# Keplerian orbits and closest approaches to Earth.

This question arose out of a discussion on Space.SE, but I think it will appeal to mathematicians more than astronomers:

Let's consider a small astronomical object following an ideal elliptic Keplerian orbit around the sun. For concreteness I'm going to call it a comet, though some of the cases I'm interested are distinctly un-cometlike orbits. The question arises: when during its orbit is the comet closest to Earth?

In some cases, such as for Pluto, the closest approach to Earth close to the time Earth passes most directly between the comet and the sun (in astronomy terms, this is when the comet is in opposition). There are also possible orbits where the closest approach to Earth will be almost simultaneous with the comet's closest approach to the sun (its perihelion) no matter where Earth is in its orbit at that time. An example of the latter would be a super-eccentric orbit whose major axis is perpendicular to the plane of the ecliptic (which is the plane that contains Earth's orbit). In the limiting case of this, the comet falls straight towards the sun along the major axis of its orbit, loops tightly around the sun in an instant and then goes back along the major axis again; then the closest approach to Earth will be exactly at perihelion.

To formalize this, let's define:

A dominant orbit is a Keplerian ellipse (in a certain orientation and position relative to Earth's orbit) such that within one orbital period (from aphelion to aphelion) the comet's closest approach to Earth will happen within three months of its perihelion, no matter when in Earth's year it passes perhelion.

In other words, for a dominant orbit, the timing of the comet will be the dominant factor in determining when it gets closest to Earth.

Conversely, a non-dominant orbit is one where one can get the closest approach to Earth to occur more than three months from the coment's perihelion, by choosing the right phase of Earth's orbit.

For non-dominant orbits, the timing of Earth will be the dominant factor in determining when the closest approach happens.

These definitions are precise, but give slightly strange results for small orbits; for example Mercury would automatically count as dominant because it takes less than six months for it to progress from aphelion to aphelion. But I'm more interested in long periods anyway. Still, Question: Is there a more convenient and well-behaved way to formalize what I'm getting at here? (Perhaps something about the timing of the closest approach varying continuously with the timing of the comet?)

In the discussion linked above, I conjectured that all elliptic orbits with sufficiently large perihelion distance are non-dominant. My heuristic argument is that for a given perihelion, there's an upper bound for how fast the comet can move at that point without the orbit becoming parabolic or hyperbolic, which means that if it reaches perihelion at the time of the year when the Earth is farthest from the perihelion point, it can't get away fast enough to be farther from the Earth half a year later. But I'm less sure of my reasoning when the comet's orbital plane may be inclined from the ecliptic.

Main question: It this conjecture true? If so, what is the largest possible perihelion distance for a dominant orbit, in AUs?

I have a few approximate napkin-and-Wolfram-Alpha calculations that suggest the threshold distance is around 5 AU, but I don't trust them very much.

Let's assume that the Earth's orbit is perfectly circular with a radius of 1 AU and an angular velocity of $\pi/6$ radians per month. The comet's orbital speed is related to this by Kepler's third law.