How high can an airplane be spotted with eyesight?

Question

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?

Answer 1

Note: There is some confusion in the comments about this answer. What I am answering here is the question:

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.

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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.

Answer 2

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.

Answer 3

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.