What Goes Around
  What goes around comes around
Version 1.0
Copyright ©2017 by Paul Niquette. All rights reserved.
orbits in metal
 Metal Wall Art: sculpture, hand forged iron loops finished in lightly burnished silver leaf.

As introduced in the Orbital Deflection puzzle, worldwide endeavors for avoiding meteor impacts can be divided into three overall tasks...

1.     Detection of threats from NEOs based on their sizes and orbital parameters.

2.     Prediction of the orbital motion of each NEO to ascertain its likely point of impact.

3.     Deflection of the threatening NEO's orbit to avoid collision with Planet Earth.

This puzzle addresses Task 2 Prediction, giving emphasis to the timing of celestial bodies in orbits, while assuming that the shapes of their respective orbits remain constant.

Let us again make use of our coplanar asteroid
Égaré, which has an orbital period τ = 2.071 years. Figure 1 shows τ divided into segments based on the time intervals between the asteroid's arrivals at two intersections with the orbit of planet Earth.


In the Rock from the Sky puzzle, solvers were warned that Égaré had been predicted to arrive at the inbound orbital intersection on the same day as planet Earth passes the same point in its orbit -- that day being Thursday, July 21, 2022.  The Prediction was regarded to be quite worrisome, and the Orbital Deflection puzzle invited solvers to invent a way to change the inclination of the asteroid's orbit to prevent its collision with our planet.

On that day, though, Earth may arrive several hours after Égaré has gone by.  Then too, with only the date given in the Prediction, solvers need not be surprised if the asteroid passes behind the earth by, say, a half-day’s time.  Inasmuch as the Earth is a ‘moving target’, to determine the probability of a collision on Thursday, July 21, 2022, one must estimate how many of those 24-hours Earth will spend blocking the path of Égaré
With an orbital radius of 150 Mkm, our planet travels through space 942.5 Mkm every year.  That's 2.6 Mkm each day or 107,600 km/h.  Earth's diameter at the Equator is about 12,800 km.  Thus, planet Earth requires only 7 minutes and 8 seconds to pass across any point on its orbit.  That’s less than 1/200th of a day.

An impact probability of 1/200 for that particular day is alarming enough for a 100-meter ‘city-smasher’.  Best that we also remind ourselves that Égaré has a coplanar orbit, making it absolutely the worst kind of NEO.  So if planet Earth lucks out on Thursday, July 21, 2022, how much time will go by before another day comes along during which that asteroid will be threatening our cities again?

Everybody knows that planet Earth returns to any given point on its orbit every 365.250 days (leap year, remember).  Meanwhile, solvers know that Égaré returns to any given point on its orbit every 2.071 x 365.250 ≈ 756.433 days.  Let m and n be quantities of orbit-passages, such that 365.250 m = 756.433 n, which signifies that upon the completion of exactly m orbits of Earth and exactly n orbits of Égaré, the two celestial bodies arrive at the intersection of their orbits on the same day.
For 365.250 m = 756.433 n, we have that m = 756.433 and n = 365.250.
Thus, Earth and Égaré will both arrive at the inbound intersection of their orbits on the same day about 756.433 years after 2022 or in the year 2778.  Similarly, both celestial bodies will meet at the outbound intersection of their orbits every 756.433 years, so potential collision days will occur twice per 756.433 years or about once every 378 years.  On each of those joint arrival days, the probability of impact is about 1/200.  Here's how to combine the day-of-intersection and the probability-of-collision during those 24 hours...
Instead of using 24-hour days, let’s quantize the planet Earth's year by a unit of time ф = 7.138 minutes, which is the time required for our planet to move along its orbit a distance equal to its diameter at the equator.  Well then, one year on Earth = 73,684.505 ф.  The orbital period for Égaré τ = 2.071 x 73,684.505 = 152,600.611 ф.  Thus, with almost certainty, Égaré is expected to collide with planet Earth some time before the year 78322 at one of the orbital intersection. 

Figure 2 in the What Goes Around puzzle depicts the fixed orbits of three other asteroids along with their respective orbital periods expressed in Earth years.  For simplicity in comparisons, the orbits have their major axes rotated to be horizontal in the sketch.
  • 1998 JY26 is 30 meters in diameter with an inclination of 1.481 degrees.
  • 2007 VK184 is 130 meters in diameter with an inclination of 1.222 degrees.
  • 2017 PZLR is a fictitious asteroid with an unknown diameter, which was studied in the Astrogating Asteroids puzzleSophisticated solvers will observe that, whatever its inclination, Puzzler appears to satisfy the collision-threat criterion (d = 2 b2 / a).


What is your Prediction for the period of time between intervals during which each asteroid threatens to collide with Earth?