35 Rockets & Rocket Propulsion

Introduction[1]

Rockets launch payloads, such as satellites and space probes, into Earth orbit. Rocket motors are also used to propel spacecraft that are already in space. By imparting a significant time rate of change of momentum to a gas flow produced by a propellant, a force is then applied from the rocket motor(s) to the vehicle. Before being expelled from the motor(s), the propellant can be stored onboard as a compressed gas, in separate fuel and oxidizer tanks, or as a solid mixture of fuel and oxidizer.

Four types of rocket motors can be used for different and diverse applications:

  1. Launch Vehicles: Rocket motors used in launch vehicles are designed to provide the necessary thrust and momentum to lift the vehicle off the ground and into space. They must be powerful enough to overcome Earth’s gravitational attraction and provide the necessary kinetic and potential energy for the payload to reach orbit.
  2. Spacecraft: Rocket motors used in spacecraft are designed to propel and maneuver them in space. They are typically smaller and less powerful than the motors used for launch vehicles. Still, they must be highly reliable, efficient, and versatile to enable the spacecraft to travel long distances and perform complex missions.
  3. Missiles: Rocket motors used in missiles are designed to provide maneuverability and speed and the ability to carry warheads or other payloads to their targets. Missiles must be highly precise and reliable and may need to operate in a wide variety of conditions and environments.
  4. Miscellaneous: There are other applications for rockets, including suborbital flights, scientific experiments, and even recreational activities like model rocketry. These motors may have specialized requirements depending on their specific use.

Learning Objectives

  • Understand the basic principles of rocket propulsion systems.
  • Distinguish the different types of rocket motors and their purpose.
  • Appreciate the concept of specific impulse as a measure of rocket efficiency.
  • Know how to derive the rocket equation and use it to solve some simple rocket problems.
  • Understand the staging process and why it is used for lunch vehicles.

Types of Rockets & Applications

Launch Vehicles

Space launch vehicles are highly specialized and tailored to specific missions and payloads. For example, some launch vehicles are designed for placing payloads into low Earth orbit, while others are intended for deep space missions. The choice of the launch vehicle will depend on various factors, including the desired orbit and payload mass and size. In addition to military and civilian applications, there is also a growing interest in commercial and tourism-related payloads. For example, companies like SpaceX and Blue Origin are developing launch vehicles that can be used for both commercial satellite launches and human spaceflight. This approach can open space to a broader range of users and applications.

Solid and liquid propellant rocket motors are commonly used for launch vehicles and can be combined to achieve specific performance characteristics. For example, solid-fuel rocket boosters can provide high initial thrust at liftoff. In contrast, liquid propellant motors can provide more precise control of the thrust generated and greater efficiency once the vehicle is in flight. In addition, the number of stages in a launch vehicle can vary depending on the mission requirements. For example, some launch vehicles have only one stage, while others have multiple stages that are separated during flight. This approach allows the vehicle to achieve higher velocities and altitudes than possible with a single-stage rocket design.

A representative launch profile of a rocket is shown in the figure below. At the moment of the initial launch, the thrust produced by the rocket motors will be greater than the vehicle’s weight, so the rocket accelerates away quickly from the pad. The rocket’s weight rapidly decreases because of the high fuel consumption, so it continues accelerating as it gains altitude. As the rocket begins to exit the atmosphere, which is about 60,000 ft (approximately 20,000 m), it will fly at supersonic speeds. It also begins to pitch to a more horizontal flight path, and the rocket gains translational velocity for the payload to reach its initial equilibrium orbital speed and altitude.

A representative launch profile for a two-stage rocket. The first stage may be recovered by parachute or burn up or crash into the sea. First-stage recovery significantly reduces the launch costs.

Several minutes into the ascent, staging will occur where the first stage is jettisoned and the rocket motor for the second stage is ignited. The first stage then falls back to the surface and either burns up in the atmosphere (depending on the staging altitude) or breaks apart and crashes into the ocean. In exceptional cases, the first stage may be recovered; solid rocket boosters are usually recovered by parachute and may be reused. The upper stage (or stages) continues to accelerate into space. As the payload reaches the required initial orbital velocity and altitude, the rocket motors will cut off at a predetermined altitude and speed.

Spacecraft

Spacecraft are used for various applications. Some are designed for specific missions, such as planetary exploration or Earth observation, while others are more general-purpose and can be used for multiple tasks. One essential function of a spacecraft is orbit insertion, which involves placing the spacecraft into a specific orbit around a planet or other celestial body. This goal requires a carefully planned trajectory and precise firings of the rocket motors to achieve the desired orbit.

Electric rocket motors, such as ion thrusters or Hall effect thrusters, are becoming increasingly attractive for use on spacecraft. These motors are highly efficient and can maintain thrust production for long periods, making them well-suited for deep space missions. However, spacecraft can be highly complex and require extensive testing and development to ensure their reliability and safety. This includes testing in simulated space environments and rigorous quality control procedures to ensure all components meet strict performance standards.

Missiles

Missiles can be broadly divided into two categories: 1. Ballistic missiles. 2. Cruise missiles. Ballistic missiles are used for long-range strikes against threat targets. They are launched high into the upper atmosphere or space, following a parabolic trajectory before reentering the atmosphere and striking their target. Ballistic missiles can be easier to intercept than cruise missiles because they follow a predetermined path that is difficult to alter once launched.

Cruise missiles are designed to fly at low altitudes and follow a more maneuverable flight path to evade enemy defenses. They can be launched from various platforms, including aircraft, ships, and ground-based launchers. They can be more challenging to detect and intercept because they fly at lower altitudes than ballistic missiles.

In addition to their propulsion system, targeting, guidance, and warhead systems, missiles require advanced sensors and communication systems to navigate to their targets and avoid obstacles accurately. As a result, missiles are highly complex weapons systems that require extensive testing and development to ensure their reliability and effectiveness. In addition, they are subject to strict regulations and controls, and their use is governed by international law.

Miscellaneous

Sounding rockets gather data on atmospheric conditions, such as temperature, pressure, and wind speed, at altitudes that are difficult to reach with aircraft or balloons. They are typically small, single-stage rockets launched on suborbital trajectories and can carry scientific instruments and sensors to collect data.

Rocket-assisted takeoff (RATO) is a technique that uses rockets to provide additional thrust during the takeoff of aircraft, particularly in situations where the aircraft is heavily loaded or taking off from a short runway. The rockets temporarily boost the aircraft’s acceleration, helping it achieve lift-off more quickly and efficiently.

Rockets can also be used to provide emergency lifelines to ships that are in distress. In this application, a rocket-powered line is fired from shore or another ship to the stranded vessel, enabling rescuers to establish a connection and provide assistance. Rockets could also deliver relief materials to inaccessible areas during natural disasters or humanitarian crises. However, this approach would require developing reliable and cost-effective rocket delivery systems, appropriate infrastructure, and logistical support.

Rocket Propulsion Fundamentals

In a rocket motor, the propellants, in the form of fuel and an oxidizer, undergo combustion at high pressure and temperature to produce thrust. Therefore, the motor needs to be designed to withstand the high pressures generated through combustion, as well as the very high temperatures.

To prevent overheating and structural failure, the combustion chamber and nozzle are actively cooled to dissipate the heat generated during combustion. This is done by circulating a coolant, usually the cold rocket fuel itself, around channels or jackets within the walls of the combustion chamber and nozzle. The coolant absorbs the heat and carries it away, allowing the rocket to operate safely for extended times.

NSA image
A rocket engine under test, in this case one that uses hydrogen and oxygen as propellants hence the almost invisible exhaust in the form of superheated steam.

The shape and length of the combustion chamber is also an important design parameter. The length must be long enough to allow for complete combustion of the propellants before the hot gases enter the nozzle to ensure efficient combustion and maximize thrust production. The length of the combustion chamber is typically determined based on numerous factors, such as the type of propellants used, combustion efficiency, and desired overall performance characteristics of the rocket motor.

Application of the Conservation Principles

Newton’s second and third laws are the basic physical principles that apply to rocket propulsion systems. Consider a typical rocket motor, as shown in the figure below. The principle of thrust generation for a rocket motor is from the reaction force associated with accelerating a mass of gases at high velocity out of an expansion nozzle, the gases being a byproduct of combustion of the fuel and the oxidizer, and so increasing the kinetic energy and momentum of the gases. Notice that, unlike an air-breathing engine, the oxidizer must be carried along with the fuel for a rocket motor. As a result, the gases and products of combustion are accelerated in one direction, and the resultant force on the vehicle is directed in the opposite direction, in accordance with Newton’s third law.

Control volume for the analysis of the thrust produced by a rocket motor.

If the flow is assumed to be steady, compressible and inviscid, then the general form of the momentum equation is

(1)   \begin{equation*} - \oiint_S p \, d\vec{S} = \oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V} \end{equation*}

The pressure integral on the left-hand side can be written as

(2)   \begin{equation*} \oiint_S p \, d\vec{S} = \iint_{A_e} p \, d\vec{S} + \iint_{\rm Nozzle} p \, d\vec{S} \end{equation*}

the throat area being considered to be much smaller that the exit area, so the pressure integral here can be assumed as zero. The force on the fluid to change its momentum, which by Newton’s 3rd law, is in the opposite direction to the thrust, T. Therefore, the momentum equation becomes

(3)   \begin{equation*} T = \oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V} + \iint_{A_e} p \, d\vec{S} \end{equation*}

If the flow can be assumed to be one-dimensional, which is a reasonable assumption for now, the thrust T produced by the rocket motor will be

(4)   \begin{equation*} T = \overbigdot{m} V_j + \left( p_e - p_a \right) A_e \end{equation*}

where p_e is the exit pressure at the nozzle, p_a is the ambient pressure, A_e is the exit area, V_j is the exit or jet velocity, and \overbigdot{m} is the propellant mass flow rate. Remember that there is no external mass flow into a rocket motor, i.e., no value of V_{\infty}, such as with an air-breathing engine.

The first term on the right-hand side of Eq. 4, i.e., the \overbigdot{m} V_j term, is a momentum flow rate or time rate of change of momentum, and the second term is the net force resulting from a pressure difference between the exit gases and the ambient pressure. For a rocket, the time rate of change of momentum is much larger than any pressure force, i.e., \overbigdot{m} V_j \gg (p_e - p_a) A_e. If p_e = p_a, the pressure force term is zero. In this case, the thrust produced is a maximum and the nozzle is said to be ideally or optimally expanded; this condition is a design goal for a rocket motor.

For a rocket motor, especially when their performance characteristics are being compared, it is often useful to write for the net thrust that

(5)   \begin{equation*} T = \overbigdot{m} V_{\rm eq} \end{equation*}

where V_{\rm eq} is called an equivalent exhaust velocity that includes the pressure term, i.e.,

(6)   \begin{equation*} V_{\rm eq} = V_{j} + \left( p_e - p_a \right) \frac{A_e}{\overbigdot{m}} \end{equation*}

In practice, however, the pressure term is relatively small because the nozzle is designed for (or close to) ideal expansion where p_e = p_a, so the value of V_{\rm eq} is very close to V_j.

Nozzle Shape

The nozzle is designed to accelerate the high-pressure, high-temperature gases generated in the combustion chamber to a very high exit or jet velocity. The nozzle typically has two sections: the convergent and divergent sections. The resulting shape is often referred to as a nozzle “bell.”

The convergent section narrows down the flow area, increasing the velocity of the supersonic gases as they pass through it. This outcome is achieved by converting the thermal energy of the gases into kinetic energy through a process of expansion. The convergent section then leads to the throat, the narrowest part of the nozzle. Beyond the throat, the divergent area widens, allowing the supersonic gases to expand and further accelerate to their final velocity, V_j. Ideally, this expansion continues smoothly and progressively until the hot gases reach the exit area of the nozzle.

Nozzle Efficiency

The design of the nozzle, including the length and shape of the divergent section, is critical in achieving optimal performance from the rocket motor. The nozzle must be long enough, and the exit area must be large enough to ensure that the pressure at the exit is close to the ambient pressure outside the nozzle. This feature is essential for maximizing the propulsion system’s efficiency and achieving maximum thrust.

Designing rocket motors for launch vehicles that can operate efficiently across a wide range of altitudes is a significant engineering challenge that requires careful consideration of many factors. These factors include the shape and size of the nozzle, the materials used for the nozzle, the propellant flow rate, the combustion characteristics of the fuel and oxidizer, and the cooling system(s). Designers aim to achieve the best possible performance of the rocket motor across the entire altitude range of a rocket’s atmospheric flight trajectory to maximize its thrust and efficiency. They also try to optimize the efficiency of the rocket motors when they reach the vacuum of space.

When the exhaust pressure at the exit of the nozzle matches the ambient pressure of the surrounding environment, it is known as ideal or optimum expansion, as shown in the figure below. In this ideal state, there is a zero pressure gradient, and all the exhaust gases are directed away from the motor, resulting in maximum thrust generation as all of the momentum of the exhaust gas is converted into thrust. This operating condition gives rocket motors their highest possible performance in terms of thrust and efficiency. However, achieving optimal expansion requires careful design and optimization of the nozzle’s shape.

The nozzles of rocket motors must be designed to balance thrust and efficiency over a wide range of ambient pressures. Under-expansion or over-expansion of the gas flow will result in a loss of thrust and efficiency.

Over-expansion means that the external (atmospheric) pressure, p_a, is higher than the exit pressure, p_e. When an over-expanded flow passes through the nozzle, the higher external pressure at the exit produces a positive pressure gradient that slows the jet flow and the jet flux converges as it exits the nozzle. The pressure difference may be high enough even to cause the flow to separate from the nozzle’s walls. Over-expansion of the gas flow reduces the thrust and efficiency. The solution in this case is to use a shorter nozzle.

The opposite situation, where the atmospheric pressure is lower than the exit pressure and gives a negative pressure gradient, is called an under-expanded flow. In this case, the flow continues to develop and expand outward after it exits the nozzle and so this process also does not contribute to thrust production. The solution in this case for thrust recovery, is a bigger and longer nozzle.

When designing rocket motors for launch vehicles that must fly in the atmosphere, the nozzle may be designed for a slight over-expansion at sea level, i.e., recognizing that the exhaust pressure at the exit of the nozzle will likely be lower than the ambient pressure of the surrounding air. This design approach can better optimize the rocket motor’s performance during more of the launch profile, allowing the motor to balance, on average, its overall thrust and efficiency throughout the atmosphere.

To this end, many rocket motors, including the Merlin used in the Falcon 9, have a nozzle that is specifically designed to work efficiently across a wide range of altitudes, from sea-level to the stratosphere where the pressure is very low. The RS-25 motors, first used for the Space Shuttle program, were optimized for sea-level operation during the initial phase of the launch but also when it transitioned to vacuum-optimized operation. To this end, the RS-25 motor has a movable nozzle extension to optimize its performance. When rocket motors operate at sea level, the nozzles are usually designed to give a slightly over-expanded condition, so they become more ideally expanded with increasing altitude.

It will be noticed that second or upper-stage “vacuum-optimized” motors of rockets have much larger nozzles than those used on sea-level (or atmospheric) optimized motors, as shown in the figure below. The Merlin second-stage motor is a good example. The second stage, “vacuum-optimized” Merlin, uses the biggest nozzle as practically possible to get an ideal expansion of the exhaust gases. The vacuum-optimized Merlin motor has a bigger exhaust section and a larger expansion nozzle ratio of 165:1, compared to the sea-level optimized version, which has a smaller 16:1 expansion ratio nozzle.  The much larger nozzle allows a more ideal and efficient expansion of the exhaust gases in the vacuum of space, hence maximizing the propulsive thrust and efficiency.

Rocket motor nozzles are designed so the pressure of the hot gases exiting the nozzle matches the external pressure at any given altitude or in the vacuum of space, giving maximum thrust and efficiency.

“Over-Expanded” Versus “Under-Expanded”?

Remember that the nozzle is designed to accelerate the exhaust gases and convert the thermal energy of combustion into kinetic energy. At low altitude, the external atmospheric pressure is higher compared to high altitude. This means that the exhaust gases from the rocket motor encounter a higher external pressure as they expand supersonically to reach the diameter of the nozzle exit. This higher external pressure resists the expansion of the flow and so the flow “over-expands” itself within the limits of the nozzle. At higher altitude, the external ambient pressure is lower, allowing the flow to expand more easily. In this case, however, as the flow reaches the nozzle exit, it may not have expanded enough, resulting in an “under-expanded” gas flow.

Steering a Rocket

A rocket must be steered along a prescribed flight path so the payload can reach the necessary altitude. A modern rocket (launch vehicle) is usually steered along its flight path by gimbaling (rotating) the motor(s) to change the direction of the thrust vector. However, as the figure below summarizes, other means may be used.

There are several methods that can be used to steer a rocket. However, the gimballed thrust design is the most common.

Early rockets used movable aerodynamic surfaces or fins at the rear of the rocket, and this technique is also used on most air-to-air missiles. These surfaces create varying aerodynamic forces and moments on the rocket, which can be used to control its trajectory. Some rockets have used additional vernier rocket motors to give control. However, vernier steered rockets are not used as much because of this system’s extra weight and the different fuel needed. On some early rockets and ballistic missiles, small thrust vanes were placed directly in the exhaust stream of the rocket exhaust to produce forces that could be used for steering.

Types of Rocket Motors

Like all propulsion systems, rocket motors are energy conversion devices. The kinetic energy of the expelled propellant (hence the eventual gain in kinetic energy of the vehicle) comes from:

  1. The work of compressing the propellant into its tank.
  2. Liberating the chemical potential energy of a fuel and an oxidizer.
  3. An electrical or thermal power supply.
  4. Some combination of these latter methods.

Rocket motors can be broadly categorized according to their thrust and thrust-producing efficiency. Rocket propulsion systems are selected according to mission objectives. Generally, there is no “one-size-fits-all” solution, and several rocket propulsion systems could be used for a given space mission. There are two primary types, namely a liquid propellant rocket and a solid propellant rocket, as shown in the figure below, the latter type often being used as a secondary booster.

There are two primary types of rockets: a liquid propellant rocket and a solid propellant rocket. Each one has relative advantages.

High Thrust Rocket Propulsion Systems

High thrust systems are used to overcome gravity, as in a planetary launch vehicle, or to quickly accelerate a vehicle already in space, i.e., for an orbital ejection maneuver. These systems store energy in the propellant so that energy can be converted at a high rate, roughly proportional to the propellant flow rate.

Bipropellant Systems

Bipropellant systems typically come to mind at the mention of rocket propulsion, i.e., one imagines flames and clouds of smoke, such as during a NASA Space Shuttle or SpaceX Falcon 9 launch. The propellant is the combustion product of a fuel and an oxidizer. Combustion is generally the fastest way to convert propellant energy. Bipropellant systems are further categorized as gas/liquid propellant systems, solid propellant systems, or hybrid systems.

Gas/Liquid

Examples of this type include the space shuttle main engine (SSME), which burned liquid hydrogen (LH_2) and liquid oxygen (LOX), and the Merlin of the SpaceX Falcon 9, which burns Rocket Propellant-One or RP-1 (a densified kerosene) and liquid oxygen or LOX. The process of mixing the fuel and oxidizer in the motor is shown in the figure below. The large volume flow rates require turbopumps, which are driven by burning a quantity of fuel and oxidizer tapped off from a bypass circuit.

The process of mixing the fuel and oxidizer in the motor requires large volume flow rates that are driven by turbopumps.

Liquid methane (CH_4) is gaining popularity as a fuel for commercial spacecraft uses because of its availability and performance. It is also far better for the environment than RP-1, which produces a lot of toxic byproducts. One advantage of a gas/liquid system is that the engine can be throttled by regulating the fuel flow. However, this capability comes at a price, including the complexity and associated weight of pumps, valves, pipes, and cryogenic fuel tanks.

Hypergolic propellants are those that combust spontaneously upon contact with one another. These are used for in-space applications rather than launch vehicles, partly because their performance is lower than a hydrocarbon and LOX system. Also, on the one hand, these chemicals tend to be highly toxic. However, on the other hand, their advantages are reliability, simplicity (no ignition system required), and ignition speed. For example, the Apollo lunar lander used Hydrazine (N_2H_4) and nitrogen tetroxide (N_2O_4), as did the Space Shuttle reaction control system (RCS).

Apollo 11, the first mission to land humans on the moon, lifts off from Kennedy Space Center in July 1969. The launch vehicle was a Saturn V rocket, and the first stage burned RP-1 and LOX.

Solid

The Space Shuttle solid rocket booster (SRB) is an excellent example of a solid propellant system. In addition, solid propellants are favored for military applications, e.g., air-to-air missiles and intercontinental ballistic missiles (ICBMs), because they require little pre-launch processing.

Solid propellant fuel is usually a powdered metal, most commonly aluminum (Al) and sometimes magnesium (Mg). Typical oxidizers are ammonium perchlorate (AP) and ammonium nitrate (AN). The fuel and oxidizer are mixed with a binder, usually plastic or synthetic rubber, such as hydroxyl-terminated polybutadiene (HTPB) or polybutadiene acrylonitrile (PBAN). The Space Shuttle SRBs used Al, AN, and PBAN. Most commercially available motors for amateur use have Al, AP, and HTPB, because this combination is more efficiently and safely processed. Unlike liquid-fueled rocket motors, solid rockets cannot be throttled, and once ignited, they must burn until the fuel is exhausted.

Hybrid

Hybrid systems have a solid fuel and a gaseous or liquid oxidizer, or rarely, the reverse. Experimentalists favor hybrids because small rocket systems can be relatively simple to construct. It is often argued that hybrid systems are safer than solid or liquid systems, although this is not true. Hybrid fuels are typically the same materials used as binders in solid propellants. Typical oxidizers are oxygen, nitrous oxide (N_2O), and hydrogen peroxide H_2O_2.

Hybrids have not found much favor in commercial applications because they have no performance advantage, and designing for optimal performance is primarily a cut-and-try process. A notable exception is Spaceship One, which uses HTPB and N_2O. Experimentalists often use polyvinyl chloride (PVC) or acrylonitrile butadiene styrene (ABS) as fuel because these are readily available and have low costs. N_2O is readily available at automotive stores for use in high-performance race engines.

This Hybridyne rocket uses a hybrid liquid-solid motor similar to that used by Virgin Galactic. It has a large liquid nitrous oxide tank with solid fuel propellant.

Monopropellant Systems

Monopropellants do not burn but decompose exothermally in contact with a catalyst. Monopropellant motors generate thrust from the propellant flowing through a valve into a catalytic decomposition chamber where the propellant goes through a highly energetic decomposition process. The hot gases then accelerate through a nozzle, as shown in the figure below. These types of thrusters generally provide thrust levels up to about 3,000 N (674 lb).

Monopropellant rocket motors are favored for small vehicles and reaction control systems because of their relative simplicity over bipropellants, although at the expense of reduced performance.

Hydrogen peroxide (H_2O_2) is often used for monopropellant motors because it decomposes into water and diatomic oxygen in contact with many metal oxides, especially silver oxide. A H_2O_2 system propelled the Apollo lunar lander trainer. Hydrazine has been used more extensively because of its higher performance and ease of reaction initiation. The rocket motor on the New Horizons spacecraft is a N_2H_4 monopropellant system.

Cold Gas Thrusters

Cold gas thrusters are a type of rocket engine that uses compressed gas, typically nitrogen or helium, as a propellant. They work by simply releasing the pressurized gas through a nozzle to generate thrust, as shown in the figure below. Because they do not involve any combustion, cold gas thrusters have a relatively low specific impulse, which means they provide less thrust per unit of propellant compared to other types of rocket motors. Overall, their simplicity makes them less efficient and less powerful than engines that use monopropellants or bipropellants.

 

Cold gas thrusters have a very simple design, consisting only of a fuel tank, a regulating valve, a propelling nozzle, and some plumbing.

Cold gas thrusters are commonly used for small spacecraft or in spacecraft subsystems that require small, precise movements or adjustments. They are appropriate for cubeSats, nanosats, and attitude control of small spacecraft. Any gas can be used as a propellant, but those with lower molecular weight, such as hydrogen (H_2) and helium (He), give better performance.

High-Efficiency Rocket Propulsion Systems

In high-efficiency systems, the energy is not stored in the propellant but is generated by an onboard system. Therefore, the energy conversion rate is not proportional to the propellant flow rate but is limited by the power supply system’s capability. For example, solar panels or a nuclear source can generate either electrical or thermal power. Considering the energy supply rate (power) to the propellant to be fixed, a lower propellant flow rate will give higher efficiency but at a lower thrust, which is insufficient for use as a launch vehicle. High-efficiency systems are then used for long-duration missions to deep space or to raise orbits of Earth satellites.

In each case, this propulsion system does not have to oppose gravity directly (that is, to “lift” the vehicle). However, it increases its velocity gradually once the vehicle is already in space. The most straightforward system heats the propellant gas, which expands to high speed through a nozzle. A solar thermal system collects and focuses the sun’s rays onto the propellant flow path. A thermal electric system heats the gas with a resistive element or an electric arc. In other systems, electrical power is used to ionize the propellant gas and produce an electric and/or magnetic field, after which the charged particles are accelerated. Several configurations exist for such systems, including ion thrusters, Hall effect thrusters, and magnetoplasmadynamic thrusters.

Total Impulse & Specific Impulse

Establishing the characteristics of a rocket motor requires some quantitative measures of performance. The total impulse is defined as the integral of the thrust over the “burnout time,” i.e.,

(7)   \begin{equation*} I = \int_0^{t_b} T \, dt = \int_0^{t_b} \overbigdot{m} V_{\rm eq} \end{equation*}

If \overbigdot{m} and V_{\rm eq} are constant, as is often a good approximation, then

(8)   \begin{equation*} I = T t_b = M_P V_{\rm eq} \end{equation*}

where M_P is the mass of propellant burned. The total impulse, therefore, is the net momentum imparted to the rocket during the burn.

The measure of efficiency used in most rocket performance calculations is the specific impulse, which is the thrust divided by the propellant flow rate. In general, one wishes to carry as little propellant as possible. The specific impulse, I_{\rm sp}, is

(9)   \begin{equation*} I_{\rm sp} = \frac{\mbox{Total~impulse}}{\mbox{Weight~of~propellant~burned}} = \frac{I}{M_P \, g_0} \end{equation*}

where g_0 is acceleration under gravity at sea level on Earth, which is used as a reference.

Standard Gravity

The standard acceleration under gravity or “standard gravity,” which is denoted by g_0 is the nominal gravitational acceleration of an object at the surface of the Earth. The value of g_0 is defined as 9.80665 m/s^2 or 32.17405 ft/s^2. The symbol “g” should not be confused with “G” for the universal gravitational constant, or “g” used as the symbol for gram.

It is further apparent using Eq. 5 that

(10)   \begin{equation*} I_{\rm sp} = \frac{V_{\rm eq}}{g_0} = \frac{T}{\overbigdot{m} \, g_0} \end{equation*}

where I_{\rm sp} is measured in units of time (seconds). Therefore, by definition, the specific impulse is the total impulse (or change in momentum delivered) per unit weight of propellant consumed, Notice then that its value is dimensionally equivalent to the generated thrust divided by the propellant flow rate in terms of weight of fuel per unit time, and so in some ways is equivalent to the inverse of the thrust specific fuel consumption used by a jet engine.

Thrust or Specific Impulse?

Remember that thrust is a force supplied by the rocket engine and its value will depend on the amount of propellant flowing through the engine. Specific impulse is not a force because it value is a measure of the impulse or momentum produced per unit of propellant and will be proportional to the exhaust velocity.

Notice that if mass (slugs or kg) is used as the unit of propellant, then the specific impulse has units of velocity. If the weight (lb or N) is used, which is much more common, then the specific impulse has units of time. Notice that these two definitions differ by a factor of g_0. The higher the specific impulse, the less propellant is needed to produce a given thrust during a given time. Some propellants and their specific impulse values are given in the table below. While it can be seen that H2/O2 (LH2/LOX) has the highest specific impulse, liquid hydrogen needs a much larger (volumetric) fuel tank than RP-1, and it is also more more expensive as well as challenging to transport and store. Liquified methane is fast becoming the fuel of choice.

Some propellants and their specific impulse values.
Propellant Specific Impulse (secs.)
H2/O2 (LH2/LOX) 445
RP-1/O2 (Kerosine/LOX) 295
H2O2 (Hydrogen Peroxide) 300
CH4/O2 (Methane/LOX) 320

Rocket Equation

The rocket equation finds considerable use in rocket sizing and propellant load estimations. The derivation of  this equation is credited to the Russian scientist Konstantin Tsiolkovsky, who published it in 1903. However, Robert Goddard and Hermann Oberth also derived the rocket equation, independently of Tsiolkovsky and of each other, during the 1920s.

Derivation

Consider an accelerating rocket vehicle where the engine’s thrust is used to propel a vehicle of mass M. Therefore, it can be written that

(11)   \begin{equation*} T = M \left( \frac{dV}{dt} \right) \end{equation*}

where dV/dt is the acceleration of the vehicle. Here, the forces from gravity and atmospheric drag have been neglected relative to the vehicle’s weight, so the equation is strictly valid for a vehicle in space. However, it is not an unreasonable approximation otherwise, including for flight in the atmosphere.

The thrust is given by

(12)   \begin{equation*} T = \overbigdot{m} V_{eq} \end{equation*}

so with Eq. 11, then

(13)   \begin{equation*} M \left( \frac{dV}{dt} \right) = \overbigdot{m} V_{eq} \end{equation*}

or

(14)   \begin{equation*} \frac{dV}{dt} = \left( \frac{\overbigdot{m}}{M} \right) V_{\rm eq} \end{equation*}

The time rate of decrease of mass is equal to the mass flow rate, i.e.,

(15)   \begin{equation*} -\frac{dM}{dt} = \overbigdot{m} \end{equation*}

so that

(16)   \begin{equation*} dV = -\left( \frac{dM}{M} \right) V_{\rm eq} \end{equation*}

Separating the variables and integrating to the limits gives

(17)   \begin{equation*} \int_{V_0}^{V_b} dV = -V_{\rm eq} \int_{M_0}^{M_b} \left( \frac{dM}{M} \right) \end{equation*}

where M_0 and V_0 are the initial mass and velocity of the rocket, respectively, and M_b and V_b are the final or burnout mass and velocity, respectively.

After integration of the equation then the change in the velocity of the vehicle is

(18)   \begin{equation*} V_b - V_0 = \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) \end{equation*}

This latter equation is called the rocket equation and is very useful in mission performance analysis and vehicle sizing. In some ways, it is analogous to the Breguet equations used for aircraft performance.

Notice that for a launch vehicle, its initial velocity on the pad is zero (V_0 = 0), so the burnout velocity for the rocket (or the first stage) will be is

(19)   \begin{equation*} V_b = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) \end{equation*}

Effects of Gravity

If gravity is included (but no aerodynamic drag) in the case of a pure vertical launch, then the rocket equation in Eq 18 is modified to

(20)   \begin{equation*} \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) - \Delta V_g \end{equation*}

The second term, \Delta V_g, is usually relatively smaller than the first term, which is usually referred to as the gravity loss.

Notice that for a rocket going up vertically, then

(21)   \begin{equation*} \Delta V_g = \int_0^{t_b} g_0 \, dt = g_0 \, t_b \end{equation*}

so the rocket equation becomes

(22)   \begin{equation*} \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) - g_0 \, t_b \end{equation*}

where t_b is the burnout time.

However, if the rocket follows a curved trajectory and pitches over at a local trajectory angle \gamma (with respect to the horizon) as it increases altitude, then

(23)   \begin{equation*} \Delta V_g = \int_0^{t_b} \sin \gamma \, g_0 \, dt \end{equation*}

where \gamma = 90^{\circ} when the rocket flies vertically and \gamma = 0 when it flies horizontally. The proper evaluation of this latter term, therefore, requires specific information about the launch profile.

Rocket Mass Breakdown

The initial mass M_0 of the rocket vehicle can be written as the sum

(24)   \begin{equation*} M_0 = M_P + M_S + M_L \end{equation*}

where M_P is the mass of the propellant, M_S is the structural mass, and M_L is the mass of the payload. The burnout mass is reached when all of the propellant is exhausted, and is given by

(25)   \begin{equation*} M_b = M_0 - M_P = M_S + M_L \end{equation*}

The initial mass to burnout mass ratio, R, is defined as

(26)   \begin{equation*} R = \frac{M_0}{M_b} \end{equation*}

The payload ratio, \lambda, is defined by

(27)   \begin{equation*} \lambda = \frac{M_L}{M_0 - M_L} = \frac{M_L}{M_P + M_S} \end{equation*}

Finally, the structural mass coefficient, \epsilon, is defined by

(28)   \begin{equation*} \epsilon = \frac{M_S}{M_P + M_S} = \frac{M_S}{M_0 - M_L} \end{equation*}

In light of these foregoing definitions, then it can be shown that the mass ratio R is given by

(29)   \begin{equation*} R = \frac{M_0}{M_b} = \frac{1 + \lambda}{\epsilon + \lambda} \end{equation*}

so in terms of the mass and payload ratios and the structural mass coefficient then

(30)   \begin{equation*} \Delta V = V_{\rm eq} \ln R = V_{\rm eq} \ln \left( \frac{1 + \lambda}{\epsilon + \lambda} \right) \end{equation*}

Payload mass ratios can vary considerably from mission to mission. However, structural mass coefficients are found to be relatively constant based on historical data for various types of launch vehicles. Also, the value of the structural mass coefficient, \epsilon, is found to fairly similar for different vehicles based on historical data, which is convenient for preliminary design purposes when this value needs to be estimated.

Worked Example #1 – Propellant Needed for a Single Stage Rocket

A single stage rocket must provide a speed of 6,000 m/s (6 km/s) to a payload mass, M_L, of 12,000 kg. The structural mass coefficient, \epsilon, of the vehicle is 0.06. The propellant used in the engine has a specific impulse, I_{\rm sp}, of 325 secs. What must be the initial mass of the rocket, M_0, and its propellant mass, M_P, to meet these requirements?

The equivalent exhaust velocity is

(31)   \begin{equation*} V_{\rm eq} = I_{\rm sp} \, g_0 = 3,188~\mbox{m/s} \end{equation*}

The initial mass to burnout mass ratio, R, is

(32)   \begin{equation*} R = \exp \left( \frac{\Delta V}{V_{\rm eq}} \right) = 6.57 \end{equation*}

and the payload ratio, \lambda, is

(33)   \begin{equation*} \lambda = \frac{1 - \epsilon R}{R - 1} = 0.109 \end{equation*}

where \epsilon = 0.06 is the structural mass coefficient. Therefore, the initial mass of the rocket, M_0, is

(34)   \begin{equation*} M_0 = \left( \frac{1+\lambda}{\lambda} \right) M_L = 122,000~\mbox{kg} \end{equation*}

and the burnout mass, M_b, is

(35)   \begin{equation*} M_b = \frac{M_0}{R} = 18,600~\mbox{kg} \end{equation*}

Finally, the propellant mass, M_P, is

(36)   \begin{equation*} M_P = M_0 - M_b = 122,000 - 18,600 = 103,400~\mbox{kg} \end{equation*}

which is a fairly large rocket.

Worked Example #2 – Burnout Velocity of a Single Stage Rocket

Use the rocket equation to determine the burnout velocity and the maximum achievable height of a simple rocket, assuming that it is launched vertically. Neglect the aerodynamic drag forces. Solve for the burnout velocity and maximum altitude if the burnout time is 60 seconds. The specific impulse is 250 seconds, the initial mass is 12,700 kg, and the propellant mass is 8,610 kg.

The rocket equation gives the change in the velocity of the vehicle \Delta V, i.e.,

    \[ \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) \]

where M_0 is the initial mass of the vehicle and M_b is the final or burnout mass. If gravity is included (but no aerodynamic drag), then

    \[ \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) - g_0 \, t_b. \]

where t_b is the burnout time, which is given as 60 seconds in this case.

The equivalent velocity V_{\rm eq} is given in terms of the specific impulse, i.e.,

    \[ V_{\rm eq} = I_{\rm sp} \, g_0 = 250 \times 9.81 = 2,452.5~\mbox{m/s} \]

and the burnout mass M_b is given by

    \[ M_b = M_0 - M_P = 12,700 - 8,610 = 4,090~\mbox{kg} \]

The burnout velocity is given by the rocket equation, i.e.,

    \[ \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) - g_0 \, t_b \]

Therefore, \Delta V at the burnout is

    \[ \Delta V = 2,778.16 - 9.81 \times 60 = 2,778.16 - 588.6 = 2,189.56~\mbox{m/s} = 2.18~\mbox{km/s} \]

Assuming the rate of fuel consumption is constant, then the mass of the rocket M varies over time as

    \[ M = M_0 - M_P \left( \frac{t}{t_b} \right) = M_0 - \left( M_0 - M_b \right) \left( \frac{t}{t_b} \right) \]

The velocity of the rocket is

    \[ V = V_{\rm eq} \ln \left( \frac{M_0}{M} \right) - g_0 \, t \]

The height achieved at the burnout time, t_b, is then

    \[ H_b = \int_0^{t_b} V dt \]

which after some rearrangement gives

    \[ H_b = V_{\rm eq} t_b \left( \frac{ \ln \left( \displaystyle{\frac{M_0}{M_b}} \right) }{\displaystyle{\frac{M_0}{M_b} - 1} } \right) - \frac{1}{2} g_0 \, t_b^2 \]

Inserting the values gives

    \[ H_b = 2,452.5 \times 60 \left( \frac{ \ln \left( \displaystyle{\frac{12,700}{4,090}} \right) }{\displaystyle{\frac{12,700}{4,090} - 1} } \right) - \frac{1}{2} \times 9.81 \times 60^2 \]

Therefore, the height achieved is

    \[ H_b = 2,452.5 \times 60 \left( \frac{1.133}{2.105} \right) - \frac{1}{2} \times 9.81 \times 60^2 = 61.54~\mbox{km} \]

The final additional coasting height of the rocket can then be determined by equating the rocket’s kinetic energy at its burnout time with its change in potential energy between that point and the maximum obtained height, which is left as an exercise.

Energy Requirements for a Launch Vehicle

The energy and propellant requirements for a rocket (booster) and its payload with the \Delta V needed to reach a specific orbital altitude h_s can be estimated using the principles of conservation of energy as well as the rocket equation, i.e., using

(37)   \begin{equation*} \Delta V = I_{\rm sp} \, g_0 \ln \left( \frac{M_0}{M_f} \right) \end{equation*}

where the value of I_{\rm sp} depends on the type of rocket motor and its propellant. and M_f is the final (burnout) mass after the burn at burnout time, t_b, during which a propellant mass M_P is consumed.

In this case it is assumed, for simplicity, that it is a single-stage launch vehicle (i.e., no staging). If the value of I_{\rm sp} is known, then this form of the rocket equation can be used to determine the propellent mass, M_P, needed to give a certain \Delta V for a satellite or spacecraft to reach an orbit at the required altitude, h_s, as shown in the figure below.

The rocket equation be used to help estimate the propellent mass needed to lift a payload into orbit.

Several factors can influence the \Delta V required, i.e.,

  1. The needed orbital altitude above the surface of the Earth, h_s.
  2. The orbital inclination relative to the Earth’s equatorial plane.
  3. The launch latitude from the Earth (this affecting the initial energy).
  4. The effects to overcome gravity (when the rocket is going vertically).
  5. The aerodynamic drag on the rocket in the lower atmosphere.

Kinetic Energy

For the satellite or spacecraft to reach the required orbital altitude, h_s, then the \Delta V required to give the needed kinetic energy is

(38)   \begin{equation*} \Delta V = \sqrt{ \frac{G \, M_E}{R_E + h_s} } = \Delta V_K \end{equation*}

where G is the universal gravitational constant where G = 6.67428 \times 10^{-11} N m^2 kg^{-2}. Notice the the result in Eq 38 does not depend on the mass of the satellite or spacecraft. This kinetic energy component, i.e., \Delta V_K, is the dominant component of the total energy needed for high orbits.

Potential Energy

The \Delta V associated with creating the needed potential energy for the same orbit can be shown to be

(39)   \begin{equation*} \Delta V = \sqrt{ \frac{ G \, M_E }{R_E} } \sqrt{ 2 - \frac{R_E}{R_E + h_s} } - \sqrt{ \frac{ G \, M_E}{R_E + h_s} } = \Delta V_U \end{equation*}

For low orbital altitudes with h \ll R_E, so then

(40)   \begin{equation*} \Delta V_U = \frac{h_s}{R_E} \sqrt{ \frac{G \, M_E}{R_E} } \end{equation*}

which is the dominant term in this case compared to the kinetic energy.

Gravity Effect

As previously discussed, there is a gravitational effect that needs to be added to the total needed \Delta V, which is usually referred to as the gravity loss. If the rocket follows a curved trajectory and pitches over at a local trajectory angle \gamma (with respect to the horizon) as it increases altitude, then the rocket equation becomes

(41)   \begin{equation*} \Delta V_g = \int_0^{t_b} \sin \gamma \, g_0 \, dt \end{equation*}

where \gamma = 90^{\circ} when the rocket flies vertically and \gamma = 0 when it flies horizontally. The proper evaluation of this latter term, therefore, requires information about the launch profile.

Aerodynamic Drag Effects

Finally, there is an aerodynamic drag on the rocket as it flies through the lower atmosphere. This drag force, D,  can be expressed as

(42)   \begin{equation*} D = \frac{1}{2} \varrho V_{\infty}^2 \, C_D \, A_{\rm ref} = q \, C_D \, A_{\rm ref} \end{equation*}

where q is the dynamic pressure, i.e., q = \frac{1}{2} \varrho V_{\infty}^2 where \varrho is the local density of the air, and V_{\infty} is the true airspeed, the latter values also requiring information about the launch profile. The density of the air can be represented by an ISA model for both the low and extended atmospheres. The reference area, A_{\rm ref}, in Eq. 42 is usually taken as the projected frontal area of the rocket. The drag coefficient C_D will also depend on the exact shape of the rocket and its flight Mach number.

It is apparent that drag of the rocket given by Eq. 42 is proportional to the density of the air and the square of the airspeed. Therefore, to minimize the aerodynamic losses the rocket should ascend vertically and as slowly as possible. However, this approach is contrary to the need to accelerate the rocket as quickly as possible to minimize gravitational effects. Nevertheless, the dynamic pressure during the launch needs particular emphasis because this quantity affects the aerodynamic-induced structural loads on the rocket. In many cases, the value of the “maximum q” on the rocket will be limited, requiring the rocket motors to be throttled down temporarily to prevent excessive aerodynamic loads.

Launch Latitude Effects

The needed \Delta V also depends on the latitude of the launch from the Earth. The initial energy given to the rocket from the Earth’s rotation is higher for launch sites closer to the equator. The needed \Delta V is also lower if the rocket is launched in the direction of the Earth’s rotation (toward the east).

The reason the Earth’s surface moves faster at the equator has to do with its solid-body rotation. It takes 24 hours for the Earth to complete one full rotation on its axis, but the equator has to cover more distance in the same amount of time than any other point. This means that anything on the surface of the Earth at the equator is already moving at a speed of about 1,670 kph from the rotation of the Earth. This speed is referred to as the Earth’s rotational velocity, and it can be harnessed to help launch spacecraft into orbit.

By launching a spacecraft from nearer to the equator, it can take advantage of the Earth’s rotational velocity, which helps it achieve the necessary speed to stay in orbit. For launches from Cape Canaveral, for example, the \Delta V_{\rm lat} boost is about 0.3 km/s, and is not insignificant in terms of reduced propellant requirements.

Total Required Velocity

Therefore, the total required \Delta V_T is the sum

(43)   \begin{equation*} \Delta V_T = \Delta V_K + \Delta V_U + \Delta V_g +\Delta V_d - \Delta V_{\rm lat} \end{equation*}

The aerodynamic contribution, \Delta V_d, is the lowest effect compared to the other three. The required propellant mass, M_P, can then be estimated from the rocket equation, i.e.,

(44)   \begin{equation*} \Delta V_T = I_{\rm sp} \, g_0 \ln \left( \frac{M_f}{M_0} \right) = I_{\rm sp} \, g_0 \ln \left( \frac{M_0 - M_P}{M_0} \right) \end{equation*}

again assuming no staging. Using the principles of logarithms, then

(45)   \begin{equation*} \frac{M_P}{M_0} = 1 - \exp \left( -\frac{\Delta V_T}{I_{\rm sp} \, g_0} \right) \end{equation*}

If staging is used, then the \Delta V for each stage would be calculated as each stage is depleted of propellant and the empty stage is then discarded. For a satellite or other spacecraft to reach an orbital altitude of 300–400 km, the needed \Delta V is about 10 km/s.

Estimating the Escape Velocity from Earth

The mass of the Earth, M_E, is 5.97 \times 10^{24} kg. The radius of an initial orbit will be the radius of the Earth R_E, which is 6.3781 \times 10^6 m, plus the orbital height, h. Also, we know that the universal gravitational constant G = 6.67428 \times 10^{-11} N m^2 kg^{-2}. Assuming that the orbital height relative to the radius of the Earth is small then it can be shown that the minimum escape velocity is given by

    \[ V_{\rm esc} = \sqrt{ \frac{2 \, G \, M_E }{R_E} } = \sqrt{ \frac{2 \times (6.67428 \times 10^{-11}) \times (5.97 \times 10^{24}) }{6.3781 \times 10^6 }} = 11.18~\mbox{km/s} \]

All spacecraft that are designed to head out into space away from the Earth must be given a velocity that is larger than 11.2 km/s.

Staged Rocket Vehicles

Staging a launch vehicle aims to maximize the payload ratio that can be launched into space. The goal is to launch the largest payload to the required burnout velocity using the least amount of non-payload mass (defined as the structural weight of the rocket plus the fuel). As shown in the figure below, there are two types of staging:

  1. Serial staging, where the stages are ignited, used, and jettisoned in serial sequence.
  2. Parallel staging, where all stages are ignited and used, but the stages are jettisoned as they burn out, e.g., solid rocket boosters.

To reach an optimal staging of the launch vehicle, there are four basic considerations:

  1. The initial stages should have lowest values of I_{\rm sp}, and later stages should have highest I_{\rm sp.
  2. The stages with the lower I_{\rm sp} should contribute more to \Delta V.
  3. Each successive stage should be smaller than the previous stage.
  4. Similar stages should provide similar increments to \Delta V.

Serial Staged Rocket

For a serial staged launch vehicle, then

(46)   \begin{equation*} R_i = \frac{M_{0_{i}}}{M_{b_{i}}} ; \quad \lambda_i = \frac{M_{L_{i}}}{M_{0_{i}} - M_{L_{i}}} ; \quad \epsilon_i = \frac{M_{0_{i}}}{M_{P_{i}} + M_{S_{i}}} \end{equation*}

where the index i refers to the stage number. Also

(47)   \begin{equation*} R_i = \frac{1 + \lambda_i}{\epsilon_i + \lambda_i} \end{equation*}

For a staged vehicle overall, then the \Delta V values are added for each stage, i.e.,

(48)   \begin{equation*} \Delta V = \sum_{i=1}^{N} \Delta V_i = \sum_{i=1}^{N} V_{{\rm eq}_{i}} \ln R_i \end{equation*}

For a staged launch vehicle then the structural mass coefficients are usually similar to those of a single stage. However, payload ratios are generally higher for a staged vehicle. In some cases, it may be desirable to find the maximum allowable structural mass to meet a certain set of payload requirements, which sets some goals and constraints for the purposes of structural design.

 

Staging is an important process for maximizing the orbital speed and height of a payload.

Below is step-by-step breakdown of the general procedure for calculating the total burnout velocity or time for a multi-stage rocket with serial staging:

    1. Divide the rocket as a system into its individual stages. Each stage is typically characterized by a specific propulsion system or motors, and will have its own set of parameters such as mass, specific impulse, thrust, and fuel weight or burn time.
    2. For each stage, calculate the initial mass (the total mass of the rocket at the beginning of the stage burn) and the final mass (the mass of the rocket at the end of the stage burn after the propellant has been burned.
    3. Use the rocket equation to calculate the burnout velocity for each individual stage. The gravitational force (weight) acting on the rocket must be considered.
    4. Add the burnout velocity of each stage to the initial velocity of the previous stage. Assuming that each stage occurs immediately after the previous one, the burnout velocity of one stage becomes the initial velocity for the next stage.
    5. Repeat steps 2–4 for the final stage of the rocket system, until the burnout time and/or burnout velocity has been calculated for the final stage carrying the payload mass.

Worked Example #3 – Two Stage Rocket Calculation

Consider a two-stage rocket with the following design characteristics. Payload mass = 60 kg. First stage: propellant mass = 7,200 kg, structural mass = 800 kg, and the mass flow rate is \overbigdot{m} = 80.0 kg s^{-1}. Second stage: propellant mass = 5,400 kg, structural mass = 600 kg, and the burn time is 100 s. The specific impulse, I_{\rm sp}, for the first and second stages is 275 s. Calculate the following:

For the first stage:

  1. The equivalent exhaust velocity.
  2. The thrust produced.
  3. The total burn time.
  4. The burnout velocity.

For the second stage:

  1. The equivalent exhaust velocity.
  2. The mass flow rate.
  3. The thrust produced.
  4. The final burnout velocity.

For the first stage:

1. The equivalent exhaust velocity, V_{{\rm eq}_{1}}, is

    \[ V_{{\rm eq}_{1}} = I_{\rm sp_{1}}} \, g_0 = 275 \times 9.81 = 2,697.75~\mbox{m/s} \]

2. The thrust, T_1, produced is

    \[ T_1 = \overbigdot{m}_1 \, V_{{\rm eq}_{1}} = 80.0\times 2,697.75 = 21,5820~\mbox{N} \]

3. The total burn time, t_{b_{1}}, is

    \[ t_{b_{1}} = \frac{M_{P_{1}}}{\overbigdot{m}_1}= \frac{7,200}{80.0} = 90.0~\mbox{s} \]

4.  The initial mass, M_{0_{1}}, is

    \[ M_{0_{1}} = M_{P_{1}} +M_{P_{2}} + M_{S_{1}} + M_{S_{2}} + M_{L} \]

and inserting the values gives

    \[ M_{0_{1}} = 7,200 +5,400 + 800 +600 +60 = 14,060~\mbox{kg} \]

The burnout mass, M_{b_{1}}, is

    \[ M_{b_{1}} = M_{0_{1}} - M_{P_{1}} = M_{P_{2}} + M_{S_{1}} + M_{S_{2}} + M_{L} \]

and with the given values leads to the burnout mass of the first stage as

    \[ M_{b_{1}} = 5,400 + 800 + 600 + 60 = 6,860~\mbox{kg} \]

The \Delta V increment for the first stage is

    \[ \Delta V_1 = V_{b_{1}} - 0 = V_{{\rm eq}_{1}} \ln \left( \frac{M_{0_{1}}}{M_{b_{1}}} \right) - g_0 \, t_{b_{1}} \]

and inserting the values gives

    \[ V_{b_{1}} = 2,697.75 \ln \left( \frac{14,060}{6,860} \right) - 9.81\times 90 = 1,053.0~\mbox{m/s} \]

Therefore, for the first stage then the burnout velocity is

    \[ V_b_{1}} = 1,053.0~\mbox{m/s} \]

For the second stage:

1. Because the I_{\rm sp} remains the same for stage 2, the equivalent exhaust velocity will also be the same, i.e.,

    \[ V_{{\rm eq}_{2}} = I_{\rm sp_{2}} \, g_0 = 275 \times 9.81 = 2,697.75~\mbox{m/s} \]

2. The mass flow rate, \overbigdot{m}_2, is

    \[ \overbigdot{m}_2 = \frac{M_{P_{2}}}{t_{b_{2}}} = \frac{5,400}{100} = 54.0~\mbox{kg/s} \]

3. The thrust T_2 produced is

    \[ T_2 = \overbigdot{m}_2 \, V_{{\rm eq}_{2}} = 54\times 2,697.75 = 145,678~\mbox{N} \]

4. For the second stage, the initial mass is

    \[ M_{0_{2}} = M_{P_{2}} + M_{S_{2}} + M_{L} = 5,400 + 600 + 60 = 6,060~\mbox{kg} \]

The burnout mass for the second stage is

    \[ M_{b_{2}} = M_{0_{2}} - M_{P_{2}} = M_{S_{2}} + M_{L} = 600 + 60 = 660~\mbox{kg} \]

The \Delta V for the second stage is

    \[ \Delta V_2 = V_{{\rm eq}_{2}} \ln \left( \frac{M_{0_{2}}}{M_{b_{2}}} \right) - g_0 \, t_{b_{2}} \]

and inserting the values gives

    \[ \Delta V_2 = 2,697.75 \ln \left( \frac{6,060}{660} \right) - 9.81 \times 100 = 5,000.5~\mbox{m/s} \]

The final value of the burnout velocity, V_f, will be

    \[ V_f = \Delta V_1 + \Delta V_2 = 1,053.0 + 5,000.5 = 6,053.5~\mbox{m/s} = 6.053~\mbox{km/s} \]

Parallel Staged Rocket

With a parallel staged launch vehicle, there are usually dissimilar rockets and rocket motors burning simultaneously, as shown in the figure below. An example would be the NASA Space Shuttle, which used LH2/LOX for the “core” main engines on the Orbiter with solid propellant rocket boosters being used to significantly augment the initial launch velocity. NASA’s SLS uses the same type of core and booster. Other types launch vehicles may be configured with different numbers of solid rocket boosters, depending on the payload mass and the desired orbital altitude. An exception is the SpaceX Falcon Heavy, which uses two additional liquid propellant boosters that are identical to the first (core) stage.

 

With a parallel staged launch vehicle, there are usually dissimilar rockets and rocket motors burning simultaneously.

The rocket equation for a core rocket system with one or more boosters can be written as

(49)   \begin{equation*} \Delta V = \overline{V}_{\rm eq} \ln \left( \frac{M_0}{M_f}\right) - g_0 \, t_{b_{1}} \end{equation*}

where M_0 is the initial mass (core plus boosters) and M_f is the final mass of the launch vehicle after booster burnout. The average or mean equivalent exhaust velocity, \overline{V}_{\rm eq}, is given by

(50)   \begin{equation*} \overline{V}_{\rm eq} = \frac{ \overbigdot{m}_b \, V_{{\rm eq}_{b}} + \overbigdot{m}_c \, V_{{{\rm eq}_{c}}} }{ \overbigdot{m}_b + \overbigdot{m}_c } \end{equation*}

where the subscripts c and b refer to the core and the boosters, respectively.

In this case, the initial mass at the launch point is

(51)   \begin{equation*} M_0 = M_{P_{c}} + M_{P_{b}} + M_{S_{c}} + M_{S_{b} + M_{L} \end{equation*}

where M_{P_{c}} and M_{P_{b}} are the propellant masses for the core and the boosters, respectively, M_{S_{c}} and M_{S_{b}} are the structural masses for the core and the boosters, respectively, and M_{L} is the payload mass.

The determination of the final mass, M_f, of a launch vehicle with parallel boosters takes some further consideration because the core launcher will still have propellant left at booster burnout. Therefore, the final mass at the time of booster burnout, say t_{b_{0}}, can be written as

(52)   \begin{equation*} M_f = \chi M_{P_{c}} + M_{S_{c}} + M_{S_{b}} + M_{L} \end{equation*}

where \chi is the fraction of propellant mass remaining in the core at booster burnout.

Notice that the mass of propellant, M_P, used in a given time, t, is M_P = \overbigdot{m} \, t, so the mean equivalent exhaust velocity, \overline{V}_{\rm eq}, can also be written as

(53)   \begin{equation*} \overline{V}_{\rm eq} = \frac{ M_{P_{b}} \, V_{{\rm eq}_{b}} + (1 - \chi) \, M_{P_{c}} \, V_{{{\rm eq}_{c}}} }{ M_{P_{b}} + (1 - \chi) M_{P_{c}}} \end{equation*}

Therefore, the launch of parallel rocket stages can be presented use the sum of pseudo-serial stages, where for stage “0” with the boosters and the core together, then

(54)   \begin{equation*} \Delta V_0 = \overline{V}_{\rm eq} \ln \left( \frac{M_{P_{c}} + M_{P_{b}} + M_{S_{c}} + M_{S_{b}} + M_{L} }{\chi M_{P_{c}} + M_{S_{c}} + M_{S_{b}} + M_{L}}\right) - g_0 \, t_{b_{0}} \end{equation*}

For stage “1” after booster separation, then the initial mass will be

(55)   \begin{equation*} M_{0_{2}} = \chi M_{P_{c}} + M_{S_{c}} + M_{L} \end{equation*}

where \chi is the fraction of propellant mass remaining in the core at booster burnout. The final mass after the stage 1 core burns out will be

(56)   \begin{equation*} M_f = M_{S_{c}} + M_{L} \end{equation*}

where \chi is the fraction of propellant mass remaining in the core at booster burnout.

Therefore, for stage 1, then

(57)   \begin{equation*} \Delta V_1 = \overline{V}_{\rm eq} \ln \left( \frac{ \chi M_{P_{c}} + M_{S_{c}} + M_{L} }{M_{S_{c}} + M_{L} }\right) - g_0 \, t_{b_{1}} \end{equation*}

The \Delta V values of the remaining stages, 2......N, are then calculated as a serial launcher, as before, i.e., the final launch velocity V_f will be

(58)   \begin{equation*} V_f = \Delta V_0 + \Delta V_1 + \Delta V_2 + ......+ \Delta V_N \end{equation*}

Worked Example #5 – Space Shuttle Launch

Consider a Space Shuttle launch, which is a parallel rocket system launch vehicle. The Shuttle used LH2/LOX for the “core” main engines on the Orbiter, with two solid rocket boosters or SRBs. It is required to estimate the final burnout velocity. Neglect the gravity term. The information available includes the following:

  • Orbiter:
    • Structural mass = 110,000 kg
    • Payload mass = 24,000 kg
    • Specific impulse = 454 seconds
    • Burnout time = 480 seconds
  • External tank
    • Structural mass = 30,000 kg
    • Propellant mass = 720,000 kg
  • SRB (for each):
    • Structural mass = 86,000 kg
    • Propellant mass = 500,000 kg
    • Specific impulse = 269 seconds
    • Burnout time = 124 seconds

For the Orbiter or the “core,” the equivalent exhaust velocity, V_{{\rm eq}_{c}}, is

    \[ V_{{\rm eq}_{c}} = I_{\rm sp_{c}}} \, g_0 = 454 \times 9.81 = 4,453.7~\mbox{m/s} \]

For the SRBs, the equivalent exhaust velocity, V_{{\rm eq}_{b}}, is

    \[ V_{{\rm eq}_{b}} = I_{\rm sp_{b}}} \, g_0 = 269 \times 9.81 = 2,638.9~\mbox{m/s} \]

The fraction of propellant mass remaining in the core at booster burnout will be

    \[ \chi = \frac{480 - 124}{480} = 0.742 \]

The mean equivalent exhaust velocity, \overline{V}_{\rm eq}, is

    \[ \overline{V}_{\rm eq} = \frac{ M_{P_{b}} \, V_{{\rm eq}_{b}} + (1 - \chi) \, M_{P_{c}} \, V_{{{\rm eq}_{c}}} }{ M_{P_{b}} + (1 - \chi) M_{P_{c}}} \]

and putting in the values gives

    \[ \overline{V}_{\rm eq} = \frac{ 2 \times 500,000 \times 2,638.9 + (1 - 0.742) \times 720,000 \times 4,453.7}{2 \times 500,000 + (1 - 0.742)\times 720,000} = 2,923.2~\mbox{m/s} \]

The initial mass at the point of launch is

    \[ M_{0_{0}} = M_{P_{c}} + M_{P_{b}} + M_{S_{c}} + M_{S_{b}} + M_{L} \]

and putting in the values gives

    \begin{eqnarray*} M_{0_{0}} & = & (720,000 + 2 \times 500,000) + (110,000 + 30,000) + 2 \times 86,000 + 24,000 \\ & = & 2,056,000~\mbox{kg} \end{eqnarray*}

The final mass of the vehicle at SRB burnout is

    \[ M_{f_{0}} = \chi M_{P_{c}} + M_{S_{c}} + M_{S_{b}} + M_{L} \]

and putting in the values gives

    \[ M_{f_{0}} = 0.742 \times 720,000 + (110,000 + 30,000) + 2 \times 86,000 + 24,000 = 870,240~\mbox{kg} \]

Therefore, the \Delta V at SRB burnout is

    \[ \Delta V_0 = \overline{V}_{\rm eq} \ln \left( \frac{M_{0_{0}}}{M_{f_{0}} } \right) = 2,923.2 \ln \left( \frac{2,056,000}{870,240} \right) = 2,512.2~\mbox{m/s} \]

The initial vehicle mass after SRB separation is

    \[ M_{0_{1}} = \chi M_{P_{c}} + M_{S_{c}} + M_{L} \]

and putting in the values gives

    \[ M_{0_{1}} = 0.742 \times 720,000 + (110,000 + 30,000) + 24,000 = 698,240~\mbox{kg} \]

The final mass at the depletion of the propellant in the external tank is

    \[ M_{f_{1}} = M_{S_{c}} + M_{L} = (110,000 + 30,000) + 24,000 = 164,000~\mbox{kg} \]

The extra \Delta V at complete burnout is

    \[ \Delta V_1 = V}_{{\rm eq}_{c}} \ln \left( \frac{M_{0_{1}}}{M_{f_{1}}} \right) = 4,453.7 \ln \left( \frac{698,240}{164,000} \right) = 6,453.1~\mbox{m/s} \]

Therefore, the final burnout velocity of the Orbiter is

    \[ V_f = \Delta V_0 + \Delta V_1 = 2,512.2 + 6,453.1 = 8,965.3~\mbox{m/s} = 32.3~\mbox{kph} \]

which is fairly close to the values quoted by NASA bearing in mind that the gravitational \Delta V has been neglected.

Summary & Closure

Gas/liquid fuel-based rotor motors have been employed for many spaceflight applications, including most launch vehicles. Such systems have good efficiency and have the advantage that the engine can be throttled by regulating the fuel flow, e.g., to limit dynamic pressure loads on the vehicle during its launch. However, this capability comes at a price, including mechanical complexity and associated weight. For some launch vehicles, the thrust from the liquid fuel engines is augmented with solid rocket boosters, which can produce more than half of the initial thrust on leaving the launch pad, e.g., the Space Shuttle concept. Solid motors are also used on missiles and for other in-space applications. The lower performance and non-throttling characteristics of solid propellant motors are acceptable because of their operational simplicity, although solid rocket boosters are by no means simple propulsion systems.

Since the dawn of human space flight in the 1960s, advancements have continued to be made by using improved propellants and rocket motor designs. Today, most space missions use a combination of different engines and fuels, which are selected to optimize the thrust capability needed at each mission stage, including the launch and when in space. Rocket motors produce extremely high thrusts and operate near their safe limits, albeit relatively for relatively short times. However, the reliability of rocket motors is still a concern, especially when they are recovered and reused to save on launch costs. In addition, the environmental compatibility of rocket fuels has become a more significant concern in recent years, and the move toward considering alternative propellants such as methane will continue.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Discuss and identify safety hazards associated with liquid, solid, and hybrid rocket propulsion systems.
  • What type of fuel is used for the SpaceX Raptor engine? What are the advantages of this type of fuel?
  • Research high specific impulse propulsion systems. What values for specific impulse are attainable?
  • Consider a staged vehicle for which all of the V_{\rm eq} values are the same. How can \Delta V be maximized?
  • What sort of mass is an astronaut?
  • What types of thrusters are used for cubesats, nanosats, and small spacecraft attitude control?

Other Useful Online Resources

To learn more about rocket propulsion, check out these helpful online resources:

  • A series of videos on how rocket motors work: The playlist is here.
  • A good video on how rockets work.
  • An article on the history of rockets by NASA.
  • A simple guide on how rockets work by NASA.
  • Learn more about solid fuel rockets from Northrop-Grumman.
  • Some good resources on solid fuel rockets by AeroJet-RocketDyne.
  • The Smithsonian Air and Space Museum entry on the X-15 a rocket-powered aircraft.
  • An article from AIAA: X-15 Propulsion System
  • Elon Musk explains how the Raptor 2 rocket engine works.
  • A video on how to start up a rocket motor!
  1. Prof. Eric "Rick" Perrell wrote initial drafts of this chapter. ↵

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Introduction to Aerospace Flight Vehicles by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n34