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title	tags	summary
Orbit manoeuvres	maneuvers, impulse, finite-burn, hohmann, delta-v	How a change in orbit is made, what it costs in propellant, and why the cheapest transfer between two circular orbits is the one everybody uses.

Orbit manoeuvres

A manoeuvre changes an orbit. The satellite is at one point on a closed curve and must end up at another point on a different closed curve, and the change has to be paid for in propellant. This is the subject: how large the change is, how much it costs, and how the cost changes when the orbit is not circular.

Definition

A delta-v is the magnitude of a change in velocity. A manoeuvre is planned as a sequence of delta-v, and the propellant cost follows from their sum by the rocket equation.

An impulse applies the whole change in zero time. It is the right model for a burn that is short compared with the orbital period, which for a low Earth orbit means anything under a minute.

A finite burn applies a thrust over a duration long enough that the satellite moves while thrusting. The distinction is not academic: a low Earth orbital transfer assisted by electric propulsion can last for hours and its gravity loss is a substantial fraction of the ideal cost.

Mathematics

The rocket equation

The velocity a vehicle can produce depends on the exhaust velocity of its propellant and on the mass ratio,


\Delta v = v_e \ln\frac{m_0}{m_f}

with v_e the exhaust velocity. Since a logarithmic function grows slowly, doubling the propellant gives far less than double the velocity. This is why the delta-v of a mission is the primary constraint on its design, and why staging exists.

The Hohmann transfer

Between two coplanar circular orbits the cheapest two-impulse transfer is the Hohmann transfer. The first burn raises or lowers the apoapsis to match the target, the satellite coasts through half a revolution, and the second burn circularises.

With r_1, r_2 the two radii and a_t = (r_1 + r_2) / 2 the transfer semi-major axis, the two burns follow from the vis-viva equation,


\begin{aligned}
\Delta v_1 &= \sqrt{\mu \left( \frac{2}{r_1} - \frac{1}{a_t} \right)} - \sqrt{\frac{\mu}{r_1}} \\
\Delta v_2 &= \sqrt{\frac{\mu}{r_2}} - \sqrt{\mu \left( \frac{2}{r_2} - \frac{1}{a_t} \right)}
\end{aligned}

and the coast takes half the period of the transfer ellipse,


t_{\text{coast}} = \pi \sqrt{\frac{a_t^3}{\mu}}

A raise from 300 km to geostationary costs about 4.1 km/s, which is a large fraction of what a launch vehicle can provide, and most of it is the first burn.

Plane changes

A transfer that also changes inclination is more expensive than a coplanar one. The combined manoeuvre, raising the apoapsis and rotating the plane at the same time, costs more than doing either alone. The optimum for a given radius change and a given inclination change is a two-burn sequence in which the plane change is made at the smaller radius, where the velocity is lowest and therefore the same angular rotation is cheapest in delta-v.


\Delta v_{\text{plane}} \sim 2 V \sin\frac{\Delta i}{2}

with V the circular speed at the radius where the rotation is performed.

Finite burns

Over a finite burn the satellite is still moving against gravity and the thrust does not act along the instantaneous velocity. The achieved velocity change is less than the ideal a t, and the shortfall is the gravity loss,


L_{\text{gravity}} = \left\lvert \Delta v_{\text{ideal}} \right\rvert - \left\lvert \Delta v_{\text{achieved}} \right\rvert

which grows with burn duration. A chemical burn is short enough to ignore it. An electric thruster is not.

Unit conventions

Delta-v and velocity in kilometres per second.
Radii in kilometres.
The gravitational parameter \mu in cubic kilometres per second squared.
Inclinations in radians; degrees hide a factor of about 57 when used in a sine.

Applicability

The Hohmann transfer is optimal only for coplanar circular orbits, and only among two-impulse transfers. Real missions are rarely that: the orbits are elliptical, inclined, or both, and a bi-elliptic transfer can beat Hohmann when the ratio of the two radii exceeds about 11.94, at the cost of a much longer flight.

The impulsive model is adequate when the burn is a small fraction of the orbital period. It is not adequate for a spiral orbit raise by electric propulsion, where the "burn" is the whole mission.

A plane change is rarely worth it. Raising the inclination to reach a different ground track costs delta-v proportional to the inclination and often exceeds the entire launch budget, which is why planes are chosen at launch and then maintained rather than changed.

Standards

Hohmann transfer: derived from the vis-viva equation; see Vallado, Fundamentals of Astrodynamics and Applications
Rocket equation: Tsiolkovsky, 1903

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