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Necessary condition for temporary asteroid capture

  • Changchun Bao EMAIL logo , Zhijie Li and Xianyu Wang
Published/Copyright: June 14, 2023
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Open Astronomy
From the journal Open Astronomy Volume 32 Issue 1

Abstract

Asteroid capture holds great significance in understanding the origins of asteroids, the source of life, and the exploration of planetary mineral resources. To ensure the success of a mission aimed at capturing asteroids, it is crucial to choose asteroids that can be naturally captured by Earth. This article focuses on the orbital characteristics of temporarily captured asteroids. Through a statistical simulation of asteroid motion and an analytical derivation of the Jacobian integral, the necessary conditions for the temporary capture of asteroids by Earth are calculated. The analysis results show an approximately linear relationship between the orbit elements of temporarily captured asteroids. This condition is used to screen the currently observed near-Earth asteroids and has identified asteroids such as 2006 RH120 and 2020 CD3, which have been temporarily captured by Earth. These findings provide important suggestions for future asteroid capture missions and target selection.

Keywords: temporary capture asteroid; near-Earth asteroids; orbital elements; numerical methods; necessary condition

1 Introduction

According to related research, thousands of near-Earth objects have been discovered, and the number continues to increase daily. Capturing asteroids has long been a popular topic in celestial studies, shedding light on the origin and evolution of the solar system as well as providing valuable insights into the formation of life. Asteroids are known to contain many valuable metals. Lewis (1997) conducted a study on a 2-km near-Earth metallic object, which could potentially contain metals and materials worth over 25 trillion dollars. The capture and study of near-Earth objects could be a crucial step toward human exploration of space. The idea of exploiting the natural resources of asteroids was proposed as early as the last century. Identifying targets among the many asteroids that can be naturally captured by Earth holds great significance for future research endeavors.

In recent years, several scholars have explored the feasibility of using optical equipment both on the ground and in space to detect temporary captures of asteroids by Earth, as well as the potential to detect these captures using Earth launch probes (Bolin et al. 2014, Granvik et al. 2013, Jedicke et al. 2015). Vieira and Winter (2009) studied the possibility of temporary captures of asteroids by Jupiter becoming its moons, taking into account the effect of thin air resistance in the restricted three-body model. Their findings indicate that the probability of a temporarily captured asteroid becoming a satellite of Jupiter based on air resistance alone is low. However, the likelihood of an asteroid becoming one of Jupiter’s satellites significantly increases if the temporary capture of small pieces of the asteroid that hit Jupiter during its orbit around the planet is taken into consideration. Fedorets et al. (2017) have also defined the orbit and size distribution of temporarily captured asteroids in the Earth–Moon system. According to Granvik et al. (2012), the relation between the probability of near-Earth asteroids (NEAs) being captured as temporarily captured asteroids and the distribution of the heliocentric orbital elements indicates that the probability of temporarily capturing NEAs in January or June is higher than at other times.

Baoyin et al. (2010) proposed the idea of capturing asteroids. In their work, one method applies an impulse to the NEAs when they are near Earth, near the zero-velocity surface within the framework of the Earth–Sun restricted three-body problem. The optimal target in their study is 2009 BD, with a velocity increment of 410 m/s. Mingotti et al. (2014) studied the method of capturing the Lagrangian point periodic orbits of the asteroid in the Earth–Moon system using the inner manifold splicing method of the Sun–Earth and Earth–Moon double trisomy models. Tan et al. (2017) studied direct and indirect methods to capture asteroids at the Earth–Moon Lagrangian point. The implementation of the direct method involves the application of two pulses in the four-body model composed of the Sun, the Earth, the Moon, and asteroids to connect the asteroids’ orbit to the stable manifold of the Earth–Moon system, finally capturing the asteroids to the periodic orbit near the Earth–Moon Lagrangian point. The indirect method captures the periodic orbit of the asteroid in the vicinity of the Lagrangian point by using the inner manifold splicing method of the Sun–Earth and Sun–Moon double three-body models. The analysis results show that the direct method takes a shorter time to capture the asteroids than the indirect method. Ceriotti and Sanchez (2016) analyzed the controllability of low-energy manifolds in capturing the periodic orbits of asteroids near the Lagrangian point. The mass of the captured asteroids plays a key role in controlling the thrust value. Capture and periodic orbits affect the stability of the main factors of mission controllability. Literature (Granvik et al. 2012) was the first to calculate the population characteristics of Earth’s irregular natural satellite, which was temporarily captured from a NEO population.

For asteroids that have been temporarily captured, Chyba et al. (2014) studied the optimal low-thrust transfer orbit from geostationary orbit to temporarily captured asteroids. Haapala and Howell (2013) studied the use of manifold detection to temporarily capture asteroid trajectories. Urrutxua et al. (2014, 2015) studied the scheme of prolonging the time of temporarily capturing asteroids. They took asteroid 2006 RH120 as the research target, applied a small thrust at its perigee, and analyzed the relationship between the duration of the small thrust and the capture time. Their results show that applying 0.27 N of low thrust to 2006 RH120 can obtain 32 m/s of velocity increment within 6 months and prolong the asteroid capture time by more than 5 years. By applying a small thrust of less than 1 N to the temporary capture asteroid of virtual structure, an increased capture time of several years or even more can be obtained when the velocity increment is 15 m/s. Anderson and Lo (2018), respectively, used the circular and elliptical restricted three-body models to analyze the chaos during the temporary capture of 2006 RH120. By comparing the calculation results in different models, they obtained the characteristic that Earth’s eccentricity plays a key role in the temporary capture process.

Screening out asteroids that may be temporarily captured by Earth from many natural satellites is especially important to the study of the temporary capture of asteroids. In this article, the zero-velocity surface is discretized using the performance of the conditions for capturing asteroids in the circular restricted three-body problem, obtaining the necessary conditions for capturing asteroids. In the second section of this article, the equations of motion and Jacobi’s integral governing the motion of asteroids in the restricted three-body model are briefly introduced. Moving on to the third section, the simulation of the temporary capture of NEAs and the corresponding captured orbits are discussed in relation to the Jacobi constant and the zero-velocity surface, as well as the orbital elements involved in temporarily capturing NEAs. The fourth section derives a formula for precisely determining the relationship between the Jacobi constant and the orbital elements. Notably, an approximately linear relationship exists between the semi-major axis and the eccentricity. The numerical solution is compared with the analytical solution to verify the approximate linear relation and provide a new condition for selecting the temporary capture of asteroids. In the last section, by substituting the currently observed NEAs into the necessary conditions obtained above, 10 asteroids that satisfy the temporary capture conditions are obtained. Using this conclusion to determine the potential temporary capture of asteroids is of great significance for future asteroid capture missions.

2 Asteroid dynamics equation

The current study on the problem of asteroid capture in the near-Earth orbit uses the circular restricted three-body problem model to approximate the real environment. In this three-body system, the masses of the Sun and Earth are, respectively, denoted as m 1 and m 2.

Dimensionless units are used in this study to analyze the problem and for calculation convenience. a 12 is the distance between the Sun and Earth (AU), and the gravitational constant G = 1. The time unit [T] is the inverse of the angular velocity (n) of Earth, i.e., [T] = 1/n. If the asteroid is moving within the gravitational range of two main celestial bodies, then its mass, length, and time in this system are calculated in units as follows:

(1) [ M ] = m 1 + m 2 [ L ] = a 12 [ T ] = a 12 3 / G ( m 1 + m 2 ) .

The dimensionless mass of the Sun and Earth is

(2) 1 − μ = m 1 m 1 + m 2 , μ = m 2 m 1 + m 2 .

The distance from the Sun and Earth to the common center of mass is

(3) r 1 ′ = μ , r 2 ′ = 1 − μ .

The centroid inertial coordinate system is denoted as C-XYZ, and its origin is set at the center of mass (C) of the Sun–Earth System. The X–Y coordinate is the plane of relative motion of the Sun and Earth, the X axis is on the Sun–Earth line at time t 0 , and points from the Sun P 1 to the Earth P 2. The coordinate system is shown in Figure 1, in which point P is the position of the asteroid, R is the position vector of the asteroid in this coordinate system, and R 1 and R 2 are the position vectors to the Sun and Earth, respectively.

Figure 1 Centroid inertial coordinate system C-XYZ and centroid rotating coordinate system C-xyz.
Figure 1

Centroid inertial coordinate system C-XYZ and centroid rotating coordinate system C-xyz.

In the centroid inertial coordinate system, the equation of motion of asteroids and Jacobi’s integral are

(4) R ̈ = ∂ U ∂ R T = − ( 1 − μ ) R 1 R 1 3 − μ R 2 R 2 3 C = 2 U − [ V 2 + 2 ( X ̇ Y − X Y ̇ ) ] + μ ( 1 − μ ) ,

where

(5) U = U ( R 1 , R 2 ) = 1 − μ R 1 + μ R 2 V 2 = R ̇ 2 = X ̇ 2 + Y ̇ 2 + Z ̇ 2 .

Another commonly used motion model is the centroid rotating coordinate system (C-xyz). The x-axis of this coordinate system is on the Sun–Earth line. The rotational angular velocity of the coordinate system is equal to the angular velocity of the relative motion of the Sun and Earth. The coordinate vectors of the asteroid and the two main celestial bodies are, respectively, recorded as r , r 1 , and r 2 . The equation of motion and Jacobi’s integral of the center of the centroid rotating coordinate system are

(6) r ̈ + 2 − y ̇ x ̇ 0 = ∂ Ω ∂ r T C = 2 Ω ( x , y , z , μ ) − ( x ̇ 2 + y ̇ 2 + z ̇ 2 ) ,

where

(7) Ω = 1 2 ( x 2 + y 2 ) + U ( r 1 , r 2 ) U ( r 1 , r 2 ) = 1 − μ r 1 + μ r 2 .

Jacobi’s integral expression indicates that this integral is determined by the initial conditions of asteroids. When the motion speed of an asteroid is zero, Jacobi’s integral corresponds to a curved surface, which is called a zero-velocity surface, and the geometric structure of the curved surface changes with the C value. The zero-velocity surface near Earth divides space into two regions. One is called Hill’s region, which is the area that asteroids can reach. The other is called the Forbidden Zone, which is the area that asteroids cannot reach under the current Jacobi’s integral.

3 Analysis of the orbital characteristics of temporarily captured asteroids

As defined previously, the two requirements for an asteroid to be temporarily captured are as follows:

  1. The planetocentric Keplerian energy must be negative.

  2. The planetocentric distance must be less than approximately three Hill radii (approximately 0.03 AU).

These two requirements are used to screen asteroids that can be temporarily captured.

Because of its temporary capture characteristics, it provides a good opportunity for ground observation or further research on asteroids. The phase space of the heliocentric orbital elements, from which captures on heliocentric orbits are possible, is restricted only to orbits near that of Earth’s: 0.87 AU < a < 1.15 AU; e < 0.12; i < 2.5° (Fedorets et al. 2017). According to the concept of zero velocity surface. When an asteroid approaches Earth, if the Jacobi’s integral of the asteroid is between the Jacobi’s integral of the second Lagrangian point and the third Lagrangian point, it may move around Earth under the influence of the Sun and Earth’s gravity and be temporarily captured.

In this section, when calculating the temporary capture of asteroids, the points of the zero-velocity surface are used as the initial points. Figure 2 shows the temporarily captured asteroid’s orbital zero-velocity surface that changes depending on Jacobi’s integral constant. The Jacobi constant (C) selected in this section varies from 3.000696928 to 3.000886928; this range is between the Jacobi’s constant of the second and third Lagrangian points and is based on many simulation tests. Considering the symmetry of the three-body problem with circular constraints, the initial value of discrete integers with y > 0 is selected on the zero-velocity surface of each Jacobi’s constant. The zero-speed surface is projected onto the x–y plane, and the two points on the curve with zero slope are taken as the initial points to discretize the zero-speed surface. Taking the Jacobi constant C = 3.000796928 as an example, the red surface in Figure 3a is a discrete zero-velocity surface, and the red curve in Figure 3b is a discrete part on a two-dimensional plane. The points obtained by these discrete zero-velocity surfaces are then used in the capture calculations. The specific calculation process is shown in Figure 4.

Figure 2 Zero-velocity surface projected on the Jacobi constant change in the x–y plane.
Figure 2

Zero-velocity surface projected on the Jacobi constant change in the x–y plane.

Figure 3 Zero-velocity surface dispersion diagram when the Jacobi integral is 3.000796928. (The b is the projection of the a in the x-y plane).
Figure 3

Zero-velocity surface dispersion diagram when the Jacobi integral is 3.000796928. (The b is the projection of the a in the x-y plane).

Figure 4 Temporary capture asteroid orbit calculation flowchart.
Figure 4

Temporary capture asteroid orbit calculation flowchart.

According to the program chart in Figure 4, the temporary capture time of asteroids is calculated in the centroid rotating coordinate system. Figure 5 shows the distribution characteristics of the initial position on the zero-velocity surface corresponding to the time of the temporary capture orbit. Different colors are used in the figure to represent different capture times, and the initial points over 1 year are marked in black.

Figure 5 Distribution characteristics of the initial capture track duration corresponding to the initial position on the zero-velocity surface.
Figure 5

Distribution characteristics of the initial capture track duration corresponding to the initial position on the zero-velocity surface.

In Figure 6, the initial points are divided into three categories based on the duration of the temporary capture. The first type marked in blue is the initial point of temporary capture for more than 300 days. The second type marked in green is the initial point of temporary capture for less than 300 days. The third type marked in dark red represents the asteroids hitting the Earth during the temporary capture period (the impact radius is Earth’s radius +300 km).

Figure 6 Distribution characteristics of three types of temporary capture orbits corresponding to the initial position on the zero-velocity curve (blue is the initial point of temporary capture for more than 300 days; green is the initial point of temporary capture for less than 300 days; red represents the asteroids hitting the Earth during the temporary capture period).
Figure 6

Distribution characteristics of three types of temporary capture orbits corresponding to the initial position on the zero-velocity curve (blue is the initial point of temporary capture for more than 300 days; green is the initial point of temporary capture for less than 300 days; red represents the asteroids hitting the Earth during the temporary capture period).

Figures 5 and 6 show that the initial points of the asteroids with a long temporary capture time take the abscissa of the small celestial body (Earth) as the vertical line and are approximately symmetrically distributed on the zero-velocity surface. The capture time of most asteroids on the zero-velocity surface is approximately 150 days. The asteroids that have been captured for more than a year are concentrated on the edge of the zero-velocity surface, corresponding to a small Jacobian constant. The initial positions of the asteroids that will impact the Earth are concentrated near the location of Earth. The closer the impact is to the location of Earth, the deeper the crimson shade that represents the impact.

In the J2000 heliocentric inertial coordinate system, the relation between the semi-major axis and the eccentricity shown in Figure 7 is obtained after calculating the orbital properties of the temporarily captured asteroid orbits on the zero-velocity surface of Figure 5. The figure below indicates that in the circular restricted three-body model, a nearly linear relation exists between the semi-major axis and the eccentricity.

Figure 7 The Kepler elements of the orbits of the temporarily captured asteroids.
Figure 7

The Kepler elements of the orbits of the temporarily captured asteroids.

4 Proof of the necessary conditions for the temporary capture of asteroids through analytical methods

In this section, the two-body equation in the centroid inertial system is introduced based on the principle of the circular restricted three-body model, and highly intuitive orbital elements (a, e, i, and f) are used to derive the Jacobi constant to further analyze the conditions for temporarily capturing asteroids.

The following is the Vis-viva formula for the Sun–asteroid two-body problem:

(8) X ̇ 2 + Y ̇ 2 + Z ̇ 2 = V 2 = 2 R 1 − 1 a μ .

The specific angular momentum (h) of the asteroid’s orbit is expressed as follows:

(9) h = R × R ̇ .

By taking the ecliptic plane as the reference plane, the vertical component of angular momentum is derived as follows:

(10) X Y ̇ − X ̇ Y = h cos i h 2 = a ( 1 − e 2 ) .

Then, Jacobi’s integral expressed by orbital elements is

(11) C = 2 U + μ ( 1 − μ ) − 2 R 1 − 1 a − 2 h cos i .

The gravitational potential energy (U) in the above equation introduces the orbital equation in the two-body system to express the distance R 1 between the Sun and the asteroid.

(12) U = ( 1 − μ ) / R 1 + μ / R 2 R 1 = ( 1 − e 2 ) / ( 1 + e cos f ) .

According to Section 3, the relation between the semi-major axis and the eccentricity of the temporarily captured asteroid was determined by simulating the capture of the asteroid in a restricted three-body model. This relation can be proven by solving Eq. (11). Without considering the phase, the true anomaly is f = 0. Through the flow of Figure 4, the temporarily captured velocity vector and the position vector of the initial point are converted into orbital elements and substituted into Eq. (11) to obtain the relation between the semi-major axis and the eccentricity.

Figure 8a is the projection of Figure 7 on the x–y plane, which is the discrete result of the zero-velocity surface, and its Jacobi constant is between 3.000696928 and 3.000886928. Figure 8b shows the relation between the semi-major axis and the eccentricity obtained by solving Eq. (11) in the same Jacobi constant interval. Given the relation of the Jacobi constant to the zero-velocity surface of the Earth and the nature of NEAs, these asteroids may be temporarily captured by the Earth. The 2006 RH120 and 2020 CD3 asteroids are the two temporarily captured asteroids that have been discovered so far. 2006 RH120 was temporarily captured by the Earth for 472 days in 2006–2007, of which 370 days it moved in the Hill’s region of Earth until June 2007. 2006 RH120 is marked with a red asterisk in Figure 8. The 2020 CD3 (Urrutxua et al. 2014, 2015) is a meter-scale asteroid discovered by Earth in 2020 and marked with a red circle in Figure 8.

Figure 8 Relation between the semi-major axis and eccentricity (a) is the result of simulation, (b) is the result of theoretical calculation).
Figure 8

Relation between the semi-major axis and eccentricity (a) is the result of simulation, (b) is the result of theoretical calculation).

Given the singularity of solving Eq. (11) in the orbit around the Earth, the Earth’s positions (at X-axis coordinate 1) in Figure 8a and b are different. Except in the orbit near Earth, the trends in Figure 8a and b are the same, and the analytical solution proves the statistical result. Therefore, the necessary conditions for Earth to temporarily capture asteroids are obtained, and the semi-major axis and eccentricity of the NEAs’ orbit satisfy a linear relation. When the semi-major axis is greater than 1, the slope of the semi-major axis is 0.7315. When the semi-major axis is less than 1, the slope is −0.7678.

5 Asteroids meeting the necessary conditions for temporary capture

In order to identify temporarily captured asteroids among NEAs, the semi-major axis and eccentricity of their orbits are plotted in Figure 9. Disregarding the phases of the asteroids, the figure reveals that the orbital elements of 10 currently observed asteroids are in close proximity to the necessary straight-line condition. Table 1 presents the corresponding orbital elements of these 10 asteroids at the time of observation, as per the ephemeris JD 2459000.5. It is noteworthy that the asteroids listed in Table 1, namely 2006 RH120 and 2020 CD3, which were naturally captured by Earth, have been observed to satisfy this necessary condition. Furthermore, Table 1 demonstrates that the two captured asteroids possess the characteristics of low eccentricity and inclination. A large speed increment is often required when the orbital inclination and eccentricity of the target asteroid are changed using a probe. Therefore, when selecting an asteroid as a capture target, a candidate target whose orbit is similar to that of Earth’s low inclination and low eccentricity, such as 2016 RD34 in Table 1, must be selected.

Figure 9 Distribution of orbits elements of asteroids that meet the conditions of temporary capture (the black points are NEAs, and the red numbered points are asteroids that can be temporarily captured).
Figure 9

Distribution of orbits elements of asteroids that meet the conditions of temporary capture (the black points are NEAs, and the red numbered points are asteroids that can be temporarily captured).

Table 1

NEAs’ orbital elements (time in the ephemeris, JD 2459000.5)

Number Asteroids a (AU) e i (deg) Ω (deg) ω (deg) M (deg)
1 2005 FJ 1.08999299 0.0649 10.0397 173.9175 316.5188 38.8101
2 2016 BS67 1.05940299 0.0437 6.5115 318.7745 113.7563 224.5117
3 2016 LA49 1.06730224 0.0503 9.1471 260.8089 28.6337 8.4004
4 2016 RD34 1.04628189 0.0347 1.9579 349.5898 11.0210 344.1049
5 2018 DB1 1.06933071 0.0522 10.8706 155.7355 7.8755 187.1844
6 2018 EB 1.01657305 0.0122 29.4360 194.5562 126.8622 104.6719
7 2019 HM 0.98730192 0.0093 5.7591 201.6338 195.8284 55.5841
8 2020 KV6 1.08416921 0.0617 22.5397 245.8158 19.9001 160.6591
9 2006 RH120 1.03315434 0.0244 0.5947 51.1815 10.0990 156.8268
10 2020 CD3 1.0226854 0.0172 0.64019 83.0538 46.8519 117.0686

6 Conclusion

The temporary capture of asteroids has significant practical implications in various ways, such as providing an opportunity for ground-based observation or further research due to their unique characteristics. Selecting asteroids that can be naturally captured by Earth is crucial for asteroid research.

To find the orbital characteristics of Earth's natural temporary capture of asteroids, the capture of asteroids is simulated in a circular restricted three-body model. A linear relation exists between the semi-major axis of the temporarily captured asteroid and the eccentricity, and the Jacobi constant expressed in orbital elements is deduced to prove the statistical relation. When the semi-major axis is greater than 1, the slope of the semi-major axis and the eccentricity are 0.7315. When the semi-major axis is less than 1, asteroids with a slope of −0.7678 can be temporarily captured by Earth. Finally, bringing the currently observed asteroids into this necessary condition results in 10 asteroids meeting the necessary conditions. Two of the asteroids are 2006 RH120 and 2020 CD3, which were temporarily captured by Earth, proving the correctness of this necessary condition.

This relationship provides new, necessary conditions for the selection of future asteroid research targets and reduces the scope of the temporary capture of asteroids. On the basis of this necessary condition and the requirements of low orbital inclination and low eccentricity similar to Earth’s orbit, a target proposal for future asteroid capture missions is proposed, i.e., asteroid 2016 RD34.

Acknowledgments

The authors would like to acknowledge the support from the National Natural Science Foundation of China Youth Fund (Grants No. 11902166) and the Natural Science Foundation of Inner Mongolia Autonomous Region (Grants No. 2021LHMS01002).

  1. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-11-25
Revised: 2023-04-09
Accepted: 2023-04-21
Published Online: 2023-06-14

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/astro-2022-0222
Keywords for this article
temporary capture asteroid; near-Earth asteroids; orbital elements; numerical methods; necessary condition
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  23. Magnetic fields with random initial conditions in discs with Kepler rotation curve
  24. Far-IR emission from bright high-redshift quasars
  25. A new version of the binary star database BDB: Challenges and directions
  26. Dust and gas in star-forming complexes in NGC 3351, NGC 5055, and NGC 5457
  27. Investigation of the halo effect in the evolution of a nonstationary disk of spiral galaxies
  28. Special Issue: Astromineralogy and cosmochemistry of the Kaba meteorite
  29. Clay minerals and other hydrous alteration products in the Kaba meteorite: Review of the literature and new XRD investigations
  30. Special Issue: The Hubble Tension: Current Research and New Perspectives
  31. Late-time dark energy and Hubble tension
  32. The history of the Cosmos: Implications for the Hubble tension
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