Chapter 30 — Key Takeaways (Launch Vehicles)

A one-page reference. Reread this before an exam, or before you choose a rocket for your mission.

The equations of this chapter

Equation Use it to find Symbols & units
$v_{\text{rot}} = 0.465\cos\phi\ \text{km/s}$ eastward rotation credit for a due-east launch $\phi$ = launch-site latitude; $0.465$ km/s = equatorial surface speed
$\cos i = \cos\phi\,\sin\beta$ orbital inclination from latitude and azimuth $i$ = inclination, $\beta$ = launch azimuth (CW from north)
$i_{\min} = \phi$ minimum inclination reachable directly (set $\beta = 90^\circ$) you cannot reach $i < \phi$ without a plane change
$\Delta v = \sqrt{v_a^2 + v_{\text{GEO}}^2 - 2 v_a v_{\text{GEO}}\cos i}$ combined circularize + plane-change at GTO apogee $v_a \approx 1.60$, $v_{\text{GEO}} \approx 3.07$ km/s
$m_p = m_f\,(e^{\Delta v/v_e} - 1)$ propellant to deliver a delta-v (rocket equation, Ch. 3) converts a delta-v penalty into kilograms

Launch-vehicle families at a glance

Axis Categories Examples
Payload class small / medium / heavy / super-heavy Electron / Falcon 9, Vulcan, Ariane 6 / Falcon Heavy, New Glenn / SLS, Starship
Propellant kerolox / hydrolox / methalox / solid Falcon 9, Soyuz / Ariane 6, SLS / Starship, Vulcan, New Glenn / SRBs
Reuse mode expendable / partial / full most of the world / Falcon 9, New Glenn / Starship (goal)

Performance rules of thumb

Fact Value
Payload to GTO vs LEO GTO ≈ ⅓ to ½ of LEO (vehicle supplies ~2.4 km/s more delta-v)
Falcon 9 to LEO ~22.8 t expendable / ~17 t recovered (Tier 2)
Reuse penalty landing propellant = payload not lifted (~25% of LEO payload for F9)
Cost/kg to LEO ~$10–20k expendable → ~$2–3k Falcon 9 → aspirationally <$1k Starship (Tier 3)

Cost per kg is the softest number in rocketry — always ask: to what orbit, how full, price or cost?

Launch-site latitude — the two effects

  • Rotation credit (helps): free eastward velocity $0.465\cos\phi$ km/s — ~0.46 at equator, ~0.41 Cape, ~0.32 Baikonur; zero for polar/southward launches (Vandenberg).
  • Minimum inclination (constrains): $i_{\min} = \phi$. Equatorial sites reach GEO ($i=0$) cheaply; high-latitude sites pay a large plane change.
  • GEO penalty audited: GTO→GEO combined burn ≈ 1.48 km/s (Kourou 5.2°), 1.83 km/s (Cape 28.5°), 2.27 km/s (Baikonur 45.9°) → for a 3,500 kg on-station comsat, ~1.6 t more mass to GTO from Baikonur than Kourou.

Two knobs that set a launch

Knob Sets Constraint
Azimuth $\beta$ orbital inclination (tilt of the plane) $\cos i = \cos\phi\sin\beta$; can't go below $\phi$
Time of day orbital node $\Omega$ (orientation of the plane) the rotating site swings into the plane ≤ 2×/day

Launch windows

  • Window = time the (inertially fixed) target plane lines up over the rotating site, ≤ twice/day.
  • Instantaneous window when you must also match phase to a target already in orbit (ISS) → miss it, recycle ~24 h.
  • Also gated by weather (winds at max-Q, lightning), range safety/availability, COLA (collision avoidance), lighting/thermal, and $J_2$ node drift (Ch. 12).
  • Interplanetary window (weeks, once per synodic period) is a different geometry — planetary alignment, Ch. 11.

Countdown & GO/NO-GO

  • Countdown: rehearsed T-minus script → propellant load → checkouts → terminal count (computer-run) → ignition → T-0 liftoff. Built-in holds are planned pauses; a scrub is unplanned.
  • Launch-commit criteria (LCC): quantitative red lines fixed before launch day.
  • GO/NO-GO poll: launch director polls every station; any single NO-GO stops the count (unanimity).
  • Solids are lit last (can't be shut down); liquids first (can be aborted after ignition).

Choosing a launch vehicle — the decision

Hard gates (pass/fail): ① payload to the target orbit/$C_3$ with ~10% margin · ② fairing volume fits · ③ inclination reachable from the site. Soft preferences (rank survivors): price · schedule/availability · reliability & heritage · loads/interfaces.

Decision aid Rule
Which performance number? to your destination ($C_3$ for escape), not the LEO headline
Mass margin capacity ÷ wet mass ≥ 1.1 (Ch. 29 margin)
Volume-limited? low-density payloads bind on fairing before mass
Reuse's effect shifts weights toward price & cadence, away from minimum-mass design (theme 5)

Common pitfalls

Pitfall Reality
"GTO payload ≈ LEO payload." GTO is ~⅓–½ of LEO; the extra delta-v eats payload.
"Min inclination can be below latitude." $i_{\min} = \phi$; below it needs a plane change.
"Cost/kg is a fixed rocket property." It depends on orbit, fill, and price-vs-cost.
"Reuse is free payload." Landing propellant lowers payload (~22.8→17 t for F9).
"Hold the window open till weather clears." Rendezvous windows are ~instantaneous; miss it, recycle ~a day.

Numbers worth memorizing

  • Equatorial rotation speed 0.465 km/s; credit $= 0.465\cos\phi$.
  • Minimum inclination = launch latitude.
  • GTO payload ≈ ⅓ of LEO payload.
  • GEO plane-change penalty, Cape vs Kourou ≈ 0.35 km/s (~hundreds of kg of satellite propellant).

Mission / astrotools additions this chapter

  • MDR: launch-vehicle selection — run the three hard gates, rank by preference, record vehicle + binding factor + mass margin.
  • astrotools/launch.py: select_launcher(payload_kg, destination, margin) — screens the Appendix-H catalog; feeds mission.py (Ch. 29/40).