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Stability of Transformation Coefficients After 4 months?

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Stability of Transformation Coefficients After 4 months?

Hello! I rechecked my transformation coefficients last night. I obtained the firsts set of coefficients about 4 months ago using M67 and some Landolt standards on a single nights run. The values were


B = 0.179    V = -0.039    I = -0.017  and B-V = 1.227


    Over the last two nights, I rechecked the coefficients using about 30 Landolt standard stars spaced every 15 degrees from 30 degrees to 90 degrees. The coefficients were the same when calculated on each individual night and when combined.


B = 0.140  V = -0.038    I = 0.016 (positive while the previous value was negative)   B-V = 1.043


    The results seem close. Are they close enough so that I would consider my system stable for 4 months, or should I obtain transformation coefficients on shorter intervals? If these look fine, can I increase the time between transformation coefficient checks?

    Thank you and best regards.



MZK's picture
Transformation Coefficient Stability

First question you might ask yourself is what was the precision (error) associated with the first set of transform coeff values (e.g., B = 0.179 + ??). This will tell you something about variablity in your measurements.

Did you really get exactly the same values on the second pair of nights? I suspect not. There will also be some uncertainty for them. Are the second set of values within the error of the first set?

I have observed significant variability in measured transform coeff depending on the quality of the sky on a given night. I'm in a suburban environment with far from "photometric" conditions.

IF the second set of values are within the error of the first set, I would not feel that it is necessary to revise your values. The system (optics, filters, ccd) has not changed significantly. If it has you may consider revising the values.

Finally, think about applying the two sets of values to some target comparison stars and determine how much the magnitudes have changed (e.g., 0.1%, 1%, 10%)? What error do you want to accept in your data?



HQA's picture
coefficient stability

Hi Mike (and Ken),

The transformation coefficients themselves should not change on long time scales (perhaps over years), as long as your system configuration remains constant.  In order for a transformation coefficient to change, you would have to have something change color in the optical train.

As Ken mentions, the real question between the two determinations is the uncertainty in each.  I'm suspicious of your (B-V) determination, for example, as those two coefficients are quite different from one another.  You should calculate your coefficients on several nights (say, 3-5), determine mean and standard deviation for each coefficient.  I'd calculate coefficients on a quarterly basis at first, and once you are sure you are getting the same result, then perhaps space out the coefficient calculations to once per year.

In general, if you are doing differential photometry using AAVSO comparison stars, the color difference between your target star and the comparison stars will be under 1 magnitude (except, perhaps, for Miras).  If you have a 3% uncertainty in a coefficient that is on the order of 0.1mag/color_index, then you will cause a shift in the standardized magnitude of perhaps 0.003mag, usually negligible.  Transforming is important, but the error in the determination of the coefficient is a second-order effect.


Hello! Thank you both for

Hello! Thank you both for your responses.

    I used Canopus for transform calculations. I used Landolt standards on 4 and 6 August. In mid-May, I used M67 and a some Landolt standards.

    On 4-August, my values were:

          B=b + 0.112(B-V) + 19.673     and s.d. = 0.102

          V=v - 0.051(B-V) + 19.673        and s.d. = 0.064

          I=i + 0.018(B-V) + 18.526         and s.d. = 0.137

          (B-V)std = 1.106(b-v)inst - 0.448     and s.d. = 0.151


    On 6-August, my values were:

          B=b + 0.188(B-V) + 18.929     and s.d. = 0.201

          V=v - 0.036(B-V) + 19.618        and s.d. = 0.057

          I=i + 0.0.014(B-V) + 18.602         and s.d. = 0.098

          (B-V)std = 1.098(b-v)inst - 0.610     and s.d. = 0.221


    Combining both gives:

          B=b + 0.154(B-V) + 19.069     and s.d. = 0.185

          V=v - 0.044(B-V) + 19.645       and s.d. = 0.065

          I=i + 0.0.015(B-V) + 18.564         and s.d. = 0.090

          (B-V)std = 1.076(b-v)inst - 0.490     and s.d. = 0.204


    It appears that the values on 6-August have a better error range.


    My values from mid-May were similar, but the error ranges were about 3 to 5 times these error ranges. I'm sorry, I did not keep the equations and s.d.s, but I can re-calculate them, if needed. What I remember is that M67 values alone had a high s.d. (about 5% of the intercept values, I believe)and the s.d.s became better when I added several Landolt stars to the data.

    If I am interpreting these values correctly, based on the intercept values, the s.d. is about 1/2% to 1%, and seem consistent between the two nights? Quarterly checks would be useful to follow the stability of the coefficients, and if stable after several checks, I can space them further apart?

    Thank you and best regards.




WBY's picture


I see a couple of potential issues with your transformation coefficients.

First, normally the filter of interest, say I, is one of the filters in the color index  

v-i = Cvi*(V-I) + C1 or

r-i = Cri*(R-I) + C2

this may not be absolutely necessary but it is usually done because you want the band you are transforming to be within the spectral range you are using to create the transform. Remember that these transformations are linear estimates for response curves and spectral distributions of light from stars that are far from linear. 

the second is that your transformation coefficients aren't in the form you need to use them

Rearranging these into a form that you want

(V-I) = 1/Cvi*(v-i) +(-C1/Cvi) = 1/Tvi*(v-i) + Zvi or 

(R-I) = 1/Cri*(r-i)+(-C2/Cri) = Tri*(r-i)+ Zri

similarly you end up with 

B-V = Tbv*(b-V) + Zbv

V-R = Tvr*(v-r) +Zvr

Also you have 

V-v = Tv*(B-V) +Cv

using s for the target star and c for the comp star you get the following in differential photometry

(B-V)s - (B-V)c = Tbv[(b-v)s-(b-v)c]  (the Zbv terms cancel) 
this expression will be used in substitution below


(V-v)s - (V-v)c = Tv*[(B-V)s - (B-V)c] (the Cv terms cancel) rearranging

Vs =vs + (V-v)c + Tv*[(B-V)s - (B-V)c]
from above we have an expression for [(B-V)s - (B-V)c] in terms of instrumental magnitudes. Substituting and doing a little rearranging you get

Vs = vs-vc + Vc +Tv*Tbv*[(b-v)s - (b-v)c]

then for each of the other color filters you have, for example, 

Bs = (B-V)s + Vs

Rs = Vs - (V-R)s

Is = Vs - (V-I)s or using Rs from the previous expression Is = Rs - (R-I)s


You don't have to worry about extinction in differential photometry unless you are doing such precise photometry that second order extinction matters because the k'X terms (X=airmass) for the target and comp cancel out since two stars in the same FOV are at essentially the same airmass. 

However, if you are calculating Transformation coefficients from standard star fields or clusters using raw magnitudes {-2.5*Log (Counts*Gain/Exptime) then you do have to convert all of your instrumental magnitudes to extra-atmospheric instrumental magnitudes (the ones you often see in the literature with a "o" subscript) because there is nothing to cancel out the k'X terms that are included in the instrumental magnitudes you measured.  In this situation you don't use a comp star. When you are imaging standard stars with a wide range of colors for the purpose of establishing transformation coefficients, why would you want to bias your transformations by using a comp star of a specific color? So you take images over a wide range of Airmass - from 3 up to 1 if you can -  and you sequence your images in IRVBUUBVRI groups and average the pairs of measurements so that the averages in a group are all as close to the same point in time (airmass) as possible. Short wavelengths are closest together since they are affected the most by airmass changes. then you establish extinction coefficients for each color using the change in raw magnitudes of the standard stars vs X. Do do a linear fit to all of the standards you measure at the various airmasses for each filter and you have the extinction coefficients. The Extra atmospheric raw magnitude for a star is the instrumental magnitude you get by extrapolating its measured magnitudes at various airmasses in a particular filter to airmass zero. There will be scatter in the results for each star. Average the resulting zero airmass values for each star.  

I hope this a help rather than a confusion. 

There was a step by step explanation of creating transformation coefficients by Bruce Gary that was linked to a version  of the AAVSO CCD Manual from about 10 years ago. I found it very helpful and have attached a copy. 

NGC 7790 is a standard cluster that is well positioned now to image over a wide range of airmasses from altitude ~27 degrees (airmass 2.2) to about 59 degrees (airmass 1.15) on the meridian. 

WBY's picture
Transformation coefficients


Did you image the same Landolt standards at each of the positions separated by 15 degree intervals, or were they different stars at the three positions? If different stars, at each position,  how did you correct for extinction?

Brad Walter, WBY

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