Accessibility of Main-belt Asteroids and Trajectory Design for Multi-target Exploration

ELSEVIER Chinese Astronomy and Astrophysics 35 (2011) 71–81

CHINESEASTRONOMYAND ASTROPHYSICS

Accessibility of Main-belt Asteroids andTrajectory Design for Multi-target

Exploration† �

XIA Yan1,2� LUO Yong-jie1,2 ZHAO Hai-bin1,3 LI Guang-yu1

1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 2100082Graduate University of Chinese Academy of Sciences, Beijing 100049

3National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012

Abstract The mission designed to explore asteroids has nowadays become a hotspot of deep space exploration, and the accessibility of the explored objects is themost important problem to make clear. The number of asteroids is large, and itneeds an enormous quantity of calculations to evaluate the accessibility for allasteroids. In this paper, based on the direct transfer strategy, we have calculatedthe accessibility for the different regions of the solar system and compared itwith the distribution of asteroids. It is found that most main-belt asteroids areaccessible by the direct transfer orbit with the launch energy of C3 = 50km2/s2,and that with an additional small velocity correction, the designed trajectory isable to realize the multi-target flyby mission. Such a kind of multi-target flybycan reach the same effect of the orbit manoeuvre in the ΔV-EGA trajectoryscheme[1,2]. Being assisted by the earth’s gravity, the accompanying flight withasteroids or the exploration of more distant asteroids can be realized with a lowerenergy. In the end, as an example, a trajectory scheme is given, in which theprobe flies by multiple main-belt asteroids at first, then with the assistance ofthe earth’s gravity, it makes the accompanying flight to a more distant asteroid.

Key words: celestial mechanics—asteroids: general

† Supported by National Natural Science FoundationReceived 2009–09–11; revised version 2009–12–07

� A translation of Acta Astron. Sin. Vol. 51, No. 2, pp. 163–172, 2010� [email protected]

0275-1062/01/$-see front matter c© 2011 Elsevier Science B. V. All rights reserved.PII:

0275-1062/11/$-see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.chinastron.2011.01.009

Chinese Astronomy and Astrophysics 35 (2011) 71–81

72 XIA Yan et al. / Chinese Astronomy and Astrophysics 35 (2011) 71–81

1. INTRODUCTION

The deep-space exploration of asteroids was started in the 90s of the last century. In thecourse of Jupiter exploration the Galileo probe of the USA flied by two asteroids, and soonafterwards, the NEAR (Near Earth Asteroid Rendezvous), the first probe mainly for thepurpose of asteroid exploration, was launched. Now, the asteroid exploration has becomeone of the hot spots of deep-space exploration. In May 2005, Japan launched the MUSES-C probe to conduct the asteroid sample return mission, and at present it is in the returncourse[3]. In September 2007, the United States launched the Dawn probe, specially forexploring the dwarf planet Ceres and the main-belt asteroid Vesta[4]. At present new asteroidexploration projects are in conduction and planning. For examples, the ANTS (AutonomousNano-Technology Swarm) project of the USA plans to launch multiple probes toward themain belt around 2020[5], and China will also conduct the asteroid exploration of multipletargets and tasks in suitable times[6].

For an exploration mission, the target selection is an important problem, in whichthe accessibility of targets is one of major specifications, and by adopting the direct transferstrategy, many accessibility studies have been made[6−9]. Under the conditions of the major-planet’s gravity assistance and different transfer strategies, to study the accessibility ofasteroids has also become a hot spot in the study of trajectory schemes. For example,References [1,2,10] propose to explore the near-earth asteroids with the gravity assistanceof earth, and Reference [11] suggests to explore the main-belt asteroids with the gravityassistance of Mars.

No matter what kind of transfer strategy is taken, to calculate one by one the accessibil-ity of every asteroid will need a large amount of calculations. Taking the direct transfer orbitas an example, it needs to try the different launching and arriving times, to derive repeat-edly the transfer orbit, and calculate the corresponding launch energy, then to evaluate theaccessibility of this asteroid according to the obtained minimum launch energy. This methodrequires too many calculations, so people tend to adopt the hereditary algorithm[10,12−14],analytical gradient searching algorithm[15−16] and other intellectual optimization algorithmsto search for the optimum launch time of a particular target. This reduces to a certain extentthe amount of computation for judging the accessibility of a singe asteroid. In resent years,along with the development of observational techniques, the number of newly discoveredasteroids increases. For example, in June 2006, the Minor Planet Center (MPC) of IAUreported that the number of the asteroids with determined orbital elements is about threehundred thousand, and in July 2009, it increases to four hundred fifty thousand, namely anincrease of 50%. The increasing number makes the quantity of calculations for one by oneevaluating the accessibility of asteroids become even more enormous.

In order to reduce the quantity of calculations, instead of evaluating one by one theaccessibility of asteroids, this paper will investigate the accessibilities of different regionsin the solar system. As soon as the region where the asteroids reside is known, then theiraccessibility can be preliminarily judged. Hence we investigate at first the regions in thesolar system which can be explored with the direct transfer strategy, then make statisticson the spatial distribution of asteroids, and finally by the comparison between the two,we find that most main-belt asteroids are accessible with the direct transfer strategy, andby the addition of a small velocity correction, the flyby mission of multiple targets can be

XIA Yan et al. / Chinese Astronomy and Astrophysics 35 (2011) 71–81 73

conducted.The further analysis indicates that such kind of multi-target flyby will be helpful for

realizing the transfer orbit assisted by the earth’s gravity, which can be used for deep-spaceexploration that can hardly be realized with the direct transfer orbit. Based on this, wehave proposed a trajectory scheme for a probe which flies by multiple main-belt asteroidsat first, then explores more distant asteroids under the assistance of the earth’s gravity, anda practical example is given.

2. REGIONS IN THE SOLAR SYSTEM, ACCESSIBLE BY DIRECTTRANSFER ORBITS

Asteroids are distributed near the ecliptic plane. Although some asteroids have rather largeinclinations, but they are rather far from the elliptic plane only when they are positionedfar apart from the ascending and descending nodes. For most of time they reside stillaround the ecliptic plane. Therefore, we discuss first the regions on the ecliptic plane whichare accessible by direct transfer orbits, then discuss the energy requirement for the regionsoutside the ecliptic plane.

If the position of the explored target is coplanar with the earth orbit, then the transferof minimum launch energy is the Hohmann transfer[17]. As shown by Fig.1, taking theexploration outside the earth orbit as an example, the perihelion of the Hohmann transferorbit is positioned on the earth orbit, and the aphelion is the target position to be explored.To set the heliocentric distance of the aphelion of the transfer orbit to be r, and takeapproximately the earth orbit as a circular orbit with the radius rE , then the semi-majorradius of the transfer orbit is a = (r+ rE)/2. By the vis-viva integration, at the initial pointof the transfer orbit the velocity VSC of the probe relative to the sun can be obtained as

V 2SC

= μ

(2rE

− 1a

), (1)

in which μ = 2.959122082855911025×10−4AU3/d2 is the gravitational constant at the centerof mass of the sun. The velocity of earth in the heliocentric ecliptic coordinate system is

V 2E

= μ

(2rE

− 1aE

), (2)

and in the geocentric coordinate system, the hyperbolic residual velocity of the probe afterescaping from the earth’s gravitational field ought to be (let rE = aE)

V∞ = VSC − VE =

√2μr

rE(r + rE)−

√μ

rE

. (3)

Consequently, the required launch energy for exploring the position of the heliocentric dis-tance r on the ecliptic plane is (let rE =1)

C3 = V 2∞ = μ

(√2 − 2

r + 1− 1

)2

, (4)


and under this kind of transfer strategy, the flight time T is a function of r:

T = π

√(r + 1)3

8μ. (5)

Fig. 1 Diagram of the Hohmann transfer orbit

If the target’s position is not on the ecliptic plane, its heliocentric distance is r, andits ecliptic latitude is β �=0, in the following we will discuss the dependence of the requiredminimum launch energy on the ecliptic latitude β for a given r.

On the ecliptic plane, the Hohmann transfer is most energy-saving, while the targetis at the upper-conjunction position of the launch point with the sun. When the target isnot on the ecliptic plane, if it still takes the upper-conjunction position, then the transferorbit will be perpendicular to the ecliptic plane and become a heliopolar orbit, the requiredlaunch energy will be very large. Hence, for exploring the positions apart from the eclipticplane, the most energy-saving target point is not the position at the upper-conjunction withthe sun.

The assemblage of the target points with the given r and β is a circle parallel to theecliptic plane. The terminal point of the transfer orbit is positioned at this circle, andthe starting point is the position of earth at the launch time. Then as soon as the flighttime is given, the transfer orbit can be determined by solving the Lambert problem, andconsequently the launch energy. According to various flight times to search various targetpositions on the circle, the minimum launch energy from the earth to this circle can beobtained.

By calculating the required minimum energy for various target points (r, β), we canobtain an iso-energy diagram of the required minimum energies for exploring different regions


in the solar system, as shown in Fig.2. In this figure, the abscissa is r cosβ, i.e., theheliocentric distance of the target point projected on the ecliptic plane, the ordinate isr sinβ, i.e., the distance of the target point from the ecliptic plane. The intersection of thedashed line and horizontal axis is the place of the heliocentric distance 1AU on the eclipticplane. Obviously, the minimum energy depends only on the heliocentric distance r and theecliptic latitude β, instead of the ecliptic longitude λ.

As an example, taking 50 km2/s2 as the maximum value of the launch energy C3, fromEq.(4) we can obtain the heliocentric distance of the accessible region by the direct transferorbit to be [0.410AU, 3.266AU], namely the region corresponding to β =0 in Fig.2 or the linesegment on the horizontal axis enclosed by the contour line of the launch energy 50 km2/s2.For any target position in this range, a suitable launch opportunity can always be found.

Fig. 2 The required minimum launch energy C3 for exploring various regions of the solar system

3. DISTRIBUTION OF ASTEROIDS IN SOLAR SYSTEM

On JD2455053.5 (10th Aug., 2009), 459433 asteroids with known orbital elements werereported by the MPC. According to the statistics on the orbital semi-major axes, periheliondistances and aphelic distances, the 340184 asteroids with aphelic distances less than 3.3AUoccupy 74% portion of the total number. These asteroids move completely in the space of theaphelic distances less than 3.3AU; the 437287 asteroids with semi-major axes in the [2 AU,3.23AU] range occupy 95.2% portion of the total number. And the statistics accordingto the heliocentric distances on other two dates indicate that on JD2455100.5, the 411168asteroids, whose heliocentric distances are less than 3.3AU, occupy 89.5% portion of thetotal number 459433; and that on JD2454400.5, the number of asteroids with heliocentricdistances less than 3.3AU is 351400, it is the 90.1% portion of the total number 391323.


After reducing to the same base, the density distribution curves are shown as Fig.3, in whichthe horizontal axis indicates the heliocentric distance, and the vertical axis represents thenumber of asteroids in each interval of 0.01AU at the corresponding heliocentric distance.These two curves are basically the same in morphology, and they can express the distributionof heliocentric distances at anytime.

Fig. 3 Statistics of the heliocentric distances of asteroids

The statistics on the distances of asteroids from the ecliptic plane is shown in Fig.4.The horizontal axis indicates the ecliptic latitude and orbital inclination, and the verticalaxis represents the corresponding number of asteroids in each interval of 0.04◦. In this figure,the solid and dashed lines are the statistics according to the ecliptic latitude on the datesJD2455100.5 and JD2454400.5, respectively, and the dot-dashed line represents the statisticsaccording to the inclination of the orbital plane relative to the ecliptic plane. Although theorbits of asteroids have generally certain inclinations, but at any time most asteroids areconcentrated around the ecliptic plane.

In order to show more apparently the distribution of asteroids in accessible regions,based on the above statistics, the surface density of asteroids is shown in Fig.5 as a functionof the projections r cosβ and r sin β of the heliocentric distance r. In the figure the surfacedensity is shown by the grey level, in units of 2500/AU2. In addition, the black curvesindicate the boundaries of the accessible regions with different launch energies, displayingthe exploration ability under the different launch energies. It is shown that when 30 km2/s2

< C3 < 50km2/s2, the accessible regions will reach or cover the densest region of asteroids.

4. EXAMPLES OF TRAJECTORY DESIGN FOR MULTI-TARGETEXPLORATIONS

To launch a probe from a point on the earth orbit, if no manoeuvre is made in the midway,then after one period of transfer orbit, the probe will return to the starting point on the earth


orbit. If the period of transfer orbit is selected to be the multiple times of the earth’s orbitalperiod, then while the probe returns to the earth orbit, the earth will return to the sameplace. At this moment, we can select to have the probe return to the earth, or under theassistance of the earth’s gravity to have the probe sent to a more distant region inaccessiblewith the direct transfer orbit. Based on this, we can find such a trajectory: departed fromthe earth, to enter at first the main belt of asteroids and fly by multiple asteroids; then toreturn to the earth orbit, and according to the requirement of the mission, to return to theearth or fly again under the gravity assistance.

Fig. 4 Statistics of the ecliptic latitudes and orbital inclinations of asteroids

For the transfer orbit whose orbital period is multiple times of the earth’s revolutionperiod (1 year), the required launch energy and the corresponding aphelic distance can becalculated with Eq.(4). If the period of transfer orbit is 2 yr, then the corresponding aphelicdistance is 2.17AU, and the launch energy is C3 =25.78km2/s2; if it is 3 yr, then the aphelicdistance and launch energy are 3.16AU and C3 =47.98km2/s2, respectively. To select thetransfer orbit of 2 yr period will require a smaller launch energy, and the relatively denseregion of asteroids can be entered (see Fig.5) to try the multi-target flyby mission.

4.1 The Orbit of 2 yr Period for a Multi-target Exploration MissionWe adopt the following procedure of trajectory design for a multi-target exploration

mission: (1) According to the given initial time, to design a Hohmann transfer orbit of2 yr period on the ecliptic plane for the probe starting from the earth’s position; (2) Tocalculate the trajectory of this orbit in these 2 years, and select along the trajectory theasteroids passing by from rather short distances and approaching at the times with certainseparations; (3) Taking these asteroids as the flyby targets, to make the velocity correctionafter approaching to each asteroid, in order to explore the next target and to redesign thetrajectory to realize the whole flyby mission of the selected targets.

Taking 1st Jan. 2015 as the initial time, the transfer orbit of C3 =25.31km2/s2 with


the 2 yr period can be obtained. To calculate the evolution of the orbit in two years, the13 asteroids encountered along the trajectory at distances less than 0.01AU are selected,then according to the principle that the approach times have certain separations, someasteroids among them are selected as the flyby targets. The reason for setting certainseparations is because that if the time of separation between two consecutively approachingasteroids is too small, the required energy of orbit manoeuvre will be excessively large.Accordingly, 4 asteroids are selected as the targets, and the trajectory is redesigned for theminimum time separation of 120 d, and the required velocity increment of the every orbitmanoeuvre is shown in Table 1. After launching, the total velocity increment required fororbit manoeuvres is 0.596km/s. For other launch times, similar trajectories can be obtainedaccording to the above procedure.

Fig. 5 Density distribution of asteroids and accessible regions with direct transfer orbits

Table 1 Velocity increments of 4 asteroids required for the orbit transfer during

flyby missions

Asteroid Close time Required dv in flybys2007 FS35 JD2457153.5 0.146 km/s2005 SR233 JD2457273.5 0.183 km/s2000 AJ59 JD2457404.5 0.267 km/s2002 PC66 JD2457550.5

In the process of screening selection and for the calculation of the distances betweenthe probe and various asteroids, one may screen out the most part of asteroids accordingto the range of ecliptic longitude of the probe’s orbit. In comparison with the one-by-onecalculation of the accessibilities of asteroids, the repeated calculations of the transfer orbitfor various assumed launch times and durations of flight, the optimal launch orbit can beselected, and this may greatly reduce the amount of computation.


4.2 Gravity AssistanceWhen the probe arrives at the target asteroid along the direct transfer orbit, the probe is

close to its aphelion, the flight velocity is rather low, but the velocity relative to the asteroidis rather large, unsuitable for performing the flyby mission. If taking further the earth’sgravity assistance when the probe returns to earth, then the mission which is infeasible forthe direct transfer orbit can be realized, for example, the accompanying flight with asteroidsor the exploration of asteroids in more distant regions.

The process of acceleration of the probe due to the gravity assistance of a major planetin the solar system is as follows[19]: Relative to the major planet, the probe moves along ahyperbolic trajectory, before entering the effective range of the planet’s gravitational forceand after escaping from it, the magnitude of the velocity relative to the major planet is notchanged, what changed is only the direction of velocity. As shown in Fig.6(a), assuming thatin the heliocentric coordinate system the velocity of the major planet is V P, and that beforethe gravity assistance is in action, the velocity of the probe relative to the major planet isvSC, then in the heliocentric coordinate system the velocity of the probe is V SC. As shownin Fig.6(b), after the gravity assistance is in action, the velocity of the probe relative to themajor planet is changed in direction, becoming v′SC, thus the velocity in the heliocentriccoordinate system becomes V ′

SC. It is changed in both magnitude and direction, thereforethe orbital major-axis is changed.

Fig. 6 Gravity assistance accelerates the probe

But for the above-mentioned Hohmann transfer orbit of 2 yr period, 2 years after theprobe is launched, when it meets again the earth, its velocity relative to earth and the velocityof earth in the heliocentric coordinate system have the same direction, hence the gravityassistance can merely reduce the heliocentric velocity of the probe, and cannot increase thevelocity of the probe, therefore the orbital semi-major axis is not enlarged.

In order to avoid this situation, References [1,2] proposed an additional deep-spacemanoeuvre at the aphelion of the 2 yr-period transfer orbit, in order to alter the velocitydirection of the probe when it meets the earth again, and obtain an acceleration after thegravity assistance is in effect. This is called the ΔV -EGA trajectory scheme.

The 2 yr-period transfer orbit can penetrate into the rather dense region of asteroidsand realize the multi-target flyby mission. As mentioned in Sec.4.1, in the course of multi-target flyby, multiple manoeuvres will be performed, and the same effect of the deep-spacemanoeuvre in the ΔV -EGA trajectory scheme can be attained. Hence the condition of theearth’s gravity assistance is improved.

Based on the direct transfer orbit designed in Sec.4.1, and after the flyby of the lastasteroid, by supplying the probe with a suitable velocity increment, the purposive gravityassistance of the earth can be realized. In Fig.7, we give an example of trajectory designs:


The probe is launched from the earth with the energy of C3 =25.31km2/s2, it flies by 4asteroids successively, then it enters the transfer orbit with the aphelic distance of 3.67AUunder the earth’s gravity assistance. Finally, at a small cost of 0.611km/s velocity increment,it realizes the accompanying flight with the asteroid 2004 VZ60. After launching, the totalvelocity increment required for the whole mission is 1.769km/s, including the flybys of 4asteroids and the accompanying flight with the asteroid of high orbital eccentricity. Thisfigure adopts the heliocentric ecliptic coordinate system of the J2000 epoch, the X-axispoints to the spring equinox, the Y -axis is positioned on the ecliptic plane and perpendicularto the X-axis. The detailed flight course is given in Table 2.

Fig. 7 The sketch of orbits on the ecliptic plane

Table 2 The flight course of multiple-target flybys and gravity assistance

Time The closed body Event2015-01-01 earth Launch from earth, C3 = 25.31 km2/s2

2015-05-11 2007 FS35 2007 FS35 flyby, dv=0.146 km/s2015-09-08 2005 SR233 2005 SR233 flyby, dv=0.183 km/s2016-01-17 2000 AJ59 2000 AJ59 flyby, dv=0.267 km/s2016-06-11 2002 PC66 2002 PC66 flyby, dv=0.562 km/s

earth gravity assistance, minimal altitude: 794 km2016-11-25 earth Relative velocity to the Sun before flyby: 35.08 km/s

after flyby: 37.64 km/s2020-01-13 2004 VZ60 Brake with dv=0.611 km/s to accompany

After the earth’s gravity assistance is adopted, the velocity of the probe relative to earthis 7.70 km/s, equivalent to the effect of the direct launch from the earth with the energyof C3 =59.24km2/s2. Using the earth’s gravity assistance, the required launch energy isgreatly reduced. This trajectory scheme provides a reference for the trajectory design toexplore other more interesting asteroids.


5. CONCLUSIONS AND PROSPECT

In this paper the accessible region in the solar system by adopting the direct transfer orbitfor the given launch energy is investigated, and this is compared with the distribution ofasteroids. It is concluded that the direct transfer orbit of 30 km2/s2 < C3 < 50 km2/s2 canreach the dense region of asteroids, and that the multi-target flyby of main-belt asteroidscan be realized by adopting the direct transfer orbit. By combining with the ΔV-EGAtrajectory scheme proposed in References [1,2], the trajectory scheme of the earth’s gravityassistance after multiple flybys is proposed. This trajectory can fulfill the mission whichis difficult to realize by the direct transfer orbit, in addition to the exploration of multipletargets. In this paper, an example of trajectory design is given. With the launch energyof C3 =25.31km2/s2 and the velocity increment of 1.769km/s after launching, the probehas realized the flybys of 4 asteroids and the accompanying flight with an asteroid of highorbital eccentricity. For this the method of trajectory design is described, probably it canprovide a reference for the trajectory design of other deep-space exploration missions.

This paper has not dealt with the physical properties of particular asteroids, the or-bit determinacy and other factors, the detailed considerations of the errors in the orbitalparameters of asteroids, sizes and other factors. The selection of the asteroids with higherexploration values and orbit accuracies, as well as the corresponding study of trajectorydesigns will be the subjects of our future works.

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Accessibility of Main-belt Asteroids and Trajectory Design for Multi-target Exploration