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With the new particle acceleration technology of plasma accelerator, electrons or protons can be accelerated to relativistic speed in a device much smaller than current Linac (According to wikipedia, electrons are accelerated to the same energy as Linac in just length of 3.3cm, instead of 64m). My proposal is that if the electron is accelerated to relativistic speed, the momentum of the particle increases exponentially without bound. By momentum conservation, the spacecraft can then be propelled. Would this be feasible?

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    $\begingroup$ For spacecraft propulsion, you can not accelerate electrons or protons alone. You have to accelerate them both to avoid accumulating charge on the spacecraft. But there is a huge difference in mass between electrons and protons, the thrust caused by the electrons alone is very small. $\endgroup$ – Uwe Mar 31 '20 at 16:05
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    $\begingroup$ yes and that's just going to form hydrogen really. $\endgroup$ – Topcode Mar 31 '20 at 16:16
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    $\begingroup$ What is the energy efficiency of that accelerator? And the particle velocities may be high, but what's the overall rate at which it accelerates them? It's no good if it turns most of the energy input into waste heat, or if it has beam current limitations that result in uselessly low thrust. $\endgroup$ – Christopher James Huff Mar 31 '20 at 17:23
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    $\begingroup$ What you're describing is an ion drive. I don't think that a plasma accelerator makes any huge difference: it might make it more practical but it depends a lot on how enegy-efficient the accelerator is. $\endgroup$ – tfb Mar 31 '20 at 17:36
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    $\begingroup$ Does this plasma accelerator move electrons only or is it useful for protons too? Is it possible to accelerate both electrons and protons? $\endgroup$ – Uwe Mar 31 '20 at 17:50
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@tfb is correct: this is another form of ion propulsion, or generally, electric propulsion. A good general reference for electric propulsion is Fundamentals of Electric Propulsion: Ion and Hall Thrusters from JPL's DESCANSO series.

The problem with electric propulsion using ultra-high exhaust velocities is the power required to drive the exhaust beam. If you assume a 100% efficient engine, then 100% of the energy into the system comes out as energy of the exhaust beam. This is never the case, if for no other reason than it takes energy to ionize the propellant in the first place. Typical electric propulsion systems are far less efficient, 50% (for really good laboratory experimental systems, not for off-the-shelf thrusters) or less, sometimes much less.

Assume you have a 100% efficient system, an exhaust velocity of c/2 — relativistic effects are there but not dominant, so I'm not going to include them — and you have a 1 GW power supply, equivalent to a large ground-based power station and many orders of magnitude larger than any space power system to date. Then $Pjet$ is given by

$$Pjet=\frac{\dot{m}}{2} V_e^2$$

Inverting,

$$\dot{m}=\frac{2Pjet}{V_e^2}$$

and you get a propellant mass flow rate of $~8.9\times 10^{-8}$ kg/s, producing a thrust of $$F=\dot{m}V_e$$ which is roughly 13 Newtons.

Now gigawatt power supplies don't weigh in at 100 or even 1000 kg; they're at least tens and probably hundreds of thousands of metric tons. Assuming $10^5$ metric tons ($10^8$ kg) and no mass in any other components, that 13 N of force produces an acceleration of ~$1.3\times10^{-7}\frac{m}{s^2}$. In a year at that acceleration you gain a whopping (well ... not really whopping) 4.2 m/s of delta-V. (Again, this doesn't include relativistic effects, but it's close enough to illustrate the point)

This is the trade with electric propulsion. If you go for extreme exhaust velocity (and thus specific impulse), then with practical power supplies you get extremely small accelerations. Yes, eventually you can reach very high velocities, but it takes half of forever to do it.

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