# If specific impulse is directly related to exhaust velocity, would a ion post-accelerator improve the Isp of a propulsion system?

I understand that ion- and hall thrusters resemble a particle accelerator, and that led me to wonder if a post-accelerator (particle accelerator) used to boost the velocity of the ions would improve the Isp.

I realize that a particle accelerator would likely have major TWR concerns if used as propulsion, but would it really be a very efficient form of propulsion in terms of propellant mass, or is there something else at play?

Is there some way that post-acceleration could become a practical way to improe ion propulsion Isp>

• To clarify, I'm wondering if a linear particle accelerator would work as propellant efficient propulsion, able to accelerate particles up to high fractions of light speed to use as exhaust. – Nick van der Kroon Jan 14 '19 at 9:50
• Welcome to Space! I like your question very much, and I've made some adjustments to the wording. Feel free to edit further or roll back if this doesn't reflect what you'd like to ask. – uhoh Jan 14 '19 at 9:55
• The piece of equipment on Earth that most resembles ion based propulsion in space would be an ion source or ion gun. The former would be used to inject charged particles into an accelerator, the latter would be used for surface modification or ion implanting. But your question about post-acceleration of ions for additional Isp is a really cool idea! – uhoh Jan 14 '19 at 9:56
• Hm, not exactly what I was going for, I was more wondering about a ordinary linear or cyclical accelerator being used as propulsion, but your earlier comment sure got me thinking. Normal ion thrusters give the ions their energy and velocity from a single anode/cathode interaction, right? Adding on a linear electromagnetic accelerator piece onto that sounds pretty interesting. – Nick van der Kroon Jan 14 '19 at 9:57
• If you have a look at those ordinary linear accelerators or cyclotrons, there is always an ion source feeding it, usually with energy of a few keV or tens of keV at least. random links: 1, 2 – uhoh Jan 14 '19 at 10:03

Since specific impulse and exhaust velocity are directly related via $$I_{SP}=\frac{V_e}{g_0}$$ anything that increases the exhaust velocity necessarily increases the specific impulse.

The issue is: do you gain anything from it? That depends on what "gain" you're looking for.

Rocket engines of any type are momentum devices. The impulse imparted to the vehicle by a small bit of exhaust being expelled (call its mass $$\Delta m$$) is proportional to the momentum of that small bit, and that bit of momentum $$\Delta p$$ is given by $$\Delta p = \Delta m \times V_e$$ where $$V_e$$ is the exhaust velocity.

But to get to that $$V_e$$ the bit of exhaust must be given an amount of kinetic energy $$E_k$$ that is proportional to the square of its velocity: $$E_k = \Delta m \times \frac{{V_e}^2}{2}$$ When you factor in the rate at which you're expelling that exhaust, i.e. the propulsion system's mass flow rate $$\dot{m}$$, you get a required power (energy per time): $$P = \dot{m} \times \frac{{V_e}^2}{2}$$ In a chemical rocket engine the chemical reaction in the combustion chamber supplies that energy. But in an ion engine, or other type of electric propulsion engine, that energy must be supplied by an electric power source (we'll ignore the power required to ionize the propellant, though that's not negligible when designing the power supply), and therein lies the rub: the mass of an electric power supply (of a given type) increases with the power it must provide.

You can increase the specific impulse of an ion engine simply by increasing the voltage across its grids, though you might have to increase the separation between the grids to prevent arcing. If you maintain the same propellant mass flow rate, voila! the thrust increases, since $$F = \dot{m} \times V_e$$ where $$F$$ is the thrust. But now the power required from the electric power supply went up as $${V_e}^2$$, so you added a non-trivial mass.

Whether you get a net increase in acceleration depends on the system scaling before uprating $$V_e$$. If the system mass before the upgrade was dominated by the sum of engine mass, tankage mass, gimbal mass, etc. (non-power-supply stuff), then the increase in power supply mass might not overwhelm the increase in thrust, and indeed you get an increase in acceleration. But if the power supply mass began as a large fraction of the system mass, the relative increase in the system mass as a result of the increase in the power supply mass might be larger than the relative increase in the thrust, and the acceleration you get actually decreases.

Back to what is meant by "gain".

Say you're trying to get a spacecraft's wet mass down to fit on a specific launch vehicle, so you're trying to minimize the propellant mass needed for this specific mission's well-defined and large $$\Delta V$$. Then you might put up with a decreased acceleration (and likely a longer trip time) to get the decrease in propellant mass from the increased $$V_e$$. But if trip time is important, then higher acceleration gets more priority. Optimizing electric propulsion systems is an exercise in balancing such factors. This includes such things as the choice of the power supply type: solar? nuclear? That specific trade can go various ways depending on such things as the heliocentric distances over which the system must operate. Mission design engineers must weigh all those factors. The optimum $$I_{SP}$$ is one of the parameters that comes out of such analyses. More is not always better!

I know, I know, there are many people who would see that last statement, widen their eyes, tilt their heads back a bit, point a shaking index finger at me, and exclaim, "BLASPHEMY!!!"

• @uhoh When power is the limiting factor in a propulsion system's performance, then energy efficiency is indeed an important consideration. It's really easy to supply 100 mg/s of propellant. It's relatively easy to supply 3000 V to an ion engine's grids (spacecraft have successfully operated instruments at thousands of V). It's not easy to supply the 220 kW beam power that requires. And if the engine is inefficient, that multiplies the power requirement. A 70% efficient engine would take 314 kW; at 50% that's 440 kW, and the extra 126 kW adds a lot of power supply mass that slows you down. – Tom Spilker Jan 17 '19 at 18:54
• @uhoh Yes, I disagree. N/kW is force per power. Force (thrust) is mdot V. Beam (or exhaust plume) power is (mdot V^2)/2. Force over power is then (mdot V)/[(mdot v^2)/2] = 2/V. To increase force over power, this says you should slow V down (decrease specific impulse). But then you have to pump lots of mdot at it to get the thrust you need, and that's obviously not an optimal propulsion system: the propellant mass goes way up. There's no single "king" metric for electric propulsion systems/missions. Optimization involves several metrics, and each mission profile yields different values. – Tom Spilker Jan 17 '19 at 19:37
• @uhoh And I am saying that "energy efficiency is a key propulsion metric". I consider a parameter to be a "key propulsion metric" when it significantly influences the optimization of a propulsion system design. I've been involved in multiple such design trades; the key metrics involved were: 1) thrust, 2) specific impulse, 3) thruster power efficiency, 4) system inert mass fraction, 5) electric power system power efficiency, and 6) specific power of the electric power source. Each takes on varying levels of importance depending on the mission profile. No one or two of them reign supreme... – Tom Spilker Jan 17 '19 at 23:03
• If $g$ is Earth acceleration, then the first equation is wrong. – Everyday Astronaut Jul 15 '19 at 9:32
• @EverydayAstronaut Yup, should be division, not multiplication. – Christopher James Huff Jul 16 '19 at 3:05

No.

An ion thruster works like this:

1. atoms are converted to ions.
2. the ions are accelerated.
3. when the ions leave the engine, loose electrons are sprayed into the exhaust to get neutral atoms again (to prevent the ions being attracted back to the engine, losing Isp.

A particle accelerator would have to be installed after 3, and needs ions to work.

• You could easily (not accounting for the mechanics and mass) install it between 2 and 3 though. – asdfex Jan 14 '19 at 11:31
• @Uwe The answer is about an ion thruster that doesn't have a a combustion chamber. But - the ion thruster does exactly the same: In the first step it generates a hot, ionized but neutral gas! The voltage is used to separate electrons from ions resulting in a net momentum. – asdfex Jan 14 '19 at 14:28

Yes.

Just like on the Earth, an ion post-accelerator would accelerate the ions from an ion source to much higher energy, and while mostly non-relativistic (up to say several tens of MeV per AMU) the momentum increases as the square of the velocity.

So increasing accelerated energy a factor of $$10$$ from say 50 keV to 500 keV without loosing current multiplies your $$I_{SP}$$ by a factor a factor of $$\sqrt{10}$$.

$$v = \sqrt{2mE}$$

However, the practical problems will stop you in your tracks before you even get to the bottom of the back of your first envelope.

Particle accelerators are fed by ion sources that use the same gizmos as ion propulsion engines use, DC and/or RF excited plasmas plus some kind of grid or orifice extraction system.

However, ignore the fact that traditional particle accelerators are extremely heavy and use huge amounts of electrical power, they required ions injected into a very small bit of phase space. You may have a several millimeter diameter hole that will only accept ions within a few milliradian cone of divergence and a few parts per thousand spread in injected energy (a few eV at a few keV).

@JohnCuster reminds us that many accelerators are RF based (linacs, cyclotrons, etc.) and so will have only tiny windows in time where they can accept ions, perhaps a few percent of the time at most. You'll either need a separate buncher or live with a huge loss in beam.

So while the ion sources look a little bit like ion propulsion systems from a functional viewpoint, they make tiny, narrow, and therefore low current beams in order to fit into the accelerators acceptance.

This is the Tyranny of the Particle Accelerator, analogous to the Tyranny of the Rocket Equation. There's no escaping phase space.

There may some day be some very unusual method for ion acceleration that could do post-acceleration on a large diameter ion propulsion engine's output, but for right now, even though in principle it's the right idea, there's no way to do this practically.

• To your last paragraph: This exists - Just make the ion thruster larger and operate it at higher voltage. Apart from its weight and technical problems there is not much that hinders you to do that. – asdfex Jan 14 '19 at 11:34
• Apart from weight and technical problems, we could build a death star, or really, anything you'd like. – Tristan Jan 14 '19 at 14:35
• At non-relativistic velocities momentum increases linearly with velocity — energy increases with the square of velocity. At relativistic velocities it gets more complicated, as that pesky SQRT[1-(v^2/c^2)] term gets non-negligible. – Tom Spilker Jan 14 '19 at 20:04
• And if the post-accel is a linac, don't forget the vary small percentage of beam time that actually gets accelerated (perhaps a percent or so). Even with a beam buncher on the input side, most of your ions from the source won't actually make it through the linac. – Jon Custer Jan 16 '19 at 18:51
• In a linac the ions have to hit at just the right time, for either an rf section or an accelerating gap. If they hit right, the ‘surf’ the rf wave, or get to zip across the gap. At the wrong time, they don’t stay up with the lucky ones (so their timing gets off more and more down the line), or they may even get de-accelerated. Depending on design, on the order of 1% of a dc beam would get accelerated through a multi-stage linac. – Jon Custer Jan 16 '19 at 19:13