Rotation of 67P. Mottola et al. (2014): Lightcurve photomerty -> Spin period of 12.40 h.

Rotation of 67P. Mottola et al. (2014): Lightcurve photomerty -> Spin period of 12.40 h. Evidences of complex rotation.

Rotation of 67P. Mottola et al. (2014): Lightcurve photomerty -> Spin period of 12.40 h. Evidences of complex rotation. Preusker et al. (2015): The accuracy of the SPG topographic model improves when a precessing spin axis is defined. Proposed rotation:

Rotation of 67P. Mottola et al. (2014): Lightcurve photomerty -> Spin period of 12.40 h. Evidences of complex rotation. Preusker et al. (2015): Spin axis moving with a period of 10.7 days ( 257 h) Jorda et al. (2015): by using the SPC method, obtain that the spin axis is moving around a central position.

Rotation of 67P. Mottola et al. (2014): Lightcurve photomerty -> Spin period of 12.40 h. Evidences of complex rotation. Preusker et al. (2015): Spin axis moving with a period of 10.7 days ( 257 h) Jorda et al. (2015): by using the SPC method, obtain that the spin axis is moving around a central position. Periodograms of the coordinates of the spin axis show a very significant periodicity at approximately 270 h.

Rotation of 67P. Mottola et al. (2014): Lightcurve photomerty -> Spin period of 12.40 h. Evidences of complex rotation. Preusker et al. (2015): Spin axis moving with a period of 10.7 days ( 257 h) Jorda et al. (2015): Spin axis moving. Periodograms of RA/Dec coordinates show a Very significant periodicity at 11.5 days (276 h) Rotation of 67P seems to have two associated periodicities: 12.40 h (Mottola et al., 2014): Rotation period. 270 h (Preusker et al., 2015; Jorda et al., 2015): ?? What is the meaning of the 270 h periodicity of the coordinates of the spin pole and what information can be extracted from it?

Rotation of 67P. In order to understand the meaning of the 270 h periodicity we performed simulations of the motion of the spin axis according to the Euler equations. Assumptions: The body is considered a rigid body under free torque conditions. The rotation period is 12.40 h (Mottola et al., 2014). The body is close to pure spin. For homogeneity, we will consider just the observational data from SPC procedure. Variables: * Inertia moments: I1 ≤ I2 ≤ I3 * Excitation level. Defining L the angular momentum and E the rotational energy, for slightly excited cases it is always true I2 < L2/2E ≤ I3 Excitation level: L2/2EI3 ≅ 1

Rotation of 67P. Notation: Under freetorque conditions and low excitation, both ω (rotation axis) and x3 (axis with the largest inertia moment) rotate around the angular momentum vector. We will describe the motion of the third axis of inertia in space. In order to do that, a convenient Euler angles set must be defined.

Rotation of 67P. Notation: Under freetorque conditions and low excitation, both ω (rotation axis) and x3 (axis with the largest inertia moment) rotate around the angular momentum vector. We will describe the motion of the third axis of inertia in space. In order to do that, a convenient Euler angles set must be defined. ϕ : Precession of the axis with the largest inertia moment around the angular momentum vector. Θ : Nutation of the axis with the largest inertia moment around the angular momentum vector. ϕ : Rotation around the axis with the largest inertia moment. xyz: Inertial frame x1x2x3: Body frame.

Rotation of 67P. Notation: Under freetorque conditions and low excitation, both ω (rotation axis) and x3 (axis with the largest inertia moment) rotate around the angular momentum vector. We will describe the motion of the third axis of inertia in space. In order to do that, a convenient Euler angles set must be defined. ϕ : Precession of the axis with the largest inertia moment around the angular momentum vector. Θ : Nutation of the axis with the largest inertia moment around the angular momentum vector. ϕ : Rotation around the axis with the largest inertia moment. Those three Euler angles, describing three different rotations of the body frame with regard to the inertial frame, have 3 associated periodicities: P_phi, P_theta, P_psi, all depending on inertia moments and excitation level. Under our conditions, it is always true that: P_psi > 2*P_phi, P_theta = P_psi/2 xyz: Inertial frame x1x2x3: Body frame.

Rotation of 67P. Simulation of the homogeneous body (Inertia moments: 1:1.83:1.98) Reasonable similarity* in the RA/Dec of the body frame z-axis in space but… * Body frame axes initial orientations are unknown, reason why distributions cannot be coincident, just compatibility of belonging to the same distribution could be estimated.

Rotation of 67P. Simulation of the homogeneous body (Inertia moments: 1:1.83:1.98) If periodograms are considered Reasonable similarity* in the RA/Dec of the body frame z-axis in space but… Simulated Observational They are quite different

Rotation of 67P. Simulation of the homogeneous body (Inertia moments: 1:1.83:1.98) Periodograms of simulated RA/Dec distributions show two significant peaks (apart from alias o artifacts) Simulated

Rotation of 67P. Simulation of the homogeneous body (Inertia moments: 1:1.83:1.98) Periodograms of simulated RA/Dec distributions show two significant peaks (apart from alias o artefacts) A periodicity corresponding exactly to P_phi (the precession period of the z-axis body frame around the angular momentum). For the homogeneous body is around 9.7 h. Simulated

Rotation of 67P. Simulation of the homogeneous body (Inertia moments: 1:1.83:1.98) Periodograms of simulated RA/Dec distributions show two significant peaks (apart from alias o artefacts) A periodicity corresponding exactly to P_phi (the precession period of the z-axis body frame around the angular momentum). For the homogeneous body is around 9.7 h. A very significant periodicity corresponding exactly to a combination of the precession periodicity, P_phi, and the nutation periodicity, particularly 1/P_c = 1/P_phi – 2/P_psi For the homogeneous body, P_c = 17.3 h. Simulated

Rotation of 67P. Performing a large number of simulations we conclude that the previous result can be generalized for any combination of inertia moments, and excitation level. Periodograms of the coordinates of the body frame z-axis in space only show two periodicities, being the most significant one a combination of the precession and nutation periodicities. The periodicity corresponding the precessional motion may eventually disappears for large inertia moments and/or if significant noise is added to the coordinates. In no single simulation, the periodicities corresponding to P_phi (the rotation around the body frame z-axis) or to P (the full rotational period) were detected in the periodograms.

Rotation of 67P. Performing a large number of simulations we conclude that the previous result can be generalized for any combination of inertia moments, and excitation level. Periodograms of the coordinates of the body frame z-axis in space only show two periodicities, being the most significant one a combination of the precession and nutation periodicities. The periodicity corresponding the precessional motion may eventually disappears for large inertia moments and/or if significant noise is added to the coordinates. In no single simulation, the periodicities corresponding to P_phi (the rotation around the body frame z-axis) or to P (the full rotational period) were detected in the periodograms. Those results allow us to interpret that the periodicity detected in the periodograms of the observationally derived coordinates of the body-frame z-axis of 67P (270 h) could actually be P_c = 1/(1/P_phi-2/P_psi), i.e. the combination of the precessional and nutational motions.

Rotation of 67P. Thus, considering 67P as a rigid-body under freetorque conditions rotating with a period of 12.40 h (Mottola et al, 2014), if it is assumed that the periodicity detected in the periodograms of the observationally derived RA/Dec coordinates of the body frame z-axis is P_c, it is possible to find the combination of inertia moments and excitation level providing us with a P_c = 270 h. It is also possible to show that P_c depends mostly on the inertia moments, and very weakly on the excitation level.

Rotation of 67P. Thus, considering 67P as a rigid-body under freetorque conditions rotating with a period of 12.40 h (Mottola et al, 2014), if it is assumed that the periodicity detected in the periodograms of the observationally derived RA/Dec coordinates of the body frame z-axis is P_c, it is possible to find the combination of inertia moments and excitation level providing us with a P_c = 270 h. It is also possible to show that P_c depends mostly on the inertia moments, and very weakly on the excitation level. Under the previous considerations, it is possible to show that if inertia moments fulfill the linear relationship I_3 = 0.963 + 0.988*I_2 The periodograms of the coordinates in space of the body frame z-axis shows the most significant periodicity at 270 h (as observationally derived data)

Rotation of 67P. Thus, considering 67P as a rigid-body under freetorque conditions rotating with a period of 12.40 h (Mottola et al, 2014), if it is assumed that the periodicity detected in the periodograms of the observationally derived RA/Dec coordinates of the body frame z-axis is P_c, it is possible to find the combination of inertia moments and excitation level providing us with a P_c = 270 h. It is also possible to show that P_c depends mostly on the inertia moments, and very weakly on the excitation level. Under the previous considerations, it is possible to show that if inertia moments fulfill the linear relationship I_3 = 0.963 + 0.988*I_2 the periodograms of the coordinates in space of the body frame z-axis shows the most significant periodicity at 270 h (as observationally derived data) It is not possible to constrain the exact (or even approximate) inertia moments ratio as many combinations fulfilling the previous relationship are compatible, statistically speaking, with the observationally derived RA/Dec distributions.

Rotation of 67P. K-S test on RA coordinates K-S test on Dec coordinates Actually, from K-S tests performed independently on both the RA and Dec coordinates point out to different best solutions. Best fit from RA coordinates I_2 = 1.90 I_3 = 2.84 Exc = 6.7e-5 Best fit from Dec coordinates I_2 = 2.25 I_3 = 3.19 Exc = 3.1e-5

Rotation of 67P. K-S test on RA coordinates K-S test on Dec coordinates Actually, from K-S tests performed independently on both the RA and Dec coordinates point out to different best solutions. Best fit from RA coordinates I_2 = 1.90 I_3 = 2.84 Exc = 6.7e-5 Best fit from Dec coordinates I_2 = 2.25 I_3 = 3.19 Exc = 3.1e-5 Still some discrepancies need to be explained, mostly the difference in spectral power of the periodicity in the periodograms. A larger error bar in observationally derived data could be a possible explanation.

Rotation of 67P. Conclusions. Considering 67P as a rigid-body under freetorque conditions rotating with a period of 12.40 h (Mottola et al, 2014), according to the results obtained from simulations considering the Euler equations, the significant peak found in the periodograms of the observationally derived coordinates of the body frame z-axis (270 h) could correspond to a combination of the precession periodicity and the nutation periodicity.

Rotation of 67P. Conclusions. Considering 67P as a rigid-body under freetorque conditions rotating with a period of 12.40 h (Mottola et al, 2014), according to the results obtained from simulations considering the Euler equations, the significant peak found in the periodograms of the observationally derived coordinates of the body frame z-axis (270 h) could correspond to a combination of the precession periodicity and the nutation periodicity. If this is confirmed, the inertia moments of the body must follow the relationship I_3 = 0.96 + 0.99 I_2 For that relationship between inertia moments, the body frame z-axis would be precessing around the angular momentum vector with a period of 6.35 h and, at the same time, the body would be spinning around its third axis of inertia with a period P_psi = 13 h, being the full rotation period 12.40 h.

Rotation of 67P. Conclusions. Considering 67P as a rigid-body under freetorque conditions rotating with a period of 12.40 h (Mottola et al, 2014), according to the results obtained from simulations considering the Euler equations, the significant peak found in the periodograms of the observationally derived coordinates of the body frame z-axis (270 h) could correspond to a combination of the precession periodicity and the nutation periodicity. If this is confirmed, the inertia moments of the body must follow the relationship I_3 = 0.96 + 0.99 I_2 For that relationship between inertia moments, the body frame z-axis would be precessing around the angular momentum vector with a period of 6.35 h and, at the same time, the body would be spinning around its third axis of inertia with a period P_psi = 13 h, being the full rotation period 12.40 h. The previous results would imply that 67P cannot be homogeneous, as the inertia moment ratio I_3/I_2 (ranging from 1.4 to 2.0) would be significantly larger than that of the homogeneous body ( I_3/I_2 = 1.98/1.83 = 1.1)