135 lines
5.0 KiB
Markdown
135 lines
5.0 KiB
Markdown
---
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title: Orbit manoeuvres
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tags: maneuvers, impulse, finite-burn, hohmann, delta-v
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summary: 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.
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---
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# Orbit manoeuvres
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A manoeuvre changes an orbit. The satellite is at one point on a closed curve and
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must end up at another point on a different closed curve, and the change has to
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be paid for in propellant. This is the subject: how large the change is, how
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much it costs, and how the cost changes when the orbit is not circular.
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## Definition
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A **delta-v** is the magnitude of a change in velocity. A manoeuvre is planned
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as a sequence of delta-v, and the propellant cost follows from their sum by the
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rocket equation.
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An **impulse** applies the whole change in zero time. It is the right model for
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a burn that is short compared with the orbital period, which for a low Earth
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orbit means anything under a minute.
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A **finite burn** applies a thrust over a duration long enough that the
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satellite moves while thrusting. The distinction is not academic: a low Earth
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orbital transfer assisted by electric propulsion can last for hours and its
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gravity loss is a substantial fraction of the ideal cost.
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## Mathematics
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### The rocket equation
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The velocity a vehicle can produce depends on the exhaust velocity of its
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propellant and on the mass ratio,
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$$
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\Delta v = v_e \ln\frac{m_0}{m_f}
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$$
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with `v_e` the exhaust velocity. Since a logarithmic function grows slowly,
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doubling the propellant gives far less than double the velocity. This is why
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the delta-v of a mission is the primary constraint on its design, and why
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staging exists.
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### The Hohmann transfer
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Between two coplanar circular orbits the cheapest two-impulse transfer is the
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Hohmann transfer. The first burn raises or lowers the apoapsis to match the
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target, the satellite coasts through half a revolution, and the second burn
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circularises.
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With `r_1`, `r_2` the two radii and `a_t = (r_1 + r_2) / 2` the transfer
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semi-major axis, the two burns follow from the vis-viva equation,
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$$
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\begin{aligned}
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\Delta v_1 &= \sqrt{\mu \left( \frac{2}{r_1} - \frac{1}{a_t} \right)} - \sqrt{\frac{\mu}{r_1}} \\
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\Delta v_2 &= \sqrt{\frac{\mu}{r_2}} - \sqrt{\mu \left( \frac{2}{r_2} - \frac{1}{a_t} \right)}
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\end{aligned}
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$$
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and the coast takes half the period of the transfer ellipse,
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$$
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t_{\text{coast}} = \pi \sqrt{\frac{a_t^3}{\mu}}
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$$
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A raise from 300 km to geostationary costs about 4.1 km/s, which is a
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large fraction of what a launch vehicle can provide, and most of it is the
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first burn.
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### Plane changes
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A transfer that also changes inclination is more expensive than a coplanar
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one. The combined manoeuvre, raising the apoapsis and rotating the plane at
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the same time, costs more than doing either alone. The optimum for a given
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radius change and a given inclination change is a two-burn sequence in which
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the plane change is made at the smaller radius, where the velocity is lowest
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and therefore the same angular rotation is cheapest in delta-v.
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$$
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\Delta v_{\text{plane}} \sim 2 V \sin\frac{\Delta i}{2}
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$$
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with `V` the circular speed at the radius where the rotation is performed.
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### Finite burns
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Over a finite burn the satellite is still moving against gravity and the
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thrust does not act along the instantaneous velocity. The achieved velocity
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change is less than the ideal `a * t`, and the shortfall is the gravity loss,
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$$
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L_{\text{gravity}} = \left\lvert \Delta v_{\text{ideal}} \right\rvert - \left\lvert \Delta v_{\text{achieved}} \right\rvert
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$$
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which grows with burn duration. A chemical burn is short enough to ignore it.
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An electric thruster is not.
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## Unit conventions
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- Delta-v and velocity in kilometres per second.
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- Radii in kilometres.
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- The gravitational parameter `mu` in cubic kilometres per second squared.
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- Inclinations in radians; degrees hide a factor of about 57 when used in a
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sine.
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## Applicability
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The Hohmann transfer is optimal only for coplanar circular orbits, and only
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among two-impulse transfers. Real missions are rarely that: the orbits are
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elliptical, inclined, or both, and a bi-elliptic transfer can beat Hohmann
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when the ratio of the two radii exceeds about 11.94, at the cost of a much
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longer flight.
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The impulsive model is adequate when the burn is a small fraction of the
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orbital period. It is not adequate for a spiral orbit raise by electric
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propulsion, where the "burn" is the whole mission.
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A plane change is rarely worth it. Raising the inclination to reach a
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different ground track costs delta-v proportional to the inclination and
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often exceeds the entire launch budget, which is why planes are chosen at
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launch and then maintained rather than changed.
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## Standards
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- Hohmann transfer: derived from the vis-viva equation; see Vallado,
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*Fundamentals of Astrodynamics and Applications*
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- Rocket equation: Tsiolkovsky, 1903
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## See also
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- [[keplerian-elements]] for the orbit being changed
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- [[sgp4]] for propagating the result of a manoeuvre
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- [[perturbations]] for what the propellant estimate omits |