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Kepler's Laws of Planetary Motion

The foundational empirical laws governing orbital motion, stating that orbits are ellipses, equal areas are swept in equal times, and the square of the period is proportional to the cube of the semi-major axis (T2∝a3T^2 \propto a^3). These laws form the basis for analyzing perturbations.
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The statement of the theorem

Let r(t)\mathbf{r}(t) be the position vector of a body in an orbit defined by the gravitational parameter μ\mu. The orbit must satisfy:\n1. **Ellipticity:** 1r=1p(1+ecos⁡(ω+f))\frac{1}{r} = \frac{1}{p} (1 + e \cos(\omega + f)), where pp is the semi-latus rectum, ee is the eccentricity, and ff is the true anomaly.\n2. **Conservation of Angular Momentum (Second Law):** 12r2θ˙=h/r\frac{1}{2} r^2 \dot{\theta} = h/r, where hh is the specific angular momentum, implying dAdt=12r2θ˙=constant\frac{dA}{dt} = \frac{1}{2} r^2 \dot{\theta} = \text{constant}.\n3. **Period-Semi-major Axis Relation:** T2=4π2μa3T^2 = \frac{4\pi^2}{\mu} a^3, where aa is the semi-major axis and TT is the orbital period.