Amor asteroids

The Amor asteroids are a class of asteroids that get very close to the Earth from the outside, usually without crossing the Earth's orbit. Some of these are classified as potential collision hazards, but most just keep their distance. I learned about them as a result of more playing with Mathematica, this time with its AstronomicalData. A particularly handy option for the package is the "Classes" argument (so AstronomicalData["Classes"]), which will return a list of possible classes, like InnerMainBeltAsteroid and DwarfSpheroidalGalaxy and so on. These classes can be used to get a list of list of astronomical objects that fall into the class.

In any case, here's a fun little diagram showing the orbits of various Amor asteroids. The thick black lines are Earth, Mars, and Jupiter's orbits, from smallest to largest. You can see that many of the Amor asteroids cross Mars's orbit, and a few even get as far away as Jupiter.

Orbits of the Amor asteroids. Black lines show
Earth, Mars, and Jupiter. Blue lines show the asteroid
orbits. The big yellow dot represents the Sun.

But overall, this isn't a terribly revealing diagram. A slightly more intriguing plot is shown below: eccentricity of the orbit as compared to the asteroid's semimajor axis.

Eccentricity of the Amor asteroids as compared to their
semimajor axes. Looks interesting, right?

It's interesting to see that as the asteroid's semimajor axis increases, its eccentricity does too. There's not exactly a concrete reason that we would physically expect this result. So what causes this trend?

It turns out that it's actually all determined by our definition of an Amor asteroid: its perihelion (closest approach to the Sun) falls between 1.0 AU (astronomical units - the average distance from Earth to the Sun) and 1.3 AU. From the geometry of ellipses, the closest an orbit gets to one of its foci (which is the perihelion by definition), can be expressed in terms of its semimajor axis and eccentricity as $r_{min}=a(1-e)$ ($a$ is the semimajor axis, $e$ the eccentricity). Plotting this, we find that the eccentricities that we observed in such a promising-looking trend above are actually just adhering to the constraints imposed by the definition.

Perihelion of Amor asteroids compared to their
semimajor axes. Looks like that trend was just a
figment of our imaginations.

Another way of seeing this is to look at the possible values of the eccentricity for various semimajor axes:

Sure enough, all those eccentricities are based on
the definition of an Amor asteroid!

The upper bound there is $e=1-\frac{1}{a}$; that is, the highest possible eccentricity (for a given semimajor axis) the asteroid can have without crossing Earth's orbit. The lower bound is $e=1-\frac{1.3}{a}$, which is the lowest possible eccentricity it can have while still getting close enough (1.3 AU) to be considered an Amor asteroid. Mystery successfully solved.