Exercises: Nuclear and Advanced Propulsion

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, $c = 2.998\times10^{8}\ \text{m/s}$, $R_u = 8.314\ \text{J/(mol·K)}$, solar constant $S = 1{,}361\ \text{W/m}^2$, and the Sun's gravity at 1 AU, $g_{\odot} = 5.9\times10^{-3}\ \text{m/s}^2$. 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. Because so many of this chapter's concepts are experimental or theoretical, several problems ask you to separate demonstrated technology from speculation and to flag which numbers are ideal versus achieved — that judgment is graded as much as the arithmetic. 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: convert and compute (⭐)

21.1 † A nuclear thermal rocket heats hydrogen ($\mathcal{M} = 0.002\ \text{kg/mol}$, $\gamma = 1.4$) to $T_c = 2{,}500\ \text{K}$. Estimate the ideal exhaust velocity $v_e$ and specific impulse $I_{sp}$. Is your answer an upper bound or a lower bound on what a real engine would achieve? Say why.

21.2 Compute the solar radiation pressure at 1 AU on (a) a perfectly absorbing surface and (b) a perfectly reflecting surface, each per square meter facing the Sun. Why does the reflector feel more?

21.3 † A square solar sail $20\ \text{m}$ on a side (area $400\ \text{m}^2$) is a perfect reflector facing the Sun at 1 AU. What force does sunlight exert on it? Express the answer in millinewtons.

21.4 An ion thruster on a nuclear-electric vehicle runs at $I_{sp} = 4{,}000\ \text{s}$. What is its effective exhaust velocity in m/s?

21.5 † Nuclear fission releases about $8\times10^{13}\ \text{J/kg}$; chemical combustion about $1.3\times10^{7}\ \text{J/kg}$. How many times more energy per kilogram does fission provide? Why does this ratio not translate into the same ratio of specific impulses for a nuclear-thermal vs. chemical rocket?

21.6 A nuclear-electric system delivers $P_{\text{elec}} = 500\ \text{kW}$ to thrusters at $I_{sp} = 3{,}000\ \text{s}$ with efficiency $\eta = 0.5$. Using $F = 2\eta P/v_e$, find the thrust.

Part B — The physics (⭐⭐)

21.7 † Starting from the fact that a photon of energy $E$ carries momentum $p = E/c$, derive the force on a perfectly reflecting sail illuminated by a beam of power $P$. Show clearly where the factor of 2 comes from.

21.8 Two propellants are heated to the same chamber temperature, one with molar mass $\mathcal{M} = 2$ and one with $\mathcal{M} = 18$. Using $v_e \propto \sqrt{T_c/\mathcal{M}}$, by what factor do their exhaust velocities differ? Explain in one sentence why this is the core reason a nuclear thermal rocket outperforms a chemical one despite a lower temperature.

21.9 † A solar sail has a characteristic acceleration (facing the Sun at 1 AU) of $a_c = 0.5\ \text{mm/s}^2$. Compute its lightness number $\beta$. Would this sail be able to "hover" against solar gravity? Explain what $\beta$ tells you.

21.10 Derive the critical areal density $\sigma$ (in $\text{g/m}^2$) at which a perfectly reflecting sail reaches $\beta = 1$ at 1 AU. Use $2S/c = 9.08\times10^{-6}\ \text{Pa}$ and $g_{\odot} = 5.9\times10^{-3}\ \text{m/s}^2$. Compare your answer to the areal density of household aluminum foil (~$40\ \text{g/m}^2$) and comment on the engineering challenge.

Part C — Implement it in Python (⭐⭐)

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

21.11 † Write ideal_ve(gamma, Tc, M) returning $\sqrt{\dfrac{2\gamma}{\gamma-1}\dfrac{R_u T_c}{\mathcal{M}}}$ with $R_u = 8.314$. Trace it for ideal_ve(1.4, 2700, 0.002).

21.12 Write sail_force(S, A, reflect=True) returning the radiation-pressure force on a sail of area A in flux S, doubling for a reflector. Use $c = 2.998\times10^{8}$. Trace it for sail_force(1361, 1e4).

21.13 † Write thrust_from_power(P, Isp, eff) returning $2\,\text{eff}\,P/(I_{sp}\,g_0)$ with $g_0 = 9.80665$. Trace it for thrust_from_power(1e6, 5000, 0.6).

Part D — Find the error (⭐⭐)

21.14 † A popular article states: "Solar sails work because the solar wind — the stream of particles blowing off the Sun — pushes on the sail like wind in a sailboat's canvas." Identify the physical error and name what actually pushes the sail. Roughly how much stronger is the real effect than the one the article names?

21.15 A student writes: "Nuclear thermal rockets get a higher $I_{sp}$ than chemical rockets because the reactor makes the gas hotter than combustion can." Both clauses contain an error. Correct them.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

21.16 † A crewed transfer stage must supply $\Delta v = 6\ \text{km/s}$. Compare the propellant mass fraction $1 - e^{-\Delta v/v_e}$ for (a) a chemical engine at $I_{sp} = 450\ \text{s}$ and (b) a nuclear thermal engine at $I_{sp} = 850\ \text{s}$. State the practical consequence of the difference for a heavy crewed vehicle.

21.17 (Mission project) Write the one-paragraph propulsion down-select for your chosen mission (Track A/B/C/D), as described in the chapter's Mission Design Checkpoint: name the propulsion you are actually using and state one requirement change that would bring an engine from this chapter into the trade. Save it to your MDR.

21.18 † Size a nuclear-electric cargo tug. It has $P_{\text{elec}} = 2\ \text{MW}$, thruster efficiency $\eta = 0.55$, and $I_{sp} = 6{,}000\ \text{s}$. (a) Find the thrust. (b) With a power-system specific mass $\alpha = 20\ \text{kg/kW}$, find the mass of the reactor-converter-radiator assembly. (c) If the fully loaded tug masses $60\ \text{t}$, find its acceleration. (d) Comment on what dominates the vehicle's mass.

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

21.19 † Why can't you just launch a Project-Orion vehicle from the ground to cheaply lift thousands of tonnes to orbit, since the physics is sound? Give the two decisive reasons, and state clearly which of them is a physics problem and which is not.

21.20 Back of the envelope. Suppose humanity could produce antimatter at $10\ \text{nanograms}$ per year. How long would it take to accumulate $1\ \text{gram}$? Compare your answer to the age of the human species (~$3\times10^{5}$ years) and comment on what this says about antimatter propulsion's timeline.

21.21 † A solar sail with characteristic acceleration $0.5\ \text{mm/s}^2$ is (idealize) held facing the Sun for a decade. What delta-v would it accumulate? Why is this an overestimate of the real gain on an outbound cruise? Compare the ideal figure to the delta-v of a good chemical stage.

Part G — Interleaved & synthesis (Chapters 3, 16, 20) (⭐⭐ / ⭐⭐⭐)

21.22 † (Ch. 3) A nuclear thermal stage has $I_{sp} = 850\ \text{s}$ ($v_e = 8.34\ \text{km/s}$) and a mass ratio of $2.0$. What delta-v does it deliver? How does this compare to a chemical stage ($v_e = 4.4\ \text{km/s}$) with the same mass ratio?

21.23 (Ch. 20) An ion thruster's very high $I_{sp}$ is an asset for an asteroid cargo mission but a liability for a fast crewed Mars mission. Explain both, using the ideas of thrust, burn time, and efficiency.

21.24 (Ch. 16) Using $F = 2\eta P/v_e$, explain why doubling a nuclear-electric system's $I_{sp}$ at fixed power halves its thrust. What does halved thrust do to the trip time of a low-thrust spiral, all else equal?

21.25 † (Ch. 3, Ch. 20) Someone claims "a solar sail has infinite specific impulse." In what sense is that literally true, and in what sense is it a meaningless statement? Tie your answer to the definition of $I_{sp}$ and to why the rocket equation does not bind a sail.

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, an ideal-versus-achieved label on every performance number, a sanity check, and — for this chapter especially — a clear statement of whether the technology in question has been demonstrated, tested, or only proposed.