The previous section looked at the kinds of accuracy that we can estimate using the Fake Star Planting and Harvesting. We found that, while it is important to have self-consistent measurements, that is really a statement about the measuring engine, not the telescope. What we showed also, was that the kind of accuracy we really want is to know that when I measure a star, my measurement will be close to the correct value for the star.

I mentioned that my planting did not include any Poisson noise for the fake star itself, and I worried that I might be misleading everyone by omission. So I reran the analyses, and included Poisson noise on every pixel of stars being planted. This was a random value computed for every pixel added to the real sky image, based on the model fake star's amplitude at that pixel location. When I did that, the apparent scatter in my multiple magnitude calibration runs actually dropped.  We still slightly underestimate the magnitudes in the raw frame, and over-estimate in the 10x10s stack. But not so much anymore. And so how can that be?

The graph above shows something similar to the earlier graphs in Parts I and II. It graphs measured SNR against star magnitude. We previously graphed measurement uncertainty versus magnitude and the curves became lines that sloped in the opposite direction - being smallest for the brightest stars. In this graph the SNR is largest for the brightest stars. SNR and uncertainty should be inversely related.

But notice that the straight green line, above, looks similar to the straight lines in the uncertainty graphs. And that is because those early models failed to include Poisson noise of the starlight being recorded in the images. When you do include the Poisson Self-Noise, you get the red curve shown above, where it begins to tip over toward the horizontal at the bright end.

Asymptoticallly, at the bright end of the red curve, you see Sqrt(Flux) behavior, where the Poisson self-noise dominates any background noise contribution.  The slope approaches 1/2. (hmm...on a magnitude scale for the flux that would be a slope of -0.2).

On the right end, at the faintest star magnitudes, the background noise begins to dominate, and the slope approaches 1. (er.., rather, -0.4 on the log-log magnitude plot.)

I used 45 DU for the Background in this modeling, taken right from my measurements of real stars in the image. That 45 DU was about the average standard deviation found among pixels in the measurement aperture ring for each star. Standard deviation is computed here as: σ = MAD / 1.48826, where we directly computed the MAD in the aperture noise ring.

And sure enough, my star measurements show SNR topping out around 300σ or so, for the brightest stars, just like you see on the left end of the red curve. It turns out that for bright stars, you really don't care much about the noise in the aperture ring. The noise of the measurement comes mostly from the starlight itself.

But for reaching the really faint stars, I set a limit before accepting a star measurement. First of all, to be a candidate star, the peak amplitude must exceed 5σ, and it must crest at the pixel under consideration. It can be equal to adjacent pixels, as sometimes happens for off-center stars. But it must definitely have higher amplitude than 2 pixels away. And that 5σ comes from the overall MAD of the whole image.

Next, I compute the median and MAD in the noise ring of the aperture, surrounding the candidate star. And the MAD is converted to standard deviation. Then I sum up the median removed flux from every pixel in the aperture core centered on the star pixel. The ratio of that sum to the RMS noise sum, from the ring plus the square root of flux,  must remain above 5σ (different sigma here...), to be an acceptable star detection and measurement. If accepted, I record the star's X and Y position in the image, its estimated magnitude, the SNR, and also the Flux from the core and the sigma from the noise ring.

But in summary, we see from the graph above, that background noise is mostly unimportant for the brighter stars. And at the faint end it is almost the only important thing. So stacking is definitely called for to beat down that noise floor and make it become less important in the magnitude estimates.

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That 5σ minimum limit also says that, even in the ultimate stacked image where the background noise goes to zero, there is still a limit in place on the integrated flux of the star. It must remain above 5σ of its own noise. So the minimum acceptable flux from a star is 25 DU above image median. For the camera in the little Seestar S50, it looks like we would not be able to "reliably" detect anything fainter than 18 mag, no matter how well stacked and beat down noise image you have.
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Well! According to my calculations, my Analyzer picked up a very faint target in my full-frame Drizzled stack of 10x10s. (I even have trouble discerning by eye)  It reports as an 8.8σ detection at 17.5 mag. I cross checked with Aladin and maaybeee (??) there is something there. Or it could be a spurious detection. But it is 8.8σ, so if the Higgs only needed 5σ, then maybe I found the most distant galaxy that I can see with the Seestar S50!

At an average ring noise level around all the detected stars of 7.5du, I have a flux of 110du. Its ring noise was 6.7du. The minimum allowable flux at this noise level should be 52du. So there might really be something there. If it looks like a duck, and walks like a duck...