Velocity Transformations

While position and magnetic field vectors are independent of the relative motion of the coordinate system this is not true for other vectors for example for the solar wind velocity vector. Usually this vector is originally given in a spacecraft reference frame. For solar wind studies it is advisable to subtract the effect of the spacecraft motion relative to a heliocentric inertial system. If the spacecraft velocity vector is not provided together with the positional data the velocity can be calculated from the temporal derivative of the position time series. The velocity vector in the transformed system is generally given by

$$\mathbf{v'} = \dot{E} \mathbf{r} + E \mathbf{v} - \mathbf{v_c},$$ (42)

where $\mathbf{v_c}$ is the relative speed of the system origins and $\dot{E}$ is the temporal derivative of the rotation matrix:

$$\dot{E}(\Omega,\theta,\phi) = A * E \dot{\Omega} + B * E \dot{\theta} + E * A \dot{\phi}$$ (43)

$$A = \dot{R_3}R_3^T = \left( \begin{array}{ccc}0&1&0\\ -1&0&0\\ 0&0&1 \end{array} \... ...in\phi & -\sin^2\phi & \cos\phi \\ \sin\phi & -\cos\phi & 0 \end{array}\right)$$

For the transformation into planetocentric systems $\phi$ is the only angle changing rapidly such that $\dot{E}(\Omega,\theta,\phi) \approx E * A \dot{\phi}$.

One of the most common transformations is the transformation from a heliocentric inertial system like $HAE_D$ to a geocentric rotating system like $GSE_D$. Since $\dot{\lambda}_E \approx 1^\circ/day \approx 2\cdot 10^{-7} rad/s$ the rotational part of the velocity transformation can be neglected for geocentric distances of less than $5\cdot 10^6$ km to keep an accuracy of $\approx 1$km/s. In that case the transformation reduces to the subtraction of the orbital velocity of the Earth which in the ecliptic system is given by

$$\mathbf{v_E}_{HAE} = v_0 * (\cos (\lambda_{geo}+90^{\circ}),\sin (\lambda_{geo}+90^{\circ}),0^\circ)$$ (44)

where $v_0 = 29.7859$ km/s is the mean orbital velocity of the Earth and $\lambda_{geo}$ the Earth longitude defined in eqn.36.