Exercises: Small Satellites, CubeSats, and Constellations

Work these with a calculator. Use $R_E = 6{,}371\ \text{km}$, solar flux $1{,}361\ \text{W/m}^2$, the speed of light $c = 299{,}792\ \text{km/s}$, and $g_0 = 9.81\ \text{m/s}^2$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each one 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.

Coverage-geometry problems use the three relations of §33.5: $$\sin\eta = \frac{R_E}{R_E + h}\cos\varepsilon, \qquad \lambda = 90^\circ - \varepsilon - \eta, \qquad f = \frac{1 - \cos\lambda}{2}, \qquad N_{\min} \gtrsim \frac{1}{f}.$$

Part A — Warm-ups: classify and compute (⭐)

33.1 † Classify each satellite into its small-sat mass class (mini / micro / nano / pico): (a) a $4\ \text{kg}$ CubeSat, (b) a $60\ \text{kg}$ imaging satellite, (c) a $300\ \text{kg}$ broadband satellite, (d) a $0.3\ \text{kg}$ pocket-sized demonstrator.

33.2 A 6U CubeSat is built from how many 1U cubes of volume, and what are its outer dimensions in centimetres?

33.3 † One large face of a 3U CubeSat measures $10\ \text{cm} \times 30\ \text{cm}$. With that face square to the Sun, solar flux $1{,}361\ \text{W/m}^2$, cell efficiency $\eta = 0.30$, and a packing/pointing factor of $0.8$, what is the peak electrical power? Comment on whether a CubeSat is limited by mass or by power.

33.4 A deployer ejects one CubeSat at $1.2\ \text{m/s}$ relative to a second CubeSat that is released with no relative velocity. Ignoring orbital mechanics (treat it as straight-line drift), how far apart are they after $30\ \text{minutes}$?

Part B — Coverage geometry (⭐⭐)

33.5 † Compute the horizon-limited Earth-central angle $\lambda_{\max}$ for a satellite at $h = 500\ \text{km}$ — that is, the footprint half-angle when the minimum elevation is $\varepsilon = 0$ (the satellite just on the horizon). (Hint: with $\varepsilon = 0$, the relations reduce to $\cos\lambda_{\max} = R_E/(R_E+h)$.)

33.6 For a satellite at $h = 500\ \text{km}$ serving a minimum elevation $\varepsilon = 20^\circ$, compute the nadir angle $\eta$, the Earth-central angle $\lambda$, the footprint fraction $f$, and the floor $N_{\min}$ on satellites for continuous global coverage.

33.7 † Repeat for $h = 1{,}000\ \text{km}$ and $\varepsilon = 25^\circ$: find $\lambda$, $f$, and $N_{\min}$. Compared with the $550\ \text{km}$ result in the chapter (~184 satellites), does going higher help, and by roughly what factor?

33.8 Demonstrate numerically that raising the required elevation shrinks the footprint. For $h = 600\ \text{km}$, compute $\lambda$ at $\varepsilon = 10^\circ$ and again at $\varepsilon = 30^\circ$, and state which needs more satellites.

33.9 † Compute the one-way ground-to-satellite-to-ground signal delay for a satellite directly overhead at (a) $h = 550\ \text{km}$ and (b) geostationary altitude $h = 35{,}786\ \text{km}$. Take their ratio, and say in one sentence why this ratio is the business case for LEO broadband.

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.

33.10 † Write coverage_fraction(alt_km, min_elev_deg) implementing the §33.5 geometry (return the fraction $f$). Trace it for coverage_fraction(550, 25).

33.11 Write min_satellites_for_global(alt_km, min_elev_deg) returning $\lceil 1/f \rceil$. Trace it for min_satellites_for_global(700, 20), and explain why it needs fewer satellites than the chapter's $550\ \text{km}$, $25^\circ$ result of $\sim 184$ (name the two knobs that changed).

33.12 † Write cost_per_kg(price_usd, payload_kg). Trace it for a $\$7.5\text{M}$ launcher carrying $300\ \text{kg}$ and for a $\$1\text{M}$ rideshare slot carrying $200\ \text{kg}$, and state the ratio.

Part D — Find the error (⭐⭐)

33.13 † A student computes $N_{\min} \approx 184$ for a $550\ \text{km}$, $25^\circ$ constellation and concludes: "So Starlink only needs 184 satellites; the four thousand they actually fly is pure marketing." Identify the two things $N_{\min}$ leaves out that make the real number far larger.

33.14 A blog claims: "CubeSats are cheap because they use radiation-hardened parts, which never fail, so you don't waste money on testing." Identify the two factual errors and state what actually makes COTS-based CubeSats affordable.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

33.15 † You must provide continuous global coverage at a minimum elevation of $\varepsilon = 25^\circ$ and are choosing between an altitude of $550\ \text{km}$ ($N_{\min} \approx 184$) and $1{,}200\ \text{km}$ ($N_{\min} \approx 57$). State which needs fewer satellites, which is more sustainable, and articulate the central trade in two or three sentences (invoke coverage, latency, and orbital decay).

33.16 (Mission project.) Add the Platform Trade note to your MDR for your chosen track (A/B/C/D): in a short paragraph, decide whether a small satellite or a constellation fits your mission better than a single traditional spacecraft, and give the driving reason. If you choose a constellation, include a first $N_{\min}$ estimate from §33.5.

33.17 † A constellation needs $600$ satellites on orbit. A Falcon 9 rideshare can carry $60$ of them per launch, and each launch costs $\$60\ \text{M}$. How many launches are needed, what is the total launch cost, and what is the launch cost per satellite? Why is this per-satellite figure only part of the total cost of a satellite?

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

33.18 † Back of the envelope. Atmospheric density falls roughly exponentially with altitude (scale height $\sim 50$–$60\ \text{km}$ in this regime), and orbital decay time scales roughly inversely with density. Argue qualitatively why a dead satellite at $550\ \text{km}$ re-enters within a few years while one at $1{,}200\ \text{km}$ lingers for centuries. (You do not need a numerical decay model — reason about the density ratio.)

33.19 Why can't you just put a broadband constellation in GEO, using three satellites instead of thousands? Frame your answer in terms of latency (use your §33.9 numbers) and coverage of high latitudes.

33.20 † Why can't you just give every CubeSat propulsion so it can deorbit itself and end the debris problem? Answer in terms of the mass, volume, power, and cost budgets of a $4\ \text{kg}$ satellite, and explain why this is the root of the tension in §33.6.

33.21 Why can't you just launch one $4\ \text{kg}$ CubeSat on its own dedicated $\$7.5\ \text{M}$ small launcher whenever you want? Compare the cost against a $\$5{,}000/\text{kg}$ rideshare slot and state the factor, then name the one thing the dedicated launch buys you.

Part G — Interleaved & synthesis (from Chapters 3, 9, 26, 29, 32)

33.22 † (Ch. 9.) A constellation must serve users at latitude $70^\circ\text{N}$ continuously. What is the approximate minimum orbital inclination required, and why does a $53^\circ$-inclination shell (like Starlink's main one) fail to serve $70^\circ\text{N}$ well? (Hint: a satellite's sub-point latitude never exceeds its inclination.)

33.23 (Ch. 29.) Write the requirements flow-down for "provide continuous global broadband," from the top-level mission requirement down through orbit, constellation size, and launch campaign — four levels, each derived from the one above.

33.24 † (Ch. 3 & 29.) A $260\ \text{kg}$ (dry) small satellite must perform an end-of-life deorbit burn of $\Delta v = 150\ \text{m/s}$ from $550\ \text{km}$ using a cold-gas COTS thruster with $I_{sp} = 220\ \text{s}$. Find the exhaust velocity, the mass ratio, and the propellant mass required. Comment on whether that is an affordable fraction of the satellite's mass.

33.25 (Ch. 26.) Explain qualitatively why a CubeSat's small antenna and few watts of transmit power force a low data rate, connecting your answer to the link-budget idea from Chapter 26. Name two ways constellations work around it.

33.26 † (Ch. 32.) A constellation has $100$ satellites, each with a probability $0.95$ of still working after one year. What is the expected number still operating after a year, and why can the service the constellation provides still be effectively $100\%$ available despite those failures?

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: explicit units, a stated minimum elevation, a sanity check on the satellite count, and — for the constellation problems — a sentence on the debris/disposal consequence of your altitude choice.