  Recent observations made with Advanced LIGO and Advanced Virgo have initiated
the era of gravitational-wave astronomy. The number of events detected by these
"2nd Generation" (2G) ground-based observatories is partially limited by noise
arising from temperature-induced position fluctuations of the test mass mirror
surfaces used for probing spacetime dynamics. The design of next-generation
gravitational-wave observatories addresses this limitation by using
cryogenically cooled test masses; current approaches for continuously removing
heat (resulting from absorbed laser light) rely on heat extraction via
black-body radiation or conduction through suspension fibres. As a
complementing approach for extracting heat during observational runs, we
investigate cooling via helium gas impinging on the test mass in free molecular
flow. We establish a relation between cooling power and corresponding
displacement noise, based on analytical models, which we compare to numerical
simulations. Applying this theoretical framework with regard to the conceptual
design of the Einstein Telescope (ET), we find a cooling power of 10 mW at 18 K
for a gas pressure that exceeds the ET design strain noise goal by at most a
factor of $\sim 3$ in the signal frequency band from 3 to 11 Hz. A cooling
power of 100 mW at 18 K corresponds to a gas pressure that exceeds the ET
design strain noise goal by at most a factor of $\sim 11$ in the band from 1 to
28 Hz.
