11.5 The Transformation Equations

The following transformation equations were used:

    

r_std = r_inst + r₁ + r₂ (B-r)_inst (11.11)

V_std = V_inst + v₁ + v₂ (B-V)_inst (11.12)

B_std = B_inst + b₁ + b₂ (B-r)_inst (11.13)

U_std = U_inst + 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 m_std-m_inst 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, $\Delta y / \Delta x = 2/15$, 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 $\sim -0.1$ mag) or red ((B-V)_std $\sim$ 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 m_std-m_inst 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	$r_1 = +0.7718 \pm 0.0049$	$r_2 = 0.1244 \pm 0.0029$	rms = 0.0185
foc	$r_1 = +0.7816 \pm 0.0012$	r2 = 0.1244 (fixed)	rms = 0.0124
def	$r_1 = +0.7624 \pm 0.0018$	r2 = 0.1244 (fixed)	rms = 0.0187
all	$v_1 = +0.9380 \pm 0.0065$	$v_2 = 0.0861 \pm 0.0038$	rms = 0.0144
foc	$v_1 = +0.9406 \pm 0.0016$	v2 = 0.0861 (fixed)	rms = 0.0164
def	$v_1 = +0.9353 \pm 0.0011$	v2 = 0.0861 (fixed)	rms = 0.0117
all	$b_1 = -0.0523 \pm 0.0055$	$b_2 = 0.1253 \pm 0.0031$	rms = 0.0219
foc	$b_1 = -0.0518 \pm 0.0029$	b2 = 0.1253 (fixed)	rms = 0.0291
def	$b_1 = -0.0526 \pm 0.0012$	b2 = 0.1253 (fixed)	rms = 0.0139
all	$u_1 = -2.2369 \pm 0.0168$	$u_2 = 0.0187 \pm 0.0058$	rms = 0.0390
foc	$u_1 = -2.2404 \pm 0.0047$	u2 = 0.0187 (fixed)	rms = 0.0461
def	$u_1 = -2.2320 \pm 0.0031$	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]
$r_1 = +0.7624 \pm 0.0018$	$r_1 = +0.7816 \pm 0.0012$	$\Delta = -0.0192 \pm 0.0022$ (8.7$\sigma$)
$v_1 = +0.9353 \pm 0.0011$	$v_1 = +0.9406 \pm 0.0016$	$\Delta = -0.0053 \pm 0.0019$ (2.8$\sigma$)
$b_1 = -0.0526 \pm 0.0012$	$b_1 = -0.0518 \pm 0.0029$	$\Delta = -0.0008 \pm 0.0031$ (0.3$\sigma$)
$u_1 = -2.2320 \pm 0.0031$	$u_1 = -2.2404 \pm 0.0047$	$\Delta = +0.0084 \pm 0.0056$ (1.5$\sigma$)

 

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.

 

Table: Adopted transformation equation coefficients

Zero point [mag]	Color term	Scatter [mag]
$r_1 = +0.7718 \pm 0.0049$	$r_2 = 0.1244 \pm 0.0029$	rms = 0.0185
$v_1 = +0.9380 \pm 0.0065$	$v_2 = 0.0861 \pm 0.0038$	rms = 0.0144
$b_1 = -0.0523 \pm 0.0055$	$b_2 = 0.1253 \pm 0.0031$	rms = 0.0219
$u_1 = -2.2369 \pm 0.0168$	$u_2 = 0.0187 \pm 0.0058$	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.