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 !
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.
For a specific treshold of L, as a funtion of LS, the star is visible
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
L is multiplied by DT^2/DP^2 but is also divided by M^2 where
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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