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The Tyranny of the Rocket Equation



 
 
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Old March 13th 18, 04:19 AM posted to alt.astronomy
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Default The Tyranny of the Rocket Equation

The Tyranny of the Rocket Equation
05.01.12

By Expedition 30/31 Flight Engineer Don Pettit

Tyranny is a human trait that we sometimes project onto Nature. This
projection is a form of rationalization, perhaps a means to cope with
matters that we cannot control. Such is the case when we invent machines
to free us from the bounds of Earth, affecting our escape into space. If
we want to expand into the solar system, this tyranny must somehow be
deposed.

Rockets are momentum machines. They spew gas out of a nozzle at high
velocity causing the nozzle and the rocket attached to it to move in the
opposite direction. Isaac Newton correctly defined the mathematics for
this exchange of momentum in 1687. Conservation of momentum applied to a
rocket was first done by Russian visionary and scientist Konstantin
Tsiolkovsky in 1903. All our rockets are governed by Tsiolkovsky’s
rocket equation.

The rocket equation contains three variables. Given any two of these,
the third becomes cast in stone. Hope, wishing, or tantrums cannot alter
this result. Although a momentum balance, these variables can be cast as
energies. They are the energy expenditure against gravity (often called
delta V or the change in rocket velocity), the energy available in your
rocket propellant (often called exhaust velocity or specific impulse),
and the propellant mass fraction (how much propellant you need compared
to the total rocket mass).

The energy expenditure against gravity is specified by where you want to
go. For human exploration, there are only a handful places we can
realistically consider at this time. The most likely candidates a
from the surface of Earth to Earth orbit, Earth orbit to surface of the
Moon, Earth orbit to surface of Mars, Earth orbit to cis-lunar space
(the region between the Earth and the Moon, including a variety of
locations such as Lagrange points, geostationary orbit, and more). Of
course there are permutations to these routes but they are the most
likely ones considering our current state of technology.

In planning an expedition into space, we first must select where we want
to go. The energy expenditure against gravity is then specified by the
starting and ending points of our journey. As humans, we are powerless
to change this number. We simply have to accept its consequences. I like
to think of this as the travel cost.

Next we need to choose the type of rocket propellant, thus specifying
the available energy. Currently, all our human rated rocket engines use
chemical reactions (combustion of a fuel and oxidizer) to produce the
energy. There are limits to the quantity of energy that can be extracted
from chemistry and thus bounds placed outside of human control on the
energy we can pack into a rocket. Some of the most energetic chemical
reactions known are chosen for rocket propulsion (e.g. like
hydrogen-oxygen combustion) and thus, the second variable is now
specified. Again, we simply have to accept the limit to what chemistry
can offer (unless we choose other energy sources, such as nuclear). I
like to think of this selection as what you have to pay for the travel
cost.

With these two variables set, the rocket mass fraction is now dictated
by the rocket equation. We must build our rocket within this mass
fraction or it will not reach its destination. This also applies to
existing rockets when new uses are contemplated. There is very little we
can do to alter this result. With some clever engineering we might be
able to shave a few percentage points off the fraction, but the basic
result is set by the gravitational environment of our solar system
(choice of where we want to go) and the chemistry of the energetic bonds
of our selected chemical components (choice of propellant).

It is constructive to put a few numbers together to illustrate the grip
that simple momentum balance places upon our rockets. Here the
approximate cost in energy has been given in terms of velocity
(kilometers per second, km/s), a common ploy engineers use to simplify
the discussion. These numbers assume ideal conditions such as no losses
for atmospheric drag or combustion but are close enough for the sake of
this illustration.

Space shuttle Endeavour launches
Space shuttle Endeavour launches in November 2008 carrying seven STS-126
crew members including Mission Specialist Don Pettit. Credit: NASA

Destination Energy Cost (km/s)
Surface of Earth to Earth orbit: 8
Earth orbit to cis-lunar locations:
Lagrange points:
Low Lunar orbit:

3.5
4.1
Earth orbit to near-Earth asteroids: 4
Earth orbit to surface Moon: 6
Earth orbit to surface Mars: 8

From this simple table, a few conclusions can be drawn. Travelling from
the surface of Earth to Earth orbit is one of the most energy intensive
steps of going anywhere else. This first step, about 400 kilometers away
from Earth, requires half of the total energy needed to go to the
surface of Mars. Destinations between the Earth and the Moon are only a
fraction of that required to simply get into Earth orbit. The cost of
this first step is due to the magnitude of Earth’s gravity. And physics
dictates that paying a penny less than the full cost will result in
Earth repossessing your spacecraft in a not so gentle way. The giant
leap for mankind is not the first step on the Moon, but in attaining
Earth orbit.

Listed next are the major categories for our chemical rocket propellants
and their energy content used for payment of the gravitational cost of
travel. These are selected from propellants with an operational history
in manned spacecraft. “Hypergols” are contact-ignited propellants, used
in the Lunar Module ascent stage to simplify the engine design and
methane-oxygen has not been used in space to date, but is under
consideration for future human missions to the Moon and Mars. The first
law of thermodynamics was used to convert the energy of combustion into
an equivalent exhaust velocity so that these units of payment are
consistent with the costs shown above.

Soyuz TMA-03M rocket launches
The Soyuz TMA-03M spacecraft launches in December 2011 carrying three
Expedition 30 crew members including Flight Engineer Don Pettit. Credit:
NASA

Propellant Payment Energy (km/s)
Solid Rocket 3.0
Kerosene-Oxygen 3.1
Hypergols 3.2
Earth orbit to near-Earth asteroids: 3.4
Methane-Oxygen 4.5

Hydrogen-oxygen is the most energetic chemical reaction known for use in
a human rated rocket. Chemistry is unable to give us any more. In the
1970’s, an experimental nuclear thermal rocket engine gave an energy
equivalent of 8.3 km/s. This engine used a nuclear reactor as the source
of energy and hydrogen as the propellant.

Since the giant leap for mankind is the first step off of Earth, our
illustration of the rocket equation uses earth orbit as the destination
with the cost of 8 kilometers per second. To pay for this cost, each of
the chemical propellants above are used with the rocket equation which
results in the following mass fractions (given as percent of the total
rocket mass):

Expedition 30/31 Flight Engineer Don Pettit
NASA astronaut Don Pettit enjoys a snack in the Unity node. Credit: NASA

Propellant Rocket Percent Propellant for Earth Orbit
Solid Rocket 96
Kerosene-Oxygen 94
Hypergols 93
Methane-Oxygen 90
Hydrogen-Oxygen 83

These are ideal numbers free from losses due to atmospheric drag,
incomplete combustion, and other factors that reduce the efficiencies of
a rocket. Such losses make these numbers even worse (moving the mass
fraction closer to a rocket being 100% propellant). However, clever
engineering constructs such as rocket staging, multiple kinds of
propellants (1st stage solids or kerosene, upper stages hydrogen), and
gravitational lean (converts radial velocity into tangential) can help
compensate. When making a rocket that is near 90% propellant (which
means it is only 10% rocket), small gains through engineering are
literally worth more than their equivalent weight in gold.

Real mass fractions from real rockets include the effect of many
engineering details. However, these machines at root are the result of
the simple application of Tsiolkovsky’s rocket equation. The ideal
results presented here are not far removed from actual rockets. The
Saturn V rocket on the launch pad was 85% propellant by mass. It had
three stages; the first using kerosene-oxygen and the second and third
stages using hydrogen-oxygen. The Space Shuttle was also 85% propellant
by mass, using a blend of solids and hydrogen-oxygen for the first stage
and hydrogen-oxygen for second. The Soyuz rocket is 91% propellant by
mass and uses kerosene-oxygen in all of its three stages. There is an
advantage to using hydrogen-oxygen as a high performance propellant;
however, it is technically more complex. Kerosene offers less
performance but gives a simpler, robust, and easier to fabricate rocket.
These numbers represent the best that our engineering can do when
working against Earth’s gravity and the energy from chemical bonds.

What are the engineering implications of fabricating a rocket that is
85% propellant and 15% rocket? The rocket must have engines, tanks, and
plumbing. It needs a structure, a backbone to support all this and it
must survive the highly dynamic environment of launch (there is fire,
shake, and force at work.) The rocket must be able to fly in the
atmosphere as well as the vacuum of space. Wings are of no use in space;
small rocket thrusters are used to control attitude. Then there are
people with their pinky flesh and their required life support machinery.
Life support equipment is complex, problematic, and heavy. You can’t
roll down the windows if the cabin gets a bit stale. If you want to
return to Earth (and most crews do), there has to be structure to
protect the crew through a fiery entry and then provide a soft landing.
Wings are heavy but allow soft landings at well equipped airfields.
Parachutes are light, giving a big splash finale. The Soyuz goes thump,
roll, roll, roll; aptly described by one of my colleagues as a series of
explosions followed by a car wreck. And finally, you want to bring some
payload – equipment with which to do something other than just be in
space. “Because it is there” (or possibly because it is not there,
depending on your definition of a vacuum) is fitting for the first time
but subsequent missions need a stronger justification. Missions into
space to do meaningful exploration require bringing significant payload.

Real payload fractions from real rockets are rather disappointing. The
Saturn V payload to Earth orbit was about 4% of its total mass at
liftoff. The Space Shuttle was only about 1%. Both the Saturn V and
Space Shuttle placed about 120 metric tons into Earth orbit. However,
the reusable part of the Space Shuttle was 100 metric tons, so its
deliverable payload was reduced to about 20 tons.

It is instructive to compare rocket mass fractions to those of other
everyday Earth vehicles. Here, the approximate numbers for propellant
(or fuel when air is used as the oxidizer) are given to illustrate the
general categories of mass fractions:

Expedition 30/31 Flight Engineer Don Pettit
NASA astronaut Don Pettit works with two still cameras mounted together
in the Destiny laboratory. Credit: NASA

Vehicle Percent Propellant (fuel)
Large Ship 3
Pickup Truck 3
Car 4
Locomotive 7
Fighter Jet 30
Cargo Jet 40
Rocket 85

The percent propellant has huge implications on the ease of fabrication
and robustness in achieving the engineering design (and cost). If a
vehicle is less than 10% propellant, it is typically made from billets
of steel. Changes to its structure are readily done without engineering
analysis; you simple weld on another hunk of steel to reinforce the
frame according to what your intuition might say. I can easily overload
my ¾ ton pickup by a factor of two. It might be moving slowly but it is
hauling the load.

Once the vehicles become airborne, the engineering becomes more serious.
Light weight structures made of aluminum, magnesium, titanium,
epoxy-graphite composites are the norm. To alter the structure takes
significant engineering; one does not simply weld on another chunk to
your airframe if you want to live (or drill a hole through some
convenient section). These vehicles cannot operate far from their
designed limits; overloading an airplane by a factor of two results in
disaster. Even though these vehicles are 30 to 40% propellant (60 to 70%
structure and payload), there is room for engineering to comfortably
operate thus there is a robust, safe, and cost effective aviation industry.

Rockets at 85% propellant and 15% structure and payload are on the
extreme edge of our engineering ability to even fabricate (and to pay
for!). They require constant engineering to keep flying. The seemingly
smallest modifications require monumental analysis and testing of
prototypes in vacuum chambers, shaker tables, and sometimes test
launches in desert regions. Typical margins in structural design are
40%. Often, testing and analysis are only taken to 10% above the
designed limit. For a Space Shuttle launch, 3 g’s are the designed limit
of acceleration. The stack has been certified (meaning tested to the
point that we know it will keep working) to 3.3 g’s. This operation has
a 10% envelope for error. Imagine driving your car at 60 mph and then
drifting to 66 mph, only to have your car self-destruct. This is life
riding rockets, compliments of the rocket equation.

Here are a few other interesting examples from container engineering to
further illustrate the extreme nature of rocket design:

A zucchini plant
A zucchini plant grows inside the International Space Station. Credit: NASA
› Read Don Pettit's Letters to Earth and the Diary of a Space Zucchini

Other Containers Percent Useful Contents
Soda Can 94
Shuttle External Tank 96
Molotov Cocktail 52

The common soda can, a marvel of mass production, is 94% soda and 6% can
by mass. Compare that to the external tank for the Space Shuttle at 96%
propellant and thus, 4% structure. The external tank, big enough inside
to hold a barn dance, contains cryogenic fluids at 20 degrees above
absolute zero (0 Kelvin), pressurized to 60 pounds per square inch, (for
a tank this size, such pressure represents a huge amount of stored
energy) and can withstand 3gs while pumping out propellant at 1.5 metric
tons per second. The level of engineering knowledge behind such a device
in our time is every bit as amazing and cutting-edge as the construction
of the pyramids was for their time.

A veteran astronaut who has been to the Moon once told me, “Sitting on
top of a rocket is like sitting on top of a Molotov cocktail”. I took
his comment to heart by first weighing a bottle of wine, emptying the
bottle, and weighing it again. Simple engineering analysis allowed me to
estimate and compensate for the density difference between wine and
gasoline (which, for this particular vintage, I am sure was not much
different). A Molotov cocktail was measured to be 52% propellant. So
sitting on top of a rocket is more dangerous than sitting on a bottle of
gasoline!

Another less recognized side effect of the rocket equation is the
sensitivity of completing the rocket burn to obtaining your goal. To
illustrate this, I will use some numbers from my Shuttle flight, STS 126
in November 2008. Our target velocity at main engine cut off was 7824
m/s (25819 ft/s). If our engines shut down at 7806 m/s (25760 ft/s),
only 18 m/s (59 ft/s) shy of the target value, we would make an orbit
but not our designated target orbit. We would not be able to rendezvous
with space station and would lose our mission objective. Like being two
pennies short of a ten dollar purchase, this is only 0.2% less than the
price of admission into space. In this case, we do have some options. We
could burn our orbital maneuvering propellant and make up this
difference. If we were 3% shy of our target, 7596 (25067 ft/s) we would
not have sufficient orbital maneuvering propellant and we would not make
any orbit. We would be forced into a trans-Atlantic abort, falling back
to Earth and landing in Spain. This final 3% of our required velocity
comes during the last 8 seconds of our burn. For astronauts and bull
riders, 8 seconds is a long time.

If the radius of our planet were larger, there could be a point at which
an Earth escaping rocket could not be built. Let us assume that building
a rocket at 96% propellant (4% rocket), currently the limit for just the
Shuttle External Tank, is the practical limit for launch vehicle
engineering. Let us also choose hydrogen-oxygen, the most energetic
chemical propellant known and currently capable of use in a human rated
rocket engine. By plugging these numbers into the rocket equation, we
can transform the calculated escape velocity into its equivalent
planetary radius. That radius would be about 9680 kilometers (Earth is
6670 km). If our planet was 50% larger in diameter, we would not be able
to venture into space, at least using rockets for transport.

Revolting against tyranny is a recurring human trait and perhaps we will
figure some way to depose the rocket equation and venture away from our
planet in a significant way. I am referring to exploration with
continuous human presence with the first step like Antarctic-type bases
(which support several thousand people) and eventually leading to
colonization, a template comparable to the expansion of western
civilization across the globe during the 17th and 18th centuries. To
call yourself a sea-faring nation in that time meant that you could set
sail on a variety of missions in a number of different types of vessels
to a myriad of destinations whenever you wanted. We have a long way to
go before anyone can claim to be a space-faring nation.

The giant leap for mankind is not the first step on the Moon but
attaining Earth orbit. If we want to break the tyranny of the rocket
equation, new paradigms of operating and new technology will be needed.
If we keep to our rockets, they must become as routine, safe, and
affordable as airplanes. One of the most rudimentary and basic skills to
master is to learn how to use raw materials from sources outside the
Earth. Our nearest planetary neighbor, the Moon is close, useful, and
interesting. Extracting and producing useful products from the raw
materials of the Moon would relieve us from the need to drag everything
required in space from the bottom of Earth’s deep gravity well,
significantly altering the consequences of the rocket equation more in
our favor. The discovery of some new physical principle could break the
tyranny and allow Earth escape outside the governance of the rocket
paradigm.

The need for new places to live and resources to use will eventually
beckon humanity off this planet. Having access to space removes the lid
from the Petri dish of Earth. And we all know what eventually happens if
the lid is not removed.

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