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I am most interested in (exo)planets with atmospheres, see e.g. my question Requirements to resolve position of Jovian Whistlers up to magnitude of Red Spot with amateur radio equipment? So if I had a 25m radio dish in my backyard, would I have the chance to observe anything useful about (exo)planetary atmospheres by pointing it at one of the known exoplanets with atmosphere?

Another cool thing to observe would be a fast radio burst. Where should I point my reflector to increase the chance to see one?

References

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On the fast radio burst side of things:

The FRB distribution is isotropic so far, which isn't surprising - you'd expect cosmological sources of this sort to be more or less everywhere in the sky. This means that you could point your telescope at random and still have a roughly equal chance of detecting one - albeit an extremely low chance. Fortunately, in the last couple of years we've discovered that at least a small subset of FRBs repeat; pointing your telescope at one of those known repeaters could substantially increase your odds of a detection.

I don't think there's been a substantial effort to look for FRBs solely with small dishes, but it's been attempted with a 25-meter dish. Spitler et al. 2018 used the 25-meter Stockert radio telescope, operated by the amateur astronomy group Astropeiler Stockert e.V., in conjunction with main observations by the 100-meter dish at Effelsberg. They were looking at FRB 121102, the first known repeating FRB. While Effelsberg detected three bursts from that source during their observations, Stockert turned up nothing - not overly surprising. As it turned out, the bursts Effelsberg picked up had flux densities much lower than you'd need for a reasonable signal-to-noise ratio with Stockert. There have been a couple other non-detections with small dishes, including the Nanshan 26-meter dish (Men et al. 2019).

On the plus side, you might have better options than FRB 121102, depending on your latitude. The ASKAP folks recently detected a bright repeat burst from the source FRB 20201124A, and say that the first burst could have been detected at SNR > 10 with a 20-meter telescope (with adequate specifications, of course). So depending on your precise setup, you might be in business - assuming, of course, you've got the proper setup.

I know less about exoplanet radio emissions, but I'd say your odds are probably much worse, partly because this is an area which has only had success even more recently than repeating FRBs. There was a detection from Tau Boötis this past year (Turner et al. 2021), and some previous tentative signs from elsewhere, but we have yet to get follow-up observations confirming things. You could point your telescope at Tau Boötis and hope for the best, but I wouldn't get your hopes up. Even that signal wasn't exactly strong.

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