antenna temp

Hello All,
When a parabolic dish is pointing towards the sky or zenith there are lot many contributing factors to antenna temperature like Sky Temp,Spillover Temp ,Ground temp etc etc.
Antenna beam has main lobes,side lobes and back lobes also. When antenna is pointing towards sky, main beam is directing towards sky temp ,side lobes will pickup spillover temp and back lobes will pick ground temp. And sum of these all factors will add up to Antenna Temperature.
Here while calculating antenna temp except spillover temp calculations all other temp calculations use Antenna beam.For spillover we use Feed Antenna radition pattern and then add that to the total sum.
Is this the right way to calculate the Antenna temp of parabolic dish? As per the defination of Antenna temp it seems to right way.
What may be the other way to find antenna temperature of parabolic dish ?
Anyone calculated Antenna temp before of dish antenna before,then pls share it here.
Thanks in advance.

My understanding is roughly in concert with your description.

Fawwaz Ulaby has a number of books on the subject and you may find them helpful (although several are expensive).