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Suppose I stand at p = (latitude,longitude) on earth. I want to manually check for any airplane flying overhead in my neighborhood with naked eye and without any equipment.

I scan air-traffic radars for every plane in the vicinity of my location meaning probe_region: latitude ± δ , longitude ± δ (we assume a region of square geometry for simplicity)

1- How far away can a man hear the jet engine of airplanes? Meaning what should be the radius of probe (or in this square case ~δ)

2- How far away can a man see the body of an air plane in clear sky with naked eye?

I'm looking for answers in kilometers or miles. (converting from radius in kilometers to (lat/lon) interest_box is another problem, like 1 deg in latitude is ~110 km)

In the diagram below, the plane inside the blue bound can be heard from p but the plane just outside of it can't be heard (I assume we can hear engines sooner than we can observe the whole plane)
diagram of problem

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    $\begingroup$ Maybe specify the engine type and number of engines? Planes are getting quieter. For a loud one, see aviation.stackexchange.com/questions/21683/…. $\endgroup$
    – Anonymous Physicist
    Commented Mar 8, 2022 at 17:13
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    $\begingroup$ I regularly hear aircraft passing overhead at FL360 or higher (approx. 6 nautical miles high), if I'm away from other sources of noise. I definitely don't hear them very far laterally laterally though. Aircraft can be seen with the unaided eye at that altitude but it's very difficult to find them in the first place unless they're creating contrails that lead to their position. $\endgroup$
    – randomhead
    Commented Mar 8, 2022 at 17:42
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    $\begingroup$ Do you need to be able to identify the shape of the object as a plane? Or is a pinprick of sunlight glinting off a fuselage sufficient? Stars, for example, are visible at distances of trillions of kilometers, although they have effectively zero angular diameter so their shape cannot be resolved. But we can still look at the sky and easily identify that something is there. Similarly, a plane could theoretically be visible at distances much farther than the distance at which you'd be able to visually identify it as a plane. $\endgroup$
    – Nuclear Hoagie
    Commented Mar 8, 2022 at 18:14
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    $\begingroup$ Whose ears and eyes are we using? Sadly, my ears don't quite hear as well as they used to (invincible teenager listening to loud music against parent's recommendation), and my eyes don't see as well as I'd like them to (despite new contacts). However, a good friend of mine can read small text at distances where I can't even make out that there is text to be read. $\endgroup$
    – FreeMan
    Commented Mar 8, 2022 at 18:48
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    $\begingroup$ ... there is so much "it depends" involved here that any answer is only good to about an order of magnitude. Something with older, loud engines at climb power will be detectable much farther than something new like a B-787 at cruise power. Then there's aircraft altitude. And ambient noise for the observer. And atmospheric conditions. And aircraft size and aspect. And ambient light. And cues like contrails, sunlight glinting off metal surfaces, and so on. Any number you get that applies to "this" case may be far off for many other cases. $\endgroup$
    – Ralph J ♦
    Commented Mar 8, 2022 at 20:12

2 Answers 2

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It depends on the altitude and orientation of the aircraft as some aircraft will have a larger cross sectional area depending on which way its facing and where you are standing. Depending on the altitude of the aircraft its eventually going to disappear over the horizon. The angular resolution of the human eye is somewhere in the area of 0.03 degrees so ultimately you will be able to see an aircraft when its relative size to your reference is in excess of 0.03 degrees (weather or not you can tell its an aircraft is likely up for debate but you should be able to physically resolve it). You can use this calculator to check it out but for reference a 737 is about 12.5m tall so with a 0.03 degree minimum resolution you should have the ability to resolve it up to 24km away however in my practical experience that very much depends on your relative position to the plane, the clarity of the air, and the time of day/night. 24Km is ~13Nm (how pilots think about distance) which you might very well be able to see on a clear day.

Sound intensity will decrease exponentially over distance but your ability to hear something will depend on other ambient noise in your local area, your hearing ability, the frequency of the engine noise (and there is dopper shift in play here) and reflective surfaces en-route. If you take a look here some sonic booms (not necessarily engine noise) can be heard as far as ~80NM from the source. Here is a fun calculator to show you the effects of sound dampening over distance (unfortunately it only works out to 1000 meters)

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  • $\begingroup$ A reasonable approach, but the limit of resolution being 0.03 degrees does not mean that things smaller than 0.03 degrees are invisible. It means that you can't distinguish between objects closer than 0.03 degrees. You certainly couldn't tell the shape of a plane at that distance, as you couldn't even distinguish one wing from another, but you still might be able to see a featureless dot in the sky at a longer distance. $\endgroup$
    – Nuclear Hoagie
    Commented Mar 9, 2022 at 15:57
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Just now, there is a Lear 35A going directly over my house at Fl36.

I could see the contrail, and just barely see the jet shape at the head of it.

But only because I knew exactly where to look, and the contrail. If it were not for that long contrail, there is no real way I would have found it before it went out of sight.

I heard nothing.

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