Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields

Marta Ceccaroni, Francesco Biscani, James Biggs

Research output: Contribution to conferencePaper

Abstract

This article provides a method for finding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the inhomogeous body is firstly stated, in polar-nodal coordinates, which takes into account the coefficients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields
perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coeffcients. The project has been developed in collaboration with the European Space Agency (ESA).

Conference

Conference2012 Analytical Methods in Celestial Mechanics
CountryRussian Federation
CitySt Petersburg
Period26/09/1229/09/12

Fingerprint

Asteroids
Gravitational Field
Analytical Methods
Orbits
Orbit
Normalization
Hamiltonians
Initial conditions
Delaunay
Spherical Harmonics
Arbitrary
Canonical Transformation
Eccentricity
Inclination
Vertex of a graph
Rigid Body
Inspection
Rotating
Degree of freedom
Trajectories

Keywords

  • orbit perturbation
  • graviational fields
  • analytical methods

Cite this

Ceccaroni, M., Biscani, F., & Biggs, J. (2012). Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields. Paper presented at 2012 Analytical Methods in Celestial Mechanics, St Petersburg, Russian Federation.
Ceccaroni, Marta ; Biscani, Francesco ; Biggs, James. / Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields. Paper presented at 2012 Analytical Methods in Celestial Mechanics, St Petersburg, Russian Federation.25 p.
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Ceccaroni, M, Biscani, F & Biggs, J 2012, 'Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields' Paper presented at 2012 Analytical Methods in Celestial Mechanics, St Petersburg, Russian Federation, 26/09/12 - 29/09/12, .

Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields. / Ceccaroni, Marta; Biscani, Francesco; Biggs, James.

2012. Paper presented at 2012 Analytical Methods in Celestial Mechanics, St Petersburg, Russian Federation.

Research output: Contribution to conferencePaper

TY - CONF

T1 - Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields

AU - Ceccaroni, Marta

AU - Biscani, Francesco

AU - Biggs, James

PY - 2012/9/26

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N2 - This article provides a method for finding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the inhomogeous body is firstly stated, in polar-nodal coordinates, which takes into account the coefficients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coeffcients. The project has been developed in collaboration with the European Space Agency (ESA).

AB - This article provides a method for finding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the inhomogeous body is firstly stated, in polar-nodal coordinates, which takes into account the coefficients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coeffcients. The project has been developed in collaboration with the European Space Agency (ESA).

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Ceccaroni M, Biscani F, Biggs J. Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields. 2012. Paper presented at 2012 Analytical Methods in Celestial Mechanics, St Petersburg, Russian Federation.