23話_軌道要素から位置ベクトルを求める数式

When $\mathbb{X} : (x, y, z)$ and parameters are given as below, resolve $\mathbb{X}$.

$$ \begin{aligned} t &= 1{,}000 \text{ hours after EPOCH.} \\[6pt] a &= 31{,}123{,}987\text{m} \quad e = 0.62343 \\[6pt] i &= 18.926\text{[deg]} \quad \Omega = 96.503\text{[deg]} \\[6pt] \omega &= 188.755\text{[deg]} \quad \text{M} = 121.221\text{[deg]} \\[6pt] \mu &= \mathbf{G} \cdot \mathbf{M}_{\text{Pl}} = 3.9860044 \times 10^{14} [\text{m}^3 / \text{s}^2] \end{aligned} $$$$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = a \begin{pmatrix} \cos \Omega & -\sin \Omega \cos i \\ \sin \Omega & \cos \Omega \cos i \\ 0 & \sin i \end{pmatrix} \begin{pmatrix} \cos \omega & -\sin \omega \\ \sin \omega & \cos \omega \end{pmatrix} \begin{pmatrix} \cos E - e \\ \sqrt{1 - e^2} \sin E \end{pmatrix} $$