What Goes Around
What goes around comes around
Version 1.1
ORBITAL METAL: BILLY MOON
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 their respective orbits are elliptical with fixed dimensions. Some realities about orbital shapes are addressed in the Plural Gravities puzzle.

 Asteroid Hazards: The View from Space, Part 3               Finding the Path Thomas Statler, Astrophysicist & Planetary Scientist

Let us again make use of our 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 Earth's orbit.  Sophisticated solvers will need no reminder that, as a coplanar asteroid, Égaré will routinely cross through those intersections time after time with no consequences for planet Earth, and then, and then...

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 day is alarming enough for a 100-meter ‘city-smasher’.  Inasmuch as Égaré has a coplanar orbit, 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 complete orbitings, such that 365.250 m = 756.433 n, which signifies that upon the completion of exactly m orbitings of Earth and exactly n orbitings 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. The common products 365.250 x 756.433 = 276,287.153 days or about 756 years.
Both Earth and Égaré will arrive at the inbound intersection of their orbits on the same day about 756 years after 2022 or in the year 2778.  Similarly, both celestial bodies will meet at the outbound intersection of their orbits every 756 years, so potential collision days will occur twice per 756 years or about once every 378 years.  On each of those joint arrival days, the probability of impact is about 1/200.  We might want to combine the day-of-intersecting and the probability-of-colliding during that day...
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.  The common products 73,684.505 x 152,600.611 = 11,244,300,484.200 ф or 152,601 years, or 76,300 years considering both intersections.

Apparently, we can estimate the interval between impact threats
by simply deriving a quantizing unit ф based on the cross-section-in-time of planet Earth and converting ф directly into years for that same unit applied to the orbital period of asteroid Égaré. This could become significant.  Here's why...
During the preparation of the What Goes Around puzzle, a headlined appeared: Small Asteroid Gives Earth a Close Shave in Highly Anticipated Flyby.
"The space rock, known as 2012 TC4 zoomed about 42,000 km above Antarctica.  That's about 11 percent of the distance between Earth and the moon, and just beyond the orbit of geostationary satellites."
The 'space rock' is approximately 15 m in diameter, which is smaller than the Chelyabinsk meteor in 2013 (20 m), and its inclination is only 0.856 degrees, which means it is a near-coplanar asteroid with an orbital period of 1.67 years.  Nota bene, during this pass, that asteroid was a potential threat to extremely valuable satellites in geostationary orbits, of which there are hundreds.
Perhaps the quantizing unit ф should be increased for the Prediction of collisions with geostationary satellites.  Thus, for ф = 10.244 minutes, the interval between threats by Égaré is reduced from 76,300 to 53,032 years.

Now, for our What Goes Around puzzle, Figure 2 depicts the fixed orbits of two real asteroids with near-coplanar orbits, and their respective orbital periods are expressed in Earth years...
• 1998 KY26 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.

...along with two fictitious asteroids,
Égaré and Puzzler. The latter was studied in the Astrogating Asteroids puzzle.   For simplicity in comparisons, all four orbits have their major axes rotated to be horizontal in the sketch.

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

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