JuanCastillo29/space-mission-intelligence-data huggingface dataset:
Context. The dynamics of a rigid cometary nucleus is described by the evolutions of its center-of-mass and of its rotation state. Solar irradiation that reaches the surface of a cometary nucleus causes the sublimation of volatiles that form the coma around the nucleus. The sublimation process transfers linear momentum and rotational angular momentum from the nucleus to the surrounding space, and thus affects the dynamics via nongravitational forces and nongravitational torques. With the exception of close approaches to planets, these torques exert the dominant influence on the rotation states of cometary nuclei. Aims. The 2014-2016 Rosetta mission accompanying the comet 67P/Churyumov-Gerasimenko provides the longest continuous observational data to track its rotation state. In particular, the data set encompasses the direction of the angular velocity, denoted by ω, and the angular frequency, |ω|, over a time period of approximately 700 days. The observed change in the rotation state is not explained by a low heat conductivity thermophysical model in combination with a homogeneous surface ice coverage of comet 67P. Spatially and/or temporally varying weights for effective active fraction with respect to a prescribed set of surface regions provide a potential solution to this problem. Methods. Here, we present a methodology for classifying the surface based on vectorial efficiency of the torque. On any cometary surface without geometric symmetry, the methodology highlights the decomposition into eight characteristic regions that encode the signs of torque efficiency with respect to all vector components. This decomposition is divided into two subsets of four regions, each of which is located in one of both hemispheric regions. Results. We analyze in detail rotation states close to lowest energy and different thermophysical models, and we discuss how the uncertainties of observations affect the model parameters. We study the occurrence of these regions for an oblate ellipsoid, a nearprolate ellipsoid, a bilobed shape, and a shape model analogous to that of comet 67P. The sensitivity analysis for comet 67P indicates that the observations constrain only one of the eight weights uniquely. The other directions are poorly constrained and show the limitation of the rotational data in determining the regional activity on comet 67P.
Key words. Comets: general
Donaldson et al. (2023) for 162P/Siding Spring and in Magri et al. (2011) for Asteroid (8567) 1996 HW1.

From spacecraft imagery the shapes of comets 1P, 9P, 19P, 67P, 81P, and 103P are constrained (see Snodgrass et al. 2022).

Knight et al. (2024) classify four of these comets as having a bilobed shape.

Observational data for the rotation state of comets are mainly extracted from light curves and radar detections.

The retrieved rotation periods are in the range between a few hours and several days.

The rotation period can change over time as is reported by Belton et al. (2011) and Chesley et al. (2013) for 9P/Tempel 1.

Even the tendency might change from increase to decrease as is reported by Farnham et al. (2021) for comet 46P/Wirtanen during one perihelion passage.

Earth-bound observations constrain the orientation of the rotation axis by studying coma morphology.

Measurements are typically associated with uncertainties on the order of 15◦, see Knight et al. (2021).

For comet 67P/CG, orientations of the rotation axis are extracted by Jorda et al. (2016) based on the method of Gaskell et al. (2008).

Models for the rotation state (see Kramer et al. 2019; Attree et al. 2019, 2023, 2024) might match complementing quantities, such as the rotation period or the production curve for global gas production.

Gas emanating from the surface carries along momentum determined by the molecular mass and velocity of the volatiles.

Thermophysical models with a monotonous response of the sublimation to the solar irradiation together with the normal direction at each surface point establish the vector fields for forces and torques on the surface.

The torque vector field can be reinterpreted as a weighted superposition of the vectorial torque efficiency.

This quantity depends on the short axis vector and local properties of the shape geometry only.

One scalar component of this torque efficiency that affects the rotation period is introduced by Keller et al. (2015b).

Jewitt (1997) and Samarasinha & Mueller (2013) consider surface integrated torque terms to reflect cancelation effects.

By integrating spatially the vector field for torque, the resulting torque vector influences the rigid body dynamics.

This depends on the body geometry, on the heliocentric distance, and on the orientation state (the illuminated cross section) of the body relative to the sun.

Already for a monotonous TPM, surface features and self-shadowing can shift the maximum sublimation activity away from the current subsolar point.

Qualitatively, a resulting force caused by irradiation pushes the nucleus into the antisolar direction.

For the torque and changes of the rotation state, such a simple relation does not persist since the shape of the nucleus and the orientation of the rotation axis have to be considered in detail.

For a lowest-energy rotation, Kramer et al. (2019) parameterize diurnal torques by the subsolar longitude and derive a torque decomposition as a product of an orbital term and a shape-dependent matrix.
a monotonous TPM, surface features and self-shadowing can shift the maximum sublimation activity away from the current subsolar point.

Qualitatively, a resulting force caused by irradiation pushes the nucleus into the antisolar direction.

For the torque and changes of the rotation state, such a simple relation does not persist since the shape of the nucleus and the orientation of the rotation axis have to be considered in detail.

For a lowest-energy rotation, Kramer et al. (2019) parameterize diurnal torques by the subsolar longitude and derive a torque decomposition as a product of an orbital term and a shape-dependent matrix.

Subsolar latitude, together with solar distance, is associated with seasonal variations. Although uniform activity on the complex shape of comet 67P/C-G explains observed changes for the rotation period, there is a mismatch between this activity and observations for the axis orientation. For this problem within a model for comet 67P/CG, a solution is to prescribe a domain decomposition consisting of surface regions and to fit the corresponding activity weights (effective active fraction). These weights are only weakly constrained and it is difficult to obtain appropriate decompositions of the surface. For any cometary body geometry, the vectorial torque efficiency (with its local characterization of the geometry) offers the option to specify a domain decomposition consisting of eight regions. For lowest-energy rotations and independent of the shape, these regions have the significant property that all points of one region share the same direction for torque formation. The manuscript describes our approach to decomposing a cometary surface into eight regions that determine the direction of the torque and the resulting evolution of the rotation state close to lowest energy. Sect. 2 introduces the dynamical equations and discusses the driving forces for rotational evolution. We consider instantaneous TPMs with a weighted activity on the cometary surface. Sect. 3 describes the formation of a torque field at each point on the cometary surface. The relation between the specified regions and the torque direction is examined for representative examples, namely, for an oblate ellipsoid, a nearprolate ellipsoid, a bilobed shape, and a shape model of comet 67P. These torques guide the evolution of the rotation state close to lowest energy in Sect. 4. Rotational data for comet 67P/C-G serve as constraints for effective active fractions with respect to our domain decomposition.
��, (6)
i (x × ν) · b1 < 0, j (x × ν) · b2 < 0, k (x × ν) · b3 < 0
|(b3 × y) × b3|, b2(y) = b3 × y
|b3 × y|, b3 = Re3. (4)
Ai,j,k =
We chose the vector y ∈R3 in two different variants that yield the local basis and the solar basis, respectively.

As shorter denotation we also use B(y) = (b1(y), b2(y), b3).

For the local basis at time t and a position vector x ∈∂Ωt on the surface, we defined y = ν(x, t).

Fig. 1 shows the outward normal vector ν on ∂Ωt.

At a fixed time t the local basis, B(ν), depends on the surface position x and on the principal axis b3.

With advancing time t at a fixed position ξ ∈∂Ω, this basis in the
and a complementing zero region, ��������
x ∈∂Ωt
.
(x × ν) · b1 = 0 or (x × ν) · b2 = 0 or (x × ν) · b3 = 0
A0 = The analysis of signs in the three plots of Fig. 3 yields these regions in Fig. 5 for the near-prolate ellipsoid.

In this case, the
Article number, page 4

-8.5 -4.3 0 4.3 8.5 (x × ) b1  (102m)
60°
0°
60°
120°
longitude

90°
120° 60° 0° 60° 120° longitude

[Figure: Fig. 4. For the bilobed shape (top) and for shape 67P (bottom), the values (x × ν) · b2(ν) (second component of the vectorial torque efficiency) are shown based on longitude, λ, and latitude, ϕ, in the body frame. b2(ν) is the local basis vector in Fig. 1.]

[Figure: Fig. 3. For the near-prolate ellipsoid, the values (x × ν) · bi(ν) (vectorial torque efficiency) are shown based on longitude, λ, and latitude, ϕ, in the body frame. bi(ν) is the local basis vector in Fig. 1.]
yields the decomposition ∂Ωt = A0 ∪ A−1,−1,−1 ∪ A1,1,−1 ∪ A1,−1,1 ∪
A−1,1,1 into four regions and A0. For the oblate ellipsoid, the ro-
tational symmetry around b3 yields (x × ν) · b1 = (x × ν) · b3 = 0
and finally ∂Ωt = A0. These two examples show that symmetries
can lead to empty regions Ai, j,k.
tum, L, directly and, as a consequence, controls the evolution of
ice sublimation close to the surface, which leads to gas expan-
sion into the surrounding space. On a surface point, we denote as
a TPM the mappings from solar irradiation, I, (and its temporal
variation) to mass generation (due to sublimation), ˙m, and to gas
ical process for material, energy and momentum on the surface
and within the volume of the nucleus below. Interactions with
subsurface layers can introduce thermal inertia, which causes
For the local evaluation of solar irradiation, I, we take into
account the shadowing of surface areas due to the concave shape
the solar irradiation is given by
-1,-1,-1 1,1,-1 1,-1,1 -1,1,1
-1,-1,-1 1,-1,-1
90°
90°
45°
45°
latitude
0°
latitude
0°
45°
45°
90°
120° 60° 0° 60° 120° longitude
90°
120° 60° 0° 60° 120° longitude
-1,-1,-1 1,-1,-1
90° 45° latitude
0° 45°
90°
120° 60° 0° 60° 120° longitude

[Figure: Fig. 5. For the near-prolate ellipsoid decomposition of ∂Ωt into the four regions Ai,j,k in Eq. (6). Region A0 is colored in white. Top: Longitude,]

[Figure: Fig. 6. For the bilobed shape (top) and for the shape 67P (bottom), decomposition of ∂Ωt into the eight regions Ai,j,k in Eq. (6). Region A0 is colored in white. Longitude, λ, and latitude, ϕ, in the body frame.]

λ, and latitude, ϕ, in the body frame. Bottom: 3D projection using the same color code.
Let be the vector that starts from the center of mass of 67P and points to a given location r on its surface. The angle between the normal surface vector ˆn and the barycenter vector r is called the tilt angle [4]. This angle helps to understand how each triangular face is oriented within the comet’s shape. Fig. 2 shows the mapping of the tilt across the surface of comet 67P. The tilt values ranges from 0◦to 140◦, with higher values over the small lobe, reaching ∼135◦. Our analysis also suggests that 98.5% of its surface has tilt values that do not exceed 100◦.
6https://github.com/a-amarante/minor-equilibria-nr The geopotential surface (the sum of gravitational potential and centrifugal potential)

[Figure: Fig. 1: 3-D polyhedral shape model in 3-D of 67P shown in six perspective views (±x, ±y, and ±z). The shape model was built with 48,420 vertices and 96,834 triangular faces. The color code indicates the distance from the centroid facet to the x−axis in kilometers (geometric height).]

[Figure: Fig. 2: Surface tilt is mapped across the comet 67P, considering the 3-D polyhedral shape model with 48,420 vertices and 96,834 faces. The color code gives the surface tilt in degrees.]
I, only. Another assumption for our TPMs is that both functions
For quantification, we consider two specific choices for in-
stantaneous TPMs, model A and model α. Model A is based on
the instantaneous energy balance of a irradiated surface covered
by pure ice. Our model A is the TPM for water of Läuter et al.
production is close to zero. For higher intensities in the range of
goes back to the observation during the 2015 perihelion pas-
sage of comet 67P/C-G, that the gas production increases more
rapidly than predicted from model A. For an exponent 1 ≤ α we
defined the TPM by
The momentum transfer coefficient applied by Rickman
Gutiérrez et al. (2005), and Attree et al. (2019) is also
open parameter. Eq. (10) expresses a relation between
ity of solar irradiation and the strength of the torque.
lar, each of both hemispheric seasons is now related to
ivity in one of the two separate subsets in Eq. (7).
taneous TPMs create torque without delay in time.
rface. For rotational dynamics, these forces lead to a
both projections to the equatorial plane coincide. The

�α
Although any choice f
on to the value α = 2
the gravitational acce
ng the body domain,
As a consequence, th
gravitational contribu
TPM, gas expansion

Article number, page 6

2016. Time sampling defines Ntime = 58 time points ti with i = 1, ..., Ntime. The sampling rate varies between a 7 day to a 33 day distance between two times with a higher sampling rate close to perihelion passage in August 2015. The uncertainty of the data at time t is estimated based on the spread of the data close to that time. This leads to uncertainties between 0.01◦and 0.11◦for dec(ω) and between 11 s and 60 s for rotation period 2π/ f. The latter range is much higher compared to uncertainties on the order of 1 s for periods from Godard et al. (2017) and Mottola et al. (2014). We chose these higher weights to distribute the uncertainties uniformly in the direction and in the length of the vector, ω. We combined these preparations into the data set, (ti, datai, σi) for i = 1, ..., Ntime, (11) of 3Ntime data points datai = data(ωi) ∈R3 and uncertainties σi ∈R3 at time ti.
The torque, T, in Eq. (10) depends on the geometric properties of the shape Ωt, on the TPM, and on the weights of the effective active fraction fEAF in Eq. (8). The sensitivity of T to Ωt and fEAF is discussed in the present section. Ωt specifies the orientation of the surface regions and, along with it, the formation of the torque density on the surface. The uniform activity with fEAF = 1 represents the dynamics of the complete shape. The combination of spatially weighted effective active fractions in Eq. (8) and complex shape geometries gives options to modify the rotational dynamics. Within this section we restrict the discussion to lowest-energy rotation states. As a consequence, two significant vectors share the same direction, the rotation axis, ω, and the principal axis, Re3 (the same as b3 in Eq. (4)). The discussion of parameters acting on torque generation is based on the torque components, T · bi(rs), with respect to the solar basis in Fig. 2. Because torque, T(t), in Eq. (1) determines the derivative ˙L, the dynamical effect of T is represented by its integral or its average T over time. Due to the rotation of point ξ ∈∂Ωwith outward normal νξ in the body frame, the surface point x(t) = R(t)ξ ∈∂Ωt, the outward normal ν(t) = R(t)νξ, the irradiation, I ∼rs · ν(t), the gas velocity, vgas, and the mass production, ˙m, change in time. Consequently, in general, the torque, T, is not constant in time. In particular this is true for the nearprolate ellipsoid and the bilobed shape holding fewer rotational symmetry than the oblate ellipsoid. At time t for T and tNG, the temporal averages over one rotation period ∆t can be expressed by

[Figure: Fig. 7. For the bilobed shape (top) and for the shape 67P (bottom), decomposition of ∂Ωt into the eight regions. The same data as in Fig. 6 but in 3D projection.]
change in time. Consequently, in general, the torque, T, is not constant in time. In particular this is true for the nearprolate ellipsoid and the bilobed shape holding fewer rotational symmetry than the oblate ellipsoid. At time t for T and tNG, the temporal averages over one rotation period ∆t can be expressed by

[Figure: Fig. 7. For the bilobed shape (top) and for the shape 67P (bottom), decomposition of ∂Ωt into the eight regions. The same data as in Fig. 6 but in 3D projection.]

ϕs = 0◦, v · b3 = 0 holds and v is in the equatorial plane. Finally, ϕs ∈{−90◦, 90◦} gives v · b1 = v · b2 = 0 and the representation v = ±|v|b3. For the special case of v = rs, ϕs is the same as SSL.
2.4. Data for the rotation state of 67P/C-G For a numerical solution of the initial value problem of Eq. (1) a set of parameters needs to be specified. These include the initial conditions, the tensor of inertia, and the mechanism for torque formation in Eq. (10). The solution of the inverse problem consists of the determination of the open parameters based on known observational data. Angular velocity, ω, was observed during the ESA mission of comet 67P/C-G during the apparition in 2015. This rotational information refers to the angular frequency, f = |ω|, of the rigid body, as well as to the directional angles ra(ω), dec(ω) of the rotation axis, ω/|ω|. Our data set is based on the measurements for ω discussed in Kramer et al. (2019). In the equatorial frame J2000 (EME2000) the angles are extracted based on the method of Gaskell et al. (2008) applied by Jorda et al. (2016) to comet 67P/C-G. Data sets data(ω) = (ra(ω), dec(ω), |ω|) are averaged along a moving time interval of 21 days and are considered between August 2014 and July
t+∆t
�
T(t) = �
tNG dσ, tNG(x, t) = 1
tNG(x(s), s) ds. (12)
∆t
t
∂Ωt
The average of tNG is evaluated along the trajectory x(s) ∈∂Ωs corresponding to ξ = R−1(t)x and νξ = R−1(t)ν. Considering constant ω and rs, Eq. (B.3) replaces the term tNG(x, t), which gives Ti = bi(rs) · T = − �
fEAF ci bi(ν) · (x × ν) dσ. (13)
∂Ωt
As is described in appendix B, the numbers ci are Fourier coefficients for the source term vgas ˙m representing the TPM. A similar idea has been introduced by Kramer et al. (2019) for the diurnal Article number, page 7

-1,-1,-1 1,-1,-1
1,1,-1 -1,1,-1
-1,-1,1 1,-1,1
1,1,1 -1,1,1
-1,-1,-1 1,-1,-1
1,1,-1 -1,1,-1
-1,-1,1 1,-1,1
1,1,1 -1,1,1
135°
135°
90°
90°
45°
45°
s for T
s for T
0°
0°
45°
45°
90°
90°
135°
135°
45° 30° 15° 0° 15° 30° 45° subsolar latitude of rs
45° 30° 15° 0° 15° 30° 45° subsolar latitude of rs
-1,-1,-1 1,-1,-1
1,1,-1 -1,1,-1
-1,-1,1 1,-1,1
1,1,1 -1,1,1
90°

[Figure: Fig. 8. Direction of torque, T, for the near-prolate ellipsoid. Torque, T,]
15° 0° 15° 30° 45° subsolar latitude of rs
45° 30° 15° 0° 15° 30° 45° subsolar latitude of rs
-1,-1,-1 1,-1,-1
1,1,-1 -1,1,-1
-1,-1,1 1,-1,1
1,1,1 -1,1,1
90°

[Figure: Fig. 8. Direction of torque, T, for the near-prolate ellipsoid. Torque, T,]

in Eq. (12) is shown with model A and fEAF = χAi, j,k for region Ai,j,k in
60°
Eq. (6). Uniform activity fEAF = 1 is shown with dotted black lines for the oblate ellipsoid and for the near-prolate ellipsoid. λs and ϕs denote longitude and latitude for T with respect to the solar basis in Fig. 2. The solar vector, rs, changes SSL from −45◦to 45◦.
30°
s for T
0° 30° 60° 90°
nents are derived for each surface region that produce the torque
movement. On the left-hand side of Eq. (13) are the components
represent directions toward and perpendicular to the sun as de-
components with respect to the local basis, B(ν), in Fig. 1. Ad-
jacent to the TPM terms, the integrand in Eq. (13) contains the

45° 30° 15° 0° 15° 30° 45° subsolar latitude of rs

[Figure: Fig. 9. Direction of torque, T, for the bilobed shape. Torque, T, in Eq. (12) is shown with model A and fEAF = χAi, j,k for region Ai,j,k in Eq. (6). Uniform activity fEAF = 1 is shown with dotted black lines. λs and ϕs denote longitude and latitude for T with respect to the solar basis in Fig. 2. The solar vector, rs, changes SSL from −45◦to 45◦.]
The properties of the bilobed shape are similar to those of
turbed. The dotted line in Fig. 9 shows the resulting longitude,
smaller for the bilobed shape compared to the other shapes. This
goes back to the directional signal caused by shadowing and is
more sensitive compared to the geometrical effects of the other
shapes.
hand side have nonzero contributions, which is shown in Fig. 3
mation. We focus our approach on the superposition of regional
choice. The definition of the regions in Eq. (6) yields the proper-
From Eq. (3) we know that the shape Ω determines the tensor of
In this section, we assume an instantaneous TPM and a constant
ferent properties. The torque, T , is constant in time and finally
values than the northern region. Then the integral in Eq. (13) is
when SSL changes from negative to positive.
The near-prolate ellipsoid is more complex, such that the
in Eq. (13), the distribution is symmetric with respect to zero,
line integrals. This term shows the same signs for points on the
ties

sgn T1(Ai,j,k) = i, sgn T2(Ai,j,k) = j, sgn T3(Ai,j,k) = k. (14) Based on Eq. (13) this extends to the signs of T(Ai, j,k) · b1,2,3(rs).
For the bilobed shape with model A, Fig. 9 shows the longitude, λs, and latitude, ϕs, of T(Ai, j,k). The same figure for the bilobed
shape with model α looks similar, leading to the same conclusion. Eq. (14) confirms the assignment of λs and ϕs to the angular quadrants. For example T1(A−1,−1,±1) and T2(A−1,−1,±1) both
Article number, page 8
the discussion for T , the vector ω can be decomposed into com-
rotation axis, and rotational angular momentum) are aligned to
each other. The presence of torque, T , causes an excitation for
For rotation states close to lowest energy, the torque formation
discussed in Sect. 3 remains valid qualitatively. The longitude,
oblate ellipsoid, the near-prolate ellipsoid, the bilobed shape and
shape 67P, we considered a second variant based on the model α
To study the evolution of the rotation state, we studied the
formation of the torque, T , according to Eq. 10. The first setup
significantly improves the analysis of the linearized equations
The shape of comet 67P is more complex than any of the ideal-
ized shapes. However, the torque formation shares some proper-
ties with simplified models, while lack of symmetry brings about
new features. According to the discussion in the last section, we
From Figs. 8, 9 for both ellipsoids and the bilobed shape, we
property for shape 67P with Eq. (13). Fig. 4 shows the compo-
predominant. For irradiation conditions coming from the north
irradiation coming from mid latitudes even from the south. For