How much further could ship-based radar see on a flat world?

Question

On our spherical planet, it is highly advantageous for modern navies to have airborne radar, because it can seem much further than even the best ship-based radar, simply due to the greater distance to the horizon.

Suppose a modern navy were operating on a flat world, with conditions otherwise similar to Earth. In particular, same atmospheric composition, density and scale height. In such circumstances, the effective range of radar would presumably be limited partly by the inverse square law and mainly by atmospheric absorption.

How much further would ship-based radar be able to see, in such circumstances?

Answer 1

Physical obstructions would become an issue before the absolute limitations of radar became an issue, but you would still probably want to avoid using long range radar

Take our world now as an example - the maximum possible distance you can sail in a straight line is just shy of 20,000 miles: https://www.popularmechanics.com/science/environment/a20114871/longest-route-sail-straight-line-without-hitting-land/

At 20,000 miles you would still be able to detect remote objects using a multi-megawatt radar antenna based on current-day technology. The power requirements would be high but serviceable (the A1B reactor can in theory produce 125 megawatts of electrical power) but you would need to scale the power appropriately in order to avoid cooking nearby objects. I think that such equipment would probably be considered uneconomical to power, uneconomical to construct, and far too hazardous.

I think it is more likely that such a navy would have specific flat world adaptations - for example long range drone swarms equipped with short range radar. The power requirements of a drone swarm would be significantly lower, and submersible drones in particular could go and loiter near the target for a very long time with the added bonus that you would not need to give away your position. The manufacturing cost would also be lower and there would be no risk of cooking anything.

Aerial radar would also still have significant advantages over surface based radar on account of being able to see over obstacles.

So that's where my money is - drone carriers equipped with hundreds of aerial and submersible drones.

Edit: Also satellites in highly elliptical orbit could still make relatively low passes over the atmosphere complementing ground based radar. You would need a lot of satellites divided into batches with different orbital characteristics to cover different regions and they could not have their periapsis near the centre of the disc so they would provide better coverage near the edge (I can't do the exact orbital dynamics for this but I believe you could only achieve atmosphere-scraping height near the edge & you would achieve lower heights over the centre with more elliptical orbits scraping one edge of the disc and flying high above the opposite edge, someone with better math-fu might be able to work it out). One satellite can only provide useful coverage for a few minutes at a time so you would need a few hundred of them to provide full coverage - very do-able if you consider that starlink aims to have over 40,000 machines in orbit. On Earth this approach is vastly inferior to aircraft-based radar (aircraft can get much closer to the target and can generate much more power) but depending on the requirements of your military and especially if the disc is very large then it could turn out to be a viable approach for long range surveillance, especially if they go with nuclear-powered satellites (Launching a constellation of nuclear-powered satellites into low earth orbit would be considered highly irresponsible and probably a violation of UNGA 47/68, but perhaps the population of your disc has a different approach to ethics or perhaps they consider appropriately re-entry protected reactor cores to be an acceptable risk - it is possible to protect an object during re-entry for later retrieval, but very awkward if it crashes on another nation's territory and you end up with the bill)

Answer 2

Not As far as light could possibly perpetrate

On earth, we can receive signals as far as the observable universe according to current physics could allow, eg CMB. This being in turn solely depends on how sensitive the equipment you have access to is. As atmospheric scattering is one of the things which throw off data of earth-based astronomy, you do have a definite limit for a radar pulse of individual frequency and amplitude, but by just,MORE POWER(If you don't mind vaporizing the immediate atmosphere) that could be overcome. Meaning that really, the limit to you're radar is really just the output of your generation and the sensitivity of your receiver, which would put them at odds and create some interesting dynamics. If you had radar power stations, power by something along the line of nuclear reactors you could probably use even just use optical radar to map the surface of you're flat earth down to the buildings present like orbital satellites do on earth. This does mean airborne radar is valuable, if only for mapping. You probably could also track large groups of humans moving with that method as well. You would have to do it in segments as optical radar can' track moving objects, but still, and that's moving objects with high accuracy, they'd probably just be something along the line of shapeless blobs, Or, In short, once you remove the curvature there is no real limit. And so, yeah.