Course compass
Key idea
Ellipse, periapsis, apoapsis and eccentricity: reading a non-circular orbit. Circular orbits are useful for learning, but many real trajectories are elliptical. An ellipse has a nearest point and a farthest point. Those simple ideas unlock transfers, capture orbits and aerobraking. The rest of the course turns that idea into an auditable line of reasoning: explicit units, stated assumptions, reproducible calculations, order-of-magnitude checks and interpretation limits. A result is useful only when the reader can explain what it measures, where every input came from and which engineering decision it can support.
Starting synthesis: derivations, examples, limitations and sources are developed in the course body.
Key concepts before you begin
orbit · periapsis · apoapsis · eccentricity · unit
1 — Build a mental picture before using a formula
Circular orbits are useful for learning, but many real trajectories are elliptical. An ellipse has a nearest point and a farthest point. Those simple ideas unlock transfers, capture orbits and aerobraking.
2 — Essential vocabulary before going further
- ellipse — a closed elongated curve; a circle is a special case.
- periapsis — closest point to the central body.
- apoapsis — farthest point.
- semi-major axis — half the ellipse’s longest diameter.
- eccentricity — dimensionless measure of elongation.
3 — Understand the mechanism step by step
Names depend on the body
Around Earth, perigee and apogee are common; around the Sun, perihelion and aphelion. Periapsis and apoapsis are generic.
Speed changes
A spacecraft moves faster near periapsis and slower near apoapsis as kinetic and potential energy trade with one another.
Eccentricity
A circle has e = 0. Bound ellipses become more elongated as e approaches 1.
4 — The formula, only now
e = (rₐ − rₚ) / (rₐ + rₚ)How to read it: e is eccentricity; rₐ is apoapsis radius; rₚ is periapsis radius. Both are measured from the body's centre in this simplified model.
Detailed calculation
With rₚ = 7,000 km and rₐ = 14,000 km: difference = 7,000 km, sum = 21,000 km, so e = 7,000/21,000 ≈ 0.333.
5 — What the units tell you
6 — Three concrete demonstrations
Example 1 — Nearly circular
rp = 7,000 km and ra = 7,100 km give e ≈ 0.0071.
Example 2 — Transfer ellipse
rp = 7,000 km and ra = 14,000 km give e = 1/3 ≈ 0.333 and semi-major axis 10,500 km.
Example 3 — Mars capture
A spacecraft may first enter a highly elliptical Mars orbit, then lower apoapsis through burns or aerobraking.
7 — Why this matters for a Mars mission
Elliptical geometry is central to transfer orbits and capture trajectories.
8 — Common traps and misleading intuitions
- mixing altitude and radius.
- assuming periapsis always means dangerously low.
- assuming constant speed in an ellipse.
- confusing eccentricity with inclination.
9 — What I should be able to explain at the end
- explain the idea in ordinary words
- read and pronounce the important symbols
- repeat at least one calculation without hidden steps
- identify what the simplified model assumes and does not prove
Reading an ellipse as a map of orbital energy
An ellipse is more than a visually stretched circle. The central body sits at one focus, so distance changes continuously. Periapsis is the closest point and apoapsis the farthest. Semi-major axis sets orbital energy in the two-body model, while eccentricity describes the shape.
The vis-viva relation v² = μ(2/r − 1/a) links position and speed directly. At fixed a, speed rises as r decreases toward periapsis and falls toward apoapsis. This is why the same impulse has different consequences depending on where it is applied.
Operationally, periapsis and apoapsis can become safety, heating, eclipse or communications constraints. Around Mars, a small periapsis error can strongly change atmospheric interaction, so these are navigation quantities, not merely vocabulary.