Intuition The one core ideaAn orbit is a speed–distance bargain : the farther you are from the planet, the slower you must go, and this trade is fixed once you know the orbit's size . A Hohmann transfer just links two circular bargains with a single stretched loop, paying the smallest possible "speed change" toll at each end.
Everything in the parent note rests on a handful of symbols. Below, each one is built from nothing: plain words → the picture → why the topic needs it. Read top to bottom; nothing is used before it is born.
M M
G G G ≈ 6.674 × 1 0 − 11 m 3 kg − 1 s − 2 mass into pull .
Why bundle them? In every orbit formula G M GM
μ μ = GM this particular central body bends paths." For Earth, μ ⊕ = 3.986 × 1 0 5 km 3 / s 2
Why the topic needs it: every speed and every burn in the derivation is a formula with μ
r r centre of the planet to the spacecraft — not from the ground, from the centre . Picture a string tied to the planet's core; r
Why "from the centre"? Gravity acts as if all of M ≈ 6371 ∼ 629
Common mistake Confusing altitude withr Feels right: "I'm 600 km up, so r = 600 Fix: r = R planet + altitude
r 1 r 2 r 1 r 2 r 1 = 7000 r 2 = 42164
Now the shape that connects them:
A "stretched circle" — an oval. Where a circle has one centre-distance, an ellipse has a closest point and a farthest point from the planet.
Definition Periapsis and apoapsis
Periapsis = the closest point of the ellipse to the planet (here it sits at r 1 Apoapsis = the farthest point (here it sits at r 2
Look at the figure: the transfer ellipse kisses the inner circle at periapsis and the outer circle at apoapsis. That double-kiss is the whole Hohmann idea.
a a half the longest diameter of an ellipse — the distance from the centre of the oval out to its most-stretched edge. For a circle (a special, un-stretched ellipse) a = r
The single most useful fact about a the closest and farthest distances add up to the full long diameter, which is 2 a r peri + r apo = 2 a
a a same total energy , even if one is a fat oval and one is a thin sliver. So a a t = 2 r 1 + r 2 r 1 r 2
v v along its path , measured in km/s. It is a magnitude here — just "how fast," direction handled separately.
The parent uses many flavoured v
Symbol
Plain words
v c 1 speed on the inner circle
v c 2 speed on the outer circle
v p speed at periapsis of the transfer ellipse
v a speed at apoapsis of the transfer ellipse
Why A square root is the tool that undoes squaring. Orbital energy naturally gives us v 2 v 2 v
This is the Vis-viva equation , the one law that ties v r a
Intuition Read it like a sentence
r 2 r bigger this term, so the faster you go — like a comet whipping around the Sun.− a 1 a a r
Put together: near = fast, far = slow, and the size a Every burn in the derivation is just this equation evaluated at two different r
Why we need this and not just v circ On the ellipse , r v circ r = a v any point of any orbit — see Orbital energy & semi-major axis .
Δ v Δ Δ v Δ v = v after − v before Δ v
Why it is the currency: rocket fuel maps almost directly onto Δ v Delta-v budget ). Less Δ v minimum total Δ v .
Intuition Why we can just subtract (not vector-add)Because Hohmann burns are done tangent — the new velocity points the same direction as the old one. Two arrows on the same line: the change is simply the difference of their lengths. That is the payoff of choosing tangent burns.
Recall The full symbol table (cover the right side)
M G μ = GM r centre to the craftr 1 , r 2 a a t = ( r 1 + r 2 ) /2 v v circ = μ / r r v p , v a Δ v ε = 2 1 v 2 − μ / r π
v_circ on the two circles
Test yourself — say the answer aloud, then reveal:
What does μ G M μ = GM
Is r From the centre of the planet: r = R planet + altitude
What is a Half the ellipse's longest diameter — the orbit's "size" and energy fingerprint.
Why is a t = ( r 1 + r 2 ) /2 Periapsis at r 1 r 2 2 a t
State vis-viva and what each term means. v 2 = μ ( 2/ r − 1/ a ) 2/ r − 1/ a
What is v circ μ / r , the speed to hold a circle — only when
r = a .
What does Δ v The size of a speed jump from a burn; it maps to fuel, so least Δ v
Why can Hohmann burns be added as plain numbers? Burns are tangent, so old and new velocities are collinear — subtract magnitudes, no vectors needed.