# Delta-v

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Delta-v (more known as "change in velocity"), symbolized as v and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such as launching from or landing on a planet or moon, or an in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle.

## Contents

As a simple example, take a conventional rocket-propelled spacecraft which achieves thrust by burning fuel. The spacecraft's delta-v is the change in velocity that spacecraft can achieve by burning its entire fuel load.

Delta-v is produced by reaction engines, such as rocket engines, and is proportional to the thrust per unit mass and the burn time. It is used to determine the mass of propellant required for the given maneuver through the Tsiolkovsky rocket equation.

For multiple maneuvers, delta-v sums linearly.

For interplanetary missions, delta-v is often plotted on a porkchop plot, which displays the required mission delta-v as a function of launch date.

## Definition

${\displaystyle \Delta {v}=\int _{t_{0}}^{t_{1}}{\frac {|T(t)|}{m(t)}}\,dt}$

where

• T(t) is the instantaneous thrust at time t.
• m(t) is the instantaneous mass at time t.

## Specific cases

In the absence of external forces:

${\displaystyle \Delta {v}=\int _{t_{0}}^{t_{1}}\left|{\dot {v}}\right|\,dt}$

where ${\displaystyle {\dot {v}}}$ is the coordinate acceleration.

When thrust is applied in a constant direction (v/|v| is constant) this simplifies to:

${\displaystyle \Delta {v}=|v_{1}-v_{0}|}$

which is simply the magnitude of the change in velocity. However, this relation does not hold in the general case: if, for instance, a constant, unidirectional acceleration is reversed after (t1t0)/2 then the velocity difference is 0, but delta-v is the same as for the non-reversed thrust.

For rockets, "absence of external forces" is taken to mean the absence of gravity and atmospheric drag, as well as the absence of aerostatic back pressure on the nozzle, and hence the vacuum Isp is used for calculating the vehicle's delta-v capacity via the rocket equation. In addition, the costs for atmospheric losses and gravity drag are added into the delta-v budget when dealing with launches from a planetary surface. [1]

## Orbital maneuvers

Orbit maneuvers are made by firing a thruster to produce a reaction force acting on the spacecraft. The size of this force will be

${\displaystyle T=v_{\text{exh}}\ \rho }$

(1)

where

• vexh is the velocity of the exhaust gas in rocket frame
• ρ is the propellant flow rate to the combustion chamber

The acceleration ${\displaystyle {\dot {v}}}$ of the spacecraft caused by this force will be

${\displaystyle {\dot {v}}={\frac {T}{m}}=v_{\text{exh}}\ {\frac {\rho }{m}}}$

(2)

where m is the mass of the spacecraft

During the burn the mass of the spacecraft will decrease due to use of fuel, the time derivative of the mass being

${\displaystyle {\dot {m}}=-\rho \,}$

(3)

If now the direction of the force, i.e. the direction of the nozzle, is fixed during the burn one gets the velocity increase from the thruster force of a burn starting at time ${\displaystyle t_{0}\,}$ and ending at t1 as

${\displaystyle \Delta {v}=-\int _{t_{0}}^{t_{1}}{v_{\text{exh}}\ {\frac {\dot {m}}{m}}}\,dt}$

(4)

Changing the integration variable from time t to the spacecraft mass m one gets

${\displaystyle \Delta {v}=-\int _{m_{0}}^{m_{1}}{v_{\text{exh}}\ {\frac {dm}{m}}}}$

(5)

Assuming ${\displaystyle v_{\text{exh}}\,}$ to be a constant not depending on the amount of fuel left this relation is integrated to

${\displaystyle \Delta {v}=v_{\text{exh}}\ \ln \left({\frac {m_{0}}{m_{1}}}\right)}$

(6)

which is the Tsiolkovsky rocket equation.

If for example 20% of the launch mass is fuel giving a constant ${\displaystyle v_{\text{exh}}}$ of 2100 m/s (a typical value for a hydrazine thruster) the capacity of the reaction control system is

${\displaystyle \Delta {v}=2100\ \ln \left({\frac {1}{0.8}}\right)\,{\text{m/s}}=460\,{\text{m/s}}.}$

If ${\displaystyle v_{\text{exh}}}$ is a non-constant function of the amount of fuel left [2]

${\displaystyle v_{\text{exh}}=v_{\text{exh}}(m)}$

the capacity of the reaction control system is computed by the integral ( 5 ).

The acceleration ( 2 ) caused by the thruster force is just an additional acceleration to be added to the other accelerations (force per unit mass) affecting the spacecraft and the orbit can easily be propagated with a numerical algorithm including also this thruster force. [3] But for many purposes, typically for studies or for maneuver optimization, they are approximated by impulsive maneuvers as illustrated in figure 1 with a ${\displaystyle \Delta {v}}$ as given by ( 4 ). Like this one can for example use a "patched conics" approach modeling the maneuver as a shift from one Kepler orbit to another by an instantaneous change of the velocity vector.

This approximation with impulsive maneuvers is in most cases very accurate, at least when chemical propulsion is used. For low thrust systems, typically electrical propulsion systems, this approximation is less accurate. But even for geostationary spacecraft using electrical propulsion for out-of-plane control with thruster burn periods extending over several hours around the nodes this approximation is fair.

## Production

Delta-v is typically provided by the thrust of a rocket engine, but can be created by other engines. The time-rate of change of delta-v is the magnitude of the acceleration caused by the engines, i.e., the thrust per total vehicle mass. The actual acceleration vector would be found by adding thrust per mass on to the gravity vector and the vectors representing any other forces acting on the object.

The total delta-v needed is a good starting point for early design decisions since consideration of the added complexities are deferred to later times in the design process.

The rocket equation shows that the required amount of propellant dramatically increases with increasing delta-v. Therefore, in modern spacecraft propulsion systems considerable study is put into reducing the total delta-v needed for a given spaceflight, as well as designing spacecraft that are capable of producing larger delta-v.

Increasing the delta-v provided by a propulsion system can be achieved by:

## Multiple maneuvers

Because the mass ratios apply to any given burn, when multiple maneuvers are performed in sequence, the mass ratios multiply.

Thus it can be shown that, provided the exhaust velocity is fixed, this means that delta-v can be summed:

When m1, m2 are the mass ratios of the maneuvers, and v1, v2 are the delta-v of the first and second maneuvers

{\displaystyle {\begin{aligned}m_{1}m_{2}&=e^{V_{1}/V_{e}}e^{V_{2}/V_{e}}\\&=e^{\frac {V_{1}+V_{2}}{V_{e}}}\\&=e^{V/V_{e}}=M\end{aligned}}}

where V = v1 + v2 and M = m1m2. This is just the rocket equation applied to the sum of the two maneuvers.

This is convenient since it means that delta-v can be calculated and simply added and the mass ratio calculated only for the overall vehicle for the entire mission. Thus delta-v is commonly quoted rather than mass ratios which would require multiplication.

## Delta-v budgets

When designing a trajectory, delta-v budget is used as a good indicator of how much propellant will be required. Propellant usage is an exponential function of delta-v in accordance with the rocket equation, it will also depend on the exhaust velocity.

It is not possible to determine delta-v requirements from conservation of energy by considering only the total energy of the vehicle in the initial and final orbits since energy is carried away in the exhaust (see also below). For example, most spacecraft are launched in an orbit with inclination fairly near to the latitude at the launch site, to take advantage of the Earth's rotational surface speed. If it is necessary, for mission-based reasons, to put the spacecraft in an orbit of different inclination, a substantial delta-v is required, though the specific kinetic and potential energies in the final orbit and the initial orbit are equal.

When rocket thrust is applied in short bursts the other sources of acceleration may be negligible, and the magnitude of the velocity change of one burst may be simply approximated by the delta-v. The total delta-v to be applied can then simply be found by addition of each of the delta-v's needed at the discrete burns, even though between bursts the magnitude and direction of the velocity changes due to gravity, e.g. in an elliptic orbit.

For examples of calculating delta-v, see Hohmann transfer orbit, gravitational slingshot, and Interplanetary Transport Network. It is also notable that large thrust can reduce gravity drag.

Delta-v is also required to keep satellites in orbit and is expended in propulsive orbital stationkeeping maneuvers. Since the propellant load on most satellites cannot be replenished, the amount of propellant initially loaded on a satellite may well determine its useful lifetime.

### Oberth effect

From power considerations, it turns out that when applying delta-v in the direction of the velocity the specific orbital energy gained per unit delta-v is equal to the instantaneous speed. This is called the Oberth effect.

For example, a satellite in an elliptical orbit is boosted more efficiently at high speed (that is, small altitude) than at low speed (that is, high altitude).

Another example is that when a vehicle is making a pass of a planet, burning the propellant at closest approach rather than further out gives significantly higher final speed, and this is even more so when the planet is a large one with a deep gravity field, such as Jupiter.

### Porkchop plot

Due to the relative positions of planets changing over time, different delta-vs are required at different launch dates. A diagram that shows the required delta-v plotted against time is sometimes called a porkchop plot. Such a diagram is useful since it enables calculation of a launch window, since launch should only occur when the mission is within the capabilities of the vehicle to be employed. [4]

### Around the Solar System

Delta-v needed for various orbital manoeuvers using conventional rockets; red arrows show where optional aerobraking can be performed in that particular direction, black numbers give delta-v in km/s that apply in either direction. [5] [6] Lower-delta-v transfers than shown can often be achieved, but involve rare transfer windows or take significantly longer, see: fuzzy orbital transfers.

C3
Escape orbit
GEO
Geosynchronous orbit
GTO
Geostationary transfer orbit
L4/5
Earth–Moon L4 L5 Lagrangian point
LEO
Low Earth orbit

## Related Research Articles

A rocket is a spacecraft, aircraft, vehicle or projectile that obtains thrust from a rocket engine. Rocket engine exhaust is formed entirely from propellant carried within the rocket. Rocket engines work by action and reaction and push rockets forward simply by expelling their exhaust in the opposite direction at high speed, and can therefore work in the vacuum of space.

Spacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. In-space propulsion exclusively deals with propulsion systems used in the vacuum of space and should not be confused with space launch or atmospheric entry.

A pulsed plasma thruster (PPT), also known as a plasma jet engine, is a form of electric spacecraft propulsion. PPTs are generally considered the simplest form of electric spacecraft propulsion and were the first form of electric propulsion to be flown in space, having flown on two Soviet probes starting in 1964. PPTs are generally flown on spacecraft with a surplus of electricity from abundantly available solar energy.

Specific impulse is a measure of how efficiently a reaction mass engine creates thrust. For engines whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to exhaust gas velocity.

In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane. The Hohmann transfer often uses the lowest possible amount of propellant in traveling between these orbits, but bi-elliptic transfers can use less in some cases.

A geosynchronous transfer orbit or geostationary transfer orbit (GTO) is a type of geocentric orbit. Satellites which are destined for geosynchronous (GSO) or geostationary orbit (GEO) are (almost) always put into a GTO as an intermediate step for reaching their final orbit.

In aerospace engineering, the propellant mass fraction is the portion of a vehicle's mass which does not reach the destination, usually used as a measure of the vehicle's performance. In other words, the propellant mass fraction is the ratio between the propellant mass and the initial mass of the vehicle. In a spacecraft, the destination is usually an orbit, while for aircraft it is their landing location. A higher mass fraction represents less weight in a design. Another related measure is the payload fraction, which is the fraction of initial weight that is payload. It can be applied to a vehicle, a stage of a vehicle or to a rocket propulsion system.

A multistage rocket or step rocket is a launch vehicle that uses two or more rocket stages, each of which contains its own engines and propellant. A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or attached next to each other. Two-stage rockets are quite common, but rockets with as many as five separate stages have been successfully launched.

The Tsiolkovsky rocket equation, classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum.

In astrodynamics and aerospace, a delta-v budget is an estimate of the total change in velocity (delta-v) required for a space mission. It is calculated as the sum of the delta-v required to perform each propulsive maneuver needed during the mission. As input to the Tsiolkovsky rocket equation, it determines how much propellant is required for a vehicle of given mass and propulsion system.

In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth an orbital maneuver is called a deep-space maneuver (DSM).

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

This is an alphabetical list of articles pertaining specifically to aerospace engineering. For a broad overview of engineering, see List of engineering topics. For biographies, see List of engineers.

A reaction engine is an engine or motor that produces thrust by expelling reaction mass, in accordance with Newton's third law of motion. This law of motion is commonly paraphrased as: "For every action force there is an equal, but opposite, reaction force."

A gravity turn or zero-lift turn is a maneuver used in launching a spacecraft into, or descending from, an orbit around a celestial body such as a planet or a moon. It is a trajectory optimization that uses gravity to steer the vehicle onto its desired trajectory. It offers two main advantages over a trajectory controlled solely through the vehicle's own thrust. First, the thrust is not used to change the spacecraft's direction, so more of it is used to accelerate the vehicle into orbit. Second, and more importantly, during the initial ascent phase the vehicle can maintain low or even zero angle of attack. This minimizes transverse aerodynamic stress on the launch vehicle, allowing for a lighter launch vehicle.

In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry.

A cold gas thruster is a type of rocket engine which uses the expansion of a pressurized gas to generate thrust. As opposed to traditional rocket engines, a cold gas thruster does not house any combustion and therefore has lower thrust and efficiency compared to conventional monopropellant and bipropellant rocket engines. Cold gas thrusters have been referred to as the "simplest manifestation of a rocket engine" because their design consists only of a fuel tank, a regulating valve, a propelling nozzle, and the little required plumbing. They are the cheapest, simplest, and most reliable propulsion systems available for orbital maintenance, maneuvering and attitude control.

This glossary of aerospace engineering terms pertains specifically to aerospace engineering, its sub-disciplines, and related fields including aviation and aeronautics. For a broad overview of engineering, see glossary of engineering.

Microwave electrothermal thruster, also known as MET, is a propulsion device that converts microwave energy into thermal energy. These thrusters are predominantly used in spacecraft propulsion, more specifically to adjust the spacecraft’s position and orbit. A MET sustains and ignites a plasma in a propellant gas. This creates a heated propellant gas which in turn changes into thrust due to the expansion of the gas going through the nozzle. A MET’s heating feature is like one of an arc-jet ; however, due to the free-floating plasma, there are no problems with the erosion of metal electrodes, and therefore the MET is more efficient.

## References

1. Sarigul-Klijn, Nesrin; Noel, Chris; Sarigul-Klijn, Martinus (2004-01-05). Air Launching Eart-to-Orbit Vehicles: Delta V gains from Launch Conditions and Vehicle Aerodynamics. doi:10.2514/6.2004-872. ISBN   9781624100789.
2. Can be the case for a "blow-down" system for which the pressure in the tank gets lower when fuel has been used and that not only the fuel rate ${\displaystyle \rho }$ but to some lesser extent also the exhaust velocity ${\displaystyle v_{\text{exh}}}$ decreases.
3. The thrust force per unit mass being ${\displaystyle {\frac {f(t)}{m(t)}}=v_{\text{exh}}(t){\frac {{\dot {m}}(t)}{m(t)}}}$ where ${\displaystyle f(t)}$ and ${\displaystyle m(t)}$ are given functions of time ${\displaystyle t}$.
4. "Mars Exploration: Features". marsprogram.jpl.nasa.gov.
5. "Rockets and Space Transportation". Archived from the original on July 1, 2007. Retrieved June 1, 2013.
6. "Delta-V Calculator". Archived from the original on March 12, 2000. Gives figures of 8.6 from Earth's surface to LEO, 4.1 and 3.8 for LEO to lunar orbit (or L5) and GEO resp., 0.7 for L5 to lunar orbit, and 2.2 for lunar orbit to lunar surface. Figures are said to come from Chapter 2 of Space Settlements: A Design Study Archived 2001-11-28 at the Library of Congress Web Archives on the NASA website (dead link).