How high can an airplane be spotted with eyesight?

How high can an airplane be spotted with eyesight? Suppose a plane has a wingspan of 15 m then at what distance it will appear as a faint dot? What is the correct way to calculate this?

• Do weather conditions at the time create contrails ? What colour is the underside of the plane? Does it have running lights? Is it day or night? Mar 27, 2020 at 22:24
• @Criggie Assuming it's day time and there are no contrails. Mar 28, 2020 at 1:20
• I'm a pilot, and I can have Air Traffic Control tell me, and see it on my ADS-B display, about a plane at some distance and altitude, so I know right where to look. White plane moving over a dart tree background are easiest to spot (the movement really helps), and given all that, it can be really hard to find that plane when it 3+ miles away (~15,000 feet). That would be near ideal conditions. Get a bright background behind and the plane above, and it gets much harder. Large commercial planes are a lot easier to spot, especially when they are east of you and the sun is west. Or west/east. Mar 28, 2020 at 1:53
• I can recall watching satellites pass overhead. Mar 28, 2020 at 13:07
• Is this really an aviation question? If I asked this question about cars, would that make it on topic for the automotive SE? Mar 29, 2020 at 1:19

Suppose a plane has a wingspan of 15 m then at what distance it will appear as a faint dot? What is the correct way to calculate this?

This does not imply that you cannot see the airplane at a greater distance any more. It just means you cannot resolve it any more, i.e. it will appear as a faint dot.

This can be estimated with the Rayleigh criterion. The angular resolution of the human eye is given by

$$\theta = 1.22 \frac{\lambda}{D}$$

where $$\lambda$$ is the wavelength of the light and $$D$$ is the pupil diameter. Using typical values of $$\lambda = 550 \, \mathrm{nm}$$ (highest sensitivity of the human eye) and $$D = 3 \, \mathrm{mm}$$ during the day, we get a resolution of

$$\theta \approx 2.24 \times 10^{-4} \, \mathrm{rad} \approx 0.013^\circ$$

If the aircraft is $$w = 15 \, \mathrm{m}$$ wide, the distance at which this corresponds to this angle is given by

$$d = \frac{w}{\tan \theta} \approx \frac{w}{\theta} \approx 67.1 \, \mathrm{km}$$

Most planes only fly up to FL410, which is about 12.5 km, so you could always distinguish them from a faint point with the naked eye. As Zeiss Ikon pointed out in the comments, the actual resolution of your eyes may be worse because your eyesight is probably not perfect. This is however an upper limit on the resolution.

• Based on my experience, human vision doesn't always live up to the Rayleigh criterion. That said, I've spotted the tiny arrowhead at the leading end of a contrail on many occasions. Mar 27, 2020 at 12:24
• @ZeissIkon That's true, this is the limitation from physics and therefore an upper limit. Human vision can definitely be worse, but how much worse will depend on the particular person's eyesight. Mar 27, 2020 at 12:30
• This seems like at best a partial answer, since it's perfectly possible to see smaller objects like satellites at a considerably greater distance, depending on how they're illuminated. Likewise, it's common to see planes around sunset (and presumably sunrise, if you're up that early), when the angle is such that sunlight reflects off it. Mar 27, 2020 at 18:00
• @jamesqf This is answering how far they can be that you can still resolve them, i.e. see them as something other than a dot. You can see satellites but only as dots. Mar 27, 2020 at 18:30
• The Rayleigh criterion only applies to resolving an object as having a shape. With sufficient contrast (eg. sunlit satellite against night sky), you can see things at a much smaller angular resolution.
– Mark
Mar 27, 2020 at 20:38

Under ideal viewing conditions (a background of thin high clouds, and an optimal color scheme on the aircraft), a radio-controlled aircraft with a 2-meter wingspan and a high aspect ratio can be seen clearly enough to actually maintain direct control unassisted by any electronic stabilization or autonomous navigation capability at 4000 feet (1.22 km) above the ground. (Source-- own experiments with well-calibrated onboard telemetry.) An aircraft with a 15-meter wingspan would have the same apparent wingspan at 30,000 feet (9.14 km) above the ground.

Note that in some circumstances it is possible to maintain control of an aircraft in this manner even if it appears to be little more than a tiny speck with few distinguishable features, because the aircraft has good intrinsic aerodynamic stability, and the aircraft's approximate heading is revealed by its direction of travel. Still, if the only criteria that is that the aircraft is perceptible at all, the corresponding altitude would be much higher.

After first being spotted with binoculars, Mississippi kites have been observed with the naked eye by observers with no more than 20/20 vision, as the birds passed in and out of the bases of cumulus clouds that were verified by consulting nearby airport ATIS/ AWOS broadcasts to be at least 4000 feet (1.22 km) above the ground. (Source-- own observations.) (The birds were not identifiable to species with the naked eye at this distance and may have appeared as little more than tiny specks; I don't have detailed notes accessable to refresh my memory at this time.) The average wingspan of a Mississippi kite is about 80 cm. The corresponding altitude for an aircraft with a 15-meter wingspan would be 75,000 feet (22.9 km).

Next time you watch an airliner fly high overhead, ask yourself "what is the smallest part of that aircraft that I can actually discern", and then do the math. You'll likely come up with a number that is at least double the aircraft's cruising altitude, and possibly much higher.

Note that the question does not appear to actually be asking at what altitude an aircraft can be visibly discerned to be something other than a tiny unknown speck. Rather, it appears to be asking at what altitude an aircraft can be visibly discerned at all.

However, since the question mentioned "spotting", keep in mind that in most daytime viewing conditions it is much more difficult to initially detect an object than to keep it in view once it is detected. Even if the object has been spotted and its approximate position is therefore known, it can be very difficult to re-acquire it if visual contact is lost. When scanning an empty blue sky, the eye tends to focus much closer than would be appropriate for detecting distant objects. Other distant features (e.g. contrails, high clouds with visible texture, the moon, etc) can be very helpful as they provide something for the eye to focus on. So the ultimate answer to the question is "the probability of spotting (within a given time interval) a given aircraft flying at a given altitude depends tremendously on the prevailing conditions such as background features and lighting conditions, and also on the aircraft's color scheme." With light-colored aircraft, brown polarized sunglasses can be extremely helpful, especially in the haze-free conditions that tend to prevail at higher ground elevations, as they can dramatically darken the blue sky.

• While piloting similar r/c sailplanes at rather lower altitudes but still "specked out" as we say, passersby have often wondered at the gizmo in my hands and my skyward gaze. More often than not, even with careful cues "that dragon-shaped cloud, two legs down and to the left, here, I'll put her in a spin so the wings glint," they can't spot it. You have to know where to look. Even in full scale aviation, spotting another airplane heading at you at your own altitude is terrifyingly difficult even just a few seconds away. Mar 28, 2020 at 3:23

20/20 vision is about 1/3000 radians. So an object that is 15m wide can be discerned about 3000*15m = 45km away. This is only a rough estimate, since there are a multitude of factors such as how good someone's eyesight is, how clear the air is, etc.

• If you apply that formula to the night sky, you'll conclude that the stars are invisible.
– Mark
Mar 29, 2020 at 22:54
• @Mark We are unable to resolve the stars. That's quite different from not being able to see them. Mar 29, 2020 at 23:11