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Abstract

One of the techniques used in the past decade to determine the shape with a good accuracy and estimate certain physical features (volume, mass, moments of inertia) of asteroids is the polyhedral model method. We rebuild the shape of the asteroid 433 Eros using data from 1998 December observations of the probe Near-Earth-Asteroid-Rendezvous-Shoemaker. In our computations, we use a code that avoids singularities from the line integrals of a homogeneous arbitrary shaped polyhedral source. This code evaluates the gravitational potential function and its first- and second-order derivatives. Taking the rotation of asteroid 433 Eros into consideration, the aim of this work is to analyse the dynamics of numerical simulations of 3D initially equatorial orbits near the body. We find that the minimum radius for direct, equatorial circular orbits that cannot impact with the Eros surface is 36 km and the minimum radius for stable orbits is 31 km despite significant perturbations of its orbit. Moreover, as the orbits suffer less perturbations due to the irregular gravitational potential of Eros in the elliptic case, the most significant result of the analysis is that stable orbits exist at a periapsis radius of 29 km for initial eccentricities ei ≤ 0.2.

gravitation, methods: numerical, celestial mechanics, minor planets, asteroids: individual: 433 Eros

INTRODUCTION

To date, few probes have been sent to study asteroids. In fact, only three probes were launched having the main goal to study asteroids. In 1996, the American probe Near Earth Asteroid Rendezvous (NEAR)-Shoemaker sent images of the asteroid 253 Mathilde and in 2002 it approached and landed on the asteroid 433 Eros. In 2005, the Japanese Spacecraft Hayabusa reached the asteroid 25143 Itokawa, and began a period of vicinity operation about the body. ESA's Rosetta spacecraft flew past 21 Lutetia, at a distance of 3162 km on 2010 July 10. The Dawn spacecraft, launched by NASA, visited one of the largest asteroids of the main-belt, 4 Vesta in 2012 and will visit 1 Ceres in 2015. The Brazilian deep space mission ASTER, as temporarily named, plans to send a small spacecraft to encounter and investigate the still unknown triple asteroid system 2001-SN263. The launch was scheduled (initially) to occur in 2015, arriving in 2018 (Sukhanov et al. 2010; Brum et al. 2011).

The irregular gravitational fields induced by small bodies add new complexity to classical celestial dynamics. Generally, these gravitational fields are non-central and asymmetrical vector fields with zero divergence and vorticity; in such cases, some trajectories can exhibit strongly unstable and chaotic motion near the surface of the small bodies (Scheeres et al. 1998). Previous studies used to start with some simple specific geometric objects, for example, triaxial ellipsoids were more frequently used for asteroid gravity approximation (Scheeres 1994). Werner (1994) and Werner & Scheeres (1997) developed a polyhedral method that describes the gravitational field of a constant density polyhedron and can evaluate with a certain precision the gravitational field around a specific asteroid. This method has been applied to the investigation of close orbit dynamics around the non-principal-axis rotator 4179 Toutatis (Scheeres et al. 1998) and the evaluation of the actual dynamical environments around 433 Eros and 25143 Itokawa (Scheeres, Williams & Miller 2000; Scheeres et al. 2006). However, singularities appear in the numerical evaluation of the closed expressions for the gravitational field of a homogeneous polyhedron. These appear in the transition from surface to line integrals when the orthogonal projection of the computation point on to the plane defined by a face of the polyhedron lies inside the polygon defining the face or falls on the line segment describing the edge. Tsoulis & Petrović (2001) refined the approach by presenting the derivation of certain singularity terms, which emerge for special locations of the computation point with respect to the attracting polyhedral source. The study of dynamics and stability around asteroids with models each time more robust and efficient has become a necessity for the planning of future missions. In this sense, the data of observations from NEAR-Shoemaker probe served Scheeres, Miller & Yeomans (2003) for reviewing the computation of the forces and model parameters of Eros published in Scheeres et al. (2000). They also made an analytical study of stability against impact of orbits in the equatorial plane (I = 0).

Our interest for 433 Eros is due to the fact that future orbital missions around asteroids will find similar conditions to those found by the NEAR-Shoemaker Spacecraft. The motivations of this paper is to evaluate the full gravity tensor of Eros using the improved method implemented by Tsoulis (2012) and study the stability of equatorial orbits in the vicinity of the body taking into account the gravitational disturbance along the three-dimensional space. First, the general properties of Eros are discussed in Section 2. In Section 3, we present the full gravity tensor of the polyhedral model. The equations of motion and the conserved quantity are discussed in Section 4 as well as the stability against impact. We also find the exact location of the equilibrium points. The dynamics equatorial close to Eros is investigated through numerical simulations and the results are presented in Section 5. Then, we discuss and conclude in Section 6.

COMPUTED PROPERTIES FROM THE SHAPE OF EROS

The first attempt to constrain the shape of 433 Eros from the inversion of Goldstone Radar Doppler Spectra, with data obtained during the asteroid’'s close approach in 1975, was made by Mitchell et al. (1998). On 1998 December 23, after an engine misfire, the NEAR-Shoemaker spacecraft flew by Eros on a high-velocity trajectory and provided the first images. Orbital operations commenced on 2000 February 14, allowed precise estimates of Eros physical parameters and topography (Thomas et al. 2002). Measurements (Miller et al. 2002) resulted in the total mass (6.69 × 1015 kg) and the bulk density (2.67 ± 0.03 g cm−3). Based on these measurements, the dimensions of Eros were established as 34.4 × 11.2 × 11.2 km and the total volume of 2503 ± 25 km3. We choose the data set of NEAR COLLECTED TARGET MODELS V1.0 (2002 April 15) to build a polyhedral model with 3897 vertices and 7790 faces as illustrated in Fig. 1.

Polyhedron model 3D of asteroid 433 Eros viewed from two perspectives. The shape was built with 7790 faces.
Figure 1.

Polyhedron model 3D of asteroid 433 Eros viewed from two perspectives. The shape was built with 7790 faces.

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We use a fast and effective algorithm (Mirtich 1996) to calculate the polyhedral mass properties of Eros and then, we find the body-fixed coordinate frame with the origin at the body centre of mass, and aligned with the principal axes of inertia of the asteroid. Thus, we estimate the polyhedron total volume as 2526 km3 with a bulk density of 2.65 g cm−3. The overall dimensions of the asteroid shape in each of the principal directions are
(1)
Using the algorithm of Mirtich (1996), we also find the moments of inertia:
(2)
where M is the total mass of the body. Miller et al. (2002) computed the principal axes moments of inertia from the measures of the laser rangefinder (NLR) of the NEAR-Shoemaker probe. As a result they found
(3)
The values agree very well, since the volume of Eros may be determined to an accuracy of 1 per cent, provided the surface is sampled randomly and systematic errors associated with the trajectory and instrument biases are about 10 m.
From the moments of inertia (Dobrovolskis 1996), we can solve for the ellipsoid, and we find a body with semimajor axes of 18.0119 × 6.9771 × 5.9033 km, which are not so close to the overall dimensions of the body given in equation (1). This is mainly due to its shape in ‘boomerang’ shown in Fig. 1. Directly related to moments of inertia are the second-degree and -order gravity coefficients. These are directly computed to be (Werner & Scheeres 1997)
(4)
where R0 is the arbitrary normalization radius. Scheeres et al. (2000) computed such coefficients but they were not recalculated after having the new data from Miller et al. (2002). A measure of the asteroid's shape and gravity field is defined in Hu & Scheeres (2004) as
(5)
For Eros, this value is σ = 0.9522. A body with σ = 1 has a prolate inertia matrix, while one with σ = 0 has an oblate matrix; thus, we see that Eros is very close to having a prolate shape index, that is compatible with Fig. 1.

FULL GRAVITY TENSOR OF THE POLYHEDRAL MODEL

Tsoulis & Petrović (2001) present a mathematical approach that overcomes the singularities, which we will define in this section, and enables the line integral formalism to be applied everywhere in space. The closed formulas for the potential, the gravitation vector and the tensor of the second-order derivatives, are respectively as follows:
(6)
(7)
(8)
Here, the additional parametrization refers to the plane defined by each polygonal surface Sp. Thus, the subscript q stands for the polyhedral segments, and it is running from 1 to m for each face p, with m denoting the variable number of maximum segments building every face. The value |$\boldsymbol {n}_{pq}$| is the normal vector of segment q that lies on the plane of polygon Sp and is pointing, per definition, outside the closed polygonal surface Sp. A critical information for each face is the relative position of the projection |$P^{^{\prime }}$| of the computation point P on the plane of each face with respect to Sp. If |$\boldsymbol {n}_{pq}$| points to the half-plane that contains |$P^{^{\prime }}$|⁠, then σpq = −1; if it points to the other half-plane, it holds that σpq = +1. The value hpq denotes the positive distance between |$P^{^{\prime }}$| and the line of each segment, whereas |${\rm cos}{(\boldsymbol {n}_{pq},\boldsymbol {e}_j)}$| is the direction cosine between |$\boldsymbol {n}_{pq}$| and the unit vector |$\boldsymbol {e}_j$| of the unit vector triad situated at P. Quantities LNpq and ANpq are abbreviations of the following transcendental expressions:
(9)
and
(10)
where l1pq and l2pq are the 3D distances between P and the end points of segment pq (e.g. fig. 1; Tsoulis & Petrović 2001). With |$P^{^{\prime \prime }}$| being the origin of a 1D local coordinate system defined on the line of segment pq, s1pq and s2pq are the 1D distances between |$P^{^{\prime \prime }}$| and the two segment end points. In these definitions, subscript 1 always stands for the end, and subscript 2 is for the beginning of the corresponding segment vector (Tsoulis 1999). Finally, the terms singAp and |${\rm sin}g_{B_{pj}}$| are the singularity terms that appear for specific locations of |$P^{^{\prime }}$| with respect to the polygonal line Gp when one attempts to apply the divergence theorem of Gauss for these cases. They express the analytical solutions of the corresponding limiting values of the line integrals that are obtained from the partial application of the divergence theorem for a small circle containing the singularity point when its radius tends towards zero. Tsoulis & Petrović (2001) showed the values of the three cases when (a) |$P^{^{\prime }}$| lies inside Sp, (b) |$P^{^{\prime }}$| is located on Gp but not at any of its vertices and (c) |$P^{^{\prime }}$| is located at one of Gp's vertices.

A program in fortran (Tsoulis 2012) computes the gravitational potential, its first-order derivatives and the full gradiometric tensor at arbitrary space points due to a general polyhedral source of constant density. We applied the program to 433 Eros and the results for the gravitational potential are shown in Fig. 2 and its gradient in Fig. 3. In the contour maps of the gravitational force, colour is used to show the intensity of the force. Figs 2 and 3 indicate the condition of the distribution of the gravitational potential and its force in the three planes. From the six figures, we note that the gravitational field of 433 Eros follows very well its shape, as expected, since it was assumed a constant density for the body.

Gravitational potential of asteroid 433 Eros in the xoy, xoz and yoz planes, respectively. The colour code gives the intensity of the potential in km2 s−2.
Figure 2.

Gravitational potential of asteroid 433 Eros in the xoy, xoz and yoz planes, respectively. The colour code gives the intensity of the potential in km2 s−2.

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Intensity of the gravitational potential gradient near the surface of asteroid 433 Eros in the planes z = 0, y = 0, x = 0, from top to bottom, respectively. The colour code gives the intensity of the gravitational force in km s−2.
Figure 3.

Intensity of the gravitational potential gradient near the surface of asteroid 433 Eros in the planes z = 0, y = 0, x = 0, from top to bottom, respectively. The colour code gives the intensity of the gravitational force in km s−2.

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The relative error between the gravitational potential derived from the homogeneous polyhedral source and more simple models, as spherical, that corresponds to a point mass, and ellipsoidal, represented by second-degree and -order gravity coefficients, is shown in Fig. 4. The gravitational potential is computed for different values of the longitude λ ∈ (0°, 360°) measured in the xy plane starting from the x-axis in the anticlockwise direction and at a distance radius from the centre of mass of Eros ranged from 20 up to 40 km. Close to Eros (20 km) the relative error is significant, especially for the spherical model (>0.2) while at 40 km, the relative error is small, but not negligible (∼1 per cent), if we choose to use the ellipsoidal model, as we can see in Fig. 4. Thus, simple shape models like ellipsoid could be chosen to analyse the dynamics of orbits around Eros with periapsis radii larger than 40 km. However, when the periapsis radii is smaller than 40 km, it becomes necessary to use the polyhedral model for a more accurate calculation of the gravitational potential.

Relative error of the gravitational potential of the polyhedral model U(r) in the latitude λ ∈ (0°, 360°) with the spherical model Us = μ/r (blue asterisks) or ellipsoidal second-degree and -order gravity coefficients U2 (red asterisks).
Figure 4.

Relative error of the gravitational potential of the polyhedral model U(r) in the latitude λ ∈ (0°, 360°) with the spherical model Us = μ/r (blue asterisks) or ellipsoidal second-degree and -order gravity coefficients U2 (red asterisks).

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EQUATIONS OF MOTION AND CONSERVED QUANTITIES

Equations of motion

In addition to the gravity and rotation models, it is also important to incorporate the perturbations from the Sun in any detailed analysis of the motion of a spacecraft about an asteroid. There are two main contributions from the Sun: gravity and radiation pressure. The effect of the solar gravity becomes pronounced as the distance of the spacecraft from the asteroid increases. The radiation pressure from the Sun has a constant direction and magnitude at a given distance of the asteroid from the Sun. Again, the effect of this force becomes more pronounced as the spacecraft flies farther away from the asteroid. As discussed by Scheeres et al. (2000), the orbital motion in the vicinity of Eros is dominated by its own gravitational field, reducing the solar effects to small perturbations, which allows us to neglect any solar effects and gravitational pull from remaining bodies in this paper. Finally, for the Eros rotation period we use the value of 5.27 h. This estimate is based on ground-based observations of Eros using both optical and radar imaging (Mitchell et al. 1998). The rotation rate of the asteroid is denoted as ω. Thus, in the body-fixed reference frame (Scheeres et al. 2000), the equations of motion are
(11)
(12)
(13)
where Ux, Uy and Uz are the first-order partial derivatives of the potential. Because these equations are time invariant, an additional integral of motion called the Jacobi constant exists. The Jacobi constant C is explicitly calculated as (Scheeres et al. 1996)
(14)
where
(15)
is the modified potential and
(16)
is the kinetic energy of the particle with respect to the rotating asteroid.

Zero-velocity surfaces and equilibrium points

Zero-velocity surfaces, defined using the Jacobi integral, provide concrete information regarding the possible motion of a particle. Since TE ≥ 0, it is possible to define an inequality
(17)
which partitions the x, y, z space into regions where the particle is allowed to be found and where it cannot be found, given a specific value for C. Note that V(x, y, z) ≥ 0 over the entire space. Thus, if C < 0, the inequality is identically satisfied and there are no a priori constraints on where the particle may be found. If C > 0, there will be regions of space where the inequality is violated, and hence where no particle may travel. When these forbidden regions separate space into disjoint regions a particle can never travel between these regions. The general situation is discussed more fully in Scheeres (1994), where the central body is assumed to be a three axial ellipsoid. Zero-velocity surfaces are defined by
(18)
This equation defines a two-dimensional surface in the three-dimensional x–y–z space.
Fig. 5 shows the zero-velocity surface corresponding to the Eros shape model in the three planes, and Fig. 6 gives the projections of the zero-velocity surfaces on to the z = 0, y = 0 and x = 0 planes. As the value of the Jacobi constant C is varied, the surfaces change. At critical values of C, the surfaces intersect or close in upon themselves at points in the x–y–z space, usually called equilibrium points. The surfaces are all evaluated close to the critical values of C, and Fig. 6 shows the location of the four equilibrium points, being that two of them E1 and E2 are saddle points and the two others (E3 and E4) centre points (Scheeres 1994). To find the exact location of these four equilibrium points, we make
(19)
and the results are shown in Table 1 including the energy value C for each point. Due to the Eros irregular shape, there is no symmetry between the saddle points and the centre points as that occurs in the case of the ellipsoidal shape model. As we know, all the four points are unstable, and then the asteroid is classified as type II (Scheeres 1994).
Zero-velocity surfaces of asteroid 433 Eros in the xoy, xoz and yoz planes, respectively. The colour code gives the intensity of the Jacobi constant C in km2 s−2.
Figure 5.

Zero-velocity surfaces of asteroid 433 Eros in the xoy, xoz and yoz planes, respectively. The colour code gives the intensity of the Jacobi constant C in km2 s−2.

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Zero-velocity curves and equilibrium points of asteroid 433 Eros in the xoy, xoz and yoz planes, respectively. The colour code gives the intensity of the Jacobi constant in km2 s−2. The equilibrium points are indicated by E1, E2, E3 and E4.
Figure 6.

Zero-velocity curves and equilibrium points of asteroid 433 Eros in the xoy, xoz and yoz planes, respectively. The colour code gives the intensity of the Jacobi constant in km2 s−2. The equilibrium points are indicated by E1, E2, E3 and E4.

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Table 1.
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Locations of equilibrium points and their respective Jacobi constant C values.

Equilibrium pointx(km)y(km)z(km)C(km2 s−2)
E119.1656−2.64940.1414−0.4842e−04
E2−19.7422−3.38300.1278−0.4943e−04
E3−0.4575−13.9529−0.0740−0.3855e−04
E40.486614.7123−0.0627−0.3963e−04
Equilibrium pointx(km)y(km)z(km)C(km2 s−2)
E119.1656−2.64940.1414−0.4842e−04
E2−19.7422−3.38300.1278−0.4943e−04
E3−0.4575−13.9529−0.0740−0.3855e−04
E40.486614.7123−0.0627−0.3963e−04
Table 1.
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Locations of equilibrium points and their respective Jacobi constant C values.

Equilibrium pointx(km)y(km)z(km)C(km2 s−2)
E119.1656−2.64940.1414−0.4842e−04
E2−19.7422−3.38300.1278−0.4943e−04
E3−0.4575−13.9529−0.0740−0.3855e−04
E40.486614.7123−0.0627−0.3963e−04
Equilibrium pointx(km)y(km)z(km)C(km2 s−2)
E119.1656−2.64940.1414−0.4842e−04
E2−19.7422−3.38300.1278−0.4943e−04
E3−0.4575−13.9529−0.0740−0.3855e−04
E40.486614.7123−0.0627−0.3963e−04

Stability against impact

Assuming a case in which the motion occurs only in the plane, to ensure stability against impact, we must choose the initial orbital conditions such that the spacecraft position resides in the outer portion of the closed zero-velocity curve. That would be when the value of the Jacobi integral is less than or equal to J0 = −4.9433 × 10−5 km2 s−2 that corresponds to the maximum value of the Jacobi constant located at the equilibrium point E2. This provides a simple check for whether or not the spacecraft might impact with the surface at some point in the future. This relation can be expressed in terms of initial osculating elements for an assumed direct, equatorial orbit specified by its periapsis radius rp, eccentricity e and initial longitude λ in the body-fixed frame (Scheeres et al. 2000):
(20)
where the gravitational potential U(r, λ) is evaluated from the actual gravity field, and μ = 4.4631 × 10−4 km3 s−2. Fig. 7 shows a plot of the limiting planar stability against impact for the Eros system (including the full effect of the irregular gravity field) in terms of initial periapsis radius and eccentricity for an equatorial orbit. Initial orbits to the left of the line in Fig. 7 may impact with Eros at some point in the future; orbits to the right of this line will not impact with Eros. To obtain this line, we set the greater value of the periapsis radius rp for each eccentricity e where the expression (20) remains negative. The difference to the one found by Scheeres et al. (2000) is due to the actual physical properties of 433 Eros measured by the second flyby of the probe NEAR-Shoemaker in 2001. The sign changed in expression 33 of Scheeres et al. (2000), compared with expression 50 of Scheeres et al. (2003), arises from the fact that R(r) = U(r) − μ/r, where R(r) represents the perturbation of the full potential implicit in expression (50).
Stability against the impact curve for equatorial, direct orbits, derived from equation (20).
Figure 7.

Stability against the impact curve for equatorial, direct orbits, derived from equation (20).

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DYNAMICS EVOLUTION CLOSE TO EROS

Numerical simulations

Our goal is to numerically test and expand the evolution of the dynamics of equatorial, direct orbits, taking into account the 3D gravitational perturbation. So as done in the previous section, we adopt the full potential tensor method implemented by Tsoulis (2012) to calculate the gravitational field of the equations of motion (equations 6–8) in the body-fixed reference frame. The Bulirsch–Stoer numerical algorithm was used to integrate the equations. The initial conditions of the spacecraft are placed in the periapsis radius of the equatorial plane (z = 0) for different values of λ ∈ (0°, 360°). The interval between each initial value of λ is 30°, being that the initial velocities are given from the two-body problem in the body-fixed reference frame. The initial periapsis radii vary from 20 up to 40 km with interval of 1 km, and the initial eccentricities of the spacecraft's orbit vary from 0 up to 0.8 with interval of 0.1. When the path of the spacecraft passes the boundaries of the ellipsoid approximation with semimajor axes (18.0119 × 6.9771 × 5.9033 km), the integration is stopped and we consider that the spacecraft impacted with the asteroid. The integration's time is about 106 s (>50 rotation periods) to determinate the collisions and 800 h to determinate the stability of the orbits (Scheeres et al. 2003).

Results

Two-dimensional analytical limits of stability against the impact on the equatorial plane of Eros were shown in the previous section. Our present interest in this section is to show the behaviour of orbits launched with initial conditions compatible with Fig. 7, but in which case we consider the full gravitational potential of Eros in the three-dimensional model, and we followed the trajectories in the 3D space to verify the outcome. The trajectories are made from various initial longitudes in the equatorial plane. We chose four of them where occur, in principle, the highest and lowest values of the potential, and we show them in Fig. 8. We see that the results are compatible with the curve of Fig. 7 but with the periapsis radii much closer to the asteroid. Thus, below 25 km, the orbits are unstable independently of the initial eccentricity. We check all the longitudes and we find that the limit radius for direct, initially equatorial circular orbits that cannot impact with the Eros surface is 35 km, a slight shift from the result (34 km) found in Scheeres et al. (2000). Moreover, orbits with initial λ = π appear to be less subject to collision.

Maximal eccentricity of the initial orbit of the particles that have impacted with 433 Eros according to their periapsis radius. We show this relation for four initial longitudes λ. The black line is the stability against impact shown in Fig. 7.
Figure 8.

Maximal eccentricity of the initial orbit of the particles that have impacted with 433 Eros according to their periapsis radius. We show this relation for four initial longitudes λ. The black line is the stability against impact shown in Fig. 7.

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We are now interested in the region very unstable under 26 km where a large number of collisions occur. The number of initial conditions that impact or not with Eros for each periapsis radius and various eccentricities is shown in Table 2. In the circular case, the collisions dominate this region. However, at a periapsis distance of 23 km, the number of initial conditions that do not collide with Eros is higher with initial eccentricity e = 0.4 and at 24 km the same occurs with e = 0.3. Thus from 25 km, the initial conditions, that do not collide, dominate even with eccentricities close or equal to zero. It shows a certain similarity with the curve of Fig. 7 but the differences with the analytical model appear when the periapsis radius pulls away from Eros. As we consider the gravitational perturbation along the z-axis, the orbits leave to be completely equatorial due to the irregular shape of the three-dimensional model of Eros. This causes the collisions which do not occur with radii lower than expected by the analytical model.

Table 2.
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Number of initial conditions that impact or not with Eros for each initial periapsis radius (first column) and eccentricities (second column).

Initial periapsis radius (km)Initial eccentricityImpactDo not impact
200102
0.284
0.4102
0.666
210102
0.266
0.484
0.666
22084
0.2102
0.475
0.666
23075
0.293
0.448
0.548
240102
0.166
0.284
0.348
250102
0.157
0.2210
0.348
26075
0.148
0.2210
0.3210
Initial periapsis radius (km)Initial eccentricityImpactDo not impact
200102
0.284
0.4102
0.666
210102
0.266
0.484
0.666
22084
0.2102
0.475
0.666
23075
0.293
0.448
0.548
240102
0.166
0.284
0.348
250102
0.157
0.2210
0.348
26075
0.148
0.2210
0.3210
Table 2.
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Number of initial conditions that impact or not with Eros for each initial periapsis radius (first column) and eccentricities (second column).

Initial periapsis radius (km)Initial eccentricityImpactDo not impact
200102
0.284
0.4102
0.666
210102
0.266
0.484
0.666
22084
0.2102
0.475
0.666
23075
0.293
0.448
0.548
240102
0.166
0.284
0.348
250102
0.157
0.2210
0.348
26075
0.148
0.2210
0.3210
Initial periapsis radius (km)Initial eccentricityImpactDo not impact
200102
0.284
0.4102
0.666
210102
0.266
0.484
0.666
22084
0.2102
0.475
0.666
23075
0.293
0.448
0.548
240102
0.166
0.284
0.348
250102
0.157
0.2210
0.348
26075
0.148
0.2210
0.3210

Taking into account only those trajectories which do not collide with the body, we generated Fig. 9 after an integration of these trajectories for 106 s or more than 50 Eros rotation periods. Here, we choose four eccentricities compatible with an observation mission (e = 0 up to e = 0.3). Largest eccentricities will hinder any accurate observation due to the fact that the passage at periapsis would be very quick. As we see, the orbits suffer less perturbations due to the irregular shape of Eros and keep their eccentricity stable at a periapsis radius of 36 km for the circular case (Fig. 9a). However, for obits with initial eccentricities (Figs 9b, c and d), this radius decreases to a value of 34 km. Below these values, most of the orbits will suffer major perturbation in the eccentricity and the tendency is that they escape. Moreover, in the circular case all the orbits have variations of their initial eccentricities while in the other cases, we find orbits that do not have or have low variations in their eccentricities. Scheeres et al. (2003) has shown that polar orbits can suffer eccentricity perturbations after 30 d and may escape. We investigated the orbits that had relatively small variations (Δe < 0.25) with their initial eccentricity after 800 h, and the results are shown in Fig. 10. Fig. 10(a) confirms that all these orbits suffered perturbations in their initial eccentricities. The minimum radius for the circular case is 31 km despite significant perturbations of its orbit. A significant result shown in Figs 10(c) and (d) is that stable orbits (Δe < 0.05) exist at a periapsis radius of 29 km for initial eccentricities e ≥ 0.2. However, we investigated orbits closer than 29 km with higher eccentricities but we did not found any stable orbits. In view of the results, we show that the stable trajectories are closer for higher eccentricities, since the initial velocity is higher (Araujo et al. 2008), especially here when the eccentricity reaches the value of 0.2. In this case, various orbits persist with the same initial eccentricity after 800 h and possibly will continue stable. Fig. 11 show that the stability of these orbits does not only depend on the distance of the periapsis radius but its position in the plane play an important role. The regions of periapsis for stable orbits are centred around 90° and 270° in the equatorial plane of asteroid 433 Eros, and these regions increase when the eccentricity increases. An example of 3D equatorial orbit that remains stable around 433 Eros after 800 h is shown in Fig. 12. We take this example to show that the perturbation of the gravitational potential of Eros disturbs the orbit making it to precess. The maximal eccentricity of this example is 0.274 and the minimal periapsis radius is 27.865 km.

Maximal eccentricity of the orbits of the particles relative to 433 Eros after 106 s. The initial eccentricity e at their periapsis radius is (a) 0, (b) 0.1, (c) 0.2 and (d) 0.3.
Figure 9.

Maximal eccentricity of the orbits of the particles relative to 433 Eros after 106 s. The initial eccentricity e at their periapsis radius is (a) 0, (b) 0.1, (c) 0.2 and (d) 0.3.

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Maximal eccentricity of the orbits of the particles relative to 433 Eros that remains stable after 800 h. The initial eccentricity e at their periapsis radius is (a) 0, (b) 0.1, (c) 0.2. and (d) 0.3.
Figure 10.

Maximal eccentricity of the orbits of the particles relative to 433 Eros that remains stable after 800 h. The initial eccentricity e at their periapsis radius is (a) 0, (b) 0.1, (c) 0.2. and (d) 0.3.

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Periapsis initial positions of the orbits that have a variation in its eccentricity Δe < 0.05 after 800 h around 433 Eros. The initial eccentricity is 0.2 on the left-hand side and 0.3 on the right-hand side.
Figure 11.

Periapsis initial positions of the orbits that have a variation in its eccentricity Δe < 0.05 after 800 h around 433 Eros. The initial eccentricity is 0.2 on the left-hand side and 0.3 on the right-hand side.

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Example of 3D equatorial orbit that remains stable around 433 Eros after 800 h. The initial eccentricity is 0.2 and its initial periapsis radius is 29 km.
Figure 12.

Example of 3D equatorial orbit that remains stable around 433 Eros after 800 h. The initial eccentricity is 0.2 and its initial periapsis radius is 29 km.

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CONCLUSIONS

In this paper, we used effective algorithms (Mirtich 1996; Tsoulis 2012) to calculate the polyhedral mass properties and the full gravitational potential of Eros. We determined the body-fixed coordinate frame with the origin at the body centre of mass and aligned with the principal axes of inertia of the asteroid. We found values of the density and moments of inertia of 433 Eros different than those reported by Miller et al. (2002). We have also shown that the line integrals of a homogeneous arbitrary shaped polyhedral source to evaluate the gravitational potential function and their first- and second-order derivatives well represent the irregular shape of Eros (as shown in Fig. 3) and should be taken into account near the body. From the pseudo-potential, we also found the location of the equilibrium points and their respective Jacobi constant. We determined a curve of stability against impact, and we found that the collisions do not occur much closer than expected with Eros by the two-dimensional analytical model (Scheeres et al. 2003). Lara & Scheeres (2002) have investigated stability bounds for three-dimensional motion close to Eros in the circular case varying the inclination orbit. They found that the transition from stable to unstable circular orbits in the equatorial plane (inclination ≅0) occurs at a radius of 33.4 km and the transition from stable to unstable circular orbits retrograde in the equatorial plane (inclination ≅180°) now occurs at a radius of 20.8 km, a slight shift from the results reported previously in Scheeres et al. (2000). Here, we only investigate the dynamics of equatorial, direct orbits for three-dimensional motion close to Eros but differently from them, we also took into account the initial eccentricity of the orbits. We found that the minimum radius for direct, equatorial circular orbits that will not impact with the Eros surface is 36 km. Moreover, in the circular case all the orbits have variations of their initial eccentricities while in the other cases, we found orbits that do not have or have low variations in their eccentricities. The most significant results of that portion of the analysis is that the minimum radius for stable orbits is 31 km for the circular case despite significant perturbations of its orbit and stable orbits exist at a periapsis radius of 29 km for initial eccentricities e ≥ 0.2. In fact, we have shown that there are stable orbits closer with initial eccentricities and do not suffer many perturbations than circular orbits. In this way, various orbits persist with the same initial eccentricity after 800 h and possibly will remain stable. Moreover, the regions of periapsis for stable orbits are centred around 90° and 270° in the equatorial plane of asteroid 433 Eros, and these regions increase when the eccentricity increases. In which regions, the values of the gravitational potential are lower due to the shape of Eros and it may also occur with other asteroids with a very flattened and non-convex shape.

This work was supported by CAPES, CNPq, INCT – Estudos do Espaço and Fapesp proc. 2011/08171-3. We thank Dr D. J. Scheeres for some helpful suggestions.

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