Can't believe nobody mentioned VNC. I have been using it for about 15 years so far. It is direct PC-to-PC connection and does not require Internet. Free and open source. Several compatible variants exist, see
UltraVNC or
TightVNC for example.
If I am not mistaken remote desktop will not allow you to use the local screen in the observatory until you close your remote session which will shut down all the running software (I am not sure if you need to use the screen in the observatory but my guess would be yes).
VNC simply exports the running screen so in your office you would have exactly the same screen as in the observatory.