  The carbon footprint of astronomical research is an increasingly topical
issue with first estimates of research institute and national community
footprints having recently been published. As these assessments have typically
excluded the contribution of astronomical research infrastructures, we
complement these studies by providing an estimate of the contribution of
astronomical space missions and ground-based observatories using greenhouse gas
emission factors that relates cost and payload mass to carbon footprint. We
find that worldwide active astronomical research infrastructures currently have
a carbon footprint of 20.3$\pm$3.3 MtCO$_2$ equivalent (CO$_2$e) and an annual
emission of 1,169$\pm$249 ktCO$_2$e yr$^{-1}$ corresponding to a footprint of
36.6$\pm$14.0 tCO$_2$e per year per astronomer. Compared with contributions
from other aspects of astronomy research activity, our results suggest that
research infrastructures make the single largest contribution to the carbon
footprint of an astronomer. We discuss the limitations and uncertainties of our
method and explore measures that can bring greenhouse gas emissions from
astronomical research infrastructures towards a sustainable level.
