Stars at Noon

Why humans can't see stars at noon

Contrarily to what you might think, the real reason for the invisibility of the stars during day is not the luminosity of the blue sky but the resolution limitations of our eyes.

Indeed, bright stars are easily visible with a telescope during day.
The only difficulty is to direct the telescope to them !

Why we can see them with a telescope

There are two reasons for that

Stars are so far away that even with the biggest telescope on earth, they look like a point.
Zooming reduces the brightness per steradian

Thus, the more we zoom, the more the sky will appear dark.
Until a certain zooming factor at which the sky is so dark that the star reappears.

A telescope has two effects. It is a funnel for light and it magnifies the view. These effects are independant and can be controlled separately. Of course, a star appears much brighter in a telescope, but the day sky does not (this is different with binoculars). For a given aperture, zooming will make the sky background dimmer.

But where does that specific zoom factor comes from ?
The star reappears when the ratio of luminosity between the retina cells touched by the light of the star and the surrounding cells touched by the light of the shy is higher that a treshold that is specific to the human eye.
This treshold is itself probably dependant on the luminosity of the star.

Imagine that the retina cells were infinitly small.
Imagine that the optics of the eye were more than perfect and would even not diffract the light.
In that situation, the light of a star is concentrated on a single cell. And the surrounding cells receive zero light. The star is perfectly visible whatever clear the sky is.

In the real world, the picture of the star is distorted by the turbulences of the atmosphere and the imperfections of the optics of the eye.

The image of the star spreads on a certain surface of the retina SS.
LS the quantity of light of the star received by the retina on SS
S the surface of a single retina cell
L the quantity of sky light received by a single retina cell.

For a specific treshold of L, as a funtion of LS, the star is visible

SS is a constant characteristic of the optics of the eye
LS is a direct function of the magnitude of the star, it is the quantity of light that gets in the eye through the pupil
S, constant, is the size of the reception area of a typical retina rod cell
L is proportional to S, the luminosity of the sky, the aperture of the pupil and is inversly proportional to the focal length of the eye.

As a first approximation, we can imagine that this treshold is such that log((L+LS)/L) = constant K
==> L = LS /(exp(K)-1)

Behind a telecope,
LS is multiplied by DT^2/DP^2 where

DT is the diameter of the telescope
DP is the diameter of the pupil

One of the consequences, is that the brightness of the star does not depend on magnification.
This makes the asumption that the human pupil takes all the light from the telescope, which is usually wrong (eye's pupillary aperture smaller that the exit pupil of the telescope).

L is multiplied by DT^2/DP^2 but is also divided by M^2 where

M is the magnification factor

So the formula becomes
==> L = LS * M^2 /(exp(K)-1)
That means that with a M magnification factor, one will still see stars even with a M^2 brigher sky.

Using magnitude scales : dMag = 2.5*log(M^2) = 5*log(M).
So with a 10 fold magnification, you gain 5 magnitudes.
With a 100 magnification, you gain 10 magnitudes.

The retina and the visual system as a whole are much more complex than a computer monitor with its regular pixelised structure. What we see is not what the first layer of retina cells receive, but the result of a complex processing. Moreover, the eye tracking of an object is never perfectly stable, it is moving slowly and randomly around a central point. The brain can get much more informations this way.
That's why the model is certainly over simplified, but that's a start.

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