Course compass
Key idea
Hohmann transfer: moving between circular orbits with two burns. 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. 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 · apoapsis · Hohmann · unit · assumption
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
- 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
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.
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
What a Hohmann transfer teaches—and simplifies
A Hohmann transfer links two circular coplanar orbits through an ellipse tangent to both. The first impulse enters the transfer ellipse; the second circularizes at the destination. It is optimal only under a particular ideal model and does not mean every real mission uses exactly two instantaneous burns.
The calculation separates three different speeds: circular departure speed, transfer-ellipse speed after the first burn, and target circular speed. The burn costs are differences between the appropriate local speeds, not simply the difference between two circular speeds.
For Earth–Mars work the Hohmann model is mainly an order-of-magnitude tool. Real planets are not perfect coplanar circles, and mission design uses ephemerides, launch dates, arrival geometry and launcher constraints.