Introduction to space systems - Appunti

Space propulsion

Introduction to space systems

Academic Year 2025/26

Parozzi Daniele

Lez 15/04/2026

The course on space propulsion is structured around two main families of propulsion systems: chemical propulsion and electric propulsion. The first part of the course focuses on chemical propulsion, while later lectures introduce more technical aspects and electric propulsion systems.

The course includes both theoretical lectures and numerical exercises. The exercises are meant to reinforce understanding and are typically scheduled weekly, although the structure may be adjusted as the course evolves.

The course assumes a basic background in aerospace engineering, but fundamental concepts are revisited when necessary. The main objective is not only to understand formulas, but to develop the ability to determine which propulsion system is appropriate for a given mission.

Role of propulsion in space

To understand why propulsion is needed, one must start from orbital mechanics. In many cases, spacecraft motion can be approximated using the two-body problem, where a spacecraft orbits a central body such as the Earth. In this idealized model, the trajectory is entirely determined by the initial conditions (position and velocity).

If no additional forces act on the system, the spacecraft will remain on the same orbit indefinitely. This leads to a fundamental conclusion: if nothing is done, the orbit does not change.

Therefore, to change the orbit, it is necessary to change the spacecraft’s velocity. This does not simply mean changing its position but modifying the way the velocity evolves over time. Since gravity alone determines the natural motion, any modification requires an additional acceleration, which is provided by propulsion.

Two main roles of propulsion can be identified. Primary propulsion is used to perform large velocity changes (ΔV), such as orbit transfers. It typically requires high thrust and significant propellant mass. Auxiliary propulsion, on the other hand, is used to counteract perturbations and maintain a desired orbit or attitude. These perturbations (as the astrodynamics’ professor said about control) include gravitational effects from the Moon and Sun, solar radiation pressure, and atmospheric drag in low Earth orbit. Auxiliary propulsion generally involves smaller thrust levels and lower propellant consumption, although this is not always strictly true.

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Physical principle of propulsion

Propulsion is fundamentally based on Newton’s third law of motion, which states that every action is accompanied by an equal and opposite reaction. In terms of mechanics, this principle is a direct consequence of the conservation of momentum: the total momentum of an isolated system cannot change unless there is an interaction with another system (quantità di moto = momento lineare). Therefore, in order to change its velocity, any body must exchange momentum with something else.

On Earth, this process is usually implicit because there is always an external medium available. When a person walks, they push the ground backward, and the ground exerts an equal and opposite force that propels them forward. Similarly, a car accelerates because its wheels push against the road, and a propeller-driven aircraft moves forward by accelerating air backward. In all these cases, the system (the person, the car, or the aircraft) exchanges momentum with an external environment (ground, air, or water), which absorbs the opposite momentum.

In space, the situation is fundamentally different. The environment is essentially a vacuum, meaning there is no surrounding medium capable of exchanging momentum with the spacecraft. As a result, a spacecraft cannot rely on any external body to generate thrust. The only way to produce acceleration is to carry onboard a mass that can be expelled. This mass is called the propellant.

The propulsion process in space therefore consists of accelerating a portion of this onboard mass and ejecting it in one direction. By conservation of momentum, the expelled propellant acquires momentum in the direction of ejection, while the spacecraft gains an equal amount of momentum in the opposite direction, resulting in a thrust force. In this sense, the momentum exchange does not occur with the external environment, but internally between the spacecraft and its own propellant.

This fundamental difference has major engineering consequences. Because the spacecraft must carry all the reaction mass it needs, the propellant often represents a large fraction of the total mass, especially in launch vehicles. This directly affects the size of the tanks, the structural design, and ultimately the cost and feasibility of the mission. For this reason, one of the primary objectives in propulsion system design is to minimize the amount of propellant required to achieve a given mission, although this objective must be balanced against other constraints such as power availability, thrust requirements, and system complexity.

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Derivation of thrust

To derive the thrust equation, we consider a system with variable mass. Let the spacecraft have mass and velocity and let it eject an infinitesimal amount of propellant (), over a small time interval. Applying conservation of momentum to an isolated system, the total momentum before and after ejection must be equal. For an infinitesimal time interval, Io: = where is the exhaust velocity relative to the spacecraft, and is the infinitesimal mass of propellant expelled (taken as a positive quantity).

To express this relation in terms of force, we divide both sides by the time interval: =

At this point, we introduce the propellant mass flow rate: ṁ =

Substituting into the previous equation, we obtain: = ṁ

Recognizing that represents the net force acting on the spacecraft (Newton’s second law), we identify this force as the thrust: = ṁ

This is the simplest expression for thrust and shows that thrust is equal to the mass flow rate multiplied by the exhaust velocity, i.e. the momentum carried away per unit time by the expelled propellant.

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Pressure correction and effective exhaust velocity

The previous derivation assumes that the expelled propellant no longer interacts with the spacecraft. In reality, at the nozzle exit plane, the exhaust gases still exert a pressure on the nozzle.

The complete expression for thrust is therefore: = ṁ + ( − ) where:

• Is the exhaust velocity at the nozzle exit,
• Is the exit pressure,
• Is the ambient pressure,
• Is the nozzle exit area.

The first term represents the momentum (dynamic) thrust, while the second term represents the pressure (static) thrust. In vacuum, the ambient pressure is negligible, and the pressure contribution is often small. However, in atmospheric conditions, this term can significantly affect the thrust and must be considered.

The pressure term in the thrust equation is positive in vacuum because but it is = 0, generally small since the exhaust pressure is also very low due to strong expansion inside the nozzle; therefore, thrust is mainly dominated by the momentum term ṁ. More generally, the pressure term represents a deviation from optimal nozzle expansion: it is negative in overexpanded conditions and positive in under expanded conditions, but in both cases it indicates a loss of efficiency compared to the ideal condition. In particular, overexpanded nozzles generate shock waves (and possibly flow = separation) to adjust to the higher ambient pressure, while underexpanded flows generate expansion fans outside the nozzle, meaning that part of the expansion occurs externally and is not fully effective in producing thrust; these phenomena also occur in vacuum, but they are much weaker because both exit and ambient pressures are very low.

Additionally, since ambient pressure varies significantly with altitude, a nozzle cannot be perfectly adapted in all operating conditions, which leads to important design trade-offs in propulsion systems.

Forse va espresso in termini di pressione relativa, dove Patm=0 così può diventare anche negativa e considerare come perdita di spinta il caso di ugello sovraespanso.

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To simplify the analysis, an effective exhaust velocity is introduced, which incorporates both the momentum and pressure contributions. It is defined as: ( − ) = + = ṁ ṁ is not the real exhaust velocity but it is the effective exhaust velocity which takes also into account the contribution of pressure of the static thrust.

Using this definition, the thrust expression can be written in a compact form as: ṁ =

This simplification is widely used because it provides a linear relation between thrust and mass flow rate. Moreover, in many space applications the pressure term is small (especially in vacuum), so the effective exhaust velocity is often very close to the actual exhaust velocity. We will call simply because the static thrust is usually really low.

Energy and power requirements

Producing thrust requires energy, since the propellant must be accelerated to the exhaust velocity. The kinetic energy per unit mass of the exhaust is: 22

If propellant is expelled at a rate, the power required to accelerate it is: ṁ ṁ=

This relationship highlights a fundamental trade-off: increasing the exhaust velocity reduces the required propellant mass, but increases the power needed to accelerate the propellant.

This explains the difference between propulsion systems. Chemical propulsion relies on chemical reactions to provide energy, resulting in high thrust but relatively moderate exhaust velocities. Electric propulsion uses electrical energy (e.g., from solar arrays) to accelerate ions to very high velocities, achieving high efficiency (high specific impulse) but typically producing low thrust.

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Performance metrics

To evaluate propulsion systems, several performance metrics are introduced.

The total impulse is defined as the integral of thrust over time: = ∫ It represents the overall effect of the propulsion system. The same total impulse can be achieved by applying a large thrust for a short duration or a small thrust for a long duration.

However, total impulse alone does not fully characterize performance, since it does not account for the spacecraft mass. A more meaningful parameter is the specific impulse, defined as: = where is the propellant mass and is standard gravity. The specific impulse is directly related to the effective exhaust velocity: = 0

Specific impulse measures how effectively a propulsion system uses propellant. A higher value means that less propellant is required to achieve a given mission objective.

Relationship between thrust and velocity change

The velocity change produced by propulsion is given by: = ∫ This expression shows that the acceleration depends on the instantaneous mass of the spacecraft. As the propellant is consumed, the mass decreases, and for the same thrust the acceleration increases. If the mass variation is small, it is possible to approximate the mass as constant and write: Δ ≍

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This approximation is acceptable for auxiliary propulsion, where the propellant mass is small compared to the total mass. However, it becomes inaccurate for primary propulsion, where mass changes are significant.

Rocket equation (Tsiolkovsky equation)

= ( )

Procedure: Starting from the definition of velocity change as the time integral of acceleration: Δ = ∫ 0 and substituting the thrust expression we obtain: = ṁ, ṁ Δ = ∫ 0

Introducing the propellant mass flow rate: ṁ = the integration variable can be changed from time to expelled mass, leading to: Δ = ∫

Since the spacecraft mass decreases as propellant is expelled, we relate the masses as: = − and the integral becomes: 0Δ = ∫

Assuming constant exhaust velocity, the integration gives: = ( )

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This is the Tsiolkovsky rocket equation, one of the most important results in space propulsion. It can also be written as: /=

The propellant mass is simply: = −

• Propellant mass, i.e. the total mass of propellant expelled during the burn
• Initial mass, i.e. the total mass of the spacecraft before the burn (including propellant, structure, and payload)
• Final mass, i.e. the total mass of the spacecraft after the burn (including structure and payload, excluding propellant)

Interpretation of the rocket equation

The key feature of the rocket equation is its exponential nature. The required propellant mass increases exponentially with the required velocity increment ΔV, for a given exhaust velocity.

If ΔV is much larger than, the propellant mass becomes extremely large (aumenta l’esponente).

For example, if the ratio ΔV/ is around 5, the initial mass can be more than 100 times the final mass. This means that to deliver a small payload, a very large amount of propellant is required.

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This has critical implications for spacecraft design:

• Propellant mass dominates the total mass,
• Tank size and structural mass increase accordingly,
• Mission cost and feasibility are strongly affected.

As a result, a major goal in propulsion system design is to maximize the exhaust velocity (or specific impulse), thereby reducing the required propellant mass.

Note: The rocket equation does not explicitly distinguish payload mass, but it is included in both the initial mass and the final mass. The payload remains after the burn, so it directly increases. Although it appears in both terms, it affects the mass ratio because the propellant mass is only present in, while the payload is present in both. As a result, increasing the payload increases the denominator more significantly, reducing the mass ratio and therefore the achievable ΔV. Consequently, a higher payload / requires more propellant to achieve the same ΔV.

Key takeaways from the lecture

Propulsion is required to modify the spacecraft velocity, and therefore its orbit. Without propulsion, orbital motion remains unchanged.

The spacecraft must carry its own propellant, since there is no external medium in space. This makes propellant mass a dominant factor in mission design.

There is a fundamental trade-off between thrust, propellant consumption, and power. High exhaust velocity reduces propellant usage but requires more energy.

Total impulse measures the integrated effect of thrust, but the most relevant quantity for mission design is the velocity increment ΔV.

The rocket equation reveals that propellant requirements grow exponentially with ΔV, making propulsion one of the main limiting factors in space missions.

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Lez 17/04/2026

Space propulsion can be analyzed starting from the concept of specific impulse, which is directly related to propellant consumption. The propellant mass is simply the difference between initial and final mass, and this relationship is governed by the Tsiolkovsky rocket equation. This equation links the mission requirement, usually expressed as a ΔV, to the propulsion system performance.

From this relation, it is clear that increasing the specific impulse reduces the amount of propellant required. However, accelerating the propellant to a given exhaust velocity requires energy, therefore power. ṁ= The source of this power defines the type of propulsion system.

In chemical propulsion, the energy comes from a chemical reaction between fuel and oxidizer. In electric propulsion, the energy comes from an external electrical source, typically solar arrays, although nuclear sources could in principle be used. Nuclear propulsion is a hybrid propulsion because the nuclear fission energy is converted into electrical energy and then in thrust power. Nuclear propulsion is not considered further because it is not yet available with sufficient power for propulsion applications.

The key difference between chemical and electric propulsion lies in how power is related to propellant flow rate. In chemical propulsion, the energy needed to accelerate the propellant is contained within the propellant itself. If a mass flow rate is burned, the generated power is proportional to the mass flow rate multiplied by the specific chemical energy. 1 2 ṁ = ṁ ℎ 2

Dove:

• = Propellant mass flow rate
• ṁ = Specific chemical energy
• ℎ Exhaust velocity (in realtà è Ce ma scriviamo C per comodità)
• =

So power generated by chemical reaction converted into kineti