Orbital Maneuvers — Hohmann Transfers, Vis-Viva and Delta-V Budgets

Moving between orbits costs velocity, not fuel directly

Every orbital manoeuvre — raising a satellite, sending a probe to Mars, docking with a station — reduces to the same accounting question: how much Δv (delta-v, a change in velocity) does the manoeuvre require? Rocket fuel only matters insofar as it buys Δv through the Tsiolkovsky rocket equation; the mission-design problem itself lives entirely in the language of velocities and orbits.

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The vis-viva equation: speed as a function of position and orbit shape

For any orbit under a single dominant gravitational body, the vis-viva equation relates a spacecraft's speed at any point to its distance from the central body and the orbit's overall size (semi-major axis a):

v² = μ(2/r − 1/a)

μ = GM, standard gravitational parameter of the central body
r   = current distance from the central body
a   = semi-major axis (a < 0 for hyperbolic escape trajectories)

This one equation is the workhorse of orbital mechanics: set a = r for a circular orbit and it reduces to v = √(μ/r); set a → ∞ and it gives escape velocity v = √(2μ/r). Every transfer manoeuvre is computed by finding the speed an object has on its current orbit and the speed it needs on the target orbit at the same point, and the difference is the Δv that engine burn must supply.

The Hohmann transfer: the cheapest way between two circular orbits

To move between two coplanar circular orbits with the least possible fuel, Walter Hohmann showed in 1925 that the optimal path is an elliptical transfer orbit tangent to the inner circle at its perigee and to the outer circle at its apogee — the ellipse just touches both circles, so the entire manoeuvre needs only two burns, each applied at exactly the point where the transfer orbit is tangent to the current orbit (so the burn only needs to change speed, not direction).

r1 = initial circular orbit radius, r2 = final circular orbit radius
a_t = (r1 + r2) / 2                     transfer ellipse semi-major axis

burn 1 (at r1): Δv1 = √(μ/r1)·[√(2r2/(r1+r2)) − 1]     enter the transfer ellipse
burn 2 (at r2): Δv2 = √(μ/r2)·[1 − √(2r1/(r1+r2))]     circularize at r2

total Δv = Δv1 + Δv2

Raising a satellite's orbit therefore means speeding up at perigee to stretch the orbit outward, coasting halfway around the new ellipse, then speeding up again at apogee to circularize; lowering an orbit reverses both burns' signs. The Hohmann transfer is provably the minimum-Δv two-impulse transfer between circular orbits when the ratio of radii is below about 11.94 — beyond that, a three-burn bi-elliptic transfer can actually use less total Δv, at the cost of taking far longer.

Timing it: the phase angle and the launch window

Getting the transfer orbit right is only half the problem — for a rendezvous (say, a probe meeting Mars) the target body has to actually be at the right place when the spacecraft's transfer ellipse arrives there, months or years later. This requires computing the correct phase angle, the angular lead the target body must have at departure so that both spacecraft and target reach the meeting point simultaneously, which is why interplanetary missions launch only during narrow launch windows that recur on a period set by the two bodies' relative orbital motion (the synodic period), not just whenever a rocket happens to be ready.

Plane changes are expensive, and why you avoid them

Changing an orbit's inclination — the tilt of its plane — costs disproportionately more Δv than changing its size, because the required velocity change scales with the sine of half the plane-change angle at a fixed speed: Δv_plane = 2v·sin(Δi/2). This is why mission designers try hard to combine a plane change with another burn (doing it at apogee, where orbital speed v is lowest, minimises the cost) rather than performing it as a separate manoeuvre, and why launch sites near the equator are preferred for equatorial-orbit missions — matching the target inclination at launch avoids paying for an expensive plane change in orbit altogether.

Frequently asked questions

Why does a Hohmann transfer need exactly two burns?

Because the transfer ellipse is tangent to both the starting and target circular orbits, the velocity vectors are aligned (same direction, different magnitude) at each tangent point — so each burn only needs to change speed, not direction, which makes two burns both necessary and sufficient, and the most fuel-efficient option in that regime.

Is the Hohmann transfer always the cheapest way to change orbits?

For circular, coplanar orbits with a moderate radius ratio, yes — it is provably the minimum two-impulse transfer. But if the ratio between the two orbit radii exceeds roughly 11.94, a three-burn bi-elliptic transfer can use less total delta-v, though it takes considerably longer to execute.

Why do plane changes cost so much more delta-v than raising or lowering an orbit?

The delta-v needed for a plane change scales with the sine of half the angle and with the current orbital speed, so it grows quickly with angle and is cheapest performed where speed is lowest, typically at apogee. This is why mission planners avoid standalone plane-change burns whenever possible and try to combine them with other manoeuvres.

Try it live

Everything above runs in your browser — open Orbital Maneuvers and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

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Orbital Maneuvers: Getting From One Orbit to Another for the Least Fuel