
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 = V_{side} = V R / XThe earlier the asteroid is spotted, the larger the value of X and the less sideways speed is required to deflect it.
The PanSTARRS 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 Fullsky Year (meters) view (seconds) survey time (deg) (days) PanSTARRS 3 3.0 60 8 2010 Hawaii LSST 8.4 3.5 15 2 2021 El Penon, Chile Flux limit Magnitude (Watts/m^{2}) limit Human eye 3.4e11 7 PanSTARRS 5e18 24 Keck 10 meter 1e19 28 Hubble 1e20 31 Webb 5e22 34
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 VMomentum conservation:
M_{cannon} V_{cannon} = M_{ball} V_{ball} E_{ball} / E_{cannon} = M_{ball}V_{ball}^{2} / (M_{cannon}V_{cannon}^{2}) = M_{cannon} / M_{ball}If we assume that the cannon is vastly heavier than the ball then
M_{ball} << M_{cannon} E_{ball} >> E_{cannon}The cannonball gets all the energy.
E = .5 M_{ball} V_{ball}^{2}The momentum of the recoiling cannon is
Q_{cannon} = M_{ball} V_{ball} = (2 M_{ball} 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.
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: Halleytype comets LPC: long period comets"NearEarth 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)
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⋅10^{13} 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 = 10^{13} Joules/kg Energy of the nuclear explosion = E Momentum imparted to the asteroid = QThe 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 U^{2}) = 50000There is vastly more energy in the nuclear explosion than in the impact.
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.
F_{i} = Fraction of the mass of the spacecraft that impacts the asteroid F_{b} = 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 spacecraftZ 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 F_{b} + F_{i} = 1 gives Fb = Fi = 1/2Hence,
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
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
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 cAll 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.
Energy (Joules) Tornado 2 ⋅10^{13} 1 km in size, 180 m/s wind Avalanche 2 ⋅10^{15} Uranium bomb 4 ⋅10^{13} .01 Megatons of TNT equivalent Hydrogen bomb 4 ⋅10^{16} 10 Megatons of TNT equivalent Asteroid, 100 meters 2 ⋅10^{17} 20 km/s, 2 g/cm^3 Krakatoa volcano, 1883 8.4⋅10^{17} Tambora Volcano, 1815 3.3⋅10^{18} Largest eruption since Lake Taupo, 180 CE Magnitude 9.5 earthquake 1.1⋅10^{19} Valdivia, Chile, 1960. Largest earthquake in the last century Asteroid, 1 km 2 ⋅10^{20} Civilization energy/year 6 ⋅10^{20} Hurricane 1 ⋅10^{21} Asteroid, 10 km 2 ⋅10^{23} Size of "dinosaur extinction" asteroid Supernova 1.2⋅10^{44} Gamma ray burst 1 ⋅10^{47}
Heat capacity of air ~ 1.0$\cdot 103$ Joules/kg/Kelvin Mass of atmosphere ~ 5.1$\cdot 1018$ 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/sA 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 dinosaurextinction asteroid was ~ 10 km in size.
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 10^{40} 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 10^{70} 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.
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 2011CQ1 1.87 1 2011 2008TS26 1.96 1 2008 2011MD 2.94 10 2011 2012KT42 3.26 ~ 7 2004 Apophis 4.9 325 2029 510 2013DA14 5.35 30 2013 2012KP24 8.99 25 2004 2012BX34 10.3 8 2012 2012TC4 14.9 17 2012 2005YU55 60.00 400 2005
If you detonate the bomb at the center and if the asteroid is too large, gravity will bring the asteroid back together. For a uniformdensity 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