Course compass
Ellipse, periapsis, apoapsis and eccentricity: reading a non-circular orbit. The lesson starts from a concrete situation, defines every term and symbol, then introduces formulas and mission use.
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
None of these words should remain mysterious. Read them once now, then return to them as the lesson progresses.
- 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
A physical equation is more than numbers. Units identify the kind of result and provide a consistency check. At every division, multiplication or square root, track what happens to the units; this catches many errors before checking the numerical value.
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.
In a real mission, operational value comes from the chain: measure, estimate, calculate, check margins, execute, then measure again. A formula by itself does not fly a spacecraft.
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
10 — Guided exercises and answers
- Restate: explain the lesson's main term aloud without a formula; define any technical word immediately.
- Units: repeat the main calculation and verify the final units represent the quantity being sought.
- Variation: change one input by 10%, predict the direction of the effect before recalculating, then check your intuition.
- Model limit: name two real effects the teaching model does not fully include.
11 — NASA / JPL sources for further study
These are primary institutional sources used to check concepts and orders of magnitude. They are more technical than this introductory lesson.