Restricted Two Body Equations

Common calculations within Restricted Two Body Problem dynamics.

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Imports

• Base
• Core
• DocStringExtensions
• LinearAlgebra
• StaticArrays

Exports

• apoapsis_radius
• argument_of_periapsis
• c3
• cartesian_to_keplerian
• cartesian_to_perifocal
• conic
• eccentricity
• eccentricity_vector
• hohmann
• inclination
• kepler
• keplerian_to_cartesian
• keplerian_to_perifocal
• lambert
• mean_motion
• orbital_period
• orbital_radius
• orbital_speed
• periapsis_radius
• perifocal_to_cartesian
• right_ascension_ascending_node
• semi_parameter
• semimajor_axis
• specific_angular_momentum
• specific_angular_momentum_vector
• specific_energy
• specific_potential_energy
• sphere_of_activity
• sphere_of_influence
• time_since_periapsis
• true_anomaly
• v_infinity

_cross(x, y, z, ẋ, ẏ, ż)

Return the cross product between two splatted three-vectors.

apoapsis_radius(a, e)

Returns apoapsis distance, rₐ.

argument_of_periapsis(x, y, z, ẋ, ẏ, ż, μ)

Returns the argument of periapsis.

c3(r, v, μ)

Returns C3 value.

cartesian_to_keplerian(x, y, z, ẋ, ẏ, ż, μ)

Returns a Keplarian representation of a Cartesian orbital state. Algorithm taught in ENAE601.

cartesian_to_perifocal(i, Ω, ω, x, y, z, ẋ, ẏ, ż)

Returns the position and velocity vectors in the Perifocal frame.

conic(e)

Returns the conic section, as specified by eccentricity e.

eccentricity(x, y, z, ẋ, ẏ, ż, μ)

Returns orbital eccentricity, e.

eccentricity_vector(x, y, z, ẋ, ẏ, ż, μ)

Returns orbital eccentricity vector e̅.

hohmann(r₁, r₂, μ)

Computes a Hohmann transfer, and returns the departure and arrival velocity vectors.

inclination(x, y, z, ẋ, ẏ, ż, μ)

Returns the orbital inclination.

kepler(x, y, z, ẋ, ẏ, ż, μ, Δt; atol, maxiter)

Solves Kepler's Problem, predicting the orbit's future state geometrically.

keplerian_to_cartesian(e, a, i, Ω, ω, ν, μ)

Returns a Cartesian representation of a Keplerian two-body orbital state in an inertial frame, centered at the center of mass of the central body. Algorithm taught in ENAE601.

keplerian_to_perifocal(a, e, ν, μ)

Returns position and velocity vectors in the Perifocal frame.

lambert(
    x₁,
    y₁,
    z₁,
    x₂,
    y₂,
    z₂,
    μ,
    Δt;
    trajectory,
    atol,
    maxiter
)

Solves Lambert's problem through the use of univeral variables.

lambert_lancaster_blanchard(
    r̲₁,
    r̲₂,
    μ,
    Δt;
    revolutions,
    branch,
    trajectory,
    atol,
    maxiter,
    output_extrema
)

The following code was converted to Julia, from a GitHub repository that hosts a MATLAB implementation. At the time of writing, this respository has a BSD license. I'm providing the copyright notice here, as instructed by the license text.

Copyright (c) 2018, Rody Oldenhuis
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this
   list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

The views and conclusions contained in the software and documentation are those
of the authors and should not be interpreted as representing official policies,
either expressed or implied, of this project.

lambert_oldenhuis(r1vec, r2vec, tf, m, muC)

The following code was converted to Julia, from a GitHub repository that hosts a MATLAB implementation. At the time of writing, this respository has a BSD license. I'm providing the copyright notice here, as instructed by the license text.

Copyright (c) 2018, Rody Oldenhuis
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this
   list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

The views and conclusions contained in the software and documentation are those
of the authors and should not be interpreted as representing official policies,
either expressed or implied, of this project.

lancaster_blanchard(x, q, m)

The following code was converted to Julia, from a GitHub repository that hosts the MATLAB implementation. At the time of writing, the respository has a BSD license. I'm providing the copyright notice here, as instructed by the license text.

Copyright (c) 2018, Rody Oldenhuis
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this
   list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

The views and conclusions contained in the software and documentation are those
of the authors and should not be interpreted as representing official policies,
either expressed or implied, of this project.

mean_motion(a, μ)

Returns mean motion, n.

minmax_distances(r̲₁, r̲₂, r₁, r₂, δₜ, a, v̲₁, v̲₂, m, μ)

The following code was converted to Julia, from a GitHub repository that hosts a MATLAB implementation. At the time of writing, this respository has a BSD license. I'm providing the copyright notice here, as instructed by the license text.

Copyright (c) 2018, Rody Oldenhuis
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this
   list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

The views and conclusions contained in the software and documentation are those
of the authors and should not be interpreted as representing official policies,
either expressed or implied, of this project.

orbital_period(a, μ)

Returns the orbital period.

orbital_radius(p, e, ν)

Returns distance, r.

orbital_speed(x, y, z, ẋ, ẏ, ż, μ)

Returns the instantaneous velocity, v, for any orbital representation.

periapsis_radius(a, e)

Returns periapsis distance, rₚ.

perifocal_to_cartesian(i, Ω, ω, x, y, z, ẋ, ẏ, ż)

Returns a spatial representation of the provied Perifocal state.

right_ascension_ascending_node(x, y, z, ẋ, ẏ, ż, μ)

Returns the right ascension of the ascending node.

semi_parameter(a, e)

Returns semilatus parameter, p.

semimajor_axis(r, v, μ)

Returns semimajor axis parameter, a.

specific_angular_momentum(x, y, z, ẋ, ẏ, ż)

Returns scalar specific angular momentum vector, h.

specific_angular_momentum_vector(x, y, z, ẋ, ẏ, ż)

Returns specific angular momentum vector, h̅.

specific_energy(a, μ)

Returns specific orbital energy, ϵ.

specific_potential_energy(r, μ)

Returns the specific potential energy: the energy per unit mass.

sphere_of_activity(a, m, M)

Sphere of activity.

sphere_of_influence(a, m, M)

Sphere of influence.

time_since_periapsis(n, e, E)

Returns time since periapsis, t.

true_anomaly(x, y, z, ẋ, ẏ, ż, μ)

Returns true anomoly, ν.

v_infinity(r, v, μ)

Returns v∞.

σmax(y)

The following code was converted to Julia, from a GitHub repository that hosts a MATLAB implementation. At the time of writing, this respository has a BSD license. I'm providing the copyright notice here, as instructed by the license text.

Copyright (c) 2018, Rody Oldenhuis
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this
   list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

The views and conclusions contained in the software and documentation are those
of the authors and should not be interpreted as representing official policies,
either expressed or implied, of this project.