3.3.43 · D5Rocket Propulsion
Question bank — FEEP, MEMS thrusters — micro-propulsion
Before you start, recall the two load-bearing formulas from the parent topic note:
- Exhaust speed of one accelerated ion: — voltage controls this.
- Thrust of the whole beam: — beam current scales this.
Keep those two knobs — and — mentally separate. Most traps below come from confusing them.
Two more real-world limits the parent note only hints at — study the figures, then use the traps:
- Knudsen number (mean free path over channel size ): at micro-scale can grow large, so the gas stops behaving as a smooth continuum and enters transitional or free-molecule (rarefied) flow — the nozzle equations above quietly break down.
- Child–Langmuir (space-charge) limit: the beam's own charge repels the ions behind it, so the maximum current you can pull is capped by voltage and geometry — and are not fully free knobs.
- Beam divergence: real ions leave along a cone, not a straight line, so only the axial component of each ion's velocity produces useful thrust — a loss captured by a divergence half-angle.
True or false — justify
Every ion in a FEEP beam leaves with the same speed regardless of ion mass, as long as the voltage is fixed.
False — , so for fixed a heavier gives a slower exit speed; only the energy per charge is fixed, not the speed.
For a fixed accelerating voltage, a heavier ion gives lower specific impulse but higher thrust per amp of beam current.
True — (lower ), yet , so each ampere of current pushes harder. It is a genuine trade, not a flaw.
Doubling the beam current doubles the thrust but leaves the exhaust speed unchanged.
True — multiplies linearly but never appears in ; more ions per second, each still the same speed.
Doubling the accelerating voltage doubles the exhaust speed.
False — , so doubling multiplies by , not by .
A higher specific impulse always means the thruster is "better" for a CubeSat.
False — high saves propellant but usually means tiny thrust and high voltage; "better" depends on whether the mission needs fine attitude control, fast maneuvers, or fuel economy.
The Taylor cone forms only when the electric field is strong enough to balance surface tension.
True — below that threshold surface tension keeps the surface rounded; the sharp cone appears exactly where outward electrostatic pressure equals the inward pull of .
Making a MEMS nozzle smaller keeps the same expansion efficiency because the ideal-gas formula has no length scale in it.
False — the ideal formula is size-blind, but real micro-nozzles have small Reynolds number (viscous losses) and large Knudsen number (rarefaction), so wall friction and free-molecule effects dominate and efficiency drops.
A FEEP thruster can, in principle, run entirely on electrical power with essentially no propellant needed for a small .
True in spirit — because and m/s, a few m/s of needs sub-gram propellant; but it is never zero — momentum still requires ejected mass.
You can always raise beam current freely just by adding more emitter voltage.
False — the Child–Langmuir space-charge limit caps the extractable current for a given voltage and geometry; the ions already in flight repel those behind them, so cannot be pushed arbitrarily high.
All the kinetic energy of a FEEP ion beam turns into forward thrust.
False — ions leave along a divergence cone, so only the axial velocity component pushes the spacecraft; the off-axis part is wasted, reducing the effective thrust coefficient.
Spot the error
"To get more thrust from FEEP, just raise the voltage — thrust grows linearly with ."
Wrong scaling — , so thrust grows only as the square root of voltage; raising is the linear knob.
"Since , we should pick the heaviest possible ion to maximize thrust."
Half-right, half-wrong — heavy ions do raise thrust-per-amp, but they crater (hence and fuel economy). The choice is a balance, not "heaviest wins."
"The Reynolds number is large in a MEMS thruster because the gas moves very fast."
Error — also shrinks with the tiny channel length ; the small wins, so is small despite high speed, making viscosity dominant.
"Continuum nozzle equations apply just fine at the micro-scale as long as the gas is moving."
Error — when the Knudsen number becomes large, the gas is rarefied (transitional or free-molecule), and the continuum assumption behind the exhaust formula fails.
"Thrust is , and is the beam current, so ."
Error — is charge per second, not mass per second. Mass flow is , giving .
"A cold-gas MEMS thruster's thrust is just ; the pressure term is negligible always."
Error — the full expression is , where is exit pressure, ambient pressure, exit area. When under- or over-expanded, or in vacuum where , the pressure term is not automatically zero.
"Because energy is conserved as , all the electrical energy becomes useful thrust."
Error — energy conservation sets the speed, but useful thrust also loses to beam divergence (off-axis ions), non-ionized neutrals, and current losses; real efficiency is below the ideal.
"FEEP needs only a few volts since we're pushing tiny ions."
Error — reaching m/s from requires kilovolt-level ; small mass does not make the required voltage small.
"Since and are independent knobs, I can pick any voltage and any current at will."
Error — the Child–Langmuir limit couples them: for a fixed geometry the space-charge-limited current rises only as , so the maximum is set once is chosen.
Why questions
Why do small satellites deliberately choose micro-newton thrust instead of more powerful thrusters?
They need precision (attitude to arc-seconds) and fuel economy, not brute force; a firehose would overshoot every tiny correction and drain scarce propellant.
Why does FEEP give such fine, tunable thrust resolution?
Thrust is set by beam current, which can be dialed electronically down to nanoamps, so can be trimmed to fractions of a micro-newton.
Why does high exhaust velocity mean almost no propellant is consumed?
The rocket equation gives for small ; a huge in the denominator shrinks the required propellant mass to near nothing.
Why does the liquid metal in FEEP organize into a cone rather than staying a smooth blob?
The outward electrostatic pressure balances the inward surface-tension pull only along a specific cone geometry, so the surface self-selects the Taylor cone.
Why can voltage and current be tuned semi-independently in a FEEP thruster to control and thrust?
Voltage sets exhaust speed (hence ) while current sets how many ions leave per second (hence thrust) — but the Child–Langmuir limit caps the current available at each voltage, so the independence is not total.
Why is caesium or indium used in FEEP rather than a very light ion?
Heavy ions carry more momentum per charge, giving more thrust per amp and steadier low-thrust control, while still reaching high at kilovolt voltages.
Why does the enthalpy-to-kinetic-energy picture explain the MEMS cold-gas exhaust formula?
Stored thermal (pressure–temperature) energy of the gas is converted to directed motion in the nozzle; the bracket (with the specific-heat ratio) is exactly the fraction of enthalpy released as the gas falls from chamber pressure to exit pressure .
Why does beam divergence reduce real FEEP thrust below the ideal ?
Because thrust is a vector: only the component of each ion's velocity along the axis pushes forward, so a nonzero divergence half-angle means the off-axis components cancel and are wasted.
Edge cases
What happens to FEEP thrust if the beam current drops to zero while voltage stays high?
Thrust vanishes — when ; the field can still exist, but with no ions leaving there is no reaction force.
What is the exhaust speed if the accelerating voltage is zero?
Zero — ; with no potential drop, ions gain no kinetic energy and are not usefully expelled.
For an under-expanded nozzle (), what sign does the pressure term contribute?
Positive — exit pressure exceeds ambient, so the term adds to thrust; the flow still had pressure "left over" it could have expanded further.
For an over-expanded nozzle (), what sign does the pressure term contribute?
Negative — ambient pressure exceeds exit pressure, so the term subtracts from thrust; the gas was expanded too far and the outside pushes back on the exit plane.
In the deep vacuum of space, what does the MEMS thrust pressure term become?
With ambient pressure , the term becomes , a positive bonus contribution from residual exit pressure on top of .
As , does the propellant-fraction formula blow up or simplify?
It simplifies — for tiny , gives , a small, well-behaved amount, not a divergence.
In the limit of a very large channel (large ), what happens to the MEMS viscous- and rarefaction-loss problems?
Both fade — larger raises (less viscous drag) and lowers (back to continuum flow), so the loss is inherently a micro-scale penalty.
What limits how high you can push the accelerating voltage in a real FEEP thruster?
Practical breakdown, insulation, and power-supply limits cap ; you cannot raise (and ) without bound because arcing and available spacecraft power intervene.
As the divergence half-angle of a FEEP beam shrinks toward zero, what happens to the useful thrust?
It approaches the ideal — a perfectly collimated (zero-divergence) beam wastes no off-axis velocity, so the thrust coefficient approaches .
Recall Two-knob summary — can you state it without looking?
Which knob sets and which sets thrust? ::: Voltage sets exhaust speed and thus ; beam current sets thrust — but Child–Langmuir couples the maximum to . Why is small and large in a MEMS micro-nozzle? ::: The tiny length scale shrinks (viscosity rules) and inflates (gas rarefies), both hurting efficiency.