compute_orbit_J2_RK4#

namespace saltro
namespace orbits

Functions

bool compute_orbit_J2_RK4(const Eigen::Vector3d &r0, const Eigen::Vector3d &v0, const Eigen::Matrix<double, 1, saltro::limits::MAX_LENGTH_TRAJ> &jtime, const int jtime_length, Eigen::Matrix<double, 3, saltro::limits::MAX_LENGTH_TRAJ> &R, Eigen::Matrix<double, 3, saltro::limits::MAX_LENGTH_TRAJ> &V)#

Propagate an orbit under the influence of the J2 perturbation using RK4 integration.

This function integrates the equations of motion for a satellite orbiting Earth, including the dominant perturbation due to Earth’s oblateness (J2 effect).

The dynamics combine the two-body gravitational acceleration with the J2 perturbation:

\[ \frac{d^2\mathbf{r}}{dt^2} = -\frac{\mu}{r^3}\mathbf{r} + \mathbf{a}_{J2} \]

where \(\mathbf{a}_{J2}\) is the acceleration due to the quadrupole moment:

\[ \mathbf{a}_{J2} = \frac{3}{2}J_2\mu\frac{R_e^2}{r^5} \left[ x\left(5\frac{z^2}{r^2} - 1\right)\hat{\mathbf{i}} + y\left(5\frac{z^2}{r^2} - 1\right)\hat{\mathbf{j}} + z\left(5\frac{z^2}{r^2} - 3\right)\hat{\mathbf{k}} \right] \]

The integration uses fourth-order Runge-Kutta (RK4) with deterministic adaptive substepping. The time step is automatically divided into smaller substeps if needed to maintain accuracy, with a configurable maximum substep size.

The J2 perturbation causes measurable orbital effects:

  • Nodal regression: The right ascension of ascending node (RAAN) decreases.

  • Apsidal precession: The argument of perigee changes.

  • Inclination conservation: Inclination remains approximately constant.

Note

  • Input time (jtime) must be in Julian centuries, not days. Multiply Julian days by \(1/36525\) to convert.

  • The initial state (r0, v0) is placed at the first output at index 0.

  • All subsequent states are propagated from the first time point.

  • The arrays R and V must have size at least (3 × MAX_LENGTH_TRAJ).

Parameters:

  • r0 – Initial position vector (3D, in meters).

  • v0 – Initial velocity vector (3D, in m/s).

  • jtime – Row vector of Julian centuries at which to compute the trajectory. The first element is used as the reference epoch \(t_0\). All subsequent times are propagated from the first element.

  • jtime_length – Number of time points to propagate (length of jtime and output arrays).

  • R – Output matrix of position vectors (3 × jtime_length). Each column \(R_i\) contains the position at time \(t_i\).

  • V – Output matrix of velocity vectors (3 × jtime_length). Each column \(V_i\) contains the velocity at time \(t_i\).

Returns:

True if integration completed successfully, false otherwise.