
A rocket generates thrust by burning fuel and channeling the exhaust with a rocket cone.
To reach orbit you need a velocity of 7.8 km/s. A onestage rocket isn't enough and so multiple stages are used.
New advances in rocketry have been made by SpaceX, which pioneered the methane rocket and a reusable first stage, and by Stratolaunch, which pioneered highaltitude launch.
The fuel that generates the fastest exhaust is hydrogen+oxygen and this is usually used for the upper stages. It can't be used for the first stage because of liquid hydrogen's low density. The first stage traditionally uses kerosene, and SpaceX's new methane rocket offers an improvement because of methane exhaust is faster than kerosene exhaust.
Fuel Exhaust Fuel Fuel boiling speed density point (km/s) (g/cm^3) (K) Hydrogen 4.4 .07 20.3 Complex because of the low boiling point of hydrogen Methane 3.7 .42 111.7 New technology Kerosene 3.3 .80 410 Simple because kerosene is a liquid at room temperature Solid fuel 2.7 1.2  Simple and cheapKerosene is a liquid consisting of hydrocarbon chains with 6 to 16 carbon atoms per chain. It is similar to gasoline.
Vulcan Aerospace is developing the Stratolaunch aircraft to launch rockets from high altitude, yielding several advantages over ground launch.
Speed (km/s) Earth rotation at equator .46 Stratolaunch aircraft .27 Earth orbit speed 7.8The rocket will be launched from near the equator so that it benefits from both the speed of the equator and from the speed of the aircraft.
The Stratolaunch will fly at 14 km where the air has 1/4 the density as at sea level, giving it an edge over ground launch. A rocket launched from the ground has to respect the atmosphere by first going up before it can go sideways. A rocket launched from an aircraft can go sideways immediately, for a big savings in fuel.
The stratolaunch aircraft has 6 engines for a total thrust of 150 tons and it can carry a 230 ton rocket.
In the future, a ramjet aircraft will be built to launch rockets from a speed of Mach 5 and an altitude of 100 km. Since ramjets only work above Mach 1, a detachable solid rocket booster will have to be used to get the aircraft to Mach 1. The aircraft fuselage will be a shell that will contain the rocket, which will be deployed from the aft of the aircraft.
Ramjets work up to Mach 5 and scramjets are required above this speed. Ramjets are a mature technology and scramjets are an emerging technology.
km/s Mach Earth rotation at equator .46 1.6 Stratolaunch aircraft .27 .9 Ramjet aircraft 1.50 5.0 Earth orbit speed 7.8 20.4 Exhaust speed of HOX fuel 4.4 14.9 Effective exhaust speed of a ramjet 12 40.7 (from specific impulse)
We compare the payloads of an Airbus A380 aircraft and a Falcon 9 rocket.
Airbus empty mass = 276.8 tons Airbus engine mass = 25.1 tons Airbus passenger mass = 100 tons Airbus fuel mass = 200 tons Airbus max takeoff = 650 tons Falcon total mass = 570 tons Falcon stage 1 full mass = 506 tons Falcon stage 1 engine mass= 5.7 tons Falcon stage 2 full mass = 52 tons Falcon stage 2 engine mass= .6 tons Falcon stage 2 empty mass = 3.1 tons Falcon payload = 13.2 tons Airbus payload fraction = .2 = 100 tons of passengers / 500 tons of aircraft Falcon payload fraction = .023 = 13.2 tons of payload / 570 tons of rocket Airbus payload cost = 4 $/kg Falcon payload cost =4100 $/kg
Reentry spacecraft:
Space shuttle empty mass = 78.0 tons, 7 crew. Launch rocket = 2030 tons SpaceX Dragon V2 = 4.2 tons, 7 crew Soyuz reentry module = 2.9 tons, 3 crew ISRO Reentry Vehicle = 3.7 tons, 3 crewThe space shuttle was senseless because there's no point in bringing unnecessary mass back to the Earth. A reentry spacecraft can be as lightweight as 1 ton/person.
If the rocket fails during launch and the crew are in a lightweight reentry spacecraft, they have the potential to survive.
The moon has ice which can be turned into hydrogen+oxygen rocket fuel using solar power. Ice can be brought into space more cheaply from the moon than from the Earth.
Ceres is the largest asteroid in the asteroid belt and it has abundant ice.
Orbit speed Atmosphere Distance from (km/s) density (kg/m^{3}) sun (AU) Earth 7.8 1.22 1.00 Mars 3.6 .020 1.52 Moon 1.68 0 1.00 Ceres .36 0 2.77A onestage rocket can easily escape the moon or Mars. Two stages are required for the Earth.
Rocket type Exhaust speed Exhaust speed (km/s) / speed of light Antimatter 150000 .5 React matter with antimatter Fission fragment 12000 .039 Nuclear fission fragments as exhaust Fusion 4900 .0163 Nuclear fusion of Deuterium + Lithium6 Ion drive 200 .00067 Uses electric power to accelerate ions Hydrogen + oxygen 4.4 .000015 Methane + oxygen 3.7 .000012 Kerosene + oxygen 3.3 .000011Chemical rockets and ion drives are proven technologies. All the other rockets could be built with present technology except for the antimatter rocket. In the distant future, antimatter rockets will be possible.
Power in space can be obtained from solar cells or from a nuclear battery. Solar cells work best at Earth orbit but they're not useful beyond Mars. Nuclear batteries work everywhere.
In a nuclear battery, radioactivity produces heat and a thermoelectric generator converts the heat to electricity.
The Voyager missions are powered by Plutonium238 nuclear batteries, which is why they are still functioning 30 years after their launch. Current plutoniumpowered missions include Cassini, Galileo, New Horizons, and Ulysses.
Plutonium238 and Strontium90 are the isotopes used for nuclear batteries in space, and Curium244 can be used as well. The possible power sources are:
Power source Generator Watts Halflife Cost /kg (years) (M$/kg) Solar cell Optic 300  .003 Power generated at Earth orbit Curium244 Thermo + Optic 40 18.1 .17 Curium244 Thermo 20 18.1 .17 Strontium90 Thermo 4 28.8 .01 Product of nuclear reactors Plutonium238 Thermo 5.4 87.7 .3 Scarce isotope Plutonium238 Stirling 4.1 87.7 .3 Scarce isotope Nuclear reactor Stirling 200  ? Data for the SAFE400 reactorThe numbers for Watts/kg are for the total system, including the isotope, the shielding, and the generator.
The following methods can convert thermal power to electric power.
Isotope Generator Electrical Fuel Total Temperature efficiency fraction efficiency (Kelvin) Plutonium238 Thermo .07 .14 .0098 1050 Plutonium238 Photo .07 .14 .0098 1050 Plutonium238 Stirling .26 .038 .0099 1050 Strontium90 Thermo .06 .1 .006 800 Strontium90 Photo .06 .1 .006 800 Electrical efficiency: Efficiency for converting heat to electricity Fuel fraction: Fuel mass / System mass Total efficiency: Electrical efficiency * Fuel fractionThe higher the temperature, the more efficient a thermoelectric or optoelectric generator is.
A thermoelectric generator and an optoelectric generator can work in tandem to produce a greater efficiency than either alone.
Watts GJoules Halflife Decay Decay Cost Produce Stockpile /kg /kg (years) (MeV) mode (M$/kg) (kg/yr) (kg) Cobalt60 27300 4533 5.27 2.82 Beta,γ 1.3 Curium244 4013 2293 18.1 5.80 Alpha .17 Tritium 1540 598 12.3 .0186 Beta 30 .4 Caesium137 864 824 30.2 1.17 Beta .01 Huge Huge Plutonium238 818 2265 87.7 5.59 Alpha 10 1 17 Strontium90 648 589 28.8 .55 Beta .01 Huge HugeThe numbers for Watts/kg and GJoules/kg are for the pure isotope and don't include the surrounding system. The energy density of gasoline is .046 GJoules/kg.
Strontium90 and Caesium137 are generated en masse as fission products in fission reactors.
For an isotope:
Atomic mass unit = M_{amu} = 1.661⋅10^{27} kg # of nucleons in nucleus = N Mass of nucleus = M_{nuc} N M_{amu} 1 MeV = 1.602⋅10^{13} Joules (1 Mega electron Volt) Nucleus decay energy = E_{decay} Nucleus energy/mass = S = E_{decay} / M_{nuc} Decay half life = T Heat power per kg = Q_{heat} = E_{decay} / T / M_{nuc} Electric power per kg = Q_{elec} Efficiency = ε = Q_{elec} / Q_{heat} (for converting heat to electric energy) Fuel mass = M_{fuel} System mass = M_{system} Fuel fraction = f_{fuel} = M_{fuel} / M_{system} System power per kg = Q_{sys} = ε f_{fuel} Q_{heat}
A pebble bed nuclear reactor doesn't melt down if the cooling system fails because it's engineered to turn off if it overheats. It's also designed so that adding and removing fuel pebbles is easy. The reactor is easy to build and it can be operated in space.
An ion drive uses electric power from a nuclear battery to accelerate ions. The values given in the table are for the Chang Diaz ion drive.
V = Ion speed = 50 km/s M = Mass of ion drive = 1000 kg m = Mass of ions ejected per second = .000096 kg/s Po = Power consumed by the ion drive = 200000 Watts Q = Efficiency of the drive = .6 For converting electric to ion power P = Power delivered to the ion beam = 120000 Watts = Q Po = .5 m V^2 F = Force generated by the ion beam = 4.8 Newtons = m V A = Acceleration of spacecraft = .0048 m/s^{2} = F / M = 2 P / (M V) Agi= Agility = Power/Mass = 120 Watts/kg = P / MAt fixed ion speed, the acceleration is determined by the powertomass ratio of the power source.
A = (2/V) * (P/M)
At fixed power there is a tradeoff between F and V:
P = .5 F VThe ion speed V can be customized. It should be at least as large as 10 km/s otherwise you might as well use a hydrogen+oxygen rocket. Increasing V decreases the fuel used, decreases the rocket force, and increases the travel time.
Suppose a spacecraft consists of
Ion Drive mass = M_{drive} = 1000 kg ChangDiaz VF200 design Solar cell mass = M_{cell} = 1000 kg To power the ion drive Argon mass = M_{argon} = 1000 kg Ions for the ion drive Scientific equipment mass = M_{equip} = 1000 kg Spacecraft total mass = M_{ship} = 4000 kg Solar cell power/mass = Q = 300 Watts/kg Solar cell power = P = M_{cell} Q Ion drive operation time = T = 10^{7} seconds Ion drive efficiency = e = .60 Ion velocity = V = 60000 (Mcell/Margon)^{½} Ion energy = E = P T e = ½ M_{argon} V^{2} Gravity constant = 6.674e11 Newton meters^{2} / kg^{2} Earthsun distance = 1.496e11 meters Sun mass = 1.989e30 kg Earth acceleration = .00593 meters/second^{2} Spacecraft recoil velocity = V M_{argon} / M_{ship} = 15 km/s Spacecraft acceleration = .00155 km/s corresponds to 1 AU/year. Using a gravity assist from Jupiter, an ion spacecraft can get anywhere in the solar system within 10 years.
When uranium fissions it produces 2 highspeed fragments, which can be herded with magnetic fields to produce thrust.
The characteristic speed of the fragments is 12000 km/s = .039 C. See the appendix for an expanded discussion.
The fuel shold have a critical mass that is as small as possible and the half life should be at least 20 years. The best candidate is Californium251.
Critical Diameter Halflife mass (cm) (Myears) (kg) Californium252 2.73 6.9 .0000026 Californium251 5 8.5 .000290 Californium249 6 9 .000351 Neptunium236 7 8.7 .154 Curium247 7.0 9.9 15.6 Curium243 8 10.5 .000029 Plutonium238 9.5 9.7 .000088 Plutonium239 10 9.9 .024 Curium245 10 11.5 .0085 Americium242 11 12 .000141 Plutonium241 12 10.5 .000014 Uranium233 15 11 .159 Uranium235 52 17 704 Neptunium237 60 18 2.14 Plutonium240 40 15 .0066
If we assume that all the energy goes into kinetic energy of exhaust, the exhaust speed is
Kinetic energy = .5 M V^{2} = .000135 M C^{2} V = 4900 km/s = .0163 CIf hydrogen bombs are used for propulsion then the spaceship has to be large to absorb the recoil.
A thermal rocket uses solar or nuclear power to heat a propellant. In space, ice is available in bulk and so either ice or hydrogen can be used for propellant.
Exhaust speed (km/s) H2 9.0 H2O 1.9In space, thin reflective material can be used to construct a large lowmass mirror to focus sunlight. Such a rocket will be able to move large objects such as asteroids. If an asteroid has its own ice then it's especially easy to move.
The Stratolaunch aircraft is subsonic. A supersonic ramjet such as the SR71 can move at Mach 5 and can launch a rocket from higher altitude than the Stratolaunch.
Launch method Speed Altitude Air density (km/s) (km) (kg/m^{3}) Ground 0 0 1.22 Conventional ground launch Subsonic aircraft .3 14 .26 Stratolaunch aircraft SR71 Blackbird 1.1 26 .038 Fastest existing ramjet Supersonic ramjet 1.5 30 .03 Maximum speed for a ramjet Electromagnetic sled 3.0 7 .4"Speed" refers to the initial speed of the launch vehicle and "Altitude" refers to the initial altitude of the launch vehicle after it has been accelerated by the launch system.
Future launch systems will use either a supersonic ramjet or an electromagnetic sled.
The Holloman Air Force Base does hypersonic research using a sled that can reach a speed of 2.88 km/s.
A launch sled can convert electrial power to sled kinetic energy with an efficiency of 90%.
Example values:
Sled acceleration = A = 50 m/s^{2} (5 g's. Maximum acceleration for humans) Sled final velocity = V = 3.0 km/s Length of the track = X = 90 km Time spent on track = T = 60 seconds V^{2} = 2 A X X = .5 A T^{2}If we launch inanimate equipment at an acceleration of 500 m/s^{2} then the track length is 9 km.
If a sled is moving at 3 km/s then a centripetal acceleration of 5 g corresponds to a radius of curvature of 180 km. The last half of the track has to be straight.
The sled only needs to reach an altitude of ~ 40 km. The rocket can do the rest. If it is launched from Everest then it needs to gain an altitude of ~ 30 km. The vertical velocity required to gain 30 km of altitude is .78 km/s. If the horizontal velocity is 3.0 km/s then the launch slope is .25.
A sled can use a heavy heat shield, which isn't possible with a rocket.
A sled launch track can use a mountain for altitude and launch angle. Possible mountains include:
Peak Height Earth Airmass Mountain range (m) rotation (tons) (km/s) Equator 0 .465 10.1 Sea level Huascaran 6768 .458 4.1 Huascaran Yerupaja 6634 .457 4.2 Huascaran Everest 8848 .41 3.1 Himalayas, Everest Kangchenjunga 8586 .41 3.3 Himalayas, Everest Aconcagua 6962 .391 4.0 Aconcagua K2 8611 .37 3.2 Himalayas, KarakoramHuascaran is the tallest peak that is close to the equator.
"Airmass" is the mass of air per meter^{2} above the given height.
The rocket has to have a mass of at least 100 tons for the airmass to not matter.
As a rocket burns through fuel it gets lighter. The "Tsoilkovsky rocket equation" relates the final rocket speed to the exhaust speed.
T = Time M(T)= Mass of rocket as a function of time M_{i} = Initial mass of rocket M_{f} = Final mass of rocket after burning its fuel V_{e} = Rocket exhaust speed V(T)= Rocket speed as a function of time. V(0)=0. V_{f} = Final rocket speed after burning its fuel F = Force generated by the rocket =  V_{e} dM/dT dV/dT = F/M = (V_{e}/M) * dM/dT V(T) = V ln(M_{i}/M) V_{f} = V ln(M_{i}/M_{f}) Tsoilkovsky rocket equation
The Oberth maneuver uses a planet's gravity to magnify a rocket impulse.
Suppose a spacecraft is on a highly elliptical orbit, with a perigee slightly larger than the Earth's radius and an apogee vastly larger than the Earth's radius.
Gravity constant = G = 6.67e11 Newton meters^{2}/kg^{2} Mass of Earth = M = 5.97e24 kg Earth radius = R = 6371 km/s Perigee radius = R_{1} Slightly larger than R Apogee radius = R_{2} R_{1} << R_{2} Escape velocity = V_{esc}= 11.2 km/s Rocket speed at perigee = V_{1} = V_{esc} Rocket speed at apogee = 0 Circular orbit speed at perigee = V_{circ}= 7.2 km/s = G M / R_{1} Circular orbit speed at apogee = 0 Rocket speed change at perigee = V_{roc} = 16.6 km/s Calculated below Final exit speed from planet = V_{exit}= 25.4 km/s Final speed after far from the planetAt apogee the energy is
E = Kinetic energy + Gravitational energy = 0 + 0At perigee the energy is
E = Kinetic energy + Gravitational energy = .5 m V_{1}^{2}  G M m / R_{1} V_{1}^{2} = 2 G M / R_{1} = 2 V_{circ}^{2} = V_{esc}^{2}V_{1} is equal to the "Escape speed", the speed required to escape the planet. The escape speed is independent of the direction of the velocity.
The escape velocity can also be obtained from the gravitational potential energy.
.5 m V_{esc}^{2} = G M m / R_{1} → V_{esc}^{2} = 2 G M / R_{1}IF the rocket fires at perigee and increases its speed by V_{roc}, the energy becomes
E = .5 m (V_{1} + V_{roc})^{2}  G M m / R_{1} = .5 m (V_{esc} + V_{roc})^{2}  .5 m V_{esc}^{2} = .5 m (V_{roc}^{2} + 2 V_{roc} V_{esc})The rocket is now on a hyperbolic orbit and will escape the Earth, As it recedes from the Earth it will approaches a constant velocity V_{exit}. When far from the Earth, the energy is
E = .5 m (V_{roc}^{2} + 2 V_{roc} V_{esc}) = .5 m V_{exit}^{2} V_{exit}= (V_{roc}^{2} + 2 V_{roc} V_{esc})^{1/2} > V_{roc}If the spacecraft starts in an elliptical orbit and changes its speed by V_{roc} at perigee, it departs the Earth at speed V_{exit}, which is larger than V_{roc}. This is the "Oberth effect".
If a rocket changes its velocity by 5 km/s at perigee, it departs the Earth with a velocity of
V_{exit}= (5^{2} + 2 * 5 * 11.2)^{1/2} = 11.7 km/sThis gets you to Mars in about 4 months.
X axis: Change in velocity at perigee (V_{roc}) Y axis: Departure velocity from the planet. V_{exit} = (V_{roc}^{2} + 2 V_{roc} V_{esc})Each curve corresponds to a different planet.
Escape velocity (km/s) Moon 2.38 Mars 5.03 Earth 11.2 Saturn 35.5 Jupiter 59.5 Sun 618
The Oberth maneuver requires a rocket with a large thrusttomass ratio. The Oberth effect is most useful when the rocket fires at Perigee, meaning the rocket has only a limited time to burn through its fuel. This restricts the rocket types that can be used for an Oberth maneuver. Chemical rockets deliver the most power, which makes them the rocket of choice for Oberth maneuvers. Nuclear rockets have a heating challenge. Ion drives and mirrorbased rockets are lowthrust and can't be used for the Oberth maneuver. The rocket engine with the largest force/mass is the Vulcain2. For this rocket,
Planet radius = R = 6371 km for the Earth Escape velocity = V_{es}= 11.2 km/s for the Earth Oberth time = T = 9.5 minutes for the Earth = R / V_{e} = Time that the rocket is near perigee Rocket exhaust speed = V_{ex}= 4.2 km/s Rocket force = F = 1359 kiloNewtons Rocket engine mass = m = 1800 kg Rocket force/mass = Z = 755 Newtons/kg = F / m Fuel mass burnt = M = T Z m / V_{ex} = 102 m Fuel mass burnt during one Oberth time Oberth velocity = V_{ob}= 16.6 km/s = 3.9 V_{ex} = [ln(M/m)  ln(2)] V_{ex} = ln(.5 T Z / V_{ex}) V_{ex} = [ln(T)  2.4] V_{ex} Momentum conservation: M V_{ex} = F TDuring one Oberth time, a Vulcain2 rocket burns 102 times its mass in fuel. The Oberth time for the Earth is long enough so that a chemical rocket can comfortably burn through all its fuel.
To calculate the Oberth velocity, we use the Tsoilkovsky rocket equation and assume that the final mass of the spaceship is twice the mass of the rocket engine.
Escape Radius Oberth Oberth Exit (km/s) time (s) velocity velocity (km/s) (km/s) Mercury 4.3 .38 563 16.5 20.4 Venus 10.5 .95 576 16.6 25.0 Earth 11.2 1.00 569 16.6 25.4 Moon 2.38 .27 723 17.6 19.8 Mars 5.03 .53 671 17.3 21.7 Jupiter 59.5 10.9 1167 19.6 52.1 Saturn 35.5 9.0 1615 20.9 43.9 Uranus 21.3 3.97 1187 19.7 35.0 Neptune 23.5 3.86 1046 19.1 35.6 Pluto 1.23 .184 953 18.7 19.9 Sun 618 109.2 1126 19.4 156.2"Exit velocity" is the maximum exit velocity from the planet using the Oberth maneuver. It is also equal to the maximum "capture velocity" for using the Oberth maneuver to be captured by a planet.
The following parameters are for a JPL design of a space mirror composed of aluminumcoated mylar.
Mylar density = 1.39 g/cm^{3} Aluminum density = 2.70 g/cm^{3} Mylar thickness = .025 mm Aluminum thickness = .010 mm Surface density = .006 kg/m^{2} (JPL design) Mirror area = 10^{4} km^{2} Mirror mass = 6⋅10^{7} kg Launch cost per kg = 1000 $/kg Launch cost = 6⋅10^{10} $
The Bigelow BA330 has as much room as the bridge of the Enterprise and the Bigelow Genesis has as much room as a Humvee. Bigelow habitats are lighter than NASA habitats and they have thicker walls. Thicker walls are helpful for defending against micrometeorites and radiation.
Volume Mass Wall thickness (m^{3}) (tons) (m) Bigelow Genesis 11.5 1.36 .15 NASA Orion 19.6 8.91 Bigelow BA330 330 23 .46 Space Station 837 450 .003
If artificial gravity is generated by spinning a spaceship, then according to en.wikipedia.org/wiki/Artificial_gravity, the spin period has to be at least 30 seconds for the inhabitants to not get dizzy. If we assume a spin period of 30 seconds and a gravity of 1 g,
Spin period = T = 2 π R / V = 30 seconds Spin radius = R = T^{2} A / (2π)^{2} = 228 meters Velocity = V = 2 π R / T = 48 meters/second Acceleration = A = V^{2} / R = 10 meters/second^{2}
Suppose we use a tether to connect a spinning spaceship. Zylon is the material with the best tensile strength to density ratio.
Tether density = D = 1520 kg/m^{3} for Zylon Tether tensile strength = P = F / Ar = 5.8 GPa Mass of spaceship = M Radius of tether = R = T^{2} A / 4 pi^{2} = V^{2} / A Tether crosssection = Ar Mass of tether = m = 2 R Ar D Centripetal acceleration = A = 10 meters/second^{2} Tether tension force = F = M A Spaceship spin period = T = 2 π R / V = 30 secondsThe mass ratio of the tether to the spaceship is
m/M = 2 T^{2} D A^{2} / P / (4 π^{2}) = 1.33e6 T^{2} = .00119To be safe, the tether can be given a mass 10 times larger then this. Even so, the tether weighs much less than the habitable module, and so the mass of the tether is not a factor in the spaceship design.
If the spaceship mass is M=1000 tons, the tether mass is m=12 tons. Such a tether can easily be launched from the Earth.
For extremely large tethers you can use iron from the moon.
Atmosphere thicknes (tons/m^{2}) Venus 1000 Titan 73 Earth, sea level 10 Earth, 12 km high 4.9 Mars .16 mSieverts Shielding thickness /year (tons/m^{2}) Terrestrial radiation 2.02 n/a Average medical radiation .60 n/a Earth surface, cosmic rays only .39 10 Earth surface, all radiation 3.5 10 Earth 2 km altitude, cosmic rays only .9 8 Earth 3 km altitude, cosmic rays only 1.7 7 Earth 4 km altitude, cosmic rays only 3.3 6 Earth, 12 km altitude, equator 20 2.5 Earth, 12 km altitude, poles 100 2.5 Space station, 420 km altitude 150 .01 1/8 inch aluminum walls Space 600 .01 Space, 4 tons/m^{2} shield 2.5 4 Mars surface 220 .16"Space" refers to interplanetary space between Earth and Mars.
At the space station, the Earth's magnetic field blocks 3/4 of the radiation from space.
The sun's magnetic field stops cosmic ray particles below 1 GeV.
The Earth's magnetic field deflects all but the highestenergy cosmic rays.
5 hour airplane flight incurs ~ .03 millisieverts.
A dose of 4800 millisieverts has a 50% risk of death.
The above data suggests that to shield against cosmic rays, you need at least 3 tons/meter^{2} of shielding.
Suppose a spherical spaceship is shielded with ice.
Radius of spaceship = r = 3 meters Radius of ice shield = R = 6 meters Density of ice shield = D = 1000 kg/meter^{3} Mass of ice shield = (4 &pi / 3) (R^{3}r^{3}) = 792 tonsThe only way to get this much ice is from the moon. This is the source of the mass for the tether calculation.
The space station life support system:
1 kWatt/person/day
1 liter of water/person/day
1 kg of food/person/day
Mass fraction in human body Oxygen .65 Carbon .18 Hydrogen .10 Nitrogen .03 Calcium .014 Phosphorus .011 Potassium .0025 Sulfur .0025 Sodium .0015 Chlorine .0015 Magnesium .0005 Iron .00006Air and water are the biggest challenges for life support in space. Water can be obtained from the moon and electrolysized to produce oxygen for air. Air also requires nitrogen, which cannot be found on the moon but is abundant in Mars' atmosphere.
The principal components of fertilizer are nitrogen, phosphorus, and potassium, with nitrogen being the heaviest component. Nitrogen can be obtained from Mars's atmosphere. If you want to grow crops on Mars you will have to bring phosphorus and potassium.
The most efficient way to grow plants in space is with aeroponics, where the roots are grown in open air. http://en.wikipedia.org/wiki/Aeroponics
Good locations for space bases are:
Earth orbit
The moon
Moon orbit
The L2 Lagrange point
Mars orbit
Ice can be shipped from the moon to the other stations.
The L2 point stays tethered to the Earth as the Earth orbits.
The L2 point is ideal for telescopes because from there you can shield the sun, the Earth, and the moon all at the same time. The Webb telescope will go there. If we had a manned space station at L2 then we could assemble telescopes onsite and build colossal telescopes.
Cosmic rays consist mostly of highenergy protons with energies > 1000 MeV. When a proton passes through matter it loses energy from collisions with electrons and with nuclei. Electron collisions subtract a small amount of energy from the proton and nuclear collisions subtract most of the energy. This is because collisions with electrons are mediated by the electromagnetic force and collisions with nuclei are mediated by the strong force.
E = Proton energy L = Distance the proton travels through matter (meters) D = Density of the matter (kg/meter^{3}) V = Proton velocity C = Speed of lightProton kinetic energy is measured in MeV. 1 MeV = 1.6e13 Joules. The rest energy of a proton is 1000 MeV.
Proton energy loss is governed by the "BetheBloch" formula. For cosmic ray protons with E > 1000 MeV, the formula may be approximated as
EnergyLoss = 200 L (D/1000) MeVIf the proton is traveling through water with a mass density of 1000 kg/meter^{3}, the energy loss rate is 200 MeV/meter. The amount of matter required to stop a proton with E = 1000 MeV is 5 meters.
Spacecraft walls are thick enough to stop lowenergy protons from the solar wind but they are of no help in stopping cosmic rays. Mars' atmosphere isn't thick enough either.
When a highenergy proton collides with a nucleus, most of the energy is lost in the collision, hence the transmission of protons through matter can be modeled as an exponential.
T = Initial intensity of protons t = Transmitted intensity of protons passing through a distance L of matter L = Distance the proton has traveled through the matter S = Characteristic stoppinglength of the matter D = Density of the matter in kg/meter^{3} A = Atomic number of the nuclei in the matter = 1 for protons = 8 for oxygen t = T exp(L/S) S = .35 A^{1/3} 1000/D metersFor oxygen, A = 8 and D=1000, hence the characteristic stopping length of protons in water is S = 0.2 meters.
Suppose you want to stop 99% of the protons.
t = .01 T L = 4.6 SIf water is used to stop cosmic ray protons, the formula predicts you need least 1 meter of it. This translates to a column density of 1 ton/meter^{2}.
This is an underestimate of the shielding required because when a highenergy proton hits a nucleus it creates a shower of secondary particles which must then be shielded. In practice, 4 meters of water are required. Muons are the biggest nuisance because they don't feel the strong force. Most of the cosmic radiation at the Earth's surface is from muons.
A Hohmann trajectory takes you from one circular orbit to another, such as from Earth's orbit to Mars' orbit.
The spacecraft starts on the cyan circular orbit.
At point "2", the spacecraft fires its rockets and increases its speed. From there, it coasts along the yellow trajectory to point "3".
When the spacecraft arrives at point "3", it fires its rockets to decrease its speed, placing it on the circular red trajectory.
In a trip from the Earth to Mars, the Earth is at point "2" and Mars is at point "3".
Departure velocity from the Earth = 2.95 km/s Arrival velocity with respect to Mars = 2.65 km/s Travel time from Earth to Mars = 8.5 months Wait time on Mars for Hohmann window = 14.9 months Travel time from Mars to Earth = 8.5 months Total mission time = 31.9 monthsThe total change in velocity that the rocket has to generate is 5.60 km/s. This is within the reach of a hydrogen+oxygen rocket, which has an exhaust speed of 4.4 km/s. This is the minimalist trajectory. If more rocket power is available then the travel time decreases.
Calculation of the EarthMars Hohmann orbit
The Oberth maneuver allows one to use a planet to magnify the impulse from a rocket.
For a Hohmann trajectory, the travel time from Earth to Mars is 289 days, which is a long time to be in zero gravity and in the radiation of space. The Oberth effect can speed up the trip.
Suppose a spacecraft is on a highly elliptical orbit, with a perigee just larger than the Earth's radius and an apogee much larger than the Earth's radius. Such an orbit would look like the Kuiper belt object "Sedna" pictured above.
An Oberth maneuver procedes as:
1) Start far from the planet at apogee 2) Coast toward the planet on a trajectory where the perigee is just above the surface of the planet. 3) At perigee, fire the rockets at maximum power 4) Coast away from the planet. The rocket escapes the planet with a speed that is enhanced by the Oberth effect Planet escape speed = V_{escape} = 11.2 km/s for the Earth Speed change from the rocket = V_{rocket} = 10 km/s for a mighty rocket Departure speed from the planet = V_{depart} = 18 km/s V^{2}_{depart} = V^{2}_{rocket} + 2 V_{rocket} V_{escape} Derivation
The Oberth effect can speed up the travel time to Mars to 3 months.
The Oberth effect can greatly magnify a small rocket boost. For example, if V_{rocket} = 1 km/s then V_{depart} = 4.8 km/s. This allows one to transport large payloads between planets, if speed isn't important.
The Oberth effect can be used on any planet or moon. The larger the mass of the object the more extreme the effect.
V_{escape} (km/s) Moon 2.38 Mars 5.03 Earth 11.2 Saturn 35.5 Jupiter 59.5 Sun 618Any mission to the outer solar system first passes by Jupiter, both for the gravity assist and for an Oberth boost. Jupiter is the hub of the solar system.
If you have a nonzero approach speed for a planet then the Oberth maneuver gives a departure speed of
Approach speed from deep space = V_{approach} V^{2}_{depart} = V^{2}_{approach} + V^{2}_{rocket} + 2 V_{rocket} V_{escape} Derivation
A mission to Mars might use the following strategy:
Mine ice on the moon.
Launch the ice from the moon into space.
Use solar energy to convert ice into hydrogen and oxygen and then liquify it. This is now rocket fuel.
Use this fuel to send supplies to Mars. The supplies will go to Mars with a slow trajectory and the astronauts will go later using a faster trajectory. Using the Oberth effect, it's possible to move a heavy spacecraft to Mars using two light nudges from the rockets, but the travel time is long.
Launch a rocket from the Earth and place it in an Oberthstyle elliptical orbit. Fuel the rocket with ice from the moon. This is the rocket that will go to Mars.
Launch the astronauts from Earth and place them on board the Mars rocket.
When the fullyfueled Mars rocket is at perigee, fire the rocket so that it escapes the Earth and heads for Mars. This is the "Oberth maneuver".
Upon arriving at Mars, use an inverse Oberth maneuver to place the rocket into an elliptical orbit around Mars.
The spacecraft must now fire its rockets again to go from an elliptical orbit to a circular lowMars orbit. It can use fuel that was sent ahead of time from the Earth for this maneuver.
Once the spacecraft is in lowMars orbit, the astronauts can drop to the surface of Mars using the atmosphere for breaking.
On Mars, ice is used to fuel the rocket that will lift the astronauts into lowMars orbit.
Once exploration is complete, the astronauts return to the spacecraft.
Using fuel sent ahead of time from the Earth, the spacecraft goes from a lowMars orbit to an elliptical orbit.
The spacecraft refuels again and uses an Oberth maneuver to depart Mars. Upon reaching Earth, an inverse Oberth maneuver is used to place the spacecraft in an Earth elliptical orbit.
With this mission plan, the manned rocket uses fuel only during the Oberth and inverse Oberth maneuvers. This minimizes the travel time.
mSieverts Shielding /year (tons/m^2) Earth surface, all radiation 3.5 10 Earth surface, cosmic rays only .39 10 Earth 2 km altitude, cosmic rays only .9 8 Earth 3 km altitude, cosmic rays only 1.7 7 Earth 4 km altitude, cosmic rays only 3.3 6 Earth, 12 km altitude, equator 20 2.5 Earth, 12 km altitude, poles 100 2.5 Space station, 420 km altitude 150 .01 1/8 inch aluminum walls Space 600 .01 Beyond the Earth's field Space, 4 tons/m^2 shield 2.5 4 Mars surface 220 .16 Mars Hellas Basin, 7 km deep ? .29To shield against cosmic rays, you need around 4 tons/meter^2 of material. This means the ship will weigh at least 500 tons, which emphasizes the need to get material from the moon. The liquid hydrogen & oxygen fuel can be used for radiation shielding.
The mass of food and water needed for the journey from Earth to Mars is much less than the mass of the radiation shielding, so you don't need to skimp on food quality.
Using Lagrange points and gravity slingshots, objects can be moved around the solar system with minimal propulsion.
Gravity assists can change a trajectory by of order the escape speed. You can use a sequence of gravity assists like a billiardsstyle trick shot to move objects around the solar system, requiring only nudges between assists. This is the "interplanetary transport network".
Mass Escape Orbit (Earth speed speed masses) (km/s) (km/s) Sun 333000 618. Mercury .0553 4.3 47.9 Venus .8150 10.46 35.0 Earth 1.0000 11.2 29.8 Mars .1074 5.03 24.1 Vesta .000045 .36 19.3 Ceres .00016 .51 17.9 Pallas .0000359 .32 17.6 Jupiter 317.83 59.5 13.1 Saturn 95.16 35.5 9.64 Uranus 14.50 21.3 6.81 Neptune 17.20 23.5 5.43 Pluto .00220 1.23 4.74 Moon .0123 2.38 1.02 Charon .000271 .23
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.
Gravity Temp Pressure Density N2 O2 CH4 CO2 Ar Xe H2S (m/s^{2}) (K) (Bar) (kg/m^{3}) (kg/m^{3}) (kg/m^{3}) Earth 9.80 287 1 1.22 .94 .209 0 .00048 .0011 0 0 Mars 3.71 210 .0063 .020 .00054 0 0 .020 Titan 1.35 94 1.46 5.3 5.22 0 .074 ? Moon 1.62 220 0 0 0 0 0 0 0 Pandora 7.8 290 1.20 1.46 1.2 .30 ? .26 ? .080 .015For humans, xenon is an anaesthetic and H2S is toxic.
Arctic scuba gear
Wingsuit (humanpowered flight is easy on Titan)
Strontium90 radioactive power source
Device for extracting nitrogen from the air.
Device for electrolyzing ice to produce oxygen.
You don't need a pressure suit because the pressure is 1.5 times Earth pressure.
Suppose a spaceship is on an elliptical orbit, such as the yellow orbit above.
G = Gravitational constant M = Mass of sun m = Mass of spacecraft R = Distance of spacecraft from sun R1 = Radius of perigee (point of closest approach to the star) R2 = Radius of apogee (point on orbit furthest from star) V = Velocity of spacecraft V1 = Velocity of spacecraft at perigee V2 = Velocity of spacecraft at apogee A = Semimajor axis of the orbital ellipse = .5 (R1 + R2) E = Energy = Kinetic energy + Gravitational potential energy = .5 m V^2  G M m / RAngular momentum is conserved. Equating angular momentum at apogee and perigee,
V1 R1 = V2 R2Equating energy at apogee and perigee,
E = .5 m V1^2  GMm/R1 = .5 m V2^2  GMm/R2Algebra gives
E =  G M m / (2A)The energy at radius R is
E = .5 m V^2  GMm/R =  GMm/(2A)For a circular orbit, V^2 = GM/R, and
E =  .5 m V^2 =  .5 GMm/RMake dimensionless:
G = 1 M = 1 m = 1 R1 = 1 R2 = r Solve for V1 and V2 .5 V1^2  1 =  1/(1+r) > V1^2 = 2r / (1+r) .5 V2^2  1/r =  1/(1+r) > V2^2 = 2/r  2/(1+r)For the EarthMars system,
r = R2/R1 = 1.524 V1 = 1.09891 V2 = .72107 U1 = Velocity of a circular orbit at radius R1 = 1. U2 = Velocity of a circular orbit at radius R2 = 1/sqrt(r) = .81004 D1 = Departure velocity from the Earth = V1  U1 = .09891 D2 = Arrival velocity at Mars = U2  V2 = .08897To restore the units, multiply the dimensionless velocities by the true value of U1.
U1 = 29.8 km/s = Velocity of Earth in its orbit U2 = 24.1 km/s = Velocity of Mars in its orbit D1 > D1 * U1 = .09891 * 29.8 = 2.948 km/s D2 > D2 * U2 = .08897 * 29.8 = 2.651 km/sThe departure velocity from the Earth is
D1 = 2.95 km/sUpon arriving Mars, the velocity with respect to Mars is
D1 = 2.65 km/sThe total change in velocity that the spacecraft must generate is
D1 + D2 = 5.60 km/sA hydrogen+oxygen rocket has an exhaust speed of 4.4 km/s and is capable of generating this change in velocity.
For a Hohmann trajectory, the travel time from Earth to Mars is 289 days, which is a long time to be in zero gravity and in the radiation of space. The Oberth effect can speed up the trip.
Suppose a spacecraft is on a highly elliptical orbit, with a perigee just larger than the Earth's radius and an apogee much larger than the Earth's radius. Such an orbit would look like the Kuiper belt object "Sedna" pictured above.
G = Gravitational constant M = Mass of Earth R1 = Perigee radius ~ Radius of Earth R2 = Apogee radius >> Radius of the Earth V1 = Velocity of spacecraft at perigee V2 = Velocity of spacecraft at apogee ~ 0 U1 = Velocity of a spacecraft on a circular orbit at radius R1 U1 = G M / R1 = 7.2 km/s for the Earth U2 = Velocity of a spacecraft on a circular orbit at radius R2 ~ 0 Ve = Escape velocity from the EarthWhen the spacecraft is at apogee, the energy is
E = Kinetic energy + Gravitational energy = 0 + 0When the spacecraft is at perigee, the energy is
E = Kinetic energy + Gravitational energy = 1/2 m V1^2  G M m / R1 V1^2 = 2 G M / R1 = 2 U1^2 = Ve^2V1 is equal to the "Escape velocity". If a spacecraft starts from the surface of the Earth and is launched directly away from the Earth, it must have a velocity of at least Ve to escape the Earth.
The escape velocity can also be obtained from the gravitational potential energy.
1/2 m Ve^2 = G M m / R1 > Ve^2 = 2 G M / R1Suppose the spacecraft fires its rockets at perigee and increases its speed by D1.
The energy is now
E = 1/2 m (V1 + D1)^2  G M m / R1 = 1/2 m (Ve + D1)^2  1/2 m Ve^2 = 1/2 m (D1^2 + 2 D1 Ve)The spacecraft is now on a hyperbolic orbit and will escape the Earth, As it recedes from the Earth, it will approach a constant velocity Q1. When it is far from the Earth, the energy is
E = 1/2 m (D1^2 + 2 D1 Ve) = 1/2 m Q1^2 Q1 = SquareRoot(D1^2 + 2 D1 Ve) Q1 > D1If the spacecraft starts in an elliptical orbit and changes its speed by D1 at perigee, it departs the Earth at speed Q1, which is larger than D1. This is the "Oberth effect".
If a rocket changes its velocity by 5 km/s at perigee, it departs the Earth with a velocity of
Q1 = SquareRoot(5^2 + 2 * 5 * 11.2) = 11.7 km/sThis gets you to Mars in about 4 months.
X: Change in velocity at perigee Y: Departure velocity from the planet Ve: Escape velocity for the planetThis is a plot of
Y^2 = X^2 + 2 X Ve Ve (km/s) Moon 2.38 Mars 5.03 Earth 11.2 Saturn 35.5 Jupiter 59.5 Sun 618.

Metallic asteroids are rich in platinum group metals, all of which are worth more than $5000 per kilogram. A typical metallic asteroid with a diameter of 100 meters has a mass of 1 billion kilograms and contains 19 tons of platinum with a value of 1 billion dollars. For a typical metallic asteroid,
Mass in Element Value of element asteroid cost/kg in the asteroid (tons) ($/kg) (B$) Platinum 19 55000 1.04 Nickel 67000 15 1.00 Rhodium 4.1 90000 .37 Iron 910000 .3 .27 Cobalt 6300 30 .19 Gold 1.8 60000 .108 Osmium 7.6 12000 .091 Germanium 37 2000 .074 Ruthenium 11 5500 .060 Palladium 3.8 14000 .053 Iridium 3.0 14000 .042 Gallium 80 280 .022 Zirconium 8 1600 .013 Rhenium .85 5000 .004The most profitable elements are platinum, nickel, rhodium, iron, cobalt, and gold.
The Sudbury basin mine in Canada is a meteor crater from as 12 km metallic meteor that struck 1.8 billion years ago. The Earth's crust is poor in these elements because they sink to the core and platinum mines tend to be at sites of metallic asteroid impacts.
The asteroid belt was formed from a planet that was shattered by collisions. The asteroid belt comsists of mostly rocky asteroids and some metallic asteroids.
Platinum fraction in early universe = .005 parts/million Platinum fraction in the sun = .009 parts/million Platinum fraction in the Earth's crust = .004 parts/million Platinum fraction in Sudbury mine ore = .5 parts/million Platinum fraction in iron asteroid = 19 parts/million Annual platinum production =500 tons/year Platinum used in mufflers =130 tons/year
The valuable elements in a metallic asteroid tend to have high boiling points. These elements can be isolated by heating the asteroid to 3200 Kelvin to boil off the lowvalue iron, nickel, and cobalt. After boiling, the asteroid has 1/5000 of its original size, has a value of 10 thousand dollars per kilogram, and is easily transported to the Earth.
Heat can be obtained by focusing sunlight with mirrors. The elements ranked by boiling point are:
Density Melt Boil $/kg ppm in metallic g/cm^{3} K K asteroid Rhenium 21.0 3459 5869 4600 .85 Tungsten 19.25 3693 5828 50 8.1 Tantalum 16.7 3290 5731 400 .06 Osmium 22.59 3306 5285 12000 7.6 Thorium 11.7 2115 5061 25 .04 Niobium 8.75 2750 5017 40 .2 Molybdenum 10.28 2896 4912 21 7.3 Hafnium 13.31 2506 4876 500 0 Iridium 22.4 2739 4701 14000 3.0 Zirconium 6.52 2128 4682 20 8 Ruthenium 12.45 2607 4423 5500 11 Uranium 19.1 1405 4404 75 .007 Platinum 21.45 2041 4098 55000 19 Rhodium 12.41 2237 3968 90000 4.1 Vanadium 6.0 2183 3680 12 6 Titanium 4.51 1941 3560 10 100 Palladium 12.02 1828 3236 13600 3.8 Cobalt 8.90 1768 3200 30 6300 Nickel 8.91 1728 3186 15 67000 Iron 7.87 1811 3134 .3 910000
Siderophile: Ironliving. Tends to sink to the core along with the iron. Lithophile: Rockloving. Tends to become included in rock and escapes sinking to the core. Chalcophile: Oreloving. Tends to combine with oxygen and sulfur and escapes sinking to the core. Atmophile: Is a gas at room temperature and tends to escape the crust into the atmosphere.In the early solar system, a small planet formed in the region that is now the asteroid belt. The planet had a hot interior and there was enough time for the dense elements to sink to the core. Then the planet was shattered by collisions and became the presentday asteroid belt. Pieces of the planet that are from the core are now metallic asteroids.
Dot size = log(Solar Abundance)Elements with a dot size of zero have no stable isotope.
Compositions are listed as parts per million by mass.
Big Bang Sun Earth Iron Core crust asteroid Amplification Hydrogen 750000 750000 1500 0 0 Helium 230000 230000 0 0 Oxygen 10000 9000 460000 0 0 Carbon 5000 3000 1800 1100 .042 Iron 1100 1000 63000 910000 1 Silicon 700 900 270000 40 .000010 Tungsten .0005 .004 1.1 8.1 .51 Platinum .005 .009 .004 19 329 Gold .0006 .001 .003 1.8 42"Core amplification" is the degree to which the element is concentrated in the core, normalized so that the core amplification of iron is 1.
Platinum is more dense than iron and is hence more likely to sink to the Earth's core than iron. This is reflected in the "core amplification factor".
Platinum abundance in the crust = c_{Pt} = .004 ppm Iron abundance in the crust = c_{Fe} = 63000 ppm Platinum abundance in the core = C_{Pt} = 19 ppm Iron abundance in the core = C_{Fe} =910000 ppm Crust platinum/iron = c_{Pt}/c_{Fe} = .000000063 Core platinum/iron = C_{Pt}/C_{Fe} = .000021 Core amplification factor = A_{Pt} = (C_{Pt}/C_{Fe}) / (c_{Pt}c_{Fe}) = 329The elements with the highest core amplification factors are:
Amplification Density (g/cm^{3}) Ruthenium 762 12.4 Siderophile, Platinum group Platinum 329 21.4 Siderophile, Platinum group Rhodium 284 12.4 Siderophile, Platinum group Osmium 263 22.6 Siderophile, Platinum group Iridium 208 22.4 Siderophile, Platinum group Nickel 52 8.9 Siderophile Palladium 44 12.0 Siderophile, Platinum group Gold 42 19.3 Siderophile Rhenium 20 21.0 Siderophile Cobalt 14.5 8.9 Siderophile Selenium 4.2 4.81 Chalcophile Germanium 1.8 5.32 Chalcophile Bismuth 1.4 9.78 Chalcophile Iron 1.0 7.9 Siderophile Tungsten .51 19.25 Lithophile Molybdenum .46 10.28 Siderophile Mercury .42 13.53 Chalcophile Lead .42 11.34 Chalcophile Arsenic .36 5.73 Chalcophile Gallium .29 5.91 Chalcophile Copper .13 8.96 Chalcophile Tin .063 7.26 Chalcophile Silver .030 10.49 Chalcophile Zinc .025 7.14 Chalcophile Thorium .00046 11.7 Lithophile Uranium .00027 19.1 LithophilePlatinum group metals are the most likely to sink to the core. Almost all of the uranium and thorium resists sinking to the core, which is why nuclear energy is cheap.
"16 Psyche" is the largest metallic asteroid and is likely the core of a failed planet that had its mantle stripped away by collisions.
Diameter Perihelion SemiMajor Value Value km AU Axis (AU) (B$) ($/kg) 16 Psyche 186 2.51 2.92 5⋅10^{8} .02 Nereus .33 .95 1.49 5 .033 Ryugu .98 .96 1.19 95 .024 2011 UW158 .45 1.01 1.62 8 .021 Didymos .8 1.01 1.64 84 .039 1989 ML .6 1.10 1.27 14 .015 1992 TC 1.1 1.11 1.57 84 .015The estimated value of each asteroid is from Wikipedia. Values are given for all asteroids except Psyche. In the table, we assume a price/mass for Psyche of .02 $/kg which leads to a value of 500000 trillion dollars, far larger than the Earth's annual gross domestic product of 75 trillion dollars.
If a 10^{10} kg asteroid is broken with a hydrogen bomb then
Mass of the asteroid = M = 10^{10} kg Energy of the hydrogen bomb = E = .5 M V^{2} = 4e16 Joules (10 megatons of TNT) Speed of the asteroid fragments = V = 3 km/sA hydrogen bomb is capable of moving an asteroid.
A space mirror can be used to vaporize a metallic asteroid, which leaves behind elements with high boiling points. These elements are the valuable ones. We assume that the asteroid is mostly iron.
Asteroid original temperature = 220 Kelvin Asteroid boiling temperature = 3200 Kelvin (Boiling point of cobalt) Melting energy of 1 kg of iron = 272 kJoules Boiling energy of 1 kg of iron = 6090 kJoules Iron solid heat capacity = .449 kJoules/kg/Kelvin Iron liquid heat capacity = .82 kJoules/kg/Kelvin Iron solid temperature change = 1591 Kelvin (From 220 Kelvin to 1811 Kelvin) Iron liquid temperature change = 1323 Kelvin (From 1811 Kelvin to 3134 Kelvin) Iron solid heating energy = 714 kJoules (Energy to heat from 220 to 1811 Kelvin) Iron liquid heating energy = 1085 kJoules (Energy to heat from 1811 to 3134 Kelvin) Iron total heating energy = 8161 kJoules (Energy to heat, melt, and vaporize) Energy to vaporize iron asteroid= 8.2⋅10^{15} Joules (Assume a mass of 1 billion kg) Power from 10 km3 space mirror = 1.3⋅10^{11} Watts Time to vaporize asteroid =60000 seconds = 17 hours Mass of a 10 km3 space mirror = 600 tonsVaporization accounts for most of the energy requirement.
A 10 km^{3} space mirror can vaporize a 1 billion kg asteroid on 1 day.
The exiting vapor can be passed through tungsten pipes heated to a temperature of 3250 Kelvin, just above the temperature of the vapor. Any platinum group metals in the vapor will condense onto the pipes. Tungsten is used because it is the element with the highest melting point (3695 Kelvin).
The iron in a metallic asteroid can be burned to produce carbonfree energy.
Fe + O_{2} → Fe_{2}O_{3}An asteroid 3 km in diameter can supply civilization's energy for one year.
Energy density of iron = 5.2 MJoules/kg World energy production = 6e20 MJoules/year Iron to produce world's energy =1.2e14 kg Iron density = 7900 kg/meter^{3} Radius of iron asteroid = 1544 metersSome of the metals in a metallic asteroid have the potential to overwhelm Earth production and reduce the price of the metal, opening new technological applications. These metals are osmium, ruthenium, iridium, rhodium, and platinum. For a 1 billion kg asteroid,
Mass in Annual Earth Mass in asteroid asteroid mining / Annual Earth mining (tons) (tons) Osmium 7.6 1 7.6 Ruthenium 11 12 .92 Germanium 37 118 .31 Gallium 80 273 .29 Iridium 3.0 12 .25 Rhodium 4.1 30 .14 Platinum 19 245 .08 Cobalt 6300 110000 .057 Nickel 67000 2100000 .032 Rhenium .85 50 .017 Palladium 3.8 250 .015 Gold 1.8 2800 .0006 Earth gold won't be eclipsed by asteroid gold Iron 910000 1700000000 .0005 Zirconium 8 900000 .000009Bringing a trillion kg asteroid to the Earth would satisfy the world's iron, nickel, and cobalt demand and then we wouldn't need to use energy for smelting.
tons/year World 161 South Africa 110 Russia 25 Zimbabwe 11 Canada 7.2 USA 3.7 Other 3.8Data for 2014
Compositions are listed as parts per million by mass.
Big Bang Milky Way Sun Earth Iron Amplification crust asteroid Hydrogen 750000 739000 750000 1500 0 0 Helium 230000 240000 230000 0 0 Oxygen 10000 10400 9000 460000 0 0 Carbon 5000 4600 3000 1800 1100 .042 Neon 1300 1340 1000 0 0 Iron 1100 1090 1000 63000 910000 1 Nitrogen 1000 960 1000 20 33 .11 Silicon 700 650 900 270000 40 .000010 Magnesium 600 580 700 29000 320 .00076 Sulfur 500 440 400 420 360 .059 Calcium 50000 500 .00069 Potassium 15000 0 0 Aluminum 50 60 82000 40 .000034 Sodium 20 40 23000 0 0 Phosphorus 7 7 1000 2200 .15 Beryllium .001 .0001 1.9 0 0 Lithium .006 .0001 17 0 0 Boron .001 .002 8.7 0 0 Fluorine .4 .5 540 0 0 Chlorine 1 8 170 0 0 Argon .04 70 0 0 Scandium .03 .04 26 0 0 Titanium 3 4 6600 100 .0010 Vanadium 1 .4 190 6 .0022 Chromium 15 20 140 15 .0074 Manganese 8 10 1100 300 .019 Iron 1100 1090 1000 63000 910000 1 Cobalt 3 4 30 6300 14.5 Nickel 60 80 90 67000 52 Copper .06 .7 68 130 .13 Zinc .3 2 79 28 .025 Gallium .01 .04 19 80 .29 Germanium .2 .2 1.4 37 1.8 Krypton .04 0 0 Strontium 360 0 0 Ytterbium .007 .01 2.8 0 0 Zirconium .05 .04 130 8 .0043 Niobium .002 .004 17 .2 .00081 Molybdenum .005 .009 1.1 7.3 .46 Ruthenium .004 .005 .001 11 762 Rhodium .0006 .002 .001 4.1 284 Palladium .002 .003 .006 3.8 44 Silver .0006 .001 .08 .035 .030 Cadmium .002 .006 .15 .02 .0092 Indium .0003 .004 .16 .01 .0043 Tin .004 .009 2.2 2 .063 Antimony .0004 .001 .34 Lutetium .0001 .001  Hafnium .0007 .001 3.3 0 0 Tantalum .0001 1.7 .06 .0024 Tungsten .0005 .004 1.1 8.1 .51 Rhenium .0002 .004 .003 .85 20 Osmium .003 .002 .002 7.6 263 Iridium .002 .002 .001 3.0 208 Platinum .005 .009 .004 19 329 Gold .0006 .001 .003 1.8 42 Mercury .001 .02 .067 0 .42 Thallium .0005 .001 0 0 Lead .01 .01 10 60 .42 Bismuth .0007 .01 .025 .5 1.4 Thorium .0004 .0003 6 .04 .00046 Uranium .0002 .001 1.8 .007 .00027 Neodymium 33 Lanthanum 34 Yttrium 29 Samarium 6 Cerium 60 Barium 340 Rubidium 60 Praseo 8.7 Gadolinium 5.2 Dysprosium 6.2 Erbium 3.0 Caesium 1.9 Europium 1.8 Arsenic 2.1 11 .36 Holmium 1.2 Terbium .94 Thulium .45 Bromine 3 Thallium .53 Antimony .2 Iodine .49 Selenium .05 3 4.2 Tellurium .001
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
Most fuels are combinations of carbon, hydrogen, and oxygen, and are collectively called "hydrocarbons".
Black: Carbon White: Hydrogen Red: Oxygen
The ATP molecule is a cannon and a phosphate ion is a cannonball. The cannonball powers enzyme action. The fact that the phosphate is large makes it easy to harness for energy. The cannon has to be substantially larger than the cannonball, which is why the ATP molecule is large.
ATP → ADP + Phosphate + Energy Use ATP to power enzymes ADP + Phosphate + Energy → ATP Creation of ATP from ADPATP is assembled by the ATPsynthase enzyme. ATP and ATPsynthase are common to all Earth life. Mitochondria convert sugar or fat into ATP and then ATP is used to power enzymes. ATP has substantially less energy/mass than sugar or fat, which is why ATP is only generated as needed.
Video of the ATPsynthase enzyme. Discussion of the physics of ATP
When ATP is depleted it can be regenerated anaerobically with creatine phosphate.
Phosphocreatine + ADP → Creatine + ATPThe reaction is reversible. If ATP isn't needed then the energy is converted back to phosphocreatine.
Creatine has half the mass of ATP and so it offers a more lightweight way to store energy.
When creatine phosphate is depleted then energy can be generated anaerobically using the lactic acid cycle. This produces less energy than aerobic respiration.
Glucose + Oxygen → 30 ATP of energy (Aerobic respiration) Glucose → 2 ATP of energy (Anaerobic respiration)During maximum exertion,
Time before ATP is exhausted = 2 seconds Time before creatine phosphate is exhausted = 10 seconds Time before lactic acid becomes uncomfortably high = 90 seconds
Energy to form ATP from ADP =.063 MJoules/mole = .653 eV = .124 MJoules/kg Energy yield for (ATP > ADP) =.029 MJoules/mole = .301 eV = .057 MJoules/kg Energy from phosphocreatine =.029 MJoules/mole = .301 eV = .137 MJoules/kg 1 MJoule/mole =10.36 eV Typical ATP cycle time when at rest = 35 seconds Human ATP content = .1 moles = .051 kg Human Phosphocreatine mass fraction =.0090 kg/kg Human ATP mass fraction = f =.0031 kg/kg ATP energy/mass = e = .057 MJoules/kg Human maximum power/mass = p = 20 Watts/kg Human time to burn through all ATP = T = fe/P = 8.8 seconds Molecular mass of ATP = 507.2 grams Molecular mass of ADP = 427.2 grams Molecular mass of Phosphate = 95.0 grams Molecular mass of H2O = 18.0 grams Molecular mass of OH = 17.0 grams Molecular mass of H+ = 1.0 grams Molecular mass of Creatine = 131.1 grams Molecular mass of Phosphocreatine = 211.1 grams
Sea level Vacuum Thrust Fuel Exhaust Exhaust Mass Thrust /mass km/s km/s kg kN N/kg Waxwing Solid 2.72 87 29.4 345 Atlas V Solid 2.70 1270 40.8 tons with fuel P230 Solid 2.80 6472 268 tons with fuel. Ariane rocket Shuttle booster Solid 2.42 2.68 12500 21200 590 tons with fuel Merlin 1D Kerosine 2.76 3.05 630 801 1300 Falcon rocket. Diameter 1.676 m Merlin 2 Kerosine 3.16 8540 In development by SpaceX. Falcon Heavy Raptor Methane 3.7 8200 In development by SpaceX Snecma HM7B HOX 4.3 165 64.8 400 Ariane rocket RL10A HOX 4.42 167 99.1 606 Atlas V. Diameter = 2.13 meters RL10B2 HOX 4.547 277 110 406 Atlas V and Delta IV rockets Mitsubishi LE5B HOX 4.38 285 137.2 490 Mitsubishi LE7A HOX 4.31 1800 1098 620 Vulcain 2 HOX 4.20 1800 1359 755 Ariane rocket. Diameter = 1.76 m Shuttle engine HOX 3.56 4.44 3500 1700 496 RS68 HOX 4.02 6600 3370 520 Most powerful HOX rocket HOX = liquid hydrogan + liquid oxygen Kerosine = kerosine + liquid oxygen Solid = aluminum + ammonium perchlorate (N H4 Cl O4) Methane = methane + liquid oxygen
Stage 1 Stage 2 Stage 3 Mass Thrust Exh Mass Thrust Exh Mass Thrust Exh Payload Payload kkg kN km/s kkg kN km/s kkg kN km/s kkg $/kg Space Shuttle 1710 25000 ~2.6 530 5100 4.44 ? 5100 4.44 93. SpaceX Falcon 9 506 6672 ~2.9 52 801 3.35    13.15 4109 SpaceX Falcon Heavy 1400 17000 ~2.9 ~480 5600 3.05 ? 445 3.35 53. 2200 Saturn V 2800 34000 2.58 710 4400 4.13 230 1000 4.13 118.00 9915 Ariane 777 12940 2.80 ? 1340 4.22 ? 64.7 4.37 16. 10500 Pegasus 23.1 .443 Stratolaunch ? 1500 n/a 230 ? ? ? ? ? 6.12 Earth rotation at equator = 463 m/s. Earth escape speed = 11.186 Earth orbit speed at 160 km = 7.58 km/s Falcon 9 stage 2 empty mass = 3.1 tons Falcon 9 Sea level thrust = 5885 kN Space shuttle: The space shuttle orbiter has a mass of 68.6 and a payload of 24.4 tons. Saturn V: Largest payload ever achieved. Launched the moon missions. Pegasus: Air launch Stratolaunch: A 6engine airplane launches the "Pegasus II" rocket.The Stratolaunch airplane is moving at ~ .3 km/s when it launches the rocket, and the launch can occur at the equator where the Earth's rotation speed is .46 km/s. This gives the rocket a total initial speed of .76 km/s.
Engine Engine Empty Max Cargo Speed Ceiling thrust mass mass takeoff mass (Mach) km (tons) (tons) (tons) (tons) (tons) Blackbird SR71 3.3 25.9 2 x 14.8 2 x 2.7 30.6 78 Spy F15 Eagle 2.5 20.0 2 x 11.3 2 x 1.70 12.7 30.8 Fighter F22 Raptor 2.25 19.8 2 x 15.9 2 x 1.77 19.7 38 Stealth Fighter Concorde 2.02 18.3 4 x 17.2 4 x 3.18 78.7 187 128 passengers Airbus A380 .96 13.1 4 x 38.2 4 x 6.27 276.8 650 853 passengers Boeing C5 Galaxy .8 4 x 19.4 4 x 3.63 172.4 381 122.5 Cargo Boeing 7478F .86 13.0 4 x 30.2 4 x 5.6 448 134.2 Cargo Antonov 224 .75 4 x 23.4 4 x 4.1 175 405 150 Cargo Antonov 225 .7 6 x 23.4 6 x 4.1 285 640 250 Cargo Stratolaunch 6 x 25.5 540 230 Orbital launch platformThe Stratolaunch (in development) is designed to launch rockets into space.
Drag force = .5 * AirDensity * CrossSection * Velocity^2 M = Rocket Mass / 400 tons A = Acceleration / 10 m/s^2 Acceleration in units of g's D = Air Density / 1 kg/m^3 Density = 1.28 kg/m^3 at sea level C = Cross section / 10 m^2 The Falcon 9 rocket has a cross section of 10 m^2 V = Velocity / 300 m/s Velocity in units of "Mach"In these units the drag equation is
10 A M ~ D C V^2 For a falcon 9 rocket, M=1 and C=1. If the rocket is at sea level, D ~ 1.If the drag acceleration is 1g, then V ~ 3 (Mach 3). This sets the speed limit for rockets in the lower atmosphere.
Fuel Exhaust Density Boil kNewtons kNewtons kNewtons Diameter Mass Rocket engine used (km/s) (g/cm^3) (K) /meter^2 /ton (meters) (kg) for data Liquid hydrogen 4.2 .07 20.3 559 755 1359 1.76 1800 Vulcain2 Liquid methane 3.7 .42 111.7 493 ? 8200 4.6 ? Raptor Kerosine 3.3 .80 410 361 1270 801 1.676 630 Merlin1D Solid fuel 2.7 1.2  673 ? 1270 1.55 ? Atlas V booster Kerosine ramjet .80 410 9.0 5.5 14.8 1.45 2.7 SR71 BlackbirdHydrogen, methane, and kerosine are all reacted with liquid oxygen that is carried by the rocket. Solid fuel contains its own oxidizer.
For the kerosine ramjet, kerosine is reacted with oxygen from the air.
"kNewtons/meter^2" is the thrust/area of the rocket.
"kNewtons/kg" is the thrusttomass ratio of the rocket engine.
The density of liquid oxygen is 1.14 g/cm^3 and the boiling point is 90.2 Kelvin.
Electricity can split H2O into H2 and O2, which can be used for rocket fuel. the maximum efficiency of this process is 0.83.
Energy to split H2O into H2 and O2 = E = 1.317e7 Joules/kg Max efficiency to split H2O into H2 and O2 = e = .83 Solar cell power per mass = Sp = 300 Watts/kg Solar cost per mass = Sc = 3000 $/kg Time for a 1 kg solar cell to form 1 kg of fuel = T = .61 days = E / e / Sp
We can calculate the maximum speed of HOX rocket exhaust from the energy required to split H2O.
V = Maximum speed of rocket exhaust for a HOX rocket 1.317e7 Joules/kg = ½ V^{2} V = 5.132 km/sIn practice, the best HOX rockets have an exhaust speed of 4.4 km/s.
Mean energies for the fission of Uranium235, in MeV:
Fission fragment kinetic energy 169.1 Prompt neutrons 4.8 Prompt gamma rays 7.0 Delayed beta rays 6.5 Delayed gamma rays 6.3 Captured neutrons 8.8 Total energy generated as heat 202.5 Prompt antineutrinos 8.8 Total energy including antineutrinos 211.3 Energy of the original U235 nucleus 218900 1 MeV = 10^6 eV = 1.6*10^13 Joules 1 Atomic mass unit = 1.6605*10^27 kg = 931.494 MeV/C^2 Mass of Uranium235 = 235.04 atomic mass units Only the kinetic energy of the fission fragments is harnessable by a rocket. C = Speed of light Mt= Mass of original nucleus E = Kinetic energy of the fission fragments F = Fraction of the mass of the original nucleus that is converted into kinetic energy. = E / (Mt C^2) = 169 MeV / (235.04 * 931.49) = .000772 Vt= Characteristic speed of the fission fragments .5 Mt Vt^2 ~ F Mt C^2 Vt = .0393 C
Fission tends to produce two fragments, one heavier than the other. The distribution is similar for all fissionable nuclei.
E = Total kinetic energy in fission fragments ~ 169 MeV F = Fraction of the mass of the original nucleus that is converted into kinetic energy. = .000772 M = Mass of heavy fragment ~ .40 * Mass of original nucleus m = Mass of light fragment ~ .58 * Mass of original nucleus V = Velocity of heavy fragment v = Velocity of light fragment Conservation of momentum: M V = m v Conservation of energy: E = .5 M V^2 + .5 m v^2 M^2 V^2 (M + m) = 2 E M m V^2 = 2 F C^2 m / M v^2 = 2 F C^2 M / m V = .0326 C v = .0473 C Critical mass Half life Americium242 .5 141 years Costs ~ 10^6 $/kg Californium251 .9 898 years Curium245 1.1 8500 years Plutonium239 5.6 241000 years Uranium235 11.0 704 million yearsFor a fission fragment rocket, the lower the critical mass the better. All of the above isotopes produce similar energy when fissioned.
Hydrogen bombs use the following reactions.
Neutron + Lithium6 > Tritium + Helium4 + 4.874 MeV Deuterium + Tritium > Helium4 + Neutron + 17.56 MeVLeaving out the neutron catalyst, this is
Deuterium + Lithium6 > Helium4 + Helium4 + 22.43 MeV Nucleons = 8 Energy / Nucleon = 22.434/8 = 2.80 MeV/Nucleon f = Fraction of mass converted to energy = (2.80 MeV/Nucleon) / (939 MeV/Nucleon) = .00298The theoretical limit for the efficiency of a hydrogen bomb is
f = .00027In practice, f is half this.
A thermal rocket uses a power source to heat the propellant. The power can come from either a nuclear reactor or from sunlight focused by mirrors.
Propellant Exhaust speed (km/s) H2 9 H2O 1.9 Energy Hydrogen + Oxygen 1.4e10 Joules/ton Uranium235 8.0e16 Joules/ton Solar energy 1.4e15 Joules. 1 km^2 collector operating for 10^6 seconds at 1 A.U.A mirrorbased thermal rocket offers a means for using H2O as propellant. Such a rocket can potentially move large asteroids.
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 ~ 1.4e15The mass of the mirror is
Surface area Thickness Density Mirror mass ~ 8*10^5 kg    1 km^2 10^4 m 8 g/cm^3A solar thermal rocket capable of delivering ~ 10^16 Watts can be built from a ~ 10 meter metallic asteroid.
If a thermal rocket can operate at a temperature high enough to dissociate H2 into elemental hydrogen then larger exhaust speeds are possible.
Suppose we use mylar film for a space mirror.
Mirror density = 1390 kg/m^3 Mirror thickness = .1 mm Mirror mass/area = .139 kg/m^2 Solar flux = 1362 Watts/m^2 H2O exhaust speed = 1.9 km/s H2O mass/time/area = .00075 kg/s/m^2 Mass of propellant per time per area Mirror acceleration = 10.3 m/sThe acceleration of a mirror rocket is limited by the strength of the mirror.
If we assume that the kinetic energy of an orbiting object comes from electricity then
Orbital speed = 7.8 km/s Energy of a 1 kg object at orbital speed = 30.4 MJoules Cost of electricity = 36.0 MJoules/$ Cost of a 1 kg object at orbital speed = .84 $For a typical hydrogen+oxygen rocket, the mass fractions are:
Payload = 1 kg Superstructure = 2 kg Hyddrogen mass = 3 kg Oxygen mass = 24 kg Total mass = 30 kg Oxygen mass / Hydrogen mass = 8 Cost of liquid hydrogen = .70 $/kg Cost of liquid oxygen = .16 $/kg Cost of liquid hydrogen = 2.1 $ Cost of liquid oxygen = 3.8 $ Typical launch cost for 1 kg = 2500 $The superstructure is everything except the payload and the fuel.
Orbit speed = 7.8 km/s Energy of 1 kg at orbit speed = 30.4 MJoule Cost of electricity = .015 $/MJoule Electricity cost of the energy= .46 $