Numerical Integrator

This module defines the Integrator class, which performs numerical integration of satellite trajectories using the Runge-Kutta 45 method (RK45) from solve_ivp. It supports both 2D (polar) and 3D (spherical) formulations, includes ground impact detection, and models optional mid-flight thrust.

def __init__(self, recorded_times=None):

  • Purpose: Initializes the satellite’s state vector, orbital constants, and optional event times for trajectory recording.

  • Parameters:

    • recorded_times: Array of times at which to record the solution.

  • Implementation:

    • Sets initial orbital elements from InitialConditions.

    • Computes initial azimuthal velocity lam_dot0 to satisfy circular orbit mechanics minus deltaV.

    • Raises an error if the initial configuration is not physically valid.

@staticmethod
def rhs_polar(t, y):
    r, r_dot, theta, theta_dot = y
    return PolarAccelerations.accelerations(r, r_dot, theta, theta_dot)

  • Purpose: Right-hand side (derivatives) of the ODE system for 2D polar motion.

  • Parameters:

    • t: Time (not used).

    • y: State vector [r, ṙ, θ, θ̇].

  • Implementation:

    • Returns [ṙ, r̈, θ̇, θ̈] from the PolarAccelerations model.

@staticmethod
def rhs_spherical(t, y):
    r, r_dot, phi, phi_dot, lam, lam_dot = y
    return SphericalAccelerations.accelerations(r, r_dot, 
    phi, phi_dot, lam, lam_dot)

  • Purpose: Right-hand side (derivatives) of the ODE system for 3D spherical motion.

  • Parameters:

    • t: Time (not used).

    • y: State vector [r, ṙ, φ, φ̇, λ, λ̇].

  • Implementation:

    • Returns [ṙ, r̈, φ̇, φ̈, λ̇, λ̈] from the SphericalAccelerations model.

def hit_ground(self):
    ...
    return events

  • Purpose: Event function for RK45 to detect when the satellite hits Earth’s surface.

  • Implementation:

    • Triggers termination when r - Re = 0 (i.e., satellite radius equals Earth’s radius).

def at_thrust(self):
    ...
    return events

  • Purpose: Event function that detects when the satellite reaches a specific altitude (h_thrust) where thrust is applied.

  • Implementation:

    • Triggers termination when r = Re + h_thrust.

def runge_kutta45_2d(self):
    ...
    return sol

  • Purpose: Solves the 2D polar equations of motion using RK45 integration.

  • Implementation:

    • Uses rhs_polar for orbit simulation.

    • Integrates until ground impact using solve_ivp, and return the solution.

def runge_kutta45_3d(self):
    ...
    return sol

  • Purpose: Solves the 3D spherical equations of motion using RK45 integration.

  • Implementation:

    • Uses self.y0 and rhs_spherical for full orbit simulation.

    • Integrates until ground impact using solve_ivp, and return the solution.

@staticmethod
def great_circle_distance(phi1, lam1, phi2, lam2):
    Δσ = np.arccos(
        np.sin(phi1) * np.sin(phi2) +
        np.cos(phi1) * np.cos(phi2) * np.cos(lam1 - lam2)
    )
    return IC.earthRadius * Δσ

  • Purpose: Computes the great-circle distance between two points on the Earth.

  • Parameters:

    • phi1, lam1: Latitude and longitude of point 1.

    • phi2, lam2: Latitude and longitude of point 2.

  • Implementation: Uses spherical law of cosines

@staticmethod
def in_populated(phi, lam):
    for pc in IC.populatedCenters:
        if Integrator.great_circle_distance(phi, lam, pc[0], pc[1]) < 
            IC.populatedRadius:
            return True
    return False

  • Purpose: Determines whether a given coordinate lies within any of the predefined populated zones.

  • Parameters:

    • phi, lam: Target coordinates.

  • Implementation:

    • Loops through all populated centers and checks distance against populatedRadius.

def get_trajectory_2d(self):
    ...
    return np.array([t_arr, r_arr, th_arr]).T

  • Purpose: Integrates 2D trajectory and returns sampled output values [time, radius, theta].

def get_trajectory_3d(self, bonus=False):
    ...
    return np.array([t_arr, r_arr, lam_arr, phi_arr]).T

  • Purpose: Integrates the 3D trajectory and applies optional thrust manually or if target would impact populated area.

  • Parameters:

    • bonus: Set the thrust manually.

  • Implementation:

    • Runs RK45 integration to ground and get the crash position.

    • If impact site is populated or bonus is enabled, simulates thrust:

      • Integrates to thrust altitude.

      • Applies deltaV vectorially to velocity.

      • Continues integration to ground.

      • Merges both trajectory segments into one.

    • Extracts and returns trajectory [time, radius, azimuthal_angle, polar_angle].