[Aavso-photometry] Calculating mag errors for filter bands from color indices

[Wolfgang Renz]

Hello


Whats the accepted standard in calculating mag errors for filter bands
from color indices ?
Just the usual error propagation that will overestimate the error ?


Calculating i' via
    (I)        i' = r' - (r'-i')            
gives a value of i' = 14.469.

Calculating rms i' with usual error propagation via
    (II)        rms i' = SQRT( SQR(rms r') + SQR(rms r'-i') )

But as nobody ever measured a color index directly the rms r'-i' must have
come from subtracing the color mags and the rms r'-i' probably from
    (III)        rms r'-i' = SQRT( SQR(rms r') + SQR(rms i') )

Inserting (III) into (II) shows that the rms r' is counted twice using usual
error propagation
    (IV)        rms i' = SQRT( SQR(rms r') + SQR(rms r') + SQR(rms i') )
leading to a too big error estimate. 

Converting (III) to rms i' gives
    (V)        rms i' = SQRT( SQR(rms r'-i') - SQR(rms r') )
numerically
    (VI)        rms i' <= SQRT( SQR(rms r'-i'[rounded up]) - SQR(rms r'[rounded down]) )
assuming all given rms were rounded up and
    (VII)        rms r'[rounded down] = rms r' - 1 significant digit

Taking G 158-100 as an example:
Usual error propagation (II) gives a value of rms i' = 0.0092 rounded up 0.010.
While (VI) leads to a alternative value of  rms i' <= 0.0049 rounded up 0.005.
Checking this value via (III) gives rms r'-i' = 0.0078 (>= 0.007 as listed in the ref).


Calculating z' via
    (VIII)        z' = r' - (r'-i') - (i'-z')
gives a value of z' = 14,377.

Calculating rms z' with usual error propagation via
    (IX)        rms z' = SQRT( SQR(rms r') + SQR(rms r'-i') + SQR(rms i'-z') )

But as again nobody ever measured a color index directly the rms i'-z' must have
come from subtracing the color mags and the rms i'-z' probably from
    (X)        rms i'-z' = SQRT( SQR(rms i') + SQR(rms z') )

Inserting (III) into (IX) shows again that the rms r' is counted twice using usual
error propagation although r' wasn't probably used at all to calc z'
    (XI)        rms z' = SQRT( SQR(rms r') + SQR(rms r') + SQR(rms i') + SQR(rms i'-z') )
leading to a too big error estimate. 

Converting (X) to rms z' gives
    (XII)        rms z' = SQRT( SQR(rms i'-z') - SQR(rms i') )
numerically
    (XIII)        rms z' <= SQRT( SQR(rms i'-z'[rounded up]) - SQR(rms i'[rounded down]) )
Inserting (V) into (XIII) gives
    (XIV)        rms z' <= SQRT( SQR(rms i'-z'[rounded up]) - ( SQR(rms r'-i'[rounded down]) - SQR(rms r'[rounded up]) ) )
or simpler
    (XV)        rms z' <= SQRT( SQR(rms i'-z'[rounded up]) - SQR(rms r'-i'[rounded down]) + SQR(rms r'[rounded up]) )
assuming all given rms were rounded up and
    (XVI)        rms r'-i'[rounded down] = rms r'-i' - 1 significant digit

Taking G 158-100 as an example:
Usual error propagation (IX) gives a value of rms z' = 0.0136 rounded up 0.014.
While (XV) leads to a value of rms z' <= 0.0100 rounded up 0.010.
Checking this alternative value and the alternative rms i' via (X) gives
rms i'-z' = 0.0111 (>= 0.010 as listed in the ref).


Simillarly would be
    (XVII)        rms g' <= SQRT( SQR(rms g'-r'[rounded up]) - SQR(rms r'[rounded down]) )
assuming all given rms were rounded up and
    (XVIII)        rms r'[rounded down] = rms r' - 1 significant digit

Simillarly would be
    (XIX)        rms u' <= SQRT( SQR(rms u'-g'[rounded up]) - SQR(rms g'-r'[rounded down]) + SQR(rms r'[rounded up]) )
assuming all given rms were rounded up and
    (XX)        rms g'-r'[rounded down] = rms g'-r' - 1 significant digit


Is it allowed to use the smaller error estimates (VI) for rms i',  (XV) for rms z', (XVII) for
rms g', and (XIX) for rms u' as long as the conditions 0 < rms r' <= rms r'-i' <= rms i'-z'
and 0 < r' <= rms g'-r' <= rms u'-g' are fullfilled ?
These conditions are true for all SDSS standard stars in the reference below !


Clear skies
  Wolfgang

-- 
Wolfgang Renz, Karlsruhe, Germany
Rz.BAV = WRe.vsnet = RWG.AAVSO



Reference:

The u'g'r'i'z' Standard Star System:
Main Page
http://home.fnal.gov/~dtucker/ugriz/tab08.dat
The u'g'r'i'z' standard star network: calibrated magnitudes and colors (Table 8 of Smith et al. 2002, AJ, 123, 2121)
====================================================================================================================
A machine-readable version of this table is also available at www.journals.uchicago.edu/AJ/journal/issues/v123n4/201445/datafile8.txt
StarName RA (J2000.0) DEC (J2000.0) r' u'-g' g'-r' r'-i' i'-z' rms r' rms u'-g' rms g'-r' rms r'-i' rms i'-z' n u' n g' n r' n i' n z' 
-------- ------------ ------------- -- ----- ----- ----- ----- ------ --------- --------- --------- --------- ---- ---- ---- ---- ----
...
G 158-100 00:33:54.60 -12:07:58.9 14.691 1.101 0.510 0.222 0.092 0.006 0.019 0.008 0.007 0.010 6 7 8 6 6 
...