  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 space time 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 fibers. As a
complementing approach, we investigate cooling via helium gas impinging on the
test mass in free molecular flow. We present analytical models for cooling
power and related displacement noise, validated by comparison 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 increases the ET design strain noise goal by at most a
factor of $\sim 3$ in a 8 Hz wide frequency band centered at 7 Hz. A cooling
power of 100 mW at 18 K corresponds to a gas pressure that increases the ET
design strain noise goal by at most a factor of $\sim 11$ in a 26 Hz wide
frequency band centered at 7 Hz.
