Rock from the Sky
Version 1.0

Typical Asteroid: photograph of Matilde 253 during NEAR flyby.

There are millions of asteroids in orbits around the sun.  An estimated 14,000 are classified as Near Earth Objects (NEOs) by the criterion that their orbits bring them within 1.3 astronomical units (AUs) of the sun.

Considering the frequency of bolides encountered by planet Earth, all populations of the world need to be concerned about NEOs larger than 100 meters in diameter becoming a Rock from the Sky.  Think back to the Tunguska Event in 1908.

For this puzzle, we will study one particular NEO, an asteroid named Égaré (French for 'stray')...

Solvers are invited to observe in Figure 1 that the orbit of Égaré has a perihelion of 0.25 AU, which is inside the orbit of both Mercury (0.387 AU) and Venus (0.723 AU), as well as Earth's orbit (1.000 AU). The orbit of Égaré has an aphelion of 3.0 AU, which is beyond the orbit of Mars (1.523 AU), placing it in the middle of the Asteroid Belt from which it 'strayed' (hence its name).

The orbital period of any object in the solar system is given by τ = 2π (a3 / μ)1/2, where a is called the 'semi-major axis' of the ellipse: a = (perihelion + aphelion)/2.  The constant μ = M×G, the mass of the sun M multiplied by the universal gravitation constant G.

In Figure 1, orbital periods are divided into two intersecting arcs.  For τ(Earth) = 0.694 + 0.306 = 1.000 year, and the orbital period of τ(Égaré) = 1.429 + 0.642 =  2.071 years (756 days).

 The most significant observation introduced by Figure 1 is that the orbit of Égaré is nearly coplanar with Earth's orbit!  That feature is emphasized in the Edge View.

The two orbital intersections are labelled 'Inbound' and 'Outbound'.  At either intersection, a collision between asteroid Égaré and the planet Earth will be possible.  Indicated in Figure 1 are the intervals measured in years between the respective arrivals at the intersections.

 Asteroid Hazards: The View from Space, Part 2            The Challenge of Detection Thomas Statler, Astrophysicist & Planetary Scientist

Telescopic observations of
Égaré from the planet Earth are limited, as indicated in Figure 2...

...which means that astronomers must rely a whole lot on orbital mechanics for predictions between sightings.  Here is a timetable
for predicted arrival dates of Égaré and Earth at the two orbital intersections based on observations beginning before the summer of 2018...

 Intersection Arrival Date Égaré Inbound Friday, June 1, 2018 Earth Inbound Sunday, July 22, 2018 Earth Outbound Sunday, November 11, 2018 Égaré Outbound Monday, January 21, 2019 Earth Inbound Monday, July 22, 2019 Earth Outbound Monday, November 11, 2019 Égaré Inbound Friday, June 26, 2020 Earth Inbound Tuesday, July 21, 2020 Earth Outbound Tuesday, November 10, 2020 Égaré Outbound Monday, February 15, 2021 Earth Inbound Wednesday, July 21, 2021 Earth Outbound Wednesday, November 10, 2021 Earth Inbound Thursday, July 21, 2022 Égaré Inbound Thursday, July 21, 2022 Earth Outbound Thursday, November 10, 2022 Égaré Outbound Monday, March 13, 2023

 The timetable indicates that Égaré is predicted to arrive at the Inbound Intersection on Thursday July 21, 2022 -- the same day that planet Earth will arrive there!

Accordingly, on that date Égaré will indeed become a Rock from the Sky.

As shown in Figure 1, planet Earth is travelling along its orbit at about 30 km/s.  Its diameter at the Equator is approximately 12,800 km.  Thus it takes 12,800/30 = 427 seconds or seven minutes and eight seconds for Earth to pass by any point along its orbit.

At least one popular reference suggests that if the arrival time of a threatening asteroid like Égaré can be delayed by merely 427 seconds, such a collision will be avoided.

Inasmuch as Égaré is indeed 100 meters in diameter its impact might destroy a city the size of Paris, so the effort to delay its arrival -- however challenging -- will be worth the cost.  Various proposals have been offered for deflecting the orbital pathway of a threatening Rock from the Sky

All such proposals require the timely delivery of an immense amount of man-made energy to the approaching NEO so as to use some of its own mass to produce vectored thrust.  Solvers will find this challenge addressed in the Orbital Deflection puzzle.

Solvers of the Trappist Orbits puzzle know that in orbital mechanics, the interactions among orbital parameters are counter-intuitive.  For our purposes here, we will will require the thrust to be vectored so as to produce a small change in orbital velocity ∆V in the prograde direction, as shown in Figure 3.  Intuitively, that would seem to speed up Égaré not slow it down.

The vis-viva equation is quite prominent in orbital mechanics and will be invoked here to explain that the increase in tangential velocity of the asteroid causes the orbit to expand, increasing the perihelion, thus increasing the length of the semi-major axis a

As mentioned above, τ = 2π (a3)1/2, so an increase in the length of a increases the orbital period τ(Égaré), thereby delaying the arrival of Égaré at the Inbound Intersection by some amount +∆t, as shown in Figure 3...

Everybody on Earth should be alarmed to observe in Figure 3 another effect of ∆V vectored in the prograde direction: Although τ(Earth) does not get changed, our planet will nevertheless take longer to arrive at the new location of the intersection by +∆t.  If by happenstance +∆t = +∆t, the collision on Thursday July 21, 2022 will not be avoided, despite our efforts to deflect Égaré!

By the way, if a collision were predicted to occur at the Outbound Intersection, delaying the arrival of Égaré by +∆t works just fine, inasmuch as Earth will arrive at the new Outbound Intersection earlier by -∆t, which may double the margin in time separation,

If delaying the arrival of Égaré by an increase in velocity ∆V in the prograde direction does not prevent the collision at the Inbound Intersection, perhaps advancing the arrival of of Égaré by vectoring velocity change ∆V in the retrograde direction as shown in Figure 4 will do the job...

As expected, the decrease in tangential velocity by the ∆V vectored in the retrograde direction does indeed decrease the perihelion and shorten the length of the semi-major axis a.  That decreases τ(Égaré) and delivers Égaré to the intersection earlier by -∆t.  However, the change in shape of the asteroid's orbit means that Earth will also arrive earlier at the new Inbound Intersection by some -∆t, which again cancels out the desired time separation for avoiding the collision with Égaré.

Here again, if a collision were predicted to occur at the Outbound Intersection, advancing the arrival of Égaré by -∆t works just fine, inasmuch as Earth will arrive at the new Outbound Intersection later by +∆t, which may double the margin in time separation,

 Discovery Alert: A search of the Internet by your puzzle-master has found proposals for vectoring of ∆V only in the tangential direction -- with no acknowledgement of this catastrophic phenomenon at the Inbound Intersection!

 What is your proposal for preventing a collision of asteroid Égaré with planet Earth?

Vis-Viva Equation

The tangential velocity V of a body in an elliptical orbit with the sun at one of its foci is given by the Vis-Viva Equation...

V = [μ (2 / r - 1 / a)]1/2

Where:

μ is the 'standard gravitational parameter':
μ = M×G
, the mass of the sun M multiplied by the universal gravitation constant G.
r is the distance from the sun to the orbiting body at any given time
a is the length of the 'semi-major axis':
a = (aphelion + perihelion)/2

 An orbit is a fixed spatial figure that does not change with time.

In the absence of external forces applied to the orbiting body itself, the value of r and therefore V are the only variables in the vis-viva equation that change with time.

Sophisticated solvers may notice that the mass of the orbiting body does not appear in the vis-viva equation, nor does the length of the 'semi-minor axis' b.

The vis-viva equation can be solved for a to produce this relationship...

a = μ / [(2 μ / r) - V2]

...from which, as one sees that for any given value of r, a change in tangential velocity V ± V by an external force results in a change in the fixed value of a ± ∆a and thus in a change to the orbital period τ ± ∆τ in compliance with...

τ = 2π (a3 / μ)1/2

Nota bene, opposites of key parameters are depicted in Figures 3 and 4.

Meanwhile, the 'eccentricity' e of a solar orbit is defined as...

e = (aphelion - perihelion) / (aphelion + perihelion)

...which decreases with the increase in perihelion resulting from an increase in a, such that at its limit (perihelion = aphelion), e = 0 for a circular orbit.

The length of 'semi-minor' axis b is given by...

b = a (1 - e2)1/2

...which implies that for a given value of eccentricity e, an increase in a widens the orbit by increasing b, making the orbit more circular -- hey, less eccentric, reducing the value of e.  Huh?  Sophisticated solvers will see right through that paradox.

Other applications of the vis-viva equation will appear throughout the Égaré series.