For two bodies interacting gravitationally, the relative motion has potential $$V(r)=-\frac{Gm_1m_2}{r}.$$ Conservation of angular momentum confines the orbit to a plane, and the inverse-square law produces trajectories that are conic sections.

Writing the orbit in polar form with $u=1/r$ gives Binet's equation. For an inverse-square attraction its solution can be written $$r(\phi)=\frac{p}{1+e\cos(\phi-\phi_0)},$$ where $p$ is the semi-latus rectum and $e$ the eccentricity. The orbit type follows from $e$:

• $0\le e<1$: ellipse, including the circular case $e=0$;
• $e=1$: parabola;
• $e>1$: hyperbola.

For a bound elliptical orbit with semi-major axis $a$, the total relative energy is $$E=-\frac{Gm_1m_2}{2a}.$$ More negative energy means a more tightly bound orbit.

Kepler's second law follows directly from angular-momentum conservation. The areal velocity is $$\frac{dA}{dt}=\frac12r^2\dot\phi=\frac{L}{2\mu},$$ which is constant: equal areas are swept in equal times. Combining the orbital geometry with Newtonian gravity gives Kepler's third law $$T^2=\frac{4\pi^2}{G(m_1+m_2)}a^3.$$ For a planet orbiting a much more massive star, this reduces to the familiar $T^2\propto a^3$.

Kepler's laws are therefore not independent empirical rules once Newtonian gravity is assumed; they emerge from central-force dynamics and conservation laws.