Deltav
Deltav (literally "change in velocity"), symbolised as ∆v and pronounced deltavee, as used in spacecraft flight dynamics, is a measure of the impulse that is needed to perform a maneuver such as launch from, or landing on a planet or moon, or inspace 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.
Deltav is produced by reaction engines, such as rocket engines, and is proportional to the thrust per unit mass, and burn time, and is used to determine the mass of propellant required for the given maneuver through the Tsiolkovsky rocket equation.
For multiple maneuvers, deltav sums linearly.
For interplanetary missions deltav is often plotted on a porkchop plot which displays the required mission deltav as a function of launch date.
Definition
where
Specific cases
In the absence of external forces:
where is the coordinate acceleration.
When thrust is applied in a constant direction (v/v is constant) 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 (t_{1} − t_{0})/2 then the velocity difference is 0, but deltav is the same as for the nonreversed thrust.
For rockets the 'absence of external forces' is taken to mean the absence of gravity, atmospheric drag as well as the absence of aerostatic back pressure on the nozzle and hence the vacuum I_{sp} is used for calculating the vehicle's deltav capacity via the rocket equation, and the costs for the atmospheric losses are rolled into the deltav budget when dealing with launches from a planetary surface.
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

(1)
where
 v_{exh} is the velocity of the exhaust gas
 ρ is the propellant flow rate to the combustion chamber
The acceleration of the spacecraft caused by this force will be

(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

(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 and ending at t_{1} as

(4)
Changing the integration variable from time t to the spacecraft mass m one gets

(5)
Assuming to be a constant not depending on the amount of fuel left this relation is integrated to

(6)
which is the Tsiolkovsky rocket equation.
If for example 20% of the launch mass is fuel giving a constant of 2100 m/s (a typical value for a hydrazine thruster) the capacity of the reaction control system is
If is a nonconstant function of the amount of fuel left^{[1]}
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.^{[2]} 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 (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 outofplane control with thruster burn periods extending over several hours around the nodes this approximation is fair.
Producing deltav
Deltav is typically provided by the thrust of a rocket engine, but can be created by other engines. The timerate of change of deltav 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 deltav 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 deltav. Therefore, in modern spacecraft propulsion systems considerable study is put into reducing the total deltav needed for a given spaceflight, as well as designing spacecraft that are capable of producing larger deltav.
Increasing the deltav provided by a propulsion system can be achieved by:
 staging
 increasing specific impulse
 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 deltav’s can be added:
When m_{1}, m_{2} are the mass ratios of the maneuvers, and v1, v2 are the deltav’s of the first and second maneuvers
Where V = v_{1} + v_{2} and M = m_{1} M_{2}.
Which is just the rocket equation applied to the sum of the two maneuvers.
This is convenient since it means that deltav’s can be calculated and simply added and the mass ratio calculated only for the overall vehicle for the entire mission. Thus deltav is commonly quoted rather than mass ratios which would require multiplication.
Deltav budgets
When designing a trajectory, deltav budget is used as a good indicator of how much propellant will be required. Propellant usage is an exponential function of deltav in accordance with the rocket equation, it will also depend on the exhaust velocity.
It is not possible to determine deltav 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 missionbased reasons, to put the spacecraft in an orbit of different inclination, a substantial deltav 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 deltav. The total deltav to be applied can then simply be found by addition of each of the deltav’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 deltav, see Hohmann transfer orbit, gravitational slingshot, and Interplanetary Transport Network. It is also notable that large thrust can reduce gravity drag.
Deltav 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 deltav in the direction of the velocity the specific orbital energy gained per unit deltav 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 deltavs are required at different launch dates. A diagram that shows the required deltav 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.^{[3]}
Deltavs around the Solar System^{[4]}^{[5]}
Deltav needed for various orbital manoeuvers using conventional rockets; red arrows show where optional aerobraking can be performed in that particular direction, black numbers give deltav in km/s that apply in either direction. Lowerdeltav transfers than shown can often be achieved, but involve rare transfer windows or take significantly longer, see: fuzzy orbital transfers. The figure 2.5 for LEO to GTO is higher than necessary^{[6]} and the figure of 30 for LEO to the sun is also too high.^{[7]}
 C3
 Escape orbit
 GEO
 Geosynchronous orbit
 GTO
 Geostationary transfer orbit
 L4/5
 Earth–Moon L_{4}L_{5} Lagrangian point
 LEO
 Low Earth orbit
See also
 Deltav budget
 Gravity drag
 Orbital maneuver
 Orbital stationkeeping
 Spacecraft propulsion
 Specific impulse
 Tsiolkovsky rocket equation
 Deltav (physics)
References
 ↑ Can be the case for a "blowdown" system for which the pressure in the tank gets lower when fuel has been used and that not only the fuel rate but to some lesser extent also the exhaust velocity decreases.
 ↑ The thrust force per unit mass being where and are given functions of time
 ↑ Mars Exploration: Features
 ↑ "Rockets and Space Transportation". Archived from the original on July 1, 2007. Retrieved June 1, 2013.
 ↑ "DeltaV Calculator". Archived from the original on Mar 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 on the NASA website (dead link).
 ↑ The sum of LEO to GTO and GTO to GEO should equal LEO to GEO. The precise figures depend on what low earth orbit is used. According to Geostationary transfer orbit, the speed of a GTO at perigee can be just 9.8 km/s. This corresponds to an LEO at about 700 km altitude, where its speed would be 7.5 km/s, giving a deltav of 2.3 km/s. Starting from a lower LEO would require more deltav to get to GTO, but then the total for LEO to GEO would have to be higher.
 ↑ The speed of the earth going around the sun is 29.78 km/s, equivalent to a specific kinetic energy of 443 km^{2}/s^{2}. One must add to this the potential energy depth of LEO, about 61 km^{2}/s^{2}, to give a kinetic energy close to Earth of 504 km^{2}/s^{2}, corresponding to a speed of 31.8 km/s. Since the LEO speed is 7.8 km/s, the deltav is only 24 km/s. It would be possible to reach the sun with less deltav using gravity assists. See Solar Probe Plus.