JuanCastillo29/space-mission-intelligence-data huggingface dataset:
90°
66.0°
oblate prolate bilobed A
67P A 67P K2019
45°
135°
65.0°
s
0°
64.0°
dec
oblate prolate bilobed A 67P A 67P
63.0°
-45°
s
-90°
62.0°

[Figure: Fig. 11. Longitude, λs, and latitude, ϕs, for angular velocity, ω, with respect to the fixed solar basis in Fig. 2, based on rs(t0) and b3 = R(t0)e3 at initial time, t0. T is evaluated with |rs| = 2 au and changing SSL from −45◦to 45◦. Considered are the oblate ellipsoid, the near-prolate ellipsoid, the bilobed shape, the shape 67P with model A/ fEAF = 1/8, and the shape 67P with model α/ fEAF = 1/4.]

31.0° 32.0° 33.0° 34.0° 35.0° ra × cos(64 deg)
×10
1.45 1.44
(γ ≈0.5◦) in Sect. 2.1 and shows the close alignment of the vectors ω and b3. The rotation remains close to a lowest-energy state. At the same time, the longitude, λs, for ω covers the entire angular range −180◦≤λs ≤180◦that describes the precessional movement of ω surrounding b3.
1)
1.43 1.42
| | (s
1.41 1.40
400 300 200 100 0 100 200 300 400
4.1. Uniform activity
time since perihelion (day)
We consider a uniformly distributed surface ice coverage. In this case, the principal axis, b3 = R(t)e3, changes in time. To plot the evolution of ω in figures with respect to an inertial frame, we fix the solar basis of Fig. 2 based on the state at initial time t0, namely with rs(t0) and b3 = R(t0)e3. Fig. 11 shows λs and ϕs for ω with respect to this fixed solar basis. Due to lowestenergy state at time t0, ω and b3 start aligned, which is reflected by ϕs = 90◦. The close alignment for b3, ω, and L implies that Fig. 11 shows the directions for the principal axis, b3(t), and for rotational angular momentum, L(t), up to the order of 10−2 degree. The torques in Sect. 3 cause directional changes on the order of 5◦for b3, ω, and L. Due to the smaller torque values for the bilobed shape, the changes in ω are smaller by an order. Qualitatively, the shapes evolve ω along the line λs = −90◦. According to Sect. 2.1, the equatorial projections of rs and ω are close to perpendicular. The torque orientations in Figs. 8, 10 confirm important properties of the evolution of ω. At SSL = 0◦for both ellipsoids the jump of torque longitude from −90◦to 90◦is reflected by corresponding turning points for the ω trajectory in Fig. 11. For shape 67P the ω trajectories (model A and model α) do not contain a turning point. This observation agrees with the smooth curve for longitude, λs, in Fig. 10. Longitude, λs, is in the range between −90◦and −70◦, which explains the deviation from λs = −90◦for ω in Fig. 11. Fig. 12 shows the evolution of ω in time from 400 days before to 400 days after the perihelion passage of comet 67P/C-G. At initial time, the evolution starts with a fixed angular velocity, ω, and a lowest-energy state. Unlike Fig. 11, the solar vector, rs, describes the trajectory of the sun during apparition 2015 relative to comet 67P/C-G. In the beginning, before the first equinox, SSL is positive and irradiation comes from the north. For both ellipsoids in Fig. 8, we know that the longitude, λs, for T is at 90◦.
400 days before to 400 days after the perihelion passage of comet 67P/C-G. At initial time, the evolution starts with a fixed angular velocity, ω, and a lowest-energy state. Unlike Fig. 11, the solar vector, rs, describes the trajectory of the sun during apparition 2015 relative to comet 67P/C-G. In the beginning, before the first equinox, SSL is positive and irradiation comes from the north. For both ellipsoids in Fig. 8, we know that the longitude, λs, for T is at 90◦.

[Figure: Fig. 12. Angular velocity, ω. R0, L0 is in lowest-energy state. T is evaluated for solar vector, rs, describing the trajectory of the sun during the apparition 2015 relative to comet 67P/C-G. Considered are the oblate ellipsoid, the near-prolate ellipsoid, the bilobed shape, the shape 67P with model A/ fEAF = 1/8, and the shape 67P with model α/ fEAF = 1/4. The markers on each curve denote the ω direction at perihelion passage in August 2015. Top: Right ascension (ra) and declination (dec) for ω in the equatorial J2000 (EME2000) reference frame. Bottom: Angular frequency, |ω|.]

For shape 67P in Fig. 10, λs for T is close to −75◦. In Fig. 12 these opposite directions are reflected in the ω trajectories. At the two equinox times (95 days before and 224 days after perihelion), the SSL is zero. For the two ellipsoids, these points are related to directional switches between −90◦and 90◦for λs of T in Fig. 8. For the corresponding ω trajectories this is related to turning points. This geometric phenomenon is known and reported; for example, from Whipple & Sekanina (1979), Sekanina (1984), and Gutiérrez & Davidsson (2007). Between the two equinox times, the SSL is negative. This setup, shown in Figs. 8, 10, leads to torque directions close to λs = −90◦for all shapes. Fig. 12 reflects this by showing similar slopes along all curves around the perihelion passage (marked by a point on the curve). For the bilobed shape, the relative small magnitude of the torque (discussed in Sect. 3) is represented by the small movement of ω. For both ellipsoids and the bilobed shape, the latitude, ϕs, of T is always zero, which yields a constant rotation period, as shown in Fig. 12. For the shape 67P, the decrease in |ω| until day -80 (followed by an increase) is directly related to the positive SSL and the corresponding torque direction (with ϕs < 0) in Fig. 10.
Article number, page 10

64.5°
0.08
bilobed A bilobed
0.07
67P A 67P
64.4°
0.06
0.05
fEAF
0.04
64.3°
0.03
bilobed A bilobed 67P A 67P K2019
0.02
64.2°
0.01
dec
0.00
-1,-1,-1 1,-1,-1 1,1,-1 -1,1,-1 -1,-1,1 1,-1,1 1,1,1 -1,1,1
region
64.1°

[Figure: Fig. 13. Activity weights fEAF,i,j,k for regions Ai,j,k in Eq. (8) for the bilobed shape and the shape 67P with model A and model α. The values]

64.0°
of fEAF,i,j,k are constrained by the ω data in Sect. 2.4.
63.9°
by the same order of magnitude as the observational data for
comet 67P. The corresponding modeled changes for the ω direc-
the order of four to six compared to the observation. This prop-
erty has been reported by Kramer et al. (2019) and Attree et al.
(2024). The second observation for the shape 67P in Fig. 12 ap-
plies to the slope along the ω trajectory at perihelion (and during
the complete pathway). Those curves seem to be rotated between
models and observation. A uniform activity on the surface of the
shape 67P with either models A or α is not able to bring into
agreement the model results and the observations in 2015. This
limitation can be resolved by the weighted activity in Eq. (8).
With the assumption of a lowest-energy state and a fixed phase
velocity (three parameters) determines the complete initial con-
tion has n = 11 open parameters p for i = 1, ..., n . We fitted

ra × cos(64 deg)
×10
1.45 1.44 1)
1.43 1.42
| | (s
1.41
1.40
400 300 200 100 0 100 200 300 400
time since perihelion (day)

[Figure: Fig. 14. Angular velocity, ω, similar to Fig. 12 for the bilobed shape and the shape 67P with model A and model α. Weighted activity from Fig. 13. Top: Right ascension (ra) and declination (dec) for ω. Bottom: Angular frequency, |ω|.]
of comet 67P, with a polyhedral shape model at its center of mass and aligned with the principal axes of inertia, is given by [31].
V (r) = −1
2ω2 � x2 + y2�
+ U(r), (1) where r represents the position of the particle in the body-fixed reference frame relative to the center of mass, ω is the angular speed, and U(r) is the gravitational potential computed numerically [22, 23] or analytically (Eq. (12)). Fig. 3 shows the geopotential computed over the surface of comet 67P. Due to its slow rotation, the gravitational potential predominates. We observed that the regions with the minimum geopotential values are located in the Hapi region (comet 67P’s neck) and the big lobe. Comparing the values of the geopotential of comet 67P with those found on the surface of the Kuiper Belt Object (KBO), Arrokoth [6], that has an irregular binary shape with similar density and rotational period of comet 67P.

We can see that comet 67P has a minimum geopotential ∼20× lower than KBO Arrokoth.

[Figure: Fig. 3: Map of the geopotential computed across the surface of the comet 67P. The color bar gives the numerical values of Eq. (1), in km2s−2.]
fits with similar accuracy, σ, and allow us to draw the same con-
clusions. The first kind of modification is to change the number
of elements and, along with it, the shape resolution that we dis-
cussed in Sect. 2.1. For the bilobed shape half and double the
to 0.01σ. The second kind refers to the bilobed shape with the
modified value α = 1.5 that we introduced in Sect. 2.3. Simi-
larly to the variations in Fig. 13, an agreement on the order of

Ntime
�2
�data j(ω(ti)) −datai,j σi, j
σ2 =
modification refers to variations in the sampling rate and uncer-
tainty weights for the data set in Sect. 2.4. A variation in the data
incorporating an equidistant time sampling of 7 days and con-
period does not change the results.
For the four model setups, the solutions in Fig. 14 solve the
mismatch between models and observations in Fig. 12. The gen-
eral changes for the ω direction and for the angular frequency,
|ω|, are qualitatively consistent with the data in Sect. 2.4. For all
than for uniform activity, on the order of three for model A
α go back to the normalization constants for both TPMs in
(2020) and locally resolved
gas activity in Läuter et al. (2022) can also be included. Any ap-
proach for a joint data set must address the difficulty of a relative
weighting of the different data sets, which introduces a bias to
the final result. With knowledge of the shape geometry, the lo-
calization of surface activity can be applied to other solar system
objects that have weaker sources of torque activity. Nongravita-
tional momenta have this potential on dark comets reported by
Seligman et al. (2023) as well as the YORP effect on asteroids
reviewed by Bottke et al. (2015).

Acknowledgements. The authors gratefully acknowledge the computing time made available to them on the high-performance computer Lise at the NHR center NHR@ZIB. This center is jointly supported by the federal ministry of education and research and the state governments participating in the NHR. This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #547 (Understanding the Activity of Comets Through 67P’s Dynamics) with N. Attree acting as PI. M.L. acknowledges the support by Johannes Kepler Universität Linz Austria for a research visit in 2024. We thank the referee for the helpful comments, which improved the presentation of the manuscript.
Small bodies in our Solar System are considered remnants of their early formation. Studying their physical and dynamic properties can provide insights into their evolution, stability, and origin. ESA’s Rosetta mission successfully landed and studied comet Churyumov-Gerasimenko (67P) for approximately two years. In this work, the aim is to analyze the surface and orbital dynamics of comet 67P in detail, using a suitable 3-D polyhedral shape model. We applied the polyhedron method to calculate dynamic surface characteristics, including geometric height, surface tilt, surface slopes, geopotential surface, acceleration surface, escape speed, equilibrium points, and zero-velocity curves. The results show that the gravitational potential is predominant on the comet’s surface due to its slow rotation. The escape speed has the maximum value in the Hapi region (the comet’s neck). The surface slopes were analyzed to predict possible regions of particle motion and accumulation. The results show that most regions of the comet’s surface have low slopes. Furthermore, we analyzed the slopes under the effects of Third-Body gravitational and Solar Radiation Pressure perturbations. Our results showed that the effects of Third-Body perturbations do not significantly affect the global behavior of slopes. Meanwhile, the Solar Radiation Pressure does not significantly affect particles across the surface of comet 67P with sizes >∼10−3 cm at apocenter and >∼10−1 cm at pericenter. We also identified four equilibrium points around comet 67P and one equilibrium point inside the body, where points E2 and E5 are linearly stable. In addition, we approximated the shape of comet 67P using the simplified Dipole Segment Model to study its dynamics, employing parameters derived from its 3-D polyhedral shape model. We found 12 families of planar symmetric periodic orbits around the body.
Keywords: celestial mechanics, comet 67P/Churyumov-Gerasimenko, methods: numerical, Solar System
from the Hatmehit depression on comet 67P, modeling the ejecta with various initial velocities using the mascon model. Finally, Scheeres [16] calculated the location and stability of equilibrium points around comet 67P using the mascon model approach. They found four unstable equilibrium points around comet 67P.
1https://www.esa.int/Enabling Support/Operations/Rosetta In this work, we analyze in detail the surface dynamics and the orbital dynamics of comet 67P/Churyumov-Gerasimenko from an adequate 3-D polyhedral shape model2 obtained by [17].

We treat the comet nucleus like the surface of an asteroid, as no drag force was implemented. Our goal is to investigate dynamic features of the surface and surroundings, particularly the slopes, incorporating Third-Body (TB) and Solar Radiation Pressure (SRP) perturbations into our analyses. We utilize an accurate 3-D polyhedral shape model of the nucleus of comet 67P for this purpose. We also compared the Dipole Segment (DS) model to the polyhedron model to identify the limitations for families of planar symmetric periodic orbits around the nucleus of comet 67P, which could support specific orbital phases of space mission planning3 [18]. Section 2 describes the methodology used in our analyses. Section 3 presents our results on the surface dynamics of comet 67P, which cover geometric height, surface tilt, geopotential surface, acceleration surface, surface slopes, escape speed, equilibrium points, and zero-velocity curves. Section 5 shows the orbital dynamics around comet 67P from the topological characterization of the families of planar symmetric periodic orbits found around it. We used the DS for this purpose. Finally, in Section 6, we present our conclusions.
It is essential to determine the physical characteristics of 67P to calculate and interpret the physical quantities associated with its surface and orbital dynamics [9]. Modeling the geopotential of small bodies requires strategies that account for their irregular shapes. While methods based on Legendre polynomials offer reasonable approximations, they can introduce discrepancies in certain regions [19]. Although computationally efficient, the mass concentration points (mascons) approach, which involves subdividing the body into small mass elements and summing their potentials [5], exhibits significant inaccuracies near the surface. In this study, we employ the polyhedron method to explore the surface dynamics. Initially developed by [20, 21] and later refined by [22, 23], the polyhedron method models the nucleus4 of comet 67P as a uniformdensity polyhedron, enabling analytical computation of its gravitational potential and its derivatives. This approach surpasses alternative models, such as mascons and spherical harmonic expansions, especially when studying regions across and close to the surface [21]. The polyhedron approach is particularly advantageous when studying surface dynamics since it adequately describes gravitational variations caused by the comet’s shape. To implement this methodology, we use the Minor-Gravity package5 [6, 24], which computes the gravitational potential of comet 67P based on the 3-D polyhedral shape model and provides its first and second derivatives accurately. We use a current density of 0.535 g cm−3 inferred by [1], from the Rosetta mission, and a rotational period of 12.4043 h derived by [25]. The polyhedral shape model of comet 67P has 48,420 vertices and 96,834 triangular faces [17]. The shape model has a volume of 18.8 km3 and an equivalent spherical diameter of 3.3 km. In the dynamic orbital analyses, we did not consider other components of the force field. As a result, we treated the comet as an asteroid since we did not implement any drag force. We emphasize that recent work shows that the activity of drag and outgassing in the form of jets when comet 67P is near perihelion may play a fundamental role in the ejection and dynamics of dust aggregates (see [13]). However, its contribution to short-period orbital solutions, far away from the comet surface, such as those investigated in this work, is negligible. For gravitational perturbations, we can safely assume that the considerable distance of the Sun, compared to the typical size of the close orbits studied by the spacecraft, makes these perturbations negligible. Furthermore, in our surface dynamic analyses, we provided a thorough
2https://pdssbn.astro.umd.edu/holdings/ro-c-multi-5-67p-shape-v2.0/data/triplate/spc lam psi/ 3https://www.asteroidmission.org/asteroid-operations/
4Without loss of generality, we will omit the word nucleus along the text. 5https://github.com/a-amarante/minor-gravity investigation of how additional forces, such as TB and SRP, affect the overall behavior of slopes on the surface of comet 67P, considering different distances and sizes of particles (sec.
we provided a thorough
2https://pdssbn.astro.umd.edu/holdings/ro-c-multi-5-67p-shape-v2.0/data/triplate/spc lam psi/ 3https://www.asteroidmission.org/asteroid-operations/
4Without loss of generality, we will omit the word nucleus along the text. 5https://github.com/a-amarante/minor-gravity investigation of how additional forces, such as TB and SRP, affect the overall behavior of slopes on the surface of comet 67P, considering different distances and sizes of particles (sec.

4). Additionally, to determine and classify equilibrium points around and inside the comet, we apply the Minor-Equilibria-NR package6 [6, 26], which computes the location and stability of the equilibria. The Minor-Equilibria-NR package uses the Newton-Raphson 3D method to find equilibria around irregular bodies with the accuracy of 10−5. Finally, we employ the Grid Search Method [27] with the DS model to compute the families of planar symmetric periodic orbits around comet 67P. The Grid Search Method is a simple and structured technique that allows the comprehensive identification of sets of families of periodic orbits in non-integrable dynamical systems, considering a delimited area of the space of initial conditions. For the planar case and symmetry about the x−axis, it works as follows:
• A grid (x0, J) is constructed, since x0 represents the x position at the initial time (t = 0) and J = (x2+y2)−2U −( ˙x2+ ˙y2) the Jacobi constant. A symmetric periodic orbit crosses the x−axis perpendicularly twice, in time P and P/2, so y = ˙x = 0, then J(x0, 0, 0, ˙y0);
• For each set of initial conditions (x0, J), integrate numerically the equation of motion (13) by TIDES (Taylor series integrator for differential equations), with absolute and relative tolerance equal 10−16 [28], compute the Poincar´e map, crossing the x−axis and ˙y > 0;
• The sign of ˙x is checked, and by continuity, if ˙x(t0 + n∆t) < 0 and ˙x(t0 + (n + 1)∆t) > 0, a root-finding process is done [29];
• The initial conditions of the symmetric periodic orbits are satisfied when convergence is achieved;
• Finally, a final check is done by integrating the initial conditions up to the total period T. It verifies if |(x, 0, 0, ˙y) −(x(mP), ˙x(mP), y(mP), ˙y(mP))| < mtol, being m multiples of periods P and mtol the tolerance. The initial conditions that do not meet the final check are disregarded.
Geometric height is defined as the distance between the centroid of each triangular face on the surface of a body and the main x−axis [6]. This feature, along with the surface’s tilt angles (see Section 3.2), is a tool for analyzing the topography of elongated and binary minor celestial bodies,such as comet 67P. Figure 1 reproduces the 3-D polyhedral shape model of comet 67P, using 48,420 vertices and 96,834 triangular faces [17]. The color scale indicates the geometric height. The colors range from purple to yellow, highlighting the shapes of the small and big lobes of comet 67P. Furthermore, El-Maarry [30] represented the topographic features of comet 67P. As we can see, the geometric height ranges from zero (purple areas in Fig. 1, such as Ma’at, Anuket, and Maftet, named in Fig. 2 from [30]) to 2.0 km (yellow areas in Fig. 1, such as Atum, Bes, and Anhur, named in Fig. 2 from [30]).
Given the geopotential at any location r on the surface of comet 67P, by calculating the gradient:
∥−∇V (r)∥, (2) we obtain the surface acceleration about comet 67P. We calculate the surface acceleration as the sum of gravitational and centrifugal accelerations. Its variation can affect the movement of loose material on the surface of comet 67P and help predict possible preferential regions for material accumulation associated with surface slopes (see Section 3.6). Fig. 4 shows the surface acceleration across the surface of comet 67P. The surface acceleration was calculated at the barycenter of each triangular face of the 3-D polyhedral shape model of the comet 67P. The color bar shows the surface acceleration. The big lobe has the maximum values of surface acceleration, which corroborates the minimum values of geopotential. In contrast, the Hapi region (comet 67P neck) has intermediary values for the surface acceleration. This behavior indicates that the gravitational potential dominates in the large lobe. Moreover, the centrifugal potential has more influence in the neck region, which is the distance that is much closer to the center of mass. The surface acceleration reaches ∼2.2 × 10−7km s−2. Comparing the results of the maximum surface acceleration with those found for KBO Arrokoth [6], we observe that comet 67P has values ∼4× lower than Arrokoth.

[Figure: Fig. 4: Surface acceleration computed over the surface of comet 67P showed in three perspective views (-x, -y, and -z). The color code gives the surface acceleration, in km s−2.]
The local normal escape speed ves was computed across the surface of comet 67P using Eq. (3) [31].
ves = −ˆn · (Ω× r) + � [ˆn · (Ω× r)]2 −2Umin −(Ω× r)2, (3) � ,
where r is the radius vector from the comet’s center of mass to the local surface, Umin = min � U(r), −GM
∥r∥
G = 6.67428 × 10−20 km3 kg−1 s−2 is the gravitational constant and M = 1.0 × 1013 kg is the total mass of comet 67P and Ωis the angular velocity vector. If the launch speed exceeds the given speed, the particle will escape. Our approach is based on the assumption that the particle is launched normally to the local surface. Fig. 5 shows the escape speed computed over comet 67P’s surface. The regions of the minimum geopotential are the areas with the maximum escape speed. We observed that the Hapi region has the maximum escape speed. This behavior is similar to the minor bodies with slow rotational periods, such as Arrokoth, Toutatis, Leucus, and Justitia, for example. The escape speed values in the neck of comet 67P can reach 1.10 × 10−3km s−1.

[Figure: Fig. 5: Local normal escape speed calculated over the surface of the comet 67P, in km s−1. The color box denotes the values of escape speed.]