astrodynx.twobody.pvpr0

astrodynx.twobody.pvpr0(r_vec, v_vec, r0_vec, v0_vec, Ft, C, mu=1)

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

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.

• Ft (DTypeLike) – The Lagrange Ft function.

• C (DTypeLike) – The C function.

• mu (DTypeLike) – The gravitational parameter.

Return type:

Array

Returns:

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

Notes

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

\[\begin{split} \begin{split} \frac{d\boldsymbol{v}}{d\boldsymbol{r}_0} = & -\frac{1}{r_0^2} (\boldsymbol{v} - \boldsymbol{v}_0) \boldsymbol{r}_0^T - \frac{1}{r^2} \boldsymbol{r} (\boldsymbol{v} - \boldsymbol{v}_0)^T \\ & + F_t \left( \boldsymbol{I} - \frac{\boldsymbol{r} \boldsymbol{r}^T}{r^2} + \frac{1}{\mu r} (\boldsymbol{r} \boldsymbol{v}^T - \boldsymbol{v} \boldsymbol{r}^T) \boldsymbol{r} (\boldsymbol{v} - \boldsymbol{v}_0)^T \right) - \frac{\mu C}{r^2 r_0^2} \boldsymbol{r} \boldsymbol{r}_0^T \end{split} \end{split}\]

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, \(F_t\) is the Lagrange Ft function, \(C\) is the C function, \(\mu\) is the gravitational parameter, and \(\boldsymbol{I}\) is the identity matrix.

References

Battin, 1999, pp.467.