Delta-v

Delta-v (more known as "change in velocity"), symbolized as ${\textstyle {\Delta v}}$ and pronounced deltah-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 said spacecraft.

For other uses, see Delta-v (disambiguation).

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 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.

$T (t)$ is the instantaneous at time $t$ .

thrust

$m (t)$ is the instantaneous at time $t$ .

mass

$v exh$ is the velocity of the exhaust gas in rocket frame

$ρ$ is the propellant flow rate to the combustion chamber

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

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

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

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 $t 1$ as

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

Assuming $v_{\text{exh}}\,$ 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 $v_{\text{exh}}$ of 2100 m/s (a typical value for a hydrazine thruster) the capacity of the reaction control system is

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

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 $\Delta {v}$ 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.

staging

increasing

specific impulse

improving

propellant mass fraction

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: