Next: 12. Data for the Up: 11. The Standard Calibration Previous: 11.4 The Standard Catalog

# 11.5 The Transformation Equations

The following transformation equations were used:

 rstd = rinst + r1 + r2 (B-r)inst (11.11) Vstd = Vinst + v1 + v2 (B-V)inst (11.12) Bstd = Binst + b1 + b2 (B-r)inst (11.13) Ustd = Uinst + u1 + u2 (U-B)inst (11.14)

std'' denotes standard magnitudes, and inst'' denotes instrumental magnitudes.

The fitting of the variables in the transformation equations to the data was done using the task fitparams (in digiphotx.photcalx).

At the very first it was tried to use transformation equations without a color term (i.e. with the subscript 2 variables set to zero). The rms scatter was so large, that it was concluded that a color term was needed. As will be seen in the following, the resulting color coefficients are indeed significantly different from zero at the 3 sigma level.

fitparams was run 3 times for the 4 filters:

1.
On all the stars, focused and defocused, with the color term coefficient as a free variable. Plots showing mstd-minst vs. instrumental color for the data points and the resulting fit are in Figure -. Note, that the aspect ratio of the 4 figures is the same, , so the color dependence can easily be compared.
2.
On the focused stars only, with the color term coefficient fixed to the value from fit .
3.
On the defocused stars only, with the color term coefficient fixed to the value from fit .

In all 3 fitting series, a number of stars were deleted (i.e. excluded from the fit) all together, that is, all the observations of the given star were deleted. These stars are listed in Table . They were deleted for a number of different reasons:

• 3 stars (PG1633+099, M67-F81, & SA110-502) were deleted because they were either very blue ((B-V)std mag) or red ((B-V)std 2.3 mag). Since we only need a transformation which is valid for the color range of E/S0 galaxies, say (B-V)std = 0.75-1.1 mag (Jørgensen, private communication), this is perfectly justifiable. The remaining stars still span a sufficient range in (B-V)std, from 0.1 (M67-F153) to 1.2 (SA110-504).
• 1 star (M67-F117) was deleted (in the Gunn r and Johnson V fits only) because it had a large scatter, i.e. the individual observations of this star were very scattered. This can be seen in Figure  for Gunn r, and Figure  for Johnson V. The reason for this was not further investigated.
• 3 stars (M67-F108, M67-F105, & M67-F141) were deleted (in the Gunn r and Johnson V fits only) because they deviated systematically from the fit. This can be seen in Figure  for Gunn r, and Figure  for Johnson V. These stars were the reddest and brightest of the defocused stars. The important question is, whether it is only these reddest and brightest defocused stars that behave differently, or if they are just the most extreme cases of a general trend that the focused and the defocused stars do not behave in the same way.

Table: Deleted stars
 Star Deleted in (B-r) (B-V) (U-B) (B-V) r V Reason for deletion GR JV JB JU instrumental, [mag] standard, [mag] PG1633+099 x x x x 0.24 0.87 1.26 -0.2 14.8 14.4 Very blue M67-F81 x x x x 0.41 0.95 1.88 -0.1 10.4 10.0 Very blue SA110-502 x x x x 3.96 3.02 4.86 2.3 11.4 12.3 Very red M67-F117 x x 1.69 1.69 0.8 12.5 12.6 Large scatter M67-F108 x x 2.44 2.16 1.4 9.3 9.7 Large res., bright M67-F105 x x 2.28 2.07 1.2 10.0 10.3 Large res., bright M67-F141 x x 2.05 1.94 1.1 10.2 10.4 Large res., bright

Notes: The instrumental colors are mean values for the individual observations of the given star. The standard color and magnitudes are taken from Table . The instrumental colors are useful for locating the stars in the mstd-minst vs. instrumental color plots, Figure -. The above stars were deleted in the same way in the 3 fitting series: all stars with free color term, focused stars with fixed color term, and defocused stars with fixed color term.

In addition, individual observations with uncertainties larger than 0.1 mag were deleted. This was only the case for Johnson U, for 9 data points, of which 4 belonged to SA110-502, which would have been deleted anyway. Also individual deviating observations were deleted. This also was only the case for Johnson U, for 3 data points.

The result from the 3 series of fit are shown in Table .

Table: Transformation equation coefficients
 Type Zero point [mag] Color term Scatter [mag] all rms = 0.0185 foc r2 = 0.1244 (fixed) rms = 0.0124 def r2 = 0.1244 (fixed) rms = 0.0187 all rms = 0.0144 foc v2 = 0.0861 (fixed) rms = 0.0164 def v2 = 0.0861 (fixed) rms = 0.0117 all rms = 0.0219 foc b2 = 0.1253 (fixed) rms = 0.0291 def b2 = 0.1253 (fixed) rms = 0.0139 all rms = 0.0390 foc u2 = 0.0187 (fixed) rms = 0.0461 def u2 = 0.0187 (fixed) rms = 0.0256

Notes: all'': all stars fitted, color term free. foc'': only focused stars fitted, color term fixed. def'': only defocused stars fitted, color term fixed. The all'' coefficients are the final ones, i.e. the ones used to standard calibrate the surface photometry.

The reason for fitting the focused and defocused stars separately with forced same color dependence, was to be able to compare their zero points. These zero point differences are shown in Table . There is a zero point difference in Gunn r at the 9 sigma level, and a zero point difference in Johnson V at the 3 sigma level.

Table: Zero point differences
 Zero point def [mag] Zero point foc [mag] Zero point difference [mag] (8.7) (2.8) (0.3) (1.5)

Notes: def'' denotes the defocused stars, foc'' denotes the focused stars. Zero point difference'' is defined as Zero point def'' minus Zero point foc''.

That the focused and defocused stars behave differently in Gunn r and Johnson V can be seen in Figure  and and Figure . The question is what kind of difference it is. The figures could indicate, that it is not only a zero point difference, but also a difference in color dependence.

However, the rms scatter when fitting all the stars together (with the deletions previously mentioned) is still sufficiently low, less than 0.02 mag in both Gunn r and Johnson V. The conclusion is, therefore, to accept the result of the fit of all the stars together (Table ), and then to watch out for small differences in the derived galaxy magnitudes when compared with literature values.

 Zero point [mag] Color term Scatter [mag] rms = 0.0185 rms = 0.0144 rms = 0.0219 rms = 0.0390

Notes: These are the all'' coefficients from Table , listed here alone for convenience.

Finally, we can check if the stars from night 4 and 6 have larger residuals than the stars from the other nights. Figure  shows mean absolute residual for the 3 fields, with the SA110 field split up into the low and the high airmass observations. No such difference is seen, except for the SA110 high airmass observations in Johnson U, but it might not be significant due to the large random scatter of the data in this filter. Otherwise, these plots do not show any correlations with night and field/airmass, except that the night 2 and 3 residual in Gunn r are somewhat higher. The conclusion is still that the standard transformation we have arrived at is acceptable.

Next: 12. Data for the Up: 11. The Standard Calibration Previous: 11.4 The Standard Catalog

Properties of E and S0 Galaxies in the Clusters HydraI and Coma
Master's Thesis, University of Copenhagen, July 1997

Bo Milvang-Jensen (milvang@astro.ku.dk)