)
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

[Figure: Fig. 2. Solar basis, B(rs) = (b1(rs), b2(rs), b3), for the choice y = rs(t) in Eq. (4) at time t.]

67P/C-G to be in lowest-energy rotation.

A free nutation motion for the angular velocity vector has not been observed within the accuracy range of a few tens of millidegrees.

A lowest-energy rotation implies ω = |ω|b3 for the angular velocity and L = J3|ω|b3 for the rotational angular momentum.

In the following, the initial states, R0 and L0, are chosen to be at the lowest energy.

With the comet becoming active, the NGTs introduce some perturbations leading to an excited state.

A torque, T = T3b3, directed along b3, is able to change the vector length |ω| (and thus the rotation period), while the directions of b3, ω, and L remain unchanged.

A torque, T, that is orthogonal to b3, namely with T · b3 = 0, breaks the initial alignment of b3 and ω in general.

For linearized Euler equations in Eq. (2), an analysis for the angle γ between b3 and ω yields the estimation tan γ ≈|T|/J3/ω2.

Assuming characteristic values for ice sublimation on comet 67P/C-G with |T| = 109 Nm, J3 = 1019 kg m2, and |ω| = 10−4 s−1 yields an upper bound |γ| ≈0.5◦.

To analyze the orientation of vector quantities (e.g., torque and angular velocity) relative to the sun and to the short axis, we defined two different basis systems with the (orthogonal) vectors
ASH = {x ∈∂Ωt | ν · b3 < 0}, ANH = {x ∈∂Ωt | 0 < ν · b3}.

Their specification is based on the sign of the b3 component for the surface normal, ν, at point x ∈∂Ωt.

For a lowest-energy rotation state, this projection controls the seasonal intensity of the solar irradiation on the surface, ∂Ωt.

For negative subsolar latitude SSL = arcsin(rs · b3/|rs|), the point of maximum irradiation is located in ASH.

For positive SSL, this point is located in ANH.

This relation expresses the mapping between ASH and ANH on the one side and both hemispheric seasons on the other side.

This concept of defining regions based on projections to the local basis in Fig. 1 can be applied to the vector x × ν.

We define the vectorial torque efficiency by the three scalar components, (x × ν) · bi, (5) for i = 1, 2, 3.

Fig. 3 shows the maps of the vectorial torque efficiency for the near-prolate ellipsoid.

Fig. 4 shows the b2 component for the more complex geometries of the bilobed shape and shape 67P.

In Sect. 3 we show that both figures represent components of the vectorial torque efficiency.

Based on these fields for i, j, k = ±1, we define eight regions,
b1(y) = (b3 × y) × b3
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x ∈∂Ωt
, (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.