  The PSI low-energy $\mu$SR spectrometer is an instrument dedicated to muon
spin rotation and relaxation measurements. Knowledge of the muon beam
parameters such as spatial, kinetic energy and arrival-time distributions at
the sample position are important ingredients to analyze the $\mu$SR spectra.
We present here the measured energy losses in the thin carbon foil of the muon
start detector deduced from time-of-flight measurements. Muonium formation in
the thin carbon foil (10 nm thickness) of the muon start detector also affect
the measurable decay asymmetry and therefore need to be accounted for. Muonium
formation and energy losses in the start detector, whose relevance increase
with decreasing muon implantation energy ($<10$ keV), have been implemented in
Geant4 Monte Carlo simulation to reproduce the measured time-of-flight spectra.
Simulated and measured time-of-flight and beam spot agrees only if a small
fraction of so called "unmoderated" muons which contaminate the mono-energetic
muon beam of the $\mu$SR spectrometer is introduced. Moreover the sensitivity
of the beam size and related upstream-downstream asymmetry for a specially
shaped "nose" sample plate has been studied for various beam line settings,
which is of relevance for the study of thermal muonium emission into vacuum
from mesoporous silica at cryogenic temperatures.
