= Kepler's laws of planetary motion
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Kepler's laws describe three connected features of an <elliptic orbit>: its shape, how fast it is traversed, and how its size determines its <orbital period>. <Kepler's first law> gives an <ellipse> with the attracting centre at one <focus of a conic>; <Kepler's second law> gives equal swept areas in equal times; <Kepler's third law> relates the squared <orbital period> to the cubed <semi-major axis>. Together they turn a drawing of a <Kepler orbit> into a prediction of where the bodies will be and when they will return. https://science.nasa.gov/learn/basics-of-space-flight/chapter3-3/[NASA, Basics of Space Flight: Kepler's laws]{ref}
Here the <mathematical model> is an isolated <two-body problem> in <Newtonian gravity>, with <point masses> $M,m>0$ and a bound, noncolliding <elliptic orbit>. Let $\mathbf r=\mathbf r_m-\mathbf r_M$ be the relative displacement, $r=|\mathbf r|$ the separation, and $\mu=G(M+m)$ the <gravitational parameter>, where $G$ is the <gravitational constant>. <Newton's law of universal gravitation> then gives
$$
\ddot{\mathbf r}=-\frac{\mu}{r^3}\mathbf r.
$$
The relative <Kepler orbit> has <semi-major axis> $a$ and <orbital eccentricity> $0\le e<1$. Both bodies move around their shared <astronomical barycenter>, the system's <centre of mass>. In its rest frame their displacements are $\mathbf r_m=M\mathbf r/(M+m)$ and $\mathbf r_M=-m\mathbf r/(M+m)$. Their individual <semi-major axes> therefore add to $a$; the relative <ellipse>'s attracting <focus of a conic> represents zero separation, rather than a body fixed in space.
One example will connect all three laws. After <nondimensionalization>, choose $a=1$, $e=0.6$ and $\mu=1$. The <semi-minor axis> is $b=a\sqrt{1-e^2}=0.8$, while the <periapsis distance> and <apoapsis distance> are
$$
r_{\rm p}=a(1-e)=0.4,\qquad r_{\rm a}=a(1+e)=1.6.
$$
Thus $a=(0.4+1.6)/2=1$ measures the <ellipse>'s size even though the separation changes fourfold. The <specific angular momentum> will be $0.8$, the two endpoint <speeds> $2$ and $0.5$, and the <orbital period> $2\pi$. These are values in the chosen scaled units, not measurements of a particular <planet>.
<Johannes Kepler> published <Kepler's first law> and <Kepler's second law> in \i[Astronomia Nova] in 1609 and <Kepler's third law> in \i[Harmonices Mundi] in 1619. These publication dates are documented by https://svs.gsfc.nasa.gov/4642[NASA Scientific Visualization Studio, Kepler's laws]{ref}. Their appeal is that three short statements organize both geometry and timing; <Newtonian gravity> supplies a common dynamical explanation.
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