  The precise measurement of the top quark mass, which is a fundamental SM
parameter, constitutes one of the main goals of the LHC top physics program.
One approach to measure this quantity uses the $\rho_\mathrm{s}$ distribution,
an observable depending on the invariant mass of the $t\bar{t}j$ system. To
fully exploit the experimental accuracy achievable in measuring top quark
production cross sections at the LHC, the theory uncertainties associated to
these measurements need to be well under control. To this end we present a
study of the effect of varying the theoretical input parameters in the
calculation of differential cross sections of the $t\bar{t}j$ process. Thereby
we studied the influence of the jet reconstruction procedure, as well as the
effect of various renormalization and factorization scale definitions and
different PDF sets. The variation of the $R$ parameter in the jet
reconstruction algorithm was found to have negligible influence on the scale
variation uncertainty. A strong reduction of scale uncertainties and a better
behaviour of the NLO/LO ratios using selected dynamical scales instead of a
static one in the high energy tails of differential distributions was observed.
This is particularly interesting in the context of the top quark mass
measurements through the $\rho_\mathrm{s}$ distribution, in which the
perturbative stability can be improved by applying the proposed dynamical scale
definition.
