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

[Wolfgang Renz]

Hello

There is really no need to blame Michael N. for me giving an
example for Mira Pro. He asked: "Can't you get that number
out of _any_  software that does photometry?" and I choose
Mira Pro for an example as he knows it best and I hoped that
he therefore might come up with an easy to use simplified/
approximation formula for this issue.

This is IMO not a beginner-only list. So there must be room also
for theoretical discussions that develop new approaches. If they
end with an improvement in some way, they also demonstrate
that the photometry is advancing. And it advanced a lot during
the last two decades, didn't it ?. Anyway, most people who are
interested in how to choose the mag value for fainter-thans will
have learned something and got to know different opinions by
reading this thread. And the ones who are not, did not read or
did delete these posts anyway.

The 'practical' and non-calculating answers are IMO OK for
visual observing and for photometry if one does want to just
guess the fainter-than mag. But when we measure things any-
way, why not determine the fainter-than mag more accuratly ?
There are enough vars with sequences whose comps that are
all brighter than the var if the var is faint and there are still se-
quences that have gaps of up to 2 mag. So there is a lot of
room to give the fainter-than mag more accuratly by using a
sufficient approximation formula.


So I'll try it myself. I see two simplified basic approaches to
calculate an (approximate) fainter-than mag by using the zero
point or by using an other star.

A) Using the zero point

The zero point is usually determined by solving the equation:
    m = -2.5 * log (f) + Zp            (I)
    m:  mag value of the standard star
    f:   flux value of the standard star
    Zp: zero point mag value of the image
for one or more standard stars.
So for f=1 the m equals the Zp. This means that the zero point
is the mag value of a star with flux 1. Many might incorrectly
have assumed that the zero point is for a star with flux 0 (= sky
background), but f=0 would actually gives always a zero point
of +indefinite mag. This also means that the zero point has a
SNR of a bit larger than 0 (depending on the noise in the image).
Here its irrelevant if the flux is given as absolute value (photons,
electrons) or as a rate (photons/sec, electrons/sec) as the zero
point will adapt accordingly when its determined.

The basic formula for the signal to noise ratio is:
    SNR = f / n            (II)
    SNR: signal to noise ratio 
    f: total flux (in photons, electrons)
    n: background noise (in photons, electrons)
I know that the noise should include more terms (like e.g. the pho-
ton noise of the star) as just the basic background noise, but as I'll
use this formula just for calculations close to the background, it is
probably a sufficient approximation.
Here its also irrelevant if the flux and noise is given as absolute
value (photons, electrons) or as a rate (photons/sec, electrons/
sec) as both will give the same SNR.

When rearranging (II), we get:
    f = SNR * n            (III)

When inserting (III) into (I), we get:
    m(SNR) = -2.5 * log (SNR * n) + Zp            (IV)
    m(SNR): fainter-than magnitude value for a choosen SNR 
and voilà, we have a first approximation formula for calculating
the fainter-than mag at a choosen SNR in an image with noise n
and a zero point Zp.
Here it will be of course important that the noise, zero point and
further used values are either calculated all for absolute values
or all for rates. With this formula, the SNR should then probably
also not be choosen with too large SNR values.

Now we just need a simple and good approximation for the noise
for faint stars just above the sky background. Here one should use
just know values (e.g. gain in e-/ADU, read noise in e-, ...) or values
that are available or can be easily determined by common photo
metry software (e.g. StdDev of the background in ADU)
Maybe Michael can help us out on this.

When rearranging (II) differently for the zero point with f=1, we get:
    n = 1 / SNR            (V)
So the reciprocal of the SNR of the zero point mag could be used
here for the noise. But this value is usually not available or must be
determined experimentally (e.g. via a SNR vs. mag scatter plot in
Mira). The SNR values are usually also just given with a too low
number of digits (e.g. 1 decimal digit in Mira) to be used for a
good approximation better than ~ 1 to 2 dmag.


B) Using an other star

The basic formula for calculating a mag difference is:
    m1 - m2 = -2.5 * log ( f1 / f2 )            (VI)
    m1: magnitude of star 1
    f1: flux of star 1
    m2: magnitude of star 2
    f2: flux of star 2

When inserting (III) for f1 and f2 into (VI), we get
    m1 - m2 = -2.5 * log ( (SNR1 * n1) / (SNR2 * n2) )            (VII)
    SNR1: SNR of star 1
    n1: background noise for star 1
    SNR2: magnitude of star 2
    n2: background noise for star 2

When assuming the noise is constant or using an averaged noise,
(VII) simplifies to:
    m1 - m2 = -2.5 * log ( SNR1 / SNR2 )            (VIII)

When rearanging and renaming (VIII) we get:
    m(SNR) = -2.5 * log ( SNR / SNRstar ) + mstar           (IX)
    m(SNR): fainter-than magnitude value for a choosen SNR 
    SNRstar: SNR of choosen star
    mstar: magnitude of choosen star
and voilà, we have a second approximation formula for calculating
the fainter-than mag at a choosen SNR in an image relative to a
star with a known mag and SNR value.
With this formula, the star should not be choosen to have a SNR too
far away from the SNR one wants to calculate the m(SNR) for.
When setting here mstar = Zp and SNRstar = f/n of the Zp = 1/n, we
get the same formula as in (IV).

I've tried this one with a few examples in Mira and it give reasonable
approximation results for the fainter-than maga in the few mmag range.

Please try it yourself and report back.
Any comments ?

Clear skies
 Wolfgang

-- 
Wolfgang Renz, Karlsruhe, Germany
Rz.BAV = WRe.vsnet = RWG.AAVSO