Intuition The ONE core ideaA spacecraft arriving at a planet carries too much energy to be trapped, so it would fly straight past — unless something steals that extra energy. Aerocapture lets the planet's own air do the stealing in a single dip, turning a "fly-by-forever" path into a "loop-around-forever" orbit.
Everything below is a tool you will need to read the parent note Aerocapture . We build each symbol from nothing — no prior notation assumed. Read top to bottom; each piece is used by the next.
Definition One consistent unit system (read this first)Physics mixes units, so we fix one rule and stick to it : for the orbit maths we work in kilometres and seconds (so r a v v ∞ μ km 3 / s 2 ε km 2 / s 2 drag maths, atmospheric data is quoted in metres and kilograms (density ρ kg/m 3 A m 2 m convert one so both match (1 km = 1000 m
Before any formula, picture the scene. A round planet sits still. A tiny spacecraft moves near it. The one quantity we measure again and again is how far the craft is from the planet's centre .
r r centre of the planet to the spacecraft, measured in kilometres (km).
Picture: the length of the yellow line in the figure above, from the dot at the planet's core to the little craft.Why we need it: gravity pulls harder when you are close and weaker when you are far. Every gravity formula must know "how close?" — that is exactly r Careful: r centre , not the surface . Altitude h
R R surface (km) — how big the ball is.
Picture: the radius of the blue disk in the figure above, from the white centre-dot to the edge.Why we need it: it links "distance from centre" r h r = R h r = R + h h = r − R
v v
Picture: the length of the blue arrow in the figure — a longer arrow means faster.Why we need it: a fast craft has lots of motion-energy; slowing it down is the whole game of aerocapture.
You have a coin. It has money-value. A moving, high-up spacecraft has energy-value made of two parts: energy from moving and energy from being high up in gravity . Aerocapture is entirely a story about this energy shrinking.
Definition Kinetic energy per unit mass,2 1 v 2 each kilogram . For motion this is 2 1 v 2
Picture: think of the blue speed-arrow squared — double the speed and this energy quadruples (the v 2 Why per-mass: the craft's actual mass cancels out of every orbit question, so physicists drop it and track "per kilogram" quantities. It keeps the maths clean.
v 2 v and a longer distance add energy, and both grow with speed — so energy grows like speed times speed, i.e. v 2
μ μ = GM G M km 3 / s 2 μ ≈ 4.28 × 1 0 4 km 3 / s 2
Picture: a "gravity dial" — a heavier planet has a bigger μ Why bundle them: in orbit problems G M
Common mistake "Slower always means captured."Not quite — capture depends on the combination v 2 /2 − μ / r v r − μ / r sign of ε , never the speed by itself.
Definition Hyperbola vs. ellipse
Hyperbola: an open path that swings once past the planet and flies off forever. This is the "escape / not captured" shape, tied to ε > 0 Hyperbolic Trajectories & Hyperbolic Excess Velocity .Ellipse: a closed oval loop that returns again and again — a bound orbit . Tied to ε < 0 Picture: the figure below — red hyperbola bends and leaves; green ellipse closes into a loop.
a a half the length of the long axis of the oval — a single number saying "how big is the orbit."
Picture: half the width of the green loop in the figure above.Why we need it: aerocapture aims for a specific target orbit, i.e. a specific a ε o u t
Now we need the tool that removes energy: drag .
ρ ρ kg/m 3 ρ ρ
Picture: dots-per-box in the figure below — dense near the ground, sparse up high.
Intuition Why exponential? Whye fraction of the pile and the pressure drops by that same fraction — a "shrink by a constant proportion per step" rule. Anything that shrinks by a fixed proportion per equal step is an exponential , and e ρ = ρ 0 e − h / H
m C D A F D = 2 1 ρ v 2 C D A
m A m 2 C D C D C D The force: F D = 2 1 ρ v 2 C D A ρ squared (v 2 C D A Ballistic Coefficient .
Δ Δ Δ v = v in − v o u t Δ ε
v e v e v e v e ≈ 3
Picture: the speed of the flame-jet blasting rearward while the craft is pushed forward.
Δ v a er o Δ v a er o = v in − v o u t for free . A rocket normally buys speed change with fuel via the rocket equation : with exhaust speed v e Δ m / m = 1 − e − Δ v / v e
sign of epsilon decides capture
ballistic coefficient beta
Cover the right side and answer aloud; reveal to check.
What is r The straight-line distance from the planet's centre to the craft (km), not from the surface.
How do r R h h = r − R
What are the two parts of ε Motion-energy 2 1 v 2 − μ / r
Which sign of ε ε < 0 ε > 0
What does μ μ = GM G M
Why is ε = − μ / ( 2 a ) Because ε a − μ / ( 2 a )
Where does a D = ρ v 2 / ( 2 β ) From a D = F D / m β = m / ( C D A )
Why does drag depend on v 2 You sweep up more air and give each bit more momentum as you go faster — two factors of v
What does β Heaviness-per-face m / ( C D A ) β β
What is v e The exhaust speed — how fast a rocket throws its propellant out; bigger v e
Why is the atmosphere exponential in altitude? Density shrinks by a fixed proportion every scale height H ρ 0 e − h / H
What is Δ v a er o v in − v o u t , the braking air gives free — fuel saved by the
rocket equation .
