JuanCastillo29/space-mission-intelligence-data huggingface dataset:
Considering the surface acceleration vector calculated on the barycenter of each triangular face of the comet 67P, using Eq. (2), let us analyze how each face is oriented about this vector. Let be the angle between the normal unit vector ˆn on the surface and the surface acceleration vector −∇V (r)

[Table: Table 1: Distribution (%) of surface slopes θ (degrees) on the comet 67P, considering a shape model of 96,834 faces and 48,420 vertices [17].
Slope Range (degrees) Number of Faces Percentage (%) 0 – 20 42,853 44.25 20 – 40 38,093 39.34 40 – 60 13,132 13.56 60 – 90 2,753 2.84 90 – 180 3 0.01 Total 96,834 100]

of a given face. The slope θ is defined as the supplement of this angle [4, 31].
θ = 180◦
� 1 −arccos �−∇V (r) · ˆn ∥−∇V (r)∥
�� . (4)
π
This dynamic characteristic provides insight into the movement of particles across the surface of comet 67P. If θ > 90◦, there is a possibility that the particles will be ejected from the surface of comet 67P, while for θ < 90◦, the particles could accumulate in some regions over the comet’s surface. Using the mascon approach, Scheeres [16] analyzed the surface slopes across the surface of comet 67P. They found that the surface slopes range from nearly 0◦in the smoothest regions to around 30◦in the steepest areas. [15], also using the mascon approach, found that most values of surface slopes are between 20◦−40◦and 60◦−90◦. The range 20◦−40◦, accounting for ∼30% of the comet 67P surface, corresponds to moderately inclined terrains, which may play a crucial role in mass-wasting processes. Meanwhile, the range 60◦−90◦covers a small portion (∼11%), representing the steepest regions, likely associated with cliff faces or highly eroded areas, where material instability is more pronounced. Finally, Vincent [32] uses the polyhedral method to calculate the slopes (θ > 60◦) to indicate erosional age. We can see from Fig. 6 that comet 67P presents regions with low values of θ. However, we can observe some areas with high values of surface slopes (θ > 140◦). This behavior suggests that a few regions on the surface of comet 67P may be ejecting particles. Table 1 presents our statistics on the surface slopes of comet 67P, utilizing a suitable 3-D polyhedral shape model, with current density of 0.535 g cm−3 from the Rosetta mission, and a rotational period of 12.4043 h [1, 25]. Compared with previous works, the most significant differences are between intervals 20◦−40◦and 60◦−90◦. From Tab. 3 of [15], the surface slopes between 20◦−40◦represent only 30% of the total surface area of comet 67P, while in Tab. 1 this represents 38%. Meanwhile, the surface slopes in the interval of 60◦−90◦cover 11% of the surface area of comet 67P. Tab. 1 indicates a coverage of only ∼3%.
complexly unstable). The region delimited by the black line intersecting the equilibrium point E1 represents the rotational Roche lobe of comet 67P.]

[Table: Table 2: Location of equilibrium points around comet 67P and their geopotential values. Equilibria are computed through the polyhedron method with the Minor-Equilibria package [6] and an accuracy of 10−5.
Point X (km) Y (km) Z (km) φ (°) λ (°) r (km) V (r) (×10−7km2s−2)
E1]

3.50342 -0.47082 0.09106 1.475 352.346 3.53541 -3.30661
E2
-0.06365 3.20769 -0.05010 359.105 91.136 3.20831 -3.08010
E3
-3.39552 -0.96706 0.06879 1.116 195.897 3.53801 -3.29503
E4
-0.05399 -3.06035 -0.03567 359.933 268.990 3.06103 -3.01194
E5
-0.23159 0.13858 -0.14763 331.320 149.102 --5.58168
(a) (b)

[Figure: Fig. 8: Surface slopes mapped on the surface of comet 67P seen from six perspectives (±x, ±y, and ±z) considering the Jupiter (a) and Sun (b) gravitational perturbations at 3.5 AU and 1.2 AU distance from comet 67P, respectively.]

� rJ/S −r ∥rJ/S −r∥3 − rJ/S ∥rJ/S∥3
� , (6)
apert(r) = GMJ/S where MJ = 1.9×1027 kg is the total mass of Jupiter, MS = 2.0×1030 kg is the total mass of Sun, rJ is the position vector of Jupiter concerning 67P, rS is the position vector of the Sun concerning 67P, ∥rJ −r∥ is the distance from the barycenter vector of each face to Jupiter, and ∥rS −r∥is the distance from the barycenter vector of each face to the Sun. The slopes θ now could be computed from Eq. (7):
θ = 180◦
� 1 −arccos �−atotal(r) · ˆn ∥−atotal(r)∥
�� , (7)
π
where atotal = apert(r) −∇V(r), and ˆn is the normal vector calculated at the centroid of each face. Figure 8 shows the global behavior of the slopes due to the gravitational effects of the TB (Jupiter/-Sun). We can observe that the TB does not influence significantly the slopes over the comet’s surface, when we take into account only the gravitational acceleration from comet 67P (Fig. 6), which corroborates with the Hill radius of comet 67P, being 318 km [35].
Another force considered for the slope analysis was the Solar Radiation Pressure (SRP). For our analyses, we once again calculated the effect of SRP on a particle located in each face of the polyhedral shape model, assuming that the particle is located at the barycenter of each face. The equation of motion for a particle with geometric cross-section A and mass m moving under the influence of the SRP acceleration is given by Eq. (8) [5, 36, 37]:
aSRP(r) = −QprS0R2
c ∥rs −r∥3 A m (rs −r), (8)

[Table: Table 3: The Solar Radiation Pressure parameters used in the mathematical model.
Parameter Value Units Comments
Qpr
1 −− Efficiency factor
S0 1.36 × 103
kg s−3
Solar constant
R0 1.495978707 × 108 km Astronomical unit distance (AU) c 2.99792 × 105
km s−1
Speed of light ρ 1.190
g cm−3
density
rp 10−4 −10−1
cm Particle sizes (spherical radius)
where rS is the position vector of the Sun relative to 67P, ∥rS−r∥is the distance from the particle’s located at the barycenter of each face to the Sun, S0 is a solar constant or radiation flux density at the distance of the astronomical unit R0 (AU), c is the speed of light, and Qpr is the dimensionless efficiency factor for radiation pressure, which depends on the properties (e.g., density, shape, size) of the particle. In the mathematical model, this value is assumed to be 1 to represent the value of an ideal material. The values of the adopted SRP parameters are presented in Tab. 3. Furthermore, the particles in the mathematical model were assumed to be fully spherical, in order to vary the area-to-mass ratio, as shown in Eq. (9):] A m = 3
,
(9)

4 1 ρrp where ρ is the particle bulk density, assumed to be equal to the comet 67P-averaged bulk density for all particles, and rp is the particle size (spherical radius) (Tab. 3). From this, we obtain the SRP acceleration vector for a particle located in each face of the 3-D polyhedral shape model and then add it to the acceleration vector used in section 3.6, considering only the gravitational acceleration from the comet 67P. Thus, we obtain the new total acceleration vector due to SRP, given in Eq. (10). atotal = aSRP(r)−∇V(r). (10) Then, we analyze the global behavior of the slopes, varying the particle size and the distance from the Sun, at pericenter rpe = 1.2 (AU) and apocenter rap = 5.7 (AU) of the comet 67P orbit7. In Figs. 9 (a) and (b), we kept the pericenter rpe = 1.2 (AU) of comet 67P’s orbit and varied the particle sizes from 10−4 cm to 10−1 cm, located at the barycenter of each face of the 3-D polyhedral model. We found that particles <∼10−4 cm are significantly influenced by SRP at the pericenter of comet 67P’s orbit, while particles on the order of ∼10−1 cm are not significantly influenced by SRP. In Figs. 9 (c) and (d) we kept the size of the particles at 10−3 cm, located at the barycenter of each face of comet 67P, and varied its distance from the Sun at pericenter rpe = 1.2 (AU) and apocenter rap = 5.7 (AU) of its orbit. We identify that particles with sizes of 10−3 cm are significantly influenced by the SRP at the pericenter. In contrast, particles of the same size do not significantly suffer the influence of the SRP at the apocenter. It is important to note that the acceleration values due to SRP were calculated based on an assumed area exposed to solar radiation of approximately 64 m2 [18], which represents the maximum contribution of the SRP. So, we can safely neglect the SRP effects in the analyses of orbital dynamics of a spacecraft around comet 67P (sec. 5).
7https://ssd.jpl.nasa.gov/tools/sbdb lookup.html#/?sstr=Churyumov-Gerasimenko
(a) Particle size 10−4 cm at pericenter rpe = 1.2 (AU). (b) Particle size 10−1 cm at pericenter rpe = 1.2 (AU).
(c) Particle size 10−3 cm at apocenter rap = 5.7 (AU). (d) Particle size 10−3 cm at pericenter rpe = 1.2 (AU).

[Figure: Fig. 9: (a) and (b): Global behavior of slopes as a function of particle sizes, keeping the comet distance from the Sun. (c) and (d): Global behavior of slopes as a function of comet distance from the Sun, keeping the particle size. The color scale shows the slopes from 0◦to 60◦.]
In this section, we use the Eq. (11) to calculate the relative error between the gravitational potentials of the polyhedron and the Dipole Segment (DS) models in percentage.
���� UDS(x, y) −Up(x, y)
����, (11)
Up(x, y) where UDS(x, y) is the gravitational potential from the DS model and Up(x, y) is the gravitational potential from the polyhedron model, both computed in points around comet 67P, in the xOy plane.

[Figure: Fig. 10: Gravitational potential relative error between Dipole Segment (DS) and polyhedron models. The]

colorbox indicates the relative error (%) around comet 67P for points in the xOy plane.

[Figure: Figure 10 shows that, for approximated distances from the comet 67P between -4 km and 4 km, about the x−axis direction, the gravitational potential relative error between DS and polyhedron models is > 5% (yellow). Also, for approximated distances from the comet 67P, between -3 km and 2 km, about the y−axis direction, the relative error is > 5%. On the other hand, for approximated distances from comet 67P, < −4 km or > 4 km about the x−axis direction and < −3 km or > 2 km about the y−axis direction, the relative error becomes < 5%, which suggests that the DS model can be used beyond this region for the study families of planar symmetric periodic orbits around comet 67P (sec. 5).]

[Table: Table 4: Parameters of comet 67P for DS model, being M1, M2, and Ms the masses of the big lobe, small lobe, and segment, respectively. The size of the segment is defined by l, and ω is the angular speed.
M1 (kg) M2 (kg) Ms (kg) l (km) ω (rad/s) κ µ
µs
5.0 × 1012 2.9 × 1012 2.1 × 1012 1.0 1.407 × 10−4 33.6104 0.3645 0.2081]
Model Approach

To investigate the evolution of the characteristic curves of comet 67P, based on the topological characterization of the orbits and families around the body [38], we use a simplified model of its shape, in which the gravitational potential is approximated by a Dipole Segment (DS). The main goal of this section is to investigate the possibility of families of planar symmetric periodic orbits (SPO) around comet 67P, utilizing the DS model for this purpose. The main advantage of this simplification is that it overcomes the difficulty of finding periodic orbits, considering the irregular gravitational potential of the body, particularly for orbits far from the comet’s surface. The DS models a massive segment connected by a mass at each end, similar to a “bone”, which the shape of comet 67P fits approximately (see Fig. 11). In this model, the masses at the ends of the segment, considered to be spherical, are approximated as a point mass. We use the mass concentrations (mascons) constrained by the 3-D polyhedral surface to infer the masses of the lobes and the segment, as well as the segment length [39, 40]. The parameters k = GM
M2
M1+M2 and µs = Ms
ω2l3 , µ =
M , being M = M1 + M2 + Ms, of the gravitational potential of the DS model (Eq. 12), in this case, were obtained from the 3-D polyhedral shape model filled with 151,906 points of mass concentration (mascons).
�� , (12)
U = −k �(1 −µ)(1 −µs) r1
+ µ(1 −µs)
r2
+ µs log �r1 + r2 + l r1 + r2 −l
where r1 = � (x + µ −µµs + µs/2)2 + y2 + z2 and r2 = � (x + 1 + µ −µµs + µs/2)2 + y2 + z2. We assumed from the comet 67P neck (Fig. 11) that the segment has ∼1.0 km (two black vertical lines) and mass Ms. So, we distributed 31,606 mascons in the neck polyhedral solid region between the two vertical lines in black (Fig. 11), 76,440 mascons for the polyhedral big lobe region (the polyhedral solid region on the left side of the first black vertical line), and 43,860 mascons for the polyhedral small lobe region (the polyhedral solid region on the right side of the second black vertical line). After that, the parameters of the DS model are defined as shown in Table 4. The dimensionless equation of motion of a particle around the body is given by Eq. 13.
¨x −2 ˙y = x −∂U
∂x ,
¨y + 2 ˙x = y −∂U
∂y ,
(13) ¨z = −∂U
∂z .
The characteristic curves are the smooth, one-parameter, continuous curves that represent the families of SPO in the initial conditions map, in this case, calculated using the Grid Search Method [27].

[Figure: Fig. 11: 3-D polyhedral shape model of the nucleus of comet 67P viewed from xOz plane, rotating 90◦ around the x−axis and 340◦around the z−axis. The color box indicates the surface distance from the center of mass (in km). The two black vertical lines indicate the considered polyhedral solid region for the comet’s neck ∼1 km (-0.2 to 0.8 km).]
]), they surround the body and all the equilibrium points, for the entire characteristic curve. The difference between them is in the looping that each one has (see Fig. 13 (b), 13 (c), and 13 (d)).
- The orbits of the Families 6 to 12 are in the zone Zr, where x0 > 0, and direct rotation. Except Family 12 which meets x0 < xP/2 and is retrograde. All of these cases have a similar common part, encircled the body (see Fig. 14 (b), 14 (c), 14 (d), 15 (a), 15 (b), 15 (c) and 15(d)).

[Table: Table 5: Topological characterization of families of SPO for the comet 67P.]
curve (D(DS) [
]) as in Fig. 14 (b), 14 (d), 15 (a), 15 (b) and 15 (c). Families 6, 8, and 9 are internal
to all equilibrium points. The orbits of families 10 and 11 are close to the collinear points, but do not
surround them.
family’s evolution, the orbits also surround the triangular points.
By comparing Figs. 10 and 13, we find that the DS model is suitable for investigating planar symmetric
periodic orbits from Family 2 (with distances greater than 5 km) around comet 67P. This approach could
be preferable to the polyhedron model, as it maintains a gravitational potential relative error of less than
of the DS model reduces the computational effort required to calculate gravitational potential compared
to the polyhedron method.
For instance, during the planning of the Rosetta mission, the distance from comet 67P was reduced to
9 km during the close approach phase. The orbital plane was shifted to the terminator plane for sufficient
confidence in the dynamical stability of the comet’s environment. This strategy was essential to favorable
observation conditions with the necessity to minimize the spacecraft’s exposure to the predominantly
radial gas flow emitted by the comet [18].
As shown in Fig. 10, we could employ the DS model to analyze the orbital dynamics around comet
67P in the close approach phase at the terminator plane, since it yields a gravitational relative error of
less than 2% compared with the polyhedron one. Furthermore, for other orbital phases, such as comet
characterization, the relative errors are even smaller [18].
Because it is a Hamiltonian system, the matrix M has six eigenvalues that, due to the symplectic structure of the system, appear in reciprocal pairs and, as a consequence of the periodicity of the orbit and energy conservation, two of them are equal to one, therefore λ1, λ2 =
λ1 , λ3, λ4 =
λ3 , λ5 = 1, λ6 = 1.
Horizontal Stability - Since horizontal stability refers to the response of the orbit to small perturbations within the orbital plane, the Jacobian matrix for this case reduces to the subspace (x, ˙x, y, ˙y) as in Eq. (14). Vxx, Vxy, Vyx, Vyy the second derivative of the effective potential (Eq. (1)).



 . (14)
Ah(t) =
Vxx Vxy 0 2 Vxy Vyy −2 0
Therefore, the characteristic polynomial is
P(λh) = λ4 h −(Sh −2)λ3 h + (2 + 2Sh)λ2 h −(2 + Sh)λh + 1, (15)
where Sh =| λh1 +
λh1 | is the horizontal stability index, a single metric for assessing the stability of the system [41]. If Sh > 2, the orbit is unstable; Sh = 2, the orbit is neutrally stable; and Sh < 2, the orbit is stable.
Vertical Stability - A planar periodic orbit in a three-dimensional system is vertically stable if, when a small out-of-plane perturbation is applied, the motion remains bounded—that is, the body oscillates around the plane and does not drift away indefinitely. Mathematically, the planar periodic orbit is vertically stable if the eigenvalues of the monodromy matrix Φv(T), λv1 and λv2, which are associated with the out-of-plane behavior, lie on the unit circle of the complex plane (i.e., | λv1 |=| λv2 |= 1 ). Otherwise, it is vertically unstable. Since Sv =| λv1 + λv2 |=| λv1 +
λv1 |, when the limit of vertical stability occurs we have Sv = 2. The Jacobian matrix reduce to the subspace (z, ˙z), as in Eq. (16), where Vzz the second derivative of the effective potential (Eq. (1)), calculated for z = 0 since the orbit is planar.
� 0 1
�
Av(t) =
. (16)
Vzz 0 Figure 16 shows the horizontal (black points) and vertical (blue points) stability index for some SPO families presented in the initial conditions map (Fig. 12). The solid red line is at a stability index equal 2.0. Note in Fig. 16, that in the families we analyzed there is a change from horizontal stability to instability, passing through index 2.0, this means that λh1 and λh2 are ±1, which implies critical points where bifurcations (exchange of horizontal stability) can occur. Regarding vertical stability, which defines the stability outside the plane of the planar orbit, note that in the Families 1, 5 and 12 there are stable and unstable orbits; in Families 2, 4 and 11 only stability; and in Family 3 the maximum index is Sv = 2.000398 (Fig. 16). When Sv = 2, the planar family of periodic orbits bifurcates into a 3D family of periodic orbits, maintaining the multiplicity (for λv1 = 1) or doubling it (λv1 = −1), depending on the sign of Sv. If 0 < Sv < 2 and Sv/2 = |cos(2πm/n) with m, n ∈Z and m < n, then the SPO family can bifurcate into a 3D family of periodic orbits with multiplicity nM.
is Sv = 2.000398 (Fig. 16). When Sv = 2, the planar family of periodic orbits bifurcates into a 3D family of periodic orbits, maintaining the multiplicity (for λv1 = 1) or doubling it (λv1 = −1), depending on the sign of Sv. If 0 < Sv < 2 and Sv/2 = |cos(2πm/n) with m, n ∈Z and m < n, then the SPO family can bifurcate into a 3D family of periodic orbits with multiplicity nM.

Validation - Since the system reduces to a 2D Hamiltonian problem for the vertical stability problem, some information can be analyzed to validate the results:
- The determinant of the monodromy matrix must be equal to 1;
- If the eigenvalues are complex conjugates of magnitude 1, the system is stable; if they are real reciprocals, the system is unstable.
- For systems with a monodromy matrix R2×2 and a determinant equal to 1, Sv =| λv1 + λv2 |=| λv1 +
λv1 |=| tr (Φv(T)) |, since tr is the trace of the monodromy matrix .
- A small vertical perturbation of z0 = 0.001 was used to evaluate vertical stability.
- The vertical stability was calculated using the complete trajectory x(t), y(t) of the periodic orbit.
The authors thank the S˜ao Paulo State University (UNESP), School of Engineering and Sciences, Campus Guaratinguet´a (grant 06/2023-PROPe - IEPe-RC), the Unesp Institutional Scientific Initiation Scholarship Program - PIBIC (grant PROPe Unesp 09/2023), the Center for Scientific Computing (NCC/GridUNESP) of the S˜ao Paulo State University (UNESP), the Center for Mathematical Sciences Applied to Industry (CeMEAI), funded by FAPESP (grant 2013/07375-0), and FAPESP, for financial support (proc. 2023/11781-5).
network: applications to modelling water activity of comets

Shunjing Zhao1, 2, 3, Xian Shi3, Hanlun Lei1, 2 1 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China 2 Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing 210023, China 3 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China e-mail: shi@shao.ac.cn e-mail: leihl@nju.edu.cn
Cometary activity is a compelling subject of study, with thermophysical models playing a pivotal role in its understanding. However, traditional numerical solutions for small body thermophysical models are computationally intensive, posing challenges for investigations requiring high-resolution or repetitive modeling. To address this limitation, we employed a machine learning approach to develop ThermoONet — a neural network designed to predict the temperature and water ice sublimation flux of comets. Performance evaluations indicate that ThermoONet achieves a low average error in subsurface temperature of approximately 2% relative to the numerical simulation, while reducing computational time by nearly six orders of magnitude. We applied ThermoONet to model the water activity of comets 67P/Churyumov-Gerasimenko and 21P/Giacobini-Zinner. By successfully fitting the water production rate curves of these comets, as obtained by the Rosetta mission and the SOHO telescope, respectively, we demonstrate the network’s effectiveness and efficiency. Furthermore, when combined with a global optimization algorithm, ThermoONet proves capable of retrieving the physical properties of target bodies.
Key words. neural network, comet, thermal physics, comaetary activity
al. 1963; Cowan & A’Hearn 1979; Weissman & Kieffer 1981). They have been applied in quantitative analysis of the impact of parameters on water ice activity (Marshall et al. 2019), as well as investigating mechanisms behind the disruption of a cometary nucleus (Steckloff et al. 2015). For interpreting observational data, more realistic thermophysical models are required, such as the inversion of 67P’s physical properties
Article number, page 1 of 13 its spin state due to the non-gravitational torque (Keller et al.

2015; Attree et al. 2024), are both complex topics, whether from the perspective of forward evolution, or from the perspective of fitting observational data. If the thermophysical model is embedded within a complex physical process as a sub-mechanism, such as the shape evolution (Zhao et al. 2021) and the coma formation (Shi et al. 2018), conducting large-scale or high-resolution studies is also challenging. In our previous work, we addressed similar issues in the thermophysical modelling for asteroids by introducing deep learning techniques (Zhao et al. 2024). We apply a similar concept in this study to develop a deep learning-based approach for the comets that can efficiently solve the thermophysical process and produce the water production rate. This paper covers: (i) how the deep learning model can be applied to model the thermal physics of comets; (ii) whether the deep learning model can serve as a unified model for simulating the water activity of the comets in large parameter spaces; (iii) how the deep learning model can be utilized in the analysis of real comets data.
normalized temperature between iterations to be below 1 × 10−4.
2.4.

Water activity The water activity of the comet is effectively represented by the sublimation flux in the model.

The global total outgassing flux of water from the comet nucleus is the integral of the sublimation flux over the entire surface.

In the polyhedral shape model, this can be expressed as follows (Hu et al. 2017):
Ztotal = �
ZkS k
(16)
k
where Zk indicates the sublimation flux of k-th facet and its area is denoted as S k.

Furthermore, we calculate the global water production rate by averaging the global outgassing flux of water Article number, page 3 of 13 over one rotational period, which quantifies the number of water molecules produced per second, given by (Hu et al.

2017)
�t0+P
nZ(t0) = 1 ˆmP
Ztotaldt (17)
t0
where P is the rotational period.