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Astronautics
Dr. Jay Maron

Rockets

Index:    Orbit    Fuel    Aircraft launch    Future rockets    Power in space     Ion drives     Fission rocket     Fusion rocket     Thermal rocket     Electromagnetic sled    Maneuvers     Chemical rocket engines    Orbital launch systems    Aircraft

Thrust

Hydrogen and oxygen are stored in liquid form and combined in the rocket.

A rocket generates thrust by burning fuel and channeling the exhaust with a rocket cone.

Orbit

To reach orbit you need a velocity of 7.8 km/s. A one-stage rocket isn't enough and so multiple stages are used.

Saturn V
Saturn V stage separation
Ariane 5

New advances in astronautics

Rocket science is undergoing a renaissance and we can soon expect things such as asteroid mining and a manned Mars mission. Advances in astronautics include:

Stratolaunch pioneered high-altitude launch, using an aircraft consisting of two 747s fused together. Article.

Bigelow Corporation developed a space module that is substantially better than the International Space Station. Article.

Dr. Chang-Diaz perfected the ion drive, which has a much greater exhaust speed than chemical rockets. Article.

The first step toward solar system exploration is to build a base on the moon and launch lunar ice into space. Ice can be used for rocket fuel, life support, and radiation shielding, and this will enable large interplanetary spaceships to be built. Article.

Asteroid mining will soon become possible and will return trillions of dollars in platinum group metals. Article.

Using lunar ice we can build a manned base station at the L2 Lagrange point and from there build colossal space telescopes. This will revolutionize astronomy. Article.

Chemical rockets

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 over kerosene.

```Fuel     Exhaust    Fuel    Fuel boiling
speed    density     point
(km/s)   (g/cm3)      (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 cheap
```
Kerosene is a liquid consisting of hydrocarbon chains with between 6 to 16 carbon atoms per chain.
Air launch

Stratolaunch
Pegasus
Pegasus

Launching a rocket from the air brings several advantages over ground launch, such as:

*) The aircraft's speed adds to the rocket speed.

*) Air at 15 km has 1/4 the density of air at sea level, meaning less air drag.

*) The rocket can be launched at the equator so that the Earth's equatorial speed adds to the rocket speed.

These advantages mean that the payload for air launch is a larger fraction of the rocket mass than for ground launch, reducing the launch cost. Current launch cost for ground launch is \$2000/kg.

Launch systems are under development by Vulcan Aerospace (the Stratolaunch aircraft) and Virgin Orbit (the LauncherOne aircraft). The Stratolaunch is constructed from two 747 fuselages and 6 747 engines and can carry a 230 ton rocket.

The Stratolaunch moves at Mach .9. Ramjet aircraft can move at Mach 5, are easy to design, and in the near future these will be used. In the distant future scramjet aircraft will be used, which can reach Mach 12.

For launch to low Earth orbit, every bit of speed helps.

```                                       Speed      Speed
(km/s)     (Mach)

Earth rotation speed at equator             .46    1.6
Stratolaunch aircraft speed                 .27     .9
Speed of low Earth orbit                   7.8    26.4
Speed of hydrogen+oxygen rocket exhaust    4.4    14.9
Ramjet speed                               1.5     5
Scramjet speed                             3.5    12
```
Turbofan, ramjet, and scramjet

Atmospheric reentry

Space shuttle
Apollo mission
Mars rover
A reentry vehicle can have a mass as low as 3 tons. The space shuttle was inefficient because it had a mass of 78 tons. There's no need to bring back to the Earth anything more than necessary.

```                   Mass (tons)   Crew

Space shuttle          78.0       7
SpaceX Dragon V2        4.2       7
Soyuz reentry module    2.9       3
ISRO Reentry Vehicle    3.7       3
```

If the rocket fails during launch and the crew are in a lightweight reentry spacecraft then they have a chance at surviving.

SpaceX Dragon

Soyuz

Moon
Earth
Mars
Moon
Ceres
Sizes to scale.

Manned solar system exploration starts by building a base on the moon to mine ice. Ice can be used for rocket fuel, life support, and radiation shielding, and because of the moon's low gravity it is easily lifted into space. Once in space it can be used to make spaceships and propel them throughout the solar system.

To make rocket fuel, a power source such as solar cells is used to split ice into hydrogen + oxygen.

The biggest hazard to humans in interplanetary space is cosmic ray radiation. 3 meters of ice are required to stop the rays, implying a spaceship mass of at least 400 tons. This much ice is difficult to obtain from the Earth and easy to obtain from the moon. Furthermore, such a massive ship requires a lot of ice fuel to move around.

Ice is present on the moon in polar craters that never see the sun. Everywhere else, the sun boils it off. In the asteroid belt, the sun is weaker and ice is everywhere. Ceres has an ocean's worth of ice.

```     Orbit speed   Gravity    Atmosphere       Distance from
(km/s)      (m/s2)    density (kg/m3)     sun (AU)

Earth    7.8         9.8       1.22             1.00
Mars     3.6         3.7        .020            1.52
Moon     1.68        1.6       0                1.00
Ceres     .36         .27      0                2.77
```
Since the moon has low gravity and no atmosphere, it's ideal for electromagnetic sled launch. This will be the method used to launch ice into space in the distant future.
Future rockets

The exhaust speed depends on the energy/mass of the fuel.

```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     .000011
```
Chemical 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.
Nuclear battery

Radioactive Plutonium-238
Solar panels on the space station

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 Plutonium-238 nuclear batteries, which is why they are still functioning 30 years after their launch. Current plutonium-powered missions include Cassini, Galileo, New Horizons, and Ulysses.

Plutonium-238 and Strontium-90 are the isotopes used for nuclear batteries in space, and Curium-244 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
Curium-244       Thermo + Optic   40    18.1     .17
Curium-244       Thermo           20    18.1     .17
Strontium-90     Thermo            4    28.8     .01    Product of uranium fission
Plutonium-238    Thermo            5.4  87.7     .3
Plutonium-238    Stirling          4.1  87.7     .3
Nuclear reactor  Stirling        200      -       ?     Data for the SAFE-400 reactor
```
The numbers for Watts/kg are for the total system, including the isotope, the shielding, and the generator.

At 1 AU from the sun, solar cells have a much larger power/mass than nuclear batteries. The distance from the sun for which solar cells and nuclear batteries have equal power/mass is 8 AU, if we assume 5 Watts/kg for the nuclear battery. Missions within Jupiter are best equipped with solar cells and missions beyond Jupiter are best equipped with nuclear batteries.

Generators

Photoelectric cell
Thermoelectric generator
Stirling engine
Stirling engine

The following methods can convert thermal power to electric power.

```Isotope        Generator  Electrical   Fuel      Total       Temperature
efficiency   fraction  efficiency  (Kelvin)

Plutonium-238  Thermo         .07        .14      .0098       1050
Plutonium-238  Photo          .07        .14      .0098       1050
Plutonium-238  Stirling       .26        .038     .0099       1050
Strontium-90   Thermo         .06        .1       .006         800
Strontium-90   Photo          .06        .1       .006         800

Electrical efficiency:  Efficiency for converting heat to electricity
Fuel fraction:          Fuel mass / System mass
Total efficiency:       Electrical efficiency * Fuel fraction
```
The 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.

The efficiency of a generator improves with temperature, and nuclear materials don't have great melting points. One has to encase them in a metal with a higher melting point.

```        Melting point (Kelvin)

Tungsten     3693
Uranium      1405
Strontium    1050
Plutonium     913
Caesium       302
```

Isotopes that are useful for generating power
```              Watts   GJoules  Halflife  Decay   Decay   Cost   Produce  Stockpile
/kg     /kg     (years)   (MeV)   mode   (M\$/kg) (kg/yr)   (kg)

Cobalt-60      27300   4533     5.27     2.82    Beta,γ    1.3
Curium-244      4013   2293    18.1      5.80    Alpha      .17
Tritium         1540    598    12.3       .0186  Beta     30       .4
Caesium-137      864    824    30.2      1.17    Beta       .01   Huge     Huge
Plutonium-238    818   2265    87.7      5.59    Alpha    10      1         17
Strontium-90     648    589    28.8       .55    Beta       .01   Huge     Huge
```
The 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.

Strontium-90 and Caesium-137 are generated en masse as fission products in fission reactors.

For an isotope:

```Atomic mass unit         =  Mamu  =  1.661⋅10-27 kg
# of nucleons in nucleus =  N
Mass of nucleus          =  Mnuc N Mamu
1 MeV                    =          1.602⋅10-13 Joules    (1 Mega electron Volt)
Nucleus decay energy     =  Edecay
Nucleus energy/mass      =  S    =  Edecay / Mnuc
Decay half life          =  T
Heat power per kg        =  Qheat =  Edecay / T / Mnuc
Electric power per kg    =  Qelec
Efficiency               =  ε     =  Qelec / Qheat    (for converting heat to electric energy)
Fuel mass                =  Mfuel
System mass              =  Msystem
Fuel fraction            =  ffuel =  Mfuel / Msystem
System power per kg      =  Qsys  =  ε ffuel Qheat
```

Ion drives

Franklin Chang-Diaz

An ion drive uses electric power from a nuclear battery to accelerate ions. The values given in the table are for the ion drive designed by Chang-Diaz.

```Ion speed                        =  V               =      50 km/s
Mass of ion drive                =  M               =    1000 kg
Mass of ions ejected per second  =  m               = .000096 kg/s
Power consumed by the ion drive  =  Po   =  200000 Watts
Drive efficiency                 =  Q               =      .6     For converting electric to ion power
Power delivered to the ion beam  =  P  = Q Po = ½ m V2 =  120000 Watts
Agility  =  Power/Mass           =  p  = P/M        =     120 Watts/kg
Force generated by the ion beam  =  F  = m V        =     4.8 Newtons
Acceleration of the ion drive    =  A  = F/M = 2p/V =   .0048 m/s2
Gravity constant                 =  G               =6.67e-11 Newton meters2 / kg2
Earth-sun distance                                  =1.496e11 meters
Sun mass                                            =1.989e30 kg
Acceleration from sun                               =  .00593 meters/second2
```

The 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 and decreases the rocket force.

Pebble bed nuclear reactor

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.

Energy and power in space

```                   kWatts/kg    MJoules/kg

Space mirror        200             -
Capacitor           100             .010
Flywheel            100             .5
Battery               1.2           .8
Solar cell, Earth      .10          -
Solar cell, Mars       .043         -
Solar cell, Ceres      .013         -
Nuclear battery        .020   589000
Hydrogen+oxygen        -          13
```
Solar cells are more powerful than nuclear batteries if closer to the sun than Ceres, and weaker if beyond Ceres.
Fission fragment rocket

Fission produces 2 fragments
Fission fragment rocket

When uranium fissions it produces 2 high-speed 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 Californium-251.

```               Critical  Diameter  Halflife
mass      (cm)    (Myears)
(kg)
Californium-252   2.73      6.9      .0000026
Californium-251   5         8.5      .000290
Californium-249   6         9        .000351
Neptunium-236     7         8.7      .154
Curium-247        7.0       9.9    15.6
Curium-243        8        10.5      .000029
Plutonium-238     9.5       9.7      .000088
Plutonium-239    10         9.9      .024
Curium-245       10        11.5      .0085
Americium-242    11        12        .000141
Plutonium-241    12        10.5      .000014
Uranium-233      15        11        .159
Uranium-235      52        17     704
Neptunium-237    60        18       2.14
Plutonium-240    40        15        .0066
```

Fusion drive
Hydrogen bombs fuse deuterium and tritium to produce energy. The maximum efficiency for converting mass to energy is .00027, and in practice the efficiency is half this.

If we assume that all the energy goes into kinetic energy of exhaust, the exhaust speed is

```Kinetic energy  =  .5 M V2  =  .000135 M C2

V  =  4900 km/s  =  .0163 C
```
If hydrogen bombs are used for propulsion then the spaceship has to be large to absorb the recoil.
Thermal rocket

Nuclear thermal rocket

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       1.9
```
In space, thin reflective material can be used to construct a large low-mass 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.
Future orbital launch systems

The Stratolaunch aircraft is subsonic. A supersonic ramjet such as the SR-71 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/m3)

Ground                 0        0       1.22      Conventional ground launch
Subsonic aircraft       .3     14        .26      Stratolaunch aircraft
SR-71 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.

Electromagnetic sled launch

An electomagnetic launch sled rides on rails like a roller coaster and can reach a speed of 3 km/s. The sled releases a rocket which accelerates to orbit. The cost of the electricity is negligible compared to the cost of the rocket and hence sled launch reduces launch cost compared to ground launch.

If energy were the only contributor to launch cost then launch cost would be tiny. It costs typically 2000 \$/kg to launch material into space with rockets. If the kinetic energy comes from electricity then the electricity cost is \$1.

```Orbit speed           =  V  =  7.8  km/s
Cost of electricity   =  q  =   36  MJoules/\$
Orbit energy/mass     =  e  =   30  MJoules/kg  =  ½ V2
Electricity cost/mass =  c  =  .85  \$/kg  =  e/q
Typical launch cost         = 2000  \$/kg       Typical cost to use rockets to launch material to orbit
```

Launch sleds are best suited for inanimate cargo that can handle large acceleration. If an acceleration low enough for humans is used then the track is excessively long. If we use a human-friendly acceleration of 5 g's,

```Sled acceleration    =  A  =  50 m/s2   (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

V2 = 2 A X                X = ½ A T2
```
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.

If we launch inanimate equipment at an acceleration of 500 m/s2 then the track length is 9 km.

Lunar launch sled

The moon is ideal for sled launch because:

*) The lunar orbit speed is 1.68 km/s, well below the practical maximum of 3 km/s for sleds. The sled can be launched directly to orbit without needing a rocket stage.

*) The moon has no atmosphere and so you can launch horizontally and none of the kinetic energy is wasted on vertical motion.

*) Horizontal launch allows the track to be arbitrarily long, enabling human-friendly low-g launch.

*) The moon has abundant iron from metallic asteroid impacts for constructing the track.

The moon has abundant ice in polar craters and it can be launched into space in bulk with sleds. If the sled power comes from solar cells then the mass launch rate is:

```Launch speed             =  V          =  1.68  km/s
Launch energy/mass       =  e  = ½ V2  =  1.41  MJoules/kg
Solar cell average power =  p          =   200  Watts/meter2
Solar cell area          =  A                   meters2
Launch power             =  P  =  p A  =  e m   Watts
Mass launch rate         =  m  =  p A / e       kg/second
```
A spaceship needs 1000 tons of ice to shield cosmic rays. Launching this much ice in one month requires .5 MegaWatts and 2500 meters2 of solar cells.
Mars launch sled

Mars is ideal for a launch sled because the orbit speed is small, the air is thin, and there is a tall mountain. Launch at near horizontal angle is possible from the mountain.

```Mars launch speed         =  3.6  km
Sled practical max speed  =  3    km/s
Mars airmass              =   .16 tons/meter2
Earth airmass             = 10.1  tons/meter2
Mars Mount Olympus height = 21.2  km
Earth Mount Everest height=  8.8  km
```

Mountains

A sled launch track can use a mountain for altitude and launch angle. Possible mountains include:

```            Peak         Height   Earth    Airmass
(m)    rotation  (tons)
(km/s)
Nepal       Everest        8848    .41       3.1
China       K2             8611    .37       3.2
Nepal       Kangchenjunga  8586    .41       3.3
Argentina   Aconcagua      6962    .39       4.0
Peru        Huascaran      6768    .46       4.1
Peru        Yerupaja       6634    .46       4.2
California  Mt. Whitney    4421    .37       5.6
Colorado    Mt. Elbert     4401    .36       5.6
-           Equator           0    .46      10.1
```
Huascaran is the tallest peak that is close to the equator.

"Airmass" is the mass of air per meter2 above the given height.

The rocket has to have a mass of at least 100 tons for the airmass to not matter.

Rocket speed

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
Mi  =  Initial mass of rocket
Mf  =  Final mass of rocket after burning its fuel
Ve  =  Rocket exhaust speed
V(T)=  Rocket speed as a function of time.  V(0)=0.
Vf  =  Final rocket speed after burning its fuel
F   =  Force generated by the rocket
=  - Ve dM/dT

dV/dT =  F/M  =  -(Ve/M) * dM/dT
V(T)  =  V ln(Mi/M)
Vf    =  V ln(Mi/Mf)        Tsoilkovsky rocket equation
```

Oberth maneuver

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  =  6.67e-11 Newton meters2/kg2
Mass of Earth                   =  M  =  5.97e24 kg
Earth radius                    =  R  =  6371 km/s
Perigee radius                  =  R1                     Slightly larger than R
Apogee radius                   =  R2                     R1 << R2
Escape velocity                 =  Vesc=  11.2 km/s
Rocket speed at perigee         =  V1  =  Vesc
Rocket speed at apogee          =  0
Circular orbit speed at perigee =  Vcirc=   7.2 km/s  =  G M / R1
Circular orbit speed at apogee  =  0
Rocket speed change at perigee  =  Vroc =  16.6 km/s      Calculated below
Final exit speed from planet    =  Vexit=  25.4 km/s      Final speed after far from the planet
```
At apogee the energy is
```E  =  Kinetic energy  +  Gravitational energy
=         0        +         0
```
At perigee the energy is
```E  =  Kinetic energy  +  Gravitational energy
=     .5 m V12     -     G M m / R1

V12 =  2 G M / R1
=  2 Vcirc2
=  Vesc2
```
V1 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 Vesc2 = G M m / R1     →    Vesc2 = 2 G M / R1
```
IF the rocket fires at perigee and increases its speed by Vroc, the energy becomes
```E  =  .5 m (V1 + Vroc)2  -  G M m / R1
=  .5 m (Vesc + Vroc)2  -  .5 m Vesc2
=  .5 m (Vroc2 + 2 Vroc Vesc)
```
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 Vexit. When far from the Earth, the energy is
```E  =  .5 m (Vroc2 + 2 Vroc Vesc)
=  .5 m Vexit2

Vexit=  (Vroc2 + 2 Vroc Vesc)1/2  >  Vroc
```
If the spacecraft starts in an elliptical orbit and changes its speed by Vroc at perigee, it departs the Earth at speed Vexit, which is larger than Vroc. 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

```Vexit=  (52 + 2 * 5 * 11.2)1/2
=  11.7 km/s
```
This gets you to Mars in about 4 months.

```X axis:  Change in velocity at perigee (Vroc)
Y axis:  Departure velocity from the planet.  Vexit = (Vroc2 + 2 Vroc Vesc)
```
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
```

Rocket power and the Oberth maneuver

The Oberth maneuver requires a rocket with a large thrust-to-mass 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 mirror-based rockets are low-thrust and can't be used for the Oberth maneuver. The rocket engine with the largest force/mass is the Vulcain-2. For this rocket,

```Planet radius           =  R  =  6371 km for the Earth
Escape velocity         =  Ves=  11.2 km/s for the Earth
Oberth time             =  T  =   9.5 minutes for the Earth  =  R / Ve
=       Time that the rocket is near perigee
Rocket exhaust speed    =  Vex=   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 / Vex  =  102 m       Fuel mass burnt during one Oberth time
Oberth velocity         =  Vob=  16.6 km/s  =  3.9 Vex  =  [ln(M/m) - ln(2)] Vex  =  ln(.5 T Z / Vex) Vex
=  [ln(T) - 2.4] Vex

Momentum conservation:    M Vex  =  F T

```
During one Oberth time, a Vulcain-2 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.
Space mirror

The following parameters are for a JPL design of a space mirror composed of aluminum-coated mylar.

```Mylar density      =  1.39 g/cm3
Aluminum density   =  2.70 g/cm3
Mylar thickness    =  .025 mm
Aluminum thickness =  .010 mm
Surface density    =  .006 kg/m2         (JPL design)
Mirror area        = 104 km2
Mirror mass        = 6⋅107 kg
Launch cost per kg = 1000 \$/kg
Launch cost        = 6⋅1010 \$
```

Payload
```               Payload  Engine  Fuel  Empty  Total  Payload  Payload   Exhaust  Thrust
tons     tons   tons  tons   tons    \$/kg    fraction    m/s    MNewton

Airbus A380      100      25    200    277    602       4      .17       -       1.24
Stratolaunch     230                          540              .43       -       1.78
Falcon stage 1   111       5.7  411     22.2  433       -      .26      3.05     8.2
Falcon stage 2    22.8      .6  107      4.0  111       -      .21      3.41      .93
Falcon total      22.8     6.3  518       -   549    4100      .042      -        -
```

Spaceship design

Bigelow BA-330 habitat
Bigelow Genesis habitat

Astronauts can expect luxuriously large spaceships. The Bigelow BA-330 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 have thicker walls. Thicker walls are helpful for defending against micrometeorites and radiation.

```                 Volume    Mass   Wall thickness
(m3)    (tons)       (m)

Bigelow Genesis    11.5     1.36    .15
NASA Orion         19.6     8.91
Bigelow BA-330    330      23       .46
Space Station     837     450       .003
```

International Space Station
NASA Orion

Spaceship configuration

Equipment on a spaceship includes:

```                     Tons

Large habitat         23       Bigelow BA-330 330 meters3
Small habitat x2       3       Bigelow Genesis, 11 meters3
Solar cells           10       1 MWatt.  For life support, generating oxygen, and generating fuel
Life support system    1       Requires 1 kWatt/person
Food                   2       1 kg/person/day
Oxygen                 1
Nitrogen               1
Computers              1       Mighty gaming system and speaker system. Movie and TV library
Battery                4       3.2 GJoules.  Lithium-ion
Rockets x4             1
Fuel tanks             2
Ice                    4       Source of water, oxygen, and fuel
Fuel generator         1       Convert ice to hydrogen and oxygen and liquefy it

Total                 50
```

Life support

Aeroponic plants
Space station life support

The space station life support system requires:

Power: 1 kWatt/person
Water: 1 kg/person/day
Food: 1 kg/person/day

Fuel

Solar cells can convert ice to hydrogen+oxygen fuel. 1 kg of fuel can be produced with 1 kg of solar cells in 3.7 days.

```Energy to convert ice to fuel =  e  =  15.9 MJoules/kg
Efficiency factor             =  Q  =    .5
Solar cell mass               =  M
Solar cell power/mass         =  p  = 100   Watts/kg
Solar cell power              =  P  =  p M
Fuel production mass/time     =  m  =  M Q p / e
Production timescale          =  T  =  M / m  =  e p-1 Q-1  =  3.7 days
```

Radiation in space

The Earth's atmosphere is thick enough to block cosmic rays from space and Mars' atmosphere isn't. The walls of spaceships are too thin to protect against cosmic rays.

```            Atmosphere thickness
(tons/m2)

Venus              1000
Titan                73
Earth, sea level     10
Earth, 12 km high     4.9
Mars                   .16

mSieverts     Shielding thickness
/year       (tons/m2)

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/m2 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 highest-energy cosmic rays.

5 hour airplane flight incurs ~ .03 millisieverts.
A dose of 4800 millisieverts has a 50% risk of death.

Hydroponics

Air 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.

```         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       .00006
```

Artificial gravity

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  =  T2 A / (2π)2 =  228 meters
Velocity       =  V  =  2 π R / T   =  48 meters/second
Acceleration   =  A  =  V2 / R      =  10 meters/second2
```

Tether

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/m3 for Zylon
Tether tensile strength  =  P  =  F / Ar  =  5.8 GPa
Mass of spaceship        =  M
Radius of tether         =  R  =  T2 A / 4 pi2  =  V2 / A
Tether cross-section     =  Ar
Mass of tether           =  m  =  2 R Ar D
Centripetal acceleration =  A  =  10 meters/second2
Tether tension force     =  F  =  M A
Spaceship spin period    =  T  =  2 π R / V  =  30 seconds
```
The mass ratio of the tether to the spaceship is
```m/M  =  2 T2 D A2 / P / (4 π2)  =  1.33e-6 T2  = .00119
```
To 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.

Radiation shielding

The above data suggests that to shield against cosmic rays, you need at least 3 tons/meter2 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/meter3
Mass of ice shield    =  (4 &pi / 3) (R3-r3)  =  792 tons
```
The only way to get this much ice is from the moon. This is the source of the mass for the tether calculation.
Space stations

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 on-site and build colossal telescopes.

Radiation shielding

Cosmic rays consist mostly of high-energy 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/meter3)
V = Proton velocity
C = Speed of light
```
Proton kinetic energy is measured in MeV. 1 MeV = 1.6e-13 Joules. The rest energy of a proton is 1000 MeV.

Proton energy loss is governed by the "Bethe-Bloch" formula. For cosmic ray protons with E > 1000 MeV, the formula may be approximated as

```EnergyLoss  =  200 L (D/1000) MeV
```
If the proton is traveling through water with a mass density of 1000 kg/meter3, 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 low-energy protons from the solar wind but they are of no help in stopping cosmic rays. Mars' atmosphere isn't thick enough either.

When a high-energy 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 stopping-length of the matter
D  =  Density of the matter in kg/meter3
A  =  Atomic number of the nuclei in the matter
=  1 for protons
=  8 for oxygen

t = T exp(-L/S)

S  =  .35 A1/3 1000/D  meters
```
For 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 S
```
If 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/meter2.

This is an underestimate of the shielding required because when a high-energy 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.

Manned Mars mission

Mars Institute base in Northern Canada

Hohmann trajectory

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  months
```
The 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 Earth-Mars Hohmann orbit

Oberth maneuver

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             =  Vescape  =  11.2 km/s for the Earth
Speed change from the rocket    =  Vrocket  =  10   km/s for a mighty rocket
Departure speed from the planet =  Vdepart  =  18   km/s

V2depart  =  V2rocket + 2 Vrocket Vescape           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 Vrocket = 1 km/s then Vdepart = 4.8 km/s. This allows one to transport large payloads between planets, if speed isn't important.

Solar system pinball

The Oberth effect can be used on any planet or moon. The larger the mass of the object the more extreme the effect.

```
Vescape (km/s)

Moon        2.38
Mars        5.03
Earth      11.2
Saturn     35.5
Jupiter    59.5
Sun       618
```
Any 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  =  Vapproach

V2depart  =  V2approach + V2rocket + 2 Vrocket Vescape           Derivation
```

Mars Mission

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 Oberth-style 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 fully-fueled 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 low-Mars orbit. It can use fuel that was sent ahead of time from the Earth for this maneuver.

Once the spacecraft is in low-Mars 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 low-Mars 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 low-Mars 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.

Radiation shielding
```                                       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           ?                .29
```
To 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.

Getting around the solar system

Lagrange points
Earth Lagrange points
Interplanetary transport network

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 billiards-style 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
```

Gravity assists

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.

Atmospheres
Earth
Titan
Veuns
Mars
```      Gravity  Temp  Pressure  Density    N2       O2        CH4      CO2       Ar     Xe     H2S
(m/s2)   (K)    (Bar)    (kg/m3)   (kg/m3)   (kg/m3)

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   .015
```
For humans, xenon is an anaesthetic and H2S is toxic.
Gear for exploring Titan

Arctic scuba gear
Wingsuit (human-powered flight is easy on Titan)
Strontium-90 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.

Appendix

Calculation of the Hohmann trajectory

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  = Semi-major axis of the orbital ellipse
= .5 (R1 + R2)
E  = Energy
= Kinetic energy  +  Gravitational potential energy
= .5 m V^2        -  G M m / R
```
Angular momentum is conserved. Equating angular momentum at apogee and perigee,
```V1 R1 = V2 R2
```
Equating energy at apogee and perigee,
```E  =  .5 m V1^2 - GMm/R1  =  .5 m V2^2 - GMm/R2
```
Algebra 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/R
```
Make 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 Earth-Mars 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
=  .08897
```
To 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/s
```
The departure velocity from the Earth is
```D1 = 2.95 km/s
```
Upon arriving Mars, the velocity with respect to Mars is
```D1 = 2.65 km/s
```
The total change in velocity that the spacecraft must generate is
```D1 + D2  =  5.60 km/s
```
A hydrogen+oxygen rocket has an exhaust speed of 4.4 km/s and is capable of generating this change in velocity.
Oberth effect

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 Earth
```
When the spacecraft is at apogee, the energy is
```E  =  Kinetic energy  +  Gravitational energy
=         0        +         0
```
When 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^2
```
V1 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 / R1
```
Suppose 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 > D1
```
If 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/s
```
This 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 planet
```
This 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.
```
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Asteroid mining

A 100 meter metallic asteroid has 1 billion dollars of platinum and we already have the technology to get it. Metallic asteroids can be mined by distillation, where a space mirror focuses sunlight onto the asteroid to boil off the iron and leave behind the platinum.

The largest metallic asteroid is 16 Psyche, which has a diameter of 186 km and contains 100 quadrillion dollars of platinum. This is the core of the planet that failed to form in the asteroid belt.

The value of the elements in a metallic asteroid is:

```         Mass in    Element   Value of element
asteroid   cost/kg   in the asteroid
(tons)    (\$/kg)    (Billions of \$)

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     .004
Total    1000000              3.3
```
The most profitable elements are platinum, nickel, rhodium, iron, cobalt, and gold.

Sudbury basin, Canada

Sudbury, Canada
Sudbury geology

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
```

Space still

Alcohol still
Laphroaig distillery

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 low-value 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/cm3    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
```

Metallic core

Moon
Titan
Ceres
Pluto

Element classification

```Siderophile:  Iron-living. Tends to sink to the core along with the iron.
Lithophile:   Rock-loving. Tends to become included in rock and escapes sinking
to the core.
Chalcophile:  Ore-loving. 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 present-day asteroid belt. Pieces of the planet that are from the core are now metallic asteroids.
Solar abundance

```Dot size  =  log(Solar Abundance)
```
Elements with a dot size of zero have no stable isotope.

Solar abundance data

Universe composition

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            .0055     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.
Core amplification

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 =  cPt  =      .004 ppm
Iron abundance in the crust     =  cFe  = 63000     ppm
Platinum abundance in the core  =  CPt  =    19     ppm
Iron abundance in the core      =  CFe  =910000     ppm
Crust platinum/iron             =  cPt/cFe  =  .000000063
Core  platinum/iron             =  CPt/CFe  =  .000021
Core amplification factor       =  APt  =  (CPt/CFe) / (cPtcFe)  =  329
```
The elements with the highest core amplification factors are:
```        Amplification   Density (g/cm3)

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     Lithophile
```
Platinum 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.
Metallic asteroids

2011 UW158

"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⋅108      .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      .015
```
The 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.
Asteroid delivery

If a 1010 kg asteroid is broken with a hydrogen bomb then

```Mass of the asteroid            =  M  =  1010 kg
Energy of the hydrogen bomb     =  E  =  .5 M V2  =  4e16 Joules   (10 megatons of TNT)
Speed of the asteroid fragments =  V  =  3 km/s
```
A hydrogen bomb is capable of moving an asteroid.
Vaporizing a metallic asteroid

A space mirror can vaporize a metallic asteroid, leaving 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
Iron melting point              = 1811 Kelvin
Iron boiling point              = 3134 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 melting energy/mass        =  247 kJoules/kg
Iron boiling energy/mass        = 6089 kJoules/kg
Iron solid heat capacity        = .449 kJoules/kg/Kelvin
Iron liquid heat capacity       = .82  kJoules/kg/Kelvin
Iron solid heating energy/mass  =  714 kJoules/kg     (Energy to heat from  220 to 1811 Kelvin)
Iron liquid heating energy/mass = 1085 kJoules/kg     (Energy to heat from 1811 to 3134 Kelvin)
Iron total heating energy/mass  = 8135 kJoules/kg     (Energy to heat, melt, and vaporize)
Asteroid mass                   =   10 billion kg
Energy to vaporize asteroid     =  1017 Joules
Power from 1 km3 space mirror   =    1 billion Watts
Mass of a 1 km3 space mirror    =    6 tons
Time to vaporize asteroid       =  108 seconds  =  3 years
```

To calculate the departure speed of evaported iron atoms,

```Iron atom mass                  =  M  =  9.27e-26 kg
Boltzmann constant              =  k  =  1.38e-23 Joules/Kelvin
Iron boiling point              =  T  =      3134 Kelvin
Iron 3D energy/mass             =  E  =  3/2 k T / M  =  700 kJoules/kg
Iron 1D speed                   =  V  =  (k T / M)½  =  683 meters/second
Iron 1D kinetic energy          =  e  =  ½ k T  =  ½ M V2
```
Vaporization accounts for most of the energy requirement.

A 10 km3 space mirror can vaporize a 1 billion kg asteroid in 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).

Economic impact

The iron in a metallic asteroid can be burned to produce carbon-free energy.

```Fe + O2 → Fe2O3
```
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/meter3
Radius of iron asteroid        =  1544 meters
```
Some 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         .000009
```
Bringing 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.
Platinum production
```           tons/year

World         161
South Africa  110
Russia         25
Zimbabwe       11
Canada          7.2
USA             3.7
Other           3.8
```
Data for 2014
Cosmic element abundance

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
```

"Amplification" is the abundance of an element in a metallic asteroid divided by its abundance in the Earth's crust, with the value normalized so that it is "1" for iron.
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Asteroid Defense
Dr. Jay Maron

Meteor Crater, Arizona
Way of the intercepting fist

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

Asteroid damage

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

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

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

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

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

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

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

Early warning

Large Synaptic Survey Telescope (LSST)

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

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

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

Flux limit    Magnitude
(Watts/m2)    limit

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

Asteroid recoil

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

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

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

The detonation should maximize Q/W

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

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

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

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

Strategy                            Q/W
(m/s)

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

Calculations

Deflection strategy

A cannon provides an example calculation.

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

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

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

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

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

Velocity distribution of near Earth asteroids

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

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

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

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

Suppose you send a spacecraft to intercept the asteroid.

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

If the spacecraft is a hydrogen bomb, then

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

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

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

From the cannonball calculation,

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

Deflecting the asteroid requires

v T  >  R

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

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

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

W  >  .2 M/T

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

W  =  77 tons
```

Propulsion

VASIMR ion drive
Nuclear thermal rocket
Orion fusion rocket

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

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

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

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

How much does an asteroid impact heat the atmosphere?

```Heat capacity of air ~ 1.0$\cdot 103$  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/s
```
A 10 km asteroid has a mass of ~ 10^15 kg. If the asteroid is less massive than this then you don't have to worry about cooking the atmosphere. The dinosaur-extinction asteroid was ~ 10 km in size.
Cosmic disasters

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

Asteroids that have passed close to the Earth

```Q = Radius of closest approach / Radius of Earth

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

Problems

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

Gravitational energy = .6 G Mass^2 / R

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

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

Mr Miyagi: Best block... not be there

Fuel
Black: Carbon    White: Hydrogen    Red: Oxygen

Methane (Natural gas)
Ethane
Propane
Butane (Lighter fluid)
Octane (gasoline)
Dodecane (Kerosene)

Hexadecane (Diesel)
Palmitic acid (fat)
Ethanol (alcohol)

Glucose (sugar)
Fructose (sugar)
Galactose (sugar)
Lactose = Glucose + Galactose
Starch (sugar chain)
Leucine (amino acid)

ATP (Adenosine triphosphate)
Phosphocreatine
Nitrocellulose (smokeless powder)
TNT
HMX (plastic explosive)

Lignin (wood)
Coal

Medival-style black powder
Modern smokeless powder
Capacitor
Lithium-ion battery
Nuclear battery (radioactive plutonium-238)
Nuclear fission
Nuclear fusion
Antimatter

ATP (Adenosine triphosphate)

Phosphate
ADP (adenosine diphosphate)
ATP (adenosine triphosphaste)

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 ADP
```
ATP is assembled by the ATP-synthase enzyme. ATP and ATP-synthase 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.

ATP synthase
ATP synthase
ATP synthase

Video of the ATP-synthase enzyme.        Discussion of the physics of ATP

Creatine

Creatine
Phosphocreatine
Creatine kinase enzyme catalyst

When ATP is depleted it can be regenerated anaerobically with creatine phosphate.

```Phosphocreatine + ADP   →   Creatine + ATP
```
The 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.

Lactic acid

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
```

ATP energy and power

```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
```

Appendix

Rocket engines

Hydrogen + Oxygen rocket

```                         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
RL-10A            HOX               4.42     167    99.1   606   Atlas V. Diameter = 2.13 meters
RL10B-2           HOX               4.547    277   110     406   Atlas V and Delta IV rockets

Mitsubishi LE-5B  HOX               4.38     285   137.2   490
Mitsubishi LE-7A  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
RS-68             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
```

Rockets for reaching low Earth orbit

Saturn V
Ariane 5
Ariane 5

Stratolaunch
Pegasus
Pegasus

```                        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 6-engine 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.

Aircraft

SR-71 Blackbird
Concorde

```                                Engine   Engine    Empty  Max    Cargo
Speed  Ceiling  thrust    mass     mass  takeoff mass
(Mach)   km     (tons)   (tons)    (tons) (tons) (tons)
Blackbird SR-71  3.3    25.9  2 x 14.8  2 x 2.7    30.6   78           Spy
F-15 Eagle       2.5    20.0  2 x 11.3  2 x 1.70   12.7   30.8         Fighter
F-22 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 C-5 Galaxy .8          4 x 19.4  4 x 3.63  172.4  381    122.5  Cargo
Boeing 747-8F     .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 platform
```
The Stratolaunch (in development) is designed to launch rockets into space.
Air drag
```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.
Rocket fuel
```
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    Vulcain-2
Liquid methane   3.7      .42     111.7    493          ?     8200       4.6        ?    Raptor
Kerosine         3.3      .80     410      361       1270      801       1.676    630    Merlin-1D
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  SR-71 Blackbird
```
Hydrogen, 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 thrust-to-mass ratio of the rocket engine.

The density of liquid oxygen is 1.14 g/cm^3 and the boiling point is 90.2 Kelvin.

Electrolysis of water into H2 and O2

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
```

Speed of HOX rocket exhaust

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 V2

V = 5.132 km/s
```
In practice, the best HOX rockets have an exhaust speed of 4.4 km/s.

Fission fragment rocket

Mean energies for the fission of Uranium-235, 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 U-235 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 Uranium-235 = 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
```
Distribution of fragment masses

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
Americium-242       .5         141 years          Costs ~ 10^6 \$/kg
Californium-251     .9         898 years
Curium-245         1.1        8500 years
Plutonium-239      5.6      241000 years
Uranium-235       11.0         704 million years
```
For a fission fragment rocket, the lower the critical mass the better. All of the above isotopes produce similar energy when fissioned.
Fusion drive

Hydrogen bombs use the following reactions.

```Neutron    +  Lithium6  ->  Tritium  +  Helium4  +   4.874 MeV
Deuterium  +  Tritium   ->  Helium4  +  Neutron  +  17.56  MeV
```
Leaving 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)
=  .00298
```
The theoretical limit for the efficiency of a hydrogen bomb is
```f = .00027
```
In practice, f is half this.
Thermal rockets

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
Uranium-235        8.0e16 Joules/ton
Solar energy       1.4e15 Joules.  1 km^2 collector operating for 10^6 seconds at 1 A.U.
```
A mirror-based 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  ~  1.4e15
```
The mass of the mirror is
```                            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.

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

Space mirror

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/s
```
The acceleration of a mirror rocket is limited by the strength of the mirror.
Launch cost

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.
Most of the launch cost is in the superstructure, not the fuel.
If the kinetic energy of the 1 kg payload comes purely from electricity, the cost of the electricity is tiny.
```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  \$
```

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