Course compass
Hohmann transfer: moving between circular orbits with two burns. 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
A Hohmann transfer uses an ellipse tangent to two coplanar circular orbits and two burns. It is not always the optimum real-world solution, but it is one of the clearest models for learning orbital transfers.
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.
- initial orbit — starting circular orbit.
- final orbit — target circular orbit.
- transfer ellipse — ellipse tangent to both circular orbits.
- first burn — maneuver that enters the transfer ellipse.
- circularization — second burn matching the final circular speed.
3 — Understand the mechanism step by step
Step 1 — raise apoapsis
A prograde burn from the lower orbit raises the far side of the new ellipse.
Step 2 — coast
The engine is off in the impulsive model while gravity carries the spacecraft along the ellipse.
Step 3 — circularize
At apoapsis, a second prograde burn matches the target circular speed.
4 — The formula, only now
Δv₁ = √(μ/r₁) [√(2r₂/(r₁+r₂)) − 1]How to read it: r₁ is starting radius, r₂ target radius, μ gravitational parameter; brackets group operations.
Detailed calculation
For r₁=7,000 km, r₂=14,000 km and Earth μ≈398,600 km³/s², initial circular speed is ≈7.546 km/s and the bracket factor ≈0.1547, giving Δv₁≈1.17 km/s.
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 — 7,000 to 14,000 km Earth radii
The ideal two-body burns are about 1.17 km/s and 0.98 km/s, total roughly 2.15 km/s.
Example 2 — Missing the second burn
Without circularization the spacecraft simply falls back along the transfer ellipse.
Example 3 — Going down
Reverse the logic using retrograde burns.
7 — Why this matters for a Mars mission
The model provides the conceptual backbone for later Earth–Mars transfer calculations.
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
- assuming continuous thrust.
- forgetting circularization.
- ignoring plane changes.
- treating real interplanetary missions as exact Hohmann copies.
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.