System temperature includes a number of contributions: receiver, atmosphere, CMB, plus antenna effects such as groundspill, surface imperfections etc. It can be measured by comparing observations of the sky and of a known load,
(Tload + TLNA) / Tsys = Y
where the measured Y-factor is given in terms of the difference in measured power (in decibels),
Y = 100.1(Pload-Psky)
Typical values are Pload-Psky = 4.5 - 6 and hence Y = 2.8 - 4.
Hence Tsys = (Tload + TLNA) / Y
Load temperature is just the ambient temperature (in Kelvin), and the LNA temperature is assumed to be 45K.
With the system temperature known, there are two ways of calculating the noise diode (cal) temperature. When measurements of the sky and load are both made with the diode included,
Tcal = (Tload + TLNA - Y Tsys) / (Y-1)
When the diode in included in the load measurement, but not in the sky-only measurement,
Tcal,2 = Y Tsys - Tload - TLNA