Chapter 20 — Key Takeaways (Electric Propulsion)

A one-page reference. Reread this before an exam, or before you choose a propulsion system for a mission.

The one idea

Electric propulsion decouples the energy source (electricity) from the reaction mass (inert gas). A chemical rocket's propellant is both fuel and reaction mass, so chemistry caps $v_e$ near 4.5 km/s. Once you power the propellant with a separate electricity supply, there is no chemical ceiling on $v_e$ — only a power limit. High $I_{sp}$ follows; tiny thrust is the price.

The equations that run the chapter

Equation Use it to find Symbols (units)
$F = \dot m\, v_e$ thrust from mass flow and exhaust velocity $F$ (N), $\dot m$ (kg/s), $v_e$ (m/s)
$F = \dfrac{2\eta P}{v_e}$ thrust from electrical power and $v_e$ $P$ (W) input power, $\eta$ efficiency (0–1)
$\dot m = \dfrac{F}{v_e}$ propellant mass flow —
$\dfrac{P}{F} = \dfrac{v_e}{2\eta}$ power-to-thrust ratio (W/N) ~25 kW/N for an ion engine
$v_e = \sqrt{\dfrac{2qV}{m}}$ ion exit speed from accelerating voltage $q$ (C), $V$ (V), $m$ (kg/ion)
$v_e = I_{sp}\,g_0$ exhaust velocity from specific impulse $g_0 = 9.81\ \text{m/s}^2$
$\Delta v_{\text{spiral}} = \lvert v_1 - v_2\rvert$ coplanar low-thrust spiral delta-v $v=\sqrt{\mu/r}$ (circular speed)

The master trade (memorize the direction)

At fixed power, thrust and specific impulse pull in opposite directions: $F \propto 1/v_e$. Higher $I_{sp}$ → less thrust → longer burns → more power-system mass per newton. There is an optimum $I_{sp}$ for each mission; for most it lands where real thrusters live (1,500–4,000 s).

The two workhorses

Gridded ion engine Hall thruster
Acceleration electrostatic, through grids crossed E×B fields in a neutral plasma
Limit space-charge (Child–Langmuir) → low thrust none of that → higher thrust density
$I_{sp}$ ~3,000–4,000 s (higher) ~1,500–2,000 s (lower)
Thrust/power ~40 mN/kW ~60 mN/kW (more)
Needs ionizer + 2 grids + neutralizer anode + magnetic circuit + cathode
Reach for it when propellant mass is everything, time is plentiful (deep space) you want speed: orbit-raising, station-keeping
Example NSTAR (Dawn): 2.3 kW, 92 mN, ~3,100 s SPT-100: 1.35 kW, 83 mN, ~1,600 s

The rest of the family

  • PPT (pulsed plasma): ablates solid Teflon; micro-impulse bits for small-sat attitude control. Simple, tiny.
  • MPD: $\mathbf{J}\times\mathbf{B}$ acceleration; whole newtons possible, but needs 100s of kW–MW. Waiting on power.
  • VASIMR: RF-heated plasma + magnetic nozzle; variable $I_{sp}$. Demonstrated on the ground (~200 kW); not flown. Waiting on power.

Why it cannot launch (numbers)

To lift 1,200 kg off Earth needs $F > mg = 11{,}800\ \text{N}$. One ion engine ≈ 0.092 N → ~128,000 engines, ~294 MW, and ~3,000 t of solar array. $T/W \ll 1$: physically hopeless. Space only.

Chemical vs electric — the payoff (Ch. 3 rocket equation)

Task Chemical ($I_{sp}=320$ s) Electric ($I_{sp}=3{,}000$ s)
5 km/s on a 1,000 kg (dry) craft ~3,920 kg propellant ~185 kg propellant (~21× less)
LEO→GEO for a 1,200 kg sat Hohmann 3.85 km/s → ~2,900 kg spiral 4.59 km/s → ~356 kg (~8× less), but ~8 months

Spiral vs Hohmann (LEO 6,778 km → GEO 42,164 km)

  • $v_\text{LEO} = 7.669$ km/s, $v_\text{GEO} = 3.075$ km/s → spiral $= 4.59$ km/s.
  • Hohmann $= 3.85$ km/s (two burns, ~2.40 and ~1.46 km/s). The spiral costs ~19% more delta-v — but far less propellant.

Numbers worth memorizing

  • Electric $I_{sp}$: 1,500–4,000+ s; thrust: tens–hundreds of milli-newtons.
  • Power-to-thrust: ~25 kW per newton (ion). Ion mass flow: ~mg/s.
  • LEO→GEO spiral ≈ 4.6 km/s, over months; GEO station-keeping ≈ 50 m/s/yr.
  • Standard propellant: xenon (krypton/argon when cost dominates, e.g. Starlink).

Common pitfalls

Pitfall Reality
"More efficient, so use it to launch." Efficiency ≠ thrust; $T/W \ll 1$. It cannot lift itself.
"Higher $I_{sp}$ is always better." It lowers thrust and needs more power/time; there's an optimum.
"The spiral saves delta-v." It costs more delta-v than Hohmann; it saves propellant (via high $I_{sp}$).
"Hall is just a worse ion engine." It's a deliberate trade: less $I_{sp}$, more thrust per watt.
"VASIMR will get us to Mars in weeks." Check the assumed power source — it doesn't exist yet.

Mission / astrotools additions this chapter

  • MDR: decide whether your mission flies electric (strong fit: GEO orbit-raising + station-keeping, deep-space high-$\Delta v$; poor fit: landing, fast maneuvers).
  • propulsion.py: ep_thrust(power, isp, eta), ep_mass_flow(thrust, isp), ep_burn_time(prop, thrust, isp).