[Aavso-photometry] Determination and use of BVRI

[Brad Walter]

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
> 
> 
>