Calibration with colour corrections

Calibration with colour corrections is usually appropriate in two cases:

• where the instrumental system is not well matched to the target standard system,

• where very high precision photometry is being carried out and even small discrepancies between the instrumental and target standard systems must be corrected for.

Calibration with colour corrections is similar to calibration without a colour correction. The calibrated magnitude is still computed from the instrumental magnitude in the corresponding band in a manner similar to Equation . However, an additional term is added corresponding to a colour index determined from an adjacent band. This term compensates for the mismatch between the instrumental and standard systems. For example, for the Johnson-Morgan UBV system the calibration formulæ are:

$$U & = & U_{\rm inst} - A_{u} + Z_{u} + C_{u} ( U - B ) + \kappa_{u} X \nonumber$$

$$B & = & B_{\rm inst} - A_{b} + Z_{b} + C_{b} ( B - V ) + \kappa_{b} X$$ (16)

$$V & = & V_{\rm inst} - A_{v} + Z_{v} + C_{v} ( B - V ) + \kappa_{v} X \nonumber$$

where:

$U$, $B$ and $V$

are the calibrated magnitudes in the three bands,

$U_{\rm inst}$, $B_{\rm inst}$ and $V_{\rm inst}$

are the instrumental magnitudes in the three bands,

$A_{x}$

is an arbitrary constant which is often added to the instrumental constants,

$C_{x}$

is the colour-correction term,

$Z_{x}$

is the photometric zero point between the standard and instrumental systems,

$\kappa_{x}$

is the atmospheric extinction coefficient,

$X$

is the air mass, and

subscripts $x = u, b, v$

refer to the individual bands.

The operational procedure is similar to that for calibration without a colour correction. For a set of observations of standard stars $Z_{x}$, $C_{x}$ and $\kappa_{x}$ (where $x = u, b, v$) are unknowns which can be solved for by least squares fitting of Equations . Once the coefficients have been determined, they can be used to compute the calibrated magnitudes of the programme objects. For really accurate work more-complex equations including higher-order terms may be introduced. For example:

$$V = V_{\rm inst} - A_{v} + Z_{v} + C_{1}(B-V) + \kappa_vX + C_{2}X(B-V) + \cdots$$ (17)

Sometimes the atmospheric-extinction coefficient is not constant, but includes a colour term. That is:

$$\kappa = \kappa ' + \kappa '' ({\rm colour~index})$$ (18)

where $\kappa '$ and $\kappa ''$ are constants. Often $\kappa ''$ is sufficiently small that $\kappa$ can be assumed to be constant. If the colour term is significant then the lines in Figure  will appear curved.

The three cautionary caveats given in the preceding section for calibrating without colour corrections are equally, if not more, applicable when colour corrections are included. Briefly: programme objects should only be calibrated with observations of standards made on the same night, when standards are fitted the residuals should be examined individually and aberrant observations discarded and the residuals should be checked for systematic trends.

Often bespoke software is used for reducing photometric observations with colour corrections, partly because the colour correction terms used will depend on the bands that were observed. There is no recipe for calibration with colour corrections in this cookbook. Further discussions are given by Massey et al.[55], Da Costa[15], Harris et al.[36] and Stetson and Harris[69].