HITEN - Computational Toolkit for the Circular Restricted Three-Body Problem

Overview

HITEN is a research-oriented Python library that provides an extensible implementation of high-order analytical and numerical techniques for the circular restricted three-body problem (CR3BP).

Installation

HITEN is published on PyPI. A recent Python version (3.9+) is required.

py -m pip install hiten

Quickstart

Full documentation is available here.

Compute a halo orbit around Earth-Moon L1 and plot a branch of its stable manifold:

from hiten import System

system = System.from_bodies("earth", "moon")
l1 = system.get_libration_point(1)

orbit = l1.create_orbit("halo", amplitude_z=0.2, zenith="southern")
orbit.correct()
orbit.propagate(steps=1000)

manifold = orbit.manifold(stable=True, direction="positive")
manifold.compute()
manifold.plot()

Examples

1. Parameterisation of periodic orbits and their invariant manifolds

   The toolkit constructs periodic solutions such as halo orbits and computes their stable and unstable manifolds.

   from hiten import System
   
   system = System.from_bodies("earth", "moon")
   l1 = system.get_libration_point(1)
   
   orbit = l1.create_orbit("halo", amplitude_z=0.2, zenith="southern")
   orbit.correct()
   orbit.propagate(steps=1000)
   
   manifold = orbit.manifold(stable=True, direction="positive")
   manifold.compute()
   manifold.plot()

   

   Figure 1 - Stable manifold of an Earth-Moon (L_1) halo orbit.

   Knowing the dynamics of the center manifold, initial conditions for vertical orbits can be computed and associated manifolds created. These reveal natural transport channels that can be exploited for low-energy mission design.

   from hiten import System, VerticalOrbit
   
   system = System.from_bodies("earth", "moon")
   l1 = system.get_libration_point(1)
   
   cm = l1.get_center_manifold(degree=6)
   cm.compute()
   
   initial_state = cm.to_synodic([0.0, 0.0], 0.6, "q3")
   
   orbit = VerticalOrbit(l1, initial_state=initial_state)
   orbit.correct()
   orbit.propagate(steps=1000)
   
   manifold = orbit.manifold(stable=True, direction="positive")
   manifold.compute()
   manifold.plot()

   

   Figure 2 - Stable manifold of an Earth-Moon (L_1) vertical orbit.

2. Generating families of periodic orbits

   The toolkit can generate families of periodic orbits by continuation.

   from hiten import System
   from hiten.algorithms.continuation.options import OrbitContinuationOptions
   from hiten.algorithms.types.states import SynodicState
   from hiten.system.family import OrbitFamily
   
   system = System.from_bodies("earth", "moon")
   l1 = system.get_libration_point(1)
   
   halo_seed = l1.create_orbit('halo', amplitude_z=0.2, zenith='southern')
   halo_seed.correct()
   halo_seed.propagate()
   
   options = OrbitContinuationOptions(
      target=(
         [halo_seed.initial_state[SynodicState.Z]], 
         [halo_seed.initial_state[SynodicState.Z] + 2.0]
      ),
      step=((0.02,),),
      max_members=100,
   )
   
   result = halo_seed.generate(options)
   
   family = OrbitFamily.from_result(result)
   family.propagate()
   family.plot()

   

   Figure 3 - Family of Earth-Moon (L_1) Halo orbits.

3. Generating Poincare maps

   The toolkit can generate Poincare maps for arbitrary sections. For example, the centre manifold of the Earth-Moon (L_1) libration point:

   from hiten import System
   
   system = System.from_bodies("earth", "moon")
   l1 = system.get_libration_point(1)
   
   cm = l1.get_center_manifold(degree=6)
   cm.compute()
   
   pm = cm.poincare_map(energy=0.7)
   pm.compute(section_coord="p3")
   pm.plot(axes=("p2", "q3"))

   

   Figure 4 - Poincare map of the centre manifold of the Earth-Moon (L_1) libration point using the (q_2=0) section.

   Or the synodic section of a vertical orbit manifold:

   from hiten import System, SynodicMap, VerticalOrbit
   
   system = System.from_bodies("earth", "moon")
   l_point = system.get_libration_point(1)
   
   cm = l_point.get_center_manifold(degree=6)
   cm.compute()
   
   ic_seed = cm.to_synodic([0.0, 0.0], 0.6, "q3") # Good initial guess from CM
   
   orbit = VerticalOrbit(l_point, initial_state=ic_seed)
   orbit.correct()
   orbit.propagate(steps=1000)
   
   manifold = orbit.manifold(stable=True, direction="positive")
   manifold.compute(step=0.005)
   manifold.plot()
   
   synodic_map = SynodicMap(manifold)
   synodic_map.compute(
      section_axis="y",
      section_offset=0.0,
      plane_coords=("x", "z"),
      direction=-1
   )
   synodic_map.plot()

   

   Figure 5 - Synodic map of the stable manifold of an Earth-Moon (L_1) vertical orbit.

4. Detecting heteroclinic connections

   The toolkit can detect heteroclinic connections between two manifolds.

    from hiten import System
    from hiten.algorithms.connections import ConnectionPipeline
    from hiten.algorithms.connections.config import ConnectionConfig
    from hiten.algorithms.connections.options import ConnectionOptions
    from hiten.algorithms.poincare import SynodicMapConfig
   
    system = System.from_bodies("earth", "moon")
    mu = system.mu
   
    l1 = system.get_libration_point(1)
    l2 = system.get_libration_point(2)
   
    halo_l1 = l1.create_orbit('halo', amplitude_z=0.5, zenith='southern')
    halo_l1.correct()
    halo_l1.propagate()
   
    halo_l2 = l2.create_orbit('halo', amplitude_z=0.3663368, zenith='northern')
    halo_l2.correct()
    halo_l2.propagate()
   
    manifold_l1 = halo_l1.manifold(stable=True, direction='positive')
    manifold_l1.compute(integration_fraction=0.9, step=0.005)
   
    manifold_l2 = halo_l2.manifold(stable=False, direction='negative')
    manifold_l2.compute(integration_fraction=1.0, step=0.005)
   
    section_cfg = SynodicMapConfig(
        section_axis="x",
        section_offset=1 - mu,
        plane_coords=("y", "z"),
    )
   
    config = ConnectionConfig(
        section=section_cfg,
        direction=-1,
    )
   
    options = ConnectionOptions(
        delta_v_tol=1,
        ballistic_tol=1e-8,
        eps2d=1e-3,
    )
    
    conn = ConnectionPipeline.with_default_engine(config=config)
   
    result = conn.solve(manifold_l1, manifold_l2, options=options)
   
    print(result)
   
    conn.plot(dark_mode=True)
   
    conn.plot_connection(dark_mode=True)

   

   Figure 6 - Heteroclinic connection between the stable manifold of an Earth-Moon (L_1) halo orbit and the unstable manifold of an Earth-Moon (L_2) halo orbit.

   We can also plot the trajectories making up the connection:

   

   Figure 7 - One of the detected connections.

5. Generating invariant tori

   Hiten can generate invariant tori for periodic orbits.

   from hiten import System, InvariantTori
   
    system = System.from_bodies("earth", "moon")
    l1 = system.get_libration_point(1)
   
    orbit = l1.create_orbit('halo', amplitude_z=0.3, zenith='southern')
    orbit.correct()
    orbit.propagate(steps=1000)
   
    torus = InvariantTori(orbit)
    torus.compute(epsilon=1e-2, n_theta1=512, n_theta2=512)
    torus.plot()

   

   Figure 7 - Invariant torus of an Earth-Moon (L_1) quasi-halo orbit.

Run the examples

Example scripts are in the examples directory. From the project root:

py -m pip install -e .
python examples\periodic_orbits.py
python examples\orbit_manifold.py
python examples\orbit_family.py
python examples\poincare_map.py
python examples\synodic_map.py
python examples\heteroclinic_connection.py
python examples\invariant_tori.py
python examples\center_manifold.py

Contributing

Issues and pull requests are welcome. For local development:

py -m pip install -e .[dev]
python -m pytest -q

License

This project is licensed under the terms of the MIT License. See LICENSE.