Orbital Deflection
  Version 1.1
Copyright ©2017 by Paul Niquette. All rights reserved.

Size comparisons of various asteroids: Barringer (50 m), Chelyabinsk (17 m), and the largest meteorite ever found Hoba (2.7 x 2.7 x 0.9 m). Courtesy of Wiki-Commons.

The near-coplanar asteroid named Égaré (French for 'stray') was informally studied in the Rock from the Sky puzzle.  Telescopic observations made before the summer of 2018 predicted its collision with Earth at an orbital intersection on Thursday July 21, 2022.  With a diameter of 100 meters, Égaré is eight times more massive than the bolide that produced the Barringer meteor crater some 50,000 years ago.  Égaré, with a mass mA = 2.4x10⁹ kg, ranks alongside the meteoroid that caused the Tunguska Event -- the largest impact in recorded history.

For general audiences, sad to say, it is customary to describe large quantities of energy in terms of the explosive power of TNT. Accordingly, the energy released that day over Tunguska has been estimated to be 15 megatons of TNT (15 MtnTNT), which is 1,000 times greater than either atomic bomb dropped in World War II.  From here on, we shall prefer Joule as our unit of energy from the International System of Units (SI), according to which 1 MtnTNT = 4.2x10¹⁵ Joules.

Figure 1 in the Rock from the Sky puzzle, indicates that when Égaré crosses Earth's orbit, it will be traveling at V = 39.4 km/s = 39.4x10³ m/s.  At impact, the asteroid's kinetic energy will be given by mA V² /2 ≈ (2.4x10⁹)(39.4x10³)² /2 ≈ 1.86x10¹⁸ joules ≈ 444 MtnTNT.  Depending on the angle that Égaré enters Earth's atmosphere, much of that energy will be dissipated in the atmosphere or exploded into fragments aloft.

In the Rock from the Sky puzzle, we discovered that some conventional proposals for protecting Earth from a collision with a near-coplanar asteroid do not work.  Specifically...

Applying a vectored thrust to delay or advance the arrival of the asteroid at an inbound intersection -- with a tangential ∆V in either the prograde or retrograde direction -- will change the shape of its orbit so that the Earth will likewise arrive earlier or later at the new location for the inbound intersection, which can defeat the avoidance altogether!

The asteroid Égaré will continue to be used in our model here in the Orbital Deflection puzzle as an informal study of some of the complexities that arise in protecting our planet from threats by a Near Earth Object (NEO) in a near-coplanar orbit (inclination ≈ 0 degrees). 

Worldwide endeavors for avoiding meteor impacts can be divided into three overall tasks...

1. Detection of potential 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.

The Orbit Deflection puzzle emphasizes Task 3, which comprises the most difficult challenges. 

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Detection

• Size of the NEO: Égaré is postulated to be 100 meters in diameter.  That is comparable to the size of the asteroid responsible for the Tunguska Event, 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 each orbital intersection 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 1.86×10¹⁸ Joules, equivalent to 444 megatons of TNT.

These topics are especially pertinent to the Detection of Near Earth Asteroids (NEAs).  Finding and cataloging NEAs has been going on since antiquity.  Worldwide efforts received a significant boost in 1992 by governmental mandates. 

Based on one discovery in the Astrogating Asteroids puzzle, the most important parameter for an NEA is the inclination of its orbit with respect to that of Earth. Nevertheless, inclination seems to be buried in listings and not mentioned at all in a key description of MOID (Minimum Orbital Intersection Distance).

There are many projects and missions now actively participating in the crucial Detection business.   Today, countless astronomers, both professional and amateur, are using advanced instruments and a growing fleet of robotic telescopes atop remote mountaintops.  They are sharing their findings on worldwide databases, listing and updating tens of thousands of entries.  Moreover, at this writing, a total of 875 named entries are being tracked as Large NEAs (> 1 km in diameter).

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Prediction

• Accuracy: The Égaré mission demands precision in Prediction, but line-of-sight tracking is limited, which necessitates placement of a transponder in orbit around the asteroid.
• Guidance for Thrust Vectoring: As sketched below in Figure 1, the transponder satellite will also provide up-close videos of Égaré to gauge its features and tumbling motion.  The satellite must relay positioning adjustments for final deployment of the Deflection tools. 

During much of its orbit Égaré is not optically visible from Earth because of daytime sky-glow. Satellites in Earth orbit, particularly Hubble, can track Égaré optically; however, the demand for frequent updates and extremes in precision must make use of a separate transponder to be put into position as soon as practicable -- but not necessarily landed on the surface.

Typically an asteroid like Égaré is rotating on its own axis normal to the orbital plane with a period of two hours.  That favors the placement of the transponder satellite in a synchronous polar orbit around Égaré, with its transceiver antenna pointed constantly to Earth.

The transponder satellite will have an orbital period given by...

τS = (rS)^3/2 / (2π G mA)^1/2, where...

τS = satellite's period in seconds
rS = satellite's orbital radius in meters
mA = mass of asteroid Égaré = 2.4×10⁹ kg

G = gravitational constant
       6.674×10^-11 m³ kg_-1 s^-2

For an orbital radius rS = 373 meters, τS ≈ 2 hours.

Meanwhile, at aphelion (3.0 AU), Égaré can be as far as 4.0 AU from Earth (600×10⁶ km), so the round-trip delay at the speed of light (300×10³ km/s) will be as long as 2×600×10⁶ / 300×10³ or more than an hour.  Some solvers may enjoy designing a protocol for 'go-round' messaging.

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Deflection

• Energy: A change in orbital inclination of 0.1 degree for asteroid Égaré requires as much as 173 ×10⁹ Joules of propulsion energy at aphelion where V = 6.7 km/s.
• Payload Mass: To deliver Deflection tools plus 173 ×10⁹ Joules of energy to a rendezvous with Égaré requires a spacecraft capable of lifting 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.
  

As discovered in the Rock from the Sky puzzle the preferred avoidance maneuver for Égaré calls for a vectored thrust of ∆V Normal to the orbital plane of the asteroid, increasing its inclination angle by ∆i.  The Right- Hand Rule is reprised here in Figure 3.  A most relevant relationship applies the law of sines...

sin (∆i / 2) = (∆V / 2) / V; thus...
∆V = 2 V sin (∆i / 2), where...

∆i = specified as increase in orbital inclination
∆V = required change in the asteroid's velocity
V = orbital velocity of Égaré at the instant the thrust is applied to produce ∆V.

In Figure 1 above, we see that the orbital intersections are about 0.75 AU (112.4 Mkm) from the major axis of the Égaré orbit.  The Deflection of the asteroid when it passes within or beyond Earth's orbit will be 0.75 sin ∆i.  Thus, if we specify that ∆i = 1/10 degree, Égaré will miss Earth by 0.75 sin (1/10) = 0.0013 AU = 262,000 km (163,000 mi) ≈ 120 Earth diameters.  Hooray for that.

The expression derived above, ∆V = 2 V sin (∆i / 2), shows that for a specified increase in orbital inclination angle ∆i, the required ∆V is directly proportional to the asteroid's tangential velocity V.  At perihelion V = 80.9 km/s, and at aphelion V = 6.7 km/s.  That gives us up to a 12-to-1 advantage in producing ∆V if we apply the vectored thrust at aphelion.  For ∆i = 1/10 degree, ∆V = 2×6.7 sin (1/20) = 0.0117 km/s = 11.7 m/s.

The change in the velocity of Égaré with its mass of 2.4×10⁹ kg by ∆V = 11.7 m/s means a change in kinetic energy of (2.4×10⁹)(11.7)²⁸ /2 = 164×10⁹ Joules.  Let's do just that.

Answering the most challenging question is left to solvers of the Orbital Deflection puzzle...

What is your proposal for how the worldwide community of nations might deploy requisite resources to deflect the orbit of asteroid Égaré thereby avoiding its collision with Planet Earth on Thursday, July 21, 2022?

Before clicking to the Solution page, you are invited to review the concepts and issues for avoiding collisions with asteroids on the worldwide web, starting at this portal.  Any avoidance strategy -- whether by destruction or deflection -- calls for the delivery by spacecraft of immense amounts of energy to Égaré.  With no atmosphere surrounding Égaré, the use of chemical energy would require us to include a large amount of oxidizer in our Deflection tool-kit as well as fuel. 

Explosives like TNT come to mind, but -- hey, to deliver 164×10⁹ Joules164 × 109 Joules for a Deflection strategy, our spacecraft needs to include 390 kilotons of TNT in its payload!

Using a thermonuclear explosion for Deflection thus may be the most 'spaceworthy' choice, inasmuch as the energy of 164×10⁹ Joules can be transported in a fission/fusion device weighing 390 kilograms -- that's one-thousandth of 390 kilotons of TNT!  However...

Explosions in space -- chemical or nuclear -- are rather tame.  Explosions in space produce heat energy not mechanical energy.  It is mechanical energy that we really need for Deflection. 

Truth be known: An explosive weapon on Earth makes full use of surrounding atmospheric mass (1.5 kg/m³).  Detonation and immense heat cause sudden over-pressures and winds capable of destroying structures, while high temperatures ignite destructive fires.  Indeed, the temperature can get so high that air itself is set ablaze (nitrogen + oxygen → NOx) in a fireball.  And a mushroom cloud.

In the vacuum of space, mechanical energy (some force f applied through through some distance x) is hard to come by.  Think rocketry -- for which one must expel a huge mass m of hot gases at high speed from a 'nozzle' to produce a reactive force f, which we call thrust.

So, does that mean we must haul some huge mass m along with immense amounts of energy from Earth all the way to Égaré in our Deflection tool-kit?   Maybe not. 

Our objective really comes down to this: Figure out a way to use part of the asteroid's own mass to produce the requisite ∆V Normal of 11.7 m/s.   That won't be easy.

In Figure 4, we see that a simple surface detonation will release most of its energy as line-of-sight radiation directly into space.  A small fraction of the explosive energy may find some usefulness in heating up a spot on the surface of Égaré.  The sudden high temperature -- thousands of degrees -- will dispatch shock waves and heat flows into the asteroid, blasting and ablating deeper and deeper, melting metal and fracturing rock.  But as for vectored thrust -- not so much.

The thermal energy will be 'processed' by physical phenomena acting on the materials encountered along the way: conduction & expansion, melting & vaporization, fracturing & ejection -- something like an inside-out volcano. 

Depending on the mass and velocity of the remedial ejecta, an appropriately vectored orbital Deflection may actually be achieved.  If not, mankind will have succeeded merely in scarring the surface of the asteroid with a glassy crater, while temporarily warming the interior of a near-earth object that is still predicted to impact Earth on Thursday, July 21, 2022.

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