## Rocket Fundamentals

It doesn't take a "rocket scientist'' to understand rocket propulsion. It's all about conservation of momentum. Modern rockets come in three basic varieties: chemical, electric, and nuclear. In all three cases, an energy source excites a reaction mass, or fuel. This is expelled from the rear of the rocket at high velocities. The reaction to this fluid release manifests as thrust and pushes the rocket forward. The momentum of the system remains unchanged. If you add the momentum of the now-moving rocket to that of its exhaust, the sum will be zero. Rockets don't create something from nothing!

FIGURE 3.2 The rocket principle.

Figure 3.2 presents a generic rocket. The total mass of the fuel is Mf and Me is the mass of the vehicle (engine, structure, fuel tanks and payload), exclusive of fuel. The ratio of fueled mass to unfueled vehicle mass [(Mf + Me)/Me] is called the Mass Ratio (MR).

If the fuel is expelled at an exhaust velocity Ve relative to the vehicle and the total change in velocity after all fuel is expelled is AV, application of Newtonian mechanics allows one to derive the famous Rocket Equation:

MR = e AV/Vex where e is the base of the natural logarithm system, which has an approximate numerical value of 2.72.

For the non-mathematician, the significant factor about this equation is the fact that it is non-linear, meaning that the Mass Ratio (and hence fuel) increases very rapidly as the velocity ratio A V/Vex increases. If this velocity ratio is exactly equal to 1, the Mass Ratio is 2.72. If it is equal to 2, the Mass Ratio becomes 7.39 and if the velocity ratio is raised to 3, the Mass Ratio becomes 20.1 To accelerate to high velocities, rocket fuels must be expelled at the highest feasible exhaust velocities and the Mass Ratio must be as high as economics and technology permit.

To gain an idea of rocket potentials and limitations, the Rocket Equation can be used to determine the Mass Ratio for the case of the best performing chemical rocket used to put a spacecraft in orbit. We assume that the exhaust velocity is 4.55 kilometers per second, about equal to that of the space shuttle's liquid fuel rocket engines in a vacuum and close to the theoretical maximum for a chemical rocket. To attain low Earth orbit (LEO), the vehicle must be accelerated to a velocity of 7.91 kilometers per second. (For readers not attuned to the metric system, this velocity is equal to 4.9 miles per second or about 17,700 miles per hour.)

The Earth rotates on its axis at a velocity of about 0.46 kilometer per second, so if the spacecraft is launched in an easterly direction from a near-equatorial site, it can gain a small velocity advantage from Earth's rotation. But its engines must still provide an incremental velocity of 7.91 - 0.46 = 7.45 kilometers per second to achieve LEO.

If it is required to place the entire mass of the vehicle at lift-off into orbit, the Rocket Equation can be used to calculate the Mass Ratio. For this case, the Mass Ratio is 5.14. More than 80% of the spacecraft's mass on the launch pad is fuel.

Let's next consider the case of direct launch to Earth escape, say on a Moon-bound or Mars-bound trajectory. The total velocity increment for this case is 11.19 kilometers per second. With an easterly launch from an equatorial site, the rocket engines must accelerate the spacecraft to 10.73 kilometers per second (or about 24,000 miles per hour). Assuming the same exhaust velocity as before, the Rocket Equation can be used to calculate that the required Mass Ratio is 10.57. In this case, about 91% of the initial spacecraft mass on the launch pad is fuel.

The non-linearity of rocket operations is evident when one compares velocity and fuel requirements for Earth-orbital and Earth-escape missions. The velocity increment required from the rocket for Earth escape is 1.44 times the velocity increment required to achieve LEO. The Mass Ratio for Earth escape is, however, 2.06 times the Mass Ratio required to achieve LEO.

At first glance, these numbers do not appear too daunting, at least for LEO operations. After all, a rocket composed of 80% fuel does not seem too ridiculous.

In practice, however, very large mass ratios are not possible for Earth-launched spacecraft. A rocket cannot consist of only fuel and payload, for instance, because of the requirement for supporting structure as it accelerates at multiples of Earth's surface gravity. A very large fraction of a rocket's fuel is exhausted, in fact, in the first few seconds after it clears the launching pad. This is why the designers of the Pegasus booster, the X-15 and Space Ship 1 have opted for air launches. Not only is a rocket dropped from a high-altitude jet moving at a high transonic velocity; it starts its flight at an altitude high enough for a significant reduction in atmospheric drag.

A more commonly improved technique to improve rocket performance is staging. As a spacecraft climbs toward the fringes of the atmosphere, it usually drops exhausted engines and fuel tanks rather than continue to carry the unneeded dead weight.

Even though staging has allowed humans or robotic probes to reach orbit, the surface of the Moon, the vicinity of other solar-system worlds, and the fringes of interstellar space, it is a horribly uneconomical process. Typical launch costs are in the vicinity of thousands of dollars per kilogram. The process has been compared to what air travel would be like if a jumbo jet discarded various components during each flight which then had to be replaced after each landing.

One attempt to apply economics has been the development of partially reusable spacecraft such as the US space shuttle. But the shuttle must be significantly refurbished after each round trip into space, it requires an army oftechnicians to maintain it, and it has not proven to be an accident-proof method of entering the space frontier.

The shuttle also has a very small payload mass fraction. Sitting on the launch pad, the fully assembled shuttle masses approximately 2 million kilograms (4.4 million pounds). The maximum payload capacity is around 30,000 kilograms (65,000 pounds). The empty mass of the orbiter (not including payload and fuel) is 75,000 kilograms (165,000 pounds)—more than 90% of the shuttle's mass on the launch pad is either dropped in the ocean or discarded as fuel.

### Rocket Varieties

A number of rocket varieties either are in the operational inventory or have been designed. By far the most common is the chemical rocket. Solar-electric rockets have flown on interplanetary voyages. Solar-thermal, nuclear-electric, and nuclear-thermal rockets have been ground tested.

Electric Rockets payload ion-fuel tank ion accelerator ion exhaust

### FIGURE 3.3 An electric rocket.

As shown in Figure 3.3, the electric rocket (also called the "ion drive") uses an external source of energy—solar arrays or an onboard nuclear reactor—to ionize reaction fuel and then to electrically accelerate the ionized fuel to a high velocity. As the high-speed ionized fuel leaves the spacecraft in one direction, the spacecraft moves in the other direction and, again, the momentum of the system remains unchanged.

Solar-electric engines are now operational, after having passed their space-qualification tests. Although capable of only very small accelerations (in the vicinity of 10-4 Earth gravities), ion-drive exhaust velocities can exceed 30 kilometers per second.

Due primarily to their low thrust, no ion-drive spacecraft will ever lift off the Earth into space, but given enough fuel and acceleration time (there is plenty of time when the destination is an asteroid many millions of kilometers away) they have demonstrated the capability of accelerating spacecraft to very high interplanetary velocities.

Solar-electric engines will find significant application in the inner solar system where sunlight is most intense, unless methods of beaming energy

o solar or nuclear energy source to ionize and accelerate fuel to interplanetary spacecraft are developed. For deep space missions far from the Sun, onboard nuclear power will almost certainly be required to power any ion engines. Today's technology falls short of that required to safely build, launch, and operate a nuclear-powered electric propulsion vehicle affordably in deep space.

Nuclear-Thermal and Solar-Thermal Rockets payload liquid-hydrogen fuel tank exhaust

### FIGURE 3.4 A nuclear-thermal rocket.

Nuclear-thermal rockets operate as shown in Figure 3.4. Here, the energy released from an onboard nuclear reactor is used to heat the fuel (usually liquid hydrogen) and expel it at high exhaust velocities.

A number of nuclear-thermal rockets were successfully ground tested during the 1960s and 1970s. If the nuclear reactor power levels are in the 100-1,000 megawatt range, these engines are capable of high-thrust operation at exhaust velocities about twice that of the space shuttle's liquid fuel chemical rockets. Nuclear-thermal rockets have the potential to be used for Earth-to-orbit applications as well as in space. However, many technical and safety issues must first be addressed.

The solar-thermal rocket replaces the onboard nuclear reactor with solar concentrators to superheat the propellant in much the same way as using a magnifying glass and sunlight can be used to burn a piece of paper or start a fire. Many of the technologies required for a nuclear-thermal rocket are applicable to a solar-themal rocket and will likely be first demonstrated using this non-nuclear propulsion system.

Ground tests have revealed that exhaust velocities as high as 10 kilometers per second are possible for the solar-thermal rocket. However, since the Sun is a diffuse energy source, solar-thermal rockets are low-acceleration devices best suited for in-space application.

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