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Circular orbits in a 1/r gravitational potential are stable: small radial or tangential perturbations lead to slightly elliptical orbits that remain bound, not runaway trajectories. The effective potential U_eff(r) = L²/(2μr²) − G M μ / r has a minimum at the stable circular orbit radius. This stability is specific to the 1/r force law; other force laws (e.g., f ∝ r) can be unstable.
A circular orbit is the simplest type of bound orbit: the radius stays constant, and the orbiting body traces a perfect circle around the central mass. It exists at the specific radius r₀ where the net radial acceleration is exactly zero — where gravitational attraction perfectly balances the centrifugal effect of circular motion. The deeper question is whether this is a stable equilibrium: if you slightly perturb the orbit — a small rocket burn, a passing body's tug — does the orbit remain nearly circular, or does it spiral inward or outward to disaster?
From your study of circular motion dynamics and orbital elements, you know the conditions for circular motion and the geometry of orbits. The effective potential framework connects both. Recall that in the central-force treatment, radial motion obeys the same energy equation as a 1D particle in U_eff(r) = L²/(2μr²) − GMμ/r. The circular orbit corresponds to the minimum of U_eff — the particle sits at the bottom of a potential energy bowl. This minimum is why circular orbits in gravity are stable: U_eff has a true minimum, not a maximum or inflection point, so a small perturbation in r causes U_eff to increase, providing a restoring force that pushes r back toward r₀. The orbit rocks gently around the circular radius rather than departing from it.
To make this quantitative, expand U_eff around r₀. At the minimum, dU_eff/dr = 0 (this is the circular orbit condition), and d²U_eff/dr² > 0 (this is the stability condition — a positive second derivative means the bottom of the bowl is concave up). The radial oscillation frequency ω_r = √(d²U_eff/dr² / μ), evaluated at r₀, describes how fast the radius oscillates around r₀ after a perturbation. For a 1/r² gravitational force, this radial frequency equals the orbital angular frequency: ω_r = ω_orbit. This exact equality is the reason bound gravitational orbits are closed ellipses — the radial oscillation completes one cycle in exactly the same time as one full orbit, so the orbit traces the same path repeatedly. This is a special, non-generic property of the 1/r² force law (and also of the harmonic oscillator potential).
For other force laws, the ratio ω_r / ω_orbit need not equal one, and bound orbits precess — the axis of the approximately elliptical orbit slowly rotates over many orbits, tracing a rosette pattern rather than a closed curve. This is not a sign of instability; the orbit can still be stable while precessing. True orbital instability occurs when the effective potential has no minimum — only a maximum or no turning point at all — meaning that any perturbation causes runaway departure. Some force laws (with n > 2, where F ∝ 1/rⁿ) produce this behavior, making stable circular orbits impossible. The stability of planetary orbits in our solar system is therefore not automatic; it depends specifically on the 1/r² character of Newtonian gravity, which gives U_eff a well-defined minimum and ensures that the planets' slightly elliptical paths remain bound for billions of years.