Model Dynamics

This module defines two classes used to compute the accelerations of a satellite in polar and spherical coordinate systems.
These accelerations are used in numerical solvers such as RK45 for trajectory simulation.

• PolarAccelerations: Provides acceleration functions in polar coordinates.

• SphericalAccelerations: Inherits from PolarAccelerations and extends the model to spherical coordinates (radial, polar, and azimuthal), enabling 3D orbital dynamics modeling.

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@staticmethod
def comb_velocity(rad, radVel, angVel):
    return np.sqrt(radVel**2 + (rad * angVel)**2)

• Purpose: Computes the total velocity magnitude in polar coordinates from radial and angular components.

• Parameters:

  • rad: Radial position (distance from Earth’s center), in meters.

  • radVel: Radial velocity (ṙ), in m/s.

  • angVel: Angular velocity (θ̇), in rad/s.

• Implementation:

  • Computes total velocity as the Euclidean norm of radial and tangential velocity: √(ṙ² + (rθ̇)²).

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@staticmethod
def drag_start(C_d, A, m, r):
    Re = 6371e3
    h = r - Re
    result = 0.5 * atmospheric_density(h) * C_d * A / m
    return result

• Purpose: Calculates the atmospheric drag force.

• Parameters:

  • C_d: Drag coefficient (dimensionless).

  • A: Cross-sectional area of the satellite (in m²).

  • m: Mass of the satellite (in kg).

  • r: Radial position from Earth’s center (in m).

• Implementation:

  • Computes the altitude above Earth’s surface: h = r - Re.

  • Queries the atmospheric density at altitude h.

  • Returns the constant multiplier part of the drag force expression.

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@staticmethod
def accelerations(u1, u2, u3, u4):
    ...
    return np.array([u1_dot, u2_dot, u3_dot, u4_dot])

• Purpose: Computes the time derivatives of the satellite’s state in polar coordinates.

• Parameters:

  • u1: Radial position (r)

  • u2: Radial velocity (ṙ)

  • u3: Angular position (θ)

  • u4: Angular velocity (θ̇)

• Implementation:

  • Computes the drag using drag_start().

  • Computes total velocity with comb_velocity().

  • Uses Newton’s second law and angular motion equations to derive accelerations.

  • Returns [ṙ, r̈, θ̇, θ̈].

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@staticmethod
def comb_velocity(rad, radVel, polAng, polVel, aziVel):
    return np.sqrt(
        radVel**2 + rad**2 * (polVel**2 + np.sin(polAng)**2 * aziVel**2))

• Purpose: Computes the total velocity magnitude in spherical coordinates using radial, polar, and azimuthal components.

• Parameters:

  • rad: Radial distance.

  • radVel: Radial velocity.

  • polAng: Polar angle.

  • polVel: Polar angular velocity.

  • aziVel: Azimuthal angular velocity.

• Implementation:

  • Calculates velocity magnitude using the spherical velocity formula:
    √(ṙ² + r²(φ̇² + sin²(φ)·λ̇²)).

---

@staticmethod
def accelerations(u1, u2, u3, u4, u5, u6):
    ...
    return np.array([u1_dot, u2_dot, u3_dot, u4_dot, u5_dot, u6_dot])

• Purpose: Computes the full state derivatives of a satellite in spherical coordinates, suitable for high-fidelity simulations.

• Parameters:

  • u1: Radial position (r)

  • u2: Radial velocity (ṙ)

  • u3: Polar angle (φ)

  • u4: Polar angular velocity (φ̇)

  • u5: Azimuthal angle (λ)

  • u6: Azimuthal angular velocity (λ̇)

• Implementation:

  • Calculates drag coefficient using drag_start().

  • Computes velocity magnitude with comb_velocity().

  • Applies equations of motion in spherical coordinates.

  • Returns [ṙ, r̈, φ̇, φ̈, λ̇, λ̈].