# Definitions of Transformation Coefficients (again...)

Some years ago I posted on "AAVSO-photometry" a question about the "right" color index to use in the equations of the trasformation coefficents of BVRcIc filters.

This was the reply: 12/28/08….. Arne to aavso-photometry group “RE Transformation Coefficients-Color Index’s The normal guideline is that the color index you choose needs to include the wavelength range of your filter. This is to remove as many systematics as possible - always interpolate, never extrapolate. Your chosen color indices will work; the ones for B and V are pretty much standard. I prefer to use (V-Ic) as the color index for Rc and Ic. (V-Ic) has more change with star color and so results in a better transformation. For example, an A0 star has (V-Ic) = (Rc-Ic) = 0.00. However, for a G2 star, (B-V) is about 0.63, while (Rc-Ic) is about 0.33, or half as large. The transformation coefficients determined using (V-Ic) are more accurate since fitting something with wide range + error is easier than fitting something with smaller range + same error. (V-Ic) is also more sensitive to interstellar extinction. Using (Rc-Ic) means you use Rc, a bandpass that has Halpha in it, a very prominent line that can be in emission or absorption, causing an error in transformation. This is very obvious with novae, where Halpha emission can dominate the continuum in the Rc bandpass.” That being said, my use of (V-Ic) is a preference, not a rule. (Rc-Ic) will work.” Arne

Since that time I'm using the following equations for transformation (simplified without extinction and zero point not important now):

B(std) = b(instr) + T(b)*(B-V)

V(std) = v(instr) + T(v)*(B-V)

Rc(std) = r(instr) + T(r)*(V-Ic)

Ic(std) = i(instr) + T(i)*(V-Ic)

Now I have another question: given the above formulas, which are the best definitions of transformation coefficients?

Tb = slope of (B-b) plotted versus (B-V) should be ok;

Tv = slope of (V-v) plotted versus (V-Rc) or versus (B-V)? Why?

Tr = slope of (R-r) plotted versus (Rc-Ic) or versus (V-Ic)? Why?

Ti = slope of (I-i) plotted versus (V-Ic)?

Again, if I'm using BVIc filters (without Rc), then:

B(std) = b(instr) + T(b)*(B-V)

V(std) = v(instr) + T(v)*(B-V) or (V-Ic)? Why?

Ic(std) = i(instr) + T(i)*(V-Ic)

and, in this case,

Tv = slope of (V-v) plotted versus (B-V) or versus (V-Ic)?

Maybe to much questions... Sorry, but I have found different solutions on this topic and I am bit confused. Thank you very much in advance for any help!

Massimiliano (Max) MMN

Dear Arne,

thank you very much to have solved my doubts on the topic.

Best regards!

Massimiliano (MMN)

Please bear with me everyone. This is a bit long, but I think it poses an interesting question that derives from Massimiliano’s question above. I hope I haven’t made any math mistakes.

In the equations below

“_t” subscript indicates the target star

“_c” subscript indicates the comp star

“raw” subscript indicates a raw magnitude = -2.5*Log(net flux) = -2.5*Log(net photons/EXPTIME)

“inst” subscript is the instrumental magnitude obtained from many differential photometry programs

Z_{b}, Z_{v} and Z_{bv} are zero points

B,V, and(B-V) are magnitudes transformed to the standard J-C system

I am confused by exactly what quantities are being called instrumental magnitudes in the previous discussion.

Assume we are working with simple differential photometry with the target star and comp star in the same field of view less than 1 degree so you can regard them as having the same air mass. Further, second order extinction is assumed to be negligible. Therefore, extinction correction terms can be left out.

The way many commonly used programs calculate the instrumental magnitude of a target star in differential photometry is simply by adding the sequence magnitude you enter for the comp star to the difference between the raw magnitudes of the target and comp stars as in the following:

b_{inst_t} = b_{raw_t} – b_{raw_c} + B_{_c}. Depending on exactly how B_{_c} was determined, this will be exactly or extremely close to

b_{inst_t} = b_{raw_t} – b_{raw_c} + V_{_c} + (B_{_c} - V_{_c}) = b_{raw_t} – b_{raw_c} + V_{_c} +(B-V)_{_c}

Also:

v_{inst_t} = v_{raw_t} – v_{raw_c} + V_{_c}

Where B_{_c} and V_{_c} are the sequence reference magnitude you enter for the comp star These are shown as capital letters since they are transformed magnitudes. Then:

(b-v)_{inst_t} = b_{inst_t }– v_{inst_t }

= b_{raw_t} – b_{raw_c} + B_{_c} – (v_{raw_t} – v_{raw_c} + V_{_c})

= (b_{raw_t} – b_{raw_c)} - (v_{raw_t} – v_{raw_c}) + (B_{_c} – V_{_c})

= (b_{raw_t} – b_{raw_c}) - (v_{raw_t} – v_{raw_c}) + (B – V)_{_c}

The equations for transformation equations give the following for any star (if extinction is an issue either because you are doing all-sky or second order extinction cannot be ignored, then the lower case letters become the extra atmospheric values after extinction correction)

(B-V) = Tbv(b_{raw} –v_{raw}) +Z_{bv}

V = v_{raw} +Tv(B-V) + Z_{v }

in particular

(B-V)_{_c }= Tbv(b_{raw_c} – v_{raw_c}) +Z_{bv}, and

V_{_c} = v_{raw_c} +Tv(B_V)_{_c} +Z_{v}

Substituting these into the equation for (b-v)_{inst_t} and v_{inst_t} above give the following:

(b-v)_{inst_t} = (b_{raw_t} – b_{raw_c}) - (v_{raw_t} – v_{raw_c}) + Tbv(b_{raw_c} – v_{raw_c}) +Z_{bv}

= ∆(b-v)_{_t-c} + Tbv(b_{raw_c} – v_{raw_c}) +Z_{bv}, and

v_{inst_t} = v_{raw_t} – v_{raw_c} + v_{raw_c} +Tv(B_V)_{_c} +Z_{v}

= v_{raw_t} +Tv(B_V)_{_c} +Z_{v}.

If I try to apply transformations to these “instrumental” values I get results that don’t seem to be the quantities you want. You get

??bv = Tbv[∆(b-v)_{_t-c} + Tbv(b_{raw_c} – v_{raw_c}) +Z_{bv},],

= Tbv[∆(b-v)_{_t-c} + Tbv( b-v)_{raw_c} + Z_{bv}], and

??v = Tv[v_{raw_t} +Tv(B_V)_{_c} +Z_{v}]

These seem to be very messed up.

The quantities you want are

(B-V)_{_t} = Tbv[∆(b-v)_{_t-c} + ( b-v)_{raw_c}] +Z_{bv},

= Tbv(b-v)_{raw_t} +Z_{bv} and

V_{_t = }v_{raw_t} + Tv(B-V)_{_t} +Z_{v}

If however, instrumental magnitudes means raw magnitudes plus the appropriate zero point then the equations in Massimiliano’s post work out for transforming single filters. However, it appears that if you are transforming a color index, like (b-v) to the corresponding standard index you have to apply the transformation coefficient to the raw color index and add the zero point afterward.

Many of these programs have been in use for decades and I haven’t seen anyone mention this issue. So I am must be missing something. Can someone show me where I have gone off track? Otherwise, you can’t simply apply transformation coefficients to the “instrumental” magnitudes that you get from doing simple photometry with programs like Maxim DL.

Brad Walter, WBY

Hi Brad,

The problem in your derivation is that you are assuming the calibration magnitudes for the comp stars are derived from raw magnitudes obtained from your images, rather than being supplied catalog values. That ends up with a double substitution. Things like Bc, Vc, (B-V)c are constants, and you cannot substitute for them. You either do everything with instrumental magnitudes, or everything using calibration magnitudes. Don't complicate the process more than necessary! Your nomenclature also is a bit inconsistent. I recommend holding off until I've finished the transformation workshop, and then see if my presentation makes more sense.

Arne

Can't Wait. I thought about the issue you mentioned but it seemed to me that it didn't make a difference since they are the transfomrd values I should obtian from my system but let's stop here and hopefully all will be clear in the workshop.

One point that would be helpful now however, is to confrim whether transformation magnitude is the same as catalog magnitude in your post.

WBY

Hi Massimiliano,

If you multiply the coefficient by some color index, then that color index is what you use to determine the slope. So in your cases, Tv = slope of (V-v) plotted vs. (B-V), Tr = slope of (Rc-rc) plotted vs. (V-Ic), Ti = slope of (Ic-ic) plotted vs. (V-Ic).

It is true that you could have

V-v = Tv*(B-V)

or

V-v = Tv*(V-Ic)

but those are two different Tv coefficients. I usually differentiate by labeling these Tv_bv and Tv_vi. Both will give transformed V magnitudes; the results will be slightly different depending on both the type of star that is being transformed, as well as the random error involved in the determination of the coefficient(s). My preference is to use (B-V) for transforming v.

Arne