Exercises: Ground Operations and Mission Control

Work these with a calculator. Use the speed of light $c = 2.998\times10^{5}\ \text{km/s}$, the astronomical unit $1\ \text{AU} = 1.496\times10^{8}\ \text{km}$, and the astronomical data of Appendix B. 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 write the result in an # Expected output: comment, exactly as the chapter does.

Part A — Latency warm-ups: convert and compute (⭐)

31.1 † A geostationary relay satellite is $35{,}786\ \text{km}$ away. What is the one-way communication latency to it, in seconds? What is the round trip?

31.2 The Moon's mean distance is $384{,}400\ \text{km}$. Compute the round-trip light time. Why do recordings of Apollo air-to-ground have a noticeable pause between a question from Houston and the crew's reply?

31.3 † Mars is $1.5\ \text{AU}$ from Earth on a particular day. Find the one-way light time in minutes. Is this closer to opposition or to conjunction? Explain.

31.4 A satellite is in a $550\ \text{km}$ low Earth orbit. Compute the one-way light time to it and compare it to a human reaction time ($\sim 0.25\ \text{s}$). What does the comparison tell you about whether light-speed delay limits real-time LEO control?

31.5 † Light from the Sun ($1\ \text{AU}$) takes how long to reach Earth? Use your answer as a "unit conversion" to state, in one sentence, how many light-minutes correspond to one AU.

Part B — The control room and the operations process (⭐⭐)

31.6 Match each console to its responsibility: (FLIGHT, CAPCOM, FIDO, EECOM, GNC) ↔ (health of the onboard guidance/control systems; trajectory and maneuvers; final authority for the mission; power, thermal, life support, and consumables; the single voice to the crew).

31.7 † In two or three sentences, explain the purpose of the flight-rules book. Why is it better to pre-decide a contingency response than to decide it live, even though the flight director can override any rule?

31.8 List, in order, the five steps of anomaly resolution given in §31.4, and state in one phrase why "safe the vehicle" comes before "diagnose the fault."

31.9 † Distinguish telemetry from commanding in one sentence each, then explain what it means to "verify a command in the telemetry" and why that step is never skipped on a distant vehicle.

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.

31.10 † Write light_time(distance_km) returning $d/c$ with $c = 2.998\times10^{5}\ \text{km/s}$. Hand-trace light_time(3.78e8) (Mars near conjunction) and express the result in minutes.

31.11 Write round_trip_light_time(distance_km) and an autonomy_level(distance_km) helper that returns "real-time" if the round trip is under $1\ \text{s}$, "supervised" if under $10\ \text{s}$, and "autonomous" otherwise. Hand-trace it for the Moon ($384{,}400\ \text{km}$).

31.12 † Write pass_duration_min(altitude_km) for the maximum horizon-to-horizon contact time of a circular-orbit satellite passing straight overhead (use $\cos\lambda = R/r$, $R = 6378\ \text{km}$, $\mu = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$). Hand-trace it for a $550\ \text{km}$ orbit.

Part D — Find the error (⭐⭐)

31.13 † A Mars-lander operations plan reads: "If anything looks wrong during the seven-minute landing, the team in the control room will command an abort." Mars is near opposition (one-way light time $\approx 4.4\ \text{min}$). Explain, with numbers, why this plan is physically impossible and what must replace it.

31.14 A controller computes that a command to a lunar orbiter "will take $1.28\ \text{s}$ to be confirmed," using the one-way light time. Where is the error, and what is the correct figure for confirmation (command received and acknowledgment returned)?

Part E — Design it (mission project) (⭐⭐ / ⭐⭐⭐)

31.15 † (Mission project) Write the operations-concept note for your chosen mission (Track A/B/C/D) and add it to your MDR. It must state: (a) the ground segment (which stations/relays/DSN and how often you get contact); (b) the one-way and round-trip latency, computed from your mission's distance; and (c) the resulting autonomy level and your safe-mode concept. Half a page is plenty; be specific.

31.16 For your mission, describe one realistic anomaly and walk it through the five-step detect–safe–diagnose–recover–document process, sizing each step to your mission's latency (how much can the ground help in the moment, and how much must the vehicle do alone?).

31.17 † A LEO Earth-observation satellite must be commanded and downlinked using ground stations that each give a usable $\sim 6$-minute pass. Estimate how many ground stations, spread in longitude, you would need for near-continuous coverage of a $400\ \text{km}$ orbit, and explain why a single GEO relay satellite can replace many of them.

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

31.18 † A deep-space probe downlinks at $500\ \text{kbit/s}$ during a single $8$-hour DSN pass per day. Estimate the data volume returned per day, in gigabytes. If the probe's camera can generate $10\ \text{GB}$ of images per day, what must the operations concept include?

31.19 Why can't you just keep a low-orbit satellite in continuous contact by pointing a single big dish at it and tracking it? Frame your answer with the pass-geometry result of §31.3.

31.20 † Why can't you just drive a Mars rover in real time with a joystick on Earth, the way a drone pilot flies a drone? Give the physical reason and the approximate numbers at both opposition and conjunction.

Part G — Interleaved & synthesis (from Chapters 13, 26, 30) (⭐⭐ / ⭐⭐⭐)

31.21 † (Ch. 13) Flight dynamics re-determines a spacecraft's orbit after every maneuver. Explain why a single tracking pass is generally not enough to pin down all six orbital elements, and why several passes across the orbit help.

31.22 (Ch. 30) A countdown includes several built-in holds. Explain how a planned hold differs from an unplanned one, and how holds protect a launch that has a fixed window.

31.23 † (Ch. 26) The Deep Space Network has only a few dozen antennas but serves dozens of active missions. Explain, in operational terms, why this makes contact a scheduled resource and how spacecraft cope with the gaps between passes.

31.24 (Ch. 26) A GEO satellite phone call has a noticeable delay. Using the GEO light time you computed in 31.1, explain why the delay is about a quarter second each way, and why a call routed up to GEO and back down feels laggy even though the people are both on Earth.

31.25 † (Synthesis — Apollo 13) The lunar module Aquarius was designed to keep two astronauts alive for about two days; it had to keep three alive for about four. Name the three consumables that actually became the binding constraints (it was not primarily oxygen), and describe the improvised fix for the carbon-dioxide problem.

31.26 (Synthesis) In one short paragraph, explain why LEO operations and Mars operations sit at opposite ends of a "real-time ↔ fully autonomous" spectrum. Tie your explanation directly to the round-trip light times of each, and name the onboard capability that deep-space missions must therefore have that LEO missions can lean on the ground for.

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 the design problems, the rubric rewards: an explicit latency calculation with units, an autonomy level that follows from it, and a safe-mode/anomaly concept that is honest about how little the ground can do in the moment at your mission's distance.