Exercises: Electric Propulsion

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, and Earth's $\mu = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$. For xenon, the ion charge is $q = 1.602\times10^{-19}\ \text{C}$ and mass $m = 2.18\times10^{-25}\ \text{kg}$. 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 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 (⭐)

20.1 † An ion engine has a specific impulse of $I_{sp} = 3{,}000\ \text{s}$. What is its effective exhaust velocity in km/s? About how many times the exhaust velocity of a good chemical engine ($v_e \approx 3.3\ \text{km/s}$) is that?

20.2 A Hall thruster expels xenon at $v_e = 16\ \text{km/s}$. What is its specific impulse in seconds?

20.3 † Using $F = 2\eta P/v_e$, find the thrust of a thruster that draws $P = 5\ \text{kW}$, runs at $v_e = 20\ \text{km/s}$, and has efficiency $\eta = 0.60$. Give the answer in milli-newtons.

20.4 A thruster produces $F = 0.15\ \text{N}$ at $v_e = 25\ \text{km/s}$. What is its propellant mass flow rate, in mg/s?

20.5 † Compute the power-to-thrust ratio $P/F = v_e/(2\eta)$ for a thruster with $I_{sp} = 2{,}000\ \text{s}$ and $\eta = 0.50$. Express it in kW per newton, and say in words what it means.

20.6 A xenon ion is accelerated through a potential difference of $V = 800\ \text{V}$. Using $v_e = \sqrt{2qV/m}$, find its exit velocity and the corresponding (ideal) specific impulse.

Part B — The power–thrust–Isp trade (⭐⭐)

20.7 † A thruster runs at fixed power and efficiency. If its designers double the exhaust velocity $v_e$ (by doubling the accelerating voltage), what happens to (a) the thrust and (b) the propellant mass flow rate? Justify each with the relevant equation.

20.8 A $10\ \text{kW}$ gridded ion engine runs at $I_{sp} = 3{,}500\ \text{s}$ with $\eta = 0.65$. Find its thrust (mN) and its xenon mass flow rate (mg/s).

20.9 † At a fixed $5\ \text{kW}$ and $\eta = 0.50$, compare the thrust of a gridded ion engine ($I_{sp} = 3{,}300\ \text{s}$) with that of a Hall thruster ($I_{sp} = 1{,}700\ \text{s}$). Which produces more thrust, and by roughly what factor? Which uses less propellant for a given delta-v, and why?

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.

20.10 † Write ep_thrust(power, isp, eta) returning $2\eta P/(I_{sp} g_0)$ in newtons (use $g_0 = 9.80665$). Trace it for ep_thrust(2300, 3100, 0.61), and report the result in mN.

20.11 Write ep_burn_time(prop_kg, thrust_n, isp_s) returning the seconds to expend prop_kg of propellant, using $\dot m = F/(I_{sp} g_0)$. Trace it for ep_burn_time(356, 0.28, 1800) and convert the result to days.

20.12 † Write power_to_thrust(isp, eta) returning $I_{sp} g_0 / (2\eta)$ in W/N. Trace it for power_to_thrust(3100, 0.61) and state the answer in kW/N.

Part D — Find the error (⭐⭐)

20.13 † A blog post argues: "Ion engines convert 60% of their electrical power into exhaust kinetic energy, while a rocket engine wastes most of its energy as heat. Since ion engines are more efficient, future launch vehicles should use them to reach orbit." Identify the flaw, in physical terms.

20.14 A student computes the LEO→GEO transfer delta-v for an electric thruster and gets a number smaller than the Hohmann transfer delta-v, concluding "the spiral is the more efficient path, so electric propulsion saves delta-v too." What did they misunderstand about low-thrust transfers, and which transfer actually has the smaller delta-v?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

20.15 † A spacecraft with a final (dry) mass of $m_f = 800\ \text{kg}$ must perform $\Delta v = 4\ \text{km/s}$ with a Hall thruster at $I_{sp} = 2{,}000\ \text{s}$. Find the xenon mass required. Then find it again for the same maneuver with a chemical engine at $I_{sp} = 320\ \text{s}$, and state the ratio.

20.16 (Mission project) For your chosen mission (Track A/B/C/D), decide whether electric propulsion is a good fit, name the thruster type and a rough $I_{sp}$ you would use, and — if electric — estimate the transfer or station-keeping time you are accepting. Record a paragraph in your MDR.

20.17 † A deep-space probe must supply $\Delta v = 8\ \text{km/s}$ and has a $500\ \text{kg}$ dry mass. You may pick either $I_{sp} = 2{,}000\ \text{s}$ (more thrust, faster) or $I_{sp} = 4{,}000\ \text{s}$ (less propellant, slower). Compute the propellant mass each requires. Then, given a fixed $5\ \text{kW}$, $\eta = 0.6$ power system, estimate each option's thrust and the burn time to expend its propellant. Discuss the trade.

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

20.18 † Why can't you just add more power to an ion engine until it has useful, chemical-scale thrust? Estimate the electrical power needed for an ion engine ($v_e = 30\ \text{km/s}$, $\eta = 0.6$) to produce $F = 100\ \text{N}$ — still a thousand times less than a small chemical engine. Comment on whether a spacecraft could carry the power system to supply it.

20.19 Why can't you just use an ion engine to land a spacecraft softly on the Moon? Frame your answer with thrust-to-weight: the Moon's surface gravity is $g_\text{Moon} \approx 1.62\ \text{m/s}^2$, and a lander masses $500\ \text{kg}$. Compare the required thrust to what an ion engine provides.

20.20 † Dawn carried $425\ \text{kg}$ of xenon and its NSTAR engine flowed roughly $3\ \text{mg/s}$ at full throttle. Estimate the total continuous-thrust time to expend all of it, in days and years. Comment on why Dawn's actual mission lasted even longer than this.

Part G — Interleaved & synthesis (Chapters 3, 6, 9, 10, 16)

20.21 † (Ch. 3) A maneuver needs $\Delta v = 6\ \text{km/s}$. Find the mass ratio and propellant fraction with a chemical engine ($v_e = 3.14\ \text{km/s}$, $I_{sp}=320$ s) and with an electric one ($v_e = 29.4\ \text{km/s}$, $I_{sp}=3000$ s). In one sentence, connect the difference to the tyranny of the rocket equation.

20.22 (Ch. 16) An ion engine flows $\dot m = 3\ \text{mg/s}$ at $v_e = 30\ \text{km/s}$; a Merlin flows $\dot m = 300\ \text{kg/s}$ at $v_e = 3\ \text{km/s}$. Compute each thrust from $F = \dot m v_e$. Which quantity — $\dot m$ or $v_e$ — accounts for the enormous thrust gap, and by what factor does it differ?

20.23 † (Ch. 6, Ch. 10) A satellite spirals with continuous low thrust from a $400\ \text{km}$ LEO ($r_1 = 6{,}778\ \text{km}$) to a circular orbit at $r_2 = 10{,}000\ \text{km}$. Using $v = \sqrt{\mu/r}$, compute the spiral delta-v $\lvert v_1 - v_2\rvert$. Is it larger or smaller than the equivalent Hohmann transfer would be, and why?

20.24 (Ch. 9) Name two features of geostationary orbit that make it the classic home of electric propulsion — one about station-keeping and one about orbit-raising. (One sentence each.)

20.25 † (Ch. 9, Ch. 12) A $3{,}000\ \text{kg}$ GEO satellite needs $50\ \text{m/s}$ of north–south station-keeping per year for a 15-year life. Compute the propellant mass with an electric thruster ($I_{sp} = 1{,}500\ \text{s}$) and with a chemical one ($I_{sp} = 300\ \text{s}$). How much propellant mass does electric station-keeping save over the life?

20.26 (Synthesis) Explain why raising a thruster's specific impulse does not always improve an electric-propulsion system, naming two costs of a higher $I_{sp}$ at fixed power. Tie your answer to the power-to-thrust ratio and to mission time.

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: a correct set-up of $m_0$ and $m_f$ or of $F = 2\eta P/v_e$, explicit units, and a sanity check on the final number (does the thrust come out in milli-newtons? does the burn take months?).