Astrogating Asteroids
astrogate: To navigate or avigate in space.
  Version 1.1
Copyright 2017 by Paul Niquette. All rights reserved.
Asteroid Belt
Permission granted by Nick Anthony Fiorenza, The Lunar Planner

There are millions of asteroids in orbits around the sun.  They appear to be concentrated in belts.  The main one is depicted above as a speckled red circle outside the orbit of Mars, far away from EarthAsteroid 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 aphhelions 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 perihelions inside Earth's orbit and aphelions outside.
  • 8,837 Apollo asteroids have aphelions outside Earth's orbit and 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.

illAn 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 put North at the top of maps and to use counter-clockwise as the prograde direction for solar orbitsSuperimposed on the Fiorenza illustration above is a representation of a particular Apollo-class asteroid with a semimajor axis about 1.8 times larger than Earth's radius.  Let's give it the name "Puzzler.Using Kepler's Third Law, one can estimate Puzzler's orbital period as 3651.83/2 = 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 northernmost location on 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 in the south-east two months later.  At that time Earth will fortunately be nearing its farthest point toward the north.
  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 in the north-east.
  6. When Puzzler arrives, Earth will have gone around 2.4 times (881/365).  It will be early-May, and our planet will have just passed the southernmost point in its orbit.
  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
catastrophic collision with Planet Earth?

Before clicking to the solution page, solvers may want to take into account the fact that Earth presents a rapidly moving target to NEOs. 
With an orbital radius of 150 Mkm (93 Mmi), our planet travels through space 942.5 Mkm (585.4 Mmi) every year.  That's 2.6 Mkm (1.6 Mmi) each day or 107,600 km/h (66,800 mph).  Earth's diameter is about 12,800 km (7,900 mi).  Therefore...
Our planet requires only 7 minutes and 8 seconds to pass across any point in space.