through the centers of the spheres and e2, e3 span the perpendicular plane.

The surface of the domain, Ω, for the bilobed shape consists of NE = 4000 triangular elements.

The last case is a shape model of comet 67P/C-G.

The NE = 3996 surface elements of the last shape are the same as those of Läuter et al. (2022), which are derived from the highresolution (meter-scale) shape model of Preusker et al. (2017).

All shapes share the same normalized volume |Ω| = 18.56 km3.

Based on these choices, all shapes share the same spatial resolution of approximately ∆x ≈150 m.

The components of the tensor of inertia are given by �
ρ(δi jξ2 −ξiξ j) dξ. (3)
JBF,i j =
Ω
For the two ellipsoids and for the bilobed shape within domain Ω, we assume the constant density ρ = 539 kg/m3.

For shape 67P, the tensor of inertia JBF is set to Eq. (A.1).

For this case, the assumption of uniform density is suspended, in good agreement with the gravitational constraints of Laurent-Varin et al. (2024).

For all shapes considered, e3 is the principal axis of JBF with the largest eigenvalue, J3.

The vector e3 in the body frame corresponds to the principal axis, Re3, in the inertial frame.

In the case of ellipsoids the moments of inertia are ordered in inverse proportion to the lengths of the semimajor axes.

The principal axis e3 corresponds to the largest moment of inertia and to the shortest axis.

Let us denote b3 = Re3 as the short axis in the inertial frame.

For a lowest-energy rotation state (represented by R and L), the directions of L, of ω, and of the short axis coincide.

During the 2015 apparition Godard et al. (2017), Preusker et al. (2017), Jorda et al. (2016), and Gutiérrez et al. (2016) consider comet Article number, page 3 body frame (namely R−1b1(ν), R−1b2(ν), R−1b3)

is constant. This means that the local basis follows the rotational movement in the inertial frame together with x = Rξ. The three vectors b1, ν, and b3 are located in a shared plane. For the solar basis, B(rs), in Eq. (4) we define y = rs(t) with the vector rs pointing toward the sun at time t. Fig. 2 shows this geometric setup. At a fixed time t the basis is constant along the surface, ∂Ωt. With advancing time t, it does not follow the rotational movement in the inertial frame and it rotates in the body frame. The vector b1 is related to the solar direction; that is, such that b1, rs, and b3 are located in a shared plane. Except for the signs for a lowest-energy state, this solar basis was already applied by Kramer & Läuter (2019) and Sekanina (1981). Kramer et al. (2019) apply this basis to represent the diurnally averaged torque affecting the rotation state. The local and solar basis share the property that b3 is aligned to the short axis so that the equatorial plane of the rotation is spanned by the vectors b1 and b2. The equatorial planes of both basis systems are the same. b1(ν), b2(ν) and b1(rs), b2(rs) are just two choices for a basis of this plane.
b3
(x, t)
b2( ) b1( )
x

[Figure: Fig. 1. Local basis, B(ν) = (b1(ν), b2(ν), b3), for the choice y = ν(x, t) in Eq. (4) at time t and at position x ∈∂Ωt with the outward normal ν.]

b3
b2(rs)
2.2. Domain decomposition of the surface For each ξ ∈∂Ωon the surface, x = Rξ, ν(x, t), and bi(ν) (the local basis vector) follow the same rotational movement in the inertial frame. Based on these quantities, we define regions on ∂Ωt that move correspondingly in the inertial frame. Their counterparts in the body frame are time-independent, so these definitions are based on geometry properties of the shape in the body frame, ∂Ω. At time t we define the southern and the northern hemisphere,
b1(rs)
rs