# Colour term coefficients and zeropoints

Chris Pritchet - June 08, 2006 (revised June 19, 2006)

Summary. Smith et al. standard stars observed from 2003 to July 20 2005 are used to derive transformation equations to put MegaCam observations on the (i) USNO and (ii) SDSS systems. Colour term coefficients can be found in Section 4 , and zeropoints can be found in Section 5. It's best to transform to the USNO system, and use the zeropoints to define a natural MegaCam system that matches the USNO system at zero colour.

## 1. Definitions of Colour Term Coefficients and Zeropoints

The MegaCam natural photometric system is close to, but not exactly equivalent to, the USNO Smith et al. (2002 AJ 123, 2121) system or the SDSS system (it is closer to the USNO system). The transformation equations to the SDSS system are approximately linear, at least over moderate range of colour, and look like

 g(SDSS) = g'(instrum) + c_g*(g-r) - k_g*(secz-1) + zp_g r(SDSS) = r'(instrum) + c_r*(g-r) - k_r*(secz-1) + zp_r i(SDSS) = i'(instrum) + c_i*(r-i) - k_i*(secz-1) + zp_i z(SDSS) = z'(instrum) + c_z*(i-z) - k_z*(secz-1) + zp_z

where c_g,c_r,c_i,c_z are colour coefficients, k_g,k_r,k_i,k_z are atmospheric extinction coefficients, and the zp_g,zp_r,zp_i,zp_z are the zeropoints. Some notes:

• CFHT uses g-z for the z' equation.
• Extinction coeffs - Note that the extinction coefficient is assumed to be positive. (In the CFHT Fits headers, it's negative!) Here we adopt the CFHT values for the extinction coefficients.
• MegaCam to USNO transform - Similar equations are used to transform MegaCam mags to the USNO (Smith et al.) system.
• USNO to SDSS transform - The transformation equations between USNO and SDSS systems can be found at this link . A recent (May 2006) message from D. Schlegel confirms these transformation equations.
• Colours on RHS are true, not instrumental - The colours on the RHS of the transformation equations are in the SDSS system, something that is not terribly useful for use on real data (where you want mags and colours on the RHS to be observables). You'll have to do some algebra to figure out the transformation equations with CFHT colours - or iterate. I'm assuming for now that Don did the same thing when comparing MegaCam mags to SDSS mags in D2/D3 (see below).
• Non-linearities - There may be small non-linearities in these transformations, but, given how close the MegaCam filters are to the standard system(s), these will be small. See the next section for more.

## 2. Importance of Colour Term Coefficients

It can be shown that the zeropoint for the "natural MegaCam system", which we're going to use for SNLS, is the same as the zeropoint of the system that you're transforming to (i.e. colour=0). If

m(SDSS) = m(instrum) + c*col - k*(secz-1) + zp

then

m(natural) = m(instrum) - k*(secz-1) + zp

where zp is the same zp computed above. Go to this link for a derivation of this result. If you're calculating transformation equations to SDSS, then this will put the MegaCam filters on the SDSS zeropoint system (not quite AB); or if you're calculating transformation equations to USNO (Smith etal.), then this will put the MegaCam filters on the USNO zeropoint system (again, not quite AB).
• USNO or SDSS? - It makes more sense to determine the zeropoints using the transform directly to Smith et al. (USNO) standards. This is because (i) the Smith et al filters are closer to the MegaCam filters; (ii) this avoids the (somewhat uncertain?) transformation from USNO to SDSS; and (iii) the correction to AB mags can be done directly from Smith et al observations of AB standards, or our observations of Calspec standards.
• Non-linearities - Non-linearities could affect the determination of the zeropoint at which zero-colour is reached. Imagine a colour term that is some function (not necessarily linear) of colour; let's assume that it's a combination of a linear function plus some additional functional dependence:

m(SDSS) = m'(instrum) + c*[col + f(col)] - k*(secz-1) + zp

You can see that this will screw up the determination of the zeropoint for the natural system, if you erroneously assume that the colour term is linear. For most of the filters this isn't too much of a problem because (i) the colour terms are small, (ii) there's no evidence of linearity, and (iii) the stars measured cover a range of colours on both sides of colour=0. But there could be a problem for g', where you have to extrapolate to reach zero colour (not much, though), and the colour term is larger.
• AB system? - The zeropoints that are determined will be on the same zp system as the USNO or SDSS systems (depending on which was used to determine the transformation). These will need a small correction to get them on the AB system, which is not the subject of this web page.

## 3. Determination of Colour Coefficients

Here is a detailed description of the various methods used to determine the colour term coefficients. A summary section can be found below at this link.
• These are colour terms for transforming MegaCam magnitudes to the Smith et al or SDSS systems. The colour used for the colour term coefficient is given in the column heading. The basic equations for the transformation are found above.
• Typical statistical errors in the colour coefficient determinations are +-0.015 to +-0.02 for gri, and +-0.03 for z. However, the true errors are better given by the differences between different methods.
• The most relevant section is the shaded area in the table: Transformation to SDSS with colour cuts. The transformation to the Smith et al system, avoiding use of the SDSS, is of interest because it doesn't use the (somewhat uncertain) Smith -> SDSS transformation, so it's more direct for getting the zeropoints at zero colour (which are the natural MegaCam system zeropoints - see above).
• Explanations and Plots for the different methods: see the links under Comments. The first 3 methods use CJP's program analstds.f.
• Differences from May 30 analysis:
• Only data up to 2005Jul20 for Smith standards. After this date there are processing problems, says J-C Jun 7, 2006.
• Also tried the limited time window Dec 2004 - Jul 2005 since Dec 2004 is when the L3 flip occurred.
• Some improvements in software, especially weighting of points.

 Type of transform Stars Comments g (g-r) r (g-r) i (r-i) z (i-z) Transformations to USNO (Smith et al.) Mega to USNO Smith stds to 2005Jul20 analstds: pairs of stars, same night (Theil/lsq) 0.090/0.085 0.016/0.007 0.051/0.038 -0.058/-0.054 Mega to USNO Smith stds to 2005Jul20 analstds: pairs of stars, same exp (Theil/lsq) 0.092/0.086 0.017/0.008 0.052/0.039 -0.044/-0.043 Mega to USNO Smith stds to 2005Jul20 analstds: single stars, Q-run corrected (Theil/lsq) 0.096/0.088 0.021/0.008 0.060/0.048 -0.039/-0.060 Transformations to SDSS Mega to SDSS Smith stds to 2005Jul20 analstds: pairs of stars, same night (Theil/lsq) 0.144/0.139 0.032/0.022 0.094/0.081 -0.077/-0.078 Mega to SDSS Smith stds to 2005Jul20 analstds: pairs of stars, same exp (Theil/lsq) 0.146/0.140 0.033/0.023 0.095/0.082 -0.063/-0.067 Mega to SDSS Smith stds to 2005Jul20 analstds: single stars, Q-run corrected (Theil/lsq) 0.147/0.140 0.036/0.022 0.097/0.083 -0.061/-0.081 Mega to SDSS SDSS DR5 D2 [from Don] 0.143/0.144 0.000/0.002 0.072/0.078 -0.075/-0.078 Mega to SDSS SDSS DR5 D3 [from Don] 0.161/0.160 0.006/0.012 0.085/0.088 -0.064/-0.073 Mega to SDSS synthetic (all) [from Mark May 24] 0.143 0.020 0.085 -0.035 Transformations to SDSS, with colour cut Mega to SDSS Smith stds to 2005Jul20 (col cut) analstds: pairs of stars, same night (Theil/lsq) 0.152/0.146 0.031/0.021 0.092/0.081 -0.077/-0.078 Mega to SDSS Smith stds to 2005Jul20 (col cut) analstds: pairs of stars, same exp (Theil/lsq) 0.156/0.149 0.031/0.022 0.092/0.080 -0.063/-0.067 Mega to SDSS Smith stds to 2005Jul20 (col cut) analstds: single stars, Q-run corrected (Theil/lsq) 0.155/0.145 0.034/0.020 0.098/0.087 -0.062/-0.083 Mega to SDSS SDSS DR5 (col cut) D2 [from Don] 0.140/0.147 0.011/0.007 0.068/0.067 -0.082/-0.089 Mega to SDSS SDSS DR5 (col cut) D3 [from Don] 0.160/0.160 0.015/0.016 0.073/0.074 -0.085/-0.014! Mega to SDSS French (col cut) 0.156 0.000 0.094 -0.050 Mega to SDSS synthetic (col cut) [from Mark May 24] 0.158 0.019 0.086 -0.038 Transformations to SDSS, with colour cut (limited time interval 2004Dec to 2005Jul) Mega to SDSS Smith stds 2004Dec-2005Jul20 (col cut) analstds: pairs of stars, same night (Theil/lsq) 0.154/0.150 0.045/0.030 0.087/0.070 -0.109/-0.098 Mega to SDSS Smith stds 2004Dec-2005Jul20 (col cut) analstds: pairs of stars, same exp (Theil/lsq) 0.158/0.153 0.046/0.036 0.092/0.079 -0.114/-0.010 Mega to SDSS Smith stds 2004Dec-2005Jul20 (col cut) analstds: single stars, Q-run corrected (Theil/lsq) 0.150/0.147 0.038/0.026 0.086/0.072 -0.104/-0.112

## 4. Summary of Colour Coefficient Results

The following is a synthesis of the above results. I'm going to use both USNO transformations and SDSS transformations with a colour cut (which is the shaded area above). I'll also use all data to 2005Jul20, and always use the Theil slope (which seems to be much more robust than least squares even with outlier rejection).
• The results are compared with the French results, Mark's synthetic values, and also the result obtained using the transform to the Smith et al. system, combined with the slope of the Smith to SDSS transform. (This was complicated for r', because the Smith to SDSS transform is in terms of r-i, whereas we do it in terms of g-r!)
• The effects of a small error in slope on the zeropoint have been investigated below - they are small.
• There are 3 nagging worries:
1. We find systematically larger colour coefficient slopes (r'i'z', but not g') using Smith et al stars, compared to the SDSS D2 and D3 stars. Not sure of the reason behind this. The effect is around 0.03 in the colour coefficient. Possibly an error in the Smith to SDSS conversion?
2. D2 and D3 give systematically different slopes in g'.
3. The data from the limited time window Dec 2004 - Jul 2005 gives somewhat different results in r' and z'. I think this demonstrates the limitations of the method: this time window has fewer stars, and so is dominated more by chip-to-chip calibration errors.

 Filt Comments SDSS transform mean slope extreme range Smith transform mean slope g' Do a straight mean of the 5 determinations (analstds and Don's D2/D3 analysis). Use only Theil slope estimator. Just average over the difference between D2 and D3. Result is 0.153, with an extreme range +0.007 - 0.013. This agrees well with the French 0.156 and synthetic 0.158. It also agrees very well with the direct MegaCam to Smith et al transform (0.093 + 0.060 = 0.153, where the last term is the quoted colour transform going from Smith to SDSS) 0.153*(g-r) +0.007 -0.013 0.093*(g-r) r' r' is more difficult than g'; there is quite a difference between SDSS D2/D3 determinations on the one hand, and Smith et al stds on the other. But there definitely seems to be a postive slope, contrary to the French (and CFHT) determination of slope=0. A straight average of the 5 methods gives 0.024, with an extreme range of +0.010 and -0.013. The transform to the Smith et al system has a slope 0.018, and correcting to SDSS (0.018 + 0.035*0.6 = 0.039, where the 0.035 is the Smith->SDSS coefficient in r-i, and the 0.6 is the rough slope d(g-r)/d(r-i) [since our colour term is in terms of g-r]), the agreement is not too bad. The agreement with the synthetic term, 0.019, is OK too. 0.024*(g-r) +0.010 -0.013 0.018*(g-r) i' Again, an offset between Smith et al stars (slope around 0.095) and SDSS stars (0.07). Straight mean give 0.085, with a range +0.013 -0.017 (better than May 30 results). Good agreement with synthetic and French values, and also transform to Smith, corrected to SDSS (0.054 + 0.041 = 0.095). 0.085*(r-i) +0.013 -0.017 0.054*(r-i) z' Large range, straight mean of 5 methods gives -0.074. Much better consistency with the 5 methods than for May 30 results. Agreement with French (-0.05) and synthetic (-0.038) is not so good however. The MegaCam to Smith et al transform coeff (-0.047) coupled with Smith to SDSS ( -0.03) gives -0.077 - spectacular agreement! -0.074*(i-z) +0.013 -0.011 -0.047*(i-z)

## 5. Zeropoints

Once the colour term coefficients are known ( see above), the zeropoints follow automatically. The following table gives zeropoints for each of g'r'i'z', for transformation to the Smith et al. USNO system, and also the SDSS (DR5 - Schlegel private communication) system. The zeropoints for the so-called "MegaCam natural system" are the same as these zeropoints (this was proven above).

 Transform to USNO Transform to SDSS g' Q run averages Individual stars Q run averages Individual stars r' Q run averages Individual stars Q run averages Individual stars i' Q run averages Individual stars Q run averages Individual stars z' Q run averages Individual stars Q run averages Individual stars

The columns marked "Q run averages" give zeropoints averaged over a queue run: The data in these files is:

• running #
• zeropoint
• assumed colour term coefficient
• assumed extinction coefficient (from CFHT)
• 1 sigma scatter of stars around zeropoint (after 5 iterations of 2.3 sigma clipping)
• # of stars surviving after sigma clipping
• Q run name

The columns marked Individual stars give zeropoints for each star observation. Note that generally there's more than on star per exposure. The data in these files is:

• CFHT exposure #
• zeropoint
• assumed colour term coefficient
• assumed extinction coefficient (from CFHT)
• Q run name
• UT day number since 2003 Jan 0 (note: to convert to modified Julian Date, add 52639)
• weight
• weight
• star name
• chip # that star fell on

## 6. Effects of Errors in Colour Coefficients on Zeropoints

Basically d(zp) = d(slope) * colmn, where colmn is the mean colour of the stars being analyzed (the "centre of gravity" of the slope fit). Typical mean colours are less than or around 0.5, so a mean slope error of +-0.01 gives a zeropoint uncertainty of +-0.005 - which is tolerable!

In more detail, I've tried jiggling the assumed colour coefficients by +-0.01 to see what the effect is on the derived zeropoints. Here are the results:

 g' r' i' z' -+0.005 -+0.005 -+0.002 -+0.001

So the effects are pretty small.

## 7. Concluding Remarks

What the table means - Column 3 gives my best estimate of the colour term for transforming MegaCam mags to the SDSS system. Column 5 gives my best estimate for transforming MegaCam mags to the Smith et al USNO system.

Calibration Procedure - Probably the best procedure is to keep all magnitudes in the MegaCam natural system, and derive the zeropoints using Smith et al stds, without ever going to the SDSS system. In that case we use the MegaCam-to-Smith colour term (column 5), and determine the zeropoint at colour=0 (as this web page explains). The SDSS system is nevertheless useful for consistency checks.

SDSS - Smith et al differences - For use of the SDSS system, the question does arise whether it's better to use the grand average colour terms of column 3 (which average together SDSS DR5 estimates and Smith et al std star estimates), or just stick with the SDSS DR5 estimates. This needs to be discussed. Personally I think that there are systematics affecting all of the determinations of the colour term slopes, and it's better to use the grand average.