009 BD 2011 MD 2013 EC20 2008 HU4
Semi-major Axis (km) 1.5207e8 1.5615e8 1.6664e8 1.6129e8
Inclination Change (deg) 0.0408 2.5836 1.3042 1.2575
Delta-V 1 (km/s) 0.4872 0.8586 1.7120 1.2936
Delta-V 2 (km/s) 0.9669 2.1450 3.3992 2.6136
Total Delta-V (km/s) 1.4541 3.0036 5.1112 3.9072
Time of Flight (days) 187.1766 194.7642 214.7004 204.4534

For all four transfer trajectories, the first delta-V maneuver was smaller than the second combined plane change maneuver. As expected, the greater inclination changes and the greater transfer trajectory semi-major axes resulted in a greater magnitude second delta-V maneuver.

For the candidates for Plan A, the asteroid for which the Hohmann transfer and combined plane change is most efficient is 2009 BD, with a total delta-V of 1.4541 km/s. This asteroid's orbit is the most similar to Earth's, and the transfer trajectory has the shortest time of flight (187.1766 days).

The least efficient transfer is for asteroid 2013 EC20, with a total delta-V of 5.1112 km/s. This asteroid candidate also has the longest time of flight of 214.7004 days.

Plan B:
Itokawa Bennu 1992 JU3 2008 EV5
Semi-major Axis (km) 2.0032e8 1.7496e8 2.7214e8 1.5122e8
Inclination Change (deg) 1.6214 6.0349 4.8259 7.4368
Delta-V 1 (km/s) 3.7554 2.3036 6.2502 0.4065
Delta-V 2 (km/s) 7.1964 5.3276 11.9216 3.9271
Total Delta-V (km/s) 10.9519 7.6312 18.1718 4.3337
Time of Flight (days) 282.9900 230.9959 448.0885 185.6058

For the Plan B transfer trajectories, the largest delta-V of 18.1718 km/s was required for the asteroid 1992 JU3. This was primarily due to the transfer orbit's very large semi-major axis with a contribution from the 4.8259 degree inclination change. This magnitude of delta-V required is very significant and would require a very large amount of fuel to perform.

The most efficient transfer trajectory was for 2008 EV5 with only 4.3337 km/s required. Even though this asteroid candidate has the greatest orbital inclination of the candidates for Plan B, its transfer semi-major axis is much smaller, resulting in the least delta-V required for both the first and second maneuvers.

Due to the large semi-major axes of these asteroids, visiting the Plan B candidates (with the exception of 2008 EV5) with a simple Hohmann transfer and combined plane change is likely not feasible due to the large delta-V requirements. However, orbital manevuers such as gravity assists or using low-thurst or low-energy trajectories with electric propulsion would significantly decrease the required delta-V. Indeed, the Asteroid Redirect Robotic Mission's baseline design includes the use of a40 kW solar electric propulsion system, rendering the transfers to visit any of these asteroid candidates much more feasible.

The delta-V requirements for the Plan A candidate asteroids as discussed previously are much smaller than those for Plan B. However, the use of solar electric propulsion, gravity assists, or both for the Plan A transfer trajectories would also result in a significant delta-V and fuel savings.

It is important to note that the above calculations are solely for the transfer trajectory to the asteroids, and do not include any delta-V for the characterization and science orbits, the asteroid proximity operations and capture, the de-spinning and de-tumbling of the asteroid if it has a significant spin or nutation rate, or the transfer of the captured asteroid to the lunar distant retrograde orbit. All of these mission phases will likely require significant amounts of fuel and delta-V to complete necessary maneuvers. Thus, it is in the best interest of the mission to minimize the delta-V needed to arrive at the asteroid. As the ARRM is a robotic mission, a longer-duration transfer trajectory should be weighed against a trajectory with minimized delta-V requirements to determine the optimum trajectory to rendezvous with a selected asteroid.

Synodic Periods and Mission Timelines

The synodic periods calculated using the osculating orbital elements for each candidate asteroid are seen in the tables below.

Plan A:
Asteroid Candidate Synodic Period (years)
2009 BD 78.2299
2011 MD 11.9087
2014 EC20 6.8377
2008 HU4 8.0087
Plan B:
Asteroid Candidate Synodic Period (years)
Itokawa 2.9097
Bennu 6.1163
1992 JU3 1.3883
2008 EV5 15.1601

The synodic periods for the candidate asteroids range from as short as 1.3 years to over 78 years. This is important to consider when comparing the asteroids for mission feasibility: For the asteroids with short synodic periods, if schedule delays were to occur the asteroid could be visited again within a couple years, however for those with longer synodic periods, the closest approach would not occur again for decades.

The magnitude of delta-V required to rendezvous with an asteroid and the synodic period between the asteroid and the Earth are related based on the semi-major axis of the asteroid trajectory. The asteroids with transfer orbits with the lowest delta-V requirements have the largest synodic periods, and vice-versa. Thus, the mission designer must weight these two parameters with respect to mission needs in order to determine which is more important and to help guide the selection of the best target asteroid.


Mission timing and delta-V requirements are two very important parameters to consider when deciding which asteroid candidate is the optimal choice to visit and capture for the Asteroid Redirect Mission. In this investigation, the orbits for eight different candidate asteroids, four for Plan A and four for Plan B, were propagated and compared to Earth's orbit. Then, transfer trajectories from the Earth to each asteroid were designed, using a Hohmann transfer and a combined plane change to increase the semi-major axis and inclination of the Asteroid Redirect Robotic Mission (ARRM) spacecraft's orbit in order to rendezvous with each asteroid. Finally, the synodic period of each asteroid was calculated with respect to the Earth to determine how often the closest approaches between the Earth and each asteroid occur.

For Plan A, the asteroid candidate with the smallest required delta-V was 2009 BD, with 1.45 km/s, while the greatest delta-V requirement was for asteroid 2014 EC20 with 5.11 km/s. 2014 EC20 had the smallest synodic period of the group at 6.8 years, and 2009 BD had the greatest at 78.2 years.

For Plan B, the asteroid candidate with the smallest required delta-V was 2008 EV5 with 4.33 km/s, and the largest delta-V requirement was for asteroid 1992 JU3 with 18.17 km/s. 1992 JU3 had the smallest synodic period of the considered asteroid candidates at 1.4 years, while the largest synodic period was for asteroid 2008 EV5 at 15.2 years.

The baseline asteroid which has been analyzed for feasibility and design purposes has been BD 2009, due to the low delta-V's needed to rendezvous with the asteroid, and the similarily of its orbit to Earth's. However, the synodic period of BD 2009 is very large (78 years), and as such the rendezvous timeline is not flexible. If funding or scheduling delays were to occur, there is a liklihood that the launch window to rendezvous with BD 2009 would be missed. The date of closest approach for BD 2009 is June 26, 2023. (Chodas) If the mission timeline dictates that the rendezvous must occur after that date, due to scheduling, funding, or other logistics, an alternate asteroid will likely be chosen.

Other important considerations relating to the selection of the optimal target asteroid include the opportunities to image and study the asteroid prior to selection. Upon on close approaches of asteroids to the Earth prior to the notional launch dates, both space-based and ground-based observatories are helpful for imaging and characterizing asteroids' sizes, shapes, compositions, and spin rates. For example, the Spitzer Space Telescope was able to ovserve the asteroid 2011 MD in February 2014, and 2008 HU4 will have an Earth flyby in April 2016, during which time observations will be made (Muirhead).

In addition to the four asteroids examined in this study, there are five other potential candidates for Plan A, with more being discovered each year. Further imaging and characterization of the candidates will contribute to the decision of which asteroids are the most feasible to capture and which are the most scientifically desireable. (Gates) It is also possible that the asteroid which will be redirected and visited for ARM has not yet been discovered, as new asteroids are being discovered every year.

Future work for this topic includes the optimization of trajectories in order to take advantage of gravity assist opportunities and electric propulsion low-thrust trajectories. Simplifying assumptions were made in light of the scope and avaliable resources for this project, however, in reality numerous perturbations have important effects on orbital trajectories for asteroid rendezvous. The gravitational effects of the Earth, the moon, and those of larger asteroids contribute to third body effects, allowing gravity assists and other special trajectories in the Earth-Moon system to be considered.