Ceres Rotation Solution under the Gravitational Torque of the Sun

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1 Ceres Rotation Solution under the Gravitational Torque of the Sun Martin Lara, Toshio Fukushima, Sebastián Ferrer (*) Real Observatorio de la Armada, San Fernando, Spain ( ) National Astronomical Observatory, Mitaka, Tokyo, Japan ( ) Dep. Applied Math., University of Murcia, Murcia, Spain Session PS6.1 European Geosciences Union General Assembly 2011 Vienna, Austria, 3 8 April 2011

2 Background Protoplanet Ceres: target of NASA s Dawn mission crucial info. on history of Solar System (Russell et al 2006) Ceres internal structure: accurate shape determination required < 0.5 km (Castillo-Rogez & McCord 2010) actual 2 km: discrepancies from different authors (Carry & al 2008, Thomas & al 2005, McCord & Sotin 2005) Available observations on the shape of Ceres: rotationally symmetric, almost spherical, oblate spheroid deviations from axisymmetry could happen at the level of observational accuracy main effect on Ceres rotational dynamics Desirable: analytical rotation theory for a triaxial Ceres 1

3 Outline Goal: analytical rotation theory for a triaxial Ceres canonical perturbation theory: perturbed spherical rotor torque-free rotation plus Sun s gravity-gradient torque Keplerian orbit about the Sun Time dependent Hamiltonian of 3-DOF Completely reduced Hamiltonian by Lie transforms (Hori 1966, Deprit 1969, Campbel & Jefferys 1970) Rotation sol. up to 1st order terms in gravity-gradient accuracy: secular trend of few mas/orbital period Outline possible planetological implications Conclusions 2

4 Torque-free rotation Classical approach integrate angular velocity in the body frame relate body and inertial frames through Euler angles Hamiltonian formulation Andoyer variables: free rigid body Hamiltonian of 1-DOF Integration by Hamilton-Jacobi reduction Sadov 1970, Kinoshita 1972, Barkin 1992 lecture notes Solution involves elliptic integrals and functions Many celest. bodies: triaxiality perturbation of axisimmetry perturbation approach in circular functions almost spherical: perturbed spherical rotor approach (Ferrer & Lara, AJ 2010) 3

5 m s 3 b 2 b 3 σ ε s 1 l λ ε µ n ν b 1 σ s 2 Andoyer variables: two sets of Euler angles invariant plane to the angular momentum vector 3-1

6 Perturbed motion: McCullag s potential Hamiltonian: torque-free motion + gravity-gradient H = M 2 N 2 ( sin 2 ) ν 2 A + cos2 ν + N 2 B 2C Gm (A + B + C 3 D) 2r3 Gm Sun gravitational constant r = r(t) Sun-Ceres distance. Keplerian approx: r = a η 2 /(1 + e cos f), η = 1 e 2 A, B, C, principal moments of inertia D = A γ B γ2 2 + C γ2 3, direction cosines γ 1, γ 2, and γ 3 Orbital and body frames related through 5 rotations (γ 1, γ 2, γ 3 ) T = R 3 (ν) R 1 (J) R 3 (µ) R 1 (I) R 3 (λ θ) (1, 0, 0) T θ polar coordinate of the orbital motion 4

7 Averaging: perturbed spherical rotor Hamiltonian ordering: H = H 0 + ɛ H 1 + ɛ2 2! H 2 + ɛ7 7! H 7 H 0 = H 0 (M) perturbed spherical rotor H 1 = H 1 (M, N) axisymmetric perturbation H 2 = H 2 (ν, M, N) triaxiality character H 7 = H 7 (µ, ν, λ, M, N, Λ; t) gravity-gradient perturbation Straightforward approach three consecutive canonical transformations elimination of: first µ, then ν, finally λ and t generating functions by quadrature, instead of PDE solution free from elliptic functions Secular Hamiltonian + transformation equations 5

8 Secular frequencies [ ω µ 1 M/C = 1 + α αβ2 ω ν N/C ω λ Λ/C = = α αβ2 2 2 [ ( ( c 2 J ) c 4 J n2 δ 3 M 2 /C 2 4η 3 (1 3c2 J ), ) ] ] n2 δ M 2 /C 2 3 4η 3 + n2 δ M 2 /C 2 3 4η 3 (1 3c2 I ) [ c 2 J + (1 6c 2 J ) c2 I ] c I = cos I, c J = cos J, η = 1 e 2, n 2 = Gm/a 3 α = 1 2 ( C A + C B ) 1, β = 1 2α ( C A C B ), δ = 1 B + A 2C 5-1

9 Accuracy: analytic vs numerical int. Secular terms: ±1 for µ and ν, and 2 arcsec for λ Μ 10 5 Λ Orbital periods Orbital periods Orbital periods Short periodic errors dominate µ- & ν-propagation Sun perturbation dominates errors in the momenta and λ 6

10 Whole theory: secular terms + transformation equations Μ 10 8 Λ Orbital periods Orbital periods Orbital periods secular trend 15 mas/orbital period (µ, ν), better for λ 6-1

11 Planetological implications (ν, J) rotation of the angular momentum vector around b 3, ω ν related to wobble, except for short-period effects Kinoshita s relation between Andoyer vars. & Euler angles φ = λ + J sin I sin µ + O(J2 ), ψ = ν + µ J cot I sin µ + O(J 2 ), Then, φ µ = λ µ ω λ, ψ µ = ν µ + µ µ ω ν + ω µ, ω λ carries the secular terms of the precession ω µ, on average, accounts for most of the rotation If ω is measured, constraint in β is obtained from our theory 7

12 Influence of the triaxiality: secular frequencies 2Π ΩΜ hours Π ΩΝ days Π Ω Λ kyrs β = 0 oblate, β = 1 prolate (Ceres: β 0.035) valid only for small β because of the assumptions made 7-1

13 Conclusions Ceres may be affected by a very small triaxiality minute deviations in axisymmetry affect the rotation not measurable from available observations NASA s Dawn: volume, shape, spin state & mass of Ceres High Res. Camera: observe landmarks rotation state observation starts 1 month prior to capture at Ceres Analytical theory for triaxial Ceres (perturb. spherical rotor) Ceres rotational state directly in Andoyer variables Dawn results may enter directly our rotation solution Alternative, spin & wobble from Dawn observations: used to solve the rotation solution for β provide a more accurate constrain on Ceres inertia 8