[Aavso-photometry] CCD 'fainter-than' question

[Jim Roe]

Wolfgang Renz wrote:
> Hi Bob
> 
> If one assumes that the noise in the sky annulus is normal distributed,
> then the following applies for the probability that a star measurement
> is still just background noise:
> <http://en.wikipedia.org/wiki/68-95-99.7_rule>
> <http://en.wikipedia.org/wiki/Standard_deviation#Rules_for_normally_distributed_data>
> <http://upload.wikimedia.org/wikipedia/commons/8/8c/Standard_deviation_diagram.svg>
> <http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/400px-Standard_deviation_diagram.svg.png>
> 
> So the probability for a measurement to lie outside the following
> ranges and to be still just a background detection are as follows:
> +/- 1 sigma   100% - 68.27% = 31.73%
> +/- 2 sigma   100% - 95.45% = 4.55%
> +/- 3 sigma   100% - 99.73% = 0.27%
> +/- 4 sigma   100% - 99.9934% = 0.0066%
> +/- 5 sigma   100% - 99.99994% = 0.00006%
> +/- 6 sigma   100% - 99.9999998% = 0,0000002%
> +/- 7 sigma   100% - 99.99999999974 % = 0.00000000026%
> But as there are no 'darker than the background' stars, one must
> half the above probabilities for just 'brighter than the background'
> stars.
> 

A precise definition, Wolfgang, but I would try to amplify it with a 
numerical example.  First off, these results must be applied pixel by pixel.

Thus, consider a background average of 100 and a sigma of 10.  The 
probability that a pixel in the measuring aperture that measures 130 (+ 
3 sigma) is just a background outlier is, as stated above, 0.0027/2. 
But how many pixels are in your measuring aperture?  The math is pi r 
squared where r is the radius of the aperture in pixels.  Suppose r = 6 
then the number of pixels is 113.  Thus, the probability that at least 
one of the background pixels in the measurement aperture is as bright as 
130 is only 15%.

But one should never claim a "detection" based upon one pixel.  There 
should be several contiguous pixels at or above the threshold before 
claiming such.  Even then, a bit of caution is warranted as cosmic ray 
strikes can produce some very star-like images.

Jim Roe [ROE]