How do planets perturb each other’s otherwise elliptical orbits?

In celestial mechanics, planets perturb each other's orbits because they exert mutual gravitational forces on one another, rather than being influenced solely by the Sun. While Kepler’s laws accurately describe a simplified two-body system, they are only approximations for the solar system because the total force acting on a planet includes the gravitational pull of other planetary bodies.1

These perturbations cause the six orbital parameters—which would remain constant in a perfect two-body system—to vary slowly over time. To analyze this, scientists often treat the orbit as an osculating ellipse, an instantaneous elliptical path that the body would follow if all perturbing forces were suddenly removed.2

Because the masses of the planets are very small compared to the Sun, these deviations remain minor, allowing for accurate predictions using perturbation theory. These mathematical models, such as the Lagrange planetary equations, help astronomers calculate how these small forces cause long-term, or secular, changes in orbital elements like the inclination or the position of the ascending node.23