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Asteroid Defense
Way of the Intercepting Fist
Dr. Jay Maron

Meteor Crater, Arizona

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.
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 be even more powerful.

Telescope   Diameter   Field of   Exposure    Full-sky    Year
            (meters)    view      (seconds)  survey time
                        (deg)                  (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      3.4e-11       7
Pan-STARRS     5e-18        24
Keck 10 meter  1e-19        28
Hubble         1e-20        31
Webb           5e-22        34

Deflection strategy

In the film "Armageddon", an asteroid is on course to hit the Earth and the plan was to deflect it with a hydrogen bomb. The scientists bickered over if it was better to detonate the bomb on the surface or from 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.

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.
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 recoil

Mass ejected by the explosion     =  m
Energy of the explosion           =  E
Momentum imparted to the asteroid =  Q  =  (2 m E)1/2
You want the nuclear explosion 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, generating 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

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.


Nuisance value of asteroids
                           Energy
                          (Joules)

Tornado                   2  ⋅1013    1 km in size, 180 m/s wind
Avalanche                 2  ⋅1015
Uranium bomb              4  ⋅1013    .01 Megatons of TNT equivalent
Hydrogen bomb             4  ⋅1016    10 Megatons of TNT equivalent
Asteroid, 100 meters      2  ⋅1017    20 km/s, 2 g/cm^3
Krakatoa volcano, 1883    8.4⋅1017
Tambora Volcano, 1815     3.3⋅1018    Largest eruption since Lake Taupo, 180 CE
Magnitude 9.5 earthquake  1.1⋅1019    Valdivia, Chile, 1960. Largest earthquake in the last century
Asteroid, 1 km            2  ⋅1020
Civilization energy/year  6  ⋅1020
Hurricane                 1  ⋅1021
Asteroid, 10 km           2  ⋅1023    Size of "dinosaur extinction" asteroid
Supernova                 1.2⋅1044
Gamma ray burst           1  ⋅1047

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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