Main site of science textbooks
Crowdfunding site for the free
online science textbooks project

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. New advances in astronautics include:

SpaceX pioneered the methane rocket, which is an improvement over traditional kerosene rockets. This improves the first stage of the rocket. SpaceX also pioneered a self-landing first stage, saving on launch cost. Article.

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.

Scramjet

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/s^2)    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 nuclear reactors
Plutonium-238    Thermo            5.4  87.7     .3     Scarce isotope
Plutonium-238    Stirling          4.1  87.7     .3     Scarce isotope
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.
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.


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

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.


Ion drives

Chang-Diaz
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 Chang Diaz ion drive.

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
Efficiency of the drive         =  Q  =    .6         For converting electric to ion power
Power delivered to the ion beam =  P  =  120000 Watts    =  Q Po  =  .5 m V^2
Force generated by the ion beam =  F  =     4.8 Newtons  =  m V
Acceleration of spacecraft      =  A  =   .0048 m/s2     =  F / M  =  2 P / (M V)
Agility  =  Power/Mass          =  Agi=     120 Watts/kg =  P / M
At fixed ion speed, the acceleration is determined by the power-to-mass ratio of the power source.
A  =  (2/V) * (P/M)

At fixed power there is a tradeoff between F and V:

P  =  .5 F V
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, decreases the rocket force, and increases the travel time.
Ion spacecraft

Suppose a spacecraft consists of

Ion Drive mass              =  Mdrive = 1000 kg   Chang-Diaz VF-200 design
Solar cell mass             =  Mcell  = 1000 kg   To power the ion drive
Argon mass                  =  Margon = 1000 kg   Ions for the ion drive
Scientific equipment mass   =  Mequip = 1000 kg
Spacecraft total mass       =  Mship  = 4000 kg
Solar cell power/mass       =  Q  =  300 Watts/kg
Solar cell power            =  P  =  Mcell Q
Ion drive operation time    =  T  =  107 seconds
Ion drive efficiency        =  e  =  .60
Ion velocity                =  V  =  60000 (Mcell/Margon)½
Ion energy                  =  E  =  P T e  =  ½ Margon V2
Gravity constant            =  6.674e-11 Newton meters2 / kg2
Earth-sun distance          =  1.496e11 meters
Sun mass                    =  1.989e30 kg
Earth acceleration          =  .00593  meters/second2
Spacecraft recoil velocity  =  Vs =  V Margon / Mship  = 15 km/s
Spacecraft acceleration     =  A  =  .0015
Time to burn through fuel   =  T  =  Vs/A  =  116 days
5 km/s corresponds to 1 AU/year. Using a gravity assist from Jupiter, an ion spacecraft can get anywhere in the solar system within 10 years.
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

The Holloman Air Force Base does hypersonic research using a sled that can reach a speed of 2.88 km/s.

A launch sled can convert electrial power to sled kinetic energy with an efficiency of 90%.

Example values:

Sled acceleration    =  A  =  50 m/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 = .5 A T2
If we launch inanimate equipment at an acceleration of 500 m/s2 then the track length is 9 km.

If a sled is moving at 3 km/s then a centripetal acceleration of 5 g corresponds to a radius of curvature of 180 km. The last half of the track has to be straight.

The sled only needs to reach an altitude of ~ 40 km. The rocket can do the rest. If it is launched from Everest then it needs to gain an altitude of ~ 30 km. The vertical velocity required to gain 30 km of altitude is .78 km/s. If the horizontal velocity is 3.0 km/s then the launch slope is .25.

A sled can use a heavy heat shield, which isn't possible with a rocket.


Mountains

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

Peak          Height   Earth    Airmass  Mountain range
               (m)    rotation  (tons)
                       (km/s)
Equator           0    .465     10.1     Sea level
Huascaran      6768    .458      4.1     Huascaran
Yerupaja       6634    .457      4.2     Huascaran
Everest        8848    .41       3.1     Himalayas, Everest
Kangchenjunga  8586    .41       3.3     Himalayas, Everest
Aconcagua      6962    .391      4.0     Aconcagua
K2             8611    .37       3.2     Himalayas, Karakoram
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

We various launch vehicles:

               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      -        -

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  $

Main page

Support the free online science textbooks project






© Jason Maron, all rights reserved.