There are millions of asteroids in orbits around the sun.  They appear to be concentrated in belts.  The main asteroid belt is depicted above as a speckled red circle outside the orbit of Mars, far away from Earth.  Asteroid orbits are not all that circular, however.   Kepler's Laws mandate that in eccentric orbits asteroids must spend more time loitering out there in the asteroid belt near aphelion than falling toward the sun, whipping around at perihelion, and sweeping back out again to await observation by astronomers.

As of this writing (2017), more than 16,000 asteroids have been classified as Near Earth Objects (NEOs), which means their eccentric orbits bring them within 195 Mkm (120 Mmi) of the sun at perihelion.  Meanwhile, the Earth's orbit is approximately circular with a radius of 150 Mkm (93 Mmi).

Some NEOs are less worrisome than others:

• 16 Apohele asteroids have orbits with aphelions less than 147 Mkm, thus they remain completely inside Earth's orbit.
• 6,144 Amor asteroids have perihelions far outside Earth's orbit. 
  

Of greatest concern to us earthlings are these NEOs:

• 1,191 Aten asteroids have their orbital centers and perihelions inside Earth's orbit and their aphelions outside.
• 8,837 Apollo asteroids have their orbital centers and aphelions outside Earth's orbit and their perihelions inside.

If an Aten or an Apollo asteroid arrives at an orbital intersection at the same time the Earth happens to be there, the result can be a shooting star or a collision, with a range of consequences for mankind that depends, of course, on the size of the asteroid.

An Apollo asteroid struck Earth as the Chelyabinsk meteor in 2013.  It was about 18 meters in diameter.  The largest Apollo asteroid, Sisyphus, is seven kilometers in diameter -- nearly half the size of the Chicxulub impactor, which brought the end of the dinosaurs 66 million years ago.

It is customary to view the solar system from a position in space that makes counter-clockwise the prograde direction for orbits.  Superimposed on the Fiorenza illustration above is a representation of a particular Apollo-class asteroid with a semimajor axis about 1.8 times larger than the radius of the Earth's orbit.  Let's give it the name "Puzzler."  Using Kepler's Third Law, one can estimate Puzzler's orbital period as 365×1.8^3/2 = 881 days.

Here is a scenario for solvers to consider based on the information at hand...

1. Let us postulate that Earth passes New Years Day at the topmost location in the map of its orbit.  There appear to be two orbital intersections, one in mid-August, about 240 days into the year, and the other in mid-October, some 60 days later.
2. We might mischievously astrogate Puzzler into its orbit inbound toward the sun and just missing Earth as it passes through that mid-October intersection.  Whew.
3. Puzzler swings past the sun and through its perihelion, returning to the outbound intersection with Earth's orbit two months later.  At that time Earth will fortunately be nearing its farthest point toward the top of the map.
4. When Earth arrives at the mid-August intersection the next year, Puzzler will have spent 240 days outbound toward the asteroid belt on its 881-day orbit. 
5. Another 641 days will go by before Puzzler will return from the asteroid belt to the inbound intersection with Earth's orbit at a location in the upper-right of the map.
6. When Puzzler arrives, Earth will have gone around 2.4 times (881/365).  It will be early-May, and our planet will be approaching the bottom-most point on the map.
7. Sixty-some days later, Puzzler will be passing through the outbound intersection, but Earth will still take another month and a half to get there.  Hooray for that.

So then, here is the challenge for the Astrogating Asteroids puzzle...

How long will it take for Puzzler to threaten the next catastrophic collision with Planet Earth?