Delta-v
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Delta-''v'' (also known as "
change Change, Changed or Changing may refer to the below. Other forms are listed at Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of t ...
in
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
"), symbolized as and pronounced , as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
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 In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...
. As used in this context, it is not the same as the physical change in velocity of said spacecraft. A simple example might be the case of a conventional rocket-propelled spacecraft, which achieves thrust by burning fuel. Such a spacecraft's delta-''v'', then, would be the change in velocity that spacecraft can achieve by burning its entire fuel load. Delta-''v'' is produced by
reaction engine A reaction engine is an engine, engine or motor that produces thrust by expelling reaction mass (reaction propulsion), in accordance with Newton's third law of motion. This law of motion is commonly paraphrased as: "For every action force there ...
s, such as
rocket engine A rocket engine is a reaction engine, producing thrust in accordance with Newton's third law by ejecting reaction mass rearward, usually a high-speed Jet (fluid), jet of high-temperature gas produced by the combustion of rocket propellants stor ...
s, and is proportional to the
thrust Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that ...
per unit mass and the burn time. It is used to determine the mass of
propellant A propellant (or propellent) is a mass that is expelled or expanded in such a way as to create a thrust or another motive force in accordance with Newton's third law of motion, and "propel" a vehicle, projectile, or fluid payload. In vehicle ...
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

\Delta = \int_^ \frac\, dt where * is the instantaneous
thrust Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that ...
at time . * is the instantaneous
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
at time . Change in velocity is useful in many cases, such as determining the change in
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
( impulse), where: \Delta \mathbf = m \Delta \mathbf, where \mathbf is momentum and m is mass.


Specific cases

In the absence of external forces: \Delta = \int_^ \left, \dot\\, dt where \dot is the coordinate acceleration. When thrust is applied in a constant direction ( is constant) this simplifies to: \Delta = , 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 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 drag and atmospheric drag, as well as the absence of aerostatic back pressure on the nozzle, and hence the vacuum I 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.


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 where * is the velocity of the exhaust gas in rocket frame * is the propellant flow rate to the combustion chamber The acceleration \dot 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 t_0\, and ending at as Changing the integration variable from time to the spacecraft mass one gets Assuming v_\text\, to be a constant not depending on the amount of fuel left this relation is integrated to which is the Tsiolkovsky rocket equation. If for example 20% of the launch mass is fuel giving a constant v_\text of 2100 m/s (a typical value for a
hydrazine Hydrazine is an inorganic compound with the chemical formula . It is a simple pnictogen hydride, and is a colourless flammable liquid with an ammonia-like odour. Hydrazine is highly hazardous unless handled in solution as, for example, hydraz ...
thruster) the capacity of the
reaction control system A reaction control system (RCS) is a spacecraft system that uses Thrusters (spacecraft), thrusters to provide Spacecraft attitude control, attitude control and translation (physics), translation. Alternatively, reaction wheels can be used for at ...
is \Delta = 2100\ \ln\left(\frac\right)\,\text = 460\,\text. If v_\text is a non-constant function of the amount of fuel left v_\text = v_\text(m) 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 \Delta 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.


Production

Delta-''v'' is typically provided by the
thrust Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that ...
of a
rocket engine A rocket engine is a reaction engine, producing thrust in accordance with Newton's third law by ejecting reaction mass rearward, usually a high-speed Jet (fluid), jet of high-temperature gas produced by the combustion of rocket propellants stor ...
, 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 increases exponentially with increasing delta-''v''. Therefore, in modern
spacecraft propulsion 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 e ...
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: * staging *increasing
specific impulse Specific impulse (usually abbreviated ) is a measure of how efficiently a reaction mass engine, such as a rocket engine, rocket using propellant or a jet engine using fuel, generates thrust. In general, this is a ratio of the ''Impulse (physics), ...
*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-''v'' can be summed: When are the mass ratios of the maneuvers, and are the delta-''v'' of the first and second maneuvers \begin m_1 m_2 &= e^ e^ \\ &= e^\frac \\ &= e^ = M \end where and . 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 The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
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 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.


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. 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 o
Space Settlements: A Design Study
on the NASA website .
Lower-delta-''v'' transfers than shown can often be achieved, but involve rare transfer windows or take significantly longer, see: . ;C3: Escape orbit ;GEO: Geosynchronous orbit ;GTO: Geostationary transfer orbit ;L4/5: Earth–Moon
Lagrangian point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium (mechanics), equilibrium for small-mass objects under the gravity, gravitational influence of two massive orbit, orbiting b ...
;LEO:
Low Earth orbit A low Earth orbit (LEO) is an geocentric orbit, orbit around Earth with a orbital period, period of 128 minutes or less (making at least 11.25 orbits per day) and an orbital eccentricity, eccentricity less than 0.25. Most of the artificial object ...


LEO reentry

For example the Soyuz spacecraft makes a de-orbit from the ISS in two steps. First, it needs a delta-''v'' of 2.18 m/s for a safe separation from the space station. Then it needs another 128 m/s for reentry.


See also

* Delta-''v'' budget * Gravity drag * Orbital maneuver * Orbital stationkeeping *
Spacecraft propulsion 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 e ...
* Orbital propellant depot *
Specific impulse Specific impulse (usually abbreviated ) is a measure of how efficiently a reaction mass engine, such as a rocket engine, rocket using propellant or a jet engine using fuel, generates thrust. In general, this is a ratio of the ''Impulse (physics), ...
* Tsiolkovsky rocket equation * Delta-''v'' (physics)


References

{{DEFAULTSORT:Delta-V Astrodynamics Spacecraft propulsion Physical quantities