Astrogating Asteroids
 astrogate: To navigate or avigate in space.
Version 2.3
Copyright ©2017 by Paul Niquette. All rights reserved.
spoiler alert...

How long will it take for Puzzler to threaten the
next catastrophic collision with Planet Earth?
In its present orbit, Puzzler can never
collide with Planet Earth.

An assumption was made in the Astrogating Asteroid puzzle -- that Puzzler's orbit is coplanar with Earth's orbit.  That assumption was not made explicit.  And -- hey, it is wrong. 

 
coplanar

The Puzzler orbit has an inclination of, say, 30 degrees
with respect to Earth's orbit. 

The sketch above shows that Earth is in danger of colliding at orbital intersections with Apollo asteroids independent of their eccentricities -- but only if their orbits are coplanar with Earth's orbital plane.  That criterion is not typically emphasized in various NEO references.

Non-Coplanar Orbits

With Earth's orbit viewed as circular above, one sees that foreshortening of each asteroid orbit creates the appearance of intersecting orbits as shown to be false in the edge-on views.
Although outside the scope of the Astrogating Asteroids puzzle, it is necessary to take into account all appropriate Keplerian elements in making orbital predictions
ifilmiAsteroid Hazards: The View from Space, Part 1
         What Makes an Asteroid a Hazard?
Thomas Statler, Astrophysicist & Planetary Scientist
 
Thus, sophisticated solvers will note that another assumption is implied by these sketches [deep breath here] -- that the argument of perihelion is at a right angle to the longitude of ascending node. Those considerations are not emphasized in some references pertaining to collisions.
Inclined Collision
What about that largest Apollo asteroid, Sisyphus, which is seven kilometers in diameter -- nearly half the size of the Chicxulub impactor, which brought the end of dinosaurs 66 million years ago?
sisyphusIn order for Earth to collide with a non-coplanar NEO, as depicted in the sketch above, the latus rectum of the asteroid's orbit must coincide with the diameter of Earth's orbit along the line of orbital intersection as shown above on the right. As for Sisyphus...
sisyphus

In the sketch, that criterion seems not to be met, so then, is Earth really safe from Sisyphus?
Sophisticated solvers must use all six orbital elements to correct the sketch and find the answer.
Astrogating Asteroids was the inspiration for the Égaré Collection.

Solvers, your comments are invited.

 
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