Foundations — Bi-elliptic transfer — when it wins over Hohmann

This page assumes nothing. Before you read Bi-elliptic transfer — when it wins over Hohmann, every letter and squiggle used there is built here from the ground up, in an order where each idea leans only on the ones before it.

1. The central body, and the letters $G$, $M$, $\mu$

Picture a planet (or the Sun) sitting still, and a tiny spacecraft looping around it. The planet is huge; the spacecraft is a speck. All the gravity comes from the planet.

• $M$ = the mass of the central body (the planet or star). Bigger $M$ → stronger pull.
• $G$ = the gravitational constant, a fixed number of nature that sets how strong gravity is everywhere.
• $\mu$ (Greek letter "mu") = the shorthand $\mu =GM$, called the standard gravitational parameter.

The picture: the planet at the centre, the spacecraft on a ring around it, and an arrow pointing inward labelled "gravity, strength set by $\mu$".

2. Radius $r$ — how far out you are

$r$ = the distance from the centre of the planet to the spacecraft. Not from the surface — from the centre.

We use $r_1$ for the small starting orbit's radius, $r_2$ for the big target orbit's radius, and later $r_b$ for a chosen far-away point of the detour.

3. A circular orbit and its speed $v_c$

A circular orbit is a spacecraft going round at a constant distance $r$ — a perfect ring. At every point it moves at the same speed. We call that speed $v_c(r)$, the ==circular speed at radius $r$==.

$v$ (just "vee") = the speed of the spacecraft: how many metres it covers per second, ignoring direction. We will always take speeds as positive numbers.

The picture shows two circular orbits: a small fast one (long velocity arrow) and a big slow one (short velocity arrow).

4. Ellipses, apsides, and the semi-major axis $a$

A circle is one special shape; the more general orbit is an ellipse — a squashed circle, like an oval racetrack. A transfer orbit is always a piece of an ellipse.

• Periapsis = the closest point of the ellipse to the planet. There you are moving fastest.
• Apoapsis = the farthest point of the ellipse. There you are moving slowest.
• Together these two are the apsides (plural of apsis).

$a$ = the semi-major axis: half of the longest diameter of the ellipse. It is the single number that says "how big" the ellipse is.

The picture: an ellipse with the planet at one focus, periapsis and apoapsis marked, and the semi-major axis $a$ drawn as the half-length of the long axis.

5. The vis-viva equation — the master speed formula

Now the single most important tool. See Vis-viva equation and Semi-major axis and orbital energy for the full story; here is what you need.

6. The burn and $\Delta v$

A rocket burn fires the engine to change speed. Because you are at some point on an orbit, changing your speed there instantly changes which orbit you are on (new $a$).

$\Delta$ (Greek "delta") means "the change in." So $\Delta v$ = the change in speed the burn produces.

The total cost of a whole maneuver is the sum of every burn's $|\Delta v|$. Hohmann adds two; bi-elliptic adds three. That sum is exactly what the parent note compares.

7. The radius ratio $R$

$R=r_2/r_1$ = the ratio of the outer radius to the inner radius: "how many times bigger is the target orbit than the start?"

8. The bi-parabolic limit ($r_b\to \infty$)

The detour point $r_b$ can be pushed farther and farther out. As $r_b\to \infty$ ("goes to infinity," i.e. grows without bound), the far ellipses stop being ellipses and become parabolas — barely-bound escape paths. This ideal is the Bi-parabolic transfer, the theoretical best case bi-elliptic can approach but never reach (it would take infinite time). The famous thresholds are computed in this limit.

How these feed the topic

Related tools you will meet later: the Oberth effect (why deep burns are efficient), Plane change maneuvers (reshaping direction), and Transfer time vs delta-v tradeoffs (the hidden time cost).

Equipment checklist

Test yourself — each line hides the answer.