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). 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. 58
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. 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. 59
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. 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/s 2. So 3g s would by 3 9.8 = 29.4 m/s 2. 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/s 2 = 2940 kg m/s 2. That unit of kg m/s 2 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/s 2 = 980 N. 60
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! 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. 61
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). 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? Propulsion speed can be quantified by the Rocket Equation 62
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. m fuel /m spacecra, 200 150 100 50 0 Fuel required to increase speed 0 1 2 3 4 5 6 Δv/v exhaust 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. 63