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A Radial Axial-symmetric Intermediary Model for the Roto-orbital Motion

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Abstract

We study the roto-orbital dynamics of a uniform sphere and a body with axial symmetry by means of a radial intermediary, which defines an integrable system. Numerical comparisons of the MacCullagh’s truncation of the gravity gradient potential and intermediary models are performed, concluding that the intermediary provides a valuable approximation with small differences when compared with the MacCullagh’s one. Our analysis includes the analytical integration and a study of the special solutions and relative equilibria.

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Abbreviations

A, B, C,:

Principal moments of inertia

A :

Rotation matrix relating vectors in the body \(\mathcal {B}_{1}\) and inertial frames

b i :

Body frame vectors

\(\mathcal {B}_{1}\) :

Body with mass m 1

\(\mathcal {B}_{2}\) :

Body with mass m 2. Main body

e :

Eccentricity

E i :

Spatial frame vectors

\(\mathcal {G}\) :

Universal gravitational constant

G, G o , G r :

Total, orbital, rotational angular momentum vectors in body frame

G, G o , G r :

Modulus of the corresponding vectors

\(\mathcal H\) :

Hamilton function of the intermediary model

\(\mathcal {H}_{K}\) :

Kepler system

\(\mathcal {H}_{E}\) :

Free rigid body

\(\mathcal {H}_{0}\) :

Zero order Hamiltonian (\(\mathcal {H}_{0}=\mathcal {H}_{K}+\mathcal {H}_{E}\))

I :

Moment of inertia tensor relative to the body \(\mathcal {B}_{1}\)

I o :

Angle between π o and π

I r :

Angle between π r and π

I :

Inclination of the total angular momentum plane

I Δ :

Inclination of the rotational plane

I Θ :

Inclination of the orbital plane

ℓ :

Node of the total angular momentum plane

ℓ I :

Body plane (see π b ) and rotational planes intersection

ℓ r ,−ℓ o :

Total, orbital and rotational planes intersection

ℓ μ δ :

Rotational and spatial planes intersection

M :

Total mass of the system

m :

Reduced mass of the system

m i :

Mass of the body \(\mathcal {B}_{i}\)

n :

Unitary vector parallel to the total angular momentum

N, N ′ :

Third component of G r

\(\mathcal {P}\) :

Perturbation

p :

Linear momentum associated to r

q :

m/m 1

r, r ′ :

Modulus of the vector joining the mass center of the bodies

r :

Vector joining the mass center of the bodies

R, R ′ :

Radial velocity of the center of mass

R 1 :

Distance between the center of mass and the furthest point of \(\mathcal {B}_{1}\)

R 2 :

Distance between the center of mass and the furthest point of \(\mathcal {B}_{2}\)

R i :

Position vector of the body \(\mathcal {B}_{i}\) in the inertial frame

U :

MacCullagh’s term

\(\mathcal {V}\) :

Potential of the intermediary model

γ i :

Direction cosines of r/r

δ, δ ′ :

Argument of the angle between ℓ r and ℓ I

Δ, Δ′ :

Magnitude of the angular momentum of the rigid body

𝜃, 𝜃 ′ :

Argument of the vector joining the center of masses of the two bodies

Θ, Θ′ :

Magnitude of the angular momentum of the center of mass

ι :

I o + I r

κ :

\(\mathcal {G}M\)

λ :

Argument of the node relative to the Andoyer plane

μ,:

Argument of the angle between ℓ μ δ and ℓ I

ν, ν ′ :

Argument of the angle between ℓ I and b 1

# #π# # :

Angular momentum of the body B 1

π:

Total angular momentum plane

π b :

Body plane, b 1 b 2-plane

π o :

Orbital plane

π r :

Andoyer plane

ρ m i n :

R 1 + R 2

σ :

Angle between π r and the body plane

ϕ, ϕ ′ :

Argument of the node of the total angular momentum plane

Φ, Φ′ :

Third component of the total angular momentum in space frame

ψ, ψ ′ :

Argument of ℓ r

Ψ, Ψ′ :

Magnitude of the total angular momentum

Ω i :

Angular velocity in the body frame

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Acknowledgments

Support from Research Agencies of Chile and Spain is acknowledged. They came in the form of the research projects 11160224, of the Chilean national agency FONDECYT and from the MTM2015-64095-P and ESP2013-41634-P, of the Ministry of Science of Spain. Partial support for one of the authors (S.F.) came from the Senior Mobility Program of the Ministry of Education of Spain. We deeply appreciate comments and suggestions of the referee which contributed to the improvement and clarity of the paper. We would also like to thank J. Cardoso for his efforts in collating numerical experiments.

Author information

Authors and Affiliations

  1. Facultad de Ciencias Departamento Matemática, Grupo GISDA, Universidad del Bío-Bío, Collao 1202, Casilla 5-C., Concepción, Chile

    F. Crespo

  2. Space Dynamic Group, DITEC, Facultad Informatica, Universidad de Murcia, Murcia, Spain

    F. J. Molero & S. Ferrer

  3. Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO, 80309, USA

    D. J. Scheeres

Authors
  1. F. Crespo
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  2. F. J. Molero
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  3. S. Ferrer
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  4. D. J. Scheeres
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Corresponding author

Correspondence to F. Crespo.

Additional information

S. Ferrer was on a sabbatical leave at the Department of Aerospace Engineering Sciences, University of Colorado at Boulder.

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Crespo, F., Molero, F.J., Ferrer, S. et al. A Radial Axial-symmetric Intermediary Model for the Roto-orbital Motion. J of Astronaut Sci 65, 1–28 (2018). https://doi.org/10.1007/s40295-017-0121-9

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