[Aavso-photometry] Differential vs. absolute magnitudes

Sat Mar 8 19:02:49 EST 2008

Jeff,

I am not disagreeing with what you say, but this brings up another point 
worth mentioning:

It is not strictly true that a B filter would reduce the precision of the 
photometry. Of course, one benefit of using a B filter would be that the 
magnitude could be transformed to a standard system (at least theoretically 
and if he also had a V filter to get the transformation coefficients). But 
let's look at the issue of magnitude precision with and without the B 
filter.

The magnitude precision is the random error the magnitude measurement which 
is simply related to the signal to noise ratio ("SNR") of the measurement. 
In equation form, this is Sigma(mag) = 1.0857 / SNR. Remember that we are 
measuring the object, not the (object+sky), so SNR is calculated using the 
net signal of the object without the sky whereas the noise includes noise 
from both the object and the sky underneath the star. The signal and noise 
are both summed inside the measuring aperture.

The B filter does not pass the Sodium D lines near 5890A that comes from 
both light pollution and the natural sky background. So if someone has a 
heavily light polluted sky caused by low pressure sodium vapor lamps, then a 
B filter would dark the sky quite a bit. But the B filter reduces the energy 
from the star by an amount that is fixed by the bandpass, and that might be 
around a factor of 4 on a typical CCD.

Two side notes: 1) The SNR is also affected by the aperture size (good 
seeing vs bad seeing). The larger the aperture, the more background is added 
to the starlight. 2) There is noise from the CCD's dark current. Even though 
the dark current is subtracted, its noise remains as part of the total noise 
underneath the star profile. In the same way that sky light is subtracted 
but leaves sky noise, the dark "light" is subtracted but leaves its dark 
noise. Therefore, observing in poor seeing or with a warm CCD will increase 
the noise and decrease the SNR. But I'm just talking about filters here.

As an example, the B filter might reduce the sky brightness by a factor of 
10, 20, or even more but the star would be reduced by about 4x. It is true 
that the sky has to be pretty bright or the seeing pretty bad for the SNR to 
actually be improved by using a B filter, but it can happen. Let's look at 
some rough numbers. I am not going to include all noise sources because the 
sky is bright and its noise swamps these other sources.

Consider a fairly light polluted urban sky so that, in the measuring 
aperture, the sky is 4 magnitudes brighter than natural sky. Suppose that, 
with the B filter, we have an exposure time and aperture size that gives 
1000e- from the star. Assume no dark current. We get the following:

B filter:
Star = 1,000e- above sky
Sky = 1,000e- (no light pollution)
Star/Sky = 1.
Noise = sqrt(1,000+1,000) = 45e-
SNR = 1000 / 45 = 22.4, Sigma(mag) = 0.048.

No filter:
Star = 4x brighter without B filter = 4,000e- above sky
Sky = 4 mag (40x) brighter = 40,000e-.
Star/Sky = 0.1
Noise = sqrt(40,000+4,000) = 210e-
SNR = 4000 / 210 = 19.1, Sigma(mag) = 0.057.

Just for grins, suppose we expose 10x longer or we combine 10 images and 
measure that. Then we get a better result for both cases:

B filter:
SNR = 70.7, Sigma(mag) = 0.015

No filter:
SNR = 60.3, Sigma(mag) = 0.018

BUT! The *ratio* of SNR obtained with and without the filter is identical to 
the case for shorter exposure. Hmmm.... what does this tell us? Answer: 
Exposing longer does not make the filtered/unfiltered comparison any 
different but it does give us higher precision in either case.

As you can see, a measurement through the B filter can win when the sky is 
rather bright. On the flip side, a brighter star will provide more photons 
so the effect is not as dramatic. The critical factor is the ratio of 
star/sky brightness inside the measuring aperture.

What I have described is the principle behind the so-called "nebular" 
filter. However it is not true that *any* filter will improve the SNR as I 
have shown. The signal in a given bandpass depends on the net transmission 
of the atmosphere, optical system, and filter combined with the response of 
the CCD and the spectrum of the star. Since the U filter is very dense and 
the CCD, atmosphere, and optics also ge worse below 4000A, the sky would 
have to be awfully bright at 5890A to win when using a U filter. But that is 
not the case for the B filter.

Michael Newberry

----- Original Message ----- 
From: "Jeff Hopkins" <phxjeff at hposoft.com>
To: <gianlucaros at gmail.com>; <aavso-photometry at mira.aavso.org>
Sent: Saturday, March 08, 2008 2:42 PM
Subject: Re: [Aavso-photometry] Differential vs. absolute magnitudes

> Hi Gianluca,
>
> You cannot increase your precision by adding a B filter. You might be
> able to increase your accuracy, however.
>
> Things seemed a  lot clearer back in the single channel photometry
> days. Most people did all-sky photometry for determining their
> transformation coefficients and for special projects and differential
> photometry for most other projects. Everyone transformed their data
> and accounted for extinction.
>
> A word of caution, absolute magnitudes are the magnitudes of stars at
> a standard distance from the star (10 parsecs or 32.6 light years).
> Naturally you cannot determine that directly using photometry. I
> think what you are referring to is the star's published or standard
> magnitude/extraterrestrial magnitude as seen from outside the Earth's
> atmosphere. Star magnitudes are usually expressed that way.
>
> With CCDs many programs let you enter a comparison star's magnitude
> and then the difference between it an the program star can produce a
> normalized magnitude for the program star. With practice you can
> increase the precision of that magnitude, but not the accuracy. To
> increase the accuracy you must use a standard filter and transform
> the data. If the stars are far apart you must also allow for
> extinction. The zero points will drop out as they are the same for
> both the comparison star and program star.
>
> Good luck.
>
> Jeff
>
> At 13:35 -0700 03/08/2008, gianlucaros at gmail.com wrote:
>>I have been doing some differential CCD photometry with a V filter.
>>Since I would like to increase precision I am thinking of buying a B
>>filter and to use transformation coefficients. Reading the AAVSO
>>manual I haven't a clear picture on how to trasform differential
>>magnitudes into absolute one. Once the data have been transformed
>>using transformation coefficients, shall I calculate zero point to get
>>the absolute magnitudes? What is the procedure? Are absolute
>>magnitudes only possible with all sky photometry or all sky photometry
>>is compulsory only if the comparisons stars are more than a field
>>apart? I am getting a bit confused.
>>
>>Gianluca (RGN)
>
> -- 
> Jeff Hopkins
> HPO SOFT
> Counting Photons
> http://www.hposoft.com/Astro/astro.html
> Hopkins Phoenix Observatory
> 7812 West Clayton Drive
> Phoenix, Arizona 85033-2439 U.S.A.
> (623)849-5889
> (623) 247-1190 (Fax)
> www.hposoft.com
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