Intro

This is a small library meant to facilitate investigation of the Circular Restricted 3 Body Problem (CR3BP)

Usage

The library consists of a System dataclass representing a 3 body system, a couple of functions which encode the equations of motion of the CR3BP, and a function for setting initial conditions to get onto a periodic trajectory around the collinear Lagrange points. Below we'll explain the System class as well as the functions.

System class

You can either use one of the provided 3 body systems (Sun-Earth, Earth-Moon, Sun-Jupiter)

from cr3bp import EarthMoon

Or you can create your own system as follows:

from cr3bp import System
mass_primary = 2  # should be in kg
mass_secondary = 1  # should be in kg
distance = 100  # should be in m
MySystem = System(mass_primary, mass_secondary, distance)

System API

Properties

mass_primary - kg
mass_seconary - kg
l - m
G - 6.67408e-11 m^3 / (kg s^2) (this is constant across all Systems)
mu
theta_dot - radians/sec
total_mass - kg
seconds - number of seconds in a non-dimensional second (s*)
s - alias for seconds
conversion_vector - provides the vector [l, l, l, l/s, l/s, l/s] for converting dimensionless state to dimensional state
L1
L2
L3
L4
L5

All of the L* functions return dimensionless values. L1-3 return a float, L4 and L5 return a tuple with x and y values (but they just return (0.5, +/- sqrt(3)/2) for all systems). L1-3 perform a root-finding calculation so it's best not to use them in a performance sensitive loop - store the returned value and use that

Functions

convert_to_dimensional_state(state)

Accepts a dimensionless 6-element state vector and returns a 6-element state vector with dimensions added in

x_eqn

This is used for calculating the location of the collinear Lagrange points. It returns a function that bakes in the mu value of the system and which can then be used for plotting or root finding.

Other functions

• DimensionalEOMs(t, X, system=EarthMoon)
  This function encodes the EOMs for the CR3BP using dimensional values (meters, seconds, etc.). It is meant to be used with a numerical integrator from scipy. For example: scipy.integrate.solve_ivp(cr3bp.DimensionalEOMs, [0, 3600*24], IC, args=[cr3bp.SunEarth], atol=1e-6, rtol=1e-12)
  In this case we are starting from some initial condition IC (a 6 element state vector) and integrating for 1 day using the SunEarth system with a tolerance down to the micrometer level (atol=1e-6)

• EOMConstructor(mu, STM=False)
  This encodes the EOMs in dimensionless form but it's a little different from DimensionalEOMs in that it takes a mu value and returns a function that can be used with numerical integrators. This just makes it a little smoother to use the numerical integrators since you specify the system when creating the EOMs as opposed to passing it in every time the EOMs are evaluated. The returned function can either integrate the State Transition Matrix or not. If integrating the STM, the returned function will expected a 42-element state vector, if not it will expect a 6-element one. Use this function like this:

EM = cr3bp.EarthMoon
eoms = cr3bp.EOMConstructor(EM.mu)
# We set the absolute tolerance to be at the micrometer level and keep the rtol small enough so that the atol dominates. rtol cannot be set lower than machine precision, so there may be some precision issues.
solution = scipy.integrate.solve_ivp(eoms, [0, 3600*24 / EM.seconds], IC, atol=0.000001/EM.l, rtol=2.3e-14)

• initial_velocity(initial_position: tuple[float, float], lpoint: float, mu: float) -> tuple[float, float]:
  This function takes a dimensionless initial position consisting of an x and y position relative to the system barycenter, the dimensionless value of the collinear Lagrange point around which one wants to oscillate, and the mu of the system and returns the necessary initial x and y velocity to get onto an oscillatory trajectory around the Lagrange point. Sample usage:

# IC (Initial Conditions) represents x, y, z, xdot, ydot, zdot
IC = [EM.L1 + 100/EM.l, 0, 0, 0, 0, 0]
IC[3:5] = cr3bp.initial_velocity(IC[:2], EM.L1, EM.mu)