Kathleen C. Howell

Three-dimensional, periodic, ‘halo’ orbits

A largely numerical study was made of families of three-dimensional, periodic, ‘halo’ orbits near the collinear libration points in the restricted three-body problem. Families extend from each of the libration points to the nearest primary. They appear to exist for all values of the mass ratio μ, from 0 to 1. More importantly, most of the families contain a range of stable orbits. Only near L1, the libration point between the two primaries, are there no stable orbits for certain values of μ. In that case the stable range decreases with increasing μ, until it disappears at μ=0.0573. Near the other libration points, stable orbits exist for all mass ratios investigated between 0 and 1. In addition, the orbits increase in size with increasing μ.A largely numerical study was made of families of three-dimensional, periodic, ‘halo’ orbits near the collinear libration points in the restricted three-body problem. Families extend from each of the libration points to the nearest primary. They appear to exist for all values of the mass ratio μ, from 0 to 1. More importantly, most of the families contain a range of stable orbits. Only near L1, the libration point between the two primaries, are there no stable orbits for certain values of μ. In that case the stable range decreases with increasing μ, until it disappears at μ=0.0573. Near the other libration points, stable orbits exist for all mass ratios investigated between 0 and 1. In addition, the orbits increase in size with increasing μ.

Numerical determination of Lissajous trajectories in the restricted three-body problem

In previous studies, Lissajous trajectories associated with the collinear libration points in the restricted three-body problem have been successfully computed analytically to at least third-order. Those approximations are utilized to determine such trajectories numerically for an arbitrary, predetermined number of revolutions in the rotating frame, for the case of circular primary motion. The numerical approach first identifies target positions at specified intervals along the trajectory and locates a continuous path through those points with velocity discontinuities. Then the Δv are simultaneously reduced in an iterative process. Such trajectories have been constructed in various primary systems, for a wide range of orbit sizes and a large number of revolutions.

Improved Corrections Process for Constrained Trajectory Design in the n-Body Problem

The general objective is the development of efficient techniques for preliminary design of trajectory arcs in nonlinear autonomous dynamic systems in which the desired solution is subject to algebraic interior and/or exterior constraints. For application to then-body problem, trajectoriesmust satisfy specific requirements, e.g., periodicity in terms of the states, interior or boundary constraints, and specified coverage. Thus, a strategy is formulated in a sequence of increasingly complex steps: 1) a trajectory isfirstmodeled as a series of arcs (analytical or numerical) and general trajectory characteristics and timing requirements are established; 2) the specific constraints and associated partials are formulated; 3) a corrections process ensures position and velocity continuity while satisfying the constraints; and finally, 4) the solution is transitioned to a full model employing ephemerides. Though the examples pertain to spacecraft mission design, the methodology is generally applicable to autonomous systems subject to algebraic constraints. For spacecraft mission design applications, an immediate advantage of this approach, particularly for the identification of periodic orbits, is that the startup solution need not exhibit any symmetry to achieve the objectives.

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional ‘halo’ periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L3-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L1-family and L2-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.

Trajectory Design in the Sun-Earth-Moon System Using Lunar Gravity Assists

Theobjectiveofthisworkisthedevelopmentofefe cienttechniquesforthepreliminarydesignoftrajectoriesthat encounterthemoonandmustsatisfyspecie ctrajectoryrequirements,suchasapogeeplacement,launchconstraints, or end-state targeting. These types of trajectories are highly applicable to mission design in the restricted threeand four-body problems. The general solution approach proceeds in three steps. In the initial analysis, conic arcs and/or other types of trajectory segments are connected at patch points to construct a e rst approximation. Next, multiconic methods are used to incorporate any additional force model effects that may have been neglected in the initial analysis. An optimization procedure is then employed to reduce the effective velocity discontinuities while satisfyinganyconstraints.Finally,anumericaldifferentialcorrectionsprocessresultsinafullycontinuousmultiplelunar-swingby trajectory that satise es the constraints and includes appropriate lunar and solar gravitational models.

Mission Design for the FIRE and PSI Missions

Mission Design for the FIRE and PSI Missions Martin W. Lo Jet Propulsion Laboratory California Institute of Technology B.T. Barden and Kathleen C. Howell School of Aeronautics and Astronautics Purdue University Sun-Earth L2 Iibration point orbits have become extremely popular for many NASA astrophysics missions due to the constant cold observation environment and low energy required for access. The Primordial Structures Investigation (PSI) and Far Infrared Explorer (FIRE) missions are currently proposed, as NASA Midex missions to study the cosmic microwave background (CMB) radiation, following in the footstep of the highly successful COBY Mission. Both PSI and FIRE have selected a relatively small amplitude Iissajous orbit about L2. The Delta-Lite class launch vehicle will place the spacecraft into an 11-day highly elliptical phasing orbit with apogee just beyond the moon. Two to four revolutions in the phasing cjrbits were considered to optimize mission performance. Near the last apogee, gravity assist from a lunar swingby transfers the spacecraft to a small amplitude (120,000 km Y, Z radius) lissajous orbit about the Sun-Earth L2 point. Lissajous orbits are smaller versions of the halo orbits used successfully by the ISEE3, S0}10, and ACE missions. The lunar swingby transfer to L2 provides greater performance and less risk than a direct launch. The single maneuver to correct the launch error is broken into several smaller maneuvers, the first of which is 11 days after launch (vs. 1 day for direct launch). Many recovery scenarios exist within our propellant budget, even if one of these correction burns were missed, For a 2.5 rev phasing orbit mission, a 5-10 day contiguous launch period is possible each month, with a 2 minute launch window claily. For a 4.5 rev phasing orbit mission, this can be extended to 15 days. once on orbit, the spacecraft rotates slowly about an axis pointed at the Sun. The optical axis of the instrument is perpendicular to this axis. The rotation sweeps out a great circle of the sky and maps the entire sky in 6 months. The mission duration is around 2 years which provides a maximum of four all-sky surveys. The mission design is forgiving and robust. In compassion to a direct launch trajectory, the lunar swingby trajectory to L2 provides many more opportunities for launch error recovery. Maneuvers are spaced no less than 2 days apart to give ample time for orbit determination and operations planning. Whether the problem is a larger launch period, degraded launch vehicle performance, or a missed critical perigee maneuver, recovery is still possible to achieve the Iissajous orbit and complete the mission. The flexibility of the mission design and the simplicity of the observational strategy make these robust and cost effective.