= Galactic-centre point-mass orbit model
{c}
{title2=$\mu=4\pi^2a^3/P^2$}
Consider a negligible-<mass> <star> moving in the stationary <Newtonian gravitational potential> of a point mass. Let its <semi-major axis> be $a$, <orbital eccentricity> $0\le e<1$, and <orbital period> $P$. Write $\mu=GM$ for the <gravitational parameter>. The <eccentric anomaly> $E$ is determined by <Kepler's equation>,
$$
E-e\sin E=\frac{2\pi(t-\tau)}P,
$$
where $\tau$ is the time of <periapsis>. Coordinates in the orbital plane, with the point mass at the origin, are
$$
\mathbf r'(E)=a\begin{pmatrix}\cos E-e\\\sqrt{1-e^2}\sin E\\0\end{pmatrix},\qquad
\dot{\mathbf r}'(E)=\frac{2\pi a}{P(1-e\cos E)}\begin{pmatrix}-\sin E\\\sqrt{1-e^2}\cos E\\0\end{pmatrix}.
$$
The focus is the central mass; the ellipse's geometric centre is displaced by $-ae$ along the first coordinate. The varying denominator makes the <star> move faster near <periapsis>, even though the <mean anomaly> increases uniformly.
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