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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 "sky temperature" 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?

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  • $\begingroup$ Fundamentally you're not trying to measure 2.7 K, you want to measure micro-kelvin anisotropies (in polarization now). Your front end doesn't need to be at 2.7 K, they were 100 mK on Planck HFI (bolometers), and 20 K for LFI (mixers). Also, at any reasonably high ell, anisotropy is dominated by foregrounds: signals from stars, dust, synchrotron, etc. Biggest one in your frequency range is dust from the galaxy. Therefore, it becomes an issue how how well you can resolve and identify the foregrounds so they can be removed in data processing (c.f. BICEP2 fail). See the Planck pre-launch papers. $\endgroup$
    – user23052
    Dec 1 '20 at 6:25
  • $\begingroup$ @user71659 I'm asking a question about noise in radio astronomy and have set up a hypothetical scenario, thus the title "Hypothetical CMB space telescope design problem, received power from extended thermal source versus receiver front end NEP?" If you can offer some information that addresses the question as asked that will be great, but answers about how to actually measure it should be placed on other questions. Thanks! $\endgroup$
    – uhoh
    Dec 1 '20 at 7:11
  • $\begingroup$ Once in a great while I forget to add the Question: indicator and today is one of those days. I'll add it back in now. $\endgroup$
    – uhoh
    Dec 1 '20 at 7:12
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    $\begingroup$ Basically you're saying you are not asking an astronomy question but need help with optics. If your question is truly on topic, it is underspecified. You need to say what the goal of your CMB telescope is. Are you trying to prove the CMB is a 2.7 K blackbody (FIRAS)? Are you trying to build the best lightly-polarimetric telescope you can (Planck)? Or are you trying to build a scientifically relevant polarimetric telescope (parameters currently unknown)? Those result in completely different designs. $\endgroup$
    – user23052
    Dec 1 '20 at 7:26
  • $\begingroup$ @user71659 I disagree in several ways with most of your comments and there is no reason to read so much extra into my question that's not there. But I have a sense that if you don't think that the question in the last sentence is on-topic here in Astronomy Stack Exchange I'm not going to be able to convince you. I also see that you don't really participate in this site very much so in this case I'm going to trust my own judgement. The question asks for a ratio of powers and it is sufficiently specified to answer with one. If there's no answer in a few days I'll try to write it myself. $\endgroup$
    – uhoh
    Dec 1 '20 at 7:32
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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 "a". 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 "a" is another question.)

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    $\begingroup$ Thanks for your answer! I think I'm getting the picture; I always get confused when thermodynamics and antennas meet. $\endgroup$
    – uhoh
    Mar 15 at 1:38

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