Circular Restricted Three-body Dynamics

Also known as CR3BP dynamics!


The Circular Restricted Three-body Problem (CR3BP) assumes a massless spacecraft which moves due to the gravity of two celestial bodies which orbit their common center of mass. This may seem like an arbitrary model, but it's actually a pretty decent approximation for how a spacecraft moves nearby the Earth and the Sun, the Earth and the Moon, the Sun and Jupiter, and other systems in our solar system! The equations of motion are provided below.

\[\begin{aligned} \frac{dx(t)}{dt} =& ẋ\left( t \right) \\ \frac{dy(t)}{dt} =& ẏ\left( t \right) \\ \frac{dz(t)}{dt} =& ż\left( t \right) \\ \frac{dẋ(t)}{dt} =& 2 ẏ\left( t \right) - \left( \frac{1}{\sqrt{\left( \mu + x\left( t \right) \right)^{2} + \left( y\left( t \right) \right)^{2} + \left( z\left( t \right) \right)^{2}}} \right)^{3} \left( 1 - \mu \right) \left( \mu + x\left( t \right) \right) - \left( \frac{1}{\sqrt{\left( -1 + \mu + x\left( t \right) \right)^{2} + \left( y\left( t \right) \right)^{2} + \left( z\left( t \right) \right)^{2}}} \right)^{3} \mu \left( -1 + \mu + x\left( t \right) \right) + x\left( t \right) \\ \frac{dẏ(t)}{dt} =& - 2 ẋ\left( t \right) - \left( \left( \frac{1}{\sqrt{\left( \mu + x\left( t \right) \right)^{2} + \left( y\left( t \right) \right)^{2} + \left( z\left( t \right) \right)^{2}}} \right)^{3} \left( 1 - \mu \right) + \left( \frac{1}{\sqrt{\left( -1 + \mu + x\left( t \right) \right)^{2} + \left( y\left( t \right) \right)^{2} + \left( z\left( t \right) \right)^{2}}} \right)^{3} \mu \right) y\left( t \right) + y\left( t \right) \\ \frac{dż(t)}{dt} =& \left( - \left( \frac{1}{\sqrt{\left( \mu + x\left( t \right) \right)^{2} + \left( y\left( t \right) \right)^{2} + \left( z\left( t \right) \right)^{2}}} \right)^{3} \left( 1 - \mu \right) - \left( \frac{1}{\sqrt{\left( -1 + \mu + x\left( t \right) \right)^{2} + \left( y\left( t \right) \right)^{2} + \left( z\left( t \right) \right)^{2}}} \right)^{3} \mu \right) z\left( t \right) \end{aligned}\]


State transition dynamics are particularly valuable for CR3BP models. Recall that the state transition matrix is simply the local linearization of a spacecraft within CR3BP dynamics. Let's look at the Jacobian (another word for "local linearization") below, evaluated at some random state.

julia> f = CR3BPFunction(; jac=true)ERROR: UndefVarError: `CR3BPFunction` not defined
julia> let x = randn(6), p = rand((0.0, 0.5)), t = 0 f.jac(x, p, t) endERROR: UndefVarError: `f` not defined

The Jacobian will always have this form (zeros in the top-left, the identity matrix in the top-right, a dense matrix in the bottom-left, and the same sparse "-2, 2" matrix in the bottom-right). We can include the state transition dynamics in our model with stm=true, initialize the state transition matrix states to the identity matrix, and propagate our spacecraft for one periodic orbit: the result is known as the Monodromy Matrix! The Monodromy Matrix provides stability characteristics for the entire periodic orbit.

julia> model = CR3BSystem(; stm=true)Model CR3BWithSTM with 42 equations
Unknowns (42):
Parameters (1):

Note that periodic orbits are not easy to find within CR3BP dynamics. Various algorithms have been developed to analytically approximate, and numerically refine, periodic CR3BP orbits. Some of those algorithms have already been implemented in Julia! See OrbitalTrajectories and GeneralAstrodynamics.