Definitions of Transformation Coefficients (again...)

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

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