[Aavso-photometry] Determination and use of BVRI

[Lionel Catalan]

Thanks Brad for taking the time to provide these thorough derivations.
It took me some time to go through the math, but it's interesting to
discover that one can in principle use Maxim DL data for
transformation coefficients.

Lionel


Two other points
 Your first equation is
mdiff (star) = minst (star) - minst (ref) + mstd (ref)

minst (star) is defined by your equation as being -2.5*Log
(Flux(star)) - See below for the reason I use Flux instead of Int -
leaving out the detail of normalizing for the integration time. minst
is an instrumental magnitude that is not referenced to a known star.
It is ofter called the "raw"
instrumental magnitude. It will always be a negative number.

However, mdiff, as you have defined it,  includes the zero point correction.
This is still called an instrumental magnitude but it isn't the value
that you need for applying transformation coefficients. I would denote
it m (star). The differential magnitude you need to apply extinction
coefficients is minst (star)- minst (ref)

Your second equation is
minst (ref) = -2.5 log (Int(ref))
This is Ok with one small technical correction. You aren't actually
mesuring intensity you are measuring flux. I know that maxim calls it
intensity. For a point source of light,intensity is power per unit of
wavelength per steradian of solid angle subtended by your telescope
aperture. Since you are not using correcting factors for the solid
angle subtended and the width of the bandpass, you are measuring flux
which is simply power received by your telescope aperture over
whatever bandpass you get from the combination of atmosphere,
telescope, filter and detector. For a given observing setup the
Intensity will be proportional to the flux reaching the telescope.
Therefore the second equation should really be

minst (ref) = -2.5 log (Flux(ref))

The problem with Maxim is that you only get the instrumental magnitude
corrected for zero point in the output table. It doesn't give you
minst(star) or minst(ref) so you don't have the basic data you need to
do transformations. You would have to record that information manually
from the information window of the photometry tool enter that data
into a spreadsheet and do the math. If you are analyzing a few hundred
images that becomes a mind numbing, time devouring chore.

In an earlier e-mail, Sat, 12 Jan 2008 13:10:49 -0500, you asked what
to do about the zero points in the color index transformation
equations. The short answer is that the zero point terms cancel out in
determining transformed Delta (B-V) between star and ref and
transformed delta V between star and ref. The correct zeropoint for
your session is added back when you add standard(B-V)ref or standard
Vref back to the differential transformed magnitudes.

The long answer is as follows.

Going back to basic equations for photometry, using spreadsheet
operators to separate variables and changing nomenclature slightly to
reduce typing v = vinst = -2.5*LOG(Flux/Integration Time) V (upper
case) = vstd

vo = extinction corrected raw magnitudes c (lower case) = a raw color
index i.e. b(inst) - v(inst), v(inst) - r(inst), etc co = extinction
corrected color index C (upper case) = the standard color, i.e. B-V,
V-R, etc.
Tc = Transformation coefficient for the color index k1c = first order
extinction coefficient for the color index k2c - second order
extinction for the color index Zc = zero point for the color Dc =
c(star)-c(ref) (difference in color index betwee star and ref)

The two basic equations for extinction correction and transformation
of color indexes (colors) are co = c*(1 - k2c*X) - k1c*X C = Tc*co
+Zc.

The difference between extinction corrected instrumental colors is Dco
= c(star) - c(ref) - k2c*X*(c(star)- c(ref))-k1c*X(star)+k1cX(ref)

In the same FOV X(star) and X(ref) are essentially equal, therefore,
Dco = Dc - Dc*k2c*X

DC = C(star) - C(ref)
DC = Tc*(co(star)-co(ref)) + Zc - Zc
   = Tc*Dco
   = Tc*(Dc - Dc*ksc*X)
   = Tc*(c(star)-c(ref)-k2c*X*(c(star) - c(ref)) or
C(star) - C(ref) = Tc*(c(star)- c(ref))*(1-k2c*X) If you are not using
second order extinction, this simplicies to
C(star) - C(ref) = Tc*(c(star) - c(ref))
C(star) = Tc*(c(star) - c(ref)) + C(ref)

The Zc term is automatically included because the expression can be rewritten as
C(star) = Tc*c(star) + C(ref)-Tc*c(ref)
And C(ref) - Tc*c(ref) includes Zc
Therefore even though you determned Tc using a particular reference
star in your M67 measurements, Tc doesn't depend on that star and the
differential photometry process adds Zc back for the reference star
you are using in any particular session.

Now we have the standard color index of interest for our target star.
Second order extinction coefficients are often omitted but should be
included for color indexes if you are trying for high accuracy or high
precision when taking data over a range of altitudes. 2nd order is
more important for color indexes than for V.

To get Vstd(star) using the same small nomenclature changes

The two basic equations for extinction and transformation in V are

vo = v - k1v*X, omitting the usually very small second order
extinction value V = vo + Tv*(B-V) + Zv Tv and Zv are the
transformation coefficients and zero point for the V band

Dvo = v(star) - v(ref) -k1v*X(star) +k1v*X(ref) where k1v = first
order extinction coefficient for V band Since X(star) and X(ref) are
essentially equal when they are in the same FOV the last two terms
cancel leaving

Dvo = v(star) - v(ref)

using
V = vo + Tv*(B-V) + Zv
and defining
D(B-V) = (B-V)(star) - (B-V)(ref)
then
DV = Dvo + Tv*D(B-V) (the Zv terms cancel out again)
   = v(star) - v(ref) + Tv*D(B-V)

Substituting for DV and vo
V(star)-V(ref) = v(star) - v(ref) + Tv*((B-V)(star) - (B-V)(ref))
V(star) = v(star) - v(ref) + Tv*((B-V)(star) - (B-V)(ref)) + V(ref)

The zero point is again automatically taken care of by the difference
between the transformed v(ref) and V(ref)
V(star) = v(star) + Tv*(B-V)(star)  + ( V(ref) - v(ref)
-Tv*(B-V)(ref)) The last term in parenthesis adds Zv

If you aren't taking data in B, you can use another color index and
transformation coefficient Tv based on the bands you are using e.g.
V-R or V-I.





> Message: 1
> Date: Tue, 15 Jan 2008 11:51:50 -0500
> From: "Lionel Catalan" <lcatalan at lakeheadu.ca>
> Subject: Re: [Aavso-photometry] Determination and use of BVRI
> 	TransformationCoefficients with Maxim DL
> To: <aavso-photometry at mira.aavso.org>
> Message-ID: <E1JEoxn-0003a6-Do at mta2.lakeheadu.ca>
> Content-Type: text/plain;	charset="us-ascii"
>
> Thanks Brad for the suggestion about including the integration time in
> the calculation of instrumental magnitudes.
>
> Lionel
>
>
> -----Original Message-----
> From: aavso-photometry-bounces at mira.aavso.org
> [mailto:aavso-photometry-bounces at mira.aavso.org] On Behalf Of Brad
> Walter
> Sent: Tuesday, January 15, 2008 5:48 AM
> To: aavso-photometry at mira.aavso.org
> Subject: Re: [Aavso-photometry] Determination and use of BVRI
> TransformationCoefficients with Maxim DL
>
>  Lionel, When computing your instrumental magnitudes I would suggest
> that you use the form
>
> minst (star) = -2.5 log (Int(star)/Tint)
>
> Where Tint is the integration time of the image. Otherwise, you will
> not be able to compare instrumental magnitudes obtained with different
> integration times.
>
> Frequently the term differential magnitude is used to denote Minst
> (star) -minst (ref) without adding mstd (ref) being added. See the CBA
> submission format in the help document for CCD/PEP Batch Upload in
> WebObs, http://www.aavso.org/bluegold/webobs.html
>
>
>
> Message: 2
> Date: Sun, 13 Jan 2008 22:49:48 -0500
> From: "Lionel Catalan" <lcatalan at lakeheadu.ca>
> Subject: Re: [Aavso-photometry] Determination and use of BVRI
> 	transformation	coefficients with Maxim DL
> To: aavso-photometry at mira.aavso.org
> Message-ID:
> 	<58b78fe40801131949r4fa5458di30f0060e3d513be0 at mail.gmail.com>
> Content-Type: text/plain; charset=WINDOWS-1252
>
> Based on comments from Arne and Gord Sarty, I thought that the most
> straightforward way of using Maxim photometry data to derive
> transformation coefficients would be to convert Maxim's differential
> magnitudes into instrumental magnitudes.
> This recognizes the fact that transformation equations are written in
> terms of instrumental magnitudes.
>
> Because the photometry analysis tool in Maxim DL only calculates
> differential magnitudes, a special procedure is required to derive
> instrumental magnitudes. Differential magnitudes in Maxim DL are
> calculated using a reference star with known standard magnitude as
> follows:
>
> mdiff (star) = minst (star) ? minst (ref) + mstd (ref)
>
> where mdiff, minst and mstd refer to differential, instrumental and
> standard magnitudes, respectively. The instrumental magnitude of the
> reference star can be calculated as follows:
>
> minst (ref) = -2.5 log (Int(ref))
>
> where Int(ref) represents the intensity of the reference star
> calculated as the sum of all pixel counts within aperture less
> background. The value of Int(ref), which is simply called "Intensity"
> in Maxim DL, can be read from the information window in aperture mode
> by centering the aperture on the reference star while using the
> photometry analysis tool. One must be careful to ensure that the
> centroid of the reference star has the same coordinates when reading
> its intensity and when calculating the differential maginutes of the
> other stars.
>
> Combining the two previous equations gives:
>
> minst (star) = mdiff (star) -2.5 log (Int(ref)) - mstd (ref)
>
> This equation is applied to magnitudes obtained with the B, V, R, and
> I filters using Excel.
>
> Lionel
>
>
>


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