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Equilibria and orbits around asteroid using the polyhedral model
Accepted Manuscript
Equilibria and Orbits around Asteroid using the Polyhedral Model Yu Jiang , Xiaodong Liu PII: DOI: Reference:
S1384-1076(18)30139-8 https://doi.org/10.1016/j.newast.2018.11.007 NEASPA 1244
To appear in:
New Astronomy
Received date: Revised date: Accepted date:
8 May 2018 2 November 2018 22 November 2018
Please cite this article as: Yu Jiang , Xiaodong Liu , Equilibria and Orbits around Asteroid using the Polyhedral Model, New Astronomy (2018), doi: https://doi.org/10.1016/j.newast.2018.11.007
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HIGHLIGHTS
The polyhedral model is used to compute the potential of the asteroid.
It is found that there are five equilibrium points for the asteroid 283 Emma.
The zero-velocity surfaces and the equilibria vary if the rotational speed vary.
The separation/primary-radius has influence on the stability of the orbit.
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Equilibria and Orbits around Asteroid using the Polyhedral Model
Yu Jiang1, 2,3 and Xiaodong Liu3 1. State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043, China
3. Astronomy Research Unit, University of Oulu, Oulu 90014, Finland
Y. Jiang () e-mail: [email protected] (corresponding author)
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2. School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
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Abstract. In the current study, we use the polyhedral model to compute the potential of the asteroid. There are five equilibrium points in the gravitational field of the asteroid 283 Emma. We concluded that the zero-velocity surfaces and the equilibrium points change with the suppositive variation of the rotational speed of the asteroid. It is found that if the rotational speed equals a half as it is in present, the number of equilibrium points is also five. However, if the rotational speed equals twice as it is in present, there are only three equilibrium points left. Four different periodic orbits are calculated using the hierarchical grid searching method. We calculated characteristic
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multipliers of periodic orbits to invistigate the stability of these periodic orbits. The orbit near the primary's equatorial plane is more likely to be stable when the separation/ primary-radius is a large number.
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Key Words: Asteroid; Polyhedral Model; Zero-Velocity Surfaces; Equilibrium Points
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1. Introductions
The moonlet (S/2003 (283) 1) of the main belt asteroid (283) Emma was discovered in 2003 (Merline et al. 2003; Marchis et al. 2008). After that, this binary asteroid system received interests to researchers. Michalowski et al. (2006) used the available lightcurves and calculated the shape model and spin parameters of the asteroid (283) 2
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Emma. Marchis et al. (2008) used 8–10-m class telescopes conducted observations of four different binary asteroid systems: (130) Elektra, (283) Emma, (379) Huenna, and (3749) Balam. Their observation data confirmed the existence of their asteroidal moonlets. That study suggested a different origin for these moonlets, and the moonlet
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of (283) Emma is more likely to be produced by tidal effects. However, they concluded that for (379) Huenna and (3749) Balam, their moonlets are likely to be generated by mutual capture. Bouvier and Horch (2011) presented the image
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reconstructions of 283 Emma of data from the data of Differential Speckle Survey Instrument (DSSI).
(283) Emma is a P-type asteroid with a significant porosity (Marchis et al. 2008).
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Previous studies also indicated the diameter, bulk density, and rotational period of
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(283) Emma. Michalowski et al. (2006) presented the primary diameter for 283 Emma to be 148±5 km from STM (Standard Thermal Model) model, and 141±6 km from
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NEATM (near-Earth asteroid Thermal Model) model. The size of the moonlet
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S/2003(283)1 is estimated to be 9 ±5 km. Thus the main belt binary asteroidal system (283) Emma and S/2003(283)1 is a large-size-ratio binary. Marchis et al. (2008)
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derived the bulk density and the rotational period for 283 Emma to be 0.7-1.0 g∙cm-3 and 6.888h, respectively. Michalowski et al. (2006) suggested the rotational period for 283 Emma is 6.895 ± 0.003 h. Strabla et al. (2011) used the Schmidt 0.32-m f/3.1 telescope at Bassano Bresciano Observatory and showed that the period of 283 Emma is 6.896±0.001h. 3
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The dynamical behavior of the binary asteroid systems is very complicated for different binary asteroid systems. For the synchronous binary asteroids (Descamps 2010), such as 809 Lundia (Birlan et al. 2014), 854 Frostia (Bernasconi et al. 2006), 1089 Tama (Bernasconi et al. 2006), etc., the two asteroids in the system
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synchronously rotate; in other words, the orbital angular speed and the attitudinal angular speed of these two asteroids are equal. Generally, two asteroids of the synchronous binary system have the same orbital angular speed, and two different
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attitudinal angular speeds. For the high-size-ratio binary asteroid systems (Ortiz et al. 2012; Bosanac et al. 2015; Walsh and Jacobson 2015), the secondary plays the role as the moonlet in the gravitational potential of the primary, such as (41) Daphne (Matter
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et al. 2011), 243 Ida (Belton et al. 1995), 283 Emma (Marchis et al. 2008), et al. The dynamical behaviors of the moonlet in the gravitational potential of the primary is
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related to the moonlet’s orbit, the primary’s shape and mass distribution, as well as the
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primary’s rotational speed.
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In this paper, we investigate the dynamical behaviors in the gravitational potential of the primary of the large-size-ratio main belt binary asteroid (283) Emma
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and S/2003 (283) 1. The irregular shape of the (283) Emma and its influence on the gravitational environment is considered. The gravitation of the moonlet S/2003(283)1 is not included in the model when calculating orbits and equilibrium points because the gravitation ratio of the moonlet and the primary is smaller than 3.0×10-4. This study is useful for understanding the dynamical behaviors in the large-size-ratio 4
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binary asteroids. This paper is organized as follows: Section 2 investigates the 3D irregular shape, the equilibrium points and zero-velocity surfaces of asteroid 283Emma. The 3D irregular shape is calculated by the homogeneous polyhedral model with the observation data from Michalowski et al. (2006). To study the variety
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of the zero-velocity surfaces and equilibrium points under the conditions of the variety of the rotational speed, we calculated the zero-velocity surfaces and equilibrium points for two cases, the first case is to assume the rotational speed twice,
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and the second case is to assume the rotational speed half. Section 3 deals with the orbits in the gravitational potential of asteroid 283 Emma. Both of the periodic orbit and the general orbit have been investigated. Four different periodic orbits near the
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surface of 283 Emma are calculated, the 3D shapes, positions, characteristic multipliers, as well as the stability of the periodic orbits are discussed. The general
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orbit is calculated near the primary's equatorial plane. The 3D shapes of the general
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orbit are plotted in the body-fixed frame and the inertia system. Both the mechanical
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energy and the Jacobi integral of the orbit are calculated.
2. Equilibrium Points and Zero-Velocity Surfaces
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In this section, we calculate the equilibrium points and zero-velocity surfaces in the gravitational potential of an asteroid. The motion equations of a massless particle in the gravitational potential of the asteroid can be expressed in Cartesian coordinates x, y, and z. Let the rotational angular velocity of the asteroid be ω , the body-fixed vector from the mass center of the asteroid to the particle be r . The parameter G = 5
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6.67 × 10-11 m3 kg-1 s-2 is the Newtonian gravitational constant, and σ is the body’s bulk density. The axes of the body-fixed frame are along the principal axis of inertia. Assume the asteroid rotates around its biggest moment of inertia. The z-axis points to the biggest moment of inertia while the x-axis points to the smallest moment of inertia.
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Equations of motion (Scheeres 2012; Aljbaae et al. 2017) for a particle orbiting around the asteroid read
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U 2 x 2 y x x 0 U 2 0, y 2 x y y U 0 z z
(1)
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where the gravitational potential U of the asteroid can be calculated by the polyhedron
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model method (Werner and Scheeres 1997):
(2)
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1 1 U G re Ee re Le + G r f Ff r f f . 2 2 eedges f faces
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Here, re and rf are body-fixed vectors from a field point to any point on the edge e and face f, respectively. Ee and Ff are body-fixed tensors of the geometric parameters of
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edge e and face f, respectively. Le is the integration factor that related to a fixed point and edge e, and ωf is the solid angle. The gravitational force acceleration U and the gravity gradient matrix U are
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eedges
Ee re Le G
U G
eedges
f faces
Ee Le +G
Ff r f f
(3)
(4)
f faces
Ff f
The Jacobi integral H follows
1 2 2 x y 2 z 2 x2 y 2 , 2 2
and the effective potential V is
2
x 2
2
y2 .
(6)
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V U
(5)
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H U
The positions of equilibrium points (Jiang et al. 2014; Scheeres 2016) in the
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potential of the asteroid is calculated by the following equations
(7)
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V x, y, z V x, y, z V x, y, z 0, x y z
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The linearised equations of motion in the vicinity of the equilibrium point (Jiang et al. 2014) around the asteroid follow
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0 2 0 Vxx Vxy Vxz d2 d 2 0 0 Vxy Vyy Vyz 0 . 2 dt dt 0 0 0 Vxz Vyz Vzz
Here,
frame,
x，y，z
(8)
is the position vector of the massless particle in the body-fixed
xE，yE，zE
is the position vector of equilibrium points in the body-fixed
2V frame, x xE， y yE， z zE , Vpq p, q x, y, z represent pq E 7
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the elements of Hessian matrix of the effective potential V . The Hessian matrix of the effective potential V can be calculated by 0 0 G Ee Le +G Ff f . eedges f faces 0
(9)
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2 0 V 0 2 0 0
Using the rotation speed and the elements of Hessian matrix, one can compute the eigenvalues (Jiang et al. 2014, 2016) of equilibrium points
Vxz
Vxz Vyz 0 . 2 Vzz
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2 Vxx 2 Vxy 2 Vxy 2 Vyy Vyz
(10)
where is the eigenvalues of the equilibrium point.
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2.1 Equilibrium Points and Zero-Velocity Surfaces for 283 Emma
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We now calculate V for 283 Emma. The density and rotational period are 0.81
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g cm3 and 6.888 h, respectively (Michałowski et al. 2006; Marchis et al. 2008). Figure 1 shows the 3D exterior shape of 283 Emma calculated by the polyhedral
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model with the observation data (Michalowski et al. 2006), where the shape is
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respectively plotted in the xy, yz, and zx planes. The model has 2032 faces and 1018 vertices. The figure indicates that the shape of 283 Emma is similar to a potato, and the shape viewed in different coordinate planes are different. The lengths of the three axes of 283 Emma are 161.34, 111.61, and 108.20 km.
Figure 2 presents equilibrium points and the contour plot of the zero-velocity 8
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surfaces for asteroid 283 Emma. The figure indicates that 283 Emma has five equilibrium points; four of them are outside and one inside. From Figure 2, one can see that the topological structures of the zero-velocity surfaces in different coordinate planes are quite different. The number of equilibrium points near the coordinate
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planes is related to the topological structures of the zero-velocity surfaces on the coordinate planes. Near the xy plane, i.e. the equatorial plane of the asteroid, there are five equilibrium points. This is because the asteroid rotates around z-axis. Near the yz
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plane, there are three equilibrium points, which are E2, E4, and E5. Near the zx plane, there are also three equilibrium points, which are E1, E3, and E5.
Figure 3 illustrates the effective potential in different coordinate planes relative
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to the shape of the body for the asteroid 283 Emma. The value of the effective
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potential is calculated using Eq. (6), and the potential in Eq. (6) is calculated by the polyhedron model method. One can see that the irregular shape of the asteroid has
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significant influence on the effective potential. The shape of the effective potential in
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the equatorial plane looks like a circular cone surface with a hemi-ellipsoidal mirror at the bottom. The shapes of the effective potential in the yz and zx planes look like
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saddle surfaces.
Table 1 gives positions and Jacobi integrals of equilibrium points, which
indicates that all the five equilibrium points are out-of-plane equilibrium points. Besides, these equilibrium points are near the equatorial plane of 283 Emma. The inner equilibrium point E5 is near the mass center of the body. The equilibrium points 9
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E2, E4, and E5 are near the yz plane of the body-fixed frame while the equilibrium points E1, E3, and E5 are near the zx plane of the body-fixed frame. The equilibrium point E5 has the largest value of Jacobi integrals. Arraying the Jacobi integral from big to small, the equilibrium points are E5, E1, E3, E4, and E2. Table 2 illustrates
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eigenvalues of these equilibrium points. From Table 2, one can see that E5 has three pairs of imaginary eigenvalues. E1 has two pairs of imaginary eigenvalues and one pair of real eigenvalues. E2 has one pair of imaginary eigenvalues and four complex
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eigenvalues. The form of eigenvalues of E3 is the same as E1, and the form of eigenvalues of E4 is the same as E2. If the equilibrium point has three pairs of imaginary eigenvalues, the equilibrium point belongs to Case O1; if the equilibrium
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point has two pairs of imaginary eigenvalues and one pair of real eigenvalues, it belongs to Case O2; and if the equilibrium point has one pair of imaginary
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(Jiang et al. 2014).
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eigenvalues and four complex eigenvalues, the equilibrium point belongs to Case O5
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Because the shape for 283 Emma seems quite spherical, so an analysis on the influence of Emma's shape to the results, as compared to that of a spherical equivalent
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of Emma, would be of interest. The mean radius of Emma is 67.35km. Thus we calculate the effective potential of a sphere which has the radius of 67.35km and the same density and rotational speed with Emma. Figure 4 shows the shape and contour plot of effective potential in xy plane of a spherical equivalent of asteroid 283 Emma. There is only one single equilibrium point for the sphere. However, between the two 10
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contour lines for the effective potential with the value of 1000 m2s-2, there is a circle; on the circle, each of the points is an equilibrium point. The point on the circle is similar with the geostationary orbit of the Earth. Now compare Figure 4 with Figure 2(a), one can find that the distances between the equilibrium point and the mass center
(a)
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of the body are both about 100 km for these two bodies with different shapes.
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(b)
(d)
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(c)
Figure 1. The 3D exterior shape of 283 Emma calculated by the polyhedral model with the
observation data. (a) 3D visual angle; (b) projection in xy plane; (c) projection in xz plane; (d) projection in yz plane.
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(b)
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(a)
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(c)
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Figure 2. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283 Emma, the unit of the values in the zero-velocity surfaces is 104m2s-2. (a) The projection of equilibrium points E1-E5 in the equatorial plane of the asteroid and the contour plot of the zero-velocity surfaces; (b) The projection of equilibrium points E2, E4, and E5 in the yz plane of the asteroid and the contour plot of the zero-velocity surfaces; (c) The projection of equilibrium
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points E1, E3, and E5 in the zx plane of the asteroid and the contour plot of the zero-velocity
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surfaces.
(a) 14
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(b)
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(c)
Figure 3. Effective potentials in different coordinate planes for asteroid 283 Emma, the unit of the
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effective potential is 1.0×1010 m2s-2. (a) The effective potential in the equatorial plane; (b) The effective potential in the yz plane; (c) The effective potential in the zx plane.
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Figure 4. Shape and contour plot of effective potential in xy plane of a spherical equivalent for
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asteroid 283 Emma, the unit of the effective potential is 1.0 m2s-2.
Table 1 Positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma x (km)
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Points E1
E3
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E4
Jacobi integral
(1.0×103m2s-2)
2.89575
0.0760579
0.908667
0.00113853
90.5246
-0.386218
0.835341
-101.833
3.62567
-0.915645
0.905864
02.54511
-91.3871
0.239091
0.839646
-0.217661
-0.274144
0.135453
1.28719
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E5
z (km)
102.494
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E2
y (km)
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Equilibrium
Table 2 Eigenvalues of the equilibrium points around asteroid 283 Emma
Equilibrium
1
2
4
3
Points(× 10-3s-1) 16
5
6
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E1
0.302826i
-0.302826i
0.287484i
-0.287484i
0.214338
-0.214338
E2
0.252690i
-0.252690i
0.0973867+
-0.0973867+
0.0973867-
-0.0973867-
0.204360i
0.204360i
0.204360i
0.204360i
0.296233i
-0.296233i
0.282719i
-0.282719i
0.198177
-0.198177
E4
0.257555i
-0.257555i
0.0910352+
-0.0910352+
0.0910352-
-0.0910352-
0.198305i
0.198305i
0.198305i
0.198305i
0.512015i
-0.512015i
0.188540i
-0.188540i
0.714186i
-0.714186i
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E5
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E3
2.2 Equilibrium Points and Zero-Velocity Surfaces with Different Rotational Speed
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In this section, we investigate equilibrium points and zero-velocity surfaces with different suppositive rotational speed of the asteroid. The rotational speed of the
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asteroid may vary due to the YORP effect. For instance, due to the YORP effect, the
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rotational acceleration of asteroids 1862 Apollo and 1620 Geographos are (5.5±1.2) ×10-8 rad∙d-2(Ďurech et al. 2008a) and (1.15±0.15)×10-8 rad∙d-2 (Ďurech et al.
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2008b). For asteroid 283 Emma, there is no data about YORP effect in previous
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literature. If the rotational acceleration of asteroid 283 Emma is 1.0×10-8 rad∙d-2, then after 6.0×106 years, the rotational speed becomes twice as it now.
Figure 5 shows equilibrium points and the contour plot of the zero-velocity
surfaces for asteroid 283 Emma when the rotational speed equals twice as it is in present. From Figure 5, one can see that there are only three equilibrium points left. 17
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The structure of the contour plot of zero-velocity surfaces in the equatorial plane, the yz plane, and the zx plane is quite different from that in Figure 2. This indicates that the structure of gravitational environment changes with the variation of the rotational speed. The equilibrium point in the middle disappears when doubling the rotational
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speed, because the rotational potentials are different for different rotational speed, which makes the structures of the effective potentials different. In Figure 5, the three equilibrium points are near the y axis of the body-fixed frame. Table 3 presents
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positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma when ω=2.0ω0. We denote the equilibrium points from the –y axis to the +y axis as F1, F2, and F3. The equilibrium points F2 is in the middle among these three equilibrium
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points, and E2 belongs to Case O2. The equilibrium points F1 and F3 belong to Case O5. This implies that all the three equilibrium points are unstable. The equilibrium
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point F3 has the minimum value of the Jacobi integral, and the equilibrium point F2
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has the maximum one.
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Figure 6 shows equilibrium points and zero-velocity surfaces when the rotational speed equals a half as it is in present. There are also five equilibrium points, and the
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structure of the contour plot of zero-velocity surfaces in the equatorial plane, the yz plane, and the zx plane are the same as it in Figure 2. Table 4 presents positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma for this case. Although the positions and eigenvalues of equilibrium points are changed, the topological cases and stability of equilibrium points keep unchanged. During the 18
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variety of the rotational speed from 1.0ω0 to 2.0ω0, the number of equilibrium points changes from 5 to 3. When 1.0ω0< ω<1.615ω0, the number is 5; when ω=1.615ω0, the number changes to 4 (see Figure 7); and when 1.615ω0< ω<2.0ω0, the number is 3.
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Although the asteroid is near a sphere, the shape of the asteroid is not symmetric. This makes the component of the equilibrium points’ position on z axis is not zero. In addition, when the rotational speed changes from 1.0ω0 to 1.615ω0, the inner
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equilibrium point moves to -x axis, and the external equilibrium point E1 moves into the body of the asteroid (becomes F2 when ω=2.0ω0). However, for a sphere, the component of the equilibrium points’ position on z axis is zero; besides, when the
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rotational speed magnifies, the inner equilibrium point is located at the mass center
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and will not move.
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Table 3 Positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma when ω=2.0ω0 Equilibrium
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Points
x (km)
y (km)
z (km)
Jacobi integral
(1.0×103m2s-2)
1.8
-43. 0
1.8
1.287
F2
0.40665
18.51 0
0.0
1.289
-1.0
53.2
-2
1.273
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F1
F3
Table 4 Positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma when ω=0.5ω0 19
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Equilibrium
x (km)
y (km)
z (km)
Jacobi integral
(1.0×103m2s-2)
Points 154.897
2.98882
-0.0548419
0.553131
F2
0.643105
147.427
-0.0880521
0.535093
F3
-154.676
3.37249
-0.328535
0.552754
F4
2.30239
-147.627
0.0737637
0.535480
F5
-0.142963
-0.220147
0.134873
(a)
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F1
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1.28719
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(b)
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Figure 5. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283 Emma when ω=2.0ω0, the unit of the values in the zero-velocity surfaces is 1.0×104m2s-2. (a) The projection of equilibrium points in the equatorial plane of the asteroid and the contour plot of the zero-velocity surfaces; (b) The detailed contour plot of (a), the discrete scale in (b) is smaller than in (a); (c) The projection of equilibrium points in the yz plane of the asteroid and the contour plot
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of the zero-velocity surfaces; (d) The projection of equilibrium points in the zx plane of the
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asteroid and the contour plot of the zero-velocity surfaces.
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(c)
Figure 6. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283
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Emma when ω=0.5ω0, the unit of the values in the zero-velocity surfaces is 1.0×104m2s-2. (a) The projection of equilibrium points in the equatorial plane of the asteroid and the contour plot of the
zero-velocity surfaces; (b) The projection of equilibrium points in the yz plane of the asteroid and the contour plot of the zero-velocity surfaces; (c) The projection of equilibrium points in the zx plane of the asteroid and the contour plot of the zero-velocity surfaces.
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Figure 7. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283
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Emma when ω=1.615ω0, the unit of the values in the zero-velocity surfaces is 1.0×104m2s-2.
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3. Orbits in the Gravitational Potential of 283 Emma
There are several kinds of orbits around a minor body, including periodic orbits,
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stable orbits, unstable orbits, resonant orbits, non-periodic orbits, etc. An orbit can
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belong to two or more kinds of the above orbits, for instance, there exist resonant periodic orbit in the gravitational potential of the primary of the triple asteroid 216
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Kleopatra (Jiang et al. 2015). In this section, we investigate the periodic orbits near
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the surface of asteroid 283 Emma. Four different periodic orbits have been studied. Furthermore, we simulated the general orbit near the primary's equatorial plane to analyze the motion stability of the moonlet.
3.1 Periodic Orbits near the Surface of 283 Emma
The equations of motion (Jiang et al. 2016) can also be expressed as 24
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(11)
0 I where I is the 3 3 unit matrix and 0 is the 3 3 zero matrix, J , I 0
p r ω r , q r , z p q , H z is the gradient of H z . Here p T
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and q are generalised momentum and generalised coordinate, respectively. J is a symplectic matrix.
Denote the solution for Eq. (11) as z t f t , z 0 , which satisfies f 0, z 0 z 0 .
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Then z t f t , z 0 is an orbit around the asteroid. Denote the orbit as p. Using the f z , one can calculate the state transition matrix of the orbit p z
as
f p d . 0 z t
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t
(12)
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For a periodic orbit with the period T, the monodromy matrix is M T .
(13)
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Eigenvalues of the matrix M are characteristic multipliers of the periodic orbit. The
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system is a Hamalitonian system, thus M is a symplectic matrix. Assuming that is an eigenvalue of M , then
1 , , and 1 are also eigenvalues of M .
Because the system is a 6-dimensional system, the matrix M has six eigenvalues. Two of the eigenvalues equal 1. Using the characteristic multipliers, one can determine the topological cases and stability of the periodic orbit. If the periodic orbit
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has a characteristic multiplier not on the unit circle, then the periodic orbit has at least one characteristic multiplier with the modulus larger than 1, and thus the periodic orbit is unstable. If all the characteristic multipliers are on the unit circle, the periodic orbit is stable.
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Using the hierarchical grid searching method (Yu and Baoyin 2012), one can search periodic orbits around asteroids. The continuation procedure follows
Xi 1 Xi Xi ,
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ω × ω ×r0i U where Xi , and the subscript i represents the i-th iteration, r0i represents the step-size, Xi represents the gradient direction of the method, is
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searched in the gradient direction such that Xi1 is the optimal results. We calculated
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different periodic orbits with the hierarchical grid searching method. Figure 8 illustrates the 3D shapes of these periodic orbits in the gravitational potential of
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asteroid 283 Emma and the distributions of characteristic multipliers. Table 5 presents
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the characteristic multipliers of these periodic orbits. We can use the positions of the periodic orbits relative to the equilibrium points and the coordinate axis of moment of
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inertia to help classify the periodic orbits. Additionally, the distribution of characteristic multipliers of the periodic orbits can be also used to classify the periodic orbits.
The projection of each periodic orbit on the equatorial plane of 283 Emma has three circles. Compare Figure 8 with Figure 2, one can see that the part of the 26
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trajectory of the periodic orbit (a) is above the equilibrium point E3 and under the equilibrium point E1. The trajectory of the periodic orbit (b) is above the equilibrium point E1 and under the equilibrium point E3. In the periodic orbit (c), the particle moves above the equilibrium point E4 and under E2. And for the periodic orbit (d),
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particle orbits above the equilibrium point E2 and under E4.
In the periodic orbit (a), two characteristic multipliers are on the unit circle and the other two are on the +x axis. This implies that this periodic orbit is unstable. The
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periodic orbit (b) has the same distribution of characteristic multipliers with the periodic orbit (a). Thus these two periodic orbits have the same topological cases and the same stability. However, the relative positions of these two periodic orbits are
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different. Both of (a) and (b) have only one characteristic multiplier larger than 1. The
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biggest characteristic multiplier for (b) is 1.63022 while the biggest one for (a) is 1.37303. Thus the periodic orbit (b) is more unstable than the periodic orbit (a). All
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the characteristic multipliers (except 1) of the periodic orbit (c) are on the +x axis, two
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are larger than 1 and the other two are smaller than 1. Thus the periodic orbit (c) is also unstable. In the periodic orbit (d), four characteristic multipliers are on the
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complex plane, they are 0.883873±0.00457382i and 1.13135±0.00585523i. The topological case of the periodic orbit (d) is different from the above three periodic orbits.
Table 5. Characteristic multipliers of periodic orbits relative to 283 Emma 27
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1 , 2
3 , 4
5 , 6
a
0.893335±0.449390i
1.37303, 0.728314
1, 1
b
0.836337±0.548214i
1.63022, 0.613414
1, 1
c
1.66757, 0.599672
1.42285, 0.702813
1, 1
d
0.883873±0.00457382i
1.13135±0.00585523i
1, 1
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Orbits
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(d)
multipliers of these periodic orbits
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Figure 8. Four different periodic orbits around 283 Emma and the distributions of characteristic
3.2 General Orbit near the Primary's Equatorial Plane
Asteroid 283 Emma has a moonlet named S/2003 (283) 1. To analyze the orbit stability around the orbital position of the moonlet, we simulate the orbit of the 31
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moonlet S/2003 (283) 1 in the gravitational potential of asteroid 283 Emma. The gravitational potential of 283 Emma is calculated by the polyhedral model method with the observation data of the irregular shape. Figure 9 shows the simulation of moonlet’s orbit in the gravitational potential of asteroid 283 Emma.
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The 3D view of the orbit relative to the body-fixed system and the inertia system are presented. We also calculated the mechanical energy and the Jacobi integral of the orbit. The integration’s time is 28.7d, and the semi-major axis of the orbit is 581 km.
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From the 3D view of the orbit relative to the body-fixed system and the inertia system, one can see that the orbit is far from the mass center of the body, which implies that the influence of the gravitational potential caused by the irregular shape of 283 Emma
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is small. The maximum length of the primary, 283 Emma, is 161.34 km, thus the
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maximum radius of the primary is 80.67km. The separation/ maximum primary radius is 581km/80.67km=7.2. Thus one can infer that the orbit stability of the moonlet. The
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orbit is more likely to be stable when the separation/ maximum primary radius is a
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large number. The mechanical energy of the orbit varies periodically, and the period of the mechanical energy equals the rotation period of 283 Emma. The maximum
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value of the mechanical energy is -46.1037 J∙kg-1, the minimum value of the mechanical energy is -46.4321 J∙kg-1, and the mean value is -46.2674 J∙kg-1. The Jacobi integral of the orbit is a constant, which equals -1471.0464 J∙kg-1. The inclination and the longitude of ascending node also vary periodically, which have a long period, an intermediate period, and a short period. The values of the intermediate 32
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period of the inclination and the longitude of ascending node are the same, and equal the long periods of semi-major axis, eccentricity, and argument of pericenter.
Compare the orbit with the periodic orbits in Section 3.1, one can see that the periodic orbits near the surface of the body are more likely to be unstable. In addition,
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from the stability of the equilibrium points investigated in Section 2, one can conclude
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the stability of periodic orbits associated with the equilibrium points.
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(b)
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(f)
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(i)
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Figure 9.The simulation of moonlet’s orbit around 283 Emma. (a) 3D view of the orbit relative to the body-fixed system; (b) 3D view of the orbit relative to the inertia system; (c) The mechanical energy of the orbit; (d) The local plot of mechanical energy; (e)semi-major axis; (f) eccentricity;
5. Conclusions
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(g) inclination; (h) longitude of ascending node; (i) argument of pericenter.
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In this paper, we use the homogeneous polyhedral model to calculate the gravitational
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environment of the asteroid 283 Emma. We used the body-fixed frame defined by the three axes of moment of inertia and the mass center of the asteroid. The equilibrium
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points, zero-velocity surfaces, periodic orbits are computed and expressed in the body-fixed frame. It is found that there are totally five equilibrium points in the gravitational potential of the asteroid 283 Emma. Four of them are outside the asteroid body and one is inside. All the outside equilibrium points are unstable, and the inside equilibrium points are linearly stable. 37
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The zero-velocity surfaces and equilibrium points are investigated with the different values of the rotational speed of the asteroid 283 Emma. The zero-velocity surfaces and the positions of equilibrium points vary when the rotational speed of the asteroid 283 Emma varies. There are only three equilibrium points left when the
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rotational speed ω=2.0ω0, and all of them are unstable. If the rotational speed ω=0.5ω0, the number of equilibrium points remains unchanged, but the positions and Jacobi integral of equilibrium points are changed.
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We calculated four different periodic orbits near the surface of 283 Emma. These periodic orbits have different shapes and characteristic multipliers. All of the four periodic orbits are unstable. The orbit near the primary's equatorial plane is simulated
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to help understand the motion stability of the moonlet S/2003 (283) 1 in the
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gravitational potential of 283 Emma caused by the irregular shape. The mechanical energy of the orbit varies periodically while the Jacobi integral keeps conservative.
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The comparison of the results indicates that if the separation/ maximum primary
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radius is a large number, the orbit near the primary's equatorial plane is more likely to be stable.
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Acknowledgements The authors would highly appreciate with the reference’s valuable and helpful comments and suggestions for improving the paper. This project is funded by China Postdoctoral Science Foundation- General Program (No. 2017M610875), the National Natural Science Foundation of China (No. 11772356), and Government-Sponsored Overseas Study for Senior Research Scholar, Visiting Scholar, and Postdoctoral Program (CSC No. 201703170036). 38
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References Aljbaae, S., Chanut, T. G. G., Carruba, V., Souchay, J., Prado, A. F. B. A. Amarante, A., 2017. Mon. Not. R. Astron. Soc. 464 (3): 3552-3560.
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Belton, M. J. S., Chapman, C. R., Thomas, P. C., Davies, M. E., Greenberg, R., Klaasen, K., et al., 1995. Nature, 374(6525), 785-788. Bernasconi, R., Behrend, L., Roy, R. et al., 2006. Astron. Astrophys. 446(3), 1177-1184.
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Birlan, M., Nedelcu, D. A., Popescu, M., Vernazza, P., Colas, F., Kryszczyńska, A., 2014. Mon. Not. R. Astron. Soc. 437(1), 176-184. Bosanac, N., Howell, K. C., Fischbach, E., 2015. Celest. Mech. Dynam. Astron. 122(1), 27-52. Bouvier, Jean-Claude, Horch, E., 2011. American Astronomical Society, AAS Meeting #218, id.409.14.
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Descamps, P., 2010. Icarus, 207(2), 758-768.
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Ďurech, J., Vokrouhlický, D., Kaasalainen, M. et al. 2008a. Astron. Astrophys. 488(1), 345-350.
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Ďurech J, Vokrouhlický D, Kaasalainen M. et al. 2008b. Astron. Astrophys. 489(2), L25-L28.
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Jiang, Y., Baoyin, H., 2016. Astron. J. 152(5), 137. Jiang, Y., Baoyin, H., Li, H., 2015. Astrophys. Space Sci. 360(2), 63.
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Jiang, Y., Baoyin, H., Li, J., Li, H., 2014. Astrophys. Space Sci. 349, 83-106. Jiang, Y., Baoyin, H., Wang, X., Yu Y., Li, H., Peng C., Zhang Z., 2016. Nonlinear Dynam. 83(1), 231-252. Matter, A., Delbo, M., Ligori, S., Crouzet, N., Tanga, P., 2011. Icarus, 215(1), 47-56. Merline, W. J., Dumas, C., Siegler, N., Close, L. M., Chapman, C. R., Tamblyn, P. M., et al., 2003. International Astronomical Union. 39
ACCEPTED MANUSCRIPT Michalowski, T., Kaasalainen, M., Polińska, M., Marciniak, A., Kwiatkowski, T., Kryszczyńska, A., Velichko, F. P., 2006. Astron. Astrophys. 459(2), 663-668. Marchis, F., Descamps, P., Berthier, J., Hestroffer, D., Vachier, F., Baek, M., et al., 2008. Icarus, 196(1), 97-118.
Scheeres, D. J., 2012. Acta. Astronaut. 7, 21-14. Scheeres, D. J., 2016. J. Nonlinear Sci. 26(5), 1445-1482.
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Ortiz, J. L., Thirouin, A., Bagatin, A. C., Duffard, R., Licandro, J., Richardson, D. C., et al., 2012. Mon. Not. R. Astron. Soc. 419(3), 2315-2324.
Strabla, L., Quadri, U., Girelli, R., 2011. Bulletin of the Minor Planets Section of the Association of Lunar and Planetary Observers, 38( 3),169-172.
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Walsh, K. J., Jacobson, S. A., 2015. pp. 375-393. Univ. of Arizona, Tucson.
Warner, B. D., Harris, A. W., Pravec, P., 2009. The asteroid lightcurve database. Icarus, 202(1), 134-146. Werner, R. A., Scheeres, D. J., 1997. Celest. Mech. Dynam. Astron. 65 (3), 313-344.
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Equilibria and Orbits around Asteroid using the Polyhedral Model Yu Jiang , Xiaodong Liu PII: DOI: Reference:
S1384-1076(18)30139-8 https://doi.org/10.1016/j.newast.2018.11.007 NEASPA 1244
To appear in:
New Astronomy
Received date: Revised date: Accepted date:
8 May 2018 2 November 2018 22 November 2018
Please cite this article as: Yu Jiang , Xiaodong Liu , Equilibria and Orbits around Asteroid using the Polyhedral Model, New Astronomy (2018), doi: https://doi.org/10.1016/j.newast.2018.11.007
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HIGHLIGHTS
The polyhedral model is used to compute the potential of the asteroid.
It is found that there are five equilibrium points for the asteroid 283 Emma.
The zero-velocity surfaces and the equilibria vary if the rotational speed vary.
The separation/primary-radius has influence on the stability of the orbit.
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Equilibria and Orbits around Asteroid using the Polyhedral Model
Yu Jiang1, 2,3 and Xiaodong Liu3 1. State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043, China
3. Astronomy Research Unit, University of Oulu, Oulu 90014, Finland
Y. Jiang () e-mail: [email protected] (corresponding author)
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2. School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
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Abstract. In the current study, we use the polyhedral model to compute the potential of the asteroid. There are five equilibrium points in the gravitational field of the asteroid 283 Emma. We concluded that the zero-velocity surfaces and the equilibrium points change with the suppositive variation of the rotational speed of the asteroid. It is found that if the rotational speed equals a half as it is in present, the number of equilibrium points is also five. However, if the rotational speed equals twice as it is in present, there are only three equilibrium points left. Four different periodic orbits are calculated using the hierarchical grid searching method. We calculated characteristic
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multipliers of periodic orbits to invistigate the stability of these periodic orbits. The orbit near the primary's equatorial plane is more likely to be stable when the separation/ primary-radius is a large number.
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Key Words: Asteroid; Polyhedral Model; Zero-Velocity Surfaces; Equilibrium Points
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1. Introductions
The moonlet (S/2003 (283) 1) of the main belt asteroid (283) Emma was discovered in 2003 (Merline et al. 2003; Marchis et al. 2008). After that, this binary asteroid system received interests to researchers. Michalowski et al. (2006) used the available lightcurves and calculated the shape model and spin parameters of the asteroid (283) 2
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Emma. Marchis et al. (2008) used 8–10-m class telescopes conducted observations of four different binary asteroid systems: (130) Elektra, (283) Emma, (379) Huenna, and (3749) Balam. Their observation data confirmed the existence of their asteroidal moonlets. That study suggested a different origin for these moonlets, and the moonlet
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of (283) Emma is more likely to be produced by tidal effects. However, they concluded that for (379) Huenna and (3749) Balam, their moonlets are likely to be generated by mutual capture. Bouvier and Horch (2011) presented the image
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reconstructions of 283 Emma of data from the data of Differential Speckle Survey Instrument (DSSI).
(283) Emma is a P-type asteroid with a significant porosity (Marchis et al. 2008).
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Previous studies also indicated the diameter, bulk density, and rotational period of
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(283) Emma. Michalowski et al. (2006) presented the primary diameter for 283 Emma to be 148±5 km from STM (Standard Thermal Model) model, and 141±6 km from
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NEATM (near-Earth asteroid Thermal Model) model. The size of the moonlet
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S/2003(283)1 is estimated to be 9 ±5 km. Thus the main belt binary asteroidal system (283) Emma and S/2003(283)1 is a large-size-ratio binary. Marchis et al. (2008)
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derived the bulk density and the rotational period for 283 Emma to be 0.7-1.0 g∙cm-3 and 6.888h, respectively. Michalowski et al. (2006) suggested the rotational period for 283 Emma is 6.895 ± 0.003 h. Strabla et al. (2011) used the Schmidt 0.32-m f/3.1 telescope at Bassano Bresciano Observatory and showed that the period of 283 Emma is 6.896±0.001h. 3
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The dynamical behavior of the binary asteroid systems is very complicated for different binary asteroid systems. For the synchronous binary asteroids (Descamps 2010), such as 809 Lundia (Birlan et al. 2014), 854 Frostia (Bernasconi et al. 2006), 1089 Tama (Bernasconi et al. 2006), etc., the two asteroids in the system
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synchronously rotate; in other words, the orbital angular speed and the attitudinal angular speed of these two asteroids are equal. Generally, two asteroids of the synchronous binary system have the same orbital angular speed, and two different
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attitudinal angular speeds. For the high-size-ratio binary asteroid systems (Ortiz et al. 2012; Bosanac et al. 2015; Walsh and Jacobson 2015), the secondary plays the role as the moonlet in the gravitational potential of the primary, such as (41) Daphne (Matter
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et al. 2011), 243 Ida (Belton et al. 1995), 283 Emma (Marchis et al. 2008), et al. The dynamical behaviors of the moonlet in the gravitational potential of the primary is
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related to the moonlet’s orbit, the primary’s shape and mass distribution, as well as the
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primary’s rotational speed.
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In this paper, we investigate the dynamical behaviors in the gravitational potential of the primary of the large-size-ratio main belt binary asteroid (283) Emma
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and S/2003 (283) 1. The irregular shape of the (283) Emma and its influence on the gravitational environment is considered. The gravitation of the moonlet S/2003(283)1 is not included in the model when calculating orbits and equilibrium points because the gravitation ratio of the moonlet and the primary is smaller than 3.0×10-4. This study is useful for understanding the dynamical behaviors in the large-size-ratio 4
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binary asteroids. This paper is organized as follows: Section 2 investigates the 3D irregular shape, the equilibrium points and zero-velocity surfaces of asteroid 283Emma. The 3D irregular shape is calculated by the homogeneous polyhedral model with the observation data from Michalowski et al. (2006). To study the variety
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of the zero-velocity surfaces and equilibrium points under the conditions of the variety of the rotational speed, we calculated the zero-velocity surfaces and equilibrium points for two cases, the first case is to assume the rotational speed twice,
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and the second case is to assume the rotational speed half. Section 3 deals with the orbits in the gravitational potential of asteroid 283 Emma. Both of the periodic orbit and the general orbit have been investigated. Four different periodic orbits near the
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surface of 283 Emma are calculated, the 3D shapes, positions, characteristic multipliers, as well as the stability of the periodic orbits are discussed. The general
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orbit is calculated near the primary's equatorial plane. The 3D shapes of the general
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orbit are plotted in the body-fixed frame and the inertia system. Both the mechanical
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energy and the Jacobi integral of the orbit are calculated.
2. Equilibrium Points and Zero-Velocity Surfaces
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In this section, we calculate the equilibrium points and zero-velocity surfaces in the gravitational potential of an asteroid. The motion equations of a massless particle in the gravitational potential of the asteroid can be expressed in Cartesian coordinates x, y, and z. Let the rotational angular velocity of the asteroid be ω , the body-fixed vector from the mass center of the asteroid to the particle be r . The parameter G = 5
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6.67 × 10-11 m3 kg-1 s-2 is the Newtonian gravitational constant, and σ is the body’s bulk density. The axes of the body-fixed frame are along the principal axis of inertia. Assume the asteroid rotates around its biggest moment of inertia. The z-axis points to the biggest moment of inertia while the x-axis points to the smallest moment of inertia.
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Equations of motion (Scheeres 2012; Aljbaae et al. 2017) for a particle orbiting around the asteroid read
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U 2 x 2 y x x 0 U 2 0, y 2 x y y U 0 z z
(1)
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where the gravitational potential U of the asteroid can be calculated by the polyhedron
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model method (Werner and Scheeres 1997):
(2)
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1 1 U G re Ee re Le + G r f Ff r f f . 2 2 eedges f faces
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Here, re and rf are body-fixed vectors from a field point to any point on the edge e and face f, respectively. Ee and Ff are body-fixed tensors of the geometric parameters of
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edge e and face f, respectively. Le is the integration factor that related to a fixed point and edge e, and ωf is the solid angle. The gravitational force acceleration U and the gravity gradient matrix U are
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eedges
Ee re Le G
U G
eedges
f faces
Ee Le +G
Ff r f f
(3)
(4)
f faces
Ff f
The Jacobi integral H follows
1 2 2 x y 2 z 2 x2 y 2 , 2 2
and the effective potential V is
2
x 2
2
y2 .
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V U
(5)
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H U
The positions of equilibrium points (Jiang et al. 2014; Scheeres 2016) in the
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potential of the asteroid is calculated by the following equations
(7)
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V x, y, z V x, y, z V x, y, z 0, x y z
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The linearised equations of motion in the vicinity of the equilibrium point (Jiang et al. 2014) around the asteroid follow
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0 2 0 Vxx Vxy Vxz d2 d 2 0 0 Vxy Vyy Vyz 0 . 2 dt dt 0 0 0 Vxz Vyz Vzz
Here,
frame,
x，y，z
(8)
is the position vector of the massless particle in the body-fixed
xE，yE，zE
is the position vector of equilibrium points in the body-fixed
2V frame, x xE， y yE， z zE , Vpq p, q x, y, z represent pq E 7
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the elements of Hessian matrix of the effective potential V . The Hessian matrix of the effective potential V can be calculated by 0 0 G Ee Le +G Ff f . eedges f faces 0
(9)
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2 0 V 0 2 0 0
Using the rotation speed and the elements of Hessian matrix, one can compute the eigenvalues (Jiang et al. 2014, 2016) of equilibrium points
Vxz
Vxz Vyz 0 . 2 Vzz
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2 Vxx 2 Vxy 2 Vxy 2 Vyy Vyz
(10)
where is the eigenvalues of the equilibrium point.
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2.1 Equilibrium Points and Zero-Velocity Surfaces for 283 Emma
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We now calculate V for 283 Emma. The density and rotational period are 0.81
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g cm3 and 6.888 h, respectively (Michałowski et al. 2006; Marchis et al. 2008). Figure 1 shows the 3D exterior shape of 283 Emma calculated by the polyhedral
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model with the observation data (Michalowski et al. 2006), where the shape is
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respectively plotted in the xy, yz, and zx planes. The model has 2032 faces and 1018 vertices. The figure indicates that the shape of 283 Emma is similar to a potato, and the shape viewed in different coordinate planes are different. The lengths of the three axes of 283 Emma are 161.34, 111.61, and 108.20 km.
Figure 2 presents equilibrium points and the contour plot of the zero-velocity 8
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surfaces for asteroid 283 Emma. The figure indicates that 283 Emma has five equilibrium points; four of them are outside and one inside. From Figure 2, one can see that the topological structures of the zero-velocity surfaces in different coordinate planes are quite different. The number of equilibrium points near the coordinate
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planes is related to the topological structures of the zero-velocity surfaces on the coordinate planes. Near the xy plane, i.e. the equatorial plane of the asteroid, there are five equilibrium points. This is because the asteroid rotates around z-axis. Near the yz
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plane, there are three equilibrium points, which are E2, E4, and E5. Near the zx plane, there are also three equilibrium points, which are E1, E3, and E5.
Figure 3 illustrates the effective potential in different coordinate planes relative
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to the shape of the body for the asteroid 283 Emma. The value of the effective
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potential is calculated using Eq. (6), and the potential in Eq. (6) is calculated by the polyhedron model method. One can see that the irregular shape of the asteroid has
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significant influence on the effective potential. The shape of the effective potential in
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the equatorial plane looks like a circular cone surface with a hemi-ellipsoidal mirror at the bottom. The shapes of the effective potential in the yz and zx planes look like
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saddle surfaces.
Table 1 gives positions and Jacobi integrals of equilibrium points, which
indicates that all the five equilibrium points are out-of-plane equilibrium points. Besides, these equilibrium points are near the equatorial plane of 283 Emma. The inner equilibrium point E5 is near the mass center of the body. The equilibrium points 9
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E2, E4, and E5 are near the yz plane of the body-fixed frame while the equilibrium points E1, E3, and E5 are near the zx plane of the body-fixed frame. The equilibrium point E5 has the largest value of Jacobi integrals. Arraying the Jacobi integral from big to small, the equilibrium points are E5, E1, E3, E4, and E2. Table 2 illustrates
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eigenvalues of these equilibrium points. From Table 2, one can see that E5 has three pairs of imaginary eigenvalues. E1 has two pairs of imaginary eigenvalues and one pair of real eigenvalues. E2 has one pair of imaginary eigenvalues and four complex
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eigenvalues. The form of eigenvalues of E3 is the same as E1, and the form of eigenvalues of E4 is the same as E2. If the equilibrium point has three pairs of imaginary eigenvalues, the equilibrium point belongs to Case O1; if the equilibrium
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point has two pairs of imaginary eigenvalues and one pair of real eigenvalues, it belongs to Case O2; and if the equilibrium point has one pair of imaginary
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(Jiang et al. 2014).
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eigenvalues and four complex eigenvalues, the equilibrium point belongs to Case O5
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Because the shape for 283 Emma seems quite spherical, so an analysis on the influence of Emma's shape to the results, as compared to that of a spherical equivalent
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of Emma, would be of interest. The mean radius of Emma is 67.35km. Thus we calculate the effective potential of a sphere which has the radius of 67.35km and the same density and rotational speed with Emma. Figure 4 shows the shape and contour plot of effective potential in xy plane of a spherical equivalent of asteroid 283 Emma. There is only one single equilibrium point for the sphere. However, between the two 10
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contour lines for the effective potential with the value of 1000 m2s-2, there is a circle; on the circle, each of the points is an equilibrium point. The point on the circle is similar with the geostationary orbit of the Earth. Now compare Figure 4 with Figure 2(a), one can find that the distances between the equilibrium point and the mass center
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of the body are both about 100 km for these two bodies with different shapes.
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(b)
(d)
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Figure 1. The 3D exterior shape of 283 Emma calculated by the polyhedral model with the
observation data. (a) 3D visual angle; (b) projection in xy plane; (c) projection in xz plane; (d) projection in yz plane.
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(c)
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Figure 2. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283 Emma, the unit of the values in the zero-velocity surfaces is 104m2s-2. (a) The projection of equilibrium points E1-E5 in the equatorial plane of the asteroid and the contour plot of the zero-velocity surfaces; (b) The projection of equilibrium points E2, E4, and E5 in the yz plane of the asteroid and the contour plot of the zero-velocity surfaces; (c) The projection of equilibrium
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points E1, E3, and E5 in the zx plane of the asteroid and the contour plot of the zero-velocity
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surfaces.
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(c)
Figure 3. Effective potentials in different coordinate planes for asteroid 283 Emma, the unit of the
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effective potential is 1.0×1010 m2s-2. (a) The effective potential in the equatorial plane; (b) The effective potential in the yz plane; (c) The effective potential in the zx plane.
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Figure 4. Shape and contour plot of effective potential in xy plane of a spherical equivalent for
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asteroid 283 Emma, the unit of the effective potential is 1.0 m2s-2.
Table 1 Positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma x (km)
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Points E1
E3
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E4
Jacobi integral
(1.0×103m2s-2)
2.89575
0.0760579
0.908667
0.00113853
90.5246
-0.386218
0.835341
-101.833
3.62567
-0.915645
0.905864
02.54511
-91.3871
0.239091
0.839646
-0.217661
-0.274144
0.135453
1.28719
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E5
z (km)
102.494
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E2
y (km)
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Equilibrium
Table 2 Eigenvalues of the equilibrium points around asteroid 283 Emma
Equilibrium
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2
4
3
Points(× 10-3s-1) 16
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E1
0.302826i
-0.302826i
0.287484i
-0.287484i
0.214338
-0.214338
E2
0.252690i
-0.252690i
0.0973867+
-0.0973867+
0.0973867-
-0.0973867-
0.204360i
0.204360i
0.204360i
0.204360i
0.296233i
-0.296233i
0.282719i
-0.282719i
0.198177
-0.198177
E4
0.257555i
-0.257555i
0.0910352+
-0.0910352+
0.0910352-
-0.0910352-
0.198305i
0.198305i
0.198305i
0.198305i
0.512015i
-0.512015i
0.188540i
-0.188540i
0.714186i
-0.714186i
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E3
2.2 Equilibrium Points and Zero-Velocity Surfaces with Different Rotational Speed
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In this section, we investigate equilibrium points and zero-velocity surfaces with different suppositive rotational speed of the asteroid. The rotational speed of the
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asteroid may vary due to the YORP effect. For instance, due to the YORP effect, the
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rotational acceleration of asteroids 1862 Apollo and 1620 Geographos are (5.5±1.2) ×10-8 rad∙d-2(Ďurech et al. 2008a) and (1.15±0.15)×10-8 rad∙d-2 (Ďurech et al.
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2008b). For asteroid 283 Emma, there is no data about YORP effect in previous
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literature. If the rotational acceleration of asteroid 283 Emma is 1.0×10-8 rad∙d-2, then after 6.0×106 years, the rotational speed becomes twice as it now.
Figure 5 shows equilibrium points and the contour plot of the zero-velocity
surfaces for asteroid 283 Emma when the rotational speed equals twice as it is in present. From Figure 5, one can see that there are only three equilibrium points left. 17
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The structure of the contour plot of zero-velocity surfaces in the equatorial plane, the yz plane, and the zx plane is quite different from that in Figure 2. This indicates that the structure of gravitational environment changes with the variation of the rotational speed. The equilibrium point in the middle disappears when doubling the rotational
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speed, because the rotational potentials are different for different rotational speed, which makes the structures of the effective potentials different. In Figure 5, the three equilibrium points are near the y axis of the body-fixed frame. Table 3 presents
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positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma when ω=2.0ω0. We denote the equilibrium points from the –y axis to the +y axis as F1, F2, and F3. The equilibrium points F2 is in the middle among these three equilibrium
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points, and E2 belongs to Case O2. The equilibrium points F1 and F3 belong to Case O5. This implies that all the three equilibrium points are unstable. The equilibrium
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point F3 has the minimum value of the Jacobi integral, and the equilibrium point F2
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has the maximum one.
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Figure 6 shows equilibrium points and zero-velocity surfaces when the rotational speed equals a half as it is in present. There are also five equilibrium points, and the
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structure of the contour plot of zero-velocity surfaces in the equatorial plane, the yz plane, and the zx plane are the same as it in Figure 2. Table 4 presents positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma for this case. Although the positions and eigenvalues of equilibrium points are changed, the topological cases and stability of equilibrium points keep unchanged. During the 18
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variety of the rotational speed from 1.0ω0 to 2.0ω0, the number of equilibrium points changes from 5 to 3. When 1.0ω0< ω<1.615ω0, the number is 5; when ω=1.615ω0, the number changes to 4 (see Figure 7); and when 1.615ω0< ω<2.0ω0, the number is 3.
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Although the asteroid is near a sphere, the shape of the asteroid is not symmetric. This makes the component of the equilibrium points’ position on z axis is not zero. In addition, when the rotational speed changes from 1.0ω0 to 1.615ω0, the inner
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equilibrium point moves to -x axis, and the external equilibrium point E1 moves into the body of the asteroid (becomes F2 when ω=2.0ω0). However, for a sphere, the component of the equilibrium points’ position on z axis is zero; besides, when the
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rotational speed magnifies, the inner equilibrium point is located at the mass center
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and will not move.
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Table 3 Positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma when ω=2.0ω0 Equilibrium
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Points
x (km)
y (km)
z (km)
Jacobi integral
(1.0×103m2s-2)
1.8
-43. 0
1.8
1.287
F2
0.40665
18.51 0
0.0
1.289
-1.0
53.2
-2
1.273
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F1
F3
Table 4 Positions and Jacobi integrals of equilibrium points in the vicinity of 283 Emma when ω=0.5ω0 19
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Equilibrium
x (km)
y (km)
z (km)
Jacobi integral
(1.0×103m2s-2)
Points 154.897
2.98882
-0.0548419
0.553131
F2
0.643105
147.427
-0.0880521
0.535093
F3
-154.676
3.37249
-0.328535
0.552754
F4
2.30239
-147.627
0.0737637
0.535480
F5
-0.142963
-0.220147
0.134873
(a)
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F1
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1.28719
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Figure 5. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283 Emma when ω=2.0ω0, the unit of the values in the zero-velocity surfaces is 1.0×104m2s-2. (a) The projection of equilibrium points in the equatorial plane of the asteroid and the contour plot of the zero-velocity surfaces; (b) The detailed contour plot of (a), the discrete scale in (b) is smaller than in (a); (c) The projection of equilibrium points in the yz plane of the asteroid and the contour plot
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of the zero-velocity surfaces; (d) The projection of equilibrium points in the zx plane of the
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asteroid and the contour plot of the zero-velocity surfaces.
(a) 22
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(c)
Figure 6. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283
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Emma when ω=0.5ω0, the unit of the values in the zero-velocity surfaces is 1.0×104m2s-2. (a) The projection of equilibrium points in the equatorial plane of the asteroid and the contour plot of the
zero-velocity surfaces; (b) The projection of equilibrium points in the yz plane of the asteroid and the contour plot of the zero-velocity surfaces; (c) The projection of equilibrium points in the zx plane of the asteroid and the contour plot of the zero-velocity surfaces.
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Figure 7. Equilibrium points and the contour plot of the zero-velocity surfaces for asteroid 283
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Emma when ω=1.615ω0, the unit of the values in the zero-velocity surfaces is 1.0×104m2s-2.
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3. Orbits in the Gravitational Potential of 283 Emma
There are several kinds of orbits around a minor body, including periodic orbits,
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stable orbits, unstable orbits, resonant orbits, non-periodic orbits, etc. An orbit can
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belong to two or more kinds of the above orbits, for instance, there exist resonant periodic orbit in the gravitational potential of the primary of the triple asteroid 216
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Kleopatra (Jiang et al. 2015). In this section, we investigate the periodic orbits near
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the surface of asteroid 283 Emma. Four different periodic orbits have been studied. Furthermore, we simulated the general orbit near the primary's equatorial plane to analyze the motion stability of the moonlet.
3.1 Periodic Orbits near the Surface of 283 Emma
The equations of motion (Jiang et al. 2016) can also be expressed as 24
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(11)
0 I where I is the 3 3 unit matrix and 0 is the 3 3 zero matrix, J , I 0
p r ω r , q r , z p q , H z is the gradient of H z . Here p T
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and q are generalised momentum and generalised coordinate, respectively. J is a symplectic matrix.
Denote the solution for Eq. (11) as z t f t , z 0 , which satisfies f 0, z 0 z 0 .
6 6 matrix f :
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Then z t f t , z 0 is an orbit around the asteroid. Denote the orbit as p. Using the f z , one can calculate the state transition matrix of the orbit p z
as
f p d . 0 z t
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t
(12)
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For a periodic orbit with the period T, the monodromy matrix is M T .
(13)
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Eigenvalues of the matrix M are characteristic multipliers of the periodic orbit. The
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system is a Hamalitonian system, thus M is a symplectic matrix. Assuming that is an eigenvalue of M , then
1 , , and 1 are also eigenvalues of M .
Because the system is a 6-dimensional system, the matrix M has six eigenvalues. Two of the eigenvalues equal 1. Using the characteristic multipliers, one can determine the topological cases and stability of the periodic orbit. If the periodic orbit
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has a characteristic multiplier not on the unit circle, then the periodic orbit has at least one characteristic multiplier with the modulus larger than 1, and thus the periodic orbit is unstable. If all the characteristic multipliers are on the unit circle, the periodic orbit is stable.
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Using the hierarchical grid searching method (Yu and Baoyin 2012), one can search periodic orbits around asteroids. The continuation procedure follows
Xi 1 Xi Xi ,
(14)
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ω × ω ×r0i U where Xi , and the subscript i represents the i-th iteration, r0i represents the step-size, Xi represents the gradient direction of the method, is
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searched in the gradient direction such that Xi1 is the optimal results. We calculated
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different periodic orbits with the hierarchical grid searching method. Figure 8 illustrates the 3D shapes of these periodic orbits in the gravitational potential of
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asteroid 283 Emma and the distributions of characteristic multipliers. Table 5 presents
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the characteristic multipliers of these periodic orbits. We can use the positions of the periodic orbits relative to the equilibrium points and the coordinate axis of moment of
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inertia to help classify the periodic orbits. Additionally, the distribution of characteristic multipliers of the periodic orbits can be also used to classify the periodic orbits.
The projection of each periodic orbit on the equatorial plane of 283 Emma has three circles. Compare Figure 8 with Figure 2, one can see that the part of the 26
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trajectory of the periodic orbit (a) is above the equilibrium point E3 and under the equilibrium point E1. The trajectory of the periodic orbit (b) is above the equilibrium point E1 and under the equilibrium point E3. In the periodic orbit (c), the particle moves above the equilibrium point E4 and under E2. And for the periodic orbit (d),
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particle orbits above the equilibrium point E2 and under E4.
In the periodic orbit (a), two characteristic multipliers are on the unit circle and the other two are on the +x axis. This implies that this periodic orbit is unstable. The
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periodic orbit (b) has the same distribution of characteristic multipliers with the periodic orbit (a). Thus these two periodic orbits have the same topological cases and the same stability. However, the relative positions of these two periodic orbits are
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different. Both of (a) and (b) have only one characteristic multiplier larger than 1. The
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biggest characteristic multiplier for (b) is 1.63022 while the biggest one for (a) is 1.37303. Thus the periodic orbit (b) is more unstable than the periodic orbit (a). All
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the characteristic multipliers (except 1) of the periodic orbit (c) are on the +x axis, two
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are larger than 1 and the other two are smaller than 1. Thus the periodic orbit (c) is also unstable. In the periodic orbit (d), four characteristic multipliers are on the
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complex plane, they are 0.883873±0.00457382i and 1.13135±0.00585523i. The topological case of the periodic orbit (d) is different from the above three periodic orbits.
Table 5. Characteristic multipliers of periodic orbits relative to 283 Emma 27
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1 , 2
3 , 4
5 , 6
a
0.893335±0.449390i
1.37303, 0.728314
1, 1
b
0.836337±0.548214i
1.63022, 0.613414
1, 1
c
1.66757, 0.599672
1.42285, 0.702813
1, 1
d
0.883873±0.00457382i
1.13135±0.00585523i
1, 1
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Orbits
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(b)
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(c)
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(d)
multipliers of these periodic orbits
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Figure 8. Four different periodic orbits around 283 Emma and the distributions of characteristic
3.2 General Orbit near the Primary's Equatorial Plane
Asteroid 283 Emma has a moonlet named S/2003 (283) 1. To analyze the orbit stability around the orbital position of the moonlet, we simulate the orbit of the 31
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moonlet S/2003 (283) 1 in the gravitational potential of asteroid 283 Emma. The gravitational potential of 283 Emma is calculated by the polyhedral model method with the observation data of the irregular shape. Figure 9 shows the simulation of moonlet’s orbit in the gravitational potential of asteroid 283 Emma.
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The 3D view of the orbit relative to the body-fixed system and the inertia system are presented. We also calculated the mechanical energy and the Jacobi integral of the orbit. The integration’s time is 28.7d, and the semi-major axis of the orbit is 581 km.
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From the 3D view of the orbit relative to the body-fixed system and the inertia system, one can see that the orbit is far from the mass center of the body, which implies that the influence of the gravitational potential caused by the irregular shape of 283 Emma
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is small. The maximum length of the primary, 283 Emma, is 161.34 km, thus the
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maximum radius of the primary is 80.67km. The separation/ maximum primary radius is 581km/80.67km=7.2. Thus one can infer that the orbit stability of the moonlet. The
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orbit is more likely to be stable when the separation/ maximum primary radius is a
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large number. The mechanical energy of the orbit varies periodically, and the period of the mechanical energy equals the rotation period of 283 Emma. The maximum
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value of the mechanical energy is -46.1037 J∙kg-1, the minimum value of the mechanical energy is -46.4321 J∙kg-1, and the mean value is -46.2674 J∙kg-1. The Jacobi integral of the orbit is a constant, which equals -1471.0464 J∙kg-1. The inclination and the longitude of ascending node also vary periodically, which have a long period, an intermediate period, and a short period. The values of the intermediate 32
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period of the inclination and the longitude of ascending node are the same, and equal the long periods of semi-major axis, eccentricity, and argument of pericenter.
Compare the orbit with the periodic orbits in Section 3.1, one can see that the periodic orbits near the surface of the body are more likely to be unstable. In addition,
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from the stability of the equilibrium points investigated in Section 2, one can conclude
(a)
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the stability of periodic orbits associated with the equilibrium points.
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(b)
(d)
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(c)
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(f)
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(h)
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Figure 9.The simulation of moonlet’s orbit around 283 Emma. (a) 3D view of the orbit relative to the body-fixed system; (b) 3D view of the orbit relative to the inertia system; (c) The mechanical energy of the orbit; (d) The local plot of mechanical energy; (e)semi-major axis; (f) eccentricity;
5. Conclusions
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(g) inclination; (h) longitude of ascending node; (i) argument of pericenter.
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In this paper, we use the homogeneous polyhedral model to calculate the gravitational
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environment of the asteroid 283 Emma. We used the body-fixed frame defined by the three axes of moment of inertia and the mass center of the asteroid. The equilibrium
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points, zero-velocity surfaces, periodic orbits are computed and expressed in the body-fixed frame. It is found that there are totally five equilibrium points in the gravitational potential of the asteroid 283 Emma. Four of them are outside the asteroid body and one is inside. All the outside equilibrium points are unstable, and the inside equilibrium points are linearly stable. 37
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The zero-velocity surfaces and equilibrium points are investigated with the different values of the rotational speed of the asteroid 283 Emma. The zero-velocity surfaces and the positions of equilibrium points vary when the rotational speed of the asteroid 283 Emma varies. There are only three equilibrium points left when the
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rotational speed ω=2.0ω0, and all of them are unstable. If the rotational speed ω=0.5ω0, the number of equilibrium points remains unchanged, but the positions and Jacobi integral of equilibrium points are changed.
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We calculated four different periodic orbits near the surface of 283 Emma. These periodic orbits have different shapes and characteristic multipliers. All of the four periodic orbits are unstable. The orbit near the primary's equatorial plane is simulated
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to help understand the motion stability of the moonlet S/2003 (283) 1 in the
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gravitational potential of 283 Emma caused by the irregular shape. The mechanical energy of the orbit varies periodically while the Jacobi integral keeps conservative.
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The comparison of the results indicates that if the separation/ maximum primary
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radius is a large number, the orbit near the primary's equatorial plane is more likely to be stable.
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Acknowledgements The authors would highly appreciate with the reference’s valuable and helpful comments and suggestions for improving the paper. This project is funded by China Postdoctoral Science Foundation- General Program (No. 2017M610875), the National Natural Science Foundation of China (No. 11772356), and Government-Sponsored Overseas Study for Senior Research Scholar, Visiting Scholar, and Postdoctoral Program (CSC No. 201703170036). 38
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