Journey Out

Starting from low Earth orbit (LEO), the journey out phase of the mission starts when a crew fires their engines to climb to the very fringe of Earth's gravity well.  The maneuver is timed so that the spacecraft's position intersects the asteroid's position at some point in the future.  Like trans-lunar injection burns of past and future missions, this maneuver places the spacecraft on a trajectory taking it away from Earth on a path that will follow one of the classical conic sections:  an ellipse, a parabola, or a hyperbola.  The path chosen will impact the amount of time spent in transit as well as the amount of propellants required for the entire mission.

The journey out orbit that requires the least amount of propellant is elliptical in shape.  This orbit is labeled 'C' in figure 1.  To minimize propellant usage on the first burn, the ellipse is aligned so that the highest point of the ellipse is the place where the spacecraft and asteroid meet, allowing some distance for safety.  The problem with such an orbital choice is that ellipses with maximum altitudes of multiple lunar radii have extremely long periods, that is, the time to complete a full trip around the Earth.  This relationship is not linear, either.  An ellipse that reaches twice the distance from the Earth to the moon has a journey-out time of 14 days, roughly equivalent to one-half of the moon's journey around the Earth.  Doubling the distance to four times the Earth-moon radius creates a one-way journey of 38 days.

Burning more propellants during the departure burn in LEO can change the shape of the departure orbit and decrease the duration of the outward journey.  Adding energy to the transfer orbit in this way will make it take the shape of a larger ellipse, a theoretical parabola ('B' in figure 1), and eventually a hyperbola ('A' in figure 1), and any of these orbits can be phased to allow intercepting an asteroid.  Enlarging the transfer orbit will decrease the travel time to the target, which will have the benefits of requiring less supplies for the crew and decreasing the loss of high-energy propellants due to boiloff, but there is a cost.  By increasing the Earth departure speed, mission planners also have to increase the propellants required to maneuver the craft into phase 2 of the mission, proximity operations.

Proximity Operations

After drifting for the necessary period of time, the spacecraft will have to fire its engines again to align its velocity with the inbound asteroid.  As mentioned before, the size of this maneuver will vary depending on the velocity of the craft relative to the asteroid at rendezvous time.  The smallest delta-v required would take place if an elliptical orbit were chosen for the departure trajectory.  Then, at maneuver time, the craft would be practically motionless relative to Earth and the burn would only have to approximate the velocity difference between the Earth and the target asteroid.  If a faster route to the rendezvous were chosen, the burn would have to combine nulling the spacecraft's motion relative to Earth and accelerating the craft's motion in the direction of the asteroid.

After firing the engines to enter proximity operations, the explorers will be ready for the first footfall on new real estate since 1969.  Many methods of carrying out the exploration of an asteroid have been suggested, and those methods are not the subject of this paper.  As the astronauts explore potentially some of the oldest material in the solar system, they, their craft and the asteroid will drift through the fringes of Earth's gravitational influence.  The amount of time spent there will depend largely on the asteroid target and its orbit relative to Earth.  Asteroids that move slower relative to Earth will stay in its vicinity longer and allow more time in the proximity phase.  At the end of the exploration period, it will be time to fire up the engines and return home.

Return to Earth

When the engines shut down, the craft starts its fall to Earth.  The amount of time spent in this mode will likely be similar to the amount of time spent traveling to the asteroid, but some adjustments to the flight path could take place and speed up the trip.  The mission ends with a high-speed reentry, testing a useful piece of hardware for future missions to Mars.

Orbit Mechanics

The mission sounds easy, right?  Theoretically, it is easy.  Finding a candidate asteroid that allows an easy mission is not.  Asteroids, like the planets or any other object traveling through space follow the laws of orbital mechanics in their paths, and those laws complicate mission planning.

Six terms are necessary to define an orbit and an object's place within it.  When describing an orbit using classic Keplerian elements, there are three major terms that affect the shape of the orbit:  inclination, eccentricity, and semi-major axis.  Another consideration is the phasing of the Earth and the asteroid in question.

Inclination

Inclination describes the angle that the orbital plane makes in comparison to another plane and is typically measured in degrees.  The Earth's equator serves as the reference plane for objects orbiting our home world, and as a satellite crosses the equator, the angle between the spacecraft's velocity vector and the equator defines the orbital inclination.  Samples of inclinations for Earth-orbiting satellites include 97 degrees for weather satellites, 52 degrees for the International Space Station (ISS), 28 degrees for the Hubble Space Telescope, and very near 0 degrees for communications satellites.

For planetary bodies, inclination is defined as the angle between the orbit in question and Earth's orbital plane.  Earth's orbital plane is also called the ecliptic.  Asteroids that pass near Earth are inclined to its orbital plane by some amount, and that amount varies greatly.  Objects in the initial data gathered for this survey had inclination values between .1 and 63 degrees.  Figure 2 shows the inclination of a sun-orbiting object and how it is measured.

While an orbit's inclination is not related to the size of that orbit, inclination plays a large role in suitability for a mission profile described here.  An asteroid in an orbit with any measurable inclination compared to the ecliptic will only be capable of a truly close approach to Earth when it crosses the ecliptic plane, a point also known as the nodal crossing.  Even on these close approaches, the differential speed of the asteroid compared to Earth is approximately 500 meters per second for each degree of inclination for the asteroid's orbit.  In an extreme case of traveling to an asteroid of 7 degrees inclination, the spacecraft would have to add velocity equal to half the low Earth orbit speed of 3.5 kilometers per second!

There may be some very special cases where an asteroid with a very low inclination makes a close approach to Earth at a time other than the nodal crossing.  In such a case, the asteroid would actually be above or below the Earth relative to the ecliptic plane, but would not require a large delta-v in order to begin or end proximity operations.

Eccentricity

Eccentricity describes the shape of an orbit, and it is a dimensionless quantity.  At the theoretical yet never achieved eccentricity of zero, an object is in a perfectly circular orbit around its parent body.  Increasing eccentricity describes a more elliptical orbit, with the near point of the orbit growing closer to the parent body and the far point growing more distant, up through an eccentricity value of 1.  An eccentricity of 1, another theoretical value, describes an infinite ellipse also called a parabola.  Eccentricities greater than 1 are reserved for objects following a hyperbolic course.  They are either leaving the parent body for the first time or simply passing through the gravitational influence of the parent body.

Semi-Major Axis

While eccentricity specifies the shape of an orbit, its semi-major axis, expressed in units of length, relates to the amount of time it takes for an object to orbit its parent.  The orbits of two objects with the same semi-major axis but different eccentricities can look very different, but take the same amount of time to make one circuit around a parent body.  Objects with a semi-major axis much smaller or larger than that of Earth can cross Earth's orbit, but doing so requires a relatively high eccentricity and these objects rapidly fall out of consideration for low delta-v missions.

Phasing

Phasing is not an orbital parameter per se, but it requires mention here.  Any asteroid that makes a close approach to Earth will, in all likelihood, make another pass at some time in the future.  The closer the orbital period is to Earth's, the more time between close approaches there will be.  The same effect can be seen in the launch windows that allow missions to other planets.  The outer planets, having orbital periods much greater than Earth's, regularly align for a minimum-energy launch window approximately once a year.  Mars, however, with an orbital period much closer to that of Earth, aligns for a mission only once every 26 months.

Near Earth asteroids, with periods even closer to Earth, can go years between close approaches that would allow the kind of missions discussed here.

Criteria for a Candidate Asteroid

A candidate asteroid must have orbital elements that when considered together allow a mission with relatively low delta-v.  For this study, initial numbers used for candidate NEOs were an inclination of less than 1 degree, eccentricity of less than .1, and a semi-major axis greater than .9 that of Earth but less than 1.1.  Further investigation showed that while this criteria helped narrow the list of potential candidates, it was not sufficient to produce a good list of targets, although it did narrow down the list of 1000+ to less than 300 candidates.

Taking the interplay between orbital mechanics into account, such as seeing the dependency of eccentricity and semi-major axis, cut the number of candidates appreciably.

Methodology

The research for this article is easy to duplicate for anyone interested.  The list of near earth asteroids and their orbital elements (a potentially large web page) was downloaded in January of 2007 and saved as a text file.  The list was converted into spreadsheet format, and only those asteroids with semi-major axes between the arbitrarily chosen .9 and 1.1 AU were used for further study.  This smaller list (still containing 300+ pieces of space rock!) was then sorted by inclination followed by eccentricity, based on an initial assumption that inclination would be a larger delta-v cost than inclination for a mission to that asteroid.

The top candidates from the list were then viewed using the orbital viewer from the JPL NEO office.  The viewer contains a disclaimer that it is for visualization only, and experience shows that this warning should be taken seriously.  One asteroid(2007 CA19), rated 1 on the Torino Scale on February 17th for an impact in 2012 was nowhere near Earth throughout the year according to the visualizer.  Since this study is related to how often such opportunities arise, and not dedicated to finding a mission date with any precision, the orbit visualizer was considered adequate.

Each candidate asteroid was observed in the viewer to find the closest approach that was less than 0.02 AU date between now and 2100 generating views such as those shown in figure 5.  The date and distance were recorded, and this information could be used as a starting point for further analysis. As research using the tool progressed, the time that the asteroid was near Earth was recorded, and used to provide an extremely rough estimate of the delta-v maneuver required for a spacecraft on a mission to explore the space rock using the profile described here.  Again, due to the limitations of the tool, the estimates must be taken as being very preliminary, but they do point out the wild differences in delta-vs required for some asteroids compared to others.  In one comparison, the delta-vs required to rendezvous with two different asteroids differed by a factor of nearly 5x.

[Update on 11 Apr 07:  There is a web page which will do most of this work for anyone interested.  It is also located on the JPL NEO site.  It can be found here.]

[Update on 26 Jan 08:  According to this site (scroll to the bottom of the list), 1991 VG may be human-made: a returning piece of space junk.  That would impact the target's interested as an asteroid to visit. There is similar speculation over SG344.]

Early Frontrunners

Based on this analysis, the following asteroids represent some of the best candidates for a survey mission until the year 2100.  They are listed in order of increasing delta-v required to rendezvous with them.  Again, a reminder that these are representative, because the visualizer used to carry out the analysis is not meant for precise orbit determination.  In fact, the best candidate based on a combination of encounter date and delta-v, 2000 SG344, is listed here as having a close encounter in 2028, while the JPL NEO website doesn't place SG344 as having a potential impact Earth (the criteria for describing a potential impact are not explained on the web page) until 2068.

A note on the repeats described in this table:  Future encounters with the asteroid do take place, as the periodicity between the Earth and the asteroid align.  In all cases examined in this study, the future encounters are not as close as the ones featured in this table.

Hopefully, new discoveries will provide a larger selection of asteroids and mission dates that require less energy to visit using the mission profile described here.  It is also possible that other candidates make themselves obvious using more exact orbital determination methods.  This will remain unknown until someone doing research within the area using better tools makes their study public.

The New Scientist article, published after submitting the final version of the TSR summary of this article, mentions an asteroid named 1998 KY26.  According to the New Scientist article, this asteroid makes close approaches with Earth in 2013 and 2024.  1998 KY26 was not considered in this work, due to its semi-major axis being 1.22, eccentricity of 0.2, and inclination of 1.4. A quick check of the orbit using the visualizer does not show approaches close enough for consideration under the criteria used here.  The 2013 close approach distance is 0.124AU (60 times lunar distance) and the 2024 encounter comes as close as .079AU (40 times lunar distance).  These distances would not correspond to the 90-day mission length described in the article without propellant expenditures to accelerate the outward bound and return leg of the mission, though the delta-v (estimated using the same procedure used for other delta-vs cited in this article) required for proximity operations appear favorable, on the order of 300 m/sec.  The New Scientist article states that 1998 KY26 may not be a good candidate for exploration due to its high rotation rate, a factor not considered for this research.

It is unclear whether the article's mention of KY26 as a candidate is due to better propogation methods on the researcher's part (KY26 is not listed on the NEO risk page, but neither are most of the asteroids considered in this paper) or through setting different criteria describing a close approach and not considering the delta-v cost of such a mission.

Conclusion

Asteroid missions are exciting for their daring, their potential for scientific return, their ability to help protect the planet, and their meaning in humankind's growth into a spacefaring species.  Opportunities to carry them out while keeping people within Earth's "neighborhood" are not common, however, and many of those instances require a lot of propellants in order to make the mission happen.  This is not necessarily a bad thing.  The fact remains that it requires a lot of propellant to get anywhere interesting in the solar system, and perhaps an asteroid mission will help kick-start architectures that will take us to those other destinations.