Exercises: Launch Vehicles

Work these with a calculator. Use $g_0 = 9.81\ \text{m/s}^2$, Earth's equatorial rotation speed $0.465\ \text{km/s}$, and the Tier-2 vehicle numbers from Appendix H unless a problem states otherwise. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. For "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.

Part A — Warm-ups: read the fleet and the tables (⭐)

30.1 † A launch vehicle lifts $23\ \text{t}$ to LEO and $8\ \text{t}$ to GTO. State, in one sentence each, (a) why the GTO number is smaller and (b) roughly what physical quantity the "missing" 15 t was converted into.

30.2 Sort these into kerolox, hydrolox, or methalox: Falcon 9, Ariane 6, Starship, Vulcan (first stage), Soyuz. (Use Appendix H if needed.)

30.3 † Compute the eastward rotation credit $0.465\cos\phi$ for a launch site at latitude $\phi = 19.6^\circ$ (Wenchang, China). Compare it to the credit at Cape Canaveral ($28.5^\circ$).

30.4 What is the minimum orbital inclination reachable by a direct launch from Baikonur ($\phi = 45.9^\circ$)? From an equatorial site ($\phi = 0^\circ$)?

30.5 † A vehicle is advertised at "\$2{,}500 per kg to LEO." List two pieces of information you must know before this number means anything.

30.6 Which comes back on a Falcon 9 launch, and which is expended: first stage, second stage, payload fairing?

Part B — Latitude, azimuth, and the plane-change penalty (⭐⭐)

30.7 † From Cape Canaveral ($\phi = 28.5^\circ$), what launch azimuth $\beta$ (clockwise from north) places a payload into a $60^\circ$-inclined orbit? Use $\cos i = \cos\phi\sin\beta$. Give both solutions and say which is a northeasterly launch.

30.8 Explain, using $\cos i = \cos\phi\sin\beta$, why no choice of azimuth from a $45.9^\circ$-latitude site can reach an orbit inclined at $30^\circ$.

30.9 † A GTO satellite reaches its apogee at GEO altitude moving at $v_a = 1.60\ \text{km/s}$; GEO circular speed is $v_{\text{GEO}} = 3.07\ \text{km/s}$. Using $\Delta v = \sqrt{v_a^2 + v_{\text{GEO}}^2 - 2 v_a v_{\text{GEO}}\cos i}$, compute the combined plane-change-plus-circularization delta-v for a launch inclination of $i = 7^\circ$ and for $i = 28.5^\circ$. How much extra delta-v does the higher-latitude launch cost the satellite?

30.10 Continuing 30.9: a satellite arrives at GEO with an on-station mass of $2{,}500\ \text{kg}$ and an apogee engine of $I_{sp} = 320\ \text{s}$ ($v_e = 3.14\ \text{km/s}$). Compute the propellant mass needed for the $i = 7^\circ$ case and the $i = 28.5^\circ$ case, and state the difference in kilograms. (Illustrative numbers — label them Tier 3.)

30.11 † Explain why dropping a satellite into a supersynchronous transfer orbit (apogee above GEO) makes its plane change cheaper. Which term in the combined-burn formula changes, and in which direction?

Part C — Implement it in Python (⭐⭐)

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

30.12 † Write eastward_credit(lat_deg) returning $0.465\cos\phi$ in km/s. Trace it for eastward_credit(5.2) and eastward_credit(45.9).

30.13 Write inclination(lat_deg, azimuth_deg) returning the inclination from $\cos i = \cos\phi\sin\beta$ (use math.degrees(math.acos(...))). Trace it for inclination(28.5, 90) and inclination(28.5, 45).

30.14 † Write select_launcher(payload_kg, destination, table, margin=0.10) that returns the sorted names of vehicles whose capacity to the destination clears payload_kg * (1 + margin). Using the table {"A": (22800, 8300), "B": (10000, 4000)} with columns (LEO, GTO), trace it for a $3{,}600\ \text{kg}$ payload to "GTO".

30.15 Write windows_per_day(lat_deg, inc_deg) that returns 2 if inc_deg > lat_deg, 1 if they are equal, and 0 if inc_deg < lat_deg (direct-launch plane openings per day). Trace it for (28.5, 51.6), (28.5, 28.5), and (28.5, 20.0).

Part D — Find the error (⭐⭐)

30.16 † A brochure claims: "Our rocket is fully reusable, so its payload to LEO is the same whether or not we recover the booster." Identify the physical error.

30.17 A student computes the minimum inclination from Kourou ($5.2^\circ$) as $0^\circ$, reasoning "it's almost on the equator, so it can reach any orbit including equatorial GEO directly." What is wrong, and what is the actual minimum inclination?

30.18 † An analyst compares two rockets by cost per kilogram: Rocket X at "\$3{,}000/kg to GTO" and Rocket Y at "\$2{,}000/kg to LEO," and concludes Y is cheaper. Why is this comparison invalid?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

30.19 † Your spacecraft has a wet mass of $5{,}200\ \text{kg}$ and must reach GTO. From Appendix H, list every vehicle that clears the mass gate with at least 10% margin, then state which single factor you would use to choose among them and why.

30.20 (Mission project) For your mission (Track A/B/C/D), run the three hard gates of §30.6 against Appendix H, choose a launch vehicle, and record the binding constraint and your mass margin in your MDR.

30.21 † A $1{,}700\ \text{kg}$ Mars orbiter must be thrown to $C_3 = 8.7\ \text{km}^2/\text{s}^2$ (from Chapter 11). A candidate vehicle is rated to deliver $2{,}300\ \text{kg}$ to that $C_3$. Compute the mass margin (capacity ÷ wet mass) and state whether it passes a 10%-margin rule. Name one non-performance factor you would still check before committing.

30.22 (Design) A low-density payload (a large deployable antenna) masses only $900\ \text{kg}$ but, folded, is $4.5\ \text{m}$ across and $9\ \text{m}$ tall. Explain why the mass gate is not the binding constraint here, and what you would check instead when choosing a vehicle.

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

30.23 † Why can't you just launch every mission from the equator to get the biggest rotation credit and the cheapest GEO plane change? Give at least three real reasons (geographic, political/logistical, and mission-dependent).

30.24 Why can't you just launch westward and use Earth's rotation to reach a retrograde orbit cheaply? Estimate the delta-v penalty at Cape Canaveral relative to an eastward launch, and explain the sign.

30.25 † Estimate, to an order of magnitude, how many kilograms of extra satellite propellant the $\sim 0.35\ \text{km/s}$ Cape-versus-Kourou GEO penalty costs for a $3{,}000\ \text{kg}$ on-station comsat ($v_e = 3.1\ \text{km/s}$). Is this a "rounding error" or a design driver? Justify.

30.26 Why can't you just hold a launch open indefinitely until the weather clears, for a mission that must rendezvous with the ISS? Frame your answer in terms of the launch window.

Part G — Interleaved & synthesis (from earlier chapters) (⭐⭐/⭐⭐⭐)

30.27 † (Ch. 22) A reusable first stage reserves propellant for boostback, entry, and landing burns. Explain, via the rocket equation, why this lowers the payload the stack can deliver to orbit — and why the trade is still usually worth it economically.

30.28 (Ch. 29) You require a launch vehicle rated for 10% more than your best current mass estimate. Name the mission-design principle and explain why spacecraft mass estimates are notorious for needing it.

30.29 † (Ch. 3 + Ch. 9) Reaching LEO is ~9.4 km/s and LEO→GTO is ~2.4 km/s more. Using the rocket equation with an overall $v_e = 3.3\ \text{km/s}$, estimate the ratio of the mass a vehicle can put in LEO to the mass it can put in GTO. Compare your ratio to the "~2–3×" rule of thumb and comment.

30.30 (Ch. 11) Why is the interplanetary launch window (measured in weeks, once every synodic period) governed by different geometry than the daily launch window of this chapter (measured in minutes, twice a day)? Name the aligning bodies in each case.

30.31 † (Ch. 32, preview) The GO/NO-GO poll gives any single console a veto, and launch-commit criteria are fixed before launch day. Explain how these two practices guard against the organizational failure mode that destroyed Challenger.

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: running the hard gates before the soft preferences, explicit units and margins, a stated Tier for every vehicle number, and a one-line sanity check on each result.