Case Study 1: Was the Saturn V Optimally Staged?

"The Saturn V is still the only launch vehicle to have carried humans beyond low Earth orbit — and it did it by tearing itself into three pieces on the way."

Executive Summary

The Saturn V, the rocket that sent Apollo to the Moon, was a three-stage serial vehicle burning two very different propellant combinations: a kerosene first stage and two hydrogen upper stages. In this case study we reconstruct each stage's delta-v from published masses, compute each stage's structural coefficient, and then ask the question this chapter equips us to answer: was the delta-v split among the three stages optimal? We will find that the vehicle's split — first stage least, upper stages most — is exactly the direction optimal-staging theory predicts for mixed propellants, but that the first stage's size is governed by a different imperative entirely (lift-off thrust), a tension that teaches more than either fact alone.

Skills applied: - Reconstructing stage delta-v with the "everything above" bookkeeping (§22.1; Chapter 3). - Computing and interpreting structural coefficients $\varepsilon = m_s/(m_s+m_p)$ (§22.1). - Applying the optimal-split rule for unequal stages (§22.2). - Distinguishing an efficiency decision (the split) from a thrust decision (first-stage size) (§22.2, §22.4).

All masses and specific impulses are approximate, widely published figures (Tier 2), rounded for a clean reconstruction; exact values vary by source and by mission.

Background

A Saturn V stood $111\ \text{m}$ tall and massed about $2{,}950\ \text{t}$ fully fueled, of which roughly $45\ \text{t}$ was the useful translunar payload (the Apollo command/service module, the lunar module, and their adapter). It reached that payload's translunar trajectory in three acts:

• S-IC (first stage): five F-1 engines burning LOX/RP-1 (kerosene), the most powerful the vehicle carried, lifting the whole stack off the pad and through the lower atmosphere.
• S-II (second stage): five J-2 engines burning LOX/LH2 (hydrogen), pushing toward orbital speed.
• S-IVB (third stage): a single J-2, which finished the climb to low Earth orbit and later reignited to send Apollo toward the Moon.

The design question we can now interrogate: with a fixed total delta-v to produce, was the work divided among these three stages the way optimal-staging theory would have divided it?

Phase 1: The vehicle in numbers

We collect the approximate masses and exhaust velocities. For the F-1 we use an averaged sea-level-to-vacuum $I_{sp} \approx 290\ \text{s}$ (its performance rises as it climbs); for the J-2 a vacuum $I_{sp} \approx 421\ \text{s}$. Exhaust velocity is $v_e = I_{sp}\,g_0$.

Translunar payload: $m_L \approx 45\ \text{t}$.

Phase 2: Reconstruct each stage's delta-v

We use the "everything above" rule: each stage must accelerate itself plus all the stages and payload stacked on top of it. Working top-down for the "above" masses: the S-IVB carries only the payload; the S-II carries the payload plus the (full) S-IVB; the S-IC carries the payload plus the full S-II and S-IVB.

import math

# (ve [m/s], m0 [kg], mf [kg]) with all mass above each stage included
stages = [
    (2845, 2_954_000, 794_000),   # S-IC:  above = payload + S-II + S-IVB
    (4130,   664_000, 208_000),   # S-II:  above = payload + S-IVB
    (4130,   168_000,  58_500),   # S-IVB: above = payload
]
for ve, m0, mf in stages:
    print(f"R = {m0/mf:.3f}   dv = {ve*math.log(m0/mf):.0f} m/s")
print("total =", round(sum(ve*math.log(m0/mf) for ve, m0, mf in stages)), "m/s")
# Expected output:
# R = 3.720   dv = 3738 m/s
# R = 3.192   dv = 4794 m/s
# R = 2.872   dv = 4357 m/s
# total = 12889 m/s

So the three stages deliver about $\Delta v_{\text{total}} \approx 12.9\ \text{km/s}$: S-IC $3.74$, S-II $4.79$, S-IVB $4.36\ \text{km/s}$. (The S-IVB figure lumps both of its burns — orbit insertion and the later translunar injection.)

Sanity check. The mission needs roughly $9.4\ \text{km/s}$ to reach low Earth orbit (Chapter 1) plus about $3.1\ \text{km/s}$ for trans-lunar injection (Chapter 3's delta-v map), totaling $\sim 12.5\ \text{km/s}$; our $12.9\ \text{km/s}$ has a small margin over that for gravity and steering losses. The reconstruction is physically sensible.

Phase 3: Structural coefficients and the ceilings

Each stage's structural coefficient tells us how good its mass fraction is:

$$ \varepsilon_{\text{S-IC}} = \frac{130}{130+2160} = 0.057, \quad \varepsilon_{\text{S-II}} = \frac{40}{40+456} = 0.081, \quad \varepsilon_{\text{S-IVB}} = \frac{13.5}{13.5+109.5} = 0.110. $$

The kerosene S-IC has the best structural coefficient ($0.057$) — dense kerosene and dense oxygen pack into small, light tanks. The hydrogen stages are structurally worse ($0.081$, $0.110$): liquid hydrogen is so low in density that its tanks are enormous, and big tanks are heavy. Each stage's mass ratio ($3.72$, $3.19$, $2.87$) sits comfortably below its ceiling $1/\varepsilon$ ($17.5$, $12.4$, $9.1$), as it must — every stage is buildable, with room to spare.

Phase 4: Was the split optimal?

Now the payoff. If the three stages were identical, §22.2 says they should split the delta-v equally: $12.9/3 = 4.30\ \text{km/s}$ each. The actual split is $3.74 / 4.79 / 4.36$ — not equal. Is that a design flaw, or exactly right?

It is (mostly) right, and for the reason §22.2's general rule gives. When stages differ, the optimum $R_i = (v_{e,i} - \mu)/(v_{e,i}\varepsilon_i)$ assigns more delta-v to stages with higher exhaust velocity and lower structural coefficient. Compare the S-IC (low $v_e = 2{,}845$) with the S-II (high $v_e = 4{,}130$): the low-$v_e$ kerosene stage should carry less of the delta-v, and it does — $3.74\ \text{km/s}$, the least of the three, well below the $4.30$ an equal split would demand. The efficient hydrogen S-II carries the most, $4.79\ \text{km/s}$. The direction of the deviation from the equal split is exactly what optimal-staging theory predicts for a vehicle that mixes a low-$v_e$ first stage with high-$v_e$ upper stages.

Phase 5: Why kerosene first, then — the tension that decides it

But the S-IC's structural coefficient ($0.057$) is the best of the three, which by the same rule would argue for it doing more, not less. The two factors — low $v_e$ (do less) and low $\varepsilon$ (do more) — pull against each other, and the real design is a compromise between them, further pushed by a constraint the delta-v optimization does not even contain: lift-off thrust.

A first stage's job is not efficiency; it is force. It must lift $2{,}950\ \text{t}$ off the pad with $T/W > 1$ and punch through the dense lower atmosphere. Kerosene wins that job decisively: it is dense (smaller tanks, so the giant propellant load fits), and the F-1 produced immense thrust. Hydrogen, for all its specific-impulse glory, is so bulky and low-thrust-density that a hydrogen first stage would need absurdly large tanks and many more engines to leave the ground. So the size and propellant of the first stage were chosen for thrust and atmosphere, and the delta-v split then fell where the mixed efficiencies dictated. The Saturn V is a textbook illustration of §22.2's warning: the first stage is big because of a thrust decision, while the delta-v split is an efficiency decision, and the two need not, and here do not, agree.

Discussion Questions

1. The S-IVB has the worst structural coefficient ($0.110$) yet does not do the least delta-v. Why might a designer accept a structurally heavy top stage anyway? (Hint: what unique capability did the S-IVB need that the others did not?)
2. If liquid hydrogen has the highest specific impulse of any practical chemical propellant, why is a hydrogen first stage rare? Frame your answer in thrust-to-weight and propellant density, not $v_e$.
3. Our reconstruction lumped the S-IVB's two separate burns. How would splitting them (LEO insertion now, trans-lunar injection hours later) change the propellant-management problems the S-IVB faces between them? (Connect to §22.3.)

Your Turn: Extensions

• Option A (analysis). Recompute the S-IC delta-v using a sea-level $I_{sp} = 263\ \text{s}$ and a vacuum $I_{sp} = 304\ \text{s}$ separately, and bracket the true value between them. How much does the first stage's delta-v depend on the $I_{sp}$ you assume?
• Option B (comparison). Repeat the whole reconstruction for a two-stage Falcon 9 (masses in Chapter 3) and compare its split to the Saturn V's. Which is closer to an equal split, and why (think about how similar its two stages' propellants are)?
• Option C (design). Suppose you had to add a small fourth stage to send $5\ \text{t}$ from trans-lunar injection into low lunar orbit ($\sim 0.8\ \text{km/s}$). Using $v_e = 3{,}100\ \text{m/s}$ and $\varepsilon = 0.12$, size it. Does adding it change your view of the "two or three stages" rule of §22.1?

Key Takeaways

• The Saturn V delivered $\sim 12.9\ \text{km/s}$ across three serial stages: $3.74 / 4.79 / 4.36\ \text{km/s}$, enough for LEO plus trans-lunar injection with margin.
• Its split is unequal because its stages are unequal: the low-$v_e$ kerosene first stage sensibly does the least delta-v, matching the direction of the optimal-staging rule for mixed propellants.
• The first stage's size and propellant were set by lift-off thrust and atmosphere, not by the delta-v optimization — a real vehicle balances efficiency against force, and the two constraints point different ways.
• Structural coefficients tell the propellant story at a glance: dense kerosene gives the best $\varepsilon$ ($0.057$); bulky hydrogen the worst ($0.110$) — which is exactly why hydrogen lives upstairs, not down.