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The rocket equation is critical in that we use that to determine number one, how big our
engines should be, and number two, how much propellant we load based on data we have
on available choices of propulsion and thrusters and engines, so that's a very critical
item….Juno was unique, though, in that it was unlike anything we’ve ever built in that
because it’s a spinner and it had to operate in this dynamic environment called the
Jovian system, normally we would have designed it with two tanks, one for the oxidizer
and one for the fuel and they’d be stacked like a launch vehicle but that’s a really bad
idea in a spacecraft that spins you get what’s called a lot of nutation and that’s the last
thing you want and it’s actually trying to turn from spinning say if you’re looking at a
finger and you’re spinning around the diameter of the finger with nutation and velocity it
can actually turn into a flat spin where it’s more like your finger is going around like the
hands of a clock which is not a good idea you should be spinning around the diameter of
your finger but by placing the six tanks around the outside that led to the configuration
right now which makes for a very stable spacecraft.
One of the more interesting things every time we do a maneuver, we saw even in the main
engine flush is again we’re spinning and the liquid is outboard of the spinning but due to
conservation of angular momentum when we fire the main engine the net acceleration
vector on the spacecraft changes and the liquid settles a little bit to the bottom but in
doing that it gets a closer to the CG and the spacecraft spins up a little bit and as soon as
you stop accelerating it spins right down…it was something like a couple hundreths of an
RPM, definitely measureable. You have this multi-thousand kilogram spacecraft but you
still have a lot of propellant onboard too and you move that maybe four or five inches, it
changes the inertia.
Tim Martin
Spacecraft Engineer, Propulsion Lead, Juno
Chapter 7 – Rocket Propulsion Physics
To move any spacecraft off the Earth, or indeed forward at all, there must be a system of
propulsion. All rocket propulsion relies on Newton’s Third Law of Motion: in order to
move the spacecraft forward, something else must be pushed backward. For most
propulsion systems, some of the material in the spacecraft is pushed out the back end of
the spacecraft, and the resulting reaction propels the craft forward. Thus, most rocket
engines are called “reaction motors”.
Rockets illustrate Newton’s Three Laws of Motion
Isaac Newton developed his Three Laws of Motion during the summer of 1666. They
would undergo various restatements, even during his lifetime, but they are the
foundations of modern physics (see the “Three Laws of Motion” box for the historical
development and the statement of the laws).
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Rocket motion demonstrates all three laws in an ideal way. For the purposes of
discussion, assume space is a vacuum (interplanetary space is a pretty good
approximation of this). We’ll start by firing the rocket.
Rocket version of the First Law: Imagine a rocket far from any source of gravity, like the
Earth or Sun, or any source of friction, like an atmosphere. If the engine on the rocket
were to fire, the rocket would be propelled in a straight line, forever. Of course, this is
never true; there is always a source of gravity
and a source of friction. But this helps
explain the motion of the rocket!
An orbit, such as LEO, is curved. This seems
to violate the First Law, but the First Law
provides an explanation: there is an outside
force acting on the rocket: the gravity of the
Earth. This force is always directed towards
the center of the Earth, so the rocket, which is
trying to move in a straight line parallel and
tangent to the Earth’s surface, is continually
yanked downward so that its path ends up a
circle around the Earth.
Rocket version of the Second Law: This law explains the effect of force on a rocket. An
engine will produce thrust, which is synonymous with force. The thrust produces an
increase in speed (an acceleration) on the rocket. Clearly, the same thrust that would
produce a high acceleration on a small rocket would produce a much lower acceleration
on a larger rocket.
To use a different example, the same force that accelerates an interplanetary probe into an
outer solar system orbit using a gravity assist trajectory will decelerate the much much
larger planet by only a tiny amount.
In fact, in a rocket, the force that a rocket can propel itself is proportional to the amount it
can accelerate the gases produced by the fuel being burned (a topic visited next chapter).
Much of the acceleration of the gas particles is due to the nozzle, the constriction between
the fuel combustion chamber and the exterior of the rocket (shown in the diagram on the
next page). Small changes in the design of the nozzle can alter greatly the acceleration of
the gas particles.
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Rocket version of the Third Law: This is the heart of how a rocket moves. As the exhaust
from the engine leaves the nozzle on the rocket, it accelerates and thus a force is exerted
on the whole exhaust plume, pointed away from the nozzle. The Third Law states that
there is an equal-magnitude but oppositely-directed force on the rocket itself. That is,
there is a force that acts on the rocket in its “forward” direction. As noted elsewhere,
there need not be a surface that the exhaust bounces off of; a rocket in the middle of
space can still propel itself forward by burning fuel in its engine.
The Third Law is the fundamental principle behind the motion of a rocket. The Third
Law is a statement of the conservation of momentum. Momentum is the quantity of an
object that is its mass multiplied by its velocity. This quantity, like energy, must be
conserved (cannot be changed in quantity) by a system. Unlike energy, momentum can’t
change form.
You may have seen movies where astronauts are marooned in their spacesuits, far from
their spacecraft. You may have seen how they use small gas canisters in their suits to
eject gas and propel themselves towards the spacecraft.
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Since the astronaut was not moving prior to opening the gas jets, her momentum was
zero; she has no velocity, though obviously she has mass. After she opens the gas jets,
some mass moves away from her at some speed. Just for the sake of directionality, let’s
place the astronaut at zero on a number line, and the gas she fires off is moving in the
negative direction.
According the principle of conservation of momentum, the total amount of momentum in
the system (astronaut plus gas) must be zero, since it was zero to begin with. Thus, since
the gas is moving in the negative direction, the astronaut is pushed in the forward
direction. The more mass of gas she ejects, the more she will move in the opposite
direction. If this sounds like Newton’s Third Law of Motion (action/reaction), it is
intentional; the conservation of momentum is a consequence of the Third Law.
Thus, a launch vehicle, like the Saturn V that took Apollo missions to the Moon, does not
require the plume of superheated gases it emits to push off of anything. Even in Earth
orbit and beyond, a rocket will move simply because there is some gas coming out of its
nozzle. But how do you quantify the force necessary to move such a rocket? To do this,
we need to introduce the concepts of thrust and impulse.
Thrust and impulse quantify the propulsion capabilities of a rocket
In Newton’s Second Law of Motion, the force felt by an object is equal to the mass of the
object multiplied by its acceleration. To give a concrete example, consider the term “g”,
as in “the astronaut experienced a g-force of about 3g’s during launch.” The primary
misconception in that statement is that whatever a “g” is, it is a measure of force.
Actually, a “g” is a measure of acceleration; in fact, on the Earth’s surface at sea level, its
value is 9.8 m/s2. So “3g”s would by 3 × 9.8 = 29.4 m/s2.
So how to translate this to a force? Since force is mass times acceleration, all we need to
know now is the mass of the astronaut. For the ease of calculation, the astronaut weighs
100 kg; so the force the astronaut experiences at launch is 100 kg × 29.4 m/s2 = 2940 kg
m/s2. That unit of “kg m/s2” is cumbersome and has been replaced by the much more
straightforward unit of force, the Newton (N). Thus, the astronaut is experiencing a force
of 2940 N, which seems like a lot. Just standing on the Earth’s surface, though, the
astronaut exerts a force of 100 kg × 9.8 m/s2 = 980 N.
Some examples of thrust: a G-class motor for a high-powered hobby rocket produces 80
Newtons of thrust; the F-1 first stage of the Saturn V rocket produced 7.75 million
Newtons of thrust.
The thrust of a rocket is the force that moves the rocket. However, this value gives you
no idea of how long the rocket is capable of sustaining such a force; clearly, it matters to
the rocket’s final velocity whether the rocket can maintain its thrust for five seconds or
for five years!
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For this reason, another rocket parameter is its impulse (often called the total impulse),
which is defined as the rocket’s force (in Newtons) multiplied by the length of time (in
seconds) of its burn. Thus, the units of total impulse are “Newton-seconds”.
Yet another parameter is the specific impulse, which is the total impulse divided by the
weight (in Newtons) of the rocket.
We can use model rocket motors as an example of how impulse is calculated and used.
Consider the following label from a model rocket engine:
The letter “C” designates the total impulse, which can be looked up on a standard table,
like the one shown below; a “C”-size motor is capable of 5 to 10 Ns of total impulse.
The next number on the label – “6” – shows the average thrust in Newtons. Dividing this
number into the total impulse range yields the length of time the motor will burn: in this
case, it will burn for 5/6 = 0.83 seconds to 10/6 = 1.67 seconds.
(The final number on the label – “4” – shows the length of the delay in seconds between
the end of the motor burn and the detonation of its ejection charge to release the
parachute).
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A C10 motor by contrast will yield the same total impulse, but the motor will only burn
between 0.5 and 1 second.
Often, a motor’s performance curve (that is, a graph that shows time along the x-axis and
the thrust of the motor along the y-axis) is not a simple straight-line graph. An example is
shown below for a C6 motor (there’s a cutaway view of the motor below the graph).
Even though the motor
averages 6 N of thrust over
the length of its burn, notice
that it starts from 0 N at the
start of the burn, quickly
peaks to nearly 15 N in less
than half a second, then
settles back to about 6 or 7
N for the remainder of its
burn.
How can you get a better
estimate for the total
impulse from this graph?
The best way of doing
exactly that is to determine
the “area under the curve”.
Shade in the area between
the performance curve of the
motor and the x-axis, and
the shaded area will represent the amount of total impulse (in calculus, this area would be
the integral of the performance curve).
This is not necessarily an easy way of determining the total impulse, given the irregular
shape of the area! Ultimately, the point of determining the thrust and the impulse of a
rocket engine is to figure out how fast it will propel the rocket and its payload. How can
this be done with the quantities we already know?
Getting through the atmosphere adds another complication
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Consider the flight of an airplane. If you ignore, for the moment, ignore all the complex
machinery needed to get a multi-ton object into the air and keep it there, the physics of
flight is relatively simple. There are only four forces acting on a plane: the lift (directed
upward), the thrust (directed in the direction of motion), the drag (directed against the
direction of motion) and, of course, gravity (also know as weight, directed down). These
forces are balanced when the plane is moving at a constant speed at a level altitude. Very
often, they are not balanced, and there is a net force (the sum of all four forces) acting on
the plane.
Now consider the flight of the rocket in the atmosphere:
the same four forces act on the rocket as they did on the
airplane. The atmosphere increases the drag on the
rocket; each molecule of nitrogen, oxygen or argon in
air, for instance, briefly interacts with the molecules on
the skin of the rocket, slowing it down by a tiny amount.
There are a huge number of such encounters every
second of the rocket’s flight, so cumulatively the effect
of these atmospheric molecular interactions is large.
This means that the more surface area of the rocket
exposed to the atmosphere, the more drag it will
experience. So trying to keep a narrow profile (with
respect to the air rushing past it) would seem to be a
good strategy; this works out to be true with one caveat.
To quantify this phenomenon of lift and drag, engineers
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developed the concept of a “center of pressure” – the point in the rocket where the
combined drag and lift forces appear to be acting. The center of pressure, abbreviated cp,
can roughly be calculated by determining the total surface area of the rocket, and figuring
out its halfway point. This is
analogous to the center of gravity
(cg), which is the point in the
rocket where the combined thrust
and weight forces appear to be
acting. The center of gravity, also
known as the center of mass, is
determined by balancing the
rocket on a knife edge; the point
along the rocket’s length at which
the rocket balances on the knife
edge is the cg for the rocket. For
the cp, about half of the surface
area of the rocket is forward
(towards the nose) and about half
back (towards the engine).
For the most stable flight through the atmosphere, a rocket should have its center of
gravity forward from its center of pressure, as illustrated in the diagram below.
Torque is the enemy of a stable flight
When a force is applied to a mass that is small (let’s say a point), then the resulting
motion is in the direction of the applied force; for instance, if you throw a ball, it
generally goes in the direction you are throwing it. However, when the mass is not that
small and the force is applied to a part of the mass away from its center of gravity, the
force becomes a torque, which causes the mass to spin around an axis that does not point
in the direction of its motion, like the “English” on a cueball in billiards when the cue
stick hits the ball off-center.
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Note that when the cp is forward from the cg, there is a force (the thrust) that, if it is
aimed slightly off the line between the cg and cp, will create a torque on the rocket which
causes it to start tumbling in its flight. If the cg is forward from the cp, then, even if the
cg and cp are not aligned, there is no torque.
So what steps can be taken to assure that the cg is, in fact, forward from the cp? There are
two traditional methods: the first is to add weight to the nosecone and reduce weight from
the engine; this moves the cg forward. The second method is to increase the surface area
near the engines by making the fins larger; this moves the cp backward.
Modern large rockets (the kind used to boost satellites into orbit) have no fins, and yet
achieve stable flight through the atmosphere. Clearly, there are other ways to achieve a
stable flight than merely making bigger fins!
The diagram to the left
shows three methods
of achieve stable flight
with fins. In the
gimbaled thrust
method, the engine can
literally move and
point in different
directions. This helps
counteract any torque
that might develop.
This is the most
commonly used
method in modern
large rockets.
Another method is to
attach small rockets
(called Vernier
rockets) in addition to
the main engine. This allows small burns that can correct for a torque.
Finally, in the oldest large rockets, there was the thrust vane, a movable piece of metal
situated in the nozzle of the engine, which allowed the thrust to be directed in different
directions. This method was discarded as gimbaled thrust rockets were developed, as the
thrust vane was subject to high temperatures and stresses, and thus could (and did) fail.
Circular motion can help stabilize the rocket’s trajectory
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Circular motion has some additional characteristics that linear motion does not. A trivial
example is that an object with purely circular motion will return to its starting point,
whereas an object with purely linear motion will never return to its starting point.
Another characteristic is angular momentum, which is like the product of the object’s
mass and its angular speed but not exactly (see the “Calculating Angular Momentum”
box). This means that a small mass being swung in a small circle will have less angular
momentum than a large mass being swung in a large circle.
An example of this is a playground merry-go-round. If the merry-go-round is not going
very fast, it is easy to slow or stop because it does not have much angular momentum. At
higher speeds, though, the merry-go-round can be quite difficult to stop due to its greater
angular momentum.
For a rocket, one of the earliest methods to stabilize its trajectory and avoid the tumbling
problem was to impart a spin on the rocket around its length. These “spin-stabilized”
rockets allowed for a more predictable flight by preventing spin along any other axis.
This prevention of spin along any other axis except the spin axis is the same principle that
endows a gyroscope its stability.
So why are the more elaborate methods of flight stabilization (mentioned in the previous
section) used, and not the seemingly-simpler spin stabilization? In the atmosphere, the
spin can be imparted on the rocket by the use of angled fins, but this increases the drag on
the rocket as well. In space, thrusters can “spin up” the rocket but this adds weight
compared to the other methods. In addition, if there is an antenna for guidance or
communication attached to the side of the rocket, then it would constantly be pointing in
different directions.
Angular momentum is conserved, in the same sense that energy is conserved: the amount
of angular momentum in a system cannot change even as the system undergoes changes.
To give the merry-go-round example an exciting and hazardous twist: place a daredevil
skater on that playground merry-go-round, then put that on an ice skating rink, and spin
up the merry-go-round. The ice skater, just like the merry-go-round itself, will have
angular momentum. Then have the ice skater leap off the merry-go-round. What will
happen?
The skater will hit the ice spinning in the same direction that the merry-go-round was
rotating. Moreover, the merry-go-round will have slowed down in its rotation, just
enough to account for the skater’s angular momentum. Thus, the total angular momentum
of the system (contained in the merry-go-round and the skater) remains the same.
Propulsion speed can be quantified by the “Rocket Equation”
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One of the consequences of the conservation of momentum was the rocket equation,
developed in its modern state by the Russia/Soviet rocket scientist Konstantin
Tsiolkovsky in 1903 (see the “Rocket Equation” box for historical development and the
equation). This equation is used to predict the velocity of a spacecraft as a function of the
exhaust gas velocity and the change in the spacecraft’s mass as a result of using its fuel to
produce the exhaust gas.
The nice result of the rocket equation is that thrust and impulse are not relevant to the
calculation. For a rocket to achieve a particular velocity, there is no particular thrust it
must have, as long as that thrust can be maintained for a long time. For instance, an ion
drive engine (more details in a couple of chapters) has a thrust of 0.02 to 0.25 N but this
can be maintained for years or decades.
The depressing result of the rocket equation is the diminishing returns of using larger
amounts of fuel. Doubling the amount of fuel burned does not double the speed the rocket
will go; indeed, the speed will increase, but by a far smaller amount. The graph below
shows this idea in detail; to increase the rocket’s speed from four to five times the speed
of the exhaust requires far more fuel than increasing the speed from two to three times the
speed of the exhaust.
mfuel/mspacecra, 200 Fuel required to increase speed 150 100 50 0 0 1 2 3 4 Δv/vexhaust
5 6 So engineers must be clever in figuring out what types of fuels work best for achieving
the speed their spacecraft requires: high or low thrust? short or long burn? These
questions will be examined in the next chapter.
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