astrodynx.twobody.prpv0

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astrodynx.twobody.prpv0#

astrodynx.twobody.prpv0(r_vec, v_vec, r0_vec, v0_vec, F, G, C, mu=1)[source]#

The State Transition Matrix for position with respect to initial velocity.

Parameters:

  • r_vec (ArrayLike) – (3,) The position vector at the current time.

  • v_vec (ArrayLike) – (3,) The velocity vector at the current time.

  • r0_vec (ArrayLike) – (3,) The position vector at the initial time.

  • v0_vec (ArrayLike) – (3,) The velocity vector at the initial time.

  • F (DTypeLike) – The Lagrange F function.

  • G (DTypeLike) – The Lagrange G function.

  • C (DTypeLike) – The C function.

  • mu (DTypeLike) – The gravitational parameter.

Return type:

Array

Returns:

The State Transition Matrix for position with respect to initial velocity.

Notes

The State Transition Matrix for position with respect to initial velocity is defined as:

\[ \frac{d\boldsymbol{r}}{d\boldsymbol{v}_0} = \frac{r_0}{\mu} (1 - F) [(\boldsymbol{r} - \boldsymbol{r}_0) \boldsymbol{v}_0^T - (\boldsymbol{v} - \boldsymbol{v}_0) \boldsymbol{r}_0^T] + \frac{C}{\mu} \boldsymbol{v} \boldsymbol{v}_0^T + G \boldsymbol{I} \]
where \(\boldsymbol{r}\) is the position vector at current time, \(\boldsymbol{v}\) is the velocity vector at the current time, \(\boldsymbol{r}_0\) is the position vector at the initial time, \(\boldsymbol{v}_0\) is the velocity vector at the initial time, \(F\) is the Lagrange F function, \(G\) is the Lagrange G function, \(C\) is the C function, \(\mu\) is the gravitational parameter, and \(\boldsymbol{I}\) is the identity matrix.

References

Battin, 1999, pp.467.

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