Chapter 22 — Key Takeaways (Staging, Propellant Management, and Reusability)

A one-page reference. Reread this before an exam, before you stage a mission design, or before you argue about whether reuse pays.

The core equations

Equation Use it to find Notes
$\Delta v_{\max} = v_e \ln(1/\varepsilon)$ a single stage's best-possible delta-v zero-payload ceiling; below orbit for real $\varepsilon$
$\Delta v_{\text{total}} = \sum_i v_{e,i}\ln R_i$ total delta-v of a staged rocket each stage's masses include everything above it
$R = \dfrac{1}{\varepsilon + \pi(1-\varepsilon)}$ mass ratio from payload ratio $\pi$ invert: $\pi = \dfrac{1-\varepsilon R}{R(1-\varepsilon)}$
$m_{\text{stage}} = m_{\text{above}}\dfrac{R-1}{1-\varepsilon R}$ a stage's loaded mass (size top-down) requires $R < 1/\varepsilon$ or it's impossible
$R_i = \dfrac{v_{e,i}-\mu}{v_{e,i}\,\varepsilon_i}$ optimal mass ratio, unequal stages one $\mu$ (velocity) solved numerically from the total-$\Delta v$ constraint
$\dot m = Q/h_{fg}$ cryogenic boiloff rate $Q$ = heat leak (W), $h_{fg}$ = latent heat (J/kg)
$C_{\text{reuse}}(N) = M/N + R + F$ cost per flight of a reused stage floor $R+F$ as $N\to\infty$

Symbols and units

Symbol Meaning Units
$\varepsilon$ structural coefficient $= m_s/(m_s+m_p)$ (Ch. 3) — (0.05–0.10 good)
$R$ stage mass ratio $m_0/m_f$ — ($< 1/\varepsilon$)
$\pi$ payload ratio $m_L/m_0$ (load above / gross) —
$v_e$ effective exhaust velocity m/s
$\mu$ optimal-staging Lagrange constant (a velocity) m/s
$Q,\ h_{fg}$ tank heat leak; latent heat of vaporization W; J/kg
$M,R,F,N$ build cost; refurb cost; fixed/expended cost; flights \$; \$; \$; —

Disambiguation: here $\varepsilon$ is the structural coefficient, not the nozzle expansion ratio (Ch. 19) or the specific orbital energy (Ch. 6).

Which idea / when — a decision aid

You want to… Use
check if a single stage can do the job compare $v_e\ln(1/\varepsilon)$ to the required $\Delta v$
decide how many stages payload fraction vs. $N$ table — two or three (diminishing returns)
split delta-v between identical stages equally ($\Delta v_{\text{total}}/N$ each)
split between unequal stages higher $v_e$ / lower $\varepsilon$ stage does more; solve $R_i=(v_{e,i}-\mu)/(v_{e,i}\varepsilon_i)$
size the vehicle top-down: $m_{\text{stage}}=m_{\text{above}}(R-1)/(1-\varepsilon R)$
judge reuse break-even $N$ on cost per kg (reuse carries less payload)

The four propellant-management demons

Demon What it is Cure
Ullage propellant floats off the outlet in coast/microgravity ullage burn (settle before restart)
Slosh lateral liquid sloshing shifts CoM, disturbs control anti-slosh baffles
POGO longitudinal structure↔feedline↔thrust feedback loop gas-charged feed-line accumulators
Boiloff heat vaporizes cryogen; vented mass lost insulation (MLI), shading, cryocoolers

Propulsive landing — the sequence and the trap

  • Sequence: stage separation → boostback burn (RTLS) → entry burn (slow before dense air) → grid-fin aerodynamic descent → landing burn → legs (or tower catch).
  • The hoverslam: minimum single-engine thrust exceeds the empty booster's weight, so $T/W_{\min}>1$ and it cannot hover. It must time one burn so velocity hits zero exactly at the ground.
  • Requires (from Ch. 17): a restartable, deeply throttleable engine — which a solid motor is not.

Common pitfalls

Pitfall Reality
"Scale the rocket up to reach orbit in one stage." The ceiling depends on $\varepsilon$, not size; scaling changes nothing.
"The biggest stage should do the most delta-v." The lowest-$v_e$ stage should do the least; first-stage size is a thrust decision, not an efficiency one.
"More engines = more reliable." More engines = more failure chances; capability needs isolation + fast detection + adaptive guidance (N1 vs. Falcon 9).
"Reuse cuts cost to propellant." Floor is $R+F$ (expended upper stage + ops), far above propellant cost.
"Reuse is cheaper from flight one." Flight one costs more (recovery hardware + payload penalty); it pays off with cadence.

Numbers worth memorizing

  • Good structural coefficient $\varepsilon \approx 0.05$–$0.10$; single-stage kerosene ceiling $\approx 8.6\ \text{km/s}$ (zero payload) — below orbit, so staging is mandatory.
  • Two-stage LEO payload fraction $\approx 3$–$4\%$; three-stage a bit more, then diminishing returns.
  • Falcon 9 lift-off $T/W \approx 1.4$; with one of nine engines out, $\approx 1.26$ (still climbs).
  • Reuse payload penalty $\approx 15$–$30\%$ (droneship); break-even $\approx 2$ flights on cost/kg.

Mission / astrotools additions this chapter

  • MDR: turned the delta-v budget into a staging plan (number of stages, split, top-down masses, lift-off mass) and added a reuse decision keyed to flight cadence.
  • rocket.py: payload_fraction(dv_total, ve, eps, n) and stage_mass(m_above, dv, ve, eps), both building on Chapter 3's stage_delta_v.