Calibration without a colour correction

Calibration without a colour correction is appropriate when the instrumental system is well matched to the target standard system. The calibrated magnitude is computed solely from the corresponding instrumental magnitude. Because magnitudes are logarithmic quantities and the standard and instrumental systems are being assumed to be well matched the principal difference between them is a zero-point correction. In this case the relation between instrumental and calibrated magnitudes is of the form:

$$m_{\rm calib} = m_{\rm inst} - A + Z + \kappa X$$ (15)

where:

$m_{\rm calib}$

is the calibrated magnitude,

$m_{\rm inst}$

is the instrumental magnitude,

$A$

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

$Z$

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

$\kappa$

is the atmospheric-extinction coefficient,

$X$

is the air mass.

For programme objects $A$ is arbitrarily chosen by you, $m_{\rm inst}$ is measured and $X$ is known (remember that the air mass depends solely on the zenith distance which, in turn, can be calculated from the celestial coordinates of the object, the location of the observatory and the time of observation; see Section  and Appendix ). $Z$ and $\kappa$ are constants which are initially unknown. Once they have been determined Equation  can be used to calculate the calibrated magnitudes.

There are various methods of determining $Z$ and $\kappa$. For example, if a single standard star⁹ is repeatedly observed throughout the night then the instrumental magnitude can be plotted against the air mass. Such a plot should be a straight line with a slope of $\kappa$. Figure  shows a schematic example of such a plot.

However, the most common method of determining the constants is to intersperse observations of your programme objects with observations of standard stars. Suitable standard stars will typically have been selected from one of the catalogues of standard stars (see Sections  and ). For each of the observations of standard stars $m_{\rm calib}$ is known in addition to $m_{\rm inst}$, $A$ and $X$ and it is possible to simply solve for $Z$ and $\kappa$ using least squares or some similar technique.

Once $Z$ and $\kappa$ have been determined Equation  can be used to simply calculate the calibrated magnitudes for the programme objects.

Thus, in essence, photometric calibration consists of making a least squares (or similar) fit to a series of observations of standard stars to determine the photometric zero point and the atmospheric extinction coefficient. However, such a fit should not be made blindly. (At least) the following caveats should be borne in mind.

1. $Z$ depends on the details of the instrumentation (CCD detector, filter, telescope etc.) and should remain fairly constant throughout an observing run. However, atmospheric extinction definitely varies from night to night¹⁰. Hence:

   observations of standard stars should only be used to calibrate observations of programme objects made on the same night.

   That is, observations made on different nights should be calibrated separately.

2. When a fit is made to the standard-star observations, the individual residuals should be examined, any observations with large residuals discarded and the remaining observations re-fitted. Passing clouds and other transient phenomena can cause temporary variations leading to aberrant and invalid observations.

3. The residuals and/or the coefficients themselves should be plotted as a function of time of observation throughout the night. Systematic variations can occur during a single night and it may be necessary to discard the observations for a portion of the night or make separate fits for different parts of the night.

Section  gives a simple recipe for calibrating photometric observations without a colour correction.