A consistent numerical integration of orbit, tides, and rotation for synchronous satellites

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Table 4

Averaged torques (in N.m) over an integer number of orbits raised by tides on the Moon and on Phobos.

	Torque from tidal theory		From coupled model
Satellite	no libration (Eq. (45), $A = 0$ $Mathematical equation: ${\cal A} = 0$$ )	with libration (Eq. (46))	$Γ_{Δ C_{22}}$ $Mathematical equation: ${\Gamma _{\Delta {C_{22}}}}$$	$Γ_{Δ S_{22}}$ $Mathematical equation: ${\Gamma _{\Delta {S_{22}}}}$$	$Γ_{S_{22}}$ $Mathematical equation: ${\Gamma _{{S_{22}}}}$$
Moon	5.6 ∙ 10¹⁴	5.6 ∙ 10¹⁴	2.8 ∙ 10¹⁴	2.8 ∙ 10¹⁴	−5.6 ∙ 10¹⁴
Phobos	1.5 ∙ 10⁸	2.0 ∙ 10⁸	1.0 ∙ 10⁸	1.0 ∙ 10⁸	−2.0 ∙ 10⁸

Notes. In our coupled propagation, the (total) averaged torque caused by the satellite’s tidal deformation matches the secular tidal torque derived from the tidal theory (Section 3.3). As discussed in Section 3.5, this residual tidal torque is, however, perfectly compensated by the triaxial torque raised by a non-zero static S₂₂ (last column).

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