Asteroid Defense: Way of the Intercepting Iceberg

Meteor Crater, Arizona
Dinosaur extinction

Velocity distribution of near Earth asteroids and comets

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)

Nuisance value of asteroids
Energy (Joules) Uranium bomb, 10 kTon TNT 4 ⋅1013 Hydrogen bomb, 10 MTon TNT 4 ⋅1016 100 meter asteroid 2 ⋅1017 20 km/s, 2 g/cm^3 Apophis asteroid, 270 meters 2 ⋅1018 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 1 km asteroid 2 ⋅1020 20 km/s, 2 g/cm^3 10 km asteroid 2 ⋅1023 20 km/s, 2 g/cm^3. Size of "dinosaur extinction" asteroid

Galileo (Jupiter)
Cassini (Saturn)
Messenger (Mercury)
Voyager 1 & 2
New Horizons (Pluto)
Dawn (Vesta & Ceres)
Each of these missions is powered by chemical rockets except for Dawn, which is powered by an ion drive. Ion drives require fewer gravitational assists than chemical rockets.
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.

Energy for propulsion
Hydrogen + Oxygen 1.4⋅1010 Joules/ton Uranium-235 8.0⋅1016 Joules/ton Solar energy 1.4⋅1015 Joules. 1 km^2 collector operating for 10^6 seconds at 1 A.U. The solar energy collected by a 1km mirror at 1 A.U. over a time of 10^6 seconds (2 weeks) is Energy ~ 1400 Watts/m^2 * 10^6 m^3 * 10^6 seconds ~ 1.4⋅1015 Nuclear power and solar power can move large objects. Chemical energy cannot.
Saturn V rocket
The Saturn V rocket, the largest ever built, can launch ~ 120 tons of uranium into low Earth orbit.
How long would it take an alien spaceship to reach the Earth?
Distance to Alpha Centauri, the nearest star 4.3 light years Milky Way diameter 0.1 million light years Distance to Andromeda 2.5 million light years Distance to the Virgo Supercluster 54 million light years Light travel time during age of universe 13750 million light years Age of the universe ~ 13.75 billion years Age of the Earth ~ 4.54 billion years An alien civilization in the Virgo Supercluster with a 1 billion year head start on us has plenty of time to get here.

Getting around the solar system
Lagrange points
Earth Lagrange points
Interplanetary transport network

Deflection strategy
Suppose a spacecraft is launched from the Earth to collide with an asteroid, and suppose that the asteroid is struck from the edge to give it a sideways momentum. V = asteroid velocity toward Earth v = velocity of spacecraft toward asteroid M = mass of asteroid m = mass of spacecraft X = initial distance of asteroid x = distance from Earth of collision S = sideways momentum gained by the asteroid D = amount of deflection when the asteroid reaches the Earth E = (1/2) m v^2 = energy used to propel the mass toward the asteroid If the spacecraft is slower than the asteroid, v << V The distance from the Earth to the collision point x is x v --- ~ --- X V The sideways momentum gained by the asteroid is S ~ m V The amount the asteroid is deflected when it reaches the Earth is D S m --- ~ ---- ~ --- x MV M The speed of the deflector (v) in terms of the propulsion energy (E) is E ~ m v^2 The deflection (D) in terms of the propulsion energy (E) is v m D ~ --- --- X ~ m^(1/2) E^(1/2) X / (M V) [Equation 1] V M For fixed propulsion energy, the deflection scales as m^(1/2). The larger the deflector mass the better. Given the difficulty of lauching material from the Earth into space, this favors a strategy of finding material from space.
Ice can serve as a deflector. Ice can serve as the exhaust for a thermal rocket. The asteroid Ceres has abundant ice. A thermal rocket can easily lift material from Ceres into space. The Dawn spacecraft will arrive at Ceres in February 2015. Mass H2O H2O density Radius Escape (kg) (kg) frac (g/cm^3) (km) speed (km/s) Earth 5.97e24 1.4e21 .00025 5.52 6378 11.2 Ceres 9.43e20 2.0e20 .21 2.08 487 .51
21 Lutetia (metallic asteroid)
Ceres (icy)

How much of the iceberg should be used for propulsion exhaust?
m_iceberg = Initial mass of iceberg m_e = Mass of iceberg used as exhaust m_f = Final mass of iceberg after operating the rocket E = Energy available for propulsion v_e = Exhaust velocity v_f = Final velocity of iceberg after operating the rocket m_iceberg = m_e + m_f Case 1: Use only a small fraction of the iceberg for propulsion. m_e << m_f Conservation of momentum: m_e v_e ~ m_f v_f v_e >> v_f m_e v_e^2 >> m_f v_f^2 The kinetic energy of the exhaust is much larger than the kinetic energy of the final iceberg. E ~ m_e v_e^2 The final speed of the iceberg is m_e v_f ~ v_e ----- ~ E^(1/2) m_e^(1/2) / m_f m_f The larger the exhaust mass, the better. Case 2: Use most of the iceberg for propulsion. m_e >> m_f The Tsoilkovsky rocket equation is Rocket speed ~ Exhaust speed * ln(Initial mass of rocket / Final mass of rocket) v_f ~ v_e * ln(m_i / m_f) The deflection imparted to the asteroid scales as (from Equation 1 above) Deflection ~ m_f v_f ~ m_f v_e * ln(m_i / m_f) Using more of the iceberg for exhaust gives you a larger final velocity but there is a point of diminishing returns reflected in the logarithm. For maximum deflection, the exhaust mass and the final mass should have the same magnitude. m_e ~ m_f The exact choice for (m_f/m_i) depends on particulars about the iceberg's orbit around the Earth. The more icebergs you have in orbit, the better.
How fast will the iceberg move?
Suppose you have have an iceberg handy in Earth orbit. Use 1/2 of the iceberg as rocket exhaust and the other half to impact the incoming asteroid. From the rocket equation, v_f ~ v_e * ln(m_i / m_f) If m_i ~ m_f v_e ~ 2 km/s then the deflector will move at ~ 1.3 km/s. Energy ~ Mass * Speed^2 With an propulsion energy of 10^16 Joules, you can move an iceberg of mass ~ 10^8 kg.
How much is the asteroid deflected?
m = Iceberg mass v = Iceberg velocity D = Amount the asteroid is deflected when it reaches Earth E = Energy for iceberg propulsion X = Distance of asteroid when the iceberg is launched From above, the iceberg velocity is v ~ 1.3 km/s D ~ m^(1/2) E^(1/2) X M^(-1) V^(-1) [Equation 1] X E 20 km/s 10^12 kg 2 km/s D ~ 0.4 * (Radius of Earth) * ------------ * ------------ * -------- * -------- * -------- 10^10 meters 10^16 Joules V M v A solar thermal rocket can deflect a 1 km asteroid if discovered at a distance of ~ 1/10 A.U.
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.
Minimum iceberg mass
To do justice to the propulsion energy, the iceberg needs a minimum mass of Propulsion energy (Max useful velocity)^2 Mass ~ 5*10^9 kg ------------------- ------------------------- 10^16 Joules (2 km/s)^2 A 200 meter iceberg can deflect a 1 km asteroid.
Fetching the iceberg
Escape speed Jupiter 59.5 Earth 10.2 Mars 5.03 Moon 2.38 Ceres .51 Vesta .36 Pallas .32 Hygiea .21 Gravity assists can change a trajectory by of order the escape speed. You can use a sequence of gravity assists like a billiards-style trick shot to move objects around the solar system, requiring only nudges between assists. This is the "interplanetary transport network". The energy required to fetch an iceberg is Mass of iceberg (Change in velocity)^2 Energy ~ 5*10^16 Joules ----------------- ------------------------ 10^10 kg (1 km/s)^2 A 1 km solar thermal rocket can move a 200 meter iceberg from the asteroid belt to the Earth.
Mirror mass
Surface area Thickness Density Mirror mass ~ 8*10^5 kg -------------- ----------- ---------- 1 km^2 10^-4 m 8 g/cm^3 A solar thermal rocket capable of delivering ~ 10^16 Watts can be built from a ~ 10 meter metallic asteroid.
Global sunscreen
The Earth's temperature is given by a balance between solar heating and blackbody radiation. I = Intensity of solar radiation T = Earth temperature dI 4 dT I ~ T^4 --> ---- ~ ------ I T Changing the Earth's temperature by 1 Kelvin requires dI/I ~ .013, corresponding to a sunscreen with area ~ .013 * 1.27e14 ~ 1.7e12 meters, or (1300 km)^2. The mass of an orbital sunscreen is Surface area Thickness Density Mass ~ 1.4*10^12 kg -------------- ----------- --------- (1300 km^2) 10^4 m 8 g/cm^3 An orbital sunscreen can be built from a ~ 1 km metallic asteroid. The energy required to retrieve this asteroid from the asteroid belt is ~ 10^18 Joules, easily achievable by a solar thermal rocket.
Climate design
A above scaling is for a brute force sunscreen. With a gyroscope-equipped fleet of orbital mirrors, finesse allows us to get by with a smaller sunscreen. For example, sunlight that would have fallen on equatorial mountains can be redirected to promote arctic agriculture, and snow can be encouraged at the poles. It may even be possible to tame hurricanes and turn them into tourist attractions. This presents an opportunity for global cooperation.
An orbital mirror can power a steam engine and the power can be beamed with microwaves to the Earth. The power used by civilization is ~ 2e13 Watts. A square mirror 100 km on a side collects ~ 1.4e13 Watts of sunlight.
Earth without CO2
If the Earth is an ideal blackbody, Solar heating ~ Blackbody radiation ~ Stefan_Boltzmann_constant * Area * Temperature^4 1361 Watts/m^2 * pi * Earth_radius^2 ~ 5.67e-8 W/m^2/K^4 * 4 pi * Earth_radius^2 * Temperature^4 --> Temperature ~ 278 Kelvin The Earth's average temperature is 287 Kelvin ( The sun's luminosity is increasing and in ~ 1 billion years the oceans will boil off even if the Earth has zero CO2. In ~ 5 billion years the sun will become a red giant and consume the Earth.
Artificial gravity
If artificial gravity is generated by spinning a spaceship, the spin period has to be at least 30 seconds for the inhabitants to not get dizzy. At 1 g this corresponds to a spin radius of ~ 250 meters.

Cosmic rays
Annual radiation on the Earth ~ 1 millisieverts. Annual radiation on the space station ~ 350 millisieverts. Annual radiation on a space station with 4 tons/meter^2 of radiation shielding ~ 2.5 millisieverts. Earth atmospheric thickness ~ 10 tons/meter^2 Mars atmospheric thickness ~ .2 tons/meter^2 To shield astronauts from cosmic rays you need ~ 10^4 tons of material. This is too much to launch into space from the Earth. Ice from space can be used for radiation shielding.
Jeet Kune Do: "Way of the intercepting fist"