Epicycle Analysis of the LISA Orbits

Paper number

IAC-07-C1.4.03

Author

Prof. Robert G. Melton, The Pennsylvania State University, United States

Coauthor

Mr. John Iannacone, United States

Year

2007

Abstract

Vincent and Bender[1] first described the required orbital configuration suitable for what would become the proposed Laser Interferometer Space Antenna (LISA) mission[2] to detect gravitational waves. This configuration produces an equilateral triangular formation, centered approximately 1 a.u. from the Sun and approximately 20 degrees behind Earth. The formation plane is tilted 60 degrees from the ecliptic and rotates once per orbital revolution. While the initial analysis provided an important baseline for mission planning, more recent concerns about detailed motion within the configuration have led to new interest in characterizing the formation. Specifically, the sides of the triangle have time-varying lengths that must be controlled to avoid exceeding the limits of the interferometric instrumentation; however, this control must be infrequent to avoid interference with the scientific mission. Sweetser[3] presented a second-order series representation of the formation dynamics in terms of epicycles. This approach has the advantage of giving a relatively simple form for the dynamics, allowing the controls analysts to predict frequency and magnitude of active control phases. Sweetser employed a novel form in which the length variation for one side of the triangle (one arm of the interferometer) is given as a series in the function C (whose elements are the radial and along-track directions):

C( r,α, φ) = {  rcos(α  t + φ), rsin(α  t + φ) }

where r is the orbital radius, α the angular rate, and φ a phase angle. This solution was developed by calculating coefficients of the series via comparison with the known solution as developed by Vincent and Bender[1].

In this paper, the motion of each satellite in the formation relative to the triangle’s center, which follows a circular path, is represented via the exact solution to the Hill-Clohessy-Wiltshire equations, modified to include epicycles:[4]

x =  b +  Acosα

y =  w −

3  2

bα − 2 Asinα

z =  Zcos(α − α N)

ẋ = − AΩsinα

ẏ =

−3  2

bΩ − 2 Acosα

ż = − ZΩsin(α − α N)

where x, y, and z are the radial, along-track, and cross-track directions, respectively, α = epicycle phase, Ω = epicycle phase rate, A = epicycle amplitude, b and w are the radial and along-track offsets, and N refers to the nodal passage. The paper will compare this formulation with that of Sweetser to characterize the tradeoff between complexity of solution and accuracy. Additionally, the paper examines the convergence properties of the series formulation, with attention to its applicability to other relative-motion problems.

References

[1]Vincent, M.A. and Bender, P.L., “The Orbital Mechanics of a Space-Borne Gravitational Wave Experiment,” AAS/AIAA Astrodynamics Specialist Conference, Kalispell, Montana, August 10-13, 1987, paper AAS 87-523.

[2]Bender, P., Ciufolini, I., Cornelisse, J., Danzmann, K., Folkner, W., Hechler, F., Hough, J., Jafry, Y., Reinhard, R., Robertson, D., Rüdiger, A., Sandford, M., Schilling, R., Schutz, B., Stebbins, R., Sumner, T., Touboul, T., Vitale, S., Ward, H., Winkler, W., “LISA: Laser Interferometer Space Antenna for the detection and observation of gravitational waves,” Pre-Phase A report, MPQ 208, Max-Planck-Institut für Quantenoptik, D-85748, Garching, Germany, February, 1996.

[3]Sweetser, T.H., “Epicycles and Oscillations: the Dynamics of the LISA Orbits,” AAS/AIAA Astrodynamics Specialists Conference, Lake Tahoe, California, August 7-11, 2005, paper AAS 05-292.

[4]Anthony, M.L., and Sasaki, F.T., “Rendezvous Problem for Nearly Circular Orbits,” AIAA Journal, Vol. 3, No. 9, 1965, pp. 1666-1673.

Abstract document

IAC-07-C1.4.03.pdf

Manuscript document

IAC-07-C1.4.03.pdf (🔒 authorized access only).

To get the manuscript, please contact IAF Secretariat.