Theoretical rocket engine based on plasma shock waves particle acceleration Credits for the article - https://www.quantamagazine.org/cosmic-map-of-ultrahigh-energy-particles-points-to-long-hidden-treasures-20210427

*

*Can this mechanism be used to speed up gas particles to super-high speeds and to create an engine that can eject mass at close to luminal speed? (Engine performance is dependent on the speed of the ejected mass).


*How to calculate the drag force for an engine that lets say eject gas at 99% of speed of light? And how much mass needs to be ejected in that case to be able to launch a rocket into orbit?

 A: Note that,

*

*The energy gain per cycle is $\Delta E\propto\beta_s$ with $\beta_s=v_s/c$ where $v_s$ is the velocity of the shock.

*The timescale for acceleration is $t_\text{acc}\propto\kappa/\beta_s^2$ where $\kappa$ is the diffusion coefficient.

*The timescale for escape1 is $t_\text{esc}\propto t_\text{acc}\beta_s$
The primary source of galactic cosmic rays are supernova remnants (SNR) which are massive (parsec length scales and $\sim10^{51}$ erg energy scales), fast ($\beta_s\sim0.01$) and old (>10k years). In these sites, the acceleration time scale is roughly a month while escape timescales are hundreds to thousands of years (hence SNR sources being old).
So since there's no way we know of to produce and contain a moving shock wave, nor can we wait the timescales needed to accelerate a particle to its escape from the accelerator, this is completely impractical.


1. Once a particle gains enough energy such that its gyroradius exceeds the Larmor radius of the accelerator, it cannot be re-accelerated & escapes; cf. this comment.
A: Just to add on to Kyle's answer, diffusive shock acceleration (DSA) is now thought to result from self-generated electromagnetic waves/fluctuations/structures [e.g., see Turner et al., 2018; Wilson et al., 2016].  A result of this is that the wavelengths/scale sizes of the resultant electromagnetic waves/fluctuations/structures tends to increase as the energy of the accelerated particles increases (e.g., see discussion at https://physics.stackexchange.com/a/618127/59023).  The initial scale sizes are already rather large in typical space plasmas, much smaller in lab plasmas where all scale sizes are much smaller.  Even so, the scale sizes would quickly exceed that of the physical container you are trying to generate such high energy particles.
Currently, the best method we have for artificially generating a collisionless shock wave is through laser ablation [e.g., see Heuer et al., 2020].  This tends to require a ton of power/energy to generate the laser pulses necessary to ablate enough material to generate a shock.  In the lab setting, the shocks are not tremendously high Mach number, like that of a supernova remnant (SNR).  Generally, as Kyle points out, high Mach numbers tend to generate higher energy particles.

How to calculate the drag force for an engine that lets say eject gas at 99% of speed of light? And how much mass needs to be ejected in that case to be able to launch a rocket into orbit?

Based on how much energy is necessary to generate a collisionless shock wave, this is a non-starter as the input to just initiate the shock is greater than any focused output.  Plus, huge capacitor banks necessary for the pulsed laser firing would be obnoxiously heavy, which would also be a non-starter (i.e., more mass = more money and fewer options for launch trajectories).
Finally, any laser ablation shock would "hit the walls" of the chamber before the particles would have enough time to get energized to sufficient energy to do anything useful like accelerating a spacecraft.  There are much more efficient options like ion thrusters that are already available and have been used on missions like Dawn.
