Delta-v

Overview

Astrodynamics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It...

a Δ

*v*or

**delta-v**(literally "change in velocity") is a scalar

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

which takes units of speed

Speed

In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

. It is a measure of the amount of "effort" that is needed to change from one trajectory

Trajectory

A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

to another by making an orbital maneuver

Orbital maneuver

In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft.For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called a deep-space maneuver .-delta-v:...

.

Delta-v is produced by the use of propellant

Propellant

A propellant is a material that produces pressurized gas that:* can be directed through a nozzle, thereby producing thrust ;...

by reaction engines to produce a thrust that accelerates the vehicle.

where is the instantaneous thrust

Thrust

Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....

is the instantaneous mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

If there are no other external force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

s than gravity, this is the integral of the magnitude of the g-force

G-force

The g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...

.

In the absence of external forces, and when thrust is applied in a constant direction this simplifies to:

which is simply the magnitude of the change in velocity

Delta-v (physics)

In general physics, delta-v is simply the change in velocity. The Greek letter delta is a standard mathematical symbol to represent change ....

.

Unanswered Questions

Encyclopedia

In astrodynamics

a Δ

which takes units of speed

. It is a measure of the amount of "effort" that is needed to change from one trajectory

to another by making an orbital maneuver

.

Delta-v is produced by the use of propellant

by reaction engines to produce a thrust that accelerates the vehicle.

where is the instantaneous thrust

is the instantaneous mass

If there are no other external force

s than gravity, this is the integral of the magnitude of the g-force

.

In the absence of external forces, and when thrust is applied in a constant direction this simplifies to:

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 then the velocity difference is , but delta-v is the same as for the non-reversed thrust.

For rockets the 'absence of external forces' usually is taken to mean the absence of 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, and the costs for the atmospheric losses are rolled into the delta-v budget

when dealing with launches from a planetary surface.

to produce a reaction force acting on the spacecraft. The size of this force will be

where

is the velocity of the exhaust gas is the propellant flow rate to the combustion chamber

The acceleration of the spacecraft caused by this force will be

where 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

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 and ending at as

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

Assuming to be a constant not depending on the amount of fuel left this relation is integrated to

which is the well known "rocket equation"

If for example 20% of the launch mass is fuel giving a constant of 2100 m/s (typical value for a hydrazine

thruster) the capacity of the reaction control system

is

m/s = 469 m/s.

If is a non-constant function of the amount of fuel left

the capacity of the reaction control system is computed by the integral

The acceleration 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. But for many purposes, typically for studies or for maneuver optimization, they are approximated by impulsive maneuvers as illustrated in figure 1 with a as given by . 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.

of a rocket engine

, but can be created by other reaction engines. The time-rate of change of delta-v is the magnitude of the acceleration

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 a large delta-v.

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

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

When M1, M2 are the mass ratios of the maneuvers, and V1, V2 are the delta-v's of the first and second maneuvers

Where V = V1 + V2 and M = M1 M2.

Which is just the rocket equation applied to the sum of the two maneuvers.

This is convenient since it means that delta-vs can be calculated and added and the mass ratio calculated only for the overall vehicle for the entire mission. Thus delta-v is commonly quoted rather than sequences of mass ratios.

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-vs 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 Superhighway. 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.

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.

See also powered slingshots.

, since launch should only occur when the mission is within the capabilities of the vehicle to be employed.

Astrodynamics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It...

a Δ

*v*or**delta-v**(literally "change in velocity") is a scalarScalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

which takes units of speed

Speed

In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

. It is a measure of the amount of "effort" that is needed to change from one trajectory

Trajectory

A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

to another by making an orbital maneuver

Orbital maneuver

In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft.For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called a deep-space maneuver .-delta-v:...

.

Delta-v is produced by the use of propellant

Propellant

A propellant is a material that produces pressurized gas that:* can be directed through a nozzle, thereby producing thrust ;...

by reaction engines to produce a thrust that accelerates the vehicle.

## Definition

where is the instantaneous thrust

Thrust

Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....

is the instantaneous mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

If there are no other external force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

s than gravity, this is the integral of the magnitude of the g-force

G-force

The g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...

.

In the absence of external forces, and when thrust is applied in a constant direction this simplifies to:

which is simply the magnitude of the change in velocity

Delta-v (physics)

In general physics, delta-v is simply the change in velocity. The Greek letter delta is a standard mathematical symbol to represent change ....

. However, this relation does not hold in the general case: If, for instance, a constant, unidirectional acceleration is reversed after then the velocity difference is , but delta-v is the same as for the non-reversed thrust.

For rockets the 'absence of external forces' usually is taken to mean the absence of 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, and the costs for the atmospheric losses are rolled into the delta-v budget

Delta-v budget

In the astrodynamics and aerospace industry, a delta-v budget is the estimated delta-v requirements for the various propulsive tasks and orbital maneuvers over one or more phases of a space mission.Sample delta-v budget will enumerate various classes of maneuvers, delta-v per maneuver, number of...

when dealing with launches from a planetary surface.

## Orbital maneuvers

Orbit maneuvers are made by firing a thrusterRocket engine

A rocket engine, or simply "rocket", is a jet engineRocket Propulsion Elements; 7th edition- chapter 1 that uses only propellant mass for forming its high speed propulsive jet. Rocket engines are reaction engines and obtain thrust in accordance with Newton's third law...

to produce a reaction force acting on the spacecraft. The size of this force will be

where

is the velocity of the exhaust gas is the propellant flow rate to the combustion chamber

The acceleration of the spacecraft caused by this force will be

where 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

If now the direction of the force, i.e. the direction of the nozzle

Nozzle

A nozzle is a device designed to control the direction or characteristics of a fluid flow as it exits an enclosed chamber or pipe via an orifice....

, is fixed during the burn one gets the velocity increase from the thruster force of a burn starting at time and ending at as

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

Assuming to be a constant not depending on the amount of fuel left this relation is integrated to

which is the well known "rocket equation"

If for example 20% of the launch mass is fuel giving a constant of 2100 m/s (typical value for a hydrazine

Hydrazine

Hydrazine is an inorganic compound with the formula N2H4. It is a colourless flammable liquid with an ammonia-like odor. Hydrazine is highly toxic and dangerously unstable unless handled in solution. Approximately 260,000 tons are manufactured annually...

thruster) the capacity of the reaction control system

Reaction control system

A reaction control system is a subsystem of a spacecraft whose purpose is attitude control and steering by the use of thrusters. An RCS system is capable of providing small amounts of thrust in any desired direction or combination of directions. An RCS is also capable of providing torque to allow...

is

m/s = 469 m/s.

If is a non-constant function of the amount of fuel left

the capacity of the reaction control system is computed by the integral

The acceleration 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. But for many purposes, typically for studies or for maneuver optimization, they are approximated by impulsive maneuvers as illustrated in figure 1 with a as given by . Like this one can for example use a "patched conics" approach modeling the maneuver as a shift from one Kepler orbit

Kepler orbit

In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...

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.

## Producing Delta-v

Delta-v is typically provided by the thrustThrust

Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....

of a rocket engine

Rocket engine

A rocket engine, or simply "rocket", is a jet engineRocket Propulsion Elements; 7th edition- chapter 1 that uses only propellant mass for forming its high speed propulsive jet. Rocket engines are reaction engines and obtain thrust in accordance with Newton's third law...

, but can be created by other reaction 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

Spacecraft propulsion

Spacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. There are many different methods. Each method has drawbacks and advantages, and spacecraft propulsion is an active area of research. However, most spacecraft today are propelled by forcing a gas from the...

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 a large delta-v.

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

- staging
- increasing specific impulseSpecific impulseSpecific impulse is a way to describe the efficiency of rocket and jet engines. It represents the derivative of the impulse with respect to amount of propellant used, i.e., the thrust divided by the amount of propellant used per unit time. If the "amount" of propellant is given in terms of mass ,...
- improving propellant mass fraction

## 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-vs can be added:

When M1, M2 are the mass ratios of the maneuvers, and V1, V2 are the delta-v's of the first and second maneuvers

Where V = V1 + V2 and M = M1 M2.

Which is just the rocket equation applied to the sum of the two maneuvers.

This is convenient since it means that delta-vs can be calculated and added and the mass ratio calculated only for the overall vehicle for the entire mission. Thus delta-v is commonly quoted rather than sequences of mass ratios.

## 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

Conservation of energy

The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

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

Inclination

Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

, 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-vs 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

Elliptic orbit

In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1; this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 . In a...

.

For examples of calculating delta-v, see Hohmann transfer orbit

Hohmann transfer orbit

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits, typically both in the same plane....

, gravitational slingshot

Gravitational slingshot

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 celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense...

, and Interplanetary Superhighway. It is also notable that large thrust can reduce gravity drag

Gravity drag

In astrodynamics and rocketry, gravity drag is a measure of the loss in the net performance of a rocket while it is thrusting in a gravitational field...

.

Delta-v is also required to keep satellites in orbit and is expended in propulsive orbital stationkeeping

Orbital stationkeeping

In astrodynamics orbital station-keeping is a term used to describe the orbital maneuvers made by thruster burns that are needed to keep a spacecraft in a particular assigned orbit.For many Earth satellites the effects of the non-Keplerian forces, i.e...

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 energySpecific orbital energy

In the gravitational two-body problem, the specific orbital energy \epsilon\,\! of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy , divided by the reduced mass...

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.

See also powered slingshots.

### 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 windowLaunch window

Launch window is a term used in spaceflight to describe a time period in which a particular launch vehicle must be launched. If the rocket does not launch within the "window", it has to wait for the next window....

, since launch should only occur when the mission is within the capabilities of the vehicle to be employed.

### Delta-vs around the Solar System

## See also

- Delta-v budgetDelta-v budgetIn the astrodynamics and aerospace industry, a delta-v budget is the estimated delta-v requirements for the various propulsive tasks and orbital maneuvers over one or more phases of a space mission.Sample delta-v budget will enumerate various classes of maneuvers, delta-v per maneuver, number of...
- Gravity dragGravity dragIn astrodynamics and rocketry, gravity drag is a measure of the loss in the net performance of a rocket while it is thrusting in a gravitational field...
- Orbital maneuverOrbital maneuverIn spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft.For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called a deep-space maneuver .-delta-v:...
- Orbital stationkeepingOrbital stationkeepingIn astrodynamics orbital station-keeping is a term used to describe the orbital maneuvers made by thruster burns that are needed to keep a spacecraft in a particular assigned orbit.For many Earth satellites the effects of the non-Keplerian forces, i.e...
- Spacecraft propulsionSpacecraft propulsionSpacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. There are many different methods. Each method has drawbacks and advantages, and spacecraft propulsion is an active area of research. However, most spacecraft today are propelled by forcing a gas from the...
- Specific impulseSpecific impulseSpecific impulse is a way to describe the efficiency of rocket and jet engines. It represents the derivative of the impulse with respect to amount of propellant used, i.e., the thrust divided by the amount of propellant used per unit time. If the "amount" of propellant is given in terms of mass ,...
- Tsiolkovsky rocket equationTsiolkovsky rocket equationThe Tsiolkovsky rocket equation, or ideal rocket equation is an equation that is useful for considering vehicles that follow the basic principle of a rocket: where a device that can apply acceleration to itself by expelling part of its mass with high speed and moving due to the conservation of...
- Delta-v (physics)Delta-v (physics)In general physics, delta-v is simply the change in velocity. The Greek letter delta is a standard mathematical symbol to represent change ....