Rock from the Sky

Version 1.1

Our initial proposals for protecting Earth from a collision with
Égaré at the Inbound Intersection turned out to be unpromising.We tried producing ∆V in both the prograde and retrograde directions, only to learn that on Thursday July 21, 2022 the collision is still likely.  So much for Tangential vectors.

According to the 'right-hand rule' in Figure 5, three choices for thrust vectoring are offered by orbital mechanics to solvers.  We have examined one of them and made an
original discovery but were unsuccessful in avoiding collisions between NEOs and our planet. Let's see if either of the other choices offer a solution to the Rock from the SkyRadial Thrust Vectors

As shown in Figure 5, the Radial direction for a ∆V vector points directly toward the sun ('radial-in') or away from the sun ('radial-out').   The effect of a radial-in ∆V is to rotate the whole orbital plane in the prograde direction without changing its shape or size.   That case is illustrated in Figure 6.

Also in Figure 6, we see that arrival times at the Inbound Intersection for both Earth and Égaré are delayed, which tends to cancel out the separation of their respective arrival times for avoiding collision.  This case strongly resembles the discouraging  results in Figure 3.  Likewise, the effect of a radial-out ∆V is to rotate the whole orbital plane in the retrograde direction without changing its shape or size.   That case strongly resembles the results in Figure 4.

Normal Thrust Vectors

The two non-pointing fingers of the right hand shown in Figure 5, represent the orbital motion of
Égaré.  The Normal direction for a ∆V vector, indicated by the thumb, points normal -- perpendicular -- to the orbital plane and produces the effect shown on the right side of Figure 7.  In particular, the orbital plane is tilted along the major axis of the ellipse causing a change in orbital inclination i.

Two orbital elements used by astronomers are shown in Figure 7.  For the case depicted in the sketch, the 'ascending node' passes through aphelion.  A ∆V vector in the other direction (opposite to the thumb) will tilt the plane so that the 'descending node' passes through aphelion.  Both cases offer an elegant solution to the Rock from the Sky puzzle...

The
Égaré orbit with a non-zero inclination does not intersect Earth's orbit. Ever!

Here then is our proposed solution...

 Deploy and maneuver a spacecraft capable of delivering a massive payload for station-keeping with asteroid Égaré, then apply its high energy detonation to produce a vectored thrust at aphelion to achieve a maximum possible vectored thrust ∆V normal to the orbital plane in the direction that assures an increase in whatever orbital inclination already exists.

...which is quite a long sentence -- but only a beginning...

The
Orbital Deflection puzzle will continue to use asteroid Égaré as a basis for formulating a mathematical model.  Puzzles with a Purpose will address some of the immense challenges made necessary for the protection of our planet from inevitable threats by Near Earth Object (NEOs), including...

• Size of the NEO: Égaré was postulated to be 100 meters in diameter.  That is comparable to the size of the asteroid responsible for the Tunguska Event in 1908, which flattened 2,000 km2.
• Mass and Velocity: Assuming a density of 5 gm/cm3, Égaré has a mass of 2.4×109 kg.  At the orbital intersections the velocity of Égaré is estimated to be V = 39.4 km/s.
• Kinetic Energy: Asteroid Égaré will have a kinetic energy of 2.0 ×109 kg-(km/s)2 at impact with planet Earth or 1,900×1015 Joules, equivalent to 450 megatons of TNT.
• Deflection Energy: A change in orbital inclination of 1/10 degree for asteroid Égaré requires as much as 173×109 Joules at aphelion where V = 6.7 km/s.
• Payload Mass: To deliver 173×109 Joules of energy to a rendezvous with Égaré at aphelion requires a spacecraft capable of lifting of tonnes of payload to escape velocity.
• Delivery Distance: The aphelion for Égaré is located at 3.0 EU = 450 Mkm from the sun.  More than a year must be allowed for delivery -- with gravity assist, say, from Mars,
• Prediction Accuracy: The mission demands extremes in precision, and line-of-sight tracking is limited.  The advanced placement of a transponder in orbit around Égaré will be necessary.
• Guidance for Thrust Vectoring: The transponder satellite will also be used to gauge features and tumbling motion on Égaré and to relay adjustments for deployment of the deflection tools.

Epilog

In his commentary on the Trampoline Deorbiting System my friend George McIlvaine added a paragraph entitled "The problem of large rocks falling from the sky."  Here is an excerpt...

Every few years we read about an astronomical near miss, usually after the fact   And the reports always mention that not all the asteroids have yet been identified.  So a sneaky pop-up asteroid intersecting Earth’s orbit with little warning when Earth happens to be at the intersection point has an estimated probability that is not reassuring.  Long term identification and deflection is not 100% effective, so are there other countermeasures?

That was the inspiration for the Astrogating Asteroids puzzle, in which one will find a minor discovery...

In order for Earth to collide with a non-coplanar NEO, the latus rectum of the asteroid's orbit must coincide with the diameter of Earth's orbit along the line of orbital intersection.

...which seems to grant immunity from orbital collisions to just about any non-coplanar NEO.

That set off a three-month web-based research effort leading to a proposal for what may be a unique deflection stratagem.  It is informally presented in the Rock from the Sky puzzle and includes this surprise...

 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!