IV-1

Round-Trip Missions to Low-Delta-V Asteroids and Implications for Material Retrieval

DAVID F. BENDER, R. SCOTT DUNBAR and DAVID J. ROSS

Low-delta- V asteroids are to be found among those which have perihelia near 1 AU. From the 50 known asteroids with perihelia less than 1.5 AU, 10 candidates for asteroid retrieval missions were selected on the basis of low eccentricity and inclination. To estimate the ranges of orbital elements for which capture in Earth orbit may be feasible, a survey was made of 180^o transfers from a set of orbits having elements near those of the Earth to the Earth. For 2 of the 10 low-delta-V asteroids and for an additional one with elements more Earth-like than any yet known, direct ballistic round trips in the 1980's were computed. A stay time of several months at the asteroid was used. The results show that the total delta V, including that for rendezvous with Earth upon return, for the known asteroids is above 14 km/sec. But if asteroids are found similar to the strawman considered, the total delta V could be as low as 10 km/sec.

To obtain trajectories with lower delta V than the direct ballistic cases, three modifications to the trajectories that use gravity-assisted flybys are considered: a lunar gravity assist at both departure and arrival at Earth, Earth-Venus-Earth flights which can greatly increase or decrease heliocentric energy, and low-thrust Earth-to-Earth transfers to increase or decrease the V at Earth.

Ballistic studies using Earth-Venus-Earth return trajectories were made for 3 of the 10 low-delta-V asteroids. These studies can form the basis for low-thrust return trajectories, but only one low-thrust-case was investigated. Low-thrust mission results are shown for a direct retrieval of the "Earth-like " asteroid and a gravity-assisted retrieval of 1977 HB Although they have the disadvantage of longer mission times, the gravity-assisted trajectories make available for possible return a much wider range of targets, as opposed to the severely restricted class of orbital elements required for direct missions; for example, the 1977 HB return requires a delta V of only 3.04 km/sec using low thrust and gravity assists.

The final phase of the return of an asteroid is capture by the Earth-Moon system, which is accomplished by a lunar gravity assist even though V at Earth arrival is as high as 1.5 km/sec. A sample trajectory showing capture into an orbit near the 2:1 Earth-Moon resonance is presented.

With the discovery of new Earth-approaching asteroids for direct missions and more detailed and exhaustive trajectory studies for gravity-assisted missions, it is felt that asteroid round trips with delta-V values as low as 6 km/sec can be found. This work serves only to demonstrate the feasibility of asteroid return missions and to indicate possible directions for future studies.

INTRODUCTION

Many types of asteroid missions, including flybys (refs. 1, 2), rendezvous (ref. 3), and sample return missions (ref. 4) have been discussed previously. Recently, the idea of retrieving all or part of an Earth-approaching asteroid as raw material for space manufacturing has been proposed. This paper addresses the problems of trajectory dynamics crucial to this scheme and proposes several techniques for conducting such a mission.

Clearly, only those asteroids accessible with the lowest possible delta V to rendezvous will be of interest, at least for the time being. Such low-delta-V asteroids will be those crossing or inside the orbit of Mars, with small eccentricity, low inclination, and a semimajor axis near 1 AU.

In addition to trajectories that travel directly from Earth to an asteroid and return, the greatest possible use will be made of gravity assists. Earth escape and capture will be facilitated by lunar-gravity assists. In the solar system, flybys of Earth and Venus will be used to shape the trajectory, with a consequent reduction in total delta V. Finally, the possibility of low-thrust trajectories between gravitational encounters will be considered briefly. (However, it was not possible to explore this concept in depth here.)

LOW-DELTA-V ASTEROIDS

A low-delta-V asteroid is one for which the total impulse to rendezvous, including Earth launch delta V, is among the lowest known. In this study, the return of a significant part of the asteroid is considered and thus the total impulse to return to Earth, including that for rendezvous with Earth, must also be as low as possible. Generally, the outbound and return thrust requirements should be essentially the same, but to return an asteroid, contrary to expectation, the outbound delta V must be minimized at the expense of the inbound delta V (ref. 5).

The low-delta-V asteroids will be found among those which come near the Earth; such a list is obtained by arranging all the asteroids by perihelion distance (Ql). Orbital elements of the 53 asteroids with perihelion under 1.5 AU are given in table 1.

TABLE 1.- ORBITAL ELEMENTS OF NEAR-EARTH OBJECTS^a

Number	Name	A	Ecc	incl	Q1	Q2
Asteroid
1566	Icarus	1.08	0.827	23.0	0.19	1.97
-	1974MA (L)	1.76	.760	37.7	.42	3.09
-	1936CA Adonis	1.87	.764	1.4	.44	3.30
-	1976UA	.84	.451	5.9	.46	1.22
1864	Daedalus	1.46	6.15	22.1	.56	2.36
1865	Cerberus	1.08	.467	16.1	.58	1.58
-	1937UB Hermes (L)	1.64	.624	6.2	.62	2.66
1981	1937EA	1.78	.650	39.8	.62	2.93
1862	Apollo	1.47	.560	6.4	.65	2.29
-	1977HB	1.08	.346	9.4	.70	1.45
1685	Toro	1.37	.436	9.4	.77	1.96
-	1976AA	.97	.182	18.9	.79	1.14
-	1977HA	1.58	.498	22.8	.80	2.37
-	6743P-L (L)	1.62	.493	7.3	.82	2.42
1620	Geographos	1.24	.335	13.3	.83	1.66
-	1976WA	2.41	.656	24.3	.83	3.99
-	1947XC (L)	2.25	.630	1.0	.83	3.67
-	1959LM (L)	1.34	.379	3.3	.83	1.85
-	1950DA (L)	1.68	.502	12.1	.84	2.53
1866	Sisyphus	1.89	.540	41.1	.87	2.92
-	1973NA (L)	2.39	.633	67.9	.88	3.91
1863	Antinous	2.26	6.06	18.4	.89	3.63
-	1975YA	1.29	.298	64.0	.91	1.67
-	6344P-L (L)	2.58	.635	4.6	.94	4.21
-	196DUA	2.26	.537	3.7	1.05	3.48
1915	QUEtzalcoatl	2.52	.583	20.5	1.05	3.99
1917	Cuyo	2.15	.505	24.0	1.06	3.23
1943	1973EC	1.43	.256	8.7	1.06	1.80
1980	1950LA	1.71	.365	26.8	1.08	2.33
1221	Amor	1.92	.436	11.9	1.08	2.76
-		2.17	.492	5.2	1.10	3.23
1580	Betulia	2.20	.490	52.0	1.12	3.27
1627	Ivar	1.86	.397	8.4	1.12	2.60
-	1972RA	2.36	.523	9.0	1.13	3.60
433	Eros	1.46	.223	10.8	1.16	1.78
887	Alinda	2.52	.544	9.1	1.15	3.88
-	4788P- L(L)	2.55	.545	10.8	1.16	3.93
719	Albert (L)	2.58	.540	10.8	1.19	3.98
1036	Ganymed	2.66	.542	26.3	1.22	4.10
-	1963UA	2.65	.530	11.1	1.24	4.05
1916	1953RA	2.27	.450	12.8	1.25	3.30
1951	Lick	1.39	.062	39.1	1.30	1.48
-	1971SC (L)	2.21	.390	12.0	1.34	3.07
-	1974UB	2.12	.359	36.3	1.36	2.89
1474	Beira	2.73	.490	26.8	1.39	4.07
-	2108P-L (L)	2.32	.385	2.6	1.43	3.21
1134	Kepler	2.68	.467	15.0	1.43	3.93
-	4548P-L (L)	2.17	.340	7.5	1.43	2.91
1009	Sirene	2.63	.454	15.8	1.44	3.82
1139	Atami	1.95	.255	13.1	1.45	2.44
-	1963RH (L)	2.38	.379	21.1	1.48	3.28
-	1975AD	2.37	.375	20.1	1.48	3.26
1198	Atlantis (L)	2.25	.335	2.7	1.49	3.00
Meteoroid
-	40503	2.02	0.642	12.6	0.72	3.31
-	Harvard 19816	2.24	.662	12.6	.76	3.72
-	Pribram	2.42	.674	10.4	.79	4.05
-	Iron Meteor	1.05	.117	13.3	.93	1.17
-	Harvard 7946	2.49	.621	18.1	.94	4.04
-	Lost City	1.66	.417	12.0	.97	2.35
-	40617	2.02	.516	3.3	.98	3.06

^aData supplied by James G. Williams of the Jet Propulsion Laboratory

The designation (L) after the name indicates that the asteroid is lost. The elements listed are semimajor axis (A), eccentricity (ECC), inclination to the ecliptic (INCL) and aphelion distance (Q2). Two of the 53 asteroids - 1977 HA and 1977 HB - were discovered in April 1977. In addition, orbital elements for seven large fireballs (meteoroids) are added to indicate a range of orbital elements for large objects that have intersected the Earth's orbit.

In selecting the low-delta-V asteroids from this list, note that a delta V of 6 km/sec for Earth escape to asteroid rendezvous or for return would allow the spacecraft to reach to 1.8 AU for e = 0.45 and i = 0, or to i = 12^o for a = 1 AU and e = 0. These values, namely a 1.8 AU, e 0.45, and i 12^o, are used as limits in selecting asteroids although by doing so regions are included for which the V would exceed 6 km/sec. Six asteroids are thus selected from table 1: 1976 UA, 1977 HB, Toro, 1959 LM, 1973 EC, and Eros. Another seven lie near but outside this boundary: Adonis, Hermes, Apollo, 6743 PL, Geographos, Amor, and Ivar. If the three lost asteroids are excluded from this set of 13, 10 may be considered as low-delta-V asteroids (see table 2).

TABLE 2.- ORBITAL ELEMENTS OF THE TEN KNOWN LOW-V ASTEROIDS

Asteroid	A	Ecc	Incl	Q1	Q2
1976 UA	0.84	0.451	5.9	0.46	1.22
1977HB	1.08	.346	9.4	.70	1.45
Toro	1.37	.436	9.4	.77	1.96
1973 EC	1.43	.256	8.7	1.06	1.80
Eros	1.46	.223	10.8	1.13	1.78
Adonis	1.87	.764	1.4	.44	3.30
Apollo	1.47	.560	6.4	.65	2.29
Geographos	1.24	.335	13.3	.83	1.66
Amor	1.92	.436	11.9	1.08	2.76
Ivar	1.86	.397	8.4	1.12	2.60

BALLISTIC TRAJECTORIES

To estimate in a general way the delta-V values for return of material from asteroid orbits, we first examined the ballistic equivalent trajectories for a general class of objects with orbital parameters near those of Earth. The range of semimajor axes was from 0.8 to 1.2; of eccentricities, from 0 to 0.4; and of inclination, from 0^o to 10^o. These limits approximate those developed above by setting a bound on the delta V at Earth of 6 km/sec to produce an elliptical orbit or an inclined orbit. This value is higher than can reasonably be provided by available thrust systems operating on an asteroidal mass, but these limits encompass the ranges of orbital elements that will be of interest.

The delta-V values are computed for 180^o ballistic transfer from the asteroid orbit to Earth. For no inclination of the asteroid orbit, the total delta V is known to be optimal. Figure 1 shows delta V at the asteroid as a function of eccentricity and semimajor axis of the asteroid orbit. The contours of constant delta V are marked with the delta V (in kilometers per second) and a symbol to indicate whether the departure is from the aphelion of the asteroid orbit (number within triangle) or from the perihelion (encircled numbers). The delta V at Earth to yield rendezvous is indicated only for 1.5 km/sec, which is taken to be the upper limit of the approach velocity for "free" capture with a lunar gravity assist (dashed lines).

For transfers from inclined orbits to Earth, the first impulse is applied at a node and is assumed to establish a transfer orbit that is coplanar with and tangent to the Earth's orbit 180^o from the node. For simplicity, only orbits with the nodes at perihelion and aphelion are considered. By departing from these nodes (which can be characterized as from perihelion or from aphelion), the same kind of curves can be presented for orbits of any single inclination as in figure 1 for coplanar orbits. Figure 2 and figure 3 show delta V at the asteroid as a function of a and e for inclinations of 2^o and 5^o, respectively. The dashed lines in figure 1 , showing the free capture region, transfer identically to figure 2 and figure 3 since the transfer orbits are the same for each figure.

For any given semimajor axis and eccentricity, one of the two transfers in figure 2 and figure 3 will have less total delta V than the other. It can be shown that for various locations of the periapse around the orbit the optimum transfer will be between these two values.

Of the 10 low-delta-V asteroids from table 2, only 1976 UA lies near the regions plotted in figures 1-3. Its position is indicated in figure 3, but, because it has an inclination of 5.9^o, the delta-V values from the graph are too small. In an effort to present an example of a moderate delta-V requirement, a fictitious asteroid, Ames SSS 77, is hypothesized with the following elements: semimajor axis, 1.08 AU; eccentricity, 0.18; inclination, 5^o; longitude of ascending node, 0^o; argument of perihelion, 45^o; and mean anomaly, 0^o, on 14 November 1979.

The position of this asteroid is also marked in figure 3, which indicates that the delta-V value to depart from the asteroid to Earth can be as low as 2.3 to 3.4 km/sec (if the relative positions of Earth and the asteroid are those required for the transfer). The capture by Earth should ideally require only a small delta V to reduce the approach speed to the 1.5 km/sec assumed as satisfactory for lunar-assisted capture.

Suppose that delta V at the asteroid is limited to 3 km/sec and the Earth approach speed is limited to the "free" capture speed of 1.5 km/sec. Then, from figure 1, a zigzag fine across the figure as shown by the arrows shows which regions of a and e are accessible. In figure 2, the corresponding fine is slightly modified from that in figure 1, but for 5^o inclination (fig. 3), the available ranges for a and e are reduced significantly.

POSSIBILITIES FOR GRAVITY-ASSISTED ASTEROID MISSIONS

Consider a spacecraft moving in the gravitational field of a central body and assume that a moderately massive secondary body is also moving in the field. The spacecraft can be directed to approach the secondary body closely, causing the spacecraft to change its direction and magnitude of velocity in the field of the central body. It is assumed to occur essentially instantaneously at the position of the secondary body. A gravity-assisted flyby uses this change in spacecraft velocity to enhance the mission under consideration.

For asteroid sample return missions, and particularly for asteroid retrieval, a gravity assist by the Moon in the Earth's gravity field would lower the delta-V requirement significantly. It has been shown that capture by the Earth-Moon system can occur for a hyperbolic approach velocity to the Earth as high as 1.85 km/sec (ref. 6). Similarly, an escape velocity of 1.85 km/sec can be achieved on a parabolic trajectory in the Earth's field by a lunar flyby on the outbound leg. These cases require a grazing flyby of the Moon. To provide a safe margin for guidance, V = 1.5 km/sec is used at Earth approach or departure. Multiple flybys of the Earth-Moon system with a lunar gravity assist at each could be used to capture at a higher velocity (ref. 6), but such trajectories considerably lengthen total mission time (1 yr per pass). For two flybys, the upper limit is 2.58 km/sec. An example of a single encounter capture is examined in a section (Lunar Gravity-Assisted Capture).

Multiple flybys of a single secondary body cannot be used to change the magnitude of the hyperbolic approach speed to that body (if its orbit is circular), assuming that no delta V is applied along the trajectory between encounters. But if the trajectory is modified by a gravity assist at a different body or by impulsive or low-thrust delta V between encounters, dramatic changes can be made. Bender and Friedlander (ref. 2) have shown that, by means of a Venus-Earth gravity-assisted trajectory, almost any asteroid in the main belt can be approached on a ballistic trajectory with lower than direct ballistic energy expenditure. In that study launch velocities of 3 to 5 km/sec from Earth were considered. The Venus flyby modifies the orbit so that the subsequent Earth flyby is at a higher relative speed. The Earth return speeds used to reach the asteroid belt and beyond were in the range of 8 to 14 km/sec. Moreover, trajectories were found for dates of the Earth return flyby over a large fraction of any synodic period of Earth and Venus.

Presumably, if any asteroid can be reached by an Earth-Venus-Earth trajectory, return trips are also possible with arrival conditions at Earth similar to the low-energy launches considered by Bender and Friedlander (ref. 2). Additionally, for an asteroid to be retrievable by means of Earth and Venus gravity assists, the asteroid must approach Earth or Venus so closely that only a small delta V is required for the first encounter.

A technique little studied to date is to shape a trajectory by low thrust between gravitational maneuvers. This concept is applied here to reduce an Earth flyby speed from 3-5 km/sec at return from Venus to 1.5 km/sec as needed for the final capture. Table 3 shows some test results obtained when the flyby speed was reduced by low-thrust propulsion between two Earth encounters. Equivalent delta V is the total delta V supplied by the low-thrust system. Low-thrust propulsion effectively reduces the relative speed at Earth arrival. Capture by the EarthMoon system can be managed in 410 days or less at approach speeds from 3 to 5 km/sec with a low-thrust delta V that provides from 1 to 2.7 km/sec.

TABLE 3.- EARTH-TO-EARTH FOR LOWERING V; FINAL V= 1.5 km/sec

Initial V	2	3	4	5
Equivalent V, km/sec	0.46	0.96	2.13	2.69
Time, days	370	390	390	410

If this capture process is combined with Venus-Earth gravity assists, a great range of asteroid orbits is opened for asteroid retrieval. The low-thrust system is required mainly to provide for the first Earth or Venus encounter and the capture phase. The only requirement is that the total velocity change required to produce a flyby of Earth (or Venus) be small. The known candidates are listed in table 2. A Venus encounter requires that the perihelion distance be less than 0.72 AU; a crossing of the Venus orbit plane at Venus's distance from the Sun is unlikely. An Earth encounter similarly requires that the perihelion distance be less than 1.0 AU. But crossing the Earth orbit plane near 1 AU is very likely since this is essentially the condition required for discovering it. Results of testing a few asteroids from table 1 indicate that a delta V of about 500 m/sec represents the lower range of values for achieving the first Earth encounter. The total delta V for the return phase of an asteroid retrieval will be from 2-4 km/sec, possibly sometimes as low as 1.5 km/sec.

If the smaller fraction of the total delta V is needed at the start of the retrieval process, it and the navigational delta-V values required to control the flybys might be supplied by a relatively modest mass driver. The final larger delta V, upon Earth return from Venus, could be supplied by a larger mass driver sent to rendezvous with the asteroid in its nearly final and much lower delta-V orbit, provided sufficient time remains to accumulate the required delta V.

BALLISTIC RESULTS

Direct Retrievals

Ballistic mission studies for asteroid sample return were performed for several asteroids, including some of interest for possible asteroid retrievals. Although there is little possibility of ballistic asteroid retrieval, ballistic studies do show the relative costs of reaching and bringing back various asteroids. The delta-V values found will not differ greatly from the low-thrust delta-V values.

Actual ballistic round trips to two asteroids 1977 HB and 1974 EC are shown in table 4. Additional data are shown for the hypothetical asteroid Ames SSS 77 (table 5) in a later opportunity, which is representative of those that might be discovered. The fact that the real departures and arrivals with a considerable stay time at the asteroid have been required means that delta V will be considerably larger than for the ideal two impulse transfer between the orbits.

Table 4 is divided into three parts: dates and time intervals, impulses needed, and orbit elements.

TABLE 4.- BALLASTIC ROUND TRIPS

Dates and Times	1977HB	1943-1973 EC	Ames SSS 77
Launch	Oct. 18,1985	May 26,1992	March 13, 1982
TF, days	242	453	221
Arrival	June 17, 1996	Aug. 20, 1993	Oct. 20, 1983
Stay time, days	202	151	394
Depart	Jan.5, 1987	Jan. 19, 1994	Nov. 18, 1983
TF, days	192	465	238
Return	July 16, 1987	Apr. 21, 1995	July 12, 1984
Total time, days	626	1069	853
Impulses needed Launch V, km/sec	7.89	5.87	3.01
Rendezvous V, km/sec	1.57	.98	1.62
Depart V, km/sec	6.78	2.57	3.76
Earth V(=V appr.) km/sec	2.83	4.62	1.68
Total V, km/sec	19.07	13.95	10.07
Launch DLA, deg	24.8	41.8	-27.2
Orbit elements Semimajor axis, AU	1.08	1.43	1.08
Eccentricity	.44	.26	.18
Inclination, deg	9.4	8.7	5
Longitude of node, deg	32.8	346.0	0
ARG of perihelion, deg	54.9	11.0	45

TABLE 5.- RETRIEVAL OF AMES SSS 77

	Equiv.V, km/sec	Mass fraction delivered
Launch Feb. 21, 1987 (VAL = 1.5km/sec) 340 days	-	-
Arrival Jan, 27, 1988 Stay time in asteroid orbit: 246 days	4.72	0.389
Departure Oct. 1, 1988 900 days	-	-
Earth arrival Ma. 19, 1991 (V = 1.5 km/sec)	3.61	.406

The impulse to remove V at Earth is included in the total delta V because the asteroid must be captured. All legs were optimized on total delta V for rendezvous. Ecliptic plane projections of the orbits of SSS 77 and 1977 HB are illustrated in figures 4 and 5 drawn to the same scale so that differences are easily observed. Ames SSS 77 is significantly more accessible than 1977 HB. Note- that in table 4 the use of lunar gravity assist on Earth departure and arrival is not included. If this "free" delta V were 1.5 km/sec, the total delta V would be reduced by 3 km/sec. Note also that the total impulse required on the retrieval phase for Ames SSS 77 is 3.76 + 1.68 - 1.5 or 3.94 km/se, which lies between 3 to 4 km/sec, postulated previously for the minimum total delta V to retrieve an asteroid by direct flight to Earth.

Earth-Venus-Earth Gravity-Assisted
(EVEGA)Retrievals

As mentioned in the preceding section, one method for reducing overall delta V during retrieval is to use gravity assists by Earth and Venus. In this technique, the orbital inclination and the high excess hyperbolic velocity at the first Earth encounter can be decreased to values that are amenable to a relatively easy capture maneuver into high Earth orbit. The cost is the flight time to Venus and back to Earth in one to three revolutions. A basic technique of searching for mission opportunities of this kind has been established but, as expected, a high velocity at the first Earth approach cannot be reduced to low values in two or three revolutions for every date. In the data below, no more than three revolutions were considered.

The scheme of using an EVEGA trajectory to retrieve an asteroid is as follows: (1) the asteroid is driven from its orbit by some sort of propulsion system (high or low thrust) so as to intercept the Earth, (2) the encounter is controlled so that the spacecraft flies on to Venus (ideally, with no intermediate thrusting during the flyby), and (3) it then makes a gravity-assisted flyby of Venus, sending it back to Earth for rendezvous. In searching for mission Opportunities of this type, each separate leg was analyzed ballistically in hopes of finding a suitable set of launch/arrival dates so that the hyperbolic excess velocities, V_in and V_out at each boundary point would be matched as closely as possible and the velocity at the second Earth approach would be minimized (i.e., from 3-5 km/sec). Given the rough boundary dates derived from this survey, the entire case could be run at once and optimized on total delta V from asteroid to Earth to Venus to Earth. The results of four such cases are given in table 6.

These results indicate that the Earth-Venus-Earth gravity-assist technique will contribute significantly to the retrieval of asteroid material. The flyby velocity at the first Earth encounter on the return trip is sometimes too fast and sometimes too slow to match the "best" value for the gravity-assist geometry that occurs. It is therefore conceivable that, when more asteroids are found, it will be possible to obtain cases for which there is a closer match of V_in and V_out at Earth. The earth return velocities are sometimes too high, but it is believed that, if only cases for which this velocity is under 5 km/sec are considered, there will still be many opportunities. These are ballistic results and the addition of low-thrust techniques to those of gravity assist will result in successful asteroid retrieval trajectories in almost every case. One such case developed from the 1977 HB results is presented in table 7.

LOW-THRUST RESULTS

It has been shown that the total delta V for retrieving an asteroid is likely to range from 2 to 4 km/sec. To estimate the capability of a mass driver in supplying such a delta V, suppose that delta V is supplied at the rate of 1 km/sec/yr to a 200-m-diameter asteroid with a mass of 10¹⁰ kg (density = 2.39 g/cm³). The acceleration is 31.7XI0^-6 m/sec² and, if the mass is expelled at 5000 m/sec, the power required is 792 MW and the mass flow rate is 63.4 kg/sec. These are strong requirements on the mass-driver system and it may not be possible to meet them. If not, the size of the asteroid would have to be reduced or the flight time increased so that the total delta V would be supplied at a slower initial rate.

Low-thrust techniques can be used to improve the ballistic retrieval trajectories in table 6. A low-thrust trajectory can be used from the asteroid to first Earth encounter in such a way as to supply the proper magnitude of V at Earth. For 1977 HB, this technique was combined with an additional revolution about the Sun, that is, by starting one revolution earlier from the asteroid orbit. The Earth flyby encounter and velocity were obtained at a total delta V of 2.04 km/sec supplied in 653 days. This slightly exceeds the acceleration value of 1 km/sec/yr adopted as a nominal upper limit. With a low-thrust Earth-to-Earth trajectory at a delta V of 1.00 km/sec to reduce V at capture from 3.04 to 1.5 km/sec (as in table 3), the return of 1977 HB is possible with a total delta V of 3.04 km/sec. As shown in table 7., the return begins from the asteroid orbit on 1 July 1985, with a 653-day low-thrust transfer to Earth; "free" flybys of Earth, Venus, and Earth in succession; and, finally, a 390-day low-thrust trajectory before final Earth encounter on 23 September 1989.

TABLE 6.- ILLUSTRATIVE EXAMPLES OF EVEGA RETRIEVAL TRAJECTORIES

Asteroid	1977HB	1976UA	1976UA	1973EC
Departure date at asteroid	Oct. 1, 1986	March 19, 1983	Feb. 8, 1990	Feb. 14, 1994
V at asteroid	1.06 km/sec	0.61 km/sec	1.18 km/sec	1.43 km/sec
Earth flyby	Apr. 15, 1987	Oct. 18, 1983	Oct. 17, 1990	May 24, 1995
Vin	12.29 km/sec	12.62 km/sec	11.90 km/sec	7.01
Vout	9.24 km/sec	11.46 km/sec	8.36 km/sec	8.95
Rad	3.14 ER	1.66 ER	3.10 ER	1.54
Venus flyby	Mar. 10, 1988	Oct. 6, 1984	July 28, 1991	Mar. 17, 1996
Vin	5.47 km/sec	7.43 km/sec	5.00	5.38
Vout	5.47 km/sec	7.43 km/sec	5.00	5.38
Rad	1.61 VR	2.92 VR	6.35	3.06
Earth arrival	Aug. 29, 1988	Mar. 27, 1986	Feb. 9, 1992	Aug. 31, 1996
Vout	3.04 km/sec	6.72 km/sec	5.24 km/sec	3.36 km/sec
Ballistic total Va to rendezvous	6.721 km/sec	8.276 km/sec	9.411 km/sec	7.432 km/sec
Total time	698 days	1104 days	731 days	929 days

^a V at Earth used at periapse, and is less than [V_in - V_out]

TABLE 7.- RETRIEVAL OF 1977 HB

Earth launch	Apr. 28 1984 214 days	(6.32 km/sec)
1977 HB (arr)	Nov. 28, 1984 216 days	Imp 2.53 km/sec
1977 HB (dep)	July 1, 1985 653 days	LT 2.04 km/sec
Earth (F.B.)	Apr. 15, 1987 330 days	(9.24 km/sec, 2.78 ER)
Venus (F.B.)	Mar. 10, 1988	(5.47 km/sec, 1.61 VR)
Earth (F.B.)	Aug. 29, 1988 390 days	(3.09 km/sec. 3.09 ER) LT 1.00 km/sec
Earth (ret)	Sept. 23, 1989	(1.5 km/sec for capture)
Total time	1,974 days 5.41 years	Vout 7.41 km/sec Vret 3.04 km/sec

Only navigational delta-V values are needed during flyby. However, the low thrust could possibly be applied during the Earth-Venus-Earth portion to further reduce the total delta-V requirement for capture. (No analysis of this variation in the technique was attempted here.)

Finally, searches were made for suitable gravity-assisted opportunities for an outbound trajectory to 1977 HB. Unfortunately, no satisfactory gravity-assisted trajectories could be found using less than three revolutions about the Sun with significantly less delta V than that of the optimum direct impulsive case. Consequently, the retrieval scenario in table 7 for 1977 HB contains the impulsive data for the Earth-to-asteroid portion. The total delta V given in table 7 reflects the assumption that V = 1.5 km/sec represents the launch or capture assistance provided by a lunar gravity assist.

It is realized that the total outbound delta V to asteroid rendezvous is more important than the return delta V in terms of providing the maximum return per unit cost (ref. 5). Unfortunately, the 1977 HB case was developed naively on the basis of minimizing the return delta V. The present outbound results remove it from serious consideration. None of the other cases in table 6 were investigated, but the outbound results for 1977 HB were instructive and indicate that satisfactory cases should be found - especially when more Earth-approaching asteroids are known.

LUNAR-GRAVITY-ASSISTED CAPTURE

As mentioned previously, it is reasonably safe to use a lunar-gravity-assisted maneuver to capture an Earth-approaching body with a V relative to the Earth of less than about 1.5 km/sec. At 1.85 km/sec, the capture orbit is almost parabolic and there is no freedom in choosing the radius of perigee. In considering the capture problem ballistically, it is desirable that the radius of perigee of the capture orbit equal the radius of perigee of the final desired orbit so that injection into the desired orbit can be accomplished in one maneuver. Because of its stability and favorable gravitational characteristics, the orbit in 2:1 resonance with the Moon, described by Heppenheimer (ref. 7), was chosen for this example (fig. 6). For a sufficient margin of safety and one that ensures capture of the approaching body into Earth orbit, a V = X lunar orbital velocity (= 1.447 km/sec) is considered. The body passes 184 km above the lunar surface, which is considered an adequate margin of safety, and has its orbit relative to the Moon bent 89.5^o and its Earth-relative velocity reduced from 2.046 to 1.316 km/sec. It is injected into an orbit having a radius of perigee of 0.3952 X lunar distance.

After the lunar-gravity-assisted flyby, an impulsive velocity change of 292.3 m/sec at the perigee is sufficient for injection into the 2:1 orbit. In practice, since the stability of the 2:1 orbit depends on the avoidance of close lunar approaches by having the body at apogee when the Moon is 90^o away, injection into the 2:1 orbit would be a two-stage process. First, a near 2:1 orbit would be achieved and allowed to coast until a suitable relationship with the Moon had been established and a second velocity change performed to lock the orbit into resonance. For a low-thrust mission, the capture orbit would be gradually changed over several periods to accomplish the same thing with a considerably higher equivalent delta V.

This simple scenario ignores two interesting possibilities. The first is the use of continuous thrust after lunar encounter to further retard the body. A body with incoming V = 2.25 km/sec injected into a hyperbola with perigee at 6 Earth radii may be captured with a continuous thrust under 5 microgee (5XI0^-5 m/sec ²) applied during the time the body is within the lunar orbit. Thus bodies with approach velocities considerably higher than the nominal 1.5 km/sec could be captured with minimal effort.

The second interesting consideration ignored by this analysis concerns multiple lunar encounters with a low-thrust interplanetary trajectory between the Earth approaches. Preliminary investigation indicates this may be very promising, but more thorough analysis is not possible at this time.

CONCLUSIONS

It has been shown that direct round-trip trajectories for asteroid retrieval missions may be possible with a total delta V of the order of 7 km/sec, assuming that lunar-gravity-assisted flybys are used both inbound and outbound. This would require a significant increase in the number of known Earth-approaching asteroids such that several with more favorable orbital elements for this type of mission are found. The total delta V for the real asteroid 1943 is about 11 km/sec.

On the other hand, gravity-assisted trajectories to rendezvous with an asteroid and to return it to Earth may have total delta-V values as low as 6 km/sec. This requires fortuitous timing and geometry so that good Earth-Venus-Earth trajectories are available for both outbound and inbound flights. However, they do not require as limited a class of orbital elements for suitable targets as are needed for direct retrieval. In fact, several candidates are known, and every asteroid discovered by virtue of a close approach to Earth becomes yet another candidate. A return trajectory using low thrust for the asteroid 1977 HB was found with a delta-V requirement of only 3.04 km/sec. Although no satisfactory outbound flight arriving at this asteroid less than 1 year before the return flight was found, it is confidently expected that, as asteroids are studied and more are discovered, cases will be found that do indeed provide retrieval for a total (round-trip) delta V of 5 km/sec.

Another technique involves the use of Earth-to-Earth low-thrust trajectories. These are used to increase the V of a spacecraft at launch from 1.5 km/sec to 3-5 km/sec (or to decrease V. for Earth approach from 3-5 km/sec to 1.5 km/sec).

Finally, the problem of capture into long-term stable orbits about the Earth was examined. A satisfactory procedure for entering resonant orbits was developed and data presented for entering the 2:1 resonant orbits for V = 1.5 km/sec at Earth approach. It appears from these studies that a significant amount of asteroidal material can be retrieved by techniques well within the grasp of present technology. More work is needed on the discovery of asteroids, the search for good mission opportunities, and possibly the development of other useful techniques to return an asteroid which will be technically and economically feasible. In addition, precursor missions to establish asteroid composition and structure are essential.

References

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