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Why are planetary orbits elliptical instead of circular?

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A diagram with labeled vector illustrations. Panel A shows vector addition with two green arrows A and B forming a parallelogram, and their sum C as a red diagonal. Panel B shows vector subtraction, with A and negative B forming resultant D. Panel C demonstrates vector components using angles and projections. Panel D uses the right-hand rule with thumbs to show the direction of the cross product E = A x B and its opposite, -E = B x A. — mechanics Figure 1: (A) The vector sum C = A + B = B + A. (B) The vector difference A + (−B) = A − B = D. (C, left) A cos θ is the component of A along B and (right) B cos θ is the component of B along A. (D, left) The right-hand rule used to find the direction of E = A × B and (right) the right-hand rule used to find the direction of −E = B × A.

In the context of classical mechanics, it is a common misconception that planetary orbits must be circular. In reality, circular orbits are merely a specific, idealized case of motion under a central force.

According to Newton’s law of gravitation, the gravitational force between the Sun and a planet varies inversely with the square of the distance between them. When a body moves under the influence of such a central force, its path is determined by its total energy and initial velocity.123

As Isaac Newton demonstrated, a body moving about a central point can follow various conic sections:1
* Elliptical orbits occur when the body has insufficient energy to escape the gravitational pull, resulting in a bound, repeating path.21
* Parabolic or hyperbolic orbits occur if the body possesses enough energy to escape the central body's gravity, resulting in an open, non-repeating path.231

Therefore, planets move in elliptical orbits because they are gravitationally bound to the Sun, and their specific orbital shapes—defined by their eccentricity—are a direct consequence of their initial conditions and total energy.41