# 8.11 Gravitational Perturbations

Newton's law of gravity is the most powerful tool we have to understand the universe. It describes the gravitational force between any two masses: planets, stars, galaxies, and even molecules. It can even explain the discrepancies we see between Kepler's laws of orbital motion and the real orbits of the planets.

It’s really only an approximation to say that the planets orbit in an ellipse with the Sun stationary at one focus. Why don’t Kepler’s Laws completely describe the motions of the planets? Newton's third law says that Jupiter exerts the same gravitational force on the Sun that the Sun exerts on Jupiter. So Jupiter tugs the Sun into moving slightly, and the Sun doesn’t stay exactly in one place at the focus. Because of the slight tugs from other planets and the resulting motions of the Sun, it’s also just an approximation to say that the planets’ orbits are ellipses. In fact, Neptune was discovered when scientists tried to find out what was perturbing Uranus from a purely elliptical orbit.

The small extra forces that planets exert on other planets are called gravitational perturbations. For gravity in the solar system:

F_{total} = F_{Sun} + Δ F

Scientists often use the Greek letter Δ “delta” to indicate a small increment. In other words, the gravitational force in the Solar System is given by the force due to the Sun, plus a little bit extra. The extra bit of gravitational force, ΔF , is a gravitational perturbation. The perturbation is much smaller than the main force, so we can write Δ F « F_{Sun}, or equivalently, Δ F / F_{Sun} « 1. The equation above can also be written:

F_{total} = F_{Sun} (1 + Δ F / F_{Sun}) ≈ F_{Sun}

The main perturbing force (Δ F) in the Solar System is the gravity of the most massive planet. Jupiter pulls the Sun into a slight motion. Jupiter tugs nearby planets away from pure elliptical orbits. Jupiter deflects comets into different orbits. Jupiter interacts with the asteroid belt to create gaps in the distribution of asteroids. All of these phenomena are caused by gravitational perturbations. As the equation above shows, it’s a good approximation to say that the Sun controls the planets’ orbits. But the extra influence of Jupiter causes some interesting and important phenomena.

Outside objects don’t cause significant perturbations within the sphere of gravitational influence of a planetary object - for example, Earth controls the motion of telecommunication satellites, Saturn controls the motion of its rings, and Pluto controls the motion of Charon. However, the spheres of gravitational influence of the planets take up a tiny fraction of the volume of the solar system. Through most of that volume, the Sun is the main influence, and Jupiter is the perturbing influence.

The idea of a perturbation occurs throughout physics. In fact, we could generalize the first equation to write X_{total} = X + Δ X, where X might be anything from a particle velocity to the strength of a magnetic field. We can use this expression anytime in nature that a small influence is added to a large influence. In mathematics, we have powerful tools for manipulating situations where a small quantity is added to a large quantity. Differential calculus is just one example.

A gravitational perturbation leads to a small deviation in an orbit. The position and velocity are altered slightly. What happens over time, as the orbit repeats? Each perturbation is a little gravitational "kick," and as the number of kicks grows, the cumulative effect on the object can become large. Let’s say a chunk of rock in the asteroid belt receives a small gravitational "kick" from Jupiter every time it orbits the Sun. We can relate the velocity before the kick to the velocity after the kick by v_{after} = v_{before} + Δ v, or equivalently:

v_{after} / v_{before} = 1 + Δ v / v_{before}

The effect of successive kicks is cumulative. After n kicks the velocity becomes:

v_{after} / v_{before} ≈ (1 + Δ v / v_{before})^{n}

Let’s say each kick increases the rock’s velocity by a tiny 1%. That means Δ v / v_{before} = 0.01. If we take 100 orbits, n = 100, and we get v_{after} / v_{before} = 2.7. So in only a few hundred years, the velocity of a perturbed rock might triple - more than enough to send it on a completely different orbit. In this simplified discussion, you can see that gravitational perturbation over time can lead to radically altered orbits.