Circular Restricted Three Body Equations

Common calculations within Circular Restricted Three Body Problem dynamics.

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Imports

• Base
• Core
• DocStringExtensions
• LinearAlgebra
• Roots
• StaticArrays

Exports

• converge
• converge!
• convergent_direction
• distance_scaling
• distance_to_primary
• distance_to_secondary
• diverge
• diverge!
• divergent_direction
• jacobi_constant
• lagrange_point
• nondimensional
• nondimensional_radii
• perturb
• perturb!
• perturbation
• potential_energy
• primary_synodic_position
• redimensioned
• reduced_mass
• richardson_halo
• richardson_ic
• secondary_synodic_position
• time_scaling
• velocity_scaling
• zero_velocity_curves

converge!(p, u, stm, Φ; eps)

Perturb halo orbit conditions in-place onto the orbit's stable manifold.

converge(u, stm, Φ; eps)

Perturb halo orbit conditions onto the orbit's unstable manifold.

convergent_direction(stm; atol)

Calculates the eigenvector associated with the stable manifold of a Monodromy matrix.

distance_scaling(a)

The length scale factor used to nondimensionalize CR3BP states.

distance_to_primary(r, μ)

Returns synodic distance to primary body.

distance_to_secondary(r, μ)

Returns synodic distance to secondary body.

diverge!(p, u, stm, Φ; eps)

Perturb halo orbit conditions in-place onto the orbit's unstable manifold.

diverge(u, stm, Φ; eps)

Perturb halo orbit conditions onto the orbit's unstable manifold.

divergent_direction(stm; atol)

Calculates the direction associated with the unstable manifold of a Monodromy matrix.

inertial_to_synodic(x, y, z, t, ω)

Given a barycentric-inertial Cartesian state, returns the state in the synodic (rotating) reference frame.

jacobi_constant(x, y, z, ẋ, ẏ, ż, μ)

Returns the Jacobi Constant, C in the Synodic frame with Normalized units.

lagrange_point(μ, L)

Returns the lagrange points for a CR3BP system.

Arguments:

• μ: Non-dimensional mass parameter for the CR3BP system.
• L: Langrange points requested, must be in range [1,5]

Outputs:

• Tuple of Lagrange points
• Throws ArgumentError if L is out of range [1,5]

References:

• Rund, 2018

nondimensional(x, y, z, ẋ, ẏ, ż, t, a, μ₁, μ₂)

Normalizes a CR3BP orbit in the rotating reference frame.

nondimensional_radii(x, y, z, xᵢ)

Returns the spacecraft's nondimensional position w.r.t. body 1 (or 2).

perturb!(p, u, stm, direction; eps)

Perturb a Cartesian state in-place along a halo orbit onto a stable or unstable manifold.

perturb(u, stm, direction; eps)

Perturb a Cartesian state along a halo orbit onto a stable or unstable manifold.

perturbation(stm, direction; eps)

Return the perturbation in Cartesian state space along a halo orbit onto the provided direction of the provided manifold.

potential_energy(x, y, z, μ)

Returns the potential energy U in the Synodic frame with Normalized units.

primary_synodic_position(μ)

Position of primary body.

redimensioned(x, y, z, ẋ, ẏ, ż, t, μ, a, T)

Redimensionalizes a CR3BP orbit in the rotating reference frame.

reduced_mass(μ₁, μ₂)

Return the reduced mass.

richardson_halo(μ, L; Z, hemisphere, ϕ, length)

Returns an analytical solution for a Halo orbit about L.

Extended Help

Arguments

• μ: Non-dimensional mass parameter for the CR3BP system.
• L: Lagrange point to orbit (L1 or L2).
• Z: Desired non-dimensional Z-amplitude for Halo orbit.
• hemisphere: Specifies northern or southern Halo orbit.
• ϕ: Desired Halo orbit phase.
• steps: Number of non-dimensional timepoints in returned state.

Outputs

• Near-periodic initial condition u
• Halo orbit period T.
• Throws ArgumentError if L is not 1 or 2.

References:

• Rund, 2018.

richardson_ic(μ, L; Z, hemisphere, ϕ)

Returns an analytical solution for a Halo orbit about L.

Extended Help

Arguments

• μ: Non-dimensional mass parameter for the CR3BP system.
• L: Lagrange point to orbit (L1 or L2).
• Z: Desired non-dimensional Z-amplitude for Halo orbit.
• hemisphere: Specifies northern or southern Halo orbit.
• ϕ: Desired Halo orbit phase.
• steps: Number of non-dimensional timepoints in returned state.

Outputs

• Near-periodic initial condition u
• Halo orbit period T.
• Throws ArgumentError if L is not 1 or 2.

References:

• Rund, 2018.

secondary_synodic_position(μ)

Position of primary body.

synodic_to_inertial(x, y, z, t, ω)

Given the Synodic frame vector, returns the vector in the barycentric-inertial reference frame.

time_scaling(a, μ₁, μ₂)

The time scale factor used to nondimensionalize CR3BP states.

velocity_scaling(a, μ₁, μ₂)

The velocity scale factor used to nondimensionalize CR3BP states.

zero_velocity_curves(r, v, μ; nondimensional_range)

Returns a Vector of Matrix values. Each Matrix contains a 3-column nondimensional position trajectory in the Synodic frame which represents a Zero Velocity Curve.