Exercises: Capstone — Your Complete Space Mission

These are the book's final exercises, and they are deliberately integrative: most touch two or more earlier chapters, because a capstone tests whether the pieces have become a whole. Work them with a calculator and $g_0 = 9.81\ \text{m/s}^2$ (use $9.80665$ where a Checkpoint or worked value must match). Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold before you peek. For the "implement it" problems, do not run the code: hand-trace it and record the result in an # Expected output: comment, exactly as the chapter does.

Many problems build on the four worked tracks of the chapter: A comsat (dry 1,500 kg; margined 2,726 m/s), B lunar lander (landed 2,200 kg; margined 5,980 m/s), C Mars orbiter (dry 900 kg), and D asteroid probe (dry 700 kg; margined ~5,300 m/s).

Part A — Warm-ups: the MDR and its spine (⭐)

40.1 † List, in order, the ten rows of a Mission Design Review document (§40.1). For each, name the one question it answers in four words or fewer.

40.2 In one sentence, what does it mean for a mission design to close? Name two budgets that must balance at once for closure.

40.3 † The MDR's load-bearing "spine" is: objective → orbit → ____ → propulsion → mass → launch vehicle → cost. Fill in the blank, and say why that item, not any other, is called the master constraint.

40.4 Match each track to the single "survival strategy" the rocket equation forced on it: (A comsat, B lunar lander, C Mars orbiter, D asteroid probe) ↔ (staging, aerobraking, electric propulsion mandatory, electric propulsion attractive).

40.5 † State the three things a good design defense leads with, in the order a review board wants to hear them (§40.6).

Part B — Sizing the four tracks (⭐⭐, Calculate it)

Use $v_e = I_{sp}\,g_0$ and $m_0 = e^{\Delta v/v_e}\,m_f$. Show units and sanity-check every mass.

40.6 † (Track A) Reproduce the comsat's chemical sizing: for $\Delta v = 2{,}726\ \text{m/s}$, $I_{sp} = 320\ \text{s}$, dry $1{,}500\ \text{kg}$, find $v_e$, the mass ratio, the wet mass, and the propellant mass. Confirm against §40.2.

40.7 (Track A) Re-size the same comsat with an electric thruster, $I_{sp} = 1{,}800\ \text{s}$. By what factor does the propellant drop, and how many electric comsats fit under one Falcon 9's ~5,500 kg GTO capacity?

40.8 † (Track B) The lunar lander sizes to mass ratio 6.72 at $I_{sp} = 320\ \text{s}$. What propellant fraction is that? Compare to the comsat's 2.38 and explain, using the exponent, why ~2× the delta-v gave ~3× the mass ratio.

40.9 (Track C) Compute the Mars orbit-insertion burn straight to a 400 km circular orbit for arrival $v_\infty = 2.65\ \text{km/s}$: find $v_{\text{hyp}}$ at $r = 3{,}790\ \text{km}$ ($\mu_{\text{Mars}} = 4.283\times10^{4}\ \text{km}^3/\text{s}^2$), the circular speed, and their difference. Confirm it is ~2.08 km/s.

40.10 † (Track D) Size the asteroid probe both ways for $\Delta v = 5{,}300\ \text{m/s}$, dry $700\ \text{kg}$: ion ($I_{sp} = 3{,}000\ \text{s}$) and chemical ($I_{sp} = 320\ \text{s}$). By what factor does the propellant differ, and why is electric "mandatory, not optional" here?

Part C — Implement it in Python (⭐⭐, Simulate it)

Write each function, hand-trace it for the given inputs, and record the result in an # Expected output: comment. Do not run it.

40.11 † Write roll_up_dv(budget) for a list of (label, dv, margin) tuples returning (ideal, margined). Trace it for the Track-A budget [("GTO->GEO", 1800, 0.05), ("SK", 750, 0.10), ("disposal", 11, 0.0)].

40.12 Write size_vehicle(dv, isp, payload) returning (m_prop, m_wet). Trace it for the lunar lander: size_vehicle(5980, 320, 2200) (use $g_0 = 9.80665$).

40.13 † Write closes(m_wet, capacity) returning (fits, n_per_launch) — whether one vehicle fits under a launcher capacity and how many identical ones fit. Trace for closes(3576, 5500) and closes(1750, 5500) (the Track-A chemical and electric comsats vs Falcon 9 to GTO).

40.14 Write size_stage(dv, isp, payload, stage_dry) (the onion-layer sizer of Case Study 2) returning the stage wet mass. Trace the two layers of the lunar sample return: ascent size_stage(2900, 320, 200, 150) then descent size_stage(5980, 320, <ascent wet>, 500).

40.15 † Write capture_burn(v_inf, mu, r, target) returning the delta-v to capture from arrival excess v_inf into either a circular (target="circ") or barely-bound elliptical (target="ellipse") orbit at radius r. Trace it for Mars, $v_\infty = 2.65$, $r = 3{,}790$, both targets.

Part D — Find the error (⭐⭐)

40.16 † A designer sizes a Mars-sample-return vehicle by adding the descent budget (5,980 m/s) and the ascent budget (2,900 m/s) into 8,880 m/s and running size_vehicle once. Explain why this under-sizes the mission and what staged/onion sizing does instead (Case Study 2).

40.17 A student computes the comsat's launch vehicle by feeding the sum of the launch leg (~11.9 km/s) and the spacecraft-own budget (2.7 km/s) into the rocket equation. What did they double-count, and which principle (from Chapter 3) forbids it?

40.18 † A reviewer is told "our delta-v budget is 2,726 m/s and we carry propellant for exactly 2,726 m/s, so we are done." Diagnose the error in terms of margin vs. estimate (Chapter 29), and say what the correct move is.

Part E — Design it (⭐⭐ / ⭐⭐⭐, Design it)

40.19 † (Your mission — the capstone.) Assemble the full ten-row MDR (§40.1) for your chosen track. For each row, give a defensible number or decision and its source chapter. Confirm the design closes: sized wet mass under launch-vehicle capacity, with margin. This is your final deliverable.

40.20 (Your mission.) Write the one-paragraph elevator summary that opens a review (§40.6): driving requirement → delta-v budget → mass ratio → wet mass → launch vehicle → cost. Keep it under 120 words and make every number defensible.

40.21 † (Your mission.) Identify your mission's driving requirement and prove it by sensitivity: show that a small change in it moves your wet mass (or closure) more than a small change in any other requirement.

40.22 (Cross-track.) Design a second track cold — one that is not yours — through the full spine (orbit → delta-v budget → sized vehicle → launch vehicle). This is the surest test that you can design a mission, not just recall one.

40.23 † (Extension.) Redesign your mission with one requirement made harder (double the comsat lifetime, double the lander cargo, halve the Mars science-orbit altitude, or raise the asteroid target's inclination by 3°). Re-size and report what broke and how you fixed it.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

40.24 † Back of the envelope. Without detailed sizing, estimate the LEO mass to return a 100 kg capsule from the lunar surface, given the round-trip amplifier of Case Study 2 was ~46:1 for a 200 kg capsule. State your assumption about how the ratio scales and your answer to one significant figure.

40.25 Why can't you just land the whole return vehicle, fueled, and launch it back — why stage (§40.3, Case Study 2)? Frame the answer in terms of what dead mass costs at each leg.

40.26 † Why can't you just use electric propulsion for the lunar lander's descent to save propellant, the way Track D uses it? Explain what physical requirement of landing rules it out (Chapter 20).

40.27 Back of the envelope. Escaping to a near-Earth asteroid at 1.2 AU needs only ~1.3 km/s of heliocentric excess, yet the mission budgeted ~4.6 km/s. Where did the other ~3.3 km/s go? Name the two biggest contributors (§40.5).

Part G — Interleaved & synthesis (Parts I–V) (⭐⭐ / ⭐⭐⭐)

40.28 † (Ch. 3, 22) The lunar lander is 85% propellant single-stage. Show that staging off the trans-lunar injection stage (so the descent stage need not haul empty TLI tanks to the surface) reduces the landed vehicle. Sketch the two-stage sizing qualitatively and say which chapter's principle you used.

40.29 (Ch. 6, 10, 11) A mission is "a route across the delta-v map." For your track, list every leg, mark which are propulsive and which (if any) are aerodynamic or gravity-assisted, and state which single leg dominates the budget.

40.30 † (Ch. 11, 12) The comsat's 15-year station-keeping (750 m/s) exists because of perturbations. Name the dominant perturbation driving north–south station-keeping at GEO, and explain why it, not east–west, dominates the budget.

40.31 (Ch. 23, 25, 29) A subsystem team saves 30 kg of dry mass but adds 120 W of power draw to your comsat. Trace the change around the mass–power–cost loop (§29.5) and argue, with the numbers you would need, whether it is a net win.

40.32 † (Ch. 16, 20, synthesis) Across the four tracks you used engines from $I_{sp} = 320\ \text{s}$ (chemical) to $3{,}000\ \text{s}$ (ion). In one paragraph, state the single trade-off that runs through all of propulsion — high thrust vs. high efficiency — and give one track where each side wins and why.

40.33 (Synthesis — the whole book.) In one page, connect the entire chain for your mission: objective → driving requirement → orbit → delta-v budget → mass ratio → wet mass → launch vehicle → operations → risk → cost. This is your MDR in prose; write it as you would present it in a review.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: a complete and margined delta-v budget, an explicit mass ratio (the amplifier), units on every number, a sanity check on every mass, a clearly named driving requirement, and — the capstone's own standard — an honest flag on every number you are not sure of.