Exercises: Mission Design

Work these with a calculator and $g_0 = 9.81\ \text{m/s}^2$ (use $9.80665$ where a Checkpoint 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 running Track-A comsat (dry mass 1,500 kg; margined budget 2,726 m/s) and the delta-v map of Chapter 3 / Appendix G.

Part A — Warm-ups: the vocabulary and the budget (⭐)

29.1 † Classify each as a functional, performance, or constraint requirement: (a) "the spacecraft shall relay data between two ground terminals"; (b) "the spacecraft shall mass no more than 1,800 kg at launch"; (c) "the downlink shall provide at least 50 Gbps."

29.2 Rewrite as a good (testable, verifiable) requirement: "the thermal system shall keep the electronics from getting too hot."

29.3 † A mission's spacecraft-own delta-v legs are: orbit insertion 1,500 m/s, station-keeping 600 m/s, disposal 40 m/s. With no margin, what is the ideal delta-v budget?

29.4 For the legs in 29.3, apply margins of 5%, 10%, and 0% respectively. What is the margined budget?

29.5 † State, in one sentence each, what the MDR, PDR, and CDR gates each confirm.

29.6 Define the driving requirement in your own words, and name a plausible driving requirement for a Mars science orbiter.

Part B — Delta-v budgets and sizing (⭐⭐, Calculate it)

29.7 † An engine has $I_{sp} = 320\ \text{s}$. A spacecraft with dry mass 1,500 kg must provide 2,726 m/s. Find (a) the exhaust velocity, (b) the mass ratio, (c) the wet mass, and (d) the propellant mass. Sanity-check (c) against the chapter.

29.8 Repeat 29.7 with an electric engine, $I_{sp} = 1{,}800\ \text{s}$. By what factor does the propellant mass drop, and why?

29.9 † A lunar cargo lander must provide a margined 5,980 m/s to a landed mass (dry + cargo) of 2,200 kg with $I_{sp} = 320\ \text{s}$. Find the mass ratio and wet mass. Compare the mass ratio to the comsat's 2.38 and explain the difference physically.

29.10 Using size_vehicle's logic, show that the amplifier — the extra wet mass per extra kilogram of dry mass — is exactly the mass ratio $m_0/m_f$. For the lander of 29.9, how much does the vehicle grow if the cargo grows by 250 kg?

29.11 † A GEO comsat is launched from an equatorial site (0° latitude) instead of Cape Canaveral (28.5°), so its GTO → GEO leg needs no plane change and costs only 1,470 m/s. Rebuild the margined budget (5% on this leg; station-keeping 750 m/s at 10%; disposal 11 m/s at 0%) and re-size the wet mass (dry 1,500 kg, $I_{sp} = 320\ \text{s}$). How much wet mass did the equatorial launch site save?

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.

29.12 † Write roll_up_dv(budget) taking a list of (label, dv, margin) tuples and returning (ideal, margined). Trace it for [("insertion", 1500, 0.05), ("SK", 600, 0.10), ("disposal", 40, 0.0)].

29.13 Write size_vehicle(dv, isp, payload) returning (m_prop, m_wet). Trace it for size_vehicle(2726, 320, 1500) (use $g_0 = 9.80665$).

29.14 † Write amplifier(dv, isp) returning the mass ratio $e^{\Delta v/v_e}$ (the wet-mass-per-dry- mass factor). Trace it for amplifier(5980, 320) and amplifier(2726, 1800).

29.15 Write close_check(m_wet, launcher_capacity) that returns how many identical spacecraft of mass m_wet fit under a launcher's capacity, and whether at least one fits. Trace it for close_check(1750, 5000) and close_check(3576, 5000).

Part D — Find the error (⭐⭐)

29.16 † A manager says: "Our best analysis gives a 2,561 m/s budget and we have exactly 2,561 m/s of propellant, so we are done — carrying more would be unscientific padding." Diagnose the error in terms of margin vs. estimate.

29.17 A student sizes the comsat's launch vehicle by adding the launch leg (~11.9 km/s) and the spacecraft-own budget (2.7 km/s) into a single 14.6 km/s and feeding that to size_vehicle. Explain why this is wrong and what they double-counted. (Hint: staging.)

29.18 † An engineer keeps mass margin constant at 25% from concept all the way to launch, arguing "uncertainty never goes away." What is wrong with holding margin constant as the design matures?

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

29.19 † (Mission project) For your chosen track, write the full delta-v budget: list every leg, its ideal delta-v (from the delta-v map / Appendix G), and a justified per-leg margin. Roll it up to an ideal and a margined total. Save it to your MDR.

29.20 (Mission project) Choose an engine for your mission (state its $I_{sp}$ and why), then use size_vehicle to compute your propellant mass and wet mass from your margined budget and your dry mass. Record the amplifier (mass ratio). This wet mass is what you bring to Chapter 30.

29.21 † Run a trade study for your mission between two architectures (e.g., chemical vs. electric propulsion, or two different orbits). Build a table with at least four weighted criteria, score each option, and state the winner and the weighting that made it win.

29.22 (Mission project) Identify your mission's driving requirement. Justify it by a sensitivity argument: show that a small change in that requirement moves the wet mass (or feasibility) more than a small change in any other requirement.

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

29.23 † Back of the envelope. Without detailed sizing, estimate the wet mass of a Mars science orbiter that must provide ~2.1 km/s for propulsive Mars orbit insertion (Ch. 11) plus ~0.3 km/s of corrections, to a 900 kg dry spacecraft with $I_{sp} = 320\ \text{s}$. State your budget, your amplifier, and your answer to one significant figure.

29.24 Why can't you just keep cutting margin to make a too-heavy spacecraft fit its launch vehicle? Frame your answer in terms of what margin is protecting against and what happens on the day the estimate turns out to be optimistic.

29.25 † Why can't you just optimize each subsystem to be as light as possible, independently, and get the lightest spacecraft? Explain using the mass–power–cost coupling why local optimization can make the system worse.

29.26 Back of the envelope. A subsystem team proposes a change that saves 20 kg of dry mass but adds 80 W of power draw. Using the coupling of §29.5 qualitatively (more power → more array → more mass → more propellant), argue whether this is obviously a win, and what number you would need to decide.

Part G — Interleaved & synthesis (Chapters 3, 10, 23) (⭐⭐ / ⭐⭐⭐)

29.27 † (Ch. 3) A spacecraft needs mass ratio 2.38 and has dry mass 1,500 kg. From the rocket equation alone, what wet mass and propellant mass does it carry? Show that this equals size_vehicle's output for 2,726 m/s at $I_{sp} = 320\ \text{s}$.

29.28 (Ch. 10) The comsat's GTO → GEO leg is 1,800 m/s largely because of a 28.5° plane change folded into the apogee burn. Using $\Delta v = 2v\sin(\Delta i/2)$ with GEO speed $v = 3.07\ \text{km/s}$, estimate the pure plane-change cost of 28.5°. Why is folding it into the apogee burn cheaper than doing it separately?

29.29 † (Ch. 23, synthesis) Your mass budget rolls up to a dry mass 8% over your target, and the rocket equation's amplifier for your mission is 3.0. By how much does the wet mass overrun, in multiples of the dry overrun? Explain to a structures engineer why their 8% is really a 24% problem for the vehicle.

29.30 (Synthesis) In one paragraph, connect the whole chain for your mission: driving requirement → delta-v budget → mass ratio (amplifier) → wet mass → launch vehicle → cost. This is the elevator summary of your MDR; write it as you would say 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 amplifier (mass ratio), units on every number, a sanity check on the wet mass, and a clearly named driving requirement.