  A reduced-speed-of-light (RSOL) approximation is a useful technique for
magnetohydrodynamic (MHD)-particle-in-cell (PIC) simulations. With a RSOL, some
"in-code" speed-of-light $\tilde{c}$ is set to much lower values than the true
$c$, allowing simulations to take larger timesteps (which are restricted by the
Courant condition given the large CR speeds). However, due to the absence of a
well-formulated RSOL implementation from the literature, with naive
substitution of the true $c$ with a RSOL, the CR properties in MHD-PIC
simulations (e.g.\ CR energy or momentum density, gyro radius) vary
artificially with respect to each other and with respect to the converged
($\tilde{c} \rightarrow c$) solutions with different choices of a RSOL. Here,
we derive a new formulation of the MHD-PIC equations with a RSOL, and show that
(1) it guarantees all steady-state properties of the CR distribution function
and background plasma/MHD quantities are {\em independent} of the RSOL
$\tilde{c}$ even for $\tilde{c} \ll c$, (2) ensures that the simulation can
{\em simultaneously} represent the real physical values of CR number, mass,
momentum, and energy density, (3) retains the correct physical meaning of
various terms like the electric field, and (4) ensures the numerical timestep
for CRs can always be safely increased by a factor $\sim c/\tilde{c}$. This new
RSOL formulation should enable greater self-consistency and reduced CPU cost in
simulations of CR-MHD interactions.
