Limits of Keplerian Orbits

(Also see Parts I and II).

III. Exceptions: large masses, high speed, long time, comets...

So far we've presented a very idealized, 16th century view of the world. But if you look at NASA's data, already you may start to see some cracks in the story. For example, why does Earth have a non-zero inclination angle? Isn't Earth's orbit, by definition, the one we measure inclination from?

Well, it turns out that Keplerian orbits are not stable. The planets all have their own gravitational fields and tug and pull on each other as they go round. Reality is not so clean as Kepler believed.

One measurable effect of all this tugging is a wobble of sorts, which causes inclination to change over time.

For this reason, we've had to introduce a new reference plane called the invariable plane, defined by the average angular momentum of all the matter in the Solar System. This value (basically) never changes. The reason why this is so is surprisingly simple: the distances between star systems are so huge that nothing ever leaves or enters the Solar System. If nothing ever leaves or comes in, then the overall momentum is conserved.

Sure, we might get a comet or two, and a Voyager probe leaves every now and then, but their masses are a rounding error next to the planets, and the planets themselves don't weigh much compared to the Sun, so this plane will remain the same for much longer than there will be life on Earth. It's all gucci.

Earth's inclination to the invariable plane is about 1.57 degrees. The inclination NASA gives for Earth's orbit is in the ecliptic coordinates, and it deviates very slightly from zero for the same reason that Earth's semi-major axis isn't precisely 1 AU: over time, it's simply drifted, and we've been able to measure it with increasing precision. The difference, though, is tiny and most likely meaningless for amateur astronomy.

Resonance

You might expect that all the planets exerting gravitational tug on each other might make the system unstable. Surprisingly, the opposite is true! You may have noticed, in the demo, that Neptune and Pluto's orbits intersect, however the two planets¹ will never collide. The mechanism preventing collision is called orbital resonance, and refers to a process where Neptune and Pluto exchange momentum in a natural oscillation, such that Neptune always completes 3 orbits for every 2 of Pluto. Similar resonances exist between other massive planets and their moons, and even between Jupiter and the asteroid belt.

Hyperbolic Comets and Extrasolar Objects

Comets and extrasolar objects might follow hyperbolic trajectories, with eccentricities exceeding 1. (We no longer call them orbits, because they don't repeat – the object simply swings around the Sun and leaves the Solar System.)

The equations stop working with hyperbolic orbits, but they're not hard to modify. Instead of the eccentric anomaly, we now need to compute a hyperbolic angle called the hyperbolic anomaly, like so:

M = e \times \sinh(H) - H

Orbital Decay, Relativistic Effects, Tidal Forces

An orbiting system is characterised by potential energy. Energy levels can decrease over time by a variety of mechanisms:

• The side of the body nearer the barycenter experiences slightly more gravitational pull than the far side. This results in a slight deformation, which generates heat by friction transferring orbital energy into heat.
• Orbiting bodies can also transfer momentum to one another through tidal forces. This effect will eventually be responsible for the Moon leaving the Earth's gravitational influence.
• Two bodies orbiting each other emit gravitational radiation, providing another mechanism for energy loss.

More Advanced Models, etc.

The instabilities, resonances and drift in orbits are not covered by Kepler's model, which is nevertheless still in use today for everything from predicting satellite positions, to planning observations. In most cases, it has held up remarkably well over the centuries, and even in the modern day, the parameters only need updating once every few decades.

As a general rule, the closer to the Sun, the less stable the model is, with Mercury's orbit being the worst of the bunch.

If you're planning a mission to Europa, the state of the art predictive model of the Solar System is probably VSOP2013. It lacks the elegance of Kepler's model, being effectively a fitted high-order polynomial. Its terms have no physical meaning, it's just a bunch of math that happens to fit.

Finally, numerical simulations can be used for exoplanets, finding Planet 9, studying star system formation and counter-factual scenarios, like seeing what would happen if a rogue extrasolar planet smashed into Jupiter. Such simulators require careful construction, but can be surprisingly precise if precision errors and integrator instability are properly handled.