The nonrelativistic thrust for a particle beam is:
⎛
⎞
F = dm
( ) Eq. 19
⎠ ⎟ Vb −Vs
⎝ ⎜
dt
Where:
dm/dt = Participle Beam Mass Flow Rate (kg/s) Vb = Particle Beam Velocity (m/s) Vs = Velocity of Spacecraft (m/s)

The power to generate that particle beam, Pgb (W), is:
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⎛
⎞
1 2
dm
⎠ ⎟ Vb
⎝ ⎜
dt
Pgb =
Eq. 20 ηB Where:
Pgb = Power to generate the beam (W) dm/dt = Participle Beam Mass Flow Rate (kg/s) Vb = Particle Beam Velocity (m/s) ηB = Power efficiency of generating the beam (%)
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The energy cost for the beam is then the sum of the energy actually used in generating the beam times the acceleration duration (beam on) (Pgb x Ta) and the “energy cost equivalent” of the beam and the beam director (discussed in Sections 8.3.1 and 8.4.2).
For a given mission concept, all these factors can be adjusted to optimize trip time and power for a given destination, so a single number does not really express the capabilities and one must explore how the system scales to destinations of different distances at different power levels, generating a set of contours. This is the plan for
pg 50 of 69

Millis 2018 Grant NNX17AE81G_for_CR.docx presenting the (deterministic)

data for the concepts so they can be meaningfully compared (see Section 9 for sample plots).
The same physics discussed above applies to “sailbeam” concepts in which, rather than launching particles or photons to the spacecraft, we use a photon or particle beam to push some kind of “nano-craft” or “chipsat” or “smart pellet” which have some ability to correct their own trajectory and thus home-in on a traveling craft [100]. This approach allows the resulting “smart pellets” to be used as propulsive mass by a suitable receiver on an accelerating spacecraft.  This effectively creates a divergence-free beam, allowing for much longer acceleration times and hence lower power levels, balanced against the potentially higher cost (converted to energy equivalent) of the pellets and the added mass of a shock-absorbing system on the spacecraft (since each ‘pellet’ imparts an impulse to the spacecraft that is large compared to a photon or elementary particle).
The last class of concept of type RP-XM is the “pellet runway” [98 p.112]. The physics of this approach is somewhat different—in a pellet runway, rather than shooting “smart pellets” at the spacecraft, the pellets are launched before the spacecraft, and the spacecraft “runs over” the pellets as it accelerates. While this is purely type RP-XM by the nomenclature, the analysis is a hybrid approach—the launching of the pellets (and the energy equivalent of that), is computed using the equations of this section. However, the pellets, containing both the energy and the reaction mass, require some modification of the ‘propeller equations’ discussed under type IP-XM in the following section. For the majority of the trajectory in which the speed at which the spacecraft encounters the pellets, Vip (m/s) is large compared to the increase in velocity provided by the ‘jet’, the thrust is controlled by the specific energy of the pellets, Esp, and their thrust conversion efficiency η (as with type IP-OM), and one can approximate the thrust as:
⎛
⎞
F ≈dm
2 −Vip ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Eq. 21
⎠ ⎟ 2Espη+Vip
⎝ ⎜
dt Where:
dm/dt = Participle Beam Mass Flow Rate into the spacecraft (kg/s) Esp = Pellet Specific Energy, where the pellets are both a reaction mass and carry energy (J/kg) η = Thrust conversion efficiency (%)