HPOP Integration

The Integrator allows you to configure the combination of the formulation of the equations of motion and the numerical integration technique to be used during orbit propagation.

Option Description
Integration The integration method to be used in propagating the orbit. Choices are:
  • RK 4 - Runge-Kutta integration method of 4th order with no error control for the integration step size.
  • RKF 7(8) - Runge-Kutta-Fehlberg integration method of 7th order with 8th order error control for the integration step size.
  • RKV 8(9) Efficient - Runge-Kutta-Verner integration method of 8th order with 9th order error control for the integration step size.

    The RKV8(9) Efficient integration method utilizes Jim Verner's coefficient, available at https://www.sfu.ca/~jverner

  • Bulirsch Stoer - Integration method based on Richardson extrapolation with automatic step size control.
  • Gauss Jackson - 12th order Gauss-Jackson integration method for second order ODEs. There is currently no error control implemented for this method meaning that a fixed step size is used.
VOP Use a variation of parameters in universal variables formulation of the equations of motion. Valid in combination with the RKF7(8) and Burlirsch-Stoer integration methods. This option is not available for missile objects.
Predictor Correction Scheme Valid for the Gauss-Jackson integration method only. Method for updating acceleration components after corrector has converged. Choices are:
  • Full Correction - Use a full evaluation of the acceleration model at the end of a Gauss-Jackson integration step. This method is more accurate than the Pseudo Correction method.
  • Pseudo Correction - Use a pseudo-evaluation where only the two body acceleration is updated. This method is more efficient than the Full Correction method.
Step Size Control Choose the method of integration step size control. Choices are:
  • Fixed Step - Step size will remain constant throughout the integration of the orbit. Error control is not employed.
  • Relative Error - Control the step size based on relative error by providing the error tolerance, and the minimum and maximum integration step sizes to be allowed via relative error control.
Time Regularization

If you use Time Regularization, integrate the orbit on regularized time. The relationship between regularized and normal time is given by the following equation:



where n is the user-provided exponent. n=1 yields steps proportional to eccentric anomaly. n=2 yields steps proportional to true anomaly for a two body orbit.

The value of steps per orbit is used to set the step size in regularized time. This option is not available for missile objects.

Interpolation

Choose the interpolation method:

  • VOP - Uses a special interpolator that deals well with ephemeris produced at a large step size, which happens frequently when using the VOP formulation. The interpolator itself uses a VOP formulation. The VOP mu value is the gravitational parameter used by the formulation. You can also specify the interpolation order.
  • Lagrange - Uses the standard Lagrange interpolation scheme, interpolating position and velocity separately. You can also specify the interpolation order.
  • Hermitian - Uses the standard Hermitian interpolation scheme, which uses the position and velocity ephemeris to interpolate position and velocity together (i.e., using a polynomial and its derivative). You can also specify the interpolation order.
Allow PosVel Covariance Interpolation When exporting a 6x6 covariance, state whether to allow covariance interpolation using a blending function based on the nominal two-body motion. Disable this option if the trajectory is not dominated by the two-body motion.
Report Ephemeris on a Fixed Time Step State whether ephemeris is to be reported on a fixed time step. The ephemeris will be reported on equal time steps, using the step size shown on the Orbit properties page. The integrator adjusts its step, even if it could take a larger step, to report on the reporting interval. The integrator may also take several smaller steps and only report the ephemeris every fixed time step.
Do not propagate below altitude of: Propagation stops if any sample of the force model occurs at an altitude less than this specified value.

The results of some integration comparison tests that were performed may help you in selecting an appropriate integrator.

  • The Bulirsch-Stoer integrator does not behave well with the integration tolerance set to 1.0e-16 or below. The problem is due to the tolerance being of the same order as the truncation error. The result is extremely long run times.
  • The combination of SRP boundary mitigation with regularized time can produce small trajectory errors due to the formulation of the mitigation process being in normal time. These errors are rarely as large as errors introduced by not using boundary mitigation.