Hypothetical CMB space telescope design problem, received power from extended thermal source versus receiver front end NEP? Consider the example of a large radio dish antenna in space equipped with a heat shield protecting it from the Sun. Let's say the amplifiers front-end effective temperature (for purposes of noise calculations (i.e. NEP) is 2.7 Kelvin which is coincidentally the temperature of the Cosmic Microwave Background that it will look at.
We'll ignore thermal radiation from the dish itself, because although it is somewhat warmer it's highly reflective metal and so the emissivity is very low. (see this answer for more on that)
The receiver system is fixed in central wavelength near the peak of the CMB, let's say 2 mm, and has a fixed 1% bandpass.
Now it's time to choose the diameter and focal length of the dish.
I know that the characteristic &quot;sky temperature&quot; of the CMB for which I'm trying to image non-uniformities and the temperature of my receiver's front end are both 2.7 K but what is the ratio of their powers that I'll be measuring?
Question: If I have a large f/no. will a given diffraction-limited feed horn receive less power and so the signal will be weaker compared to thermal noise of the receiver, or will they end up somehow being roughly equal no matter the diameter or focal length of the dish?
The feed horn is an antenna which has an antenna pattern  Source
However, not all feeds are horns.  Here is an image of a parabolic dish with a Yagi feed:  Source
Ideally, you want the beam width of the horn antenna to cover the entire dish and only the dish.  So if the beam width of the horn antenna is, say, 30 degrees, you want the dish to subtend 30 degrees of the view from the center of the entrance of the feed.
Since the dish is parabolic, it can be described by the equation y=a(x-h)^2 Source.  Assuming the vertex of the parabola is at the origin, the feed antenna will be 1/(4a) units away from the vertex.  This is the focal length of the dish.  The aperture of the dish is its diameter.  Using the values from the above equation, the f-number of a dish antenna is (1/(4a))/x or 1/(4ax).
Now that we know where the focus is, we can compute the beam width of the feed required to cover the dish, then design a feed antenna meets that requirement.
Notice that the f-number doesn't contribute to the solution to the problem as we would expect if this were an optical system.  The most important thing is to match the beam width of the feed antenna to the radius of the dish, which is set by the value of &quot;a&quot;.  So a feed antenna will receive the same amount of power regardless of f-number as long as the feed antenna is at the focus of the aperture.  (Whether one can design a feed antenna for a given &quot;a&quot; is another question.)