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Asteroid Defense
Dr. Jay Maron

Meteor Crater, Arizona
Way of the intercepting fist

The most effective way to deflect an asteroid is a hydrogen bomb and all other methods are far weaker. Hydrogen bombs are the best because they have the most energy per mass. A hydrogen bomb can deflect a 1 km asteroid if given a year's warning. To defend against asteroids, we should have a rocket with a hydrogen bomb in Earth orbit, ready to go if an asteroid is found. We also need wide-angle telescopes for detecting asteroids, such as the Pan-STARRS and LSST telescopes.


Asteroid damage

The smallest asteroid that we need to worry about is 50 meters, the minimum for getting through the atmosphere. Such an asteroid has the energy of a 10 Megaton fusion bomb. The minimum for creating a megatsunami is 200 meters. The LSST telescope will find all 200 meter and larger asteroids that are in near-Earth orbits, and then hydrogen bombs can redirect any that will impact the Earth. For asteroids from more distant regions of the solar system it won't find them soon enough to deflect them. For this we need more powerful telescopes.

The following table shows impact damage as a function of asteroid size.

Asteroid   Energy    Tsunami   Crater    Impact   Equivalent energy
diameter             height   diameter  interval
 meters    EJoules   meters      km      years

     8         .0001     0       0           5    Fission bomb, 25 kton TNT equivalent
    80         .100      0       1        3000    Fusion bomb, 25 Mton TNT equivalent
   200        1         10       3       20000    Krakatoa Volcano, 1883
   400       10         20       5      100000    Mag 9.5 quake. Chile, 1960.
  2000     1000        200      40     1000000    Hurricane
 10000   100000       4000     200   100000000    Asteroid that killed the dinosaurs
1 EJoule = 1018 Joules.
"Crater diameter" is for if the asteroid hits land and "Tsunami height" is for if the asteroid hits ocean.
"Impact interval" is the average number of years between asteroids strikes of that size.
Deflection strategy

In the film "Armageddon", an asteroid is on course to hit the Earth and the astronauts deflected it with a hydrogen bomb. The scientists bickered over if it was better to detonate the bomb on the surface or underground. The answer is that for fixed explosion energy, the asteroid recoil momentum increases with the mass ejected by the explosion, and so one should detonate the bomb underground.

If you have more than a year of warning before impact then there is enough time to send a manned spaceship to match speeds with the asteroid, land on it, and bury a hydrogen bomb underneath the surface.

If there isn't enough time to match speeds with the asteroid then one has to settle for detonating the bomb at the surface. A rocket is sent to intercept the asteroid and the bomb is detonated just before it hits. The detonation is delivered to the side of the asteroid to give it a sideways deflection. The deflection can be amplified by crashing an object into the asteroid just before the bomb detonates, which kicks up a cloud of material and increases the mass ejected by the bomb. In the appendix below we calculate the momentum delivered to the asteroid as a function of bomb energy.


Early warning

Large Synaptic Survey Telescope (LSST)

The Pan-STARRS telescope specializes in finding asteroids. It has a wide field of view and takes short exposures, allowing it to cover the entire sky in 8 days. The upcoming Large Synaptic Survey Telescope (LSST) will cover the sky every 2 days.

Telescope   Diameter   Field of     Exposure    Sky survey    Year
            (meters)   view (deg)   (seconds)   time (days)

Pan-STARRS    3          3.0          60           8          2010    Hawaii
LSST          8.4        3.5          15           2          2021    El Penon, Chile


             Flux limit    Magnitude
             (Watts/m2)    limit

Human eye      3e-11         7
Pan-STARRS     5e-18        24
LSST           2e-18        25
Keck 10 meter  1e-19        28
Hubble         1e-20        31
Webb           5e-22        34

Asteroid recoil

We calculate the asteroid recoil momentum as a function of hydrogen bomb energy.

Mass ejected by the explosion     =  m
Energy of the explosion           =  E
Momentum imparted to the asteroid =  Q  =  (2 m E)½
You want the bomb to eject as much mass from the asteroid as possible. If you detonate the bomb from above the surface, most of the explosion energy goes into space without ejecting any mass. You can increase the mass ejected by arranging for an impactor to hit the asteroid just before the bomb arrives. The impactor blasts material into space and the nuclear explosion heats that material, giving an impulse on the asteroid.

Fi = Fraction of the mass of the spacecraft that impacts the asteroid
Fb = Fraction of the mass of the spacecraft that is a hydrogen bomb
Z  = Mass of material ejected by the impactor / Mass of the impactor
e  = Energy per mass of the hydrogen bomb
W  = Total mass of spacecraft
Z is a dimensionless number that parameterizes how effective the impactor is at ejecting material. We can expect that Z > 100 and it can conceivably be much larger.

The detonation should maximize Q/W

Explosion energy                = W Fb e
Mass ejected from the asteroid  = W Fi Z

Q/W  =  (2 W Fb e W Fi Z)^(1/2) / W
     =  (2 e Z Fb Fi)^(1/2)

Maximizing  (Fb Fi)1/2  subject to the constraint  Fb + Fi = 1  gives

Fb = Fi = 1/2
Hence,
Q/W ~ (e Z)^(1/2)
Setting e = 1e13 and Z = 100,
Q/W ~ 3e7  meters/second


Strategy                            Q/W
                                   (m/s)

Hydrogen bomb with impactor         3e7
Hydrogen bomb without impactor      3e6
Impactor without hydrogen bomb      3e5

Calculations

Deflection strategy

A cannon provides an example calculation.

Energy   =  E
Mass     =  M
Velocity =  V
Momentum =  Q  =  M V
Momentum conservation:
Mcannon Vcannon  =  Mball Vball

Eball / Ecannon  =  MballVball2 / (McannonVcannon2)  =  Mcannon / Mball
If we assume that the cannon is vastly heavier than the ball then
Mball << Mcannon

Eball >> Ecannon
The cannonball gets all the energy.
E  =  .5 Mball Vball2
The momentum of the recoiling cannon is
Qcannon  =  Mball Vball  =  (2 Mball E)1/2
For fixed gunpowder energy, cannon recoil increases with cannonball mass. This suggests that if you want to deflect an asteroid by detonating a hydrogen bomb that you should arrange for as much material to be ejected from the asteroid as possible. If you can land on the asteroid then you want to bury the bomb before detonating it. If you can't land on the asteroid then you can arrange for an impactor to eject material before detonating the bomb.
Deflection

Suppose an asteroid is on a collision course with the Earth and we deflect it by giving it sideways speed. To estimate the required speed,

Distance from Earth when the asteroid is spotted   =  X
Velocity of the asteroid                           =  V
Time for the asteroid to reach Earth               =  T  =  X / V
Radius of the Earth                                =  R
Sideways speed required to deflect the asteroid    =  Vside =  V R / X
The earlier the asteroid is spotted, the larger the value of X and the less sideways speed is required to deflect it.
Asteroid impact speed

Velocity distribution of near Earth asteroids

Near Earth asteroids (NEA) approach the Earth at a characteristic speed of ~ 20 km/s. Retrograde comets can approach the Earth as fast as 75 km/s.

NEA: near Earth asteroids
SPC: short period comets
HTC: Halley-type comets
LPC: long period comets
"Near-Earth object velocity distributions and consequences for the Chicxulub impactor" S. V. Jeffers, S. P. Manley, M. E. Bailey, D. J. Asher, Mon. Not. R. Astron. Soc. 327, 126–132 (2001)
Impact energy

If you want to land a spaceship on the asteroid you need at least a year of maneuvering to match speeds with it. If you don't have this much time, then the next best strategy is to send out a hydrogen bomb from the Earth and arrange for the bomb to detonate just before it hits the asteroid. The detonation is arranged to give the asteroid sideways momentum.

The theoretical maximum energy density of a fusion bomb is 2.4⋅1013 Joules/kg, and in practice the energy density is half this.

Suppose you send a spacecraft to intercept the asteroid.

Speed of asteroid                  =  U  = 20 km/s
Speed of the spacecraft            =  u
Mass of the spacecraft             =  m
Energy density of a hydrogen bomb  =  e  = 1013 Joules/kg
Energy of the nuclear explosion    =  E
Momentum imparted to the asteroid  =  Q
The spacecraft hits the asteroid at a speed of u+U. Typically, u << U, and so we can approximate the collision velocity as U.

If the spacecraft is a hydrogen bomb, then

Energy of the hydrogen bomb / Kinetic energy of the spacecraft
  = e m / (.5 m U2)
  = 50000
There is vastly more energy in the nuclear explosion than in the impact.
Asteroid deflection

Suppose an asteroid is on course for a direct hit on the Earth and we're going to deflect it with a hydrogen explosion.

V  =  Speed of the asteroid on its way to the Earth
   =  20 km/s  (a typical value)
v  =  Sideways speed delivered to the asteroid by the hydrogen bomb
M  =  Mass of asteroid
m  =  Mass of material ejected by the hydrogen bomb
R  =  Radius of the Earth
   =  6.371e6 meters
E  =  Energy provided by the hydrogen bomb
e  =  Energy/mass of the hydrogen bomb
   =  1e13 Joules/kg
Z  =  Mass of material ejected / Mass of spacecraft
   ~  100
D  =  Distance of the asteroid from the Earth when the hydrogen bomb detonates
T  =  Time between the hydrogen bomb detonation and when the asteroid reaches the Earth
      detones
   =  D / V

From the cannonball calculation,

M v = W (e Z)^(1/2)

Deflecting the asteroid requires

v T  >  R

W (e Z)^(1/2) T / (M R) >  1

(eZ)^(1/2) / R  ~  5


To deflect the asteroid the spacecraft must have a mass of at least

W  >  .2 M/T


If an asteroid has
Size  =  1 km
Mass  =  1e12 kg
T     =  1 month
then

W  =  77 tons

Propulsion

VASIMR ion drive
Nuclear thermal rocket
Orion fusion rocket

                                     Exhaust
                                     (km/s)
Hydrogen+oxygen rocket                  5
Dawn ion drive                         31
VASIMR ion drive                       50
Nuclear thermal rocket, H2 exhaust      9       NERVA design
Nuclear thermal rocket, H2O exhaust     1.9     NERVA design
Solar thermal rocket, H2 exhaust        9
Solar thermal rocket, H2O exhaust       1.9
Orion fusion rocket                 10000
Antimatter rocket                 ~ 1/2 c
All of these rockets are possible with current technology except for the antimatter rocket.

Ion drives cannot move heavy objects because of their low thrust.

If a thermal rocket can operate at a temperature high enough to dissociate H2 into elemental hydrogen, larger exhaust speeds are possible.

The performance of a solar thermal rocket depends on its proximity to the sun. Nuclear thermal rockets work everywhere.


How much does an asteroid impact heat the atmosphere?

Heat capacity of air ~ 1.0⋅103  Joules/kg/Kelvin
Mass of atmosphere  ~  5.1⋅1018 kg

Let F = the fraction of the asteroid's kinetic energy that goes into heating the
atmosphere. The atmospheric heating is

                                Mass of asteroid      Speed of asteroid
Heating  ~  40 kelvin  *  F  *  ----------------  * ( ----------------- )^2
                                    10^15 kg               20 km/s
A 10 km asteroid has a mass of ~ 10^15 kg. If the asteroid is less massive than this then you don't have to worry about cooking the atmosphere. The dinosaur-extinction asteroid was ~ 10 km in size.
Cosmic disasters

The collision betwen the Milky Way and Andromeda galaxies, 4 billion years from now.
A maurading star disrupts the orbit of the Earth.
In 5 billion years, the sun will explode in a nova and consume the Earth.
The moon is spiraling outward and in 1 billion years will be stolen by the sun. After this, the Earth will have no defense against angular momentum and its spin orientation will start to drift.
The ice caps melt and the Earth overheats.
Protons probably decay and the half life has been theorized to be in the range of 1040 years. After this, the universe will consist of electrons, positrons, and neutrinos.
If the masses of the Higgs boson and the top quark have unfavorable values, then the universe is unstable to vacuum decay. This would destroy the entire universe without warning.
Black holes emit radiation by the Hawking mechanism. In 1070 years they will have radiated all their mass and will end their lives in an explosion of gamma rays.
In 10 billion years the dark energy will expand the universe, leaving behind only the galaxies of the local group.
If dark energy has an unfavrable equation of state then the universe will end in a "big rip", where all matter is shredded into its fundamental particles.


Asteroids that have passed close to the Earth

Q = Radius of closest approach / Radius of Earth

                  Q    Diameter  Date    Energy
                       (meters)         (Mtons TNT)
Chelyabinsk      1.0      19     2013      .44
Tunguska         1.0      50     1908    12         Flattened a forest
Arizona asteroid 1.0      50   -50000    10         1 km crater
1972 Fireball    1.0089  ~ 6     1972               Skimmed the upper atmosphere
2011-CQ1         1.87      1     2011
2008-TS26        1.96      1     2008
2011-MD          2.94     10     2011
2012-KT42        3.26    ~ 7     2004
Apophis          4.9     325     2029   510
2013-DA14        5.35     30     2013
2012-KP24        8.99     25     2004
2012-BX34       10.3       8     2012
2012-TC4        14.9      17     2012
2005-YU55       60.00    400     2005

Problems

If you detonate the bomb at the center and if the asteroid is too large, gravity will bring the asteroid back together. For a uniform-density sphere,

Gravitational energy = .6 G Mass^2 / R

Suppose the hydrogen bomb has the energy of 10 megatons of TNT, which is 4*10^16 Joules. What would you estimate is the largest value for the radius of an asteroid that this bomb can shatter?

In "Star Wars", a Death Star shatters a planet. If the planet is identical to the Earth, how much energy does this take? If the energy were provided by a sphere of antimatter with the density of iron, what is the radius of this sphere?


Mr Miyagi: Best block... not be there


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