Exercises: Power Systems

Work these with a calculator. Use the solar constant $S = 1361\ \text{W/m}^2$, space-cell efficiency $\eta = 0.30$ unless told otherwise, and the path efficiencies $X_d = 0.85$, $X_e = 0.65$ from the chapter. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each cold first. 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 — Warm-ups: flux, cosine, and conversions (⭐)

25.1 † Compute the solar flux at Saturn's distance ($9.54\ \text{AU}$). Comment on whether solar power is practical there.

25.2 A $30\%$-efficient array of $8\ \text{m}^2$ faces the Sun square-on at $1\ \text{AU}$. What is its beginning-of-life electrical output?

25.3 † An array's normal points $40^\circ$ away from the Sun. What fraction of its square-on power does it produce?

25.4 An RTG produces $2{,}000\ \text{W}$ of heat and converts it at $6.5\%$ efficiency. What is its electrical output?

25.5 † A battery of capacity $500\ \text{W·h}$ delivers $150\ \text{W·h}$ each cycle. What is its depth of discharge?

25.6 At what distance from the Sun (in AU) does the solar flux fall to one-quarter of its value at Earth?

Part B — Array sizing and degradation (⭐⭐)

25.7 † A spacecraft needs an orbit-average load of $1{,}500\ \text{W}$ (in sunlight and eclipse alike). Its orbit gives $60\ \text{min}$ of sunlight and $36\ \text{min}$ of eclipse. Using the chapter's path efficiencies, find the array output power $P_{sa}$ it must supply.

25.8 The array of 25.7 uses $30\%$ cells at a worst-case sun angle with $\cos\theta = 0.90$. Find its required beginning-of-life area.

25.9 † The array of 25.8 degrades at $3\%$ per year over a $10$-year mission. Find the life degradation factor $L_d$ and the area you must actually build so it still meets the load at end of life.

25.10 A Mars-surface array (flux $586\ \text{W/m}^2$, $30\%$ cells, low-sun $\cos\theta = 0.70$) must make $900\ \text{W}$. Find its area, and name one Mars-specific effect that would force it larger still.

Part C — Batteries and depth of discharge (⭐⭐)

25.11 † A geostationary comsat draws $1{,}200\ \text{W}$ through its longest eclipse of $72\ \text{min}$. With $\text{DoD} = 0.70$ and discharge efficiency $\eta = 0.90$, find the required battery capacity and its mass at $150\ \text{W·h/kg}$.

25.12 Now size a battery for the same $1{,}440\ \text{W·h}$ of eclipse energy, but as a low-Earth-orbit battery at $\text{DoD} = 0.25$. Compare its mass to the GEO battery of 25.11. Why the difference, given the eclipse energy is identical?

25.13 † Explain, in terms of cycle count, why a GEO battery may be cycled to $70\%$ DoD but a LEO battery to only $\sim25\%$. Roughly how many cycles does each accumulate per year?

Part D — RTGs and fission (⭐⭐)

25.14 † An MMRTG begins life at $110\ \text{W}$ electrical. Using thermal decay only (Pu-238 half-life $87.7\ \text{yr}$), estimate its output after $14$ years. Why is the real number lower?

25.15 How much pure Pu-238 (heat output $0.54\ \text{W/g}$) is needed to generate $300\ \text{W}$ of electricity at $7\%$ efficiency? Compare to the GPHS-RTG's $\sim10.9\ \text{kg}$ and explain the gap.

25.16 † A deep-space mission needs $5\ \text{kW}$ of electrical power. (a) How much Pu-238 would an RTG need at $6\%$ efficiency? (b) Given US production of $\sim1.5\ \text{kg/yr}$, why is this infeasible, and what power source solves it?

Part E — Implement it in Python (⭐⭐)

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

25.17 † Write solar_flux(au) returning $1361/\text{au}^2$. Trace solar_flux(1.52).

25.18 Write eclipse_fraction(alt_km) returning $\frac{1}{\pi}\arcsin!\left(\frac{6378}{6378 + \text{alt}}\right)$. Trace eclipse_fraction(400).

25.19 † Write array_area(p, flux, eff, cos_theta, life) returning $p/(\text{flux}\cdot\text{eff}\cdot\cos\theta\cdot\text{life})$. Trace array_area(3149, 1361, 0.30, 0.90, 0.7374).

Part F — Find the error, back of the envelope, and "why can't you just…" (⭐⭐⭐)

25.20 † Find the error. A designer writes: "Our cells make $400\ \text{W/m}^2$ at $1\ \text{AU}$, and we need $2{,}000\ \text{W}$, so we need $2000/400 = 5\ \text{m}^2$ of array — done." List the worst-case factors this estimate ignores, and estimate how much bigger the real array must be.

25.21 Back of the envelope. Estimate the array area a spacecraft would need at Jupiter ($5.2\ \text{AU}$) to run a $400\ \text{W}$ load, ignoring eclipse and degradation. Then say why the real Juno arrays are close to $60\ \text{m}^2$.

25.22 † Why can't you just fly a bigger battery instead of an RTG for a mission to Saturn? Answer in terms of the difference between a power source and power storage.

Part G — Design it and interleave (⭐⭐ / ⭐⭐⭐)

25.23 (Mission project) Build the power budget for your mission (Track A/B/C/D) across at least two modes (an active mode and an eclipse/quiet mode), including harness loss and margin. Size the source (array or RTG) and the battery, choose a bus voltage, and record all of it in your MDR.

25.24 † (Interleaved — Ch. 3, Ch. 23) Your power system masses $60\ \text{kg}$ (array $20$, battery $25$, electronics $15$). Explain how this mass propagates through the mass budget and the rocket equation. If a launcher delivers $2.4\%$ of its lift-off mass as payload, what lift-off mass does the power system's share alone imply?

25.25 (Interleaved — Ch. 24) The array converts $30\%$ of incident sunlight to electricity. Where does the other $70\%$ go, and which subsystem must handle it?

25.26 † (Interleaved — Ch. 20) An ion thruster needs $5\ \text{kW}$ of electrical power. Explain why solar-electric propulsion works well near Earth but an electric-propulsion mission to the outer planets may require a fission reactor.

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: sizing to end-of-life and worst case (not beginning-of-life square-on), explicit units, and a sanity check on every final number.