How did Newton’s laws of motion explain the orbits of planets and comets?

Isaac Newton explained the orbits of planets and comets by applying his three laws of motion and his law of universal gravitation to celestial mechanics.12

Newton demonstrated that a body in motion would continue in a straight line unless acted upon by an external force. By quantifying this force, he showed that a planet is continuously diverted from its inertial path by a central gravitational attraction toward the Sun. He determined that this force must vary inversely with the square of the distance between the bodies.123

Using these principles, Newton proved that the path of a body moving under such a central force depends on its total energy and initial velocity:14
* Elliptical orbits: Occur when the initial velocity is not too high, resulting in bound paths like those of planets.14
* Parabolic or hyperbolic orbits: Occur at higher velocities, allowing objects like comets to enter the solar system and potentially leave without returning.14

Edmond Halley played a crucial role in this development by encouraging Newton to publish these findings in the Principia.25