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Choosing the equations of motion

There are a number of equations of motion implemented in HighFidelityEphemerisModel.jl.

  • functions starting with eom_ integrates the translational state ([x,y,z,vx,vy,vz])
  • functions starting with eom_stm_ integrates both the translational state and the flattened 6-by-6 STM.
Note

The STM is flattened row-wise, so to extract the state & STM from the ODESolution, make sure to reshape then tranapose; for example,

x_stm_tf = sol.u[end]                       # concatenated state & STM (flattened)
x_tf     = x_stm_tf[1:6]                    # final state [x,y,z,vx,vy,vz]
STM_tf   = reshape(sol.u[end][7:42],6,6)'   # final 6-by-6 STM

The dynamics model is of the form

\[\dot{\boldsymbol{x}} = \begin{bmatrix} \dot{\boldsymbol{r}} \\ \dot{\boldsymbol{v}} \end{bmatrix} = \begin{bmatrix} \boldsymbol{v} \\ -\dfrac{\mu}{\| \boldsymbol{r} \|_2^3}\boldsymbol{r} \end{bmatrix} + \sum_{i} \begin{bmatrix} \boldsymbol{0}_{3 \times 1} \\ \boldsymbol{a}_{\mathrm{3bd},i} \end{bmatrix} + \begin{bmatrix} \boldsymbol{0}_{3 \times 1} \\ \boldsymbol{a}_{\mathrm{SH},n_{\max}} \end{bmatrix}\]

where:

  • Nbody: central gravity term + third-body perturbations ($\boldsymbol{a}_{\mathrm{3bd},i}$)
  • NbodySH: central gravity term + third-body perturbations + spherical harmonics perturbations up to nmax degree ($\boldsymbol{a}_{\mathrm{SH},n_{\max}}$)

List of equations of motion in HighFidelityEphemerisModel.jl

The STM is integrated with the Jacobian, which is computed either analytically (using symbolic derivative) or via ForwardDiff (functions containing _fd)

eomeom + STM (analytical)eom + STM (ForwardDiff)EnsembleThreads compatibility
eom_Nbody_SPICE!eom_stm_Nbody_SPICE!no
eom_Nbody_Interp!eom_stm_Nbody_Interp!eom_stm_Nbody_Interp_fd!yes
eom_NbodySH_SPICE!eom_stm_NbodySH_SPICE_fd!no
eom_NbodySH_Interp!eom_stm_NbodySH_Interp_fd!yes
Note

In order to use Julia's dual numbers, make sure to use a function that does not contain SPICE calls (i.e. use ones with _interp in the name); this is enabled by interpolating ahead of time ephemerides/transformation matrices.

Warning

The accuracy of interpolated equations of motion (with _interp in the name) depends on the interpolation_time_step; if high-accuracy integration is required, it is advised to directly use the equations of motion that internally call SPICE (i.e. with _SPICE in the name)