Editing Delta-v (section)

==Orbital maneuvers==
{{main article|orbital maneuver|rocket equation}}
Orbit maneuvers are made by firing a [[Rocket engine|thruster]] to produce a reaction force acting on the spacecraft. The size of this force will be
{{block indent|{{NumBlk||<math>T = v_\text{exh}\ \rho</math>|{{EquationRef|1}}}}}}
where
*{{math|{{var|v<sub>exh</sub>}}}} is the velocity of the exhaust gas in rocket frame
*{{math|{{var|ρ}}}} is the propellant flow rate to the combustion chamber

The acceleration <math>\dot{v}</math> of the spacecraft caused by this force will be
{{block indent|{{NumBlk||<math>\dot{v} = \frac{T}{m} = v_\text{exh}\ \frac{\rho}{m}</math>|{{EquationRef|2}}}}}}
where {{math|{{var|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
{{block indent|{{NumBlk||<math>\dot{m} = -\rho\,</math>|{{EquationRef|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 <math>t_0\,</math> and ending at {{math|{{var|t}}{{sub|1}}}} as
{{block indent|{{NumBlk||<math>\Delta{v} = -\int_{t_0}^{t_1} {v_\text{exh}\ \frac{\dot{m}}{m}}\, dt</math>|{{EquationRef|4}}}}}}

Changing the integration variable from time {{math|{{var|t}}}} to the spacecraft mass {{math|{{var|m}}}} one gets
{{block indent|{{NumBlk||<math>\Delta{v} = -\int_{m_0}^{m_1} {v_\text{exh}\ \frac{dm}{m}}</math>|{{EquationRef|5}}}}}}

Assuming <math>v_\text{exh}\,</math> to be a constant not depending on the amount of fuel left this relation is integrated to
{{block indent|{{NumBlk||<math>\Delta{v} =  v_\text{exh}\ \ln\left(\frac{m_0}{m_1}\right)</math>|{{EquationRef|6}}}}}}

which is the [[Tsiolkovsky rocket equation]].

If for example 20% of the launch mass is fuel giving a constant <math>v_\text{exh}</math> of 2100&nbsp;m/s (a typical value for a [[hydrazine]] thruster) the capacity of the [[reaction control system]] is
<math display="block">\Delta{v} = 2100\ \ln\left(\frac{1}{0.8}\right)\,\text{m/s} = 460\,\text{m/s}.</math>

If <math>v_\text{exh}</math> is a non-constant function of the amount of fuel left<ref>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 <math>\rho</math> but to some lesser extent also the exhaust velocity <math>v_\text{exh}</math> decreases.</ref>
<math display="block">v_\text{exh} = v_\text{exh}(m)</math>
the capacity of the reaction control system is computed by the integral ({{EquationNote|5}}).

The acceleration ({{EquationNote|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.<ref>The thrust force per unit mass being <math>\frac{f(t)}{m(t)} = v_\text{exh}(t) \frac{\dot{m}(t)}{m(t)}</math> where <math>f(t)</math> and <math>m(t)</math> are given functions of time <math>t</math>.</ref> But for many purposes, typically for studies or for maneuver optimization, they are approximated by impulsive maneuvers as illustrated in figure 1 with a <math>\Delta{v}</math> as given by ({{EquationNote|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.

[[File:Impulsive maneuver.svg|thumb|Figure 1: Approximation of a finite thrust maneuver with an impulsive change in velocity having the delta-''v'' given by ({{EquationNote|4}}).]]
This approximation with impulsive maneuvers is in most cases very accurate, at least when chemical propulsion is used. For low thrust systems, typically [[Electrically powered spacecraft propulsion|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.