Why stars don't shine at daytime ?

As often on Internet forums, at the question "why stars are not visible at daytime" posted by an amateur it is not enough to answer "because the sun shines, guy... !". 

The eye and CCD

It is in reading technical notes about CCD's, that I had the idea to compare the signal-to-noise of this electronic device with the reponse of the human eye, what helped me to answer to the above question.

In summary the problem of seeing a star in daylight in linked to the sensitivity of the eye fovea and to the level of influence of the parasitic light presents in the observer near-environment.

What, at first sight, can be considered as a "filter" caused by the light spread in the atmosphere can be translated by the "noise" generated by the sky background, in this case the presence of the intense sunlight. How to explain this phenomenon in technical and mathematical terms and link it to the signal-to-noise ratio ? Here is a explanation.

In the past I was pro photographer and user of image processing technics to enhance my B/W or colors snapshots (including astronomy pics) as my teachers always asked me to do (at least in the beginning) : "balance this color, reduce this dominence, reveales more details in the light please",... I learnt the lesson ! I used masks, sandwitches of negatives, etc to reach this objective. Now thing changed. I am pro computist, my interest in astrophotography remains but the problem became more technical. We use computers, image intensifiers, CCD detectors and we try to reduce their "quantum defaults" in order to get good pictures, well balanced for colors, saturated, with the lesser noise of all kind we can... In short, a true challenge !

Made a comparison between the human eye and a CCD device. As you know both are light detectors that have the ability to process a continue analog signal. The CCD transforms photons in digital data (electrons are converted in binary digits) that we can easily process using a computer as they "speak" the same binary language.

The sensitivity of our eye, as the one of a CCD detector depends,  besides its specifications, on how we (our brain or the image acquisition software) process the "noises" present in the raw original signal.

As we explained in the chapter dealing with CCD cameras, these "noises" could be the readout noise (parasits introduced during the A/D conversion), the dark current and bias caused by the electron agitation (temperature) in a CCD and the background noise coming from natural and artificial sources (skyglow, moonlight, light pollution...). The eye is not a electronic device but it takes advantage of the electromagnetic properties of electrons when it has to transmit its information to the brain via the optic nerve. Now what is the relation between the eye and the CCD in this matter ?

I'am not a neuro-physicist (the eyes are "only" brain extensions) nor a cybernetician, but the "readout noise" and the "dark current" are probably reduced to nothing in our eyes (the brain). Without stimulus, we detect no visual signal and we stay like blinded... But whatever their form, if these signals exist you will quickly understand by reading the remain explanation their contribution is negligible.

Rest the "background noise". At night it is surely not negligible even if we substact the liners and other jets that cross through the firmament. This "noise" can for example appear when a cosmic ray hits one of our sensitive eye cells or a neural one. In these rare occasions we could see a flash light in our eyes like an instantaneous nova lighting the night. But it really appears in skyglow (excited atoms emitting light) and light pollution which limit our ability to see faint objects (in urban sites the signal and noise can be on par, offering us few chance to see stars). 

To be complete we have to add the weather factor too. But if the seeing and transparency are at their best, the source signal (planet, moon, star, DSO) is stronger than the noise. This explanation with ordinary words help us to understand why we can see stars in the night; "noises" of all kind have intensities much lower than signal coming from stars.

But what's the matter at daytime ?

Maths and physics

Trying to find a star in daylight, we have to admit that conditions are completely different and opposite to the night conditions. At first sight we can say now the Sun and the sky brightness add much "noises" over the star signal, and not simply a "filter" we could substract from raws images. Yes, this is the explanation ! Now the mathematical explanation.

Imagine a thinking experiment (all numbers are fictive, just for the demonstration) in which we want to look at a star in the daylight. The sunlight, the potential star light and noises of all kind hit our sensitives eye cells at a rate of 10000 counts per second per cell. This value represents the source signal without discriminating the star one. Remember too the field of view of the eye is around 120 degrees wide, so potentially catching a wide area of the sky and much light, probably adding a brightness factor we'll confirm later^[1].

Such a signal is a complex entity, composed of wavefronts of photon, quanta of energy "rule" by the quantum theory. The noise is thus a random quantum event according this theory. As for a CCD detector, the total noise (N) presents in the signal we search for naked eye represents the quantum uncertainty or the standard deviation from the average brightness. Its expression is defined as the square root of the sum of the squares of the individual noises values (oops !) :

N = Ö^¯( noise₁² + noise₂² +...)     <Equation 1>

What becomes the star signal now ? Comparing with a night measurement focused on this peculiar star, we know after integration that in the incoming daylight signal, our sensitive eye cells count to say, 50 hits due to the star. We know the star signal is defined as the input signal without the sun and other noises contributions. All being an affair of random quantum events, the contribution of all noises (equation 1) produces : Ö^¯10000, or 100 hits per visual cell.

Now knowing that the 50 hits from the star are mixed with the 9950 others due to the combination of sunlight, skylow and other light pollution spread in the field, we can determine the power of the star signal, the signal-to-noise ratio as one say, or S/N. Applying our relations to the star, S/N = 50/100, or 0.5. What means this number ?

In this example 0.5 means that the star magnitude (or brightness) can not be seen with a precision better than 1/0.5, or a 2 factor !, a value at least 10 times less than the accepted thresold of detectability for a CCD detector (which S/N may reaches 20 or more per pixel, able to detect a 20st magnitude star per arcsecond squared), but our eye is not exactly a CCD as our sensitives eye cells cannot accumulate light ! This value of 0.5 means also the star brightness is embedded is the sky brightness which considerably decreases the signal-to-noise ratio of the star we try to detect in the daylight. Exposed on a monitor, the image of our star is nearly invisible in the sky brightness given the feeling of looking at an old TV screen full of parasites.

Now we can solve our problem of trying to see a star in presence of the Sun. How to proceed ? Simply in increasing the S/N ratio of the star. How ? If we reduce our field of view using for example a scope or a long tube providing a true field of view of around 10' or less we reduce drastically the noise contribution of a ten factor or more and increase our star S/N ratio. Using this construction, the S/N ratio of the star is now enhanced while the background brightness is strongly reduced. On our control monitor the image is became more soft, the star appears now like a small bright dot circled by a dim blue-grey skyglow, typical of a lower signal deviation from the average or, in other words, a smaller S/N uncertainty.

By this thinking construction we have reduced noises contributions in our star S/N ratio to 50/Ö^¯1000 or less in place of 50/Ö^¯10000, so around 15 or better in place of 0.5. With this "sampling" value 30 times higher we can theoretically discriminate more easly the star signal, thus see some stars at daytime and estimate their magnitude with a precision of 1/15 or around 7% in place of a 2 factor !

Conclusion

With words borrowed to the CCD world, we can say that at start our star image was "undersampled" and cannot be recorded by our visual detector (eye). To enhance the "sampling" and discrimate our star in the daylight, we have had to increase the "scope" focal length to get a larger image scale. But the comparison stops here as we cannot use terms as focal reducer or binning (combination of pixels) to reach this specific goal, as we do using a CCD. But I never say we cannot substrat the sky brightness and increase this way the star signal using a CCD and image processing software. Don't interpret my words…

Thanks maths !

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[1] We have to temper performances of our eyes in reminding that the maximum diameter of the human iris is really tiny, limited to an opening of about 7-8 mm of diameter. No question in these conditions to get a better image that the one displayed by a dark room made of a pinhead. If we "clearly" see the world, this is the fact of our brain that interpretes and processes information transmitted by eyes and rearrange them according our learning of visual phenomena and the world surrounding us